Abstract
In this work, a new existence result is established for a nonlocal hybrid boundary value problem which contains one left Caputo and one right Riemann–Liouville fractional derivatives and integrals. The main result is proved by applying a new generalization of Darbo’s theorem associated with measures of noncompactness. Finally, an example to justify the theoretical result is also presented.
1. Introduction
In the past years, fractional differential equations have attracted a lot of attention from many research studies as they have played a key role in many basic sciences such as chemistry, control theory, biology, and other arenas [1–3]. In addition, boundary conditions of differential models are the strongest tools to extend applications of those equations [4–6]. In fact, fractional differential equations can be extended by creating different types of boundary conditions. Newly, many authors have studied various types of boundary conditions to obtain new results of differential models.
The following hybrid differential equation was studied by Dhage and Lakshmikantham [7]:where and are continuous functions from into and , respectively. Based on the above work, the Caputo hybrid boundary value problem of the form was studied by Hilal and Kajouni [8] in which , and are continuous functions from into and , respectively, and are real constants with . For some recent results on hybrid fractional differential equations, see [9–12].
In [13], the authors proved the following integro-differential equation: where and indicate right Caputo and left Riemann–Liouville fractional derivatives of orders and , respectively, are continuous functions, and the symbols and denote both right and left Riemann–Liouville fractional integrals of orders , respectively. Ahmad et al. [13] applied Banach and Krasnosel’skiĭ fixed point theorems as well Leray–Schauder nonlinear alternative to obtain main results. We point out that fractional differential equations containing mixed fractional derivatives appear in the study of variational principles [14].
For some recent results for boundary value problems involving left or/and right fractional derivatives, we refer to the papers [15–31] and references therein.
In the present paper, we combine mixed fractional derivatives and hybrid fractional differential equations. More precisely, we investigate the existence of solutions for the following hybrid boundary value problem which contains both left Caputo and right Riemann–Liouville fractional derivatives and integrals and nonlocal hybrid conditions of the form: where and are right Caputo and left Riemann–Liouville fractional derivatives of orders and , respectively, and the symbols and denote both right and left Riemann–Liouville fractional integrals of orders , respectively, , , and . An existence result is obtained via a new extension of Darbo’s theorem associated with measures of noncompactness.
The structure of the paper has been organized as follows. Section 2 presents some basic definitions and lemmas which will be applied in the future. In Section 3, we prove an existence result for problem (4). Finally, we present an example to illustrate the obtained result.
2. Preliminaries
Now, some basic notations are recalled from [2].
Definition 1. For an integrable function , we define the left and right Riemann–Liouville fractional integrals of order , respectively, by
Definition 2. For the function in which , we define the left Riemann–Liouville fractional derivative and the right Caputo fractional derivative of order , respectively, by
Lemma 1. If and , then the following relations hold almost everywhere on :
As the technique of measure of noncompactness will be applied to obtain our main result, we recall some basic facts about the notion of measure of noncompactness.
Assume that is the real Banach space with the norm and zero element . For a nonempty subset of , the closure and the closed convex hull of will be denoted by and , respectively. Also, and denote the family of all nonempty and bounded subsets of and its subfamily consisting of all relatively compact sets, respectively.
Definition 3 (see [32]). We say that a mapping is a measure of noncompactness, if the following conditions hold true:(1)The family Ker is nonempty and Ker (2)(3)(4)(5) for (6)For the sequence of closed sets from in which for and , we have
In [33], some generalizations of Darbo’s theorem have been proved by Samadi and Ghaemi. Also, in [34], Darbo’s theorem was extended, and the following result was presented which is basis for our main result.
Theorem 1. Let be a continuous self-mapping operator on the set , where denotes a nonempty, bounded, closed, and convex subset of a Banach space . Assume that, for all nonempty subset of , we havewhere is an arbitrary measure of noncompactness defined in and . Then, has a fixed point in .
In Theorem 1, let indicate the set of all pairs where the following conditions hold true: () for each strictly increasing sequence () is strictly increasing function () If be a sequence of positive numbers, then () Let be a decreasing sequence in which and , then we have
Next, the definition of a measure of noncompactness in the space is recalled which will be applied later. Fix , and for and , we define
Banas and Goebel [32] proved that is a measure of noncompactness in the space .
Lemma 2 (see [32]). The measure of noncompactness on satisfies the following condition:for all .
3. Main Existence Result
In this section, an existence result of problem (4) is investigated. In view of [13], Lemma 2, we present the following lemma which is an essential tool in our consideration.
Lemma 3. Let , , and . Then, the solution of the problemhas the form:
where
Now, the hypotheses which will be applied to prove the main result of this section are presented. is a continuous function, and there exists a positive real number provided that where and . Moreover, assume that . are continuous functions provided that where and . The inequality has a positive solution . Also, assume that where
Theorem 2. Suppose that the hypotheses are true. Then, the hybrid boundary value problem (4) has at least one solution on .
Proof. Due to Lemma 3, assume that the operator has been defined on as follows:whereFirst, we show that in which . In view of assumption , we conclude that , . Consequently, by proving , the claim is obtained. Let be a sequence in such that . Then, due to our assumptions, we getHence, . To obtain that , by the definitions of and , we have and . Hence, we have . Consequently, for all .
Now, we prove that the ball is mapped into itself by the operator . Let us fix . Hence, due to existence assumptions, for , we haveConsequently, according to assumption we conclude that maps the ball into itself.
Now, the continuity property of the operator is considered on the ball . To do this, fix and take such that . Then, for , we haveThen, we haveConsequently, the continuity property of is obtained on the ball .
To finish the proof, condition (8) of Theorem 1 is proved. Consider as a nonempty subset of the ball and assume that , be arbitrarily constant. Choose such that and . Taking into account our assumptions, we getwhereConsequently,As is uniformly continuous on , we have as . Thus, from (27), we conclude thatNext, we estimate and . In view of (21), since is uniformly continuous on , then for fixed , there exists such that, for with , we haveBesides, since and are uniformly continuous on , for with , we have and also . Consequently, we conclude that . Now, we estimate for . By applying (28) and (29) and Lemma 2 and using the fact that , we getConsequently, we derive thatThus, we conclude the contractive condition in Theorem 1 with and . Thus, by Theorem 1, at least one solution is obtained for the operator in which is a solution of problem (4) and the proof is completed.
Now, the following example is investigated to show the applicability of the obtained result.
Example 1. Consider the following hybrid boundary value problem:By puttingin problem (4), we conclude the above hybrid boundary value problem as a special case of problem (4). Now, the conditions of Theorem 2 are checked. For all and , we haveMoreover, we have . Besides, the functions and are continuous, and for all and , we haveIn this example , , , , , , , and . Hence, , and . Consequently, the existent inequality in condition has the form:Obviously, the above inequality has the positive solution , for example . Moreover, according to the obtained values we haveThus, we conclude all conditions of Theorem 2, and hence at least one solution of the mapping is obtained on which is a solution of problem (32).
4. Conclusion
We have studied a nonlocal hybrid boundary value problem which contains both left Caputo and right Riemann–Liouville fractional derivatives and integrals and nonlocal hybrid conditions. An existence result is proved by applying a new generalization of Darbo’s fixed point theorem associated with measures of noncompactness. The result obtained in this paper is new and significantly contributes to the existing literature on the topic.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research was funded by King Mongkut’s University of Technology North Bangkok (contract no. KMUTNB-62-KNOW-28).