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Arab. J. Math. https://doi.org/10.1007/s40065-023-00430-3 Arabian Journal of Mathematics Fatima Zahra Arioui Weighted fractional stochastic integro-differential equation with inﬁnite delay Received: 19 July 2022 / Accepted: 12 May 2023 © The Author(s) 2023 Abstract In this paper, we consider a weighted fractional stochastic integro-differential equation with inﬁnite delay and nonzero initial values involving a Riemann–Liouville fractional derivative of order 1/2 <α < 1. The existence of a mild solution is investigated using fractional calculus, stochastic analysis, and the ﬁxed point theorem. An example is also provided to illustrate the obtained result. Mathematics Subject Classiﬁcation 35A 01 · 60 H 10 · 60 H 20 1 Introduction Fractional calculus provides an excellent set of tools for describing memory and hereditary properties of many materials and processes, allowing for the modeling of non-local and diffuse effects that frequently occur in natural phenomena. The application of fractional calculus tools and techniques is widespread in almost all areas of engineering and science in general. Viscoelasticity, robotics, control theory, and other ﬁelds have numerous applications (see [2,4,12,13,21]). Phenomena are precisely modeled when the inﬂuences that processes may encounter along their course are considered. For example, it is well known that time delays may exist in a variety of technical systems where the derivative of a state variable depends not only on its current state but also on information from the past. When investigating fractional differential equations, the properties of the derivative type, such as the condition at the initial moment, are of interest. In a fractional differential equation with the Caputo derivative, the initial condition is the same as in an ordinary differential equation, whereas for the Riemann–Liouville derivative, the initial condition must be well taken. In this regard, before addressing our problem, we will go over some papers that deal with Riemann–Liouville fractional differential equations with delay. Benchohra et al. [4] studied the following model with a zero initial condition: L α D [y(t ) − g(t, y )]= f (t, y ), t ∈ (0, b], 0 <α < 1, t t (1.1) y(t ) = ϕ(t ), t ∈ (−∞, 0], L α where D is the standard Riemann–Liouville derivative, ϕ ∈ B with ϕ(0) = 0, where B is the phase space deﬁned axiomatically by Hale and Kato. Several other papers have investigated the case where y(0) = 0 with a ﬁnite or inﬁnite delay (see [1,2,4,10,11,13,16,21]). It is well known that when the initial condition is nonzero, i.e., y(0) = 0, the solution to the equations discussed in the preceding papers may not be bounded at some neighborhood of the initial point t = 0. As a result, some researchers have considered weighted fractional Fatima Zahra Arioui (B) Laboratory of Statistics and Stochastic Processes, Djillali Liabes University, P.O. Box 89, 22000 Sidi-Bel-Abbes, Algeria E-mail: ariouifatimazahra@gmail.com; fatimazahra.arioui@univ-sba.dz 123 Arab. J. Math. differential equations; for example, Dong [10] studied the weighted functional fractional differential equation with inﬁnite delay of the form L α ⎨ D y(t ) = f (t, y ), t ∈ (0, b], 0 <α 1, (1.2) y = ϕ ∈ B, t ∈ (−∞, 0], where the existence and continuous dependence results of solutions are obtained, Dong et al. [11] investigated the model (1.2) in an abstract Banach space and obtained the existence and uniqueness results using the measure of non-compactness and the ﬁxed point theorems, Abdo et al. [1] studied the weighted fractional neutral functional differential equation of the form ⎨ D [y(t ) − g(t, y )]= f (t, y ), t ∈ (0, b], 0 <α 1, t t (1.3) y = ϕ, t ∈ (−∞, 0]. The Banach and Schauder ﬁxed point theorems were used to achieve the required results. Since the presence of the stochastic term (random term) is important due to the possibility of unpredictability in the characteristics of natural systems, stochastic differential equations are more adaptable to real-world phenomena than deterministic ones. Fractional stochastic differential equations, in particular, are used to study the memory and genetic properties of stochastic systems and have become a hot topic in recent decades. Numerous academics have shown interest in these equations due to their applications in many ﬁelds of research, such as disease transmission, option pricing, quantitative ﬁnance, and so on (see [8,9,15,16,18,20]). Inspired by the works mentioned above, we consider the weighted fractional stochastic integro-differential equation of the form L α dw(t ) D u(t ) = f (t, u , a(t, s, u )ds) + g(t, u ) , t ∈ (0, b], t s t 0 dt (1.4) u = ϕ ∈ B, t ∈ (−∞, 0], L α where D denotes the Riemann–Liouville derivative of order 1/2 <α < 1, t ∈ (0, b], u(·) is a random variable takes its values in a separable real Hilbert space H. f , a and g are appropriate functions to be speciﬁed later. B is the phase space of functions mapping from (−∞, 0] into H described axiomatically in Sect. 2.The 1−α notation u represents the function deﬁned by u (θ ) = u(t + θ) for θ ∈ (−∞, 0]. u(t ) = t u(t ) and t t ϕ(t ) = ϕ(t ) for t ∈ (−∞, 0]. The purpose of this study is to prove the existence of a solution to the model (1.4) with nonzero initial values using the Kuratowski measure of non-compactness and the Mönch ﬁxed point theorem. The paper is organized as follows: Section 2 introduces some fundamental notations and preliminaries. Section 3 provides some sufﬁcient conditions for the existence of a mild solution for the model (1.4). Section 4 includes an example to demonstrate the obtained result. 2 Preliminaries In this section, we introduce some basic deﬁnitions, lemmas, and notations that will be used to answer the existence problem for our model (1.4). Let H and K two real separable Hilbert spaces, and L(K, H) the space of all linear and bounded operators from K to H,weuse thesamenotation · to denote the norms in K, H and L(K, H), and we use (·, ·) to denote the inner product of K and H. L ((0, b), H) is the space of Bochner integrable functions from (0, b) loc into H.Let (, F , {F } , P) be a complete probability space equipped with a normal ﬁltration {F } t t 0 t t 0 satisfying the usual conditions. An H-valued random variable is an F-measurable function u(t, ·) : → H; in the rest of the paper, we write u(t ) instead of u(t, ) for all ∈ . Let {w(t )} be a Q-Wiener process deﬁned on (, F , {F } , P) with the covariance operator Q such t 0 t t 0 that Tr (Q)< ∞. It is assumed that there exists a complete orthonormal system ς in K, and positive real numbers λ such that Qς = λ ς , k = 1, 2,..., and a sequence of independent Brownian motions such that k k k k (w(t ), e) = λ (ς , e)β (t ), e ∈ K, t 0. k k k k=1 123 Arab. J. Math. The space of all strongly measurable, square integrable, H-valued random variables, denoted by L (, H),is a Banach space equipped with the norm u(·) = E u(·) ,where Eu = 2 L (,H) u(w)dP. The family of all F -measurable, H-valued random variables u(0) is denoted by L (, H). Let C ((0, b]; L (, H)) be the Banach space of all continuous F -adapted measurable process from (0, b] 1−α 2 t 2 2 1−α 1−α into L (, H) such that lim t u(t ) exists, with the norm u = sup E t u(t ) . 2 t →0 C t ∈(0,b] 1−α In this paper, we consider the phase space B, which fulﬁlls the following fundamental axioms, which are similar to those introduced by Hale and Kato [12] and Hino et al. [14]: Deﬁnition 2.1 [1] B is a linear space of F -measurable functions mapping (−∞, 0] into H endowed with the semi-norm · , which satisﬁes the following axioms: (i) if x : (−∞, b]→ H (b > 0) is continuous on (0, b] and x ∈ B, then for each t ∈ (0, b] the following conditions hold (a) x ∈ B; (b) x (t ) L x ,where L > 0 is a constant; t B { } (c) x K (t ) sup x (s) : 0 s t + N (t ) x ,where K , N :[0, ∞) →[0, ∞), K is contin- t 0 B B uous, N is locally bounded and K , N are independent of x (·). Denote K = sup {K (t ) : t ∈ (0, b]} and N = sup {N (t ) : t ∈ (0, b]} ; (ii) for the function x (·) in (i ), the function t → x is continuous from (0, b] into B; (iii) the space B is complete. Now, consider the following space P: P ={u : (−∞, b]→ H : u| ∈ B, u| ∈ C ((0, b]; L (, H))}, (−∞,0] (0,b] 1−α 2 where u| is the restriction of u over (0, b]. (0,b] σ σ σ Lemma 2.2 [20] For σ ∈ (0, 1] and 0 < a b, we have |a − b | (b − a) . Lemma 2.3 [5] Let the space M(K, H) ={(·, ·) : is an L(K, H)-valued process on [0, b]× such that (t ) is F -measurable for all t ∈[0, b]}.If ∈ M(K, H) with E (t ) dt < ∞,then b b 2 2 E (t,w)dw(t ) Tr(Q) E(t ) dt. 0 0 Let us now present some fundamental fractional calculus deﬁnitions and lemmas (see [7,17]). Deﬁnition 2.4 [7]Let E be a Banach space. The Riemann–Liouville fractional integral of order α> 0of a function u :[a, b]→ E is deﬁned by α α−1 I u(t ) = (t − s) u(s)ds, t ∈[a, b], 0+ (α) provided that the right hand-side is point-wise deﬁned on [0, ∞),where (·) denotes the Euler’s Gamma α−1 −t function deﬁned by (α) = t e dt. Deﬁnition 2.5 [7]Let α> 0be ﬁxed and n =[α]+ 1. The Riemann–Liouville fractional derivative of order α of u :[0, ∞) → E at the point t is deﬁned by n t 1 d L α n−α−1 D u(t ) = (t − s) u(s)ds, t ∈[a, b], 0+ (n − α) dt provided the right side is point-wisely deﬁned, where [α] denotes the integer part of the real number α. 123 Arab. J. Math. Lemma 2.6 [7] Let 0 <α < 1, then the unique solutions to the equation D u(t ) = 0 are given by the formula α−1 u(t ) = ct 1 1 for t > 0,where c ∈ R is a constant, provided u ∈ C (0, b]∩ L (0, b].Further,if f ∈ C (0, b]∩ L (0, b] loc loc α 1 such that D f ∈ C ((0, b]) ∩ L (0, b],then 0 loc α α α−1 I D f (t ) = f (t ) + ct 0 0 for t > 0 and some constant c ∈ R. Next, we introduce the concept of a measure of non-compactness. Deﬁnition 2.7 [3] The Kuratowski measure of non-compactness μ(·) deﬁned on a bounded subset A of a Banach space E is: μ(A) = inf > 0 : A = A , and diam(A ) . i i i =1 Deﬁnition 2.8 [3]Let μ denote the Kuratowski measure of non-compactness on the real Banach space X and A, B ⊆ X be bounded. The following properties are satisﬁed: (i) μ(A) = 0 if and only if A is relatively compact, ¯ ¯ (ii) μ(A) = μ(A),where A denotes the closure of A, (iii) μ(A ∪ B) = max(μ(A), μ(B)), (iv) If A ⊆ B,then μ(A) μ(B), (v) μ(λA) =|λ|μ(A), with λ ∈ R, (vi) μ(A + B) μ(A) + μ(B),where A + B ={x + y : x ∈ A, y ∈ B}, (vii) μ(A + x ) μ(A) for all x ∈ E, 0 0 (viii) μ(conv(A)) = μ(A), where conv(A) is the closed convex hull of A. Lemma 2.9 [3] If D ⊂ C ([0, b], E ) is bounded, then μ(D(t )) μ (D) for all t ∈[0, b],where D(t ) ={u(t ) : u ∈ D}⊆ E. Furthermore, if D is equicontinuous on [0, b],then μ(D(t )) is continuous on [0, b], and μ (D) = sup μ(D(t )). t ∈[0,b] Lemma 2.10 [8] Let D ={u }⊂ C ([0, b], E ) be a bounded and countable set. Then μ(D(t )) is the Lebesgue integral on [0, b], and b b μ u (t )dt |n ∈ N 2 μ (D(t )) dt. 0 0 Theorem 2.11 [19] Let D be a bounded, closed and convex subset of E and x ∈ D. Let A : D → Dbe a continuous mapping. If the implication C = conv({x ∪ A(C )}) ⇒ μ(C ) = 0 holds for every subset C ⊂ D, then A has a ﬁxed point in D. 123 Arab. J. Math. 3 Existence result The mild solution of model (1.4) can be deﬁned as follows: Deﬁnition 3.1 A stochastic process u ∈ P is called a mild solution of model (1.4)if dw(t ) L α (i) u satisﬁes the equation D u(t ) = f (t, u , a(t, s, u )ds) + g(t, u ) , t ∈ (0, b], with initial t s t 0 0 dt condition u = ϕ ∈ B; (ii) u(t ) is measurable and F -adapted for each t ∈ (0, b]. Using Lemma 2.6, the solution u to our model (1.4) satisﬁes the following stochastic integral equation t s t 1 1 α−1 α−1 α−1 u(t ) = t ϕ(0) + (t − s) f (s, u , a(s,τ, u )dτ)ds + (t − s) g(s, u )dw(s). s τ s (α) (α) 0 0 0 Before demonstrating our existence result for the model (1.4), we will make a list of hypotheses that will be enforced in our main theorem. (H1) The function f : (0, b]× B × H → H satisﬁes the following conditions: (i) f (t, ·, ·) : B × H → H is continuous for each t ∈ (0, b], and for every (φ, ψ ) ∈ B × H, the function t → f (t,φ,ψ) is strongly measurable; (ii) there exists p ∈ L ((0, b], [0, ∞)) and a continuous non-decreasing function θ :[0, ∞) →[0, ∞) f f such that for all (t, x , y) ∈ (0, b]× B × H,wehave 2 2 2 E f (t, x , y) p (t )θ (x + E y ); f f (iii) there exists a function ζ ∈ L ((0, b], [0, ∞)), α ∈ (0,α) such that for each bounded set D ∈ B f 1 1 and D ∈ H, μ ( f (t, D , D )) ζ (t ) sup μ(D (θ )) + μ(D ) . 1 2 f 1 2 −∞<θ 0 (H2) The function g : (0, b]× B → L(K, H) satisﬁes the conditions: (i) g(t, ·) : B → L(K, H) is continuous for each t ∈ (0, b], and for every φ ∈ B, the function t → g(t,φ) is strongly measurable; (ii) there exists p ∈ L ((0, b], [0, ∞)), α ∈ (0, 2α − 1) and a continuous non-decreasing function g 2 θ :[0, ∞) →[0, ∞) such that for all (t, x ) ∈ (0, b]× B,wehave 2 2 E g(t, x ) p (t )θ (x ); g g 2α−1 (iii) there exists a function ζ ∈ L ((0, b], [0, ∞)), α ∈ (0, ) such that for each bounded set D ∈ B, g 3 3 μ (g(t, D )) ζ (t ) sup μ(D (θ )). 3 g 3 −∞<θ 0 (H3) The function a : (0, b]×[0, b]× B → H satisﬁes: (i) for each (t, s) ∈ (0, b]×[0, b], the function a(t, s, ·) : B → H is continuous and for each x ∈ B,the function (t, s) → a(t, s, x ) is strongly measurable; (ii) there exists L > 0 such that E a(t, s, x )ds L (1 + x ); (iii) there exists L > 0 such that for each bounded set D ∈ B, 2 4 μ (a(t, s, D )) L sup μ(D (θ )). 4 2 4 −∞<θ 0 123 Arab. J. Math. Theorem 3.2 Assume that the hypotheses (H1)–(H3) hold, then the model (1.4) has a solution on (−∞, b] if the following condition holds: 1−2α √ 1−2α 3 3 1−α 1−α 1 1 2 2 2b (1 + 2bL ) ζ (1 − α ) b Tr (Q) ζ (1 − 2α ) 1 1 2 f 1 g 3 α α 1 3 L L l = + < 1. 1−2α 1−α 1 3 (α)(α − α ) (α)(2α − 1 − 2α ) (3.1) Proof According to the Deﬁnition 3.1, u is a mild solution to (1.4)if t s α−1 1 α−1 ⎪ t ϕ(0) + (t − s) f (s, u , a(s,τ, u )dτ)ds s τ (α) 0 0 u(t ) = 1 α−1 ⎪ + (t − s) g(s, u )dw(s), t ∈ (0, b], (α) ⎪ 0 ϕ(t ), t ∈ (−∞, 0]. Let ϕ be a function deﬁned by 0, t ∈ (0, b], ϕ(t ) = ϕ(t ), t ∈ (−∞, 0]. We extend z to (−∞, b], which is deﬁned as 1−α t z(t ), t ∈ (0, b], z(t ) = 0, t ∈ (−∞, 0]. It is obvious that if u satisﬁes the integral equation t s α−1 α−1 u(t ) = t ϕ(0) + (t − s) f (s, u , a(s,τ, u )dτ)ds s τ (α) 0 0 α−1 + (t − s) g(s, u )dw(s), t ∈ (0, b], (α) then we can decompose u(·) as u(t ) = z(t )+ϕ(t ), which implies that u(t ) = z(t )+ ϕ(t ), and thus u = z + ϕ , t t t where z(t ) satisﬁes, t s α−1 α−1 z(t ) = t ϕ(0) + (t − s) f (s, z + ϕ , a(s,τ, z + ϕ )dτ)ds s s τ τ (α) 0 0 α−1 + (t − s) g(s, z + ϕ )dw(s), t > 0. s s (α) Set P ={z : (−∞, b]→ H, z| ∈ C ((0, b]; H), z = 0}.For z ∈ P ,and let z be the seminorm 0 (0,b] 1−α 0 0 P in P deﬁned by 1−α z = z + z = sup E t z(t ) . P C B 0 1−α t ∈(0,b] Then (P , · ) is a Banach space. Let B deﬁned by B ={z ∈ P ; z q}.The set B is clearly 0 q q 0 q 0 P bounded, closed, and convex. Choose q satisﬁes 1−α 1−α 2 3b 3b Tr(Q) 1 − α q 3E ϕ(0) + θ (M ) p + θ M p 1 , f 1 f 1 g 2 g 2 L 2 (2α − 1) (α) (α) 2α − 1 − α 2 where 2 2 2 2 M = 2K q + 2N ϕ + L (1 + 2K q + 2N ϕ ), 1 1 B B 2 2 M = 2K q + 2N ϕ . 123 Arab. J. Math. Deﬁne the operator : P → P as follows: 0 0 t s α−1 α−1 (z)(t ) = t ϕ(0) + (t − s) f (s, z + ϕ , a(s,τ, z + ϕ )dτ)ds s s τ τ (α) 0 0 α−1 + (t − s) g(s, z + ϕ )dw(s). s s (α) We will prove by Mönch’s ﬁxed point theorem that has a ﬁxed point. Step 1: maps B into itself. Using Hölder’s inequality and Lemma 2.3, we obtain for t ∈ (0, b], 1−α E t (z)(t ) 1−α t s 2 α−1 3E ϕ(0) + 3E (t − s) f (s, z + ϕ , a(s,τ, z + ϕ )dτ)ds s s τ τ (α) 0 0 1−α t α−1 +3E (t − s) g(s, z + ϕ )dw(s) s s (α) 2(1−α) t s 3t 2 α−1 3E ϕ(0) + E (t − s) f (s, z + ϕ , a(s,τ, z + ϕ )dτ)ds s s τ τ (α) 0 0 2(1−α) t 3t α−1 + E (t − s) g(s, z + ϕ )dw(s) s s (α) 2(1−α) t t s 3t 2 2(α−1) 3E ϕ(0) + (t − s) ds E f (s, z + ϕ , a(s,τ, z + ϕ )dτ) ds s s τ τ (α) 0 0 0 2(1−α) t 3t Tr (Q) 2(α−1) 2 + (t − s) E g(s, z + ϕ ) ds. s s (α) Moreover, we have 2 2 2 z + ϕ 2 z + 2 ϕ s s s s B B B 2 2 2 2 2 2 2 2K sup E z(τ ) + 2N z + 2K sup E ϕ(τ ) + 2N ϕ 0 0 B B τ ∈[0,s] τ ∈[0,s] 2 2 2 2K sup E z(τ ) + 2N ϕ τ ∈[0,s] 2 2 2K q + 2N ϕ . (3.2) As a result of hypotheses (H1, H2, H3)-(ii) and inequality (3.2), we get 1−α E t (z)(t ) t s 3t 2 2 3E ϕ(0) + p (s)θ z + ϕ + E a(s,τ, z + ϕ )dτ ds f f s s s s (2α − 1) (α) 0 0 2(1−α) t 3t Tr (Q) 2(α−1) 2 + (t − s) p (s)θ z + ϕ ds g g s s (α) 3t 2 2 2 2 2 3E ϕ(0) + p (s)θ 2K q + 2N ϕ + L (1 + 2K q + 2N ϕ ) ds f f 1 B B (2α − 1) (α) 2(1−α) t 3t Tr (Q) 2(α−1) 2 2 + (t − s) p (s)θ 2K q + 2N ϕ ds g g (α) 3t 3E ϕ(0) + θ (M ) p (s)ds f 1 f (2α − 1) (α) 1−α α 2(1−α) t 2 t 2 2(α−1) 3t Tr (Q) 1−α α 2 2 + θ M (t − s) ds ( p (s)) ds g 2 g (α) 0 0 123 Arab. J. Math. 3t 3E ϕ(0) + θ (M ) p f 1 f 1 (2α − 1) (α) 2(1−α) 1−α 3t Tr (Q) 1 − α 2α−1−α + θ M t p . g 2 g 2 α (α) 2α − 1 − α 2 Finally, we conclude, 2 1−α z = sup E t (z)(t ) t ∈(0,b] 3b 3E ϕ(0) + θ (M ) p f 1 f 1 (2α − 1) (α) 1−α 1−α 2 2 3b Tr (Q) 1 − α + θ M p 1 g 2 g (α) 2α − 1 − α 2 q. Step 2: (B ) is equicontinuous. Let z ∈ B and t , t ∈ (0, b] such that 0 < t < t b,thenwehave q 1 2 1 2 1−α 1−α E t (z)(t ) − t (z)(t ) 2 1 2 1 1−α 1−α t t s 2 1 t t 2 α−1 1 α−1 2E (t − s) − (t − s) f (s, z + ϕ , a(s,τ, z + ϕ )dτ)ds 2 1 s s τ τ (α) (α) 0 0 0 1−α 1−α t t 2 1 t t 2 α−1 1 α−1 +2E (t − s) − (t − s) g(s, z + ϕ )dw(s) 2 1 s s (α) (α) 0 0 1−α α−1 1−α α−1 4E t (t − s) − t (t − s) f (s, z + ϕ , a(s,τ, z + ϕ )dτ)ds 2 1 s s τ τ 2 1 (α) 0 0 t s 2 2 1−α α−1 +4E t (t − s) f (s, z + ϕ , a(s,τ, z + ϕ )dτ)ds 2 s s τ τ (α) t 0 1−α α−1 1−α α−1 +4E t (t − s) − t (t − s) g(s, z + ϕ )dw(s) 2 1 s s 2 1 (α) 1−α α−1 +4E t (t − s) g(s, z + ϕ )dw(s) 2 s s (α) I . (3.3) i =1 First, we calculate I and I using hypotheses (H1, H3)-(ii), inequality (3.2), and Hölder’s inequality. 1 2 t s 1−α α−1 1−α α−1 I = 4E t (t − s) − t (t − s) f (s, z + ϕ , a(s,τ, z + ϕ )dτ)ds 1 2 1 s s τ τ 2 1 (α) 0 0 t t s 1 1 1−α α−1 1−α α−1 t (t −s) − t (t − s) ds E f (s, z+ϕ , a(s,τ, z + ϕ )dτ) ds 2 1 s s τ τ 2 1 (α) 0 0 0 1−α α−1 α−1 α−1 1−α 1−α t (t − s) − (t − s) + (t − s) t − t ds 2 1 1 2 2 1 (α) t s × E f (s, z + ϕ , a(s,τ, z + ϕ )dτ) ds s s τ τ 0 0 1 1 8 2 2(1−α) 1−α 1−α α−1 α−1 2(α−1) t (t − s) − (t − s) ds + t − t (t − s) ds 2 1 1 2 2 1 (α) 0 0 t s × E f (s, z + ϕ , a(s,τ, z + ϕ )dτ) ds s s τ τ 0 0 123 Arab. J. Math. 1 1 2(1−α) 1−α 1−α 2(α−1) 2(α−1) 2(α−1) t (t − s) − (t − s) ds + t − t (t − s) ds 1 2 1 2 2 1 (α) 0 0 × p (s)θ (M )ds f f 1 ⎛ ⎞ 2(1−α) 2α−1 2α−1 2α−1 2α−1 1−α 1−α 2 t t + (t − t ) − t 2 1 8 2 1 2 t (t − t ) 1 2 1 ⎝ ⎠ + p θ (M ) f 1 f 1 2 L (α) 2α − 1 2α − 1 8 p θ (M ) f 1 f 1 L 2(1−α) 2α−1 2α−1 2α−1 2α−1 1−α 1−α 2 b t + (t − t ) − t + b (t − t ) , 2 1 1 2 2 1 (2α − 1) (α) and t s 1−α α−1 I = 4E t (t − s) f (s, z + ϕ , a(s,τ, z + ϕ )dτ)ds 2 2 s s τ τ (α) t 0 2(1−α) t t s 2 2 4t 2(α−1) (t − s) ds E f (s, z + ϕ , a(s,τ, z + ϕ )dτ) ds 2 s s τ τ (α) t t 0 1 1 2(1−α) 2α−1 4t (t − t ) 2 1 p θ (M ) f f 1 2 L (2α − 1) (α) 2(1−α) 2α−1 4b (t − t ) 2 1 p θ (M ). f f 1 2 L (2α − 1) (α) Second, we calculate I and I using hypothesis (H2)-(ii), Lemma 2.3, and inequality (3.2). 3 4 1−α α−1 1−α α−1 I = 4E t (t − s) − t (t − s) g(s, z + ϕ )dw(s) 3 2 1 s s 2 1 (α) 4Tr (Q) 1−α 1−α α−1 α−1 2 t (t − s) − t (t − s) E g(s, z + ϕ ) ds 2 1 s s 2 1 (α) 4Tr (Q) 1−α α−1 α−1 α−1 1−α 1−α 2 t (t −s) −(t − s) + (t − s) t − t E g(s, z + ϕ ) ds 2 1 1 s s 2 2 1 (α) 2(1−α) 8Tr(Q)t 2 α−1 α−1 2 (t − s) − (t − s) E g(s, z + ϕ ) ds 1 2 s s (α) 1−α 1−α 8Tr(Q) t − t t 2 1 2(α−1) 2 + (t − s) E g(s, z + ϕ ) ds 1 s s (α) 2(1−α) 1−α 8Tr(Q)t 1 − α 2 2α−1−α 2α−1−α 2α−1−α 2 2 t + (t − t ) − t × p 1 θ (M ) 2 1 g g 2 1 2 (α) 2α − 1 − α L 1−α 1−α 1−α 8Tr(Q) t − t 2 2 1 1 − α 2α−1−α + t p 1 θ (M ) g g 2 (α) 2α − 1 − α L 2 1−α 8Tr(Q)(1 − α ) p θ (M ) 2 g g 2 2 1−α (α)(2α − 1 − α ) 2α−1−α 2α−1−α 1−α 1−α 2(1−α) 2 2α−1−α 2 2α−1−α 2 2 × b t + (t − t ) − t + b t − t , 2 1 1 2 2 1 and 1−α α−1 I = 4E t (t − s) g(s, z + ϕ )dw(s) 4 2 s s (α) 2(1−α) 4Tr (Q)t 2 2(α−1) 2 (t − s) E g(s, z + ϕ ) ds 2 s s (α) 123 Arab. J. Math. 2(1−α) 1−α 4Tr(Q)t 1 − α 2 2α−1−α (t − t ) p θ (M ) 2 1 g g 2 2 α (α) 2α − 1 − α L 2(1−α) 1−α 2α−1−α 2 2 4Tr(Q)b (1 − α ) (t − t ) 2 2 1 p 1 θ (M ). g g 2 2 1−α (α)(2α − 1 − α ) 2 Substituting (I ) − (I ) into (3.3), we get 1 4 1−α 1−α Et (z)(t ) − t (z)(t ) 2 1 2 1 8 p θ (M ) f 1 f 1 L 2(1−α) 2α−1 2α−1 2α−1 2α−1 1−α 1−α 2 b t + (t − t ) − t + b (t − t ) 2 1 1 2 2 1 (2α − 1) (α) 2(1−α) 2α−1 4b (t − t ) 2 1 + p θ (M ) f 1 f 1 2 L (2α − 1) (α) 1−α 8Tr(Q)(1 − α ) p 1 θ (M ) 2 g g 2 2 1−α (α)(2α − 1 − α ) 2(1−α) 2α−1−α 2α−1−α 2α−1−α 2α−1−α 1−α 1−α 2 2 2 2 × b t + (t − t ) − t + b t − t 2 1 1 2 2 1 2(1−α) 1−α 2α−1−α 2 2 4Tr(Q)b (1 − α ) (t − t ) 2 2 1 + p θ (M ). g g 2 2 1−α α (α)(2α − 1 − α ) L As a result, according to Lemma 2.2, the right hand side of the previous inequality tends to zero as t → t . 1 2 Thus, (B ) is equicontinuous. Step 3: is continuous. Let {z } be a sequence such that {z } converges to z in B as n →∞. It is clear from axiom (i) in the n n∈N n q Deﬁnition 2.1 that {z } → z uniformly for s ∈ (0, b] as n →∞,and then { z } → z as n →∞, ns n∈N s ns n∈N s we can deduce from hypotheses (H1, H2, H3)-(i), a(s,τ, z + ϕ ) → a(s,τ, z + ϕ ), as n →∞, nτ τ τ τ s s f (s, z + ϕ , a(s,τ, z + ϕ )dτ) → f (s, z + ϕ , a(s,τ, z + ϕ )dτ), as n →∞, ns s nτ τ s s τ τ 0 0 g(s, z + ϕ ) → g(s, z + ϕ ), as n →∞. ns s s s From hypotheses (H1, H2, H3)-(ii), Hölder’s inequality, Lemma 2.3, and the dominated convergence theorem, we have 1−α 1−α E t (z )(t ) − t (z)(t ) s s 1−α α−1 2E (t −s) f (s, z +ϕ , a(s,τ, z +ϕ )dτ)− f (s, z+ϕ , a(s,τ, z +ϕ )dτ) ds ns s τ τ s s τ τ (α) 0 0 0 1−α t α−1 + 2E (t − s) (g(s, z + ϕ ) − g(s, z + ϕ )) dw(s) ns s s s (α) t s s E f (s, z +ϕ , a(s,τ, z +ϕ )dτ)− f (s, z +ϕ , a(s,τ, z +ϕ )dτ) ds ns s nτ τ s s τ τ (2α−1) (α) 0 0 0 2(1−α) t 2Tr (Q)t 2(α−1) 2 + (t − s) E g(s, z + ϕ ) − g(s, z + ϕ ds −→0as n −→ ∞. ns s s s (α) As a result, is continuous. Step 4: satisﬁes Mönch’s condition. Assume that D ⊆ B be countable and D ⊆ conv({0}∪ (D)). We prove that D is relatively compact, that is, μ(D) = 0. 1−α From the above steps, it is easy to see that D = (·) D is bounded and equicontinuous. According to Lemma 2.9 and Deﬁnition 2.8, we can derive for t ∈ (0, b] the following, 1−α μ(D(t )) sup μ(D(t )) = sup μ(t D(t )) t ∈(0,b] t ∈(0,b] 123 Arab. J. Math. 1−α sup μ(t ( D)(t )) t ∈(0,b] t 1−α s α−1 μ (ϕ(0)) + sup μ (t − s) f (s, D + ϕ , a(s,τ, D + ϕ )dτ)ds s s τ τ (α) 0 0 t ∈(0,b] t 1−α α−1 + sup μ (t − s) g(s, D + ϕ )dw(s) s s (α) t ∈(0,b] 0 I + I . (3.4) 1 2 From Lemma 2.10, hypotheses (H1, H3)-(iii), we get t 1−α s α−1 I = sup μ (t − s) f (s, D + ϕ , a(s,τ, D + ϕ )dτ)ds 1 s s τ τ (α) t ∈(0,b] 0 0 1−α t α−1 2sup (t − s) μ f (s, D + ϕ , a(s,τ, D + ϕ )dτ) ds s s τ τ (α) t ∈(0,b] 0 0 1−α t α−1 2sup (t − s) ζ (s) (α) t ∈(0,b] 0 × sup μ(D (θ ) + ϕ (θ )) + μ a(s,τ, D (θ ) + ϕ (θ ) dτ) ds s s τ τ −∞<θ 0 1−α t α−1 2sup (t − s) ζ (s) (α) t ∈(0,b] 0 × sup μ(D (θ ) + ϕ (θ )) + 2 L sup μ(D (θ ) + ϕ (θ ))dτ ds s s 2 τ τ −∞<θ 0 −∞<θ 0 1−α t α−1 2sup (t − s) ζ (s) sup μ(D(η)) + 2 L sup μ(D(ϑ ))dτ ds f 2 (α) t ∈(0,b] 0 −∞<ηs 0 −∞<ϑ τ 1−α 2b α−1 sup sup μ(D(η)) + sup sup μ(D(ϑ )) × 2bL sup (t − s) ζ (s)ds 2 f (α) 0<t b 0<ηt 0<t b 0<ϑ t t ∈(0,b] 0 1−α 1−α 1 2b 1 − α α−α sup μ(D(ν))(1 + 2bL ) b ζ 1 2 f (α) α − α L 1 0<νb 1 1−α 2b (1 + 2bL ) ζ 1 1−α 2 f 1 α 1 − α 1 1 μ(D). (3.5) (α) α − α For any u,v ∈ B , applying Lemma 2.3, we obtain 1−α t α−1 E (t − s) (g(s, u + ϕ ) − g(s, v + ϕ )) dw(s) s s s s (α) 2(1−α) t t Tr (Q) 2(α−1) 2 (t − s) E g(s, u + ϕ ) − g(s, v + ϕ ) ds. s s s s (α) By the properties of the measure of the stochastic integral (see [6]) and (H2)-(iii), we have t 1−α α−1 I = sup μ (t − s) g(s, D + ϕ )dw(s) 2 s s (α) 0<t b 0 1−α t 2 2 2(α−1) sup Tr(Q) (t − s) μ(g(s, D + ϕ )) ds s s (α) 0<t b 0 √ 2 1−α t t Tr(Q) 2(α−1) sup (t − s) ζ (s) sup μ(D (θ ) + ϕ (θ )) ds g s s (α) 0<t b 0 −∞<θ 0 123 Arab. J. Math. √ 2 1−α t t Tr(Q) 2(α−1) sup (t − s) ζ (s) sup μ(D(μ)) ds (α) 0<t b 0 0<μs 1−α b Tr(Q) 2(α−1) 2 sup sup μ(D(μ)) sup (t − s) ζ (s)ds (α) 0<t b 0<μt 0<t b 0 1−2α 1−α 2α−1−2α b Tr(Q) 1 − 2α 3 sup μ(D(μ)) b ζ 1 (α) 2α − 1 − 2α 3 0<μb 3 1−2α √ 1−2α b Tr(Q) ζ 1 α 1 − 2α μ(D). (3.6) (α) 2α − 1 − 2α Substituting (3.5)and (3.6)into(3.4), we get for t ∈ (0, b], sup μ(D(t )) t ∈(0,b] ⎛ ⎞ 1−2α √ 1−2α 3 3 1−α 1−α 1 1 2 2 2b (1 + 2bL ) ζ 1 (1 − α ) b Tr(Q) ζ 1 (1 − 2α ) 2 f 1 g 3 α α ⎜ 1 3 ⎟ L L ⎝ + ⎠ μ(D). 1−2α 1−α 3 (α)(α − α ) (α)(2α − 1 − 2α ) Hence μ(D) l μ(D). Since l < 1in(3.1), we get μ(D) = 0, which implies that μ(D) = 0, proving that the set D is relatively compact. We conclude from Theorem 2.11 that has a ﬁxed point that is a mild solution to the model (1.4). 4 Example Consider the following weighted fractional stochastic integro-differential equation with inﬁnite delay: t t s L α ⎨ D u(t ) = ρ(s − t ) u(s)ds + (t − s) ϑ(τ − s) u(τ )dτ ds 0 −∞ 0 −∞ dw(s) (4.1) + ξ(s − t ) u(s)ds , t ∈ (0, b], −∞ ds u (t ) = ϕ(t ), t ∈ (−∞, 0], 1 p where <α < 1. We denote by PC × L (h, H) the space of all functions ϕ : (−∞, 0]→ H such that ϕ| ∈ PC([−r, 0], H), ϕ(·) is Lebesgue measurable on (−∞, −r ),and h ϕ is Lebesgue integrable on [−r,0] (−∞, −r ) (see [14] for details). The seminorm is given by ϕ = sup ϕ(s) + h(s) ϕ ds . −r s0 −∞ In addition, we assume that the following conditions hold: The functions ρ, ξ, ϑ : R → R are continuous, with 0 2 0 2 0 2 ρ (s) ξ (s) ϑ (s) c = ds < ∞, c = ds < ∞, c = ds < ∞, ρ ξ ϑ ˜ ˜ ˜ h(s) h(s) h(s) −∞ −∞ −∞ We choose B = PC × L (h, H).For t ∈ (0, b] and ϕ ∈ B,wegive t 0 t 0 f t,ϕ, a(t, s,ϕ)ds = ρ(s)ϕ(s)ds + (t − s) ρ(τ)ϕ(τ)dτ ds, 0 −∞ 0 −∞ g(t,ϕ) = ρ(s)ϕ(s)ds, −∞ 123 Arab. J. Math. Obviously, θ (t ) = θ (t ) = t. Under the above conditions, we can represent the Example (4.1)bythe f g model (1.4). Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Funding No funding is availed for this research work. Data availability statement Not applicable. 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Arabian Journal of Mathematics – Springer Journals
Published: Dec 1, 2023
Keywords: 35A 01; 60 H 10; 60 H 20
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