VMSTA Modern Stochastics: Theory and Applications 2351-60542351-60462351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania VMSTA295 10.15559/26-VMSTA295 Research Article Viability for time fractional functional differential equations driven by the fractional Brownian motion https://orcid.org/0009-0006-5035-516X HanJingqijingqi_han@sues.edu.coma SunYaqinsyq958420434@163.comb YanLitanlitan-yan@hotmail.comb∗ School of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, 333 Longteng Rd., Songjiang, Shanghai 201620, P.R. China Department of Statistics, College of Science, Donghua University, 2999 North Renmin Rd., Songjiang, Shanghai 201620, P.R. China Corresponding author. 2026 1420260013797202515220262122026 © 2026 The Author(s). Published by VTeX2026 Open access article under the CC BY license.

Let B H be a fractional Brownian motion with Hurst index 1 2 < H < 1. In this paper, we consider the time fractional functional differential equation of the form C D t γ x ( t ) = f ( t , x t ) + G ( t , x t ) d d t B H ( t ) , t ∈ ( 0 , T ] , x 0 ( t ) = η ( t ) , t ∈ [ − r , 0 ] , where 3 2 − H < γ < 1, C D t γ denotes the Caputo derivative, and x t ∈ C r = C ( [ − r , 0 ] , R ) with x t ( u ) = x ( t + u ), u ∈ [ − r , 0 ]. We prove the global existence and uniqueness of the solution of the equation and study its viability. As an application, we also discuss the existence of positive solutions.

 Fractional Brownian motion Caputo derivative stochastic functional differential equation time delay viability 60G22 60H10 45R05 Jingqi Han is supported by NSFC (12201393), Litan Yan and Yaqin Sun are supported by NSFC (11971101, 12171081) and the Natural Science Foundation of Shanghai (24ZR1402900). Introduction

Over the last decade, there has been considerable interest in studying fractional Brownian motion due to its compact properties such as long/short range dependence, self-similarity, stationary increments and Hölder’s continuity. Its applications can be found in various scientific areas including telecommunications, turbulence, image processing and finance. Fractional Brownian motion is a generalization of standard Brownian motion and it is a relatively simple stochastic process which possesses the above good properties. Some surveys and complete literatures on fractional Brownian motion could be found in Alós et al. [2], Biagini et al [6], Hu [11], Mishura [15], Nourdin [17], Nualart [18], Tudor [21], and the references therein. Recall that a fractional Brownian motion (in short, fBm) B H = { B H ( t ) , t ≥ 0 } with Hurst index H ∈ ( 0 , 1 ) is a zero mean Gaussian process with the covariance function E B H ( t ) B H ( s ) = 1 2 t 2 H + s 2 H − | t − s | 2 H for all t , s ≥ 0. For H = 1 / 2, B H coincides with the standard Brownian motion B. Since B H is neither a semimartingale nor a Markov process unless H = 1 / 2, many of the powerful techniques from classical stochastic analysis are not applicable to B H . However, as a Gaussian process, one can develop the stochastic calculus of variations with respect to B H .

On the other hand, the viability property has been extensively studied for deterministic differential equations and inclusions, starting with Nagumo’s pioneering work in 1943 (see Aubin [3]). Girejko et al [9] first considered the viability of the fractional differential equations with the Caputo derivative and Carja et al [7] introduced the viability of fractional differential inclusions. Viability theory is a mathematical theory that offers mathematical metaphors of evolution of macrosystems arising in biology, economics, cognitive sciences, games, and similar areas, as well as in nonlinear systems of control theory. The characterization of the viability property in a stochastic framework was first given by Aubin and Da Prato [4] in 1990. The key point of their work consists in defining a suitable Bouligand’s stochastic tangent cone which generalizes the cone used in the study of the viability property for deterministic systems. Then, Nie-Rascanu [16] established the characterization of viability for some particular sets K. Recently, Melnikov et al [14] considered stochastic viability and comparison theorems for mixed stochastic differential equations. Ciotir and Rascanu [8] proved a viability result for multidimensional, time dependent, stochastic differential equations driven by fractional Brownian motion, and moreover, in [22, 23] Xu and Luo extended this to the viability for stochastic functional differential equations associated with fractional Brownian motion. In a more recent development, Li et al [13] addressed the problem of a class of coupled multidimensional stochastic differential equations driven by fractional Brownian motion.

Motivated by these results, in this paper, we consider the viability result for the time fractional functional differential equation of the form C D t γ x ( t ) = f ( t , x t ) + G ( t , x t ) d d t B H ( t ) , t ∈ ( 0 , T ] , x 0 ( t ) = η ( t ) , t ∈ [ − r , 0 ] with 1 2 < H < 1 and 3 2 − H < γ < 1, where

B H = { B H ( t ) , 0 ≤ t ≤ T } is a fractional Brownian motion with Hurst index H ∈ ( 1 2 , 1 );

C D t γ denotes the Caputo derivative;

x t ∈ C r with x t ( u ) = x ( t + u ) for u ∈ [ − r , 0 ], C r = C ( [ − r , 0 ] ) is the space of continuous functions f from [ − r , 0 ] to R endowed by the uniform norm ‖ · ‖ C r ;

f , G : [ 0 , T ] × C r → R are two Borel functions;

η : [ − r , 0 ] → R is a smooth function.

 It is important to note that the strongest motivation to study such equations comes for the following reasons. Compared with the classical differential equation, the fractional order models are better to describe the hereditary character of various kinds of materials and processes. The advantages of fractional derivatives become apparent in various fields of science and engineering such as control theory, porous media, viscoelasticity, image and signal processing. (see e.g. [1, 5, 12, 20]). As mentioned before, fractional Brownian motion is a Gaussian process with long-memory properties and a relatively simple structure. Therefore, it is more reasonable to introduce fractional Brownian motion into the noise term of fractional order differential equations to characterize systems with memory.

We know that the equation (1.1) can be written as the following integral form: x ( t ) = η ( 0 ) + 1 Γ ( γ ) ∫ 0 t ( t − s ) γ − 1 f ( s , x s ) d s + 1 Γ ( γ ) ∫ 0 t ( t − s ) γ − 1 G ( s , x s ) d B H ( s ) , t ∈ ( 0 , T ] , x 0 ( t ) = η ( t ) , t ∈ [ − r , 0 ] , where Γ ( · ) denotes the classical Gamma function and the integral with respect to B H is a pathwise Riemann-Stieltjes (R-S) integral in the sense of Zähle [25, 26]. Our approach is inspired by Nualart and Rascanu [19], while we consider the time fractional cases. This paper is organized as follows. In Section 2, we present the assumptions on the coefficients and briefly review the basic definitions and key results on the fractional integrals and derivatives. In Section 3, we derive some useful estimates for the indefinite integrals and show the existence and uniqueness for the solution of the equation (1.1). In Section 4, we state our main results and obtain the sufficient and necessary condition for a closed subset being a viable domain of the equation (1.1). In Section 5, as an application, we discuss the existence of positive solution when K = [ 0 , ∞ ).

 Preliminaries

Throughout this paper we fix a time interval [ 0 , T ] and a complete probability space ( Ω , F , P ). For some given real numbers a, b with a < b and κ ∈ ( 0 , 1 ], we will denote by C κ ( [ a , b ] ) the space of κ-Hölder continuous functions f : [ a , b ] → R, endowed with the norm ‖ f ‖ κ ; [ a , b ] : = ‖ f ‖ + sup a ≤ s < t ≤ b | f ( t ) − f ( s ) | ( t − s ) κ , where ‖ f ‖ = sup t ∈ [ a , b ] | f ( t ) |. For any λ ≥ 0, we introduce the following equivalent norm: ‖ f ‖ κ , λ ; [ a , b ] = sup t ∈ [ a , b ] e − λ t | f ( t ) | + sup a ≤ s < t ≤ b e − λ t | f ( t ) − f ( s ) | ( t − s ) κ .

Denote by C r = C ( [ − r , 0 ] ) the space of continuous functions f from [ − r , 0 ] to R endowed by the uniform norm ‖ · ‖ C r .

The assumptions on the equation (<xref rid="j_vmsta295_eq_003">1.1</xref>)

To study the equation (1.1), we introduce the following assumptions:

The function f is local Lipschitz continuous and has linear growth. That is, there exists a constant L 1 and for every R ≥ 0 there exists L R ( 1 ) such that | f ( t , ξ ) − f ( s , η ) | ≤ L R ( 1 ) | t − s | + ‖ ξ − η ‖ C r ( ∀ ‖ ξ ‖ C r , ‖ η ‖ C r ≤ R ) and | f ( t , ξ ) | ≤ L 1 ( 1 + ‖ ξ ‖ C r ) for all ξ , η ∈ C r and s , t ∈ [ 0 , T ].

The function G ( t , x ) is Fréchet differentiable in x. Moreover, there exist constants μ ∈ ( 2 − H − γ , 1 ], L 2 and for every R ≥ 0 there exists L R ( 2 ) such that | G ( t , ξ ) − G ( s , η ) | ≤ L 2 | t − s | μ + ‖ ξ − η ‖ C r and | D G ( t , ξ ) − D G ( s , η ) | L ( C r , R ) ≤ L R ( 2 ) | t − s | μ + ‖ ξ − η ‖ C r ( ∀ ‖ ξ ‖ C r , ‖ η ‖ C r ≤ R ) for all ξ , η ∈ C r and s , t ∈ [ 0 , T ].

 It is important to note that the assumption (H-G) implies the linear growth property, i.e., there exists a constant L 3 > 0 such that | G ( t , ξ ) | ≤ L 3 ( 1 + ‖ ξ ‖ C r ) for all ξ ∈ C r and t ∈ [ 0 , T ].

 The fractional derivative

Now, we recall some definitions and notions of fractional calculus.

Let α > 0. The fractional integral of order α for a Borel function f : [ 0 , ∞ ) → R is defined as I α f ( t ) = 1 Γ ( α ) ∫ 0 t f ( s ) ( t − s ) 1 − α d s t > 0 , provided the right side is point-wise defined on [ 0 , ∞ ), where Γ ( · ) is the classical Gamma function defined by Γ ( x ) = ∫ 0 ∞ t x − 1 e − t d t.

Let n > 0 be an integer number and let n − 1 < α < n. The Caputo derivative of order α for a function f ∈ C n ( [ 0 , ∞ ) ) is defined as C D t α f ( t ) = 1 Γ ( n − α ) ∫ 0 t f ( n ) ( s ) ( t − s ) 1 + α − n d s = I n − α f ( n ) ( t ) t > 0 .

Based on fractional integrals and derivatives, Zähle [25] has introduced the Riemann-Stieltjes integral. We refer the reader to Young [24], Zähle [25, 26] and Nualart and Rascanu [19] for the general theory of this integral. Fix a parameter 0 < α < 1 2 , and denote by W α , 1 ( 0 , T ; R ) the space of measurable functions f : [ 0 , T ] → R such that ‖ f ‖ α , 1 : = ∫ 0 T | f ( s ) | s α + ∫ 0 s | f ( s ) − f ( t ) | ( s − t ) α + 1 d t d s < ∞ . We also denote by W 1 − α , ∞ ( 0 , T ; R ) the space of measurable functions g : [ 0 , T ] → R such that ‖ g ‖ 1 − α , ∞ : = sup 0 < s < t < T | g ( t ) − g ( s ) | ( t − s ) 1 − α + ∫ s t | g ( u ) − g ( s ) | ( u − s ) 2 − α d u < ∞ . Clearly, C 1 − α + ε ( 0 , T ; R ) ⊂ W 1 − α , ∞ ( 0 , T ; R ) ⊂ C 1 − α ( 0 , T ; R ) for any ε > 0. Given two functions f ∈ W α , 1 ( 0 , T ; R ) and g ∈ W 1 − α , ∞ ( 0 , T ; R ), the generalized Stieltjes integral ∫ 0 T f ( s ) d g ( s ) is defined by ∫ 0 T f ( s ) d g ( s ) = ( − 1 ) α ∫ 0 T D 0 + α f ( t ) D T − 1 − α g T − ( t ) d t , where ( − 1 ) α = e α π i , D 0 + α f ( t ) = 1 Γ ( 1 − α ) f ( t ) t α + α ∫ 0 t f ( t ) − f ( s ) ( t − s ) α + 1 d s and D T − 1 − α g T − ( t ) = ( − 1 ) α Γ ( 1 − α ) g ( t ) − g ( T ) ( T − t ) α + α ∫ t T g ( t ) − g ( s ) ( s − t ) α + 1 d s . Furthermore, we have the estimate ∫ 0 t f d g ≤ Λ α ( g ) ‖ f ‖ α , 1 , where Λ α ( g ) : = 1 Γ ( 1 − α ) sup 0 < s < t < T | D t − 1 − α g t − ( s ) | ≤ 1 Γ ( 1 − α ) Γ ( α ) ‖ g ‖ 1 − α , ∞ .

Fractional Brownian motion

In this subsection, we briefly review some basic results of fBm. As we have pointed out before, fBm B H = B H ( t ) , 0 ≤ t ≤ T , on the probability space ( Ω , F H , P ) with Hurst index H ∈ ( 0 , 1 ) is a central Gaussian processes such that B H ( 0 ) = 0 and E B H ( t ) B H ( s ) = 1 2 t 2 H + s 2 H − | t − s | 2 H for all s , t ≥ 0. The process is H-self similar and its paths are Hölder continuous of order ν ∈ ( 0 , H ). Even though it is not a semimartingale, we may define the stochastic integrals with respect to it by using some other methods. Throughout this paper, we assume that 1 2 < H < 1. Thus, we can handle its stochastic calculus and related stochastic differential equations by using Young’s integration. By the Hölder continuity of t ↦ B t H , we have known that the trajectories of B H belong to the space W 1 − β , ∞ ( 0 , T ; R ) for all β ∈ 1 − H , 1 2 . As a consequence, if u = { u ( t ) , t ∈ [ 0 , T ] } is a stochastic process whose trajectories belong a.s. to the space W β , 1 ( 0 , T ; R ) with β ∈ 1 − H , 1 2 , the Riemann-Stieltjes integral ∫ 0 T u ( s ) d B H ( s ) exists and ∫ 0 T u ( s ) d B H ( s ) ≤ Λ β ( B H ) ‖ u ‖ β , 1 . Thus, one study the time fractional functional differential equation (1.2) essentially is equivalent to handle the deterministic time fractional differential equation of the form x ( t ) = η ( 0 ) + 1 Γ ( γ ) ∫ 0 t ( t − s ) γ − 1 f ( s , x s ) d s + 1 Γ ( γ ) ∫ 0 t ( t − s ) γ − 1 G ( s , x s ) d g ( s ) , t ∈ ( 0 , T ] x 0 ( t ) = η ( t ) , t ∈ [ − r , 0 ] , where the coefficients f and G are as in the equation (1.1), g ∈ C ν ( [ 0 , T ] ) and x ∈ C 1 − α ( [ − r , T ] ). In fact, we are considering about the above issues in this paper.

 The existence and uniqueness

In this section, we show the existence and uniqueness of the solution of the equation (1.1). As mentioned above, our analysis starts from the equivalent deterministic equation (2.3) to first establish some basic estimates. Throughout this section we assume that 1 2 < ν < 1 is arbitrary but fixed and we let 3 2 − ν < γ < 1 , 2 − ν − γ < α < min 1 2 , μ .

There are several key parameters whose admissible ranges are interdepent. We fix an arbitrary ν ∈ ( 1 2 , 1 ). Given ν, the constant γ is chosen from 3 2 − ν < γ < 1. Then, this choice determines the range of α as 2 − ν − γ < α < min 1 2 , μ . Furthermore, we can also obtain that these parameters should satisfy ν + γ > 3 2 and 1 < 2 − ν < α + γ < 3 2 . Meanwhile, in subsequent proofs, we will also use the auxiliary parameters α 0 and β where α 0 = min α + γ − 1 , μ and β ∈ ( 1 − ν , α 0 ). Note that α 0 < 1 2 .

Define the processes I ( x ) = { I ( x ) ( t ) , 0 ≤ t ≤ T } and J ( x ) = { J ( x ) ( t ) , 0 ≤ t ≤ T } as follows I ( x ) ( t ) = ∫ 0 t ( t − s ) γ − 1 f ( s , x s ) d s and J ( x ) ( t ) = ∫ 0 t ( t − s ) γ − 1 G ( s , x s ) d g ( s ) . Unless specified otherwise, we consider the setting where g ∈ C ν ( [ 0 , T ] ), and the coefficients f and G satisfy conditions (H-f) and (H-G), respectively. In addition, we simplify the norm ‖ x ‖ 1 − α ; [ − r , T ] as ‖ x ‖ 1 − α when the domain of definition of x is explicit. We will also use M to denote a generic constant which may change from line to line.

Let x ∈ C 1 − α ( [ − r , T ] ). We have that J ( x ) ∈ C 1 − α ( [ 0 , T ] ) ; and there exist some constants M 1 , M 1 ( λ ) > 0 such that ‖ J ( x ) ‖ 1 − α , λ ≤ M 1 + M 1 ( λ ) ‖ x ‖ 1 − α , λ . Moreover, M 1 is independent of λ and M 1 ( λ ) → 0 as λ → ∞.

For the proof we mimic the ideas from [10]. To prove the first statement, we let α 0 = min α + γ − 1 , μ and fix β ∈ ( 1 − ν , α 0 ). Since α 0 < 1 2 , then Λ β ( g ) < ∞. Clearly, we have | J ( x ) ( t ) − J ( x ) ( s ) | ≤ ∫ s t ( t − u ) γ − 1 G ( u , x u ) d g ( u ) + ∫ 0 s ( s − u ) γ − 1 − ( t − u ) γ − 1 G ( u , x u ) d g ( u ) ≡ J 11 + J 12 for s , t ∈ [ 0 , T ] with s < t. From (2.1) we can write J 11 ≤ Λ β ( g ) ∫ s t ( t − u ) γ − 1 G ( u , x u ) d u ( u − s ) β + ∫ s t ∫ s u ( t − u ) γ − 1 G ( u , x u ) − ( t − v ) γ − 1 G ( v , x v ) d v d u ( u − v ) β + 1 for s , t ∈ [ 0 , T ] with s < t. By the condition (H-G) and γ − β > 1 − α, we get ∫ s t ( t − u ) γ − 1 G ( u , x u ) d u ( u − s ) β ≤ ∫ s t L 3 ( t − u ) γ − 1 1 + ‖ x u ‖ C r d u ( u − s ) β ≤ M ( t − s ) γ − β 1 + ‖ x ‖ 1 − α ≤ M ( t − s ) 1 − α 1 + ‖ x ‖ 1 − α for s , t ∈ [ 0 , T ] with s < t. Using the condition (H-G) again and the fact ∫ s t ∫ s u ( t − u ) γ − 1 − ( t − v ) γ − 1 d v d u ( u − v ) β + 1 = M γ , β ( t − s ) γ − β , where M γ , β = ∫ 0 1 ( 1 − x ) γ − 1 + x γ d x γ x β + 1 < ∞, we get that ∫ s t ∫ s u ( t − u ) γ − 1 G ( u , x u ) − ( t − v ) γ − 1 G ( v , x v ) d v d u ( u − v ) β + 1 ≤ ∫ s t ∫ s u ( ( t − u ) γ − 1 − ( t − v ) γ − 1 G ( v , x v ) + ( t − u ) γ − 1 [ G ( u , x u ) − G ( v , x v ) ] ) d v d u ( u − v ) β + 1 ≤ ∫ s t ∫ s u L 3 ( t − u ) γ − 1 − ( t − v ) γ − 1 1 + ‖ x v ‖ C r d v d u ( u − v ) β + 1 + ∫ s t ∫ s u L 2 ( t − u ) γ − 1 ( u − v ) μ + ‖ x u − x v ‖ C r d v d u ( u − v ) β + 1 ≤ ∫ s t ∫ s u M ( t − u ) γ − 1 − ( t − v ) γ − 1 1 + ‖ x ‖ 1 − α d v d u ( u − v ) β + 1 + ∫ s t ∫ s u M ( t − u ) γ − 1 ( u − v ) μ + ( u − v ) 1 − α ‖ x ‖ 1 − α d v d u ( u − v ) β + 1 ≤ M ( t − s ) γ − β 1 + ‖ x ‖ 1 − α + M ( t − s ) γ + μ − β + M ( t − s ) γ + 1 − α − β ‖ x ‖ 1 − α ≤ M ( t − s ) 1 − α 1 + ‖ x ‖ 1 − α for s , t ∈ [ 0 , T ] with s < t. Combining this with (3.1) and (3.2), we see that J 11 ≤ M ( t − s ) 1 − α 1 + ‖ x ‖ 1 − α for s , t ∈ [ 0 , T ] with s < t. For J 12 , we have J 12 ≤ Λ β ( g ) ∫ 0 s ( t − u ) γ − 1 − ( s − u ) γ − 1 G ( u , x u ) d u u β + ∫ 0 s ∫ 0 u ( t − u ) γ − 1 − ( s − u ) γ − 1 G ( u , x u ) − G ( v , x v ) d v d u ( u − v ) β + 1 + ∫ 0 s ∫ 0 u | [ ( t − u ) γ − 1 − ( t − v ) γ − 1 − ( s − u ) γ − 1 + ( s − v ) γ − 1 ] G ( v , x v ) | d v d u ( u − v ) β + 1 by (2.1). We now estimate the three terms in (3.4). For the first term, by the condition (H-G) we get that ∫ 0 s ( t − u ) γ − 1 − ( s − u ) γ − 1 G ( u , x u ) d u u β ≤ ∫ 0 s ( t − u ) γ − 1 − ( s − u ) γ − 1 L 3 ( 1 + ‖ x u ‖ C r ) d u u β ≤ M 1 + ‖ x ‖ 1 − α ∫ 0 s ( s − u ) γ − 1 − ( t − u ) γ − 1 u − β d u = M 1 + ‖ x ‖ 1 − α ∫ 0 s ( s − u ) γ − 1 u − β d u − ∫ 0 t ( t − u ) γ − 1 u − β d u + ∫ s t ( t − u ) γ − 1 u − β d u ≤ M 1 + ‖ x ‖ 1 − α ∫ s t ( t − u ) γ − 1 u − β d u ≤ M 1 + ‖ x ‖ 1 − α ( t − s ) γ − β ≤ M ( t − s ) 1 − α ( 1 + ‖ x ‖ 1 − α ) . For the second term, we also have that ∫ 0 s ∫ 0 u ( t − u ) γ − 1 − ( s − u ) γ − 1 G ( u , x u ) − G ( v , x v ) d v d u ( u − v ) β + 1 ≤ ∫ 0 s ∫ 0 u ( t − u ) γ − 1 − ( s − u ) γ − 1 L 2 ( u − v ) μ + ‖ x u − x v ‖ C r d v d u ( u − v ) β + 1 ≤ M ∫ 0 s ( s − u ) γ − 1 − ( t − u ) γ − 1 d u ∫ 0 u ( u − v ) μ + ( u − v ) 1 − α ‖ x ‖ 1 − α d v ( u − v ) β + 1 ≤ M [ ( t − s ) γ + μ − β + ‖ x ‖ 1 − α ( t − s ) γ + 1 − α − β ] ≤ M ( 1 + ‖ x ‖ 1 − α ) ( t − s ) 1 − α . Finally, some elementary calculations may show that ∫ 0 s ∫ 0 u ( t − u ) γ − 1 − ( t − v ) γ − 1 − ( s − u ) γ − 1 + ( s − v ) γ − 1 d v d u ( u − v ) β + 1 = ∫ 0 s ∫ 0 u ( s − u ) γ − 1 − ( s − v ) γ − 1 d v d u ( u − v ) β + 1 − ∫ 0 t ∫ 0 u ( t − u ) γ − 1 − ( t − v ) γ − 1 d v d u ( u − v ) β + 1 + ∫ s t ∫ 0 u ( t − u ) γ − 1 − ( t − v ) γ − 1 d v d u ( u − v ) β + 1 ≤ ∫ s t ∫ 0 s + ∫ s t ∫ s u ( t − u ) γ − 1 − ( t − v ) γ − 1 d v d u ( u − v ) β + 1 ≤ ∫ s t ∫ 0 s ( t − u ) γ − 1 ( u − v ) β + 1 d v d u + M ( t − s ) γ − β ≤ M ( t − s ) γ − β ≤ M ( t − s ) 1 − α for s , t ∈ [ 0 , T ] with s < t. It follows from the condition (H-G) that ∫ 0 s ∫ 0 u ( t − u ) γ − 1 − ( t − v ) γ − 1 − ( s − u ) γ − 1 + ( s − v ) γ − 1 G ( v , x v ) d v d u ( u − v ) β + 1 ≤ ∫ 0 s ∫ 0 u | [ ( t − u ) γ − 1 − ( t − v ) γ − 1 − ( s − u ) γ − 1 + ( s − v ) γ − 1 ] L 3 1 + ‖ x v ‖ C r | d v d u ( u − v ) β + 1 ≤ M 1 + ‖ x ‖ 1 − α ( t − s ) 1 − α for s , t ∈ [ 0 , T ] with s < t. Combining this with (3.4), (3.5), (3.6) and (3.8), we get that J 12 ≤ M ( t − s ) 1 − α 1 + ‖ x ‖ 1 − α for s , t ∈ [ 0 , T ] with s < t. Thus, we have showed that | J ( x ) ( t ) − J ( x ) ( s ) | = J 11 + J 12 ≤ M ( t − s ) 1 − α 1 + ‖ x ‖ 1 − α for s , t ∈ [ 0 , T ] with s < t, which implies that J ( x ) ∈ C 1 − α ( [ 0 , T ] ).

For the second assertion, fix β ∈ ( 1 − ν , α 0 ) as above. Then, by (2.1) we have that J ( x ) ( t ) − J ( x ) ( s ) e − λ t ( t − s ) 1 − α ≤ e − λ t ( t − s ) 1 − α ∫ 0 s ( t − u ) γ − 1 G ( u , x u ) − ( s − u ) γ − 1 G ( u , x u ) d g ( u ) + e − λ t ( t − s ) 1 − α ∫ s t ( t − u ) γ − 1 G ( u , x u ) d g ( u ) ≤ Λ β ( g ) e − λ t ( t − s ) 1 − α ∫ 0 s ( t − u ) γ − 1 G ( u , x u ) − ( s − u ) γ − 1 G ( u , x u ) u − β d u + ∫ 0 s ∫ 0 u | [ ( t − u ) γ − 1 − ( s − u ) γ − 1 − ( t − v ) γ − 1 + ( s − v ) γ − 1 ] G ( v , x v ) | d v d u ( u − v ) β + 1 + ∫ 0 s ∫ 0 u ( t − u ) γ − 1 − ( s − u ) γ − 1 G ( u , x u ) − G ( v , x v ) d u d v ( u − v ) β + 1 + ∫ s t ( t − u ) γ − 1 G ( u , x u ) d u ( u − s ) β + ∫ s t ∫ s u ( t − u ) γ − 1 G ( u , x u ) − ( t − v ) γ − 1 G ( v , x v ) d v d u ( u − v ) β + 1 ≡ ∑ i = 1 5 J 2 i for t , s ∈ [ 0 , T ] with s < t. We need to estimate J 2 i , i = 1 , 2 , … , 5. From the condition (H-G), we get J 24 ≤ M Λ β ( g ) e − λ t ( t − s ) 1 − α ∫ s t ( t − u ) γ − 1 1 + ‖ x u ‖ C r ( u − s ) β d u ≤ M Λ β ( g ) T γ − β + α − 1 + M Λ β ( g ) ‖ x ‖ 1 − α , λ ( t − s ) 1 − α ∫ s t ( t − u ) γ − 1 e − λ ( t − u ) ( u − s ) β d u ≤ M Λ β ( g ) T γ − β + α − 1 + M Λ β ( g ) ‖ x ‖ 1 − α , λ ∫ s t e − λ ( t − u ) ( t − u ) 2 − α − γ ( u − s ) β d u ≤ M Λ β ( g ) T γ − β + α − 1 + M Λ β ( g ) ‖ x ‖ 1 − α , λ λ γ − 1 + α − β sup k > 0 ∫ 0 k e − z z 2 − α − γ ( k − z ) β d z and J 25 ≤ M Λ β ( g ) e − λ t ( t − s ) 1 − α ∫ s t ∫ s u ( t − u ) γ − 1 − ( t − v ) γ − 1 G ( v , x v ) d v d u ( u − v ) β + 1 + ∫ s t ∫ s u ( t − u ) γ − 1 G ( u , x u ) − G ( v , x v ) d v d u ( u − v ) β + 1 ≤ M Λ β ( g ) e − λ t ( t − s ) 1 − α { ∫ s t ∫ s u ( t − u ) γ − 1 − ( t − v ) γ − 1 1 + e λ v ‖ x ‖ 1 − α , λ d v d u ( u − v ) β + 1 + ∫ s t ∫ s u ( t − u ) γ − 1 ( u − v ) μ d v d u ( u − v ) β + 1 + ∫ s t ∫ s u ( t − u ) γ − 1 e λ u ‖ x ‖ 1 − α , λ d v d u ( u − v ) β + α } ≤ M Λ β ( g ) T γ − β + α − 1 + T γ − β + α − 1 + μ + M Λ β ( g ) ‖ x ‖ 1 − α , λ ( t − s ) 1 − α ∫ s t ∫ s u ( t − u ) γ − 1 − ( t − v ) γ − 1 e − λ ( t − v ) d v d u ( u − v ) β + 1 + ∫ s t ∫ s u ( t − u ) γ − 1 e − λ ( t − u ) d v d u ( u − v ) β + α ≤ M Λ β ( g ) 1 + ‖ x ‖ 1 − α , λ ( t − s ) 1 − α ∫ 0 t − s ∫ 0 t − s − w r γ − 1 − ( r + w ) γ − 1 e − λ ( r + w ) w β + 1 d r d w + ‖ x ‖ 1 − α , λ ( t − s ) 1 − α ∫ s t ( t − u ) γ − 1 e − λ ( t − u ) ( u − s ) 1 − β − α d u ≤ M Λ β ( g ) 1 + ‖ x ‖ 1 − α , λ λ α + γ − β − 1 ∫ 0 ∞ e − z z α + γ − β − 2 d z + ‖ x ‖ 1 − α , λ λ α + γ − 1 ∫ 0 ∞ z α + γ − 2 e − z d z for t , s ∈ [ 0 , T ] with s < t. For the term J 21 , we have e − λ t ( t − s ) 1 − α ∫ 0 s ( s − u ) γ − 1 − ( t − u ) γ − 1 e λ u u β d u = e − λ t ( t − s ) 1 − α ∫ 0 s ( s − u ) γ − 1 e λ u u β d u − ∫ 0 t − ∫ s t ( t − u ) γ − 1 e λ u u β d u ≤ e − λ t ( t − s ) 1 − α ∫ s t ( t − u ) γ − 1 e λ u u β d u ≤ 1 λ γ − 1 + α − β sup k > 0 ∫ 0 k e − z z 2 − α − γ ( k − z ) β d z for all s , t ∈ [ 0 , T ] with s < t. It follows that J 21 ≤ Λ β ( g ) e − λ t ( t − s ) 1 − α ∫ 0 s ( s − u ) γ − 1 − ( t − u ) γ − 1 L 3 1 + ‖ x u ‖ C r u − β d u ≤ M Λ β ( g ) e − λ t ( t − s ) 1 − α ∫ 0 s ( s − u ) γ − 1 − ( t − u ) γ − 1 1 + e λ u ‖ x ‖ 1 − α , λ u − β d u ≤ M Λ β ( g ) T γ − 1 + α − β + M Λ β ( g ) ‖ x ‖ 1 − α , λ e − λ t ( t − s ) 1 − α · ∫ 0 s ( s − u ) γ − 1 − ( t − u ) γ − 1 e λ u u − β d u ≤ M Λ β ( g ) T γ − 1 + α − β + M Λ β ( g ) ‖ x ‖ 1 − α , λ λ γ − 1 + α − β sup k > 0 ∫ 0 k e − z z 2 − α − γ ( k − z ) β d z for t , s ∈ [ 0 , T ] with s < t. For the term J 22 , we have e − λ t ( t − s ) 1 − α ∫ 0 s ∫ 0 u { [ ( s − u ) γ − 1 − ( s − v ) γ − 1 ] − [ ( t − u ) γ − 1 − ( t − v ) γ − 1 ] } e λ u d v d u ( u − v ) β + 1 = e − λ t ( t − s ) 1 − α ∫ 0 s ∫ 0 u ( s − u ) γ − 1 − ( s − v ) γ − 1 e λ u d v d u ( u − v ) β + 1 − ∫ 0 t ∫ 0 u − ∫ s t ∫ 0 u ( t − u ) γ − 1 − ( t − v ) γ − 1 e λ u d v d u ( u − v ) β + 1 ≤ e − λ t ( t − s ) 1 − α ∫ s t ∫ s u + ∫ s t ∫ 0 s ( t − u ) γ − 1 − ( t − v ) γ − 1 e λ u d v d u ( u − v ) β + 1 ≤ M λ α + γ − β − 1 ∫ 0 ∞ e − z z α + γ − β − 2 d z + M λ γ − β + α − 1 sup k > 0 ∫ 0 k e − z z 2 − α − γ ( k − z ) β d z for t , s ∈ [ 0 , T ] with s < t, which implies that J 22 ≤ M Λ β ( g ) e − λ t ( t − s ) 1 − α · ∫ 0 s ∫ 0 u [ ( s − u ) γ − 1 − ( s − v ) γ − 1 ] − [ ( t − u ) γ − 1 − ( t − v ) γ − 1 ] ( u − v ) β + 1 ( 1 + ‖ x v ‖ C r ) d v d u ≤ M Λ β ( g ) T γ − β + α − 1 + M Λ β ( g ) e − λ t ‖ x ‖ 1 − α , λ ( t − s ) 1 − α · ∫ 0 s ∫ 0 u ( s − u ) γ − 1 − ( s − v ) γ − 1 − ( t − u ) γ − 1 − ( t − v ) γ − 1 e λ v ( u − v ) β + 1 d v d u ≤ M Λ β ( g ) { 1 + ‖ x ‖ 1 − α , λ λ α + γ − β − 1 ∫ 0 ∞ e − z z α + γ − β − 2 d z + ‖ x ‖ 1 − α , λ λ γ − β + α − 1 sup k > 0 ∫ 0 k e − z z 2 − α − γ ( k − z ) β d z } for t , s ∈ [ 0 , T ] with s < t. For the last term J 23 , we have J 23 ≤ Λ β ( g ) e − λ t ( t − s ) 1 − α · ∫ 0 s ∫ 0 u ( s − u ) γ − 1 − ( t − u ) γ − 1 L 2 ( u − v ) μ + ‖ x u − x v ‖ C r ( u − v ) β + 1 d v d u ≤ M Λ β ( g ) e − λ t ( t − s ) 1 − α · ∫ 0 s ∫ 0 u ( s − u ) γ − 1 − ( t − u ) γ − 1 ( u − v ) μ + e λ u ( u − v ) 1 − α ‖ x ‖ 1 − α , λ ( u − v ) β + 1 d v d u ≤ M Λ β ( g ) T μ − β + γ − 1 + α + M Λ β ( g ) e − λ t ‖ x ‖ 1 − α , λ ( t − s ) 1 − α · ∫ 0 s ( s − u ) γ − 1 − ( t − u ) γ − 1 e λ u u 1 − α − β d u ≤ M Λ β ( g ) T μ − β + γ − 1 + α + M Λ β ( g ) ‖ x ‖ 1 − α , λ λ α + γ − 1 ∫ 0 ∞ e − z z α + γ − 2 d z for t , s ∈ [ 0 , T ] with s < t. Thus, we have showed that there exist some constants M 1 , M 1 ( λ ) > 0 such that ‖ J ( x ) ‖ 1 − α , λ ≤ M 1 + M 1 ( λ ) ‖ x ‖ 1 − α , λ , where M 1 is independent of λ and M 1 ( λ ) → 0 as λ → ∞. This completes the proof.  □

Let x , y ∈ C 1 − α ( [ − r , T ] ) with ‖ x ‖ ≤ R and ‖ y ‖ ≤ R. There exists M R ( 1 ) ( λ ) > 0 such that M R ( 1 ) ( λ ) → 0 as λ → ∞, and ‖ J ( x ) − J ( y ) ‖ 1 − α , λ ≤ M R ( 1 ) ( λ ) 1 + ‖ x ‖ 1 − α + ‖ y ‖ 1 − α ‖ x − y ‖ 1 − α , λ .

Let β ∈ ( 1 − ν , α 0 ) as in the proof of Proposition 3.1. We have e − λ t ( t − s ) 1 − α J ( x ) ( t ) − J ( y ) ( t ) − J ( x ) ( s ) + J ( y ) ( s ) ≤ e − λ t ( t − s ) 1 − α ∫ 0 s ( t − u ) γ − 1 − ( s − u ) γ − 1 G ( u , x u ) − G ( u , y u ) d g ( u ) + ∫ s t ( t − u ) γ − 1 G ( u , x u ) − G ( u , y u ) d g ( u ) ≡ J 31 + J 32 for t , s ∈ [ 0 , T ] with s < t. From (2.1), we have J 31 ≤ Λ β ( g ) e − λ t ( t − s ) 1 − α { ∫ 0 s ( t − u ) γ − 1 − ( s − u ) γ − 1 G ( u , x u ) − G ( u , y u ) d u u β + ∫ 0 s ∫ 0 u | ( t − u ) γ − 1 − ( s − u ) γ − 1 − ( t − v ) γ − 1 + ( s − v ) γ − 1 | | G ( v , x v ) − G ( v , y v ) | d v d u ( u − v ) β + 1 + ∫ 0 s ∫ 0 u ( t − u ) γ − 1 − ( s − u ) γ − 1 · | G ( u , x u ) − G ( v , x v ) − G ( u , y u ) + G ( v , y v ) | d v d u ( u − v ) β + 1 } ≡ Λ β ( g ) ∑ i = 1 3 J 31 ( i ) for t , s ∈ [ 0 , T ] with s < t. Then, we have J 31 ( 1 ) ≤ L 2 e − λ t ( t − s ) 1 − α ∫ 0 s ( s − u ) γ − 1 − ( t − u ) γ − 1 ‖ x u − y u ‖ C r d u u β ≤ M ‖ x − y ‖ 1 − α , λ λ γ − β + α − 1 sup k > 0 ∫ 0 k e − z z 2 − α − γ ( k − z ) β d z , ( Similar to (3.12) ) , and J 31 ( 2 ) ≤ L 2 e − λ t ( t − s ) 1 − α ∫ 0 s ∫ 0 u | ( t − u ) γ − 1 − ( s − u ) γ − 1 − ( t − v ) γ − 1 + ( s − v ) γ − 1 | ‖ x v − y v ‖ C r d v d u ( u − v ) β + 1 ≤ M ‖ x − y ‖ 1 − α , λ λ γ − β + α − 1 { ∫ 0 ∞ e − z z α + γ − β − 2 d z + sup k > 0 ∫ 0 k e − z z 2 − α − γ ( k − z ) β d z } , ( Similar to (3.13) ) . By the mean value theorem and the condition (H-G), we have for all u , v ∈ [ 0 , T ], ‖ x ‖ ≤ R and ‖ y ‖ ≤ R, | G ( u , x u ) − G ( u , y u ) − G ( v , x v ) + G ( v , y v ) | = | ∫ 0 1 D G ( v , r x v + ( 1 − r ) y v ) ( x u − y u ) d r − ∫ 0 1 D G ( v , r x v + ( 1 − r ) y v ) ( x v − y v ) d r + ∫ 0 1 D G ( u , r x u + ( 1 − r ) y u ) ( x u − y u ) d r − ∫ 0 1 D G ( v , r x v + ( 1 − r ) y v ) ( x u − y u ) d r | ≤ | ∫ 0 1 D G ( v , r x v + ( 1 − r ) y v ) ( x u − y u − x v + y v ) d r | + | ∫ 0 1 [ D G ( u , r x u + ( 1 − r ) y u ) − D G ( v , r x v + ( 1 − r ) y v ) ] ( x u − y u ) d r | ≤ L 2 ‖ x u − y u − x v + y v ‖ C r + L R ( 2 ) ‖ x u − x v ‖ C r + ‖ y u − y v ‖ C r + ( u − v ) μ ‖ x u − y u ‖ C r . It follows that J 31 ( 3 ) ≤ e − λ t ( t − s ) 1 − α ∫ 0 s ∫ 0 u ( t − u ) γ − 1 − ( s − u ) γ − 1 · L 2 ‖ x u − y u − x v + y v ‖ C r L 2 d v d u ( u − v ) β + 1 + ∫ 0 s ∫ 0 u ( t − u ) γ − 1 − ( s − u ) γ − 1 · L R ( 2 ) ‖ x u − x v ‖ C r + ‖ y u − y v ‖ C r ‖ x u − y u ‖ C r L R ( 2 ) d v d u ( u − v ) β + 1 + ∫ 0 s ∫ 0 u ( t − u ) γ − 1 − ( s − u ) γ − 1 L R ( 2 ) ( u − v ) μ ‖ x u − y u ‖ C r L R ( 2 ) d v d u ( u − v ) β + 1 ≡ J 31 ( 3 , 1 ) + J 31 ( 3 , 2 ) + J 31 ( 3 , 3 ) for t , s ∈ [ 0 , T ] with s < t. Some elementary calculations, such as those shown in (3.14), can illustrate that J 31 ( 3 , 1 ) ≤ M e − λ t ( t − s ) 1 − α ∫ 0 s ∫ 0 u ( s − u ) γ − 1 − ( t − u ) γ − 1 ‖ x − y ‖ 1 − α , λ e λ u d v d u ( u − v ) β + α ≤ M ‖ x − y ‖ 1 − α , λ λ α + γ − 1 ∫ 0 ∞ e − z z α + γ − 2 d z , J 31 ( 3 , 2 ) ≤ M R e − λ t ( t − s ) 1 − α ∫ 0 s ∫ 0 u ( s − u ) γ − 1 − ( t − u ) γ − 1 · ‖ x u − y u ‖ C r ‖ x ‖ 1 − α + ‖ y ‖ 1 − α e λ u d v d u ( u − v ) β + α ≤ M R ‖ x − y ‖ 1 − α , λ λ α + γ − 1 ‖ x ‖ 1 − α + ‖ y ‖ 1 − α ∫ 0 ∞ e − z z α + γ − 2 d z , and J 31 ( 3 , 3 ) ≤ M R e − λ t ( t − s ) 1 − α ∫ 0 s ∫ 0 u ( s − u ) γ − 1 − ( t − u ) γ − 1 · ‖ x − y ‖ 1 − α , λ e λ u d v d u ( u − v ) β + 1 − μ ≤ M R ‖ x − y ‖ 1 − α , λ λ α + γ − 1 ∫ 0 ∞ e − z z α + γ − 2 d z for all s , t ∈ [ 0 , T ] with s < t. Combining this with inequalities (3.15)-(3.18), we get J 31 ≤ M R ( λ ) ( 1 + ‖ x ‖ 1 − α + ‖ y ‖ 1 − α ) ‖ x − y ‖ 1 − α , λ for all s , t ∈ [ 0 , T ] with s < t and lim λ → ∞ M R ( λ ) = 0.

Now, let us estimate the term J 32 . From (2.1), we have that J 32 ≤ Λ β ( g ) e − λ t ( t − s ) 1 − α ∫ s t ( t − u ) γ − 1 G ( u , x u ) − G ( u , y u ) d u ( u − s ) β + ∫ s t ∫ s u ( t − u ) γ − 1 − ( t − v ) γ − 1 | G ( v , x v ) − G ( v , y v ) | d v d u ( u − v ) β + 1 + ∫ s t ∫ s u ( t − u ) γ − 1 G ( u , x u ) − G ( v , x v ) − G ( u , y u ) + G ( v , y v ) d v d u ( u − v ) β + 1 ≡ Λ β ( g ) ∑ i = 1 3 J 32 ( i ) for all s , t ∈ [ 0 , T ] with s < t. Obviously, we have J 32 ( 1 ) ≤ L 2 e − λ t ( t − s ) 1 − α ∫ s t ( t − u ) γ − 1 ‖ x u − y u ‖ C r d u ( u − s ) β ≤ M ‖ x − y ‖ 1 − α , λ λ γ − β + α − 1 sup k > 0 ∫ 0 k e − z z 2 − α − γ ( k − z ) β d z , ( Similar to (3.10) ) , and J 32 ( 2 ) ≤ L 2 e − λ t ( t − s ) 1 − α ∫ s t ∫ s u ( t − u ) γ − 1 − ( t − v ) γ − 1 ‖ x v − y v ‖ C r d v d u ( u − v ) β + 1 ≤ M ‖ x − y ‖ 1 − α , λ λ γ − β + α − 1 ∫ 0 ∞ e − z z α + γ − β − 2 d z , ( Similar to (3.11) ) , for all s , t ∈ [ 0 , T ] with s < t. In the same arguments as in the term J 31 ( 3 ), we decompose it into three separate terms and evaluate each through direct calculation. Then, we get J 32 ( 3 ) ≤ e − λ t ( t − s ) 1 − α ∫ s t ∫ s u ( t − u ) γ − 1 L 2 ‖ x u − y u − x v + y v ‖ C r L 2 d v d u ( u − v ) β + 1 + ∫ s t ∫ s u ( t − u ) γ − 1 L R ( 2 ) ‖ x u − x v ‖ C r + ‖ y u − y v ‖ C r + ( u − v ) μ · ‖ x u − y u ‖ C r d v d u ( u − v ) β + 1 ≤ M R ‖ x − y ‖ 1 − α , λ λ α + γ − 1 1 + ‖ x ‖ 1 − α + ‖ y ‖ 1 − α ∫ 0 ∞ z α + γ − 2 e − z d z for all s , t ∈ [ 0 , T ] with s < t. It follows from (3.20)-(3.23) that J 32 ≤ M R λ ) ( 1 + ‖ x ‖ 1 − α + ‖ y ‖ 1 − α ‖ x − y ‖ 1 − α , λ for all s , t ∈ [ 0 , T ] with s < t and lim λ → ∞ M R ( λ ) = 0. Thus, we have showed that the desired estimate ‖ J ( x ) − J ( y ) ‖ ≤ M R ( 1 ) ( λ ) 1 + ‖ x ‖ 1 − α + ‖ y ‖ 1 − α ‖ x − y ‖ 1 − α , λ holds and the constant M R ( 1 ) ( λ ) satisfied lim λ → ∞ M R ( 1 ) ( λ ) = 0.  □

We can also get similar estimates for I ( x ).

Let x ∈ C 1 − α ( [ − r , T ] ). We have that I ( x ) ∈ C 1 − α ( [ 0 , T ] ) and there exist some constants M 2 , M 2 ( λ ) > 0 such that ‖ I ( x ) ‖ 1 − α , λ ≤ M 2 + M 2 ( λ ) ‖ x ‖ 1 − α , λ . If x , y ∈ C 1 − α ( [ − r , T ] ) with ‖ x ‖ ≤ R and ‖ y ‖ ≤ R, there exists M R ( 2 ) ( λ ) > 0 such that ‖ I ( x ) − I ( y ) ‖ 1 − α , λ ≤ M R ( 2 ) ( λ ) ‖ x − y ‖ 1 − α , λ Moreover, M 2 is independent of λ and M 2 ( λ ) , M R ( 2 ) ( λ ) → 0 as λ → ∞.

Define the operator Ψ : C 1 − α ( [ − r , T ] ) → C 1 − α ( [ − r , T ] ) by Ψ ( x ) ( t ) = x ( t ) for t ∈ [ − r , 0 ] and Ψ ( x ) ( t ) = x ( 0 ) + 1 Γ ( γ ) ∫ 0 t ( t − s ) γ − 1 f ( s , x s ) d s + 1 Γ ( γ ) ∫ 0 t ( t − s ) γ − 1 G ( s , x s ) d g ( s ) for t ∈ [ 0 , T ], where the coefficients f and G are as in the equation (1.1), g ∈ C ν ( [ 0 , T ] ) and x ∈ C 1 − α ( [ − r , T ] ). According to Proposition 3.2 and Proposition 3.3, we get the following result.

Let x , y ∈ C 1 − α ( [ − r , T ] ) with ‖ x ‖ ≤ R and ‖ y ‖ ≤ R. There exist some constants M 3 , M 3 ( λ ) , M R ( 3 ) ( λ ) > 0 such that M 3 ( λ ) , M R ( 3 ) ( λ ) → 0 as λ → ∞, and ‖ Ψ ( x ) ‖ 1 − α , λ ≤ M 3 + ‖ x ( 0 ) ‖ 1 − α , λ + M 3 ( λ ) ‖ x ‖ 1 − α , λ and ‖ Ψ ( x ) − Ψ ( y ) ‖ 1 − α , λ ≤ ‖ x ( 0 ) − y ( 0 ) ‖ 1 − α , λ + M R ( 3 ) ( λ ) 1 + ‖ x ‖ 1 − α + ‖ y ‖ 1 − α ‖ x − y ‖ 1 − α , λ .

Consider the following form of the deterministic time fractional differential equation (2.3): x ( t ) = η ( 0 ) + 1 Γ ( γ ) ∫ 0 t ( t − s ) γ − 1 f ( s , x s ) d s + 1 Γ ( γ ) ∫ 0 t ( t − s ) γ − 1 G ( s , x s ) d g ( s ) , t ∈ ( 0 , T ] x ( t ) = η ( t ) , t ∈ [ − r , 0 ] , where the coefficients f and G are as in the equation (1.1), g ∈ C ν ( [ 0 , T ] ) and x ∈ C 1 − α ( [ − r , T ] ).

If η ∈ C 1 − α ( [ − r , 0 ] ), then the equation (2.3) admits a unique solution in C 1 − α ( [ − r , T ] ).

We first prove the existence of the solution. Let H 1 − α ( [ − r , T ] , η ) = x ∈ C 1 − α ( [ − r , T ] ) | x = η on [ − r , 0 ] and define the operator Φ : H 1 − α ( [ − r , T ] , η ) → H 1 − α ( [ − r , T ] , η ) by Φ ( x ) ( t ) = η ( t ) with t ∈ [ − r , 0 ] and Φ ( x ) ( t ) = η ( 0 ) + 1 Γ ( γ ) ∫ 0 t ( t − s ) γ − 1 f ( s , x s ) d s + 1 Γ ( γ ) ∫ 0 t ( t − s ) γ − 1 G ( s , x s ) d g ( s ) with t ∈ [ 0 , T ] for g ∈ C ν ( [ 0 , T ] ) and x ∈ C 1 − α ( [ − r , T ] ). Choose R such that ‖ x ‖ ≤ R. Then, Corollary 3.1 implies that ‖ Φ ( x ) ‖ 1 − α , λ ≤ M 3 + ‖ η ‖ 1 − α , λ + M 3 ( λ ) ‖ x ‖ 1 − α , λ , where M 3 is a constant and lim λ → ∞ M 3 ( λ ) = 0. Let λ 0 be sufficiently large such that M 3 ( λ 0 ) ≤ 1 2 and denote M 0 = 2 ( ‖ η ‖ 1 − α , λ 0 + M 3 ) and B λ 0 = x ∈ H 1 − α ( [ − r , T ] , η ) | ‖ x ‖ 1 − α , λ 0 ≤ M 0 . We then have that Φ maps B λ 0 into itself. Thus, to give the existence we need to show that there exists λ > λ 0 such that Φ is a contraction on B λ 0 under the norm ‖ · ‖ 1 − α , λ . Choose R such that ‖ x ‖ ≤ R and ‖ y ‖ ≤ R. In fact, by Corollary 3.1, we see that ‖ Φ ( x ) − Φ ( y ) ‖ 1 − α , λ ≤ M R ( 3 ) ( λ ) ( 1 + ‖ x ‖ 1 − α + ‖ y ‖ 1 − α ) ‖ x − y ‖ 1 − α , λ for all x , y ∈ H 1 − α ( [ − r , T ] , η ). It follows that ‖ Φ ( x ) − Φ ( y ) ‖ 1 − α , λ ≤ M R ( 3 ) ( λ ) ( 1 + 2 R 0 ) ‖ x − y ‖ 1 − α , λ for all x , y ∈ B λ 0 , where R 0 = sup x ∈ B λ 0 ‖ x ‖ 1 − α . Taking λ > λ 0 with M R ( 3 ) ( λ ) ( 1 + 2 R 0 ) < 1 / 2 to lead ‖ Φ ( x ) − Φ ( y ) ‖ 1 − α , λ ≤ 1 2 ‖ x − y ‖ 1 − α , λ for x , y ∈ B λ 0 . This means that Φ is a contraction on the B λ of the complete metric space C 1 − α ( [ − r , T ] ), which implies that Φ has a fixed point x in B λ 0 . From the definition of Φ, the fixed point x is a solution of (2.3) in C 1 − α ( [ − r , T ] ).

We now prove the uniqueness. Let x an y be two solutions of (2.3) in C 1 − α ( [ − r , T ] ), we can choose R such that ‖ x ‖ ≤ R and ‖ y ‖ ≤ R. Then Corollary 3.1 implies that ‖ x − y ‖ 1 − α , λ ≤ 1 2 ‖ x − y ‖ 1 − α , λ for λ large enough, which shows that x = y. This completes the proof.  □

By the properties of fractional Brownian motion, as a simple consequence of the above facts, we have expounded and proved the existence and uniqueness of the solution of (1.1).

Let 1 2 < H < 1 and 3 2 − H < γ < 1. Assume that the coefficients f and G satisfy the assumptions (H-f) and (H-G), respectively. If 2 − H − γ < α < min 1 2 , μ and η ∈ C 1 − α ( [ − r , 0 ] ) almost surely, then there exists a unique solution x of (1.1) with paths in C 1 − α ( [ − r , T ] ) almost surely.

In [10], Han-Yan showed that the existence and uniqueness of the solution for (1.1) with the coefficients f and G are independent of t and under global Lipschitz conditions. But here we obtain the results for more general equations under local Lipschitz conditions.

 Viability

Keep the notations in Section 3. For g ∈ C ν ( [ 0 , T ] ), we consider the deterministic time fractional differential equation of the form x s η ( t ) = η ( 0 ) + 1 Γ ( γ ) ∫ s t ( t − r ) γ − 1 f ( r , x r s η ) d r + 1 Γ ( γ ) ∫ s t ( t − r ) γ − 1 G ( r , x r s η ) d g ( r ) , t ∈ ( s , T ] with the initial condition x s = η ∈ C r , where 0 ≤ s ≤ T.

If x s η is a solution of (4.1), then x s η is ( 1 − α )-Hölder continuous and ‖ x s η ‖ 1 − α ; [ s , T ] ≤ M ( 1 + ‖ η ‖ C r ) for all 0 ≤ s ≤ T.

By Proposition 3.1 and 3.3, we have ‖ x s η ‖ 1 − α , λ ; [ s , T ] ≤ ‖ η ‖ C r + 1 Γ ( γ ) ∫ s t ( t − r ) γ − 1 f ( r , x r s η ) d r 1 − α , λ ; [ s , T ] + 1 Γ ( γ ) ∫ s t ( t − r ) γ − 1 G ( r , x r s η ) d g ( r ) 1 − α , λ ; [ s , T ] ≤ M 0 ( 1 + ‖ η ‖ C r ) + M ( λ ) ‖ x s η ‖ 1 − α , λ ; [ s , T ] for all λ ≥ 1 and 0 ≤ s < T, where the constant M 0 > 0 is independent of λ, and M ( λ ) converge to zero, as λ → ∞. Thus, we can choose λ ‾ sufficiently large such that M ( λ ) ≤ 1 2 for all λ > λ ‾. It follows that ‖ x s η ‖ 1 − α , λ ; [ s , T ] ≤ 2 M 0 ( 1 + ‖ η ‖ C r ) . and ‖ x s η ‖ 1 − α ; [ s , T ] ≤ 2 e λ ‾ T M 0 ( 1 + ‖ η ‖ C r ) = M ( 1 + ‖ η ‖ C r ) for all 0 ≤ s < T and λ ≥ λ ‾. This completes the proof.  □

If X is a Hölder continuous function with ‖ X ‖ 1 − α ; [ s , T ] ≤ R, then there exist some positive constants M R i , i = 1 , 2 , 3 , 4 such that ∫ τ t ( t − r ) γ − 1 [ f ( r , X r ) − f ( s , X s ) ] d r ≤ M R 1 ( t − τ ) γ ( t − s ) 1 − α , ∫ τ t ( t − r ) γ − 1 [ G ( r , X r ) − G ( s , X s ) ] d g ( r ) ≤ M R 2 ( t − τ ) 1 − α ( t − s ) min { μ , 1 − α } with 0 ≤ s ≤ τ ≤ t ≤ T, and ∫ s τ 1 [ ( τ 2 − r ) γ − 1 − ( t − r ) γ − 1 ] [ f ( r , X r ) − f ( τ 1 , X τ 1 ) ] d r ≤ M R 3 ( τ 2 − t ) γ , ∫ s τ 1 [ ( τ 2 − r ) γ − 1 − ( t − r ) γ − 1 ] [ G ( r , X r ) − G ( τ 1 , X τ 1 ) ] d g ( r ) ≤ M R 4 ( τ 2 − t ) 1 − α with 0 ≤ s ≤ τ 1 ≤ t ≤ τ 2 ≤ T.

By condition (H-f), we have that ∫ τ t ( t − r ) γ − 1 [ f ( r , X r ) − f ( s , X s ) ] d r ≤ ∫ τ t ( t − r ) γ − 1 f ( r , X r ) − f ( s , X s ) d r ≤ ∫ τ t ( t − r ) γ − 1 d r sup r ∈ [ τ , t ] f ( r , X r ) − f ( s , X s ) ≤ ( t − τ ) γ γ L R ( 1 ) sup r ∈ [ τ , t ] | r − s | + ‖ X r − X s ‖ C r ≤ L R ( 1 ) ( t − τ ) γ γ sup r ∈ [ τ , t ] [ ( T α + R ) ( r − s ) 1 − α ] ≤ M R 1 ( t − τ ) γ ( t − s ) 1 − α which gives (4.2). Noting the condition (H-G) implies that | G ( r , X r ) − G ( s , X s ) | ≤ L 2 ( | r − s | μ + ‖ X r − X s ‖ C r ) ≤ L 2 ( | r − s | μ + R | r − s | 1 − α ) , fix β ∈ ( 1 − ν , α 0 ) and we get that ∫ τ t ( t − r ) γ − 1 [ G ( r , X r ) − G ( s , X s ) ] d g ( r ) ≤ Λ β ( g ) { ∫ τ t ( t − r ) γ − 1 [ G ( r , X r ) − G ( s , X s ) ] d r ( r − τ ) β + ∫ τ t ∫ τ r | ( t − r ) γ − 1 [ G ( r , X r ) − G ( s , X s ) ] − ( t − u ) γ − 1 [ G ( u , X u ) − G ( s , X s ) ] | d u d r ( r − u ) β + 1 } ≤ Λ β ( g ) ∫ τ t ( t − r ) γ − 1 | G ( r , X r ) − G ( s , X s ) | d r ( r − τ ) β + ∫ τ t ∫ τ r [ ( t − r ) γ − 1 − ( t − u ) γ − 1 | G ( u , X u ) − G ( s , X s ) | + ( t − r ) γ − 1 | G ( r , X r ) − G ( u , X u ) | ] d u d r ( r − u ) β + 1 ≤ Λ β ( g ) ∫ τ t ( t − r ) γ − 1 L 2 [ ( r − s ) μ + R ( r − s ) 1 − α ] d r ( r − τ ) β + ∫ τ t ∫ τ r { ( t − r ) γ − 1 − ( t − u ) γ − 1 L 2 [ ( u − s ) μ + R ( u − s ) 1 − α ] + ( t − r ) γ − 1 L 2 [ ( r − u ) μ + R ( r − u ) 1 − α ] } d u d r ( r − u ) β + 1 ≤ M R [ ( t − τ ) γ − β ( t − s ) min { μ , 1 − α } + ( t − τ ) γ − β ( t − s ) min { μ , 1 − α } + ( t − τ ) γ − β + min { μ , 1 − α } ] ≤ M R 2 ( t − τ ) 1 − α ( t − s ) min { μ , 1 − α } , and (4.3) follows. Moreover, the inequality (4.4) follows from the next estimates: ∫ s τ 1 ( τ 2 − r ) γ − 1 − ( t − r ) γ − 1 [ f ( r , X r ) − f ( τ 1 , X τ 1 ) ] d r ≤ ∫ s τ 1 ( t − r ) γ − 1 − ( τ 2 − r ) γ − 1 f ( r , X r ) − f ( τ 1 , X τ 1 ) d r ≤ ∫ s τ 1 ( t − r ) γ − 1 − ( τ 2 − r ) γ − 1 d r sup r ∈ [ s , τ 1 ] f ( r , X r ) − f ( τ 1 , X τ 1 ) ≤ ( τ 2 − t ) γ γ L 1 2 + ‖ X r ‖ C r + ‖ X τ 1 ‖ C r ≤ M R 3 ( τ 2 − t ) γ . For the inequality (4.5) with 0 ≤ s ≤ τ 1 ≤ t ≤ τ 2 ≤ T, fix β ∈ ( 1 − ν , α 0 ). Since ∫ s τ 1 ( t − r ) γ − 1 − ( τ 2 − r ) γ − 1 ( r − s ) − β d r = ∫ s t − ∫ τ 1 t ( t − r ) γ − 1 ( r − s ) − β d r − ∫ s τ 2 − ∫ τ 1 τ 2 ( τ 2 − r ) γ − 1 ( r − s ) − β d r ≤ − ∫ τ 1 t ( τ 2 − r ) γ − 1 ( r − s ) − β d r + ∫ τ 1 τ 2 ( τ 2 − r ) γ − 1 ( r − s ) − β d r = ∫ t τ 2 ( τ 2 − r ) γ − 1 ( r − s ) − β d r and similarly ∫ s τ 1 ∫ s r ( τ 2 − r ) γ − 1 − ( t − r ) γ − 1 − ( τ 2 − u ) γ − 1 + ( t − u ) γ − 1 d u d r ( r − u ) β + 1 = ∫ s t ∫ s r − ∫ τ 1 t ∫ s r [ ( t − r ) γ − 1 − ( t − u ) γ − 1 ] d u d r ( r − u ) β + 1 − ∫ s τ 2 ∫ s r − ∫ τ 1 τ 2 ∫ s r [ ( τ 2 − r ) γ − 1 − ( τ 2 − u ) γ − 1 ] d u d r ( r − u ) β + 1 ≤ ∫ t τ 2 ∫ s r ( τ 2 − r ) γ − 1 − ( τ 2 − u ) γ − 1 d u d r ( r − u ) β + 1 , we also have that ∫ s τ 1 [ ( τ 2 − r ) γ − 1 − ( t − r ) γ − 1 ] [ G ( r , X r ) − G ( τ 1 , X τ 1 ) ] d g ( r ) ≤ Λ β ( g ) ∫ s τ 1 ( τ 2 − r ) γ − 1 − ( t − r ) γ − 1 | G ( r , X r ) − G ( τ 1 , X τ 1 ) | d r ( r − s ) β + ∫ s τ 1 ∫ s r ( τ 2 − r ) γ − 1 − ( t − r ) γ − 1 − ( τ 2 − u ) γ − 1 + ( t − u ) γ − 1 · | G ( u , X u ) − G ( τ 1 , X τ 1 ) | d u d r ( r − u ) β + 1 + ∫ s τ 1 ∫ s r ( τ 2 − r ) γ − 1 − ( t − r ) γ − 1 | G ( u , X u ) − G ( r , X r ) | d u d r ( r − u ) β + 1 ≤ M Λ β ( g ) ∫ s τ 1 [ ( t − r ) γ − 1 − ( τ 2 − r ) γ − 1 ] [ ( τ 1 − r ) μ + R ( τ 1 − r ) 1 − α ] d r ( r − s ) β + ∫ s τ 1 ∫ s r ( τ 2 − r ) γ − 1 − ( t − r ) γ − 1 − ( τ 2 − u ) γ − 1 + ( t − u ) γ − 1 · ( τ 1 − u ) μ + R ( τ 1 − u ) 1 − α d u d r ( r − u ) β + 1 + ∫ s τ 1 ∫ s r ( τ 2 − r ) γ − 1 − ( t − r ) γ − 1 ( r − u ) μ + R ( r − u ) 1 − α d u d r ( r − u ) β + 1 ≤ M Λ β ( g ) ∫ s τ 1 ( t − r ) γ − 1 − ( τ 2 − r ) γ − 1 T μ + R T 1 − α d r ( r − s ) β + ∫ s τ 1 ∫ s r ( τ 2 − r ) γ − 1 − ( t − r ) γ − 1 − ( τ 2 − u ) γ − 1 + ( t − u ) γ − 1 · T μ + R T 1 − α d u d r ( r − u ) β + 1 + ∫ s τ 1 ∫ s r ( τ 2 − r ) γ − 1 − ( t − r ) γ − 1 d u d r ( r − u ) β + 1 − μ + R d u d r ( r − u ) α + β ≤ M Λ β ( g ) { ∫ t τ 2 ( τ 2 − r ) γ − 1 d r ( r − s ) β + ∫ t τ 2 ∫ s r ( τ 2 − r ) γ − 1 − ( τ 2 − u ) γ − 1 d u d r ( r − u ) β + 1 + ∫ s τ 1 ( τ 2 − r ) γ − 1 − ( t − r ) γ − 1 T μ − β + R T 1 − α − β d r } ≤ M R ( τ 2 − t ) γ − β + ( τ 2 − t ) γ − β + ( τ 2 − t ) γ ≤ M R 4 ( τ 2 − t ) 1 − α . Thus, we have completed the proof.  □

If X is a Hölder continuous function with ‖ X ‖ 1 − α ; [ s , T ] ≤ R, then there exist some positive constants M ‾ R , M _ R such that ∫ s τ [ ( τ − u ) γ − 1 − ( t − u ) γ − 1 ] f ( u , X u ) d u ≤ M ‾ R ( t − τ ) γ , ∫ s τ [ ( τ − u ) γ − 1 − ( t − u ) γ − 1 ] G ( u , X u ) d g ( u ) ≤ M _ R ( t − τ ) 1 − α with 0 ≤ s ≤ τ ≤ t ≤ T.

The proof of (4.6) is easy, so we omit it here. For (4.7), fix β ∈ ( 1 − ν , α 0 ) and we have ∫ s τ [ ( τ − u ) γ − 1 − ( t − u ) γ − 1 ] G ( u , X u ) d g ( u ) ≤ Λ β ( g ) { ∫ s τ ( τ − u ) γ − 1 − ( t − u ) γ − 1 | G ( u , X u ) | d u ( u − s ) β + ∫ s τ ∫ s u ( τ − u ) γ − 1 − ( t − u ) γ − 1 | G ( u , X u ) − G ( v , X v ) | d v d u ( u − v ) β + 1 + ∫ s τ ∫ s u | ( τ − u ) γ − 1 − ( τ − v ) γ − 1 − ( t − u ) γ − 1 + ( t − v ) γ − 1 | | G ( v , X v ) | d v d u ( u − v ) β + 1 } ≤ Λ β ( g ) { ∫ s τ ( τ − u ) γ − 1 − ( t − u ) γ − 1 L 3 [ 1 + ‖ X u ‖ C r ] d u ( u − s ) β + ∫ s τ ∫ s u ( τ − u ) γ − 1 − ( t − u ) γ − 1 L 2 [ ( u − v ) μ + ‖ X u − X v ‖ C r ] d v d u ( u − v ) β + 1 + ∫ s τ ∫ s u | ( τ − u ) γ − 1 − ( τ − v ) γ − 1 − ( t − u ) γ − 1 + ( t − v ) γ − 1 | L 3 [ 1 + ‖ X v ‖ C r ] d v d u ( u − v ) β + 1 } ≤ M Λ β ( g ) { ∫ s τ ( τ − u ) γ − 1 − ( t − u ) γ − 1 ( 1 + ‖ X ‖ 1 − α ; [ s , T ] ) d u ( u − s ) β + ∫ s τ ∫ s u ( τ − u ) γ − 1 − ( t − u ) γ − 1 · ( u − v ) μ + ( u − v ) 1 − α ‖ X ‖ 1 − α ; [ s , T ] d v d u ( u − v ) β + 1 + ∫ s τ ∫ s u | ( τ − u ) γ − 1 − ( τ − v ) γ − 1 − ( t − u ) γ − 1 + ( t − v ) γ − 1 | ( 1 + ‖ X ‖ 1 − α ; [ s , T ] ) d v d u ( u − v ) β + 1 } ≤ M R [ ( t − τ ) γ − β + ( t − τ ) γ − β + min { μ , 1 − α } + ( t − τ ) γ − β ] ≤ M R ( t − τ ) γ − β ≤ M _ R ( t − τ ) 1 − α for 0 ≤ s ≤ τ ≤ t ≤ T and the lemma follows.  □

Let K = { K ( t ) : t ∈ [ 0 , T ] } be a family of subsets of R. We say that K is viable for (4.1) if for each s ∈ [ 0 , T ] and each η ∈ C r with η ( 0 ) ∈ K ( s ), there exists at least one mild solution x s η = { x s η ( t ) , t ∈ [ s − r , T ] } such that x s η ( t ) ∈ K ( t ) for all t ∈ [ s , T ].

The family K is said to be invariant for (4.1) if for each s ∈ [ 0 , T ] and each η ∈ C r with η ( 0 ) ∈ K ( s ), all mild solutions { x s η } of (4.1) have the property x s η ( t ) ∈ K ( t ) for all t ∈ [ s , T ].

Note that when the equation has a unique mild solution, viability is equivalent with invariance. That is, if the conditions (H-f) and (H-G) hold, viability and invariance for (4.1) is coincident.

(Contingency Set).

Let s ∈ [ 0 , T ] and η ∈ C r with η ( 0 ) ∈ K ( s ). The ( 1 − α ) −fractional g −contingent set to K ( s ) in ( s , η ), denoted by C K ( s ) ( s , η ), is the set of pairs ( ξ , ζ ) ∈ R × R such that there exist h ¯ > 0, a function Q : [ s , s + h ¯ ] → R, and moreover for every R > 0 satisfying ‖ η ‖ C r ≤ R there exist some constants ω = ω R ∈ ( 0 , 1 ), G _ R > 0 and G ‾ R > 0 independent of ( s , h ¯ ) such that | Q ( ρ ) − Q ( τ ) | ≤ G _ R | ρ − τ | 1 − α , | Q ( τ ) | ≤ G ‾ R | τ − s | 1 + ω , for all ρ , τ ∈ [ s , s + h ¯ ] and η ( 0 ) + ( t − s ) γ Γ ( γ + 1 ) ξ + ζ Γ ( γ ) ∫ s t ( t − r ) γ − 1 d g ( r ) + Q ( t ) ∈ K ( t ) for all t ∈ [ s , s + h ¯ ].

 (Tangency Set).

Let s ∈ [ 0 , T ] and η ∈ C r with η ( 0 ) ∈ K ( s ). The ( 1 − α ) −fractional g −tangent set to K ( s ) in ( s , η ), denoted by T K ( s ) ( s , η ), is the set of pairs ( ξ , ζ ) ∈ R × R, such that there exist h ˜ > 0, and two functions U : [ s , s + h ˜ ] → R and V : [ s , s + h ˜ ] → R with U ( s ) = 0 and V ( s ) = 0, and moreover for every R > 0 satisfying ‖ η ‖ C r ≤ R there exist two constants D _ R , D ‾ R independent of ( s , h ˜ ) such that | U ( ρ ) − U ( τ ) | ≤ D _ R | ρ − τ | 1 − α , | V ( ρ ) − V ( τ ) | ≤ D ‾ R | ρ − τ | m i n { μ , 1 − α } for all ρ , τ ∈ [ s , s + h ˜ ] and η ( 0 ) + 1 Γ ( γ ) ∫ s t ( t − r ) γ − 1 [ ξ + U ( r ) ] d r + 1 Γ ( γ ) ∫ s t ( t − r ) γ − 1 [ ζ + V ( r ) ] d g ( r ) ∈ K ( t ) for all t ∈ [ s , s + h ˜ ].

Given two stochastic process U, V satisfying the conditions in Definition 4.4. Then for all s ≤ τ ≤ t ≤ s + h ˜, there exist constants M _ R , M ‾ R such that ∫ τ t ( t − r ) γ − 1 U ( r ) d r ≤ M _ R ( t − s ) 1 − α ( t − τ ) γ , ∫ τ t ( t − r ) γ − 1 V ( r ) d g ( r ) ≤ M ‾ R ( t − s ) min { μ , 1 − α } ( t − τ ) 1 − α .

For (4.8), ∫ τ t ( t − r ) γ − 1 U ( r ) d r = ∫ τ t ( t − r ) γ − 1 [ U ( r ) − U ( s ) ] d r ≤ D _ R ∫ τ t ( t − r ) γ − 1 ( r − s ) 1 − α d r ≤ M _ R ( t − s ) 1 − α ( t − τ ) γ . Indeed, fix β ∈ ( 1 − ν , α 0 ) and we have ∫ τ t ( t − r ) γ − 1 V ( r ) d g ( r ) ≤ Λ β ( g ) ∫ τ t { | ( t − r ) γ − 1 [ V ( r ) − V ( s ) ] | ( r − τ ) β + ∫ τ r | ( t − r ) γ − 1 V ( r ) − ( t − u ) γ − 1 V ( u ) | ( r − u ) β + 1 d u } d r ≤ Λ β ( g ) ∫ τ t { | ( t − r ) γ − 1 [ V ( r ) − V ( s ) ] | ( r − τ ) β + ∫ τ r | [ ( t − r ) γ − 1 − ( t − u ) γ − 1 ] [ V ( u ) − V ( s ) ] | + | ( t − r ) γ − 1 [ V ( r ) − V ( u ) ] | ( r − u ) β + 1 d u } d r ≤ M R ( t − s ) min { μ , 1 − α } ( t − τ ) γ − β ≤ M ‾ R ( t − s ) min { μ , 1 − α } ( t − τ ) 1 − α , which gives (4.9).  □

We can now state the main result in this section.

Let K = { K ( t ) : t ∈ [ 0 , T ] } be a family of nonempty closed subsets of R. Assume (H-f) and (H-G) are satisfied and 2 − ν − γ < α < min 1 2 , μ . Then the following assertions are equivalent:

K is C 1 − α −viable for the fractional differential equation (4.1);

For t ∈ [ 0 , T ] and η ∈ C r with η ( 0 ) ∈ K ( t ), f ( t , η ) , G ( t , η ) ∈ T K ( t ) ( t , η );

For t ∈ [ 0 , T ] and η ∈ C r with η ( 0 ) ∈ K ( t ), f ( t , η ) , G ( t , η ) ∈ C K ( t ) ( t , η ).

We first show that (i) implies (ii). Let s ∈ [ 0 , T ], η ∈ C r with η ( 0 ) ∈ K ( s ) and x s η ∈ C 1 − α ( [ s , T ] ; R ) be a solution of (4.1) such that x s η ( t ) ∈ K ( t ) for all t ∈ [ s , T ]. Set R > 0 such that ‖ η s ‖ C r ≤ R. Then, Lemma 4.1 implies that x s η 1 − α ; [ s , T ] ≤ R 0 = M ( 1 + R ) . Taking h ˜ = min T − t , 1 , we then have x s η ( t ) = η ( 0 ) + 1 Γ ( γ ) ∫ s t ( t − r ) γ − 1 f ( r , x r s η ) d r + 1 Γ ( γ ) ∫ s t ( t − r ) γ − 1 G ( r , x r s η ) d g ( r ) for all t ∈ [ s , s + h ˜ ]. Clearly, for all t ∈ [ s , s + h ˜ ] x s η ( t ) = η ( 0 ) + 1 Γ ( γ ) ∫ s t ( t − r ) γ − 1 f ( s , η ) + U ( r ) d r + 1 Γ ( γ ) ∫ s t ( t − r ) γ − 1 G ( s , η ) + V ( r ) d g ( r ) , where U ( r ) = f ( r , x r s η ) − f ( s , η ) and V ( r ) = G ( r , x r s η ) − G ( s , η ). Obviously, U and V satisfy the conditions described in Definition 4.4, and the assertion (ii) follows.

We nextly show that (ii) implies that (iii). Let ‖ η ‖ C r ≤ R. We need to verify that Q ( t ) = 1 Γ ( γ ) ∫ s t ( t − r ) γ − 1 U ( r ) d r + 1 Γ ( γ ) ∫ s t ( t − r ) γ − 1 V ( r ) d g ( r ) satisfies the Hölder condition from the definition of the contingency property. According to Lemma 4.4(the inequalities (4.8) and (4.9) with τ = s), we have | Q ( t ) | ≤ G ˜ R ( t − s ) 1 + ω . Let s ≤ τ ≤ t ≤ s + h ¯. Fix β ∈ ( 1 − ν , α 0 ) and some elementary calculation may show that ∫ s τ [ ( t − r ) γ − 1 − ( τ − r ) γ − 1 ] V ( r ) d g ( r ) ≤ Λ β ( g ) Γ ( γ ) { ∫ s τ | [ ( t − r ) γ − 1 − ( τ − r ) γ − 1 ] [ V ( r ) − V ( s ) ] | ( r − s ) β d r + ∫ s τ ∫ s r | [ ( t − r ) γ − 1 − ( τ − r ) γ − 1 ] V ( r ) − [ ( t − u ) γ − 1 − ( τ − u ) γ − 1 ] V ( u ) | ( r − u ) β + 1 d u d r } ≤ Λ β ( g ) Γ ( γ ) { ∫ s τ D _ R | [ ( t − r ) γ − 1 − ( τ − r ) γ − 1 ] ( r − s ) min { μ , 1 − α } | ( r − s ) β d r + ∫ s τ ∫ s r | [ ( t − r ) γ − 1 − ( τ − r ) γ − 1 − ( t − u ) γ − 1 + ( τ − u ) γ − 1 ] [ V ( r ) − V ( s ) ] | ( r − u ) β + 1 d u d r + ∫ s τ ∫ s r | [ ( t − u ) γ − 1 − ( τ − u ) γ − 1 ] [ V ( r ) − V ( u ) ] | ( r − u ) β + 1 d u d r } ≤ M ( t − τ ) γ − β + min { μ , 1 − α } ≤ M ( t − τ ) 1 − α . Combining this with Lemma 4.4, we obtain that | Q ( t ) − Q ( τ ) | = 1 Γ ( γ ) ∫ s t ( t − r ) γ − 1 U ( r ) d r + ∫ s t ( t − r ) γ − 1 V ( r ) d g ( r ) − ∫ s τ ( τ − r ) γ − 1 U ( r ) d r − ∫ s τ ( τ − r ) γ − 1 V ( r ) d g ( r ) ≤ 1 Γ ( γ ) { | ∫ s τ ( t − r ) γ − 1 − ( τ − r ) γ − 1 [ U ( r ) − U ( s ) ] d r | + ∫ τ t ( t − r ) γ − 1 [ U ( r ) − U ( s ) ] d r + ∫ s τ ( t − r ) γ − 1 − ( τ − r ) γ − 1 V ( r ) d g ( r ) + ∫ τ t ( t − r ) γ − 1 V ( r ) d g ( r ) } ≤ G ‾ R ( t − τ ) 1 − α . This shows that Q is ( 1 − α )-Hölder continuous on [ s , s + h ¯ ].

We finally show that the assertion (iii) implies (i). Fix s ∈ [ 0 , T ], η ∈ C r with η ( 0 ) ∈ K ( s ) and 0 < ε ≤ 1. Let R > 0 satisfy ‖ η ‖ C r ≤ R and denote by A ε ( s , η ) the set of pairs ( T x , x ), where T x ∈ [ 0 , T ] and x : [ s , T x ] → R is a Hölder continuous function satisfying the following conditions:

x s = η, x ( t ) ∈ K ( t ) for all t ∈ [ s , T x ] and there exists a positive constant M 0 ≥ R depending only on R, T, α, γ and independent of ε, such that ‖ x ( · ) ‖ 1 − α ; [ s , T x ] ≤ M 0 ;

The error function ξ : [ s , T x ] → R defined by ξ ( t ) = x ( t ) − η ( 0 ) − 1 Γ ( γ ) ∫ s t ( t − u ) γ − 1 f ( u , x u ) d u − 1 Γ ( γ ) ∫ s t ( t − u ) γ − 1 G ( u , x u ) d g ( u ) satisfies

| ξ ( t ) | ≤ ε ( t − s ) γ − α for all 0 ≤ s < t ≤ T x ,

| ξ ( t ) − ξ ( τ ) | ≤ M 1 | t − τ | 1 − α for all 0 ≤ s < τ < t ≤ T x , where the constant M 1 depending only on R, T, α, γ and independent of ε.

 It is easy to check that the set A ε ( s , η ) is non-empty. Define T x ε : = sup { θ ∈ [ s , T ] : ∃ ( θ , x ) ∈ A ε ( s , η ) } , and construct a trajectory x ε on the maximal interval [ s , T x ε ) by concatenating admissible trajectories on subintervals converging to T x ε . The uniform Hölder bound in condition (1) ensures that x ε is uniformly continuous on [ s , T x ε ), which implies that lim r → T x ε − x ε ( r ) exists. Then, x ε can be continuously extended to the closed interval [ s , T x ε ]. The extended function still satisfies conditions (1)–(2), hence ( T x ε , x ε ) ∈ A ε ( s , η ). We shall prove that T x ε = T by reductio ad absurdum.

Assume that T x ε < T. Denote x T x ε ε = η ε . We have ‖ x ε ‖ 1 − α ; [ s , T x ε ] ≤ M 0 and in particular ‖ η ε ‖ C r ≤ M 0 . From the hypotheses we have known that f ( T x ε , η ε ) , G ( T x ε , η ε ) is ( 1 − α )-fractional g −contingent to K in ( T x ε , η ε ), i.e. there exist h ‾ ε > 0 sufficiently small (for the moment 0 < h ‾ ε < min T − T x ε , 1 ), and a Borel function Q ε : [ T x ε , T x ε + h ‾ ε ] → R , and some positive constants ω = ω M 0 ∈ ( 0 , 1 ), G ‾ 0 = G ‾ M 0 , G _ 0 = G _ M 0 > 0 independent of ( T x ε , h ‾ ε ) such that | Q ε ( s ) − Q ε ( τ ) | ≤ G ‾ 0 | s − τ | 1 − α and | Q ε ( s ) | ≤ G _ 0 | s − T x ε | 1 + ω for all s , τ ∈ [ T x ε , T x ε + h ‾ ε ], and η ε ( 0 ) + ( t − T x ε ) γ Γ ( γ + 1 ) f ( T x ε , η ε ) + G ( T x ε , η ε ) Γ ( γ ) ∫ T x ε t ( t − r ) γ − 1 d g ( r ) + Q ε ( t ) ∈ K ( t ) for all t ∈ [ T x ε , T x ε + h ‾ ε ]. We set T ε = T x ε + h ‾ ε and define x ˆ ε : [ s , T ε ] → R as an extension of x ε by x ˆ ε ( t ) = x ε ( t ) , t ∈ [ s , T x ε ] η ε ( 0 ) + ( t − T x ε ) γ Γ ( γ + 1 ) f ( T x ε , η ε ) + G ( T x ε , η ε ) Γ ( γ ) ∫ T x ε t ( t − r ) γ − 1 d g ( r ) + Q ε ( t ) , t ∈ ( T x ε , T ε ] . We will prove that the extension ( T ε , x ˆ ε ) ∈ A ε ( s , η ) in two steps.

Step I. Clearly x ˆ s ε = η ε and x ˆ ε ( t ) ∈ K ( t ) for all t ∈ [ s , T ε ]. Now let we show that ‖ x ˆ ε ‖ 1 − α ; [ s , T ε ] ≤ M 0 ( 1 ) , where M 0 ( 1 ) ≥ R and M 0 ( 1 ) depends only on T, α, γ. For T x ε ≤ τ ≤ t ≤ T ε , we have | x ˆ ε ( t ) − x ˆ ε ( τ ) | = ( t − T x ε ) γ Γ ( γ + 1 ) f ( T x ε , η ε ) − ( τ − T x ε ) γ Γ ( γ + 1 ) f ( T x ε , η ε ) + | G ( T x ε , η ε ) | Γ ( γ ) ∫ T x ε t ( t − r ) γ − 1 d g ( r ) − ∫ T x ε τ ( τ − r ) γ − 1 d g ( r ) + | Q ε ( t ) − Q ε ( τ ) | ≤ | t − τ | γ Γ ( γ + 1 ) ( 1 + ‖ η ε ‖ C r ) + M | t − τ | 1 − α Γ ( γ ) ( 1 + ‖ η ε ‖ C r ) + M | t − τ | 1 − α ≤ M R | t − τ | 1 − α , and | x ˆ ε ( t ) | ≤ | x ˆ ε ( t ) − x ˆ ε ( T x ε ) | + ‖ η ε ‖ C r ≤ M R T 1 − α + M 0 . Hence, ‖ x ˆ ε ( · ) ‖ 1 − α ; [ s , T ε ] ≤ 2 ‖ x ˆ ε ( · ) ‖ 1 − α ; [ s , T x ε ] + 2 ‖ x ˆ ε ( · ) ‖ 1 − α ; [ T x ε , T ε ] ≤ 2 M 0 + 2 M R T 1 − α + 2 M 0 + 2 M R : = M 0 ( 1 ) .

Step II. Let the error functions ξ ε : [ s , T x ε ] → R and ξ ˆ ε : [ s , T ε ] → R as follows ξ ε = x ε ( t ) − η ( 0 ) − 1 Γ ( γ ) ∫ s t ( t − u ) γ − 1 f ( u , x u ε ) d u − 1 Γ ( γ ) ∫ s t ( t − u ) γ − 1 G ( u , x u ε ) d g ( u ) , ξ ˆ ε = x ˆ ε ( t ) − η ( 0 ) − 1 Γ ( γ ) ∫ s t ( t − u ) γ − 1 f ( u , x ˆ u ε ) d u − 1 Γ ( γ ) ∫ s t ( t − u ) γ − 1 G ( u , x ˆ u ε ) d g ( u ) . Clearly, | ξ ε ( t ) | = | ξ ˆ ε ( t ) | ≤ ε ( t − s ) γ − α for all t ∈ [ s , T x ε ]. Let t ∈ [ T x ε , T ε ]. Using Lemma 4.2(the inequalities (4.2),(4.3) with τ = s = T x ε , X s = η ε , X u = x ˆ u ε ) and Lemma 4.3(the inequalities (4.6),(4.7) with τ = T x ε , X u = x u ε ), we have | ξ ˆ ε ( t ) | ≤ | η ε ( 0 ) − η ( 0 ) − 1 Γ ( γ ) ∫ s T x ε ( T x ε − u ) γ − 1 f ( u , x u ε ) d u − 1 Γ ( γ ) ∫ s T x ε ( T x ε − u ) γ − 1 G ( u , x u ε ) d g ( u ) | + | ( t − T x ε ) γ Γ ( γ + 1 ) f ( T x ε , η ε ) + G ( T x ε , η ε ) Γ ( γ ) ∫ T x ε t ( t − r ) γ − 1 d g ( r ) − 1 Γ ( γ ) ∫ s t ( t − u ) γ − 1 f ( u , x ˆ u ε ) d u − 1 Γ ( γ ) ∫ s t ( t − u ) γ − 1 G ( u , x ˆ u ε ) d g ( u ) + 1 Γ ( γ ) ∫ s T x ε ( T x ε − u ) γ − 1 f ( u , x u ε ) d u + 1 Γ ( γ ) ∫ s T x ε ( T x ε − u ) γ − 1 G ( u , x u ε ) d g ( u ) | + | Q ε ( t ) | ≤ ε ( T x ε − s ) γ − α + G _ 0 ( t − T x ε ) 1 + ω + 1 Γ ( γ ) ∫ T x ε t ( t − u ) γ − 1 f ( T x ε , η ε ) − f ( u , x ˆ u ε ) d u + ∫ T x ε t ( t − u ) γ − 1 G ( T x ε , η ε ) − G ( u , x ˆ u ε ) d g ( u ) + 1 Γ ( γ ) ∫ s T x ε [ ( T x ε − u ) γ − 1 − ( t − u ) γ − 1 ] f ( u , x u ε ) d u + ∫ s T x ε [ ( T x ε − u ) γ − 1 − ( t − u ) γ − 1 ] G ( u , x u ε ) d g ( u ) ≤ ε ( T x ε − s ) γ − α + G _ 0 ( t − T x ε ) 1 + ω + M R 1 ( t − T x ε ) 1 + γ − α + M R 2 ( t − T x ε ) 1 − α + min { μ , 1 − α } + M ‾ R ( t − T x ε ) γ + M _ R ( t − T x ε ) 1 − α ≤ ε ( t − s ) γ − α for h ‾ ε sufficiently small. Hence | ξ ˆ ε ( t ) | ≤ ε ( t − s ) γ − α for all t ∈ [ s , T ε ] . Let T x ε ≤ τ ≤ t ≤ T ε . Using Lemma 4.2 again (the inequalities (4.2),(4.3) with s = T x ε , X s = η ε , X u = x ˆ u ε , and the inequalities (4.4),(4.5) with τ 1 = T x ε , X τ 1 = η ε , X u = x ˆ u ε , T x ε ≤ τ ≤ t ≤ T ε ), we have | ξ ˆ ε ( t ) − ξ ˆ ε ( τ ) | ≤ 1 Γ ( γ ) { ∫ τ t ( t − u ) γ − 1 [ f ( T x ε , η ε ) − f ( u , x ˆ u ε ) ] d u + ∫ τ t ( t − u ) γ − 1 [ G ( T x ε , η ε ) − G ( u , x ˆ u ε ) ] d g ( u ) + ∫ T x ε τ [ ( t − u ) γ − 1 − ( τ − u ) γ − 1 ] f ( T x ε , η ε ) − f ( u , x ˆ u ε ) d u + ∫ T x ε τ [ ( t − u ) γ − 1 − ( τ − u ) γ − 1 ] G ( T x ε , η ε ) − G ( u , x ˆ u ε ) d g ( u ) + ∫ τ T x ε [ ( t − u ) γ − 1 − ( τ − u ) γ − 1 ] f ( T x ε , η ε ) − f ( u , x ˆ u ε ) d u + ∫ τ T x ε [ ( t − u ) γ − 1 − ( τ − u ) γ − 1 ] [ G ( T x ε , η ε ) − G ( u , x ˆ u ε ) ] d g ( u ) | } + | Q ε ( t ) − Q ε ( t ) | ≤ M R ( t − τ ) 1 − α . From the definition of A ε ( s , η ) it follows that | ξ ˆ ε ( t ) − ξ ˆ ε ( τ ) | = | ξ ε ( t ) − ξ ε ( τ ) | ≤ M 1 ( t − τ ) 1 − α . for s ≤ τ < t ≤ T x ε .

On the other hand, we also have | ξ ˆ ε ( t ) − ξ ˆ ε ( τ ) | ≤ | ξ ˆ ε ( t ) − ξ ˆ ε ( T x ε ) | + | ξ ˆ ε ( T x ε ) − ξ ˆ ε ( τ ) | ≤ M R ( t − T x ε ) 1 − α + M 1 ( T x ε − τ ) 1 − α ≤ ( M R ∨ M 1 ) ( t − τ ) 1 − α for s ≤ τ < T x ε < t ≤ T ε . Therefore, we have constructed an extension ( T ε , x ˆ ε ) with T x ε < T ε ≤ T such that ( T ε , x ˆ ε ) ∈ A ε ( s , η ). This contradicts the definition of T x ε as the supremum of admissible existence times. So, we have T x ε = T.

Let x ε be the solution defined on [ s , T ] of A ε ( s , η ). Then from the definition of A ε ( s , η ), we see that x s ε = η, x ε ( t ) ∈ K ( t ) for all t ∈ [ s , T ], and there exists a positive constant M 0 ≤ R depending only on α, γ, T, R, such that ‖ x ε ( · ) ‖ 1 − α ; [ s , T ] ≤ M 0 . Consider the error function ξ ε : [ s , T ] → R defined by ξ ε = x ε ( t ) − η ( 0 ) − 1 Γ ( γ ) ∫ s t ( t − u ) γ − 1 f ( u , x u ε ) d u − 1 Γ ( γ ) ∫ s t ( t − u ) γ − 1 G ( u , x u ε ) d g ( u ) such that

| ξ ε ( t ) | ≤ ε ( t − s ) γ − α for all 0 ≤ s < t ≤ T,

| ξ ε ( t ) − ξ ε ( τ ) | ≤ M 1 | t − τ | 1 − α for all 0 ≤ s < τ < t ≤ T,

 where the constant M 1 depends only on α, γ, T, R independent of ε.

To show that (1) is true, we need to estimate ‖ ξ ε ‖ α , λ ; [ s , T ] and ‖ x ε − x κ ‖ α , λ ; [ s , T ] . We have ‖ ξ ε ‖ α , λ ; [ s , T ] ≤ ‖ ξ ε ‖ α ; [ s , T ] = sup t ∈ [ s , T ] | ξ ε ( t ) | + sup s ≤ τ < t ≤ T | ξ ε ( t ) − ξ ε ( τ ) | ( t − τ ) α ≤ ε ( T − s ) γ − α + sup s ≤ τ < t ≤ T | ξ ε ( t ) − ξ ε ( τ ) | 1 2 − α | ξ ε ( t ) − ξ ε ( τ ) | 1 2 + α ( t − τ ) α ≤ ε ( T − s ) γ − α + [ 2 ε ( T − s ) γ − α ] 1 2 − α M 1 1 2 + α ( T − s ) ( 1 2 − α ) ( 1 + α ) ≤ M R ε 1 2 − α for any λ ≥ 0. It remains to prove that the limit of the sequence x ε exists as ε → 0 and this limit is a solution to the equation (4.1). Let 0 < ε, κ ≤ 1. Using the Proposition 3.2 and Proposition 3.3, we get ‖ x ε − x κ ‖ α , λ ; [ s , T ] ≤ ‖ I ( x ε ) − I ( x κ ) ‖ α , λ ; [ s , T ] + ‖ J ( x ε ) − J ( x κ ) ‖ α , λ ; [ s , T ] + ‖ ξ ε − ξ κ ‖ α , λ ; [ s , T ] ≤ M ( λ ) ‖ x ε − x κ ‖ α , λ ; [ s , T ] + M R ( λ ) ( 1 + ‖ x ε ‖ α , λ ; [ s , T ] + ‖ x ε ‖ α , λ ; [ s , T ] ) · ‖ x ε − x κ ‖ α , λ ; [ s , T ] + M R ε 1 2 − α + M R κ 1 2 − α ≤ [ M ( λ ) + M R ( λ ) ( 1 + 2 M 0 ) ] ‖ x ε − x κ ‖ α , λ ; [ s , T ] + M R ε 1 2 − α + M R κ 1 2 − α . Let λ = λ ¯ sufficiently large such that M ( λ ) + M R ( λ ) ( 1 + 2 M 0 ) ≤ 1 2 . Then ‖ x ε − x κ ‖ ∞ ; [ s , T ] ≤ e λ ¯ T ‖ x ε − x κ ‖ α , λ ; [ s , T ] ≤ 2 M R e λ ¯ T ε 1 2 − α + κ 1 2 − α . Hence there exists x s η such that x ε → x s η in C ( [ s , T ] ; R ) and x ε → x s η in W α , ∞ ( [ s , T ] ; R ). Noting that ‖ x ε ‖ ∞ ; [ s , T ] + | x ε ( t ) − x ε ( τ ) | | t − τ | 1 − α ≤ M 0 for all t , τ ∈ [ s , T ], we obtain ‖ x s η ( · ) ‖ 1 − α ; [ s , T ] ≤ M 0 by setting ε → 0. By x ε ( t ) ∈ K ( t ) for all t ∈ [ s , T ] it follows that x s η ( t ) ∈ K ( t ) for all t ∈ [ s , T ], which implies that x s η ( t ) is a solution of (4.1). Thus, we have prove that the assertion (1) is true and the theorem follows.  □

Let K = { K ( t ) : t ∈ [ 0 , T ] } be a family of nonempty closed subsets of R. Assume (H-f) and (H-G) are satisfied and 2 − H − γ < α < min 1 2 , μ . Then the following assertions are equivalent:

K is C 1 − α -viable for the fractional differential equation (1.2), i.e. for each s ∈ [ 0 , T ] and each η ∈ C r with η ( 0 ) ∈ K ( s ), there exists a mild solution x s η ( ω , · ) ∈ C 1 − α ( s , T ) of the equation x s η ( t ) = η s ( 0 ) + 1 Γ ( γ ) ∫ s t ( t − r ) γ − 1 f ( r , x r s η ) d r + 1 Γ ( γ ) ∫ s t ( t − r ) γ − 1 G ( r , x r s η ) d B H ( r ) , t ∈ ( s , T ] , a . s . ω ∈ Ω x s = η ∈ C r , and x s η ( t ) ∈ K ( t ), for all t ∈ [ s , T ];

for each s ∈ [ 0 , T ] and each η ∈ C r with η ( 0 ) ∈ K ( s ), f ( s , η ) , G ( s , η ) is ( 1 − α )-fractional B H -contingent to K ( s ) in ( s , η ).

As mentioned before, the result follows directly from the deterministic Theorem 4.1.  □

Moreover, it follows:

Assume (H-f) and (H-G) are satisfied and 2 − ν − γ < α < min 1 2 , μ . If K = [ a , b ] ⊂ R is independent of t, the following assertions are equivalent:

K is C 1 − α −viable for the fractional differential equation (4.1);

For t ∈ [ 0 , T ] and η ∈ C r with η ( 0 ) = a or η ( 0 ) = b, f ( t , η ) , G ( t , η ) ∈ T K ( t , η );

For t ∈ [ 0 , T ] and η ∈ C r with η ( 0 ) = a or η ( 0 ) = b, f ( t , η ) , G ( t , η ) ∈ C K ( t , η ).

Since K is independent of t, from Theorem 4.1, it is obvious that ( j ) ⇒ ( j j ) ⇒ ( j j j ). It is sufficient to prove ( j j j ) ⇒ ( j ). Let t ∈ [ 0 , T ] and η ∈ C r with η ( 0 ) ∈ ( a , b ) be chosen arbitrary. Since x s η ( t ) is continuous, there exists a random viable h ˜ such that for all t ∈ [ s , s + h ˜ ]„ x s η ( t ) = η ( 0 ) + 1 Γ ( γ ) ∫ s t ( t − r ) γ − 1 f ( r , x r s η ) d r + 1 Γ ( γ ) ∫ s t ( t − r ) γ − 1 G ( r , x r s η ) d g ( r ) ∈ K , which can be written as x s η ( t ) = η ( 0 ) + 1 Γ ( γ ) ∫ s t ( t − r ) γ − 1 [ f ( s , η ) + U ( r ) ] d r + 1 Γ ( γ ) ∫ s t ( t − r ) γ − 1 [ G ( s , η ) + V ( r ) ] d g ( r ) ∈ K , where U ( r ) = f ( r , x r s η ) − f ( s , η ) and V ( r ) = G ( r , x r s η ) − G ( s , η ). Obviously ( f ( t , η ) , G ( t , η ) ) ∈ T K ( t , η ). By ( j j j ) and taking into account that K is independent of t, we can get ( i i i ) of Theorem 4.1. Then the corollary follows.  □

 Positive solution

An application of viability is the existence of positive solution. Firstly we prove some preliminary results.

For every s ∈ [ 0 , + ∞ ) and 1 − H < γ, we have lim sup t ↓ s 1 ( t − s ) γ ∫ s t ( t − r ) γ − 1 d B H ( r ) = ∞ with probability one.

Given s ≥ 0 and consider Gaussian random variables X t ( s ) = ∫ s t ( t − r ) γ − 1 d B H ( r ) ( t − s ) γ + H − 1 , t ≥ s . Notice that E ∫ s t ( t − r ) γ − 1 d B H ( r ) ( t − s ) γ + H − 1 2 = H ( 2 H − 1 ) ( t − s ) 2 γ + 2 H − 2 ∫ s t ∫ s t ( t − r 1 ) γ − 1 ( t − r 2 ) γ − 1 | r 1 − r 2 | 2 H − 2 d r 1 d r 2 = H ( 2 H − 1 ) ( t − s ) 2 γ + 2 H − 2 ∫ 0 t − s ∫ 0 t − s x γ − 1 y γ − 1 | x − y | 2 H − 2 d x d y = H ( 2 H − 1 ) γ + H − 1 B ( γ , 2 H − 1 ) ≡ σ 2 , where B ( · , · ) is the classical Beta function. We see that X t ( s ) = σ ζ in distribution for any 0 ≤ s < t, where ζ ∼ N ( 0 , 1 ). It remains to prove that the convergence (5.1) holds with probability one. To end this, for the non-negative sequence { δ n } with δ n ↓ 0 ( n → ∞) we define the events A s ( δ n ) = sup 0 ≤ t − s ≤ δ n 1 ( t − s ) γ ∫ s t ( t − r ) γ − 1 d B H ( r ) > d = sup 0 ≤ t − s ≤ δ n 1 ( t − s ) 1 − H X t ( s ) > d , n = 1 , 2 , … for any d > 0 and s ≥ 0. Then, A s ( δ n ) ⊃ A s ( δ n + 1 ) for all n ≥ 1 and P A s ( δ n ) ≥ P 1 δ n 1 − H X δ n + s ( s ) > d = P | ζ | > d σ δ n 1 − H ⟶ P | ζ | > 0 = 1 for any d > 0 and s ≥ 0, as n tends to infinity. This shows that P A s ( δ n ) ⟶ 1 , as n tends to infinity. By the Borel-Cantelli lemma and the arbitrariness of d, the convergence (5.1) follows.  □

Since 1 ( t − s ) γ ∫ s t ( t − r ) γ − 1 d B H ( r ) is centered Gaussian and symmetric in distribution, the events A = lim sup t ↓ s 1 ( t − s ) γ ∫ s t ( t − r ) γ − 1 d B H ( r ) = ∞ , B = lim inf t ↓ s 1 ( t − s ) γ ∫ s t ( t − r ) γ − 1 d B H ( r ) = − ∞ have the same probability. As t → ∞, the variance of 1 ( t − s ) γ ∫ s t ( t − r ) γ − 1 d B H ( r ) diverges, and the trajectories will almost surely be unbounded and therefore cross any fixed threshold. Consequently, these events are not disjoint. Lemma 5.1 implies that P ( A ∪ B ) = 1. Then, the following convergence hold: P lim sup t ↓ s 1 ( t − s ) γ ∫ s t ( t − r ) γ − 1 d B H ( r ) = ∞ = 1 and P lim inf t ↓ s 1 ( t − s ) γ ∫ s t ( t − r ) γ − 1 d B H ( r ) = − ∞ = 1 .

Assume (H-f) and (H-G) are satisfied and 2 − H − γ < α < min 1 2 , μ . Let K = [ a , b ] independent of t. Then for t ∈ [ 0 , T ] and η ∈ C r with η ( 0 ) = a, f ( t , η ) , G ( t , η ) ∈ T K ( t , η ) ⟺ f ( t , a ) ≥ 0 , G ( t , a ) = 0 , and η ∈ C r with η ( 0 ) = b, f ( t , η ) , G ( t , η ) ∈ T K ( t , η ) ⟺ f ( t , b ) ≤ 0 , G ( t , b ) = 0 .

It is convenient to prove only for the case: η ∈ C r with η ( 0 ) = a, the other case is similar.

To prove the necessity. If f ( t , a ) ≥ 0, G ( t , a ) = 0, taking U ( r ) ≡ 0, V ( r ) ≡ 0, we can find h ¯ small enough, such that for all t ∈ [ s , s + h ¯ ], a ≤ η ( 0 ) + 1 Γ ( γ ) ∫ s t ( t − r ) γ − 1 [ f ( s , a ) + U ( r ) ] d r + 1 Γ ( γ ) ∫ s t ( t − r ) γ − 1 [ G ( s , a ) + V ( r ) ] d B H ( r ) ≤ b . This means that f ( t , η ) , G ( t , η ) ∈ T K ( t , η ).

To prove the sufficiency. Since ( f ( t , η ) , G ( t , η ) ) ∈ T K ( t , η ), there exist h ˜ > 0 and two functions U and V satisfying the conditions in Definition 4.4, and for all t ∈ [ s , s + h ˜ ] and η ∈ C r with η ( 0 ) = a, a ≤ η ( 0 ) + 1 Γ ( γ ) ∫ s t ( t − r ) γ − 1 [ f ( s , a ) + U ( r ) ] d r + 1 Γ ( γ ) ∫ s t ( t − r ) γ − 1 [ G ( s , a ) + V ( r ) ] d B H ( r ) ≤ b . Then, we have ( t − s ) γ γ f ( s , a ) + G ( s , a ) ∫ s t ( t − r ) γ − 1 d B H ( r ) + ∫ s t ( t − r ) γ − 1 U ( r ) d r + ∫ s t ( t − r ) γ − 1 V ( r ) d B H ( r ) ≥ 0 . By Remark 5.1, for P-almost every ω ∈ Ω, (5.2) is satisfied and there exists a sequence s ≤ t n = t n ( ω ) ≤ s + h ¯ ( ω ), t n ↓ s, such that lim t n ↓ s 1 ( t n − s ) γ ∫ s t n ( t n − r ) γ − 1 d B H ( r , ω ) = + ∞ . According to Lemma 4.4 ∫ s t n ( t n − s ) γ − 1 U ( r ) d r + ∫ s t n ( t n − s ) γ − 1 V ( r ) d B H ( r , ω ) ( t n − s ) γ → 0 , we have G ( s , a ) ≤ 0.

Similarly we can prove that G ( s , a ) ≥ 0 by P lim inf t ↓ s 1 ( t − s ) γ ∫ s t ( t − r ) γ − 1 d B H ( r ) = − ∞ = 1 . Consequently, G ( s , a ) = 0 . Then by (5.2), we deduce that f ( s , a ) + ∫ s t ( t − r ) γ − 1 U ( r ) d r + ∫ s t ( t − r ) γ − 1 V ( r ) d B H ( r ) ( t − s ) γ ≥ 0 . Via Lemma 4.4, it follows that f ( s , a ) ≥ 0 by taking t → s.  □

Assume (H-f) and (H-G) are satisfied and 2 − H − γ < α < min 1 2 , μ . Then, for any t ∈ [ 0 , T ] and η ∈ C r with η ( 0 ) ≥ 0, the fractional equation (1.1) has a positive solution if and only if f ( t , 0 ) ≥ 0 , G ( t , 0 ) = 0 for all t ∈ [ 0 , T ].

Taking K = [ 0 , + ∞ ). By Lemma 5.2, we have that ( f ( t , 0 ) , G ( t , 0 ) ) ∈ T K ( t , 0 ) if and only if f ( t , 0 ) ≥ 0 and G ( t , 0 ) = 0. Taking into account of Corollary 4.1, the result follows.  □

 Acknowledgments

The authors would like to thank anonymous referees for their constructive remarks and suggestions, which greatly improved the presentation of our paper.

 References Alfifi, H.Y., Saad, I.B., Turki, S.: Existence and asymptotic behavior of positive solutions for a coupled system of semilinear fractional differential equations. Result. Math. 71, 705–730 (2017) MR3648441. https://doi.org/10.1007/s00025-016-0528-9 Alós, E., Mazet, O., Nualart, D.: Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29, 766–801 (2001) MR1849177. https://doi.org/10.1214/aop/1008956692 Aubin, J.P.: Viability Theory. Springer (2009) Aubin, J.P., Da Prato, G.: Stochastic viability and invariance. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 27, 595–694 (1990) MR1093711 Bai, C., Fang, J.: The existence of a positive solution for a singular coupled system of nonlinear fractional differential equations. Appl. Math. Comput. 150, 611–621 (2004) MR2039662. https://doi.org/10.1016/S0096-3003(03)00294-7 Biagini, F., Hu, Y., Øksendal, B., Zhang, T.: Stochastic Calculus for Fractional Brownian Motion and Applications. Probab. Appl.. Springer (2008) MR2387368. https://doi.org/10.1007/978-1-84628-797-8 Carja, O., Donchev, T., Rafaqat, M., Ahmed, R.: Viability of fractional differential inclusions. Appl. Math. Lett. 38, 48–51 (2014) MR3258200. https://doi.org/10.1016/j.aml.2014.06.012 Ciotir, I., Rascanu, A.: Viability for differential equations driven by fractional Brownian motion. J. Differ. Equ. 247, 1505–1528 (2009) MR2541419. https://doi.org/10.1016/j.jde.2009.06.002 Girejko, E., Mozyrska, D., Wyrwas, M.: A sufficient condition of viability for fractional differential equations with the caputo derivative. J. Math. Anal. Appl. 381, 146–154 (2011) MR2796198. https://doi.org/10.1016/j.jmaa.2011.04.004 Han, J., Yan, L.: A time fractional functional differential equation driven by the fractional Brownian motion. J. Appl. Anal. Comput. 9, 547–567 (2019) MR3937564. https://doi.org/10.11948/2156-907X.20180068 Hu, Y.: Integral Transformations and Anticipative Calculus for Fractional Brownian Motions. Mem. Amer. Math. Soc., vol. 175 (2005) MR2130224. https://doi.org/10.1090/memo/0825 Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier (2006) MR2218073 Li, Z., Xu, L., Zhou, J.: Viability for coupled sdes driven by fractional Brownian motion. Appl. Math. Optim. 84(Suppl 1), 55–98 (2021) MR4316780. https://doi.org/10.1007/s00245-021-09761-z Melnikov, A., Mishura, Y., Shevchenko, G.: Stochastic viability and comparison theorems for mixed stochastic differential equations. Methodol. Comput. Appl. Probab. 17, 169–188 (2015) MR3306678. https://doi.org/10.1007/s11009-013-9336-9 Mishura, Y.S.: Stochastic Calculus for Fractional Brownian Motion and Related Processes. Springer (2008) MR2378138. https://doi.org/10.1007/978-3-540-75873-0 Nie, T., Rascanu, A.: Deterministic characterization of viability for stochastic differential equation driven by fractional Brownian motion. ESAIM Control Optim. Calc. Var. 18, 915–929 (2012) MR3019464. https://doi.org/10.1051/cocv/2011188 Nourdin, I.: Selected Aspects of Fractional Brownian Motion. Springer (2012) MR3076266. https://doi.org/10.1007/978-88-470-2823-4 Nualart, D.: Malliavin Calculus and Related Topics. Springer (2006) MR2200233 Nualart, D., Rascanu, A.: Differential equations driven by fractional Brownian motion. Collect. Math. 53, 55–82 (2002) MR1893308 Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach (1993) MR1347689 Tudor, C.A.: Analysis of Variations for Self-similar Processes. Springer (2013) MR3112799. https://doi.org/10.1007/978-3-319-00936-0 Xu, L.: Viability for stochastic functional differential equations with infinite memory driven by a fractional Brownian motion. Phys. A 527, 121076 (2019) MR3952203. https://doi.org/10.1016/j.physa.2019.121076 Xu, L., Luo, J.: Viability for stochastic functional differential equations in Hilbert spaces driven by fractional Brownian motion. Appl. Math. Comput. 341, 93–110 (2019) MR3861742. https://doi.org/10.1016/j.amc.2018.08.016 Young, L.C.: An inequality of Hölder type connected with stieltjes integration. Acta Math. 67, 251–282 (1936) MR1555421. https://doi.org/10.1007/BF02401743 Zähle, M.: Integration with respect to fractal functions and stochastic calculus I. Probab. Theory Relat. Fields 111, 333–374 (1998) MR1640795. https://doi.org/10.1007/s004400050171 Zähle, M.: Integration with respect to fractal functions and stochastic calculus II. Math. Nachr. 225, 145–183 (2001) MR1827093. https://doi.org/10.1002/1522-2616(200105)225:1<145::AID-MANA145>3.3.CO;2-S