Open Access

Existence of Periodic Solution for a Nonlinear Fractional Differential Equation

  • Mohammed Belmekki1,
  • JuanJ Nieto2 and
  • Rosana Rodríguez-López2Email author
Boundary Value Problems20092009:324561

DOI: 10.1155/2009/324561

Received: 2 February 2009

Accepted: 4 June 2009

Published: 14 July 2009

Abstract

We study the existence of solutions for a class of fractional differential equations. Due to the singularity of the possible solutions, we introduce a new and proper concept of periodic boundary value conditions. We present Green's function and give some existence results for the linear case and then we study the nonlinear problem.

1. Introduction

Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary noninteger order. The subject is as old as the differential calculus, and goes back to time when Leibnitz and Newton invented differential calculus. The idea of fractional calculus has been a subject of interest not only among mathematicians but also among physicists and engineers. See, for instance, [16].

Fractional-order models are more accurate than integer-order models, that is, there are more degrees of freedom in the fractional-order models. Furthermore, fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes due to the existence of a "memory" term in a model. This memory term insures the history and its impact to the present and future. For more details, see [7].

Fractional calculus appears in rheology, viscoelasticity, electrochemistry, electromagnetism, and so forth. For details, see the monographs of Kilbas et al. [8], Kiryakova [9], Miller and Ross [10], Podlubny [11], Oldham and Spanier [12], and Samko et al. [13], and the papers of Diethelm et al. [1416], Mainardi [17], Metzler et al. [18], Podlubny et al. [19], and the references therein. For some recent advances on fractional calculus and differential equations, see [1, 3, 2024].

In this paper we consider the following nonlinear fractional differential equation of the form
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq1_HTML.gif is the standard Riemann-Liouville fractional derivative, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq2_HTML.gif is continuous, and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq3_HTML.gif .

This paper is organized as follows. in Section 2 we recall some definitions of fractional integral and derivative and related basic properties which will be used in the sequel. In Section 3, we deal with the linear case where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq4_HTML.gif is a continuous function. Section 4 is devoted to the nonlinear case.

2. Preliminary Results

In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq5_HTML.gif the Banach space of all continuous real functions defined on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq6_HTML.gif with the norm https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq7_HTML.gif Define for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq8_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq9_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq10_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq11_HTML.gif be the space of all functions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq12_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq13_HTML.gif which turn out to be a Banach space when endowed with the norm https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq14_HTML.gif

By https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq15_HTML.gif we denote the space of all real functions defined on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq16_HTML.gif which are Lebesgue integrable.

Obviously https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq17_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq18_HTML.gif

Definition 2.1 (see [11, 13]).

The Riemann-Liouville fractional primitive of order https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq19_HTML.gif of a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq20_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ2_HTML.gif
(2.1)

provided the right side is pointwise defined on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq21_HTML.gif , and where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq22_HTML.gif is the gamma function.

For instance, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq23_HTML.gif exists for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq24_HTML.gif , when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq25_HTML.gif ; note also that when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq26_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq27_HTML.gif and moreover https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq28_HTML.gif

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq29_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq30_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq31_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq32_HTML.gif , with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq33_HTML.gif If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq34_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq35_HTML.gif is bounded at the origin, whereas if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq36_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq37_HTML.gif , then we may expect https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq38_HTML.gif to be unbounded at the origin.

Recall that the law of composition https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq39_HTML.gif holds for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq40_HTML.gif

Definition 2.2 (see [11, 13]).

The Riemann-Liouville fractional derivative of order https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq41_HTML.gif of a continuous function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq42_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ3_HTML.gif
(2.2)

We have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq43_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq44_HTML.gif .

Lemma 2.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq45_HTML.gif . If one assumes https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq46_HTML.gif , then the fractional differential equation
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ4_HTML.gif
(2.3)

has https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq47_HTML.gif , as unique solutions.

From this lemma we deduce the following law of composition.

Proposition 2.4.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq48_HTML.gif is in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq49_HTML.gif with a fractional derivative of order https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq50_HTML.gif that belongs to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq51_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ5_HTML.gif
(2.4)

for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq52_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq53_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq54_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq55_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq56_HTML.gif .

3. Linear Problem

In this section, we will be concerned with the following linear fractional differential equation:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ6_HTML.gif
(3.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq57_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq58_HTML.gif is a continuous function.

Before stating our main results for this section, we study the equation
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ7_HTML.gif
(3.2)
Then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ8_HTML.gif
(3.3)

for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq59_HTML.gif .

Note that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq60_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq61_HTML.gif . However, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq62_HTML.gif since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq63_HTML.gif has a singularity at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq64_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq65_HTML.gif

It is easy to show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq66_HTML.gif . Hence we should look for solutions, not in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq67_HTML.gif but in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq68_HTML.gif . We cannot consider the usual initial condition https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq69_HTML.gif , but https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq70_HTML.gif Hence, to study the periodic boundary value problem, one has to consider the following boundary condition of periodic type
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ9_HTML.gif
(3.4)
From (3.3), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ10_HTML.gif
(3.5)

that leads to the following.

Theorem 3.1.

The periodic boundary value problem (3.2)–(3.4) has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq71_HTML.gif if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ11_HTML.gif
(3.6)
The previous result remains true even if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq72_HTML.gif . In this case, (3.2) is reduced to the ordinary differential equation
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ12_HTML.gif
(3.7)
with the periodic boundary condition
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ13_HTML.gif
(3.8)
and the condition (3.6) is reduced to the classical one:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ14_HTML.gif
(3.9)
Now, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq73_HTML.gif different from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq74_HTML.gif , consider the homogenous linear equation
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ15_HTML.gif
(3.10)
The solution is given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ16_HTML.gif
(3.11)
Indeed, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ17_HTML.gif
(3.12)

since the series representing https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq75_HTML.gif is absolutely convergent.

Using the identities
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ18_HTML.gif
(3.13)
we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ19_HTML.gif
(3.14)
Then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ20_HTML.gif
(3.15)
Note that the solution can be expressed by means of the classical Mittag-Leffler special functions [8]. Indeed
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ21_HTML.gif
(3.16)
The previous formula remains valid for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq76_HTML.gif . In this case,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ22_HTML.gif
(3.17)
Then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ23_HTML.gif
(3.18)
which is the classical solution to the homogeneous linear differential equation
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ24_HTML.gif
(3.19)
Now, consider the nonhomogeneous problem (3.1). We seek the particular solution in the following form:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ25_HTML.gif
(3.20)
It suffices to show that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ26_HTML.gif
(3.21)
Indeed
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ27_HTML.gif
(3.22)

Using the change of variable

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ28_HTML.gif
(3.23)

we get

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ29_HTML.gif
(3.24)
Then,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ30_HTML.gif
(3.25)
Hence, the general solution of the nonhomogeneous equation (3.1) takes the form
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ31_HTML.gif
(3.26)
Now, consider the periodic boundary value problem (3.1)–(3.4). Its unique solution is given by (3.26) for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq77_HTML.gif . Also https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq78_HTML.gif is in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq79_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ32_HTML.gif
(3.27)
From (3.26), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ33_HTML.gif
(3.28)
which leads to
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ34_HTML.gif
(3.29)
since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq80_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq81_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ35_HTML.gif
(3.30)
Then the solution of the problem (3.1)–(3.4) is given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ36_HTML.gif
(3.31)

Thus we have the following result.

Theorem 3.2.

The periodic boundary value problem (3.1)–(3.4) has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq82_HTML.gif given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ37_HTML.gif
(3.32)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ38_HTML.gif
(3.33)

For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq83_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq84_HTML.gif given, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq85_HTML.gif is bounded on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq86_HTML.gif .

For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq87_HTML.gif , (3.1) is
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ39_HTML.gif
(3.34)
and the boundary condition (3.4) is
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ40_HTML.gif
(3.35)
In this situation Green's function is
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ41_HTML.gif
(3.36)

which is precisely Green's function for the periodic boundary value problem considered in [25, 26].

4. Nonlinear Problem

In this section we will be concerned with the existence and uniqueness of solution to the nonlinear problem (1.1)–(3.4). To this end, we need the following fixed point theorem of Schaeffer.

Theorem 4.1.

Assume https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq88_HTML.gif to be a normed linear space, and let operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq89_HTML.gif be compact. Then either

(i)the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq90_HTML.gif has a fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq91_HTML.gif , or

(ii)the set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq92_HTML.gif is unbounded.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq93_HTML.gif is a solution of problem (1.1)–(3.4), then it is given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ42_HTML.gif
(4.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq94_HTML.gif is Green's function defined in Theorem 3.2.

Define the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq95_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ43_HTML.gif
(4.2)

Then the problem (1.1)–(3.4) has solutions if and only if the operator equation https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq96_HTML.gif has fixed points.

Lemma 4.2.

Suppose that the following hold:

(i)there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq97_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ44_HTML.gif
(4.3)
(ii)there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq98_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ45_HTML.gif
(4.4)

Then the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq99_HTML.gif is well defined, continuous, and compact.

Proof.
  1. (a)
    We check, using hypothesis (4.3), that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq100_HTML.gif , for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq101_HTML.gif . Indeed, for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq102_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq103_HTML.gif , we have
    https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ46_HTML.gif
    (4.5)
     
From the previous expression, we deduce that, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq104_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ47_HTML.gif
(4.6)
Indeed, note that the integral https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq105_HTML.gif is bounded by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ48_HTML.gif
(4.7)
A similar argument is useful to study the behavior of the last three terms of the long inequality above. On the other hand, if we denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq106_HTML.gif the second term in the right-hand side of that inequality, then it is satisfied that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ49_HTML.gif
(4.8)
Note that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ50_HTML.gif
(4.9)
and, concerning https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq107_HTML.gif , we distinguish two cases. If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq108_HTML.gif is such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq109_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ51_HTML.gif
(4.10)
and, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq110_HTML.gif is such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq111_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ52_HTML.gif
(4.11)
In consequence,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ53_HTML.gif
(4.12)
The first term in the right-hand side of the previous inequality clearly tends to zero as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq112_HTML.gif . On the other hand, denoting by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq113_HTML.gif the integer part function, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ54_HTML.gif
(4.13)
The finite sum obviously has limit zero as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq114_HTML.gif . The infinite sum is equal to
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ55_HTML.gif
(4.14)

and its limit as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq115_HTML.gif is zero. Note that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq116_HTML.gif is bounded above by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq117_HTML.gif .

The previous calculus shows that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq118_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq119_HTML.gif , hence we can define https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq120_HTML.gif .
  1. (b)

    Next, we prove that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq121_HTML.gif is continuous.

     
Note that, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq122_HTML.gif and for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq123_HTML.gif , we have, using hypothesis (4.4),
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ56_HTML.gif
(4.15)
Using the definition of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq124_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ57_HTML.gif
(4.16)
Moreover,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ58_HTML.gif
(4.17)
Using that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq125_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq126_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq127_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq128_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq129_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq130_HTML.gif we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ59_HTML.gif
(4.18)
Note that the Beta function, also called the Euler integral of the first kind,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ60_HTML.gif
(4.19)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq131_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq132_HTML.gif , satisfies that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq133_HTML.gif . In particular, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq134_HTML.gif . On the other hand, using the change of variable https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq135_HTML.gif , we deduce that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ61_HTML.gif
(4.20)
This proves that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ62_HTML.gif
(4.21)
Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ63_HTML.gif
(4.22)
In consequence,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ64_HTML.gif
(4.23)
Finally, we check that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq136_HTML.gif is compact. Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq137_HTML.gif be a bounded set in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq138_HTML.gif .
  1. (i)

    First, we check that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq139_HTML.gif is a bounded set in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq140_HTML.gif .

     
Indeed,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ65_HTML.gif
(4.24)
Hence
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ66_HTML.gif
(4.25)
and then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ67_HTML.gif
(4.26)
which implies that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq141_HTML.gif is a bounded set in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq142_HTML.gif .
  1. (ii)

    Now, we prove that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq143_HTML.gif is an equicontinuous set in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq144_HTML.gif . Following the calculus in (a), we show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq145_HTML.gif tends to zero as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq146_HTML.gif .

     

Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq147_HTML.gif is equicontinuous in the space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq148_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq149_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq150_HTML.gif .

As a consequence of (i) and (ii), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq151_HTML.gif is a bounded and equicontinuous set in the space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq152_HTML.gif .

Hence, for a sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq153_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq154_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq155_HTML.gif has a subsequence converging to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq156_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ68_HTML.gif
(4.27)
Taking https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq157_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ69_HTML.gif
(4.28)

which means that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq158_HTML.gif , which proves that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq159_HTML.gif is compact.

Theorem 4.3.

Assume that (4.3) and (4.4) hold. Then the problem (1.1)–(3.4) has at least one solution in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq160_HTML.gif

Proof.

Consider the set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq161_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq162_HTML.gif be any element of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq163_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq164_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq165_HTML.gif . Thus for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq166_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ70_HTML.gif
(4.29)
As in Lemma 4.2, (i), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ71_HTML.gif
(4.30)

which implies that the set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq167_HTML.gif is bounded independently of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq168_HTML.gif . Using Lemma 4.2 and Theorem 4.1, we obtain that the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq169_HTML.gif has at least a fixed point.

Remark 4.4.

In Lemma 4.2, condition (4.3) is used to prove that the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq170_HTML.gif is continuous. Hence, in Lemma 4.2 and, in consequence, in Theorem 4.3, we can assume the weaker condition.

(i)For each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq171_HTML.gif fixed, there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq172_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ72_HTML.gif
(4.31)

instead of (4.3).

However, to prove the existence and uniqueness of solution given in the following theorem, we need to assume the Lipschitzian character of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq173_HTML.gif (condition (4.3).

Theorem 4.5.

Assume that (4.4) holds. Then the problem (1.1)–(3.4) has a unique solution in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq174_HTML.gif provided that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ73_HTML.gif
(4.32)

Proof.

We use the Banach contraction principle to prove that the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq175_HTML.gif has a unique fixed point.

Using the calculus in (b) Lemma 4.2, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq176_HTML.gif is a contraction by condition (4.32). As a consequence of Banach fixed point theorem, we deduce that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq177_HTML.gif has a unique fixed point which gives rise to a unique solution of problem (1.1)–(3.4).

Remark 4.6.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_IEq178_HTML.gif , condition (4.32) is reduced to
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F324561/MediaObjects/13661_2009_Article_841_Equ74_HTML.gif
(4.33)

Declarations

Acknowledgment

The research of J. J. Nieto and R. Rodríguez-López has been partially supported by Ministerio de Educacion y Ciencia and FEDER, project MTM2007-61724, and by Xunta de Galicia and FEDER, project PGIDIT06PXIB207023PR.

Authors’ Affiliations

(1)
Département de Mathématiques, Université de Saïda
(2)
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela

References

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© Mohammed Belmekki et al. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.