Nonlinear Analysis: Theory, Methods & Applications
Existence of the mild solution for fractional semilinear initial value problems
Introduction
Fractional differential equations have gained importance and popularity during the past three decades or so, mainly due to its demonstrated applications in numerous seemingly diverse fields of science and engineering. For example, the nonlinear oscillation of earthquake can be modeled with fractional derivatives [2], and the fluid-dynamic traffic model with fractional derivatives [4] can eliminate the deficiency arising from the assumption of continuum traffic flow. Based on experimental data fractional partial differential equations for seepage flow in porous media are suggested in Ref. [5], and differential equations with fractional order have recently proved to be valuable tools to the modeling of many physical phenomena [6], [7].
Our aim in this paper is to consider the existence and uniqueness of mild solution for the semilinear initial value problem of non-integer order where is the generator of a strongly continuous semigroup on a Banach space and is continuous in , and satisfies the following condition where
Here we recall the following known definition and properties, for more details see Refs. [1], [6], [7], [8], [9].
Definition 1.1 The Riemann–Liouville fractional integral operator of order , of a function is defined as The Riemann–Liouville derivative has certain disadvantages when trying to model real-world phenomena with fractional differential equations. Therefore, we shall introduce a modified fractional differential operator proposed by M. Caputo in his work on the theory of viscoelasticity [9]
Definition 1.2 The fractional derivative of in the Caputo sense is defined as Also, we need here two of its basic properties.
Definition 1.3 A continuous solution of the integral equation will be called a mild solution of the initial value problem
Lemma 1.1 The initial value problem(1)is equivalent to the nonlinear integral equationwhere . In other words, every solution of the integral equation(2)is also solution of our original initial value problem(1)and vice versa.
Proof It can be proved by applying the integral operator (2) to both sides of (1), as we did in [3], and using some classical results from fractional calculus in [1] to get (3). In order to proceed, we need the following hypotheses: (H1) are continuous and there exist functions such that ; Let be the semigroup generated by the unbounded operator . Let be the Banach space of all linear and bounded on . Let . (H2) The function satisfies the inequality , where , and . □
Section snippets
The main theorems
In this section, we shall prove our main results, we begin by proving a theorem concerned with the existence and uniqueness of mild solution for the semilinear initial value problem (1).
Theorem 2.1 Let be a continuous in on and uniformly Lipschitz continuous (with constant ) on . If A is the generator of a strongly continuous semigroup on a Banach space , then for every the initial value problemwhere has a
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