Existence of Mild Solutions to Fractional Integrodifferential Equations of Neutral Type with Infinite Delay
© Fang Li and Jun Zhang. 2011
Received: 5 December 2010
Accepted: 30 January 2011
Published: 24 February 2011
We study the solvability of the fractional integrodifferential equations of neutral type with infinite delay in a Banach space . An existence result of mild solutions to such problems is obtained under the conditions in respect of Kuratowski's measure of noncompactness. As an application of the abstract result, we show the existence of solutions for an integrodifferential equation.
The fractional differential equations are valuable tools in the modeling of many phenomena in various fields of science and engineering; so, they attracted many researchers (cf., e.g., [1–6] and references therein). On the other hand, the integrodifferential equations arise in various applications such as viscoelasticity, heat equations, and many other physical phenomena (cf., e.g., [7–10] and references therein). Moreover, the Cauchy problem for various delay equations in Banach spaces has been receiving more and more attention during the past decades (cf., e.g., [7, 10–15] and references therein).
Neutral functional differential equations arise in many areas of applied mathematics and for this reason, the study of this type of equations has received great attention in the last few years (cf., e.g., [12, 14–16] and references therein). In [12, 16], Hernández and Henríquez studied neutral functional differential equations with infinite delay. In the following, we will extend such results to fractional-order functional differential equations of neutral type with infinite delay. To the authors' knowledge, few papers can be found in the literature for the solvability of the fractional-order functional integrodifferential equations of neutral type with infinite delay.
where , , is a phase space that will be defined later (see Definition 2.5). is a generator of an analytic semigroup of uniformly bounded linear operators on . Then, there exists such that . , , ( ), and defined by , for , belongs to and . The fractional derivative is understood here in the Caputo sense.
The aim of our paper is to study the solvability of (1.1) and present the existence of mild solution of (1.1) based on Kuratowski's measures of noncompactness. Moreover, an example is presented to show an application of the abstract results.
Throughout this paper, we set and denote by a real Banach space, by the Banach space of all linear and bounded operators on , and by the Banach space of all -valued continuous functions on with the uniform norm topology.
Let us recall the definition of Kuratowski's measure of noncompactness.
This measure of noncompactness satisfies some important properties.
Lemma 2.2 (see ).
The following lemmas will be needed.
Lemma 2.3 (see ).
Lemma 2.4 (see ).
The following definition about the phase space is due to Hale and Kato .
The following result will be used later.
We list the following basic assumptions of this paper.
3. Main Result
Now, we present and prove our main result.
Clearly, the first term on the right-hand side of (3.20) tends to 0 as . The second term on the right-hand side of (3.20) tends to 0 as as a consequence of the continuity of in the uniform operator topology for .
Then (4.1) can be reformulated as the abstract (1.1).
then (4.1) has a mild solution by Theorem 3.1.
The authors are grateful to the referees for their valuable suggestions. F. Li is supported by the NSF of Yunnan Province (2009ZC054M). J. Zhang is supported by Tianyuan Fund of Mathematics in China (11026100).
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