The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces

doi:10.1016/j.na.2010.02.035

Nonlinear Analysis: Theory, Methods & Applications

Volume 72, Issue 12, 15 June 2010, Pages 4587-4593

https://doi.org/10.1016/j.na.2010.02.035 Get rights and content

Abstract

In this paper we prove the existence of solutions of certain kinds of nonlinear fractional integrodifferential equations in Banach spaces. Further, Cauchy problems with nonlocal initial conditions are discussed for the aforementioned fractional integrodifferential equations. At the end, an example is presented.

Introduction

The use of fractional differential equations has emerged as a new branch of applied mathematics, which has been used for constructing many mathematical models in science and engineering. In fact fractional differential equations are considered as models alternative to nonlinear differential equations [1] and other kinds of equations [2], [3], [4]. The theory of fractional differential equations has been extensively studied by many authors [5], [6], [7], [8], [9], [10], [11], [12]. In [13], [14] the authors proved the existence of solutions of abstract fractional differential equations by using semigroup theory and the fixed point theorem. Many partial fractional differential or integrodifferential equations can be expressed as fractional differential or integrodifferential equations in some Banach spaces [15].

Byszewski [16] initiated the study of nonlocal Cauchy problems for abstract evolution differential equations. Subsequently several authors discussed the problem for different kinds of nonlinear differential equations and integrodifferential equations including functional differential equations in Banach spaces [17], [18], [19]. Balachandran et al. [20], [21], [22] established the existence of solutions of quasilinear integrodifferential equations with nonlocal conditions. In these papers the quasilinear operator is unbounded. Recently N’Guerekata [23] and Balachandran and Park [24] investigated the existence of solutions of fractional abstract differential equations with nonlocal initial condition. Benchohra and Seba [25] studied the existence problem for impulsive fractional differential equations in Banach spaces. Balachandran and Kiruthika [26] discussed the nonlocal Cauchy problem with an impulsive condition for semilinear fractional differential equations, whereas Chang and Nieto [27] studied the same problem for neutral integrodifferential equations via fractional operators. Belmekki et al. [28] studied the existence of periodic solutions of nonlinear fractional differential equations. Cuevas and Cesar de Souza [29] discussed $ω$ -periodic solutions of fractional integrodifferential equations. In this paper we study the existence of solutions of fractional quasilinear integrodifferential equations in Banach spaces by using the fractional calculus and the Banach fixed point theorem.

Section snippets

Preliminaries

We need some basic definitions and properties of fractional calculus which are used in this paper.

Let $X$ be a Banach space and $R_{+} = [0, \infty)$ . Suppose $f \in L_{1} (R_{+})$ .

Definition 2.1

The Riemann–Liouville fractional integral operator of order $α > 0$ of function $f \in L_{1} (R_{+})$ is defined as $I_{a +}^{α} f (t) = \frac{1}{Γ (α)} \int_{a}^{t} {(t - s)}^{α - 1} f (s) d s,$ where $a \in R$ and $Γ (\cdot)$ is the Euler gamma function.

Definition 2.2

The Riemann–Liouville fractional derivative order $α > 0$ , $n - 1 < α < n$ , $n \in N$ , is defined as $^{R - L} D_{a +}^{α} f (t) = \frac{1}{Γ (n - α)} {(\frac{d}{d t})}^{n} \int_{a}^{t} {(t - s)}^{n - α - 1} f (s) d s,$ where the function $f (t)$ has

A quasilinear integrodifferential equation

Consider the fractional quasilinear integrodifferential equation of the form $^{C} D^{q} u (t) = A (t, u) u (t) + f (t, u (t), \int_{0}^{t} h (t, s, u (s)) d s), 0 \leq t \leq T,$ $u (0) = u_{0},$ where $A (t, u)$ is a bounded linear operator on $X$ and $f : J \times X \times X \to X, h : Δ \times X \to X$ are continuous. Nonlinear functions $f$ of this type with integral term $h$ occur in mathematical problems concerned with heat flow in materials with memory and viscoelastic problems in which the integral term represents the viscosity part of the problem [14]. Here $Δ = {(t, s) : 0 \leq s \leq t \leq T}$ . For brevity

A nonlocal Cauchy problem

In this section we discuss the existence of a solution of fractional quasilinear integrodifferential equation (3) with a nonlocal condition of the form $u (0) + g (u) = u_{0},$ where $g : C (J, X) \to X$ is a given function. We assume the following conditions:

(H6)
$g : C (J, X) \to X$ is continuous and there exists a constant $G > 0$ such that $‖ g (u) - g (v) ‖ \leq G ‖ u - v ‖, for u, v \in C (J, X) .$
(H7)
$‖ u_{0} ‖ + G r + ‖ g (0) ‖ + (M r + K) r γ + γ M_{0} \leq r$ .
(H8)
$p^{'} = G + γ (2 M r + K + L + L L^{*} T)$ is such that $0 \leq p^{'} < 1$ .

Theorem 4.1

If the hypotheses (H1)–(H3), (H6)–(H8) are satisfied, then the fractional quasilinear

A semilinear integrodifferential equation

Now consider the following semilinear fractional integrodifferential evolution equation with nonlocal condition (6): $^{C} D^{q} u (t) = A (t) u (t) + f (t, u (t), H u (t)), 0 \leq t \leq T,$ where $A (t)$ is a bounded linear operator and $f : J \times X \times X \to X$ , $H : X \to X$ are continuous. Assume the following additional condition:

(H9)
$A (t)$ is a bounded linear operator on $X$ for each $t \in J$ . The function $t \to A (t)$ is continuous in the uniform operator topology.

The equation is equivalent to the integral equation

u (t) = u_{0} - g (u) + \frac{1}{Γ (q)} \int_{0}^{t} {(t - s)}^{q - 1} A (s) u (s) d s + \frac{1}{Γ (q)} \int_{0}^{t} (t

Example

Consider the following fractional integrodifferential equation: $^{C} D^{q} u (t) = \frac{1}{10} sin u (t) u (t) + \frac{e^{- t} u (t)}{(9 + e^{t}) (1 + u (t))} + \frac{1}{10} \int_{0}^{t} e^{- \frac{1}{2} u (s)} d s, t \in J,$ $u (0) = u_{0},$ where $0 < q \leq 1$ . Take $X = R^{+}$ , $J ≔ [0, 1]$ and so $T = 1$ .

Set $A (t, u) = \frac{1}{10} sin u (t) I, H u (t) = \int_{0}^{t} e^{- \frac{1}{2} u (s)} d s$ , $f (t, u, H u) = \frac{e^{- t} u}{(9 + e^{t}) (1 + u)} + H u, t \in J, u \in X .$ Let $u, v \in X$ and $t \in J$ . Then we have $‖ H u - H v ‖ = | \int_{0}^{t} e^{- \frac{1}{2} u (s)} d s - \int_{0}^{t} e^{- \frac{1}{2} u (s)} d s | \leq \frac{1}{2} | u - v |$ $‖ f (t, u, H u) - f (t, v, H v) ‖ = | \frac{e^{- t}}{(9 + e^{t})} \frac{u}{(1 + u)} - \frac{v}{(1 + v)} + \frac{1}{10} (H u - H v) | \leq \frac{e^{- t} | u - v |}{(9 + e^{t}) (1 + u) (1 + v)} + \frac{1}{10} ‖ H u - H v ‖ \leq \frac{e^{- t}}{(9 + e^{t})} | u - v | + \frac{1}{10} ‖ H u - H v ‖ \leq \frac{1}{10} (| u - v | + ‖ H u - H v ‖) .$ Hence the

Acknowledgement

The authors are grateful to the referee for improvements to the paper.

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^☆: This work was supported, in part, by MICINN (MTM2007-60246).

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The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces☆

Abstract

Introduction

Section snippets

Preliminaries

A quasilinear integrodifferential equation

A nonlocal Cauchy problem

A semilinear integrodifferential equation

Example

Acknowledgement

Applied Mathematics and Computation

Nonlinear Analysis: Real World Applications

Nonlinear Analysis

Computers and Mathematics with Applications

Nonlinear Analysis

Journal of Mathematical Analysis and Applications

Nonlinear Analysis

Nonlinear Analysis

Applied Mathematics Letters

Journal of Mathematical Analysis and Applications

Applied Mathematics and Computation

Nonlinear Analysis

Journal of Mathematical Analysis and Applications

Nonlinear Analysis

Nonlinear Analysis

Nonlinear Analysis

Nonlinear Analysis