Abstract
In this article, by using the Schauder fixed point theorem, we first study the existence of solutions for a new coupled system of Caputo fractional differential equations with multipoint boundary value conditions under the assumption that the nonlinear term satisfies two types of the Carathéodory conditions. Using this result, we provide the existence of solutions of the system with infinite points and Riemann–Stieltjes integral boundary conditions, respectively, instead of doing it directly. Finally, we give three examples to illustrate the feasibility of main results.
1. Introduction
The fractional differential equation is an important branch of differential equations, which is widely used in mathematics, physics, engineering, and other fields, and it solves robotics, signal processing, and conversion problems [1–4]. In fact, the fractional differential equation is an important tool for mathematical modeling due to its memorability and genetic properties. Recently, nonlocal integer and fractional orders fractional differential equations have attracted attention of a large number of scholars, see [5–24]. Indeed, the nonlocal problems for (fractional) differential equations have received great attention, see[5–14] and the references therein. In this paper, we consider the coupled system of Caputo fractional differential equationswith the multipoint conditionswhere and are the Caputo fractional derivatives, , , . Furthermore, we consider system (1) with the following Riemann–Stieltjes integral conditions:where is an increasing function, , . We also discuss system (1) with infinite-point boundary conditionswhere , .
There are many methods for searching the existence of solutions of fractional boundary value problems, such as partial order methods [15], topology degree theory [10–14, 16–18], the critical point theorem [19], and so on. Recently, the authors in [10] studied the existence and continuous dependence of solutions by the Schauder fixed point theorem for the following coupled system:
Moreover, the authors studied the following Riemann–Stieltjes integral conditionsand also discussed the following infinite-point conditions:By using the Schauder fixed point theorem and Banach contraction mapping principle, the existence and uniqueness of solutions of the system are proved.
By applying the Guo-Krasnosel’skii fixed point theorem of the cone expansion-compression type, Zhang in [11] considered the several local existence and multiplicity of positive solutions for the following problem:where is Riemann–Liouville fractional derivative, is a fixed integer, , and may be singular with respect to both the time and space variables.
In [12], the existence of solutions for the following problem was obtained by using the Schauder’s fixed point theoremwhere is the Caputo fractional derivative and , is an increasing function.
In this paper, we discuss the coupled system (1), in which the nonlinear terms include two space variables and the integral terms. By using the Schauder fixed point theorem, the existence of solutions for system (1) with the multipoint boundary conditions (2) is proved under the assumption that the nonlinear term satisfies two types of the Carathéodory conditions, which consists sublinear and linear conditions. Furthermore, based on the above results, we give the existence of solutions of system (1) with infinite point and Riemann–Stieltjes integral boundary conditions, respectively, which avoids proving again. Compared with the articles mentioned above, system (1) is more general. For example, the nonlinear terms in [10] contain a single space variable, while we consider two space variables.
The outline of this paper is organized as follows. In Section 2, we state some preliminary settings. In Section 3, we show the proof of existence of solutions. In Section 4, three examples are given in order to illustrate our results. In Section 5, we give the conclusions of this paper.
2. Preliminaries and Lemmas
In this section, we will provide some definitions and lemmas to be used in the following proofs.
Definition 1. (see [1]) Let . Then, the Caputo fractional derivative of order of is given by the following:which exists almost everywhere on .
Definition 2. (see [1]) For a function , the Riemann–Liouville fractional integral of order of is given by the following:provided that the right-hand side is pointwise defined on .
Lemma 1. (see [1]) For , the fractional differential equation has a general solution
Lemma 2. (see [1]) Assume that , then
Lemma 3. (see [4]) Let , , where , . Then, , andwhere .
Lemma 4. Let , . Then, the systemwith the boundary conditionshas a unique solution , which can be expressed by
Proof. According to the known conditions and Lemma 3, we havewhere and , it can be seen .Then,in the same wayTherefore, it is concluded that the integral exists.
Next, using Lemma 2 to integrate both sides at the same time, we haveBy the boundary conditions (2), the expression of can be obtained as follows:Consequently, substituting into formula (21), we obtain the expression of the solution as follows (17).
In order to prove later, we make some assumptions here as follows:
satisfies the Carathéodory condition, i.e.,(i) is measurable for any .(ii) is continuous for almost all .(iii)There exist functions with , , such that satisfies the Carathéodory condition, i.e.,(i) is measurable for any .(ii) is continuous for almost all .(iii)There exist functions with , such that satisfies the Carathéodory condition, i.e.,(i) is measurable for any .(ii) is continuous for almost all .(iii)There exist functions with such that satisfies the Carathéodory condition, i.e.,(i) is measurable for any .(iii) is continuous for almost all .(iii)There exist functions with such thatwhere .
3. Main Results
Let be the Banach space equipped with the norm . Let be the Banach space of ordered pairs equipped with the norm . In the following part, from Lemma 3, we define the operator as follows:where
In the following part, for convenience, we set
Theorem 1. Assume that the assumptions - hold. Then, systems (1) and (2) have at least one solution.
Proof. Let us set up , , then . Accordingly, we infer that, there exists large enough such thatwithout loss of generality, we may assume that .
First, we prove that , where . It should be noted that for any ,Thus, we havewhich means . In the same way, we get thatBased on (35) and (36), we conclude thatThis proves that .
Second, we prove that operator is continuous. Let , and in . Utilizing the assumptions -, we haveOn the other hand, we haveBased on , we know . Thus, by the Lebesgue dominated convergence theorem, we haveAccordingly, we conclude that in as , that is, the operator is continuous.
In the following part, we prove that is equicontinuous. Based on , we know , , , , . Thus, for any , , we haveSince the function is continuous on , we get that, for any , there exists such that, for any , ,Consequently, for any , , we havewhich means that is equicontinuous on , and also, is equicontinuous on . Based on (35) and (36), we know that the set is uniformly bounded. According to the Arzal -Ascoli theorem, we prove that operator is a completely continuous operator. Thus, by the Schauder fixed point theorem, we deduce that systems (1) and (2) have at least one solution , which can be expressed as follows:
Theorem 2. Assume that the assumptions - hold. Then, systems (1) and (3) have at least one solution.
Proof. First, we make an interval partition: such that . Let . Then, we haveBy Theorem 1, we know, systems (1) and (2) have a solution , which can be expressed in (44) . Let in (45), we then havethat is, the boundary conditions (3). Therefore, we take the limit to the solution (44), and we then get the solution of systems (1) and (3), which can be expressed as follows:
Theorem 3. Assume that the assumptions - hold. Then, systems (1) and (4) have at least one solution.
Proof. Based on Theorem 1, we know, for any , the solution of systems (1) and (2) can be expressed as follows:Let in (48), we then haveBased on the convergence of and , it is easy to check thatare convergent. Thus, by using the Lebesgue-dominated convergence theorem, we know that the existence of the limits for the right-hand side in (44). On the other hand, we notice that the boundary conditions (2) can be transformed to the boundary conditions (4) as , we obtain that the limits of (44) is the solution of systems (1) and (4), which can be expressed as follows:
Theorem 4. Assume that the assumptions - and hold. Then, systems (1) and (2) have at least one solution.
Proof. It follows from that . Thus, there exists large enough such thatIn the following part, we prove that , where . Similarly as (35), we obtain that, for any ,Thus, we obtain that . By the same way as Theorem 1, we get that is a completely continuous operator. Then, by the Schauder fixed point theorem, we deduce that there exists at least one solution of systems (1) and (2).
Theorem 5. Assume that the assumptions -, hold. Then, systems (1) and (3) have at least one solution.
Proof. First, we make an interval partition: such that . Let . In this case, we obtain as follows:So, the assumption implies that is true. Thus, based on Theorem 4, we know that systems (1) and (2) have a solution in . Accordingly, by Theorem 2, we take the limit to the solution of systems (1) and (2), the solution of systems (1) and (3) is then obtained.
Theorem 6. Assume that the assumptions -, hold. Then, systems (1) and (4) have at least one solution.
Proof. The proof is similar to that of Theorem 3. So, we omit details.
4. Examples
Example 1. In this section, we consider the following system:with the following boundary conditions:Let , ,Then, systems (55) and (56) can be transformed to systems (1) and (2). In addition, we obtain thatLet , , , , , , , , , , , , , , , , , , , , , , . It then holds that , , , , , , , , , , , , , andThus, we get that the conditions - are satisfied. Therefore, by Theorem 1, we infer that systems (55)-(56) have at least one solution.
Example 2. We discuss the following system:with the following boundary condition:whereLet , ,Then, systems (60) and (61) can be transformed to systems (1) and (3). In addition, we obtain thatLet , , , , , , , , , , , , , , , , , , , , , , . It then holds that , , , , , , , , , , , , andThus, we get that the conditions - are satisfied. Therefore, by Theorem 2, we infer that systems (60)-(61) have at least one solution.
Example 3. We discuss the following system:with the following boundary condition:Let , ,Then, systems (66) and (67) can be transformed to systems (1) and (4). In addition, we obtain thatLet , , , , , , , , , , , , , , , , , , , , , , . It then holds that , , , , , , , , , , , , , andThus, we infer that the conditions - are satisfied. Therefore, by Theorem 3, we infer that systems (66)-(67) have at least one solution.
5. Conclusions
In this paper, by using the Schauder fixed point theorem, the existence of solutions for the coupled system (1) with multipoint boundary conditions (2) is proved, in which the nonlinear term satisfies two types of the Carathéodory condition. Under these two conditions, the power exponent of the control function of the nonlinear term includes sublinear and linear cases, which makes the problem we study more general. Furthermore, we consider the Riemann–Stieltjes integral and infinite-point boundary conditions around it. Based on the results of systems (1)-(2), the existence of solutions for the last two systems is obtained directly. Finally, three examples are given to illustrate our theoretical results.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Acknowledgments
This research was supported by the National Natural Science Foundation of China (11871302)