Existence of Solutions for Integrodifferential Equations of Fractional Order with Antiperiodic Boundary Conditions
We discuss the existence of solutions for a nonlinear antiperiodic boundary value problem of integrodifferential equations of fractional order . The contraction mapping principle and Krasnoselskii's fixed point theorem are applied to establish the results.
Recently, the subject of fractional differential equations has emerged as an important area of investigation. Fractional differential equations arise in many engineering and scientific disciplines as the fractional derivatives describe numerous events and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, and so forth. For some recent development on the subject, see [1–15] and the references therein.
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. For details, see [16–20] and the references therein.
In this paper, we prove some existence and uniqueness results for the following antiperiodic fractional boundary value problem: where denotes the Caputo fractional derivative of order , and for ,
with . 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 .
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 that the right-hand side is pointwise defined on .
Lemma 2.4 (see ). For the general solution of the fractional differential equation is given by where , ().
In view of Lemma 2.4, it follows that for some , ().
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 we find that Thus, the unique solution of (2.6) is where is given by (2.8). This completes the proof.
3. Main Results
To prove the main results, we need the following assumptions:(), for all , ;(), for all and .
Theorem 3.1. Let be a jointly continuous function satisfying the assumption with . Then the antiperiodic boundary value problem (1.1) has a unique solution.
Proof. Define by Setting and choosing , we show that where For we haveNow, 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 (Banach fixed point theorem).
Theorem 3.2. Let be a jointly continuous function mapping bounded subsets of into relatively compact subsets of and the assumptions - hold with . Then the antiperiodic 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 Now we prove the compactness of the operator In view of 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 antiperiodic boundary value problem (1.1) has at least one solution on .
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