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Existence of Solutions for Nonlinear Fractional IntegroDifferential Equations with ThreePoint Nonlocal Fractional Boundary Conditions
Advances in Difference Equations volume 2010, Article number: 691721 (2010)
Abstract
We prove the existence and uniqueness of solutions for nonlinear integrodifferential equations of fractional order with threepoint nonlocal fractional boundary conditions by applying some standard fixed point theorems.
1. Introduction
Fractional calculus (differentiation and integration of arbitrary order) is proved to be an important tool in the modelling of dynamical systems associated with phenomena such as fractal and chaos. In fact, this branch of calculus has found its applications in various disciplines of science and engineering such as mechanics, electricity, chemistry, biology, economics, control theory, signal and image processing, polymer rheology, regular variation in thermodynamics, biophysics, blood flow phenomena, aerodynamics, electrodynamics of complex medium, viscoelasticity and damping, control theory, wave propagation, percolation, identification, and fitting of experimental data [1–4].
Recently, differential equations of fractional order have been addressed by several researchers with the sphere of study ranging from the theoretical aspects of existence and uniqueness of solutions to the analytic and numerical methods for finding solutions. For some recent work on fractional differential equations, see [5–11] and the references therein.
In this paper, we study the following nonlinear fractional integrodifferential equations with threepoint nonlocal fractional boundary conditions
where is the standard RiemannLiouville fractional derivative, : is continuous, for :
and satisfies the condition Here, is a Banach space and denotes the Banach space of all continuous functions from endowed with a topology of uniform convergence with the norm denoted by
We remark that fractional boundary conditions result in the existence of both electric and magnetic surface currents on the strip and are similar to the impedance boundary conditions with pure imaginary impedance, and in the physical optics approximation, the ratio of the surface currents is the same as for the impedance strip. For the comparison of the physical characteristics of the fractional and impedance strips such as radiation pattern, monostatic radar crosssection, and surface current densities, see [12]. The concept of nonlocal multipoint boundary conditions is quite important in various physical problems of applied nature when the controllers at the end points of the interval (under consideration) dissipate or add energy according to the censors located at intermediate points. Some recent results on nonlocal fractional boundary value problems can be found in [13–15].
2. Preliminaries
Let us recall some basic definitions [1–3] on fractional calculus.
Definition 2.1.
The RiemannLiouville fractional integral of order is defined as
provided the integral exists.
Definition 2.2.
The RiemannLiouville fractional derivative of order for a function is defined by
provided the righthand side is pointwise defined on
Lemma 2.3 (see [16]).
For let . Then
where ( is the smallest integer such that ).
Lemma 2.4 (see [2]).
Let . Then
Lemma 2.5.
For a given the unique solution of the boundary value problem
is given by
Proof.
In view of Lemma 2.3, the fractional differential equation in (2.4) is equivalent to the integral equation
where are arbitrary constants. Applying the boundary conditions for (2.4), we find that and
Substituting the values of and in (2.6), we obtain (2.5). This completes the proof.
3. Main Results
To establish the main results, we need the following assumptions.

(A_{1}) There exist positive functions such that
(3.1)Further,
(3.2) 
(A_{2}) There exists a number such that , where
(3.3) 
(A_{3}) for all
Theorem 3.1.
Assume that is a jointly continuous function and satisfies the assumption Then the boundary value problem (1.1) has a unique solution provided , where is given in the assumption .
Proof.
Define by
Let us set and choose
where is such that Now we show that where For we have
Now, for and for each we obtain
where we have used the assumption . As therefore is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle.
Now, we state Krasnoselskii's fixed point theorem [17] which is needed to prove the following result to prove the existence of at least one solution of (1.1).
Theorem 3.2.
Let be a closed convex and nonempty subset of a Banach space Let be the operators such that (i) whenever ; (ii) is compact and continuous; (iii) is a contraction mapping. Then there exists such that
Theorem 3.3.
Let be jointly continuous, and the assumptions and hold with
Then there exists at least one solution of the boundary value problem (1.1) on
Proof.
Let us fix
and consider We define the operators and on as
For we find that
Thus, It follows from the assumption that is a contraction mapping for
In order to prove that is compact and continuous, we follow the approach used in [6, 7]. Continuity of implies that the operator is continuous. Also, is uniformly bounded on as
Now, we show that is equicontinuous. Since is bounded on the compact set , therefore, we define . Consequently, for , we have
which is independent of So, is relatively compact on . Hence, By ArzelaAscoli's Theorem, is compact on . Thus all the assumptions of Theorem 3.2 are satisfied and the conclusion of Theorem 3.2 implies that the boundary value problem (1.1) has at least one solution on
Example 3.
Consider the following boundary value problem:
Here, With we find that
Thus, by Theorem 3.1, the boundary value problem (3.14) has a unique solution on
4. Conclusions
This paper studies the existence and uniqueness of solutions for nonlinear integrodifferential equations of fractional order with threepoint nonlocal fractional boundary conditions involving the fractional derivative . Our results are based on a generalized variant of Lipschitz condition given in , that is, there exist positive functions and such that
In case , and are constant functions, that is, , and ( and are positive real numbers), then Lipschitzgeneralized variant reduces to the classical Lipschitz condition and in the assumption takes the form
In the limit , our results correspond to a secondorder integrodifferential equation with fractional boundary conditions:
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Acknowledgment
The authors are grateful to the referees for their careful review of the manuscript.
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Alsaedi, A., Ahmad, B. Existence of Solutions for Nonlinear Fractional IntegroDifferential Equations with ThreePoint Nonlocal Fractional Boundary Conditions. Adv Differ Equ 2010, 691721 (2010). https://doi.org/10.1155/2010/691721
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Keywords
 Fractional Order
 Fractional Calculus
 Fractional Differential Equation
 Impedance Boundary Condition
 Surface Current Density