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The Existence and Uniqueness of Solutions for a Class of Nonlinear Fractional Differential Equations with Infinite Delay
We prove the existence and uniqueness of solutions for two classes of infinite delay nonlinear fractional order differential equations involving Riemann-Liouville fractional derivatives. The analysis is based on the alternative of the Leray-Schauder fixed-point theorem, the Banach fixed-point theorem, and the Arzela-Ascoli theorem in such that is continuous and is a phase space.
Fractional derivatives and integrals have been vastly used in different fields, facing a huge development especially during the last few decades (see, e.g., [1–9] and the references therein). The approaches based on fractional calculus establish models of engineering systems better than the ordinary derivatives approaches [1–6].
In particular, fractional differential equations as an important research branch of fractional calculus attracted much more attention (see, e.g., [10–20] and the references therein). Also varieties of schemes for numerical solutions of fractional differential equations are reported (see, e.g., [6, 21–23] and the references therein). We notice that some investigations have been done on the existence and uniqueness of solutions for fractional differential equations with delay (see, e.g., [24, 25] and the references therein).
Having all the aforementioned facts in mind, in this paper we study the existence and uniqueness of solutions for a class of delayed fractional differential equations, namely, where is a positive integer, is a given function satisfying some assumptions that will be specified later, with , and?? is called a phase space that will be defined later. and are the standard Riemann-Liouville fractional derivatives. , which is an element , is defined as any function on as follows:
Here represents the preoperational state from time up to time . We also consider the following nonlinear fractional differential equation: where , , , , and are as (1) and is a given function which satisfies .
The notion of the phase space plays an important role in the study of both qualitative and quantitative theories for functional differential equations. A common choice is a seminormed space satisfying suitable axioms, which was introduced by Hale and Kato .
In this section, we present some basic notations and properties which are used throughout this paper. First of all, we will explain the phase space introduced by Hale and Kato . Let , and let be a Banach space with norm . Further, let be a linear space of functions mapping into with seminorm having the following axioms,(B1)If is continuous on and , then and are continuous for any .(B2)There exist functions and with the following properties. ?(i) is continuous for . ?(ii) is locally bounded for . ?(iii) For every function, has the properties of and , holds that . (B3)There exists a positive constant such that for all .(B4)The quotient space is a Banach space. We notice that in this paper, we select and ; thus (iii) can be converted to , ?for?all?.
See  for examples of the phase space satisfying all axioms –.
Let and be the space of all continuous real function on . Consider also the space of all continuous real functions on which later identifies with the class of all such that . By , we denote the Banach space of all continuous functions from into with the norm , where is a suitable complete norm on .
The most common notation for th order derivative of a real-valued function , which is defined in an interval denoted by , is . Here, the negative value of corresponds to the fractional integral.
Definition 1. For a function defined on an interval , the Riemann-Liouville fractional integral of of order is defined by [1, 6] and the Riemann-Liouville fractional derivative of of order reads as Also, we denote as and as . Further, and are referred to as and , respectively. If the fractional derivative is integrable, then we have [4, page 71] If is continuous on , then is integrable, , and
Proposition 2. Let be continuous on and a nonnegative integer, then where
Lemma 3 (see ). Let be a real function and a nonnegative, locally integrable function on . If there exist positive constants and such that , then there exists a constant such that , for all .
3. Existence and Uniqueness
In this section, we prove the existence results for (1) and (3) by using the alternative of Leray-Schauder’s theorem. Further, our results for the unique solution is based on the Banach contraction principle. Let us start by defining what we mean by a solution of (1). Let the space A function is said to be a solution of (1) if satisfies (1).
For the existence results on (1), we need the following lemma.
Lemma 4. Equation (1) is equivalent to the Volterra integral equation
Proof. The proof is an immediate consequence of Proposition 2.
To study the existence and uniqueness of solutions for (1), we transform (1) into a fixed-point problem. Consider the operator defined by where, Let be the function defined as Then, we get . For each with , we denote by the function defined as follows: If satisfies the integral equation , then we can decompose as , , which implies for every , and the function satisfies set , and let be the seminorm in defined by . is a Banach space with norm . Let the operator be defined by where . The operator has a fixed point equivalent to that has a fixed point too.
Theorem 5. Assume that is a continuous function, and there exist such that . Then, (1) has at least one solution on .
Proof. It is enough to show that the operator defined as (18) satisfies the following: (i) is continuous, (ii) maps bounded sets into bounded sets in , (iii) maps bounded sets into equicontinuous sets of , and (iv) is completely continuous. (i) Let converges to in , then
Hence, as , and thus is continuous.(ii) For any , let be a bounded set. We show that there exists a positive constant such that . Let , since is a continuous function, we have for each ,
where , and . Hence, we obtain .(iii) Let and . Let be a bounded set of as in and , then given choose
and . If , then
where . Hence, is equicontinuous.(iv) It is an immediate consequence from (i)–(iii), together with the Arzela-Ascoli theorem.
We show in the following that there exists an open set with for and . Let and for some . Then, for each , we have . It follows by assumption of the theorem On other hand, we have If we let the right-hand side of (25), then and, therefore, Using the aforementioned inequality and the definition of , we get Then, using Lemma 3, there exists a constant such that where is mentioned in (22), and Hence, and then . Therefore, Set . Then, is continuous and completely continuous. From the choice of , there is no such that , for ; therefore, by the nonlinear alternative of the Leray-Schauder theorem, the proof is complete.
Theorem 6. Let be a continuous function. If there exists a positive constant such that , and then (1) has a unique solution in the interval , where,
Proof. The solution of (1) is equivalent to the solution of the integral equation (17). Hence, it is enough to show that the operator , satisfies the Banach fixed-point theorem. Consider and for each , we have Hence, , and then is a contraction. Therefore, has a unique fixed point by Banach’s contraction principle.
Theorem 7. Let be a continuous function, and let the following assumptions hold.(H1) There exist such that for each and and .(H2) The function is continuous and completely continuous. For any bounded set in , the set is equicontinuous in . There exist positive constants and such that for each and .If , then (3) has at least one solution on , where .
Proof. Consider the operator defined by
In analog to Theorem 5, we consider the operator defined by
By using (H2) and Theorem 5, the operator is continuous and completely continuous. Now, it is sufficient to show that there exists an open set with for and .
Let and for some . Then, for each ,. Hence, where is named the in right-hand side of (25) such that . Since , if we let , then Then, using Lemma 3, there exists a constant such that and, therefore, , where and . Then, and, hence, Set . From the choice of , there is no such that for . As a consequence of the nonlinear alternative of the Leray-Schauder theorem, we deduce that has a fixed-point in , which is a solution of (3).
The unique solution of (3), under some conditions, is studied in the following theorem which is the result of the Banach contraction mapping.
Theorem 8. Let be a continuous function, and there exist positive constants , such that where and . Then, (3) with the following conditions has a unique solution in the interval such that is defined in Theorem 6.
Proof. The proof is a similar process Theorem 6.
In this paper, the existence and the uniqueness of solutions for the nonlinear fractional differential equations with infinite delay comprising standard Riemann-Liouville derivatives have been discussed in the phase space. Leray-Schauder’s alternative theorem and the Banach contraction principle were used to prove the obtained results. Further generalizations can be developed to some other class of fractional differential equations such as , where , and is nonnegative integer.
The authors would like to thank the referee for helpful comments and suggestions. This paper was funded by King Abdulaziz University, under Grant no. (130-1-1433/HiCi). The authors, therefore, acknowledge technical and financial support of KAU.
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