A nonlocal multi-point multi-term fractional boundary value problem with Riemann-Liouville type integral boundary conditions involving two indices
© Alsaedi et al.; licensee Springer. 2013
Received: 6 November 2013
Accepted: 1 December 2013
Published: 13 December 2013
In this paper, we study the existence of solutions for fractional differential equations of arbitrary order with multi-point multi-term Riemann-Liouville type integral boundary conditions involving two indices. The Riemann-Liouville 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.
Keywordsfractional differential equations nonlocal boundary conditions fixed point theorems
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 [1–6]. A fractional-order differential operator distinguishes itself from the integer-order 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 fractional-order models more realistic and practical than the classical integer-order 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 [7–25] and the references therein. In particular, the authors studied nonlinear fractional differential equations and inclusions of arbitrary order with multi-strip boundary conditions in , while a boundary value of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multi-strip boundary conditions was investigated in . Sudsutad and Tariboon  obtained some existence results for an integro-differential equation of fractional order with m-point multi-term fractional-order integral boundary conditions.
Here we emphasize that Riemann-Liouville type integral boundary conditions involving two indices give rise to a more general situation in contrast to the case of a single index . Furthermore, the present work dealing with an arbitrary-order problem generalizes the results for the problem of order obtained in . 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
where denotes the integer part of the real number q.
provided the integral exists.
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 (Leray-Schauder alternative [, p.4])
is bounded. Then T has a fixed point in X.
Theorem 3.2 
Then T has a fixed point in .
Observe that problem (1.1) has a solution if and only if the associated fixed point problem has a fixed point.
Theorem 3.3 Assume that there exists a positive constant such that for , . Then problem (1.1) has at least one solution.
This implies that is equicontinuous on . Thus, by the Arzela-Ascoli theorem, the operator is completely continuous.
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 3.4 Let there exist a small positive number τ such that for , with , where ϑ is given by (3.2). Then problem (1.1) has at least one solution.
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 Leray-Schauder nonlinear alternative.
Lemma 3.5 (Nonlinear alternative for single-valued maps [, p.135])
F has a fixed point in , or
there are (the boundary of U in C) and with .
Theorem 3.6 Assume that
(A1) there exist a function and a nondecreasing function such that , ;
Then boundary value problem (1.1) has at least one solution on .
Obviously the right-hand side of the above inequality tends to zero independently of as . As satisfies the above assumptions, therefore it follows by the Arzelá-Ascoli theorem that is completely continuous.
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 Leray-Schauder 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 3.7 Suppose that is a continuous function and satisfies the following assumption:
(A3) , , , .
where ϑ is given by (3.2).
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 . □
- (a)As a first example, let us take(4.2)
- (b)Let us consider(4.3)
- (d)For the illustration of the existence-uniqueness result, we choose(4.5)
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).
This research was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.
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