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Existence of solutions for integro-differential equations of fractional order with nonlocal three-point fractional boundary conditions
Advances in Difference Equations volume 2013, Article number: 128 (2013)
In this paper, we study the existence of solutions for Riemann-Liouville type integro-differential equations of fractional order with nonlocal three-point fractional boundary conditions via Sadovskii’s fixed point theorem for condensing maps. An illustrative example is also presented.
MSC:34A08, 34B10, 34B15.
Fractional calculus has recently evolved as an interesting and important field of research. The much interest in the subject owes to its extensive applications in the mathematical modeling of several phenomena in many engineering and scientific disciplines such as physics, chemistry, biophysics, biology, blood flow problems, control theory, aerodynamics, nonlinear oscillation of earthquake, the fluid-dynamic traffic model, polymer rheology, regular variation in thermodynamics, economics, fitting of experimental data, etc. [1, 2]. A significant feature of a fractional order differential operator, in contrast to its counterpart in classical calculus, is its nonlocal behavior. It means that the future state of a dynamical system or process based on the fractional differential operator depends on its current state as well its past states. It is equivalent to saying that differential equations of arbitrary order are capable of describing memory and hereditary properties of certain important materials and processes. This aspect of fractional calculus has contributed towards the growing popularity of the subject.
Nonlocal initial and boundary value problems of nonlinear fractional order differential equations have recently been investigated by several researchers. The domain of study ranges from the theoretical aspects (like existence, uniqueness, periodicity, asymptotic behavior, etc.) to the analytic and numerical methods for fractional differential equations. In fact the theory of differential equations of fractional order (parallel to the well-known theory of ordinary differential equations) has been growing independently for the last three decades. For some recent development of the subject, we refer, for instance, to a series of papers [3–14] and references cited therein.
In this paper, we discuss the existence of solutions for a boundary value problem of integro-differential equations of fractional order with nonlocal three-point fractional boundary conditions
where , , , , denotes the Riemann-Liouville fractional derivative of order , f, g are given continuous functions, and A, B, a are real constants.
The objective of the present work is to establish the existence of solutions for the given problem by applying Sadovskii’s fixed point theorem for condensing maps. It is imperative to note that the application of Sadovskii’s fixed point theorem for condensing maps in the present scenario is new. Moreover, the right-hand side of the fractional differential equation in (1) provides a liberty to fix it in terms of non-integral and integral terms. Observe that the integral term in (1) is a Riemann-Liouville integral of order , which reduces to the classical integral term () in the limit . The nature of the nonlinearity considered in the problem (1) becomes of non-integral type if we take in (1) and corresponds to integral type for in (1). Furthermore, the given boundary conditions are also interesting and important from a physical point of view [15, 16] as the condition is a fractional analogue of the classical flux condition (the difference of flux values at the right end point and at an intermediate point of the interval remains constant).
2 Auxiliary results
We recall here the following definitions.
Definition 2.1 The Riemann-Liouville fractional integral of order q for a continuous function g is defined as
provided the integral exists.
Definition 2.2 The Riemann-Liouville derivative of fractional order q for a continuous function is defined as
where denotes the integer part of the real number q.
By the substitution , where is a suitable continuous function, the problem (1) takes the form
Lemma 2.1 For any , the unique solution of the linear fractional boundary value problem
Proof It is well known that the solutions of the fractional differential equation in (3) can be written as 
where are arbitrary constants. Using the given boundary conditions, we find that , and
Substituting these values in (4), we get
This completes the proof. □
Thus, the solution of a linear variant of the problem (1) can be written as
where we have used the substitution in the integral of the last term. Using the relation for beta function ,
The solution of the original nonlinear problem (1) can be obtained by replacing h with the right-hand side of the fractional equation of (1) in (5).
Let denote the Banach space of all continuous functions from endowed with the norm defined by .
Definition 2.3 Let M be a bounded set in a metric space , then the Kuratowskii measure of noncompactness is defined as .
Definition 2.4 
Let be a bounded and continuous operator on a Banach space X. Then Φ is called a condensing map if for all bounded sets , where α denotes the Kuratowski measure of noncompactness.
Lemma 2.2 [, Example 11.7]
The map is a k-set contraction with , and thus also condensing, if
are operators on the Banach space X;
K is k-contractive, i.e.,
for all and fixed ;
C is compact.
Theorem 2.1 
Let B be a convex, bounded and closed subset of a Banach space X and let be a condensing map. Then Φ has a fixed point.
3 Main result
In the following we denote by the space of -Lebesgue measurable functions from to with the norm .
Theorem 3.1 Let be continuous functions satisfying the following conditions:
() f satisfies the Lipschitz condition
where , , , ;
() there exist a function and a nondecreasing function such that
Then the boundary value problem (1) has at least one solution, provided
Proof Let be a closed bounded and convex subset of , where r will be fixed later. Using (5), we define a map as
Observe that the problem (1) is equivalent to a fixed point problem .
Let us decompose Φ as , where
Step 1. .
For that, we select , where
Using for , , we get
In a similar manner, we have that
which implies that .
Step 2. is continuous and γ-contractive.
To show the continuity of for , let us consider a sequence converging to x. Then, by the assumption (), we have
Next, we show that is γ-contractive. For , we get
By the given assumption,
it follows that is γ-contractive.
Step 3. is compact.
In Step 1, it has been shown that is uniformly bounded. Now we show that maps bounded sets into equicontinuous sets of . Let with and . Then we obtain
Obviously, the right-hand side of the above inequality tends to zero independently of as . Therefore it follows by the Arzelá-Ascoli theorem that is completely continuous. Thus is compact on .
Step 4. Φ is condensing.
Since is continuous, γ-contraction and is compact, therefore, by Lemma 2.2, with is a condensing map on .
Consequently, by Theorem 2.1, the map Φ has a fixed point which implies that the problem (1) has a solution. □
In the special case when , L a constant, we have the following.
Corollary 3.1 Let be continuous functions. Assume that g satisfies () and f satisfies the following condition:
()′ for all , is a constant.
then the boundary value problem (1) has at least one solution.
Example 3.1 Consider a boundary value problem of integro-differential equations of fractional order with nonlocal fractional boundary conditions given by
where , , , , .
Then we have
Also, for all and each , we have
and with () and . Selecting and using the given data, we find that
As , therefore, by the conclusion of Theorem 3.1, the problem (6) has a solution.
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The authors gratefully acknowledge the referees for their constructive comments that led to the improvement of the original manuscript. 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.
The authors declare that they have no competing interests.
Each of the authors, RPA, SKN, BA and MSA, contributed to each part of this study equally and read and approved the final version of the manuscript.
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Agarwal, R.P., Ntouyas, S.K., Ahmad, B. et al. Existence of solutions for integro-differential equations of fractional order with nonlocal three-point fractional boundary conditions. Adv Differ Equ 2013, 128 (2013). https://doi.org/10.1186/1687-1847-2013-128
- fractional differential equations
- nonlocal integral boundary conditions
- fixed point theorems