Discrete Dynamics in Nature and Society
Volume 2017 (2017), Article ID 8607946, 7 pages
https://doi.org/10.1155/2017/8607946

Research Article

Weak Solutions for Partial Random Hadamard Fractional Integral Equations with Multiple Delays

Saïd Abbas,¹ Mouffak Benchohra,² Yong Zhou,3,4 and Ahmed Alsaedi⁴

¹Laboratory of Mathematics, University of Saïda, P.O. Box 138 ENNASR, 20000 Saïda, Algeria
²Laboratory of Mathematics, University of Sidi Bel-Abbès, P.O. Box 89, 22000 Sidi Bel-Abbès, Algeria
³Faculty of Mathematics and Computational Science, Xiangtan University, Hunan 411105, China
⁴Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Correspondence should be addressed to Yong Zhou; yzhou@xtu.edu.cn

Received 23 June 2017; Accepted 14 September 2017; Published 15 October 2017

Academic Editor: Jorge E. Macías-Díaz

Copyright © 2017 Saïd Abbas et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We present some results concerning the existence of weak solutions for some functional integral equations of Hadamard fractional order with random effects and multiple delays by applying Mönch’s and Engl’s fixed point theorems associated with the technique of measure of weak noncompactness.

1. Introduction

Random differential equations arise in many applications and have been studied in the literature on bounded as well as unbounded internals of the real line for different aspects of the solution. See, for example, [1]. We refer the reader to the monograph [2] and the papers [1, 3] and the references therein. There are real-world phenomena with anomalous dynamics such as signals transmissions through strong magnetic fields, atmospheric diffusion of pollution, network traffic, and the effect of speculations on the profitability of stocks in financial markets, where the classical models are not sufficiently good to describe these features. In this case, the theory of fractional differential equations is a good tool for modeling such phenomena. Therefore, the study of the fractional differential equations with random parameters seems to be a natural one. For some fundamental results in the theory of fractional calculus and fractional differential equations, we refer the reader to the monographs of Abbas et al. [4, 5], Baleanu et al. [6], and Kilbas et al. [7] and a series of recent research articles [8–12] and the references therein.

The measure of weak noncompactness is introduced by De Blasi [13]. The strong measure of noncompactness was developed first by Banaś and Goebel [14] and subsequently developed and used in many papers; see, for example, Akhmerov et al. [15], Alvàrez [16], and Guo et al. [17] and the references therein. In [18], the authors considered some existence results by applying the techniques of the measure of noncompactness. Recently, several researchers obtained other results by application of the technique of measure of weak noncompactness; see [5]. Existence of random solutions for functional differential and integral equations has extensively been studied in various papers; see [19, 20] and the references therein.

In this paper, we discuss the existence of random solutions for the partial Hadamard fractional integral equation of the following form:

where , , , (), , , , , (), and are given continuous functions, is a measurable space, and is a real (or complex) Banach space with norm and dual , such that is the dual of a weakly compactly generated Banach space , is the left-sided mixed Hadamard integral of order , and is a given continuous and measurable function such that

2. Preliminaries

Let be the Banach space of all continuous functions from into with the supremum (uniform) norm . By , we denote the Banach space of measurable functions which are essentially bounded equipped with the norm

Denote by the Banach space with its weak topology.

Definition 1. A Banach space is called weakly compactly generated (WCG, in short) if it contains a weakly compact set whose linear span is dense in

Definition 2. A function is said to be weakly sequentially continuous if takes each weakly convergent sequence in to a weakly convergent sequence in (i.e., for any in with in , one has in ).

Definition 3 (see [21]). The function is said to be Pettis integrable on if and only if there is an element corresponding to each such that for all , where the integral on the right-hand side is assumed to exist in the sense of Lebesgue (by definition,

Let be the space of all -valued Pettis integrable functions on , and let be the Banach space of Lebesgue measurable functions Define the class by

The space is normed by

where stands for a Lebesgue measure on

The following result is due to Pettis (see [21], Theorem  3.4 and Corollary  3.41).

Proposition 4 (see [21]). If and is a measurable and essentially bounded -valued function, then

For all what follows, the sign “” denotes the Pettis integral.

Let us recall the definitions of Pettis integral and Hadamard integral of fractional order.

Definition 5 (see [7]). The left-sided mixed Pettis Hadamard integral of order , for a function , is defined as

Remark 6. Let For every , one has

Definition 7. Let , , and For , define the left-sided mixed Pettis Hadamard partial fractional integral of order by the expression

Let be the -algebra of Borel subsets of . A mapping is said to be measurable if, for any , one has

To define integrals of sample paths of random process, it is necessary to define a jointly measurable map.

Definition 8. A function is called jointly measurable if, for any , one has

where is the direct product of the -algebras and , those defined in and , respectively.

Lemma 9 (see [22]). A function is jointly measurable if is measurable for all and is continuous for all

Definition 10. A function is called random Carathéodory if the following conditions are satisfied:

(i)The map is jointly measurable for all .(ii)The map is continuous for all and

Let be a mapping. Then is called a random operator if is measurable in for all and it is expressed as In this case, we also say that is a random operator on A random operator on is called continuous (resp., compact, totally bounded, and completely continuous) if is continuous (resp., compact, totally bounded, and completely continuous) in for all The details of completely continuous random operators in Banach spaces and their properties appear in Itoh [23].

Definition 11 (see [24]). Let be the family of all nonempty subsets of and let be a mapping from into A mapping is called random operator with stochastic domain if is measurable (i.e., for all closed , is measurable) and for all open and all , is measurable. will be called continuous if every is continuous. For a random operator , a mapping is called random (stochastic) fixed point of if for almost all , , and and for all open , is measurable.

Definition 12 (see [13]). Let be a Banach space, let be the bounded subsets of , and let be the unit ball of De Blasi measure of weak noncompactness is the map defined by

De Blasi measure of weak noncompactness satisfies the following properties:

(a).(b) is weakly relatively compact.(c).(d) ( denotes the weak closure of ).(e).(f).(g).(h).

The next result follows directly from the Hahn-Banach theorem.

Proposition 13. Let be a normed space, and with Then, there exists with and .

For a given set of functions , let us denote that

Lemma 14 (see [17]). Let be bounded and equicontinuous. Then the function is continuous on , and

where ; , and is De Blasi measure of weak noncompactness defined on the bounded sets of

We will need the following fixed point theorems.

Theorem 15 (see [25]). Let be a nonempty, closed, convex, and equicontinuous subset of a metrizable locally convex vector space such that . Suppose that is weakly, sequentially continuous. If the implication

holds for every subset , then the operator has a fixed point.

Theorem 16 (see [24]). Let be measurable with closed, convex, and solid (i.e., ) for all We assume that there exists measurable with for all Let be a continuous random operator with stochastic domain such that, for every . Then has a stochastic fixed point.

3. Existence Results

Let us start by defining what we mean by a random solution of problem (1).

Definition 17. By a random solution of problem (1), we mean a measurable function that satisfies the integral equation on , as well as on

The following hypotheses will be used in the sequel:

The functions , , are bounded for a.e. , and The function is random Carathéodory on for each For a.e. , as well as all , the function is weakly, sequentially continuous.There exist functions with , , for each , such that, for all , one has

 for all and a.e. For all , there exists a continuous function with , such that, for each and any , one has

There exists a function with for each such that, for any bounded ,

 where There exists a random function such that

 where

Set

Theorem 18. Assume that hypotheses hold. If

then problem (1) has a random solution defined on

Proof. Define the operator by

The functions , , are continuous for all Again, as the function is continuous on , defines a mapping Thus, is a solution for problem (1) if and only if . We shall show that the operator satisfies all conditions of Theorem 16. The proof will be given in several steps.
Step  1 ( is a random operator with stochastic domain on ). Since is random Carathéodory, the map is measurable in view of Definition 10. Therefore, the map

is measurable. As a result, is a random operator on into
Let be defined by

Clearly, the subset is closed, convex, end equicontinuous for all Then is measurable by Lemma  17 in [24]. Therefore, is a random operator with stochastic domain
Step  2 ( is continuous). Let be a sequence such that in Then, there exists such that , and
Thus, for each and , one has

Using the Lebesgue Dominated Convergence Theorem, we get

As a consequence of Steps 1 and 2, we can conclude that is a continuous random operator with stochastic domain , and is bounded.
Step  3 (for every ). For this, we apply Theorem 15. The proof will be given in several claims.
Claim  1 ( maps into itself). Let be fixed, and let Assume that Then there exists such that Thus, we get

Next, for any fixed , let such that and , and let , where Then there exists such that and Thus, one has

Hence Therefore, maps into itself.
Claim  2 ( is weakly, sequentially continuous). Let be a sequence in and let in for any , and each Fix , and since satisfies assumption , one has that converges weakly uniformly to

Hence, the Lebesgue Dominated Convergence Theorem for Pettis integral implies that converges weakly uniformly to in We do it for any , and each , so Then is weakly, sequentially continuous.
Claim  3 (implication (14) holds). Let be a subset of such that Obviously, ). Further, as is bounded and equicontinuous, by Lemma  3 in [26], the function is continuous on Since the function is continuous on , the set is compact. From Lemma 14 and the properties of the measure , for any , as well as each , one has

Thus,

From (21), we get , that is; , for any , and each Hence, Theorem  2 in [27] shows that is weakly relatively compact in .
As consequence of Claims 1–3 and from Theorem 15, it follows that, for every , Apply now Theorem 16; Steps 1–3 show that, for each has at least one fixed point in Since , and a measurable selector of exists, the operator has a stochastic fixed point; that is, problem (1) has at least one random solution defined on

4. An Example

Let

be the Banach space with norm and let be equipped with the usual -algebra consisting of Lebesgue measurable subsets of . Given a measurable function , consider the functional random integral problem of the following form:

where , , and ; , , , and

, , , and is a continuous and measurable function such that , where

Set

Then, For each , , and , one has

Hence condition is satisfied with and

A simple computation shows that all other conditions of Theorem 18 are satisfied. Consequently, Theorem 18 implies that problem (33) has a random solution defined on .

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research is supported by the National Natural Science Foundation of China (11671339).

References

J. L. Strand, “Random ordinary differential equations,” Journal of Differential Equations, vol. 7, pp. 538–553, 1970.

A. T. Bharucha-Reid, “Random algebraic equations,” in Probabilistic Methods in Applied MAThematics, Vol. 2, pp. 1–52, Academic Press, New York, 1970.

V. Lupulescu, D. O'Regan, and G. ur Rahman, “Existence results for random fractional differential equations,” Opuscula Mathematica, vol. 34, no. 4, pp. 813–825, 2014.

S. Abbas, M. Benchohra, and G. M. N'Guérékata, Topics in Fractional Differential Equations, Springer, New York, NY, USA, 2012.

S. Abbas, M. Benchohra, and G. M. N'Guérékata, Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, NY, USA, 2015.

D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus Models and Numerical Methods, vol. 3 of Series on Complexity, Nonlinearity and Chaos, World Scientific, 2012.

A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, New York, NY, USA, Elsevier, 2006.

Y. Zhou and L. Peng, “Weak solutions of the time-fractional Navier-Stokes equations and optimal control,” Computers & Mathematics with Applications. An International Journal, vol. 73, no. 6, pp. 1016–1027, 2017.

Y. Zhou and L. Peng, “On the time-fractional Navier-Stokes equations,” Computers & Mathematics with Applications. An International Journal, vol. 73, no. 6, pp. 874–891, 2017.

Y. Zhou, V. Vijayakumar, and R. Murugesu, “Controllability for fractional evolution inclusions without compactness,” Evolution Equations and Control Theory, vol. 4, no. 4, pp. 507–524, 2015.

Y. Zhou and L. Zhang, “Existence and multiplicity results of homoclinic solutions for fractional Hamiltonian systems,” Computers & Mathematics with Applications. An International Journal, vol. 73, no. 6, pp. 1325–1345, 2017.

Y. Zhou, B. Ahmad, and A. Alsaedi, “Existence of nonoscillatory solutions for fractional neutral differential equations,” Applied Mathematics Letters, vol. 72, pp. 70–74, 2017.

F. S. De Blasi, “On a property of the unit sphere in a Banach space,” vol. 21(69), no. 3-4, pp. 259–262, 1977.

J. Banaś and K. Goebel, Measures of Noncompactness, vol. 60 of Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1980.

R. R. Akhmerov, M. I. Kamenskii, A. S. Potapov, A. E. Rodkina, and B. N. Sadovskii, Measures of Noncompactness and Condensing Operators, vol. 55 of Operator Theory: Advances and Applications, Birkhäauser, Basel, Switzerland, 1992.

J. C. Alvàrez, “Measure of noncompactness and fixed points of nonexpansive condensing mappings in locally convex spaces,” Real Academia de Ciencias Exactas, Fisicas y Naturales de Madrid. Revista, vol. 79, no. 1-2, pp. 53–66, 1985.

D. Guo, V. Lakshmikantham, and X. Liu, Nonlinear Integral Equations in Abstract Spaces, vol. 373 of Mathematics and its Applications, Kluwer Academic Publishers Group, Dordrecht, The Netherlands, 1996.

D. O'Regan, “Weak solutions of ordinary differential equations in Banach spaces,” Applied Mathematics Letters, vol. 12, no. 1, pp. 101–105, 1999.

S. d. Abbas, W. A. Albarakati, M. Benchohra, and J. Henderson, “Existence and Ulam stabilities for Hadamard fractional integral equations with random effects,” Electronic Journal of Differential Equations, Paper No. 25, 12 pages, 2016.

B. C. Dhage, S. V. Badgire, and S. K. Ntouyas, “Periodic boundary value problems of second order random differential equations,” Electronic Journal of Qualitative Theory of Differential Equations, pp. 1–14, 2009.

B. J. Pettis, “On integration in vector spaces,” Transactions of the American Mathematical Society, vol. 44, no. 2, pp. 277–304, 1938.

A. S. Kechris, Classical Descriptive Set Theory, vol. 156 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1995.

S. Itoh, “Random fixed-point theorems with an application to random differential equations in Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 67, no. 2, pp. 261–273, 1979.

H. W. Engl, “A general stochastic fixed-point theorem for continuous random operators on stochastic domains,” Journal of Mathematical Analysis and Applications, vol. 66, no. 1, pp. 220–231, 1978.

D. O'Regan, “Fixed-point theory for weakly sequentially continuous mappings,” Mathematical and Computer Modelling, vol. 27, no. 5, pp. 1–14, 1998.

D. Bugajewski and S. a. Szufla, “Kneser's theorem for weak solutions of the Darboux problem in Banach spaces,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, vol. 20, no. 2, pp. 169–173, 1993.

A. R. Mitchell and C. h. Smith, “Nonlinear Equations in Abstract Spaces,” in An existence theorem for weak solutions of differential equations in Banach spaces, V. Lakshmikantham, Ed., pp. 387–403, Academic Press, New York, NY, USA, 1978.