Abstract
In this paper, we study the existence of solutions for fractional differential equations of arbitrary order with multipoint multiterm RiemannLiouville type integral boundary conditions involving two indices. The RiemannLiouville type integral boundary conditions considered in the problem address a more general situation in contrast to the case of a single index. Our results are based on standard fixed point theorems. Some illustrative examples are also presented.
MSC: 26A33, 34A08.
Keywords:
fractional differential equations; nonlocal boundary conditions; fixed point theorems1 Introduction
In the last few decades, the subject of fractional differential equations has become a hot topic for the researchers due to its intensive development and applications in the field of physics, mechanics, chemistry, engineering, etc. For a reader interested in the systematic development of the topic, we refer the books [16]. A fractionalorder differential operator distinguishes itself from the integerorder differential operator in the sense that it is nonlocal in nature, that is, the future state of a dynamical system or process involving fractional derivative depends on its current state as well its past states. In other words, differential equations of arbitrary order describe memory and hereditary properties of various materials and processes. As a matter of fact, this characteristic of fractional calculus makes the fractionalorder models more realistic and practical than the classical integerorder models. There has been a great surge in developing the theoretical aspects such as periodicity, asymptotic behavior and numerical methods for fractional equations. For some recent work on the topic, see [725] and the references therein. In particular, the authors studied nonlinear fractional differential equations and inclusions of arbitrary order with multistrip boundary conditions in [21], while a boundary value of nonlinear fractional differential equations of arbitrary order with RiemannLiouville type multistrip boundary conditions was investigated in [22]. Sudsutad and Tariboon [26] obtained some existence results for an integrodifferential equation of fractional order with mpoint multiterm fractionalorder integral boundary conditions.
In this paper, we study a boundary value problem of fractional differential equations of arbitrary order , with mpoint multiterm RiemannLiouville type integral boundary conditions involving two indices given by
where denotes the Caputo fractional derivative of order q, f is a given continuous function, , is the RiemannLiouville fractional integral of order , , , , , and is such that
Here we emphasize that RiemannLiouville type integral boundary conditions involving two indices give rise to a more general situation in contrast to the case of a single index [22]. Furthermore, the present work dealing with an arbitraryorder problem generalizes the results for the problem of order obtained in [26]. Several examples are considered to show the worth of the results established in this paper.
We develop some existence results for problem (1.1) by using standard techniques of fixed point theory. The paper is organized as follows. In Section 2 we recall some preliminary facts that we need in the sequel, and Section 3 contains the main results. Section 4 provides some examples for the illustration of the main results.
2 Preliminaries from fractional calculus
Let us recall some basic definitions of fractional calculus [24].
Definition 2.1 For an at least ntimes continuously differentiable function , the Caputo derivative of fractional order q is defined as
where denotes the integer part of the real number q.
Definition 2.2 The RiemannLiouville fractional integral of order q is defined as
provided the integral exists.
Lemma 2.3For, the fractional boundary value problem
has a unique solution given by
where
Proof The general solution of fractional differential equations in (2.1) can be written as [4]
Using the given boundary conditions, it is found that , , …, . Applying the RiemannLiouville integral operator on (2.4), we get
Using the concept of beta function, we find that
Now using the condition
we obtain
which yields
where δ is given by (2.3). Substituting the values of in (2.4), we obtain (2.2). This completes the proof. □
3 Main results
Let denote the Banach space of all continuous functions defined on endowed with a topology of uniform convergence with the norm .
To prove the existence results for problem (1.1), we need the following known results.
Theorem 3.1 (LeraySchauder alternative [[27], p.4])
LetXbe a Banach space. Assume thatis a completely continuous operator and the set
is bounded. ThenThas a fixed point inX.
Theorem 3.2[28]
LetXbe a Banach space. Assume that Ω is an open bounded subset of Xwith, and letbe a completely continuous operator such that
By Lemma 2.3, we define an operator as
Observe that problem (1.1) has a solution if and only if the associated fixed point problem has a fixed point.
For the sake of convenience, we set
Theorem 3.3Assume that there exists a positive constantsuch thatfor, . Then problem (1.1) has at least one solution.
Proof First of all, we show that the operator is completely continuous. Note that the operator is continuous in view of the continuity of f. Let be a bounded set. By the assumption that , for , we have
which implies that . Further, we find that
This implies that is equicontinuous on . Thus, by the ArzelaAscoli theorem, the operator is completely continuous.
Next, we consider the set
and show that the set V is bounded. Let , then , . For any , we have
Thus, for any . So, the set V is bounded. Thus, by the conclusion of Theorem 3.1, the operator has at least one fixed point, which implies that (1.1) has at least one solution. □
Theorem 3.4Let there exist a small positive numberτsuch thatfor, with, whereϑis given by (3.2). Then problem (1.1) has at least one solution.
Proof Let us define and take such that , that is, . As before, it can be shown that is completely continuous and
which, in view of the given condition , gives , . Therefore, by Theorem 3.2, the operator has at least one fixed point, which in turn implies that problem (1.1) has at least one solution. □
Our next result is based on LeraySchauder nonlinear alternative.
Lemma 3.5 (Nonlinear alternative for singlevalued maps [[27], p.135])
LetEbe a Banach space, Cbe a closed, convex subset ofE, Ube an open subset ofCand. Suppose thatis a continuous, compact (that is, is a relatively compact subset ofC) map. Then either
(ii) there are (the boundary ofUinC) andwith.
Theorem 3.6Assume that
(A_{1}) there exist a functionand a nondecreasing functionsuch that, ;
(A_{2}) there exists a constantsuch that
Then boundary value problem (1.1) has at least one solution on.
Proof Consider the operator defined by (3.1). We show that maps bounded sets into bounded sets in. For a positive number r, let be a bounded set in . Then
Next we show that Fmaps bounded sets into equicontinuous sets of. Let with and , where is a bounded set of . Then we obtain
Obviously the righthand side of the above inequality tends to zero independently of as . As satisfies the above assumptions, therefore it follows by the ArzeláAscoli theorem that is completely continuous.
Let x be a solution. Then, for , and following the similar computations as before, we find that
In consequence, we have
Thus, by (A_{2}), there exists M such that . Let us set
Note that the operator is continuous and completely continuous. From the choice of V, there is no such that for some . Consequently, by the nonlinear alternative of LeraySchauder type (Lemma 3.5), we deduce that has a fixed point which is a solution of problem (1.1). This completes the proof. □
Finally we prove an existence and uniqueness result by means of Banach’s contraction mapping principle.
Theorem 3.7Suppose thatis a continuous function and satisfies the following assumption:
Then boundary value problem (1.1) has a unique solution provided
whereϑis given by (3.2).
Proof With , we define , where and ϑ is given by (2.3). Then we show that . For , by means of the inequality , it can easily be shown that
Now, for and for each , we obtain
Note that ϑ depends only on the parameters involved in the problem. As , therefore is a contraction. Hence, by Banach’s contraction mapping principle, problem (1.1) has a unique solution on . □
4 Examples
In this section, we present some examples for the illustration of the results established in Section 3 by choosing the nonlinear function appropriately. Let us consider the following nonlocal boundary value problem:
where , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , . Using the given data, we find that
and
(a) As a first example, let us take
Observe that with . Thus the hypothesis of Theorem 3.3 is satisfied. Hence, by the conclusion of Theorem 3.3, problem (4.1) with given by (4.2) has at least one solution.
(b) Let us consider
For sufficiently small x (ignoring and higher powers of x), we have
Choosing , all the assumptions of Theorem 3.4 hold. Therefore, the conclusion of Theorem 3.4 implies that problem (4.1) with given by (4.3) has at least one solution.
(c) Consider
with and . Using , , we find by condition (A_{2}) that , where . Thus all the assumptions of Theorem 3.6 are satisfied. Hence, it follows by Theorem 3.6 that problem (4.1) with defined by (4.4) has at least one solution.
(d) For the illustration of the existenceuniqueness result, we choose
Clearly, as and . Therefore all the conditions of Theorem 3.7 hold, and consequently there exists a unique solution for problem (4.1) with given by (4.5).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
Each of the authors, AA, SKN, RPA and BA, contributed to each part of this work equally and read and approved the final version of the manuscript.
Acknowledgements
This research was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.
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