On the solvability of a general class of a coupled system of stochastic functional integral equations

Abstract The main objective in this work is to find some weaker conditions that guarantee the existence of continuous solutions of some stochastic coupled systems of the Urysohn–Volterra–Itô–Doob type. The uniqueness of the solution for these systems is investigated as well. It is worth mentioning that the results in this work include many previously published results as special cases.


Introduction
Integral equations are considered significant tools in the analysis, and formulation of many phenomena arising in most applied sciences.In other words, integral equations can successfully describe the evolution of many real-world systems, such as systems in quantum mechanics, Newtonian mechanics, electrical engineering, electromagnetic theory, and many other fields, (see e.g.Poljak et al., 2018).Also, in ordinary and partial differential equations we can convert several initial and boundary value problems to an equivalent integral equations.At the same time examining coupled systems associated with integral equations is important as well, because such systems model many physical problems, (see e.g.El-Sayed & Al-Fadel, 2018;Hashem & El-Sayed, 2017;Zhang, 2018).Recently, A.M.A. El-Sayed et al. (El-Sayed & Kenawy, 2014b) used the fixed point principle, under sufficient conditions, to prove the existence of continuous solutions, (x, y), of the coupled system xðtÞ ¼ a 1 ðtÞ þ ð t 0 f 1 ðt, s, yð/ 1 ðsÞÞÞds, t 2 0, a ½ : (1.1) where x, and y are real-valued continuous functions defined on ½0, a, a < 1, and endowed with the Chebyshev's norm.The functions / j , j ¼ 1, 2, are continuous selfmaps on ½0, a: Also, A.M.A. El-Sayed et al. utilized in (El-Sayed & Kenawy, 2014a) the same previous technique to study the existence of solution of the coupled system defined by xðtÞ ¼ a 1 ðtÞ þ ð 1 0 f 1 ðt, s, J b 1 yðsÞÞds, t 2 0, 1 ½ : (1.2) where 0 < b j < 1, j ¼ 1, 2, and J b j is the Riemann-Liouville fractional order integral operator.The uniqueness of the solution is investigated for Equations (1.1), and (1.2) as well.But due to some unexpected disturbance which may effect the accuracy of the system, we believe that stochastic models give more accurate results than the proposed models in (El-Sayed & Kenawy, 2014a, 2014b), see for more details (Elborai, Abdou, & Youssef, 2013a, 2013b;Elborai & Youssef, 2019;Klyatskin, 2015;Umamaheswari, Balachandran, & Annapoorani, 2017, 2018).Therefore, our aim in this article is to generalize the deterministic models 1.1, and 1.2 to the stochastic form in a suitable sense.So, we shall study the existence of solutions of the coupled system of stochastic functional integral equations of the Urysohn-Volterra-Itô-Doob type defined by g 2 ðt, s, xðsÞ, AðsÞxðsÞÞdWðsÞ: where t 2 I :¼ ½0, a, a < 1: The coupled system 1.3 will be studied in the space CðI, L 2 ðX, F , PÞÞ Â CðI, L 2 ðX, F , PÞÞ, which contains all ordered pairs, (x, y), of continuous stochastic processes in the mean square sense which are defined from I into L 2 ðX, F , PÞ and adapted to the proposed filtration fF t g t2I , see the next section for more explanations.(Elborai & Youssef, 2019).The functions h j ðtÞ, f j ðtÞ, and g j ðtÞ, j ¼ 1, 2, are F t -measurable scalar functions satisfying some proposed conditions will be mentioned in the next section.It is clear that Equations (1.3) are more general than Equations (1.1), and (1.2), see Lemma 2.8 in (Kilbas, Srivastava, & Trujillo, 2006).The following sections are organized as follows.Section 2 provides the necessary definitions, and auxiliary theorems.New results are introduced in Section 3. Eventually, a conclusion and suggested future work are presented in Section 4.

Definitions and auxiliary results
This section includes some preliminaries which will be utilized to prove our main results.For more explanations, we refer to (Klyatskin, 2015;Pavliotis, 2014).Let ðX, F , fF t g t2I , PÞ, where F :¼ F a , be a filtered probability space satisfying the usual conditions, where X is known as the sample space and contains a collection abstract points x, F is a sigma-algebra on X and includes all possible events which take place through the time interval I, P is a probability measure defined on F : Let L 2 :¼ L 2 ðX, F , PÞ be the space of all real stochastic processes fxðtÞ : t 2 Ig which have finite second moments ði:e: EfjxðtÞj 2 g < 1Þ, for all t 2 I: Let the norm on the space L 2 be defined as kxðtÞk Theorem 2.1 (Kreyszig, 1978).Let X and Y be two normed spaces.Let T be a linear operator from DðTÞ & X into Y.Then, the operator T is continuous if and only if it is bounded.
Theorem 2.2 (Kreyszig, 1978).A linear operator defined from a Banach space X into a Banach space Y is continuous if and only if it is closed.
Theorem 2.3 (Hochstadt, 1988).A completely continuous operator T defined on a closed bounded convex subset S in a Banach space X, such that TS & S has at least one fixed point in the set S.

Main results
Suppose the following coupled system of stochastic functional integral equations.
where, the first integral in each equation is the Riemann integral in the mean square sense while the second integral will be studied in the Itô-Doob sense.The stochastic process W(t) is assumed to be F t -adapted real martingale.The real random functions f j , and g j , j ¼ 1, 2, are defined on I Â I Â C Â C into the space L 2 and will be specified in the conditions below.The given random forcing functions h j , j ¼ 1, 2, are defined from I into the space C and called the stochastic perturbing terms.Now, let us propose some sufficient conditions.H 1 : Suppose there exists a non-decreasing continuous real function H 2 : The random functions f j ðt, s, x, yÞ, and g j ðt, s, x, yÞ, j ¼ 1, 2, are measurable in s 2 I for each ðt, x, yÞ 2 I Â C Â C, continuous in the mean square with respect to (t, x, y) for each s 2 I: H 3 : There exist two real random functions m j , j ¼ 1, 2, defined on I Â I, and four positive constants where the symbol E represents the expectation with respect to m j , j ¼ 1, 2: H 4 : There exist two positive constants b j , j ¼ 1, 2, such that Now, we define, for each t 2 I, the integral operator T by Tðx, yÞðtÞ :¼ ðT 1 yðtÞ, T 2 xðtÞÞ: (3.1) Where, for each t 2 I, the integral operators T 1 , and T 2 are defined by (i) 8 ðx, yÞ 2 X, we have Tðx, yÞ 2 X (i:e: T maps X into itself).(ii) T is a mean square continuous operator on X.
Proof.Clearly, the conditions H 1 , H 3 , and H 4 guarantee that the functions T 1 y, T 2 x are F t -adapted for each t 2 I, and their second moments are finite, see (Elborai & Youssef, 2019), and hence the operator T makes sense.Now, to complete the proof of part (i) in Lemma 3.1, it remains to prove that T(x, y) is continuous in mean square 8ðx, yÞ 2 X: That is, we need to prove T 1 y 2 C, and T 2 x 2 C, 8t 2 I: Let t 1 2 I, t 2 2 I, and assume, without loss of generality, that t 2 > t 1 : From the Cauchy-Schwartz inequality, and condition H 1 we have The operators A(t) are bounded on C 8t 2 I from applying the closed graph theorem.So, there exists a non-negative real constant, c 1 , such that kAðtÞyk C cðtÞkyk C c 1 kyk C , where c 1 :¼ max t2I cðtÞ È É : Also, the operators B(t) are bounded on C. Therefore, there exists a non-negative real constant, n 1 , such that kBðtÞyk C nðtÞkyk C n 1 kyk C , where Also, applying condition H 4 , and the continuity of and hence the function T 1 y is continuous in the mean square sense on I. Using an arguments similar to the one used above, we can prove that kT So, T(x, y) is continuous in the mean square sense and thereby Tðx, yÞ 2 X 8ðx, yÞ 2 X: Therefore, the integral operator T maps the space X into itself.Let ðx n , y n Þ !ðx, yÞ in X almost surely, as n ! 1, with ðx n , y n Þ È É 1 n¼1 X: we need to prove Tðx n , y n Þ !Tðx, yÞ 2 X, as n ! 1, 8t 2 I, where Tðx n , y n ÞðtÞ ¼ ðT 1 y n ðtÞ, T 2 x n ðtÞÞ: That is, we will prove that T 1 y n !T 1 y in C, and T 2 x n !T 2 x in C 8t 2 I: Using an arguments similar to the one used above yields Therefore, kT 1 y n À T 1 yk C ! 0 as ky n À yk C ! 0 (that is, the operator T 1 is continuous on C).By the same way, we can prove that kT 2 x n À T 2 xk C ! 0 as kx n À xk C ! 0, and thereby 8t 2 I we have That is, the operator T is continuous on X 8t 2 I: w Remark 3.1.In the sequel, let r j :¼ 3b j ½FðaÞ À Fð0Þ, and g j :¼ 3a 2 ½b j þ b Ã j f 2 , j ¼ 1, 2: Now, Let the sets S r 1 , S r 2 , and S r be defined as follows Theorem 3.1.Suppose conditions H 1 À H 4 are satisfied.Let the operator T : S r !X.Then the stochastic coupled system 1.3 has at least one solution in S r .
Proof. it is clear that the set S r is a closed bounded convex nonempty set contained in X.From Lemma 3.1, we deduce that T : S r !X is a continuous operator.We suppose the sequence Tðx n , y n Þ È É 1 n¼1 whose elements are continuous functions in the set TS r .Let t 1 2 I, t 2 2 I, and assume, without loss of generality, that t 2 > t 1 : Applying similar argument to those used in Lemma 3.1 yields !0 as t 2 !t 1 8n 2 N: Therefore, the sequence T 1 y n f g 1 n¼1 is a mean square equicontinuous.Applying similar arguments we can show that T 2 x n f g 1 n¼1 is, also, a mean square equicontinuous, and thereby we have So, the sequence Tðx n , y n Þ È É 1 n¼1 is a mean square equicontinuous in the set TS r .It is easy to prove that the sequences T 1 y n f g 1 n¼1 , and T 2 x n f g 1 n¼1 are uniformly bounded in the mean-square sense, see (Elborai & Youssef, 2019).So, the sequence Tðx n , y n Þ È É 1 n¼1 is also uniformly bounded in the mean-square sense.From the Arzela Ascoli theorem, we can find a convergent subsequence whose convergence is uniformly in TS r and thereby the set TS r is compact.The previous discussion shows that the operator T is completely continuous.It is easy to show TS r & S r because for every y 2 S r 1 , we have r and simplifying yields kT 1 yk C r 1 for each y 2 S r 1 : Also, using similar arguments gives kT 2 xk C r 2 for each x 2 S r 2 : Now for every ðx, yÞ 2 S r , we have So, for each ðx, yÞ 2 S r , we have Tðx, yÞ 2 S r , and therefore TS r & S r : From the fixed point theorem due to Schauder there exists at least one fixed point for the operator T in the set S r and hence the coupled system 1.3 has at least one solution in S r .w Corollary 3.1.1.Suppose conditions H 1 À H 4 are satisfied.Consider the operator T 1 : S r 1 !C. Then the stochastic functional integral equation Proof.The proof follows directly from theorem 3.1 when Corollary 3.1.2.Suppose conditions H 1 À H 3 are satisfied.Let the operator T : S r !X.Then the following stochastic coupled system has at least one solution in S r .
Proof.The proof comes directly from theorem 3.1 when g j ¼ 0, and j ¼ 1, 2. Proof.The proof comes directly from theorem 3.1 when f j ¼ 0, and j ¼ 1, 2. w Theorem 3.2.Suppose conditions H 1 , H 3 and H 4 are satisfied.let the operator T : S r !X.Let the functions f j and g j , j ¼ 1, 2, satisfy the following conditions: H 5 : There exists two real deterministic functions l j defined on I Â I, such that 8 0 s t a, x j 2 C, y j 2 C, j ¼ 1, 2, we have jf j ðt, s, x 2 , y 2 Þ À f j ðt, s, x 1 , y 1 Þj l j ðt, sÞ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðjx 2 À x 1 j 2 þ jy 2 À y 1 j 2 Þ q : ð a 0 jl j ðt, sÞj 2 ds N j , N j > 0, j ¼ 1, 2: H 6 : There exist two constants q j > 0, j ¼ 1, 2 such that: 8 0 s t a, x j 2 C, y j 2 C, j ¼ 1, 2: Then the stochastic system 1.3 has a unique solution in S r , provided that 0 where, q :¼ max q 1 , q 2 f g , N :¼ max N 1 , N 2 f g: Proof.It is obvious the set S r is a closed subspace of the Banach space X.So, it is also a Banach space.
Using an argument similar to those in theorem 3.1 it is easy to show that, TS r & S r : Therefore, it remains to show the contraction property of the operator T. Let (x, y) and ðx Ã , y Ã Þ belong to the set S r .Using the Cauchy-Schwartz inequality, conditions H 5 , H 6 , and taking the supremum over t 2 I gives Adding inequalities 3.3, 3.4 and using the definition of the norm in X gives So, T is a contraction operator, and hence has a unique fixed point in S r from applying the Banach fixed point theorem.Therefore, the stochastic coupled system 1.3 has a unique solution in S r .

Conclusion
We have derived some new results on the existence, and uniqueness of continuous solution of a general class of stochastic coupled systems of the Urysohn-Volterra-Itô-Doob type.The main techniques which we used are the fixed point principle, the Carath eodory conditions, and the stochastic analysis due to Itô and Doob.We believe that our results are important in their own right.Our results offer a generalization to the results developed by A.M.A. El-Sayed et al. in (El-Sayed & Kenawy, 2014a, 2014b).In the future, many models can be investigated, such as fractal integral systems (Volterra or Fredholm) with more concentration on the singular types (He, 2020).
yðsÞ, AðsÞyðsÞÞdWðsÞ: has at least one solution in S r 1 :

w
Corollary 3.2.1.Suppose conditions H 1 , and H 3 À H 6 are satisfied.Define an operator T 1 : S r 1 !C. Then the stochastic functional integral equation yðtÞ ¼ h 1 ðtÞ þ , s, yðsÞ, AðsÞyðsÞÞdWðsÞ: has a unique solution in S r 1 .provided that , s, xðsÞ, BðsÞxðsÞÞds: has a unique solution in S r .Provided that PÞÞ be the space of all stochastic processes defined on I, continuous in the mean square sense and adapted to the filtration fF t g t2I : The norm on C is defined as kxkC ¼ sup t2I kxðtÞk L 2 È É : Let X :¼ CðI,L 2 ðX,F ,PÞÞ Â CðI,L 2 ðX,F ,PÞÞ ¼ fðx,yÞ : x 2 C, y 2 Cg:That is, the space X is the Banach space of all ordered pairs (x, y) of stochastic processes which are defined on I, continuous in the mean square sense, and adapted to the filtration fF t g t2I : Let the space X be equipped with the norm kðx,yÞk X :¼ kxk C þ kyk C , 8ðx,yÞ 2 X: the stochastic coupled system 1.3 if the process ðx, yÞ 2 CðI, L 2 ðX, F , PÞÞ Â CðI, L 2 ðX, F , PÞÞ, and satisfies Equations (1.3) almost surely.