This paper deals with some existence results for a boundary value problem involving a nonlinear integrodifferential equation of fractional order with integral boundary conditions. Our results are based on contraction mapping principle and Krasnosel'skiĭ's fixed point theorem.
1. Introduction
In the last few decades, fractionalorder models are found to be more adequate than integerorder models for some real world problems. Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. This is the main advantage of fractional differential equations in comparison with classical integerorder models. Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modelling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, and so forth, involves derivatives of fractional order. In consequence, the subject of fractional differential equations is gaining much importance and attention. For examples and details, see [1–22] and the references therein. However, the theory of boundary value problems for nonlinear fractional differential equations is still in the initial stages and many aspects of this theory need to be explored.
Integrodifferential equations arise in many engineering and scientific disciplines, often as approximation to partial differential equations, which represent much of the continuum phenomena. Many forms of these equations are possible. Some of the applications are unsteady aerodynamics and aero elastic phenomena, visco elasticity, visco elastic panel in super sonic gas flow, fluid dynamics, electrodynamics of complex medium, many models of population growth, polymer rheology, neural network modeling, sandwich system identification, materials with fading memory, mathematical modeling of the diffusion of discrete particles in a turbulent fluid, heat conduction in materials with memory, theory of lossless transmission lines, theory of population dynamics, compartmental systems, nuclear reactors, and mathematical modeling of a hereditary phenomena. For details, see [23–29] and the references therein.
Integral boundary conditions have various applications in applied fields such as blood flow problems, chemical engineering, thermoelasticity, underground water flow, population dynamics, and so forth. For a detailed description of the integral boundary conditions, we refer the reader to a recent paper [30]. For more details of nonlocal and integral boundary conditions, see [31–37] and references therein.
In this paper, we consider the following boundary value problem for a nonlinear fractional integrodifferential equation with integral boundary conditions
where is the Caputo fractional derivative, for
and are real numbers. 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
2. Preliminaries
First of all, we recall some basic definitions [15, 18, 20].
Definition 2.1.
For a function the Caputo derivative of fractional order is defined as
where denotes the integer part of the real number
Definition 2.2.
The RiemannLiouville fractional integral of order is defined as
provided the integral exists.
Definition 2.3.
The RiemannLiouville fractional derivative of order for a function is defined by
provided the right hand side is pointwise defined on
In passing, we remark that the definition of RiemannLiouville fractional derivative, which did certainly play an important role in the development of theory of fractional derivatives and integrals, could hardly produce the physical interpretation of the initial conditions required for the initial value problems involving fractional differential equations. The same applies to the boundary value problems of fractional differential equations. It was Caputo definition of fractional derivative which solved this problem. In fact, the Caputo derivative becomes the conventional th derivative of the function as and the initial conditions for fractional differential equations retain the same form as that of ordinary differential equations with integer derivatives. Another difference is that the Caputo derivative for a constant is zero while the RiemannLiouville fractional derivative of a constant is nonzero. For more details, see [20].
Lemma 2.4 (see [22]).
For the general solution of the fractional differential equation is given by
where ().
In view of Lemma 2.4, it follows that
for some ().
Now, we state a known result due to Krasnosel'skiĭ [38] which is needed to prove the existence of at least one solution of (1.1).
Theorem 2.5.
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
Lemma 2.6.
For any the unique solution of the boundary value problem
is given by
where is the Green's function given by
Proof.
Using (2.5), for some constants we have
In view of the relations and for we obtain
Applying the boundary conditions for (2.6), we find that
Thus, the unique solution of (2.6) is
where is given by (2.8). This completes the proof.
3. Main Results
Theorem 3.1.
Assume that is jointly continuous and maps bounded subsets of into relatively compact subsets of is continuous with and are continuous functions. Further, there exist positive constants such that
(A_{1}) for all
(A_{2}) with
Then the boundary value problem (1.1) has a unique solution provided
with
Proof.
Define by
Setting (by the assumption on ) and Choosing
we show that where For we have
Now, for and for each we obtain
where
which depends only on the parameters involved in the problem. As therefore is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle.
Theorem 3.2.
Assume that (A_{1})(A_{2}) hold with where and
Then the boundary value problem (1.1) has at least one solution on
Proof.
Let us fix
and consider We define the operators and on as
For we find that
Thus, It follows from the assumption (A_{1}), (A_{2}) that is a contraction mapping for
Continuity of implies that the operator is continuous. Also, is uniformly bounded on as
Now we prove the compactness of the operator In view of (A_{1}), we define and consequently we have
which is independent of So is relatively compact on Hence, By Arzela Ascoli Theorem, is compact on Thus all the assumptions of Theorem 2.5 are satisfied and the conclusion of Theorem 2.5 implies that the boundary value problem (1.1) has at least one solution on
Example 3.3.
Consider the following boundary value problem:
Here, As therefore, (A_{1}) and (A_{2}) are satisfied with Further,
Thus, by Theorem 3.1, the boundary value problem (3.15) has a unique solution on
Acknowledgments
The authors are grateful to the anonymous referee for his/her valuable suggestions that led to the improvement of the original manuscript. The research of J. J. Nieto has been partially supported by Ministerio de Educacion y Ciencia and FEDER, Project MTM200761724, and by Xunta de Galicia and FEDER, project PGIDIT05PXIC20702PN.
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