Existence of Solutions to Fractional Mixed Integrodifferential Equations with Nonlocal Initial Condition
© A. Anguraj et al. 2011
Received: 14 November 2010
Accepted: 6 January 2011
Published: 11 January 2011
We study the existence and uniqueness theorem for the nonlinear fractional mixed Volterra-Fredholm integrodifferential equation with nonlocal initial condition , where , , and is a given function. We point out that such a kind of initial conditions or nonlocal restrictions could play an interesting role in the applications of the mentioned model. The results obtainded are applied to an example.
Recently it have been proved that the differential models involving nonlocal derivatives of fractional order arise in many engineering and scientific disciplines as the mathematical modeling of systems and processes in many fields, for instance, physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, and so forth (see [1–6]). In fact, such models can be considered as an efficient alternative to the classical nonlinear differential models to simulate many complex processes (see ). For instance, fractional differential equations are an excellent tool to describe hereditary properties of viscoelastic materials and, in general, to simulate the dynamics of many processes on anomalous media. Theory of fractional differential equations has been extensively studied by several authors as Delbosco and Rodino , Kilbas et al. , Lakshmikantham et al. [9–11], and also see [2, 12–16].
Subsequently several authors have investigated the problem for different types of nonlinear differential equations and integrodifferential equations including functional differential equations in Banach spaces.
Very recently N'Guérékata [2, 18] discussed the existence of solutions of fractional abstract differential equations with nonlocal initial condition. The nonlocal Cauchy problem is discussed by authors in  using the fixed-point concepts. Tidke  studied the nonfractional mixed Volterra-Fredholm integrodifferential equations with nonlocal conditions using Leray-Schauder Theorem.
where , , , , is a continuous function on with values in the Banach space and , and , , and are continuous -valued functions. Here , and . The operator denotes the Caputo fractional derivative of order .
The paper is organized as follows. In Section 2, some definitions, lemmas and preliminary results are introduced to be used in the sequel. Section 3 will involve the assumptions, main results and proofs of existence problem of (1.2), together with a nonlocal initial condition. Finally an example is presented.
Let be a real Banach space and the zero element of . Let be the Banach space of measurable functions which are Lebesgue integrable, equipped with the norm . We will use the following notation and . A function is called a solution of (1.2) if it satisfies (1.2).
It is obvious that the Caputo's derivative of a constant is equal to 0.
Theorem 2.7 (Krasnoselkii).
Let be a Banach space, let be a bounded closed convex subset of and let , be maps of into such that for every pair . If A is completely continuous and B is a contraction then the equation has a solution on S.
3. Main Results
Firstly, we obtain the following lemmas to prove the main results on the existence of solutions to (1.2).
If (A1)–(A5) are satisfied, then the fractional integrodifferential equation (1.2) has a unique solution continuous in .
This paper has been partially supported by MICINN (project MTM2010-16499) to which the authors are very thankful.
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