Abstract
We study some existence results in a Banach space for a nonlocal boundary value problem involving a nonlinear differential equation of fractional order given by , , , , , , , . Our results are based on the contraction mapping principle and Krasnoselskii's fixed point theorem.
1. Introduction
Fractional differential equations involve derivatives of fractional order. They arise in many engineering and scientific disciplines such as the mathematical modeling of systems and processes in the fields of physics, chemistry, aerodynamics, electro-dynamics of complex medium, and polymer rheology. In consequence, the subject of fractional differential equations is gaining much importance and attention. For examples and details, see [1–17] 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.
The subject of multipoint nonlocal boundary value problems, initiated by Ilin and Moiseev [18, 19], has been addressed by many authors, for instance, [20–26]. The multipoint boundary conditions appear in certain problems of thermodynamics, elasticity, and wave propagation, see [27] and the references therein. The multipoint boundary conditions may be understood in the sense that the controllers at the end points dissipate or add energy according to censors located at intermediate positions.
For and we consider the following nonlinear fractional differential equation of order with nonlocal boundary conditions: where is the Caputo fractional derivative and is continuous. 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
By a solution of (1.1), we mean a function of class which satisfies the nonlocal fractional boundary value problem (1.1).
2. Preliminaries
Let us recall some basic definitions [12, 15, 17] on fractional calculus. 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 Riemann-Liouville fractional integral of order is defined as provided that the integral exists.
Definition 2.3. The Riemann-Liouville fractional derivative of order for a function is defined by
provided the right hand side is pointwise defined on
We remark that 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-order derivatives. On the other hand, the Riemann-Liouville fractional derivative 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). Moreover, the Caputo derivative for a constant is zero while the Riemann-Liouville fractional derivative of a constant is nonzero. For more details, see [17].
Lemma 2.4 (see [28]). 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 Krasnoselskii [29] 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
To study the nonlinear problem (1.1), we first consider the associated linear problem and obtain its solution.
Lemma 2.6. For a given the unique solution of the boundary value problem, is given by
Proof. Using (2.5), we have where are arbitrary constants. In view of the relations and for we obtain Applying the boundary conditions for (2.6), we find that and Substituting the values of in (2.8), we obtain This completes the proof.
3. Main Results
For the forthcoming analysis, we need the following assumptions:
;In relation to the nonlocal problem (1.1), we define the constants: 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 by (3.1).
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 Clearly depends 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. Let be a jointly continuous function mapping bounded subsets of into relatively compact subsets of Further, the assumptions hold with where is given by (3.1). 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 that is a contraction mapping for Continuity of implies that the operator is continuous. Also, is uniformly bounded on as To show that the operator is compact, we use the classical Arzela-Ascoli theorem. Let be a bounded subset of We have to show that is equicontinuous and for each the set is relatively compact in In view of we define and consequently we have which is independent of Thus, is equicontinuous. Using the fact that maps bounded subsets into relatively compact subsets, we have that is relatively compact in for every . Therefore, 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, is satisfied with Further, Thus, by Theorem 3.1, the boundary value problem (3.11) has a unique solution on
Acknowledgments
The authors thank the reviewers for their useful comments. The research of J. J. Nieto has been partially supported by Ministerio de Educacion y Ciencia and FEDER, project MTM2007-61724, and by Xunta de Galicia and FEDER, project PGIDIT06PXIB207023PR.