Open Access

Existence Results for Nonlinear Boundary Value Problems of Fractional Integrodifferential Equations with Integral Boundary Conditions

Boundary Value Problems20092009:708576

DOI: 10.1155/2009/708576

Received: 9 December 2008

Accepted: 23 January 2009

Published: 12 February 2009

Abstract

This paper deals with some existence results for a boundary value problem involving a nonlinear integrodifferential equation of fractional order https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq1_HTML.gif with integral boundary conditions. Our results are based on contraction mapping principle and Krasnosel'skiĭ's fixed point theorem.

1. Introduction

In the last few decades, fractional-order models are found to be more adequate than integer-order models for some real world problems. Fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various materials and processes. This is the main advantage of fractional differential equations in comparison with classical integer-order models. Fractional differential equations arise in many engineering and scientific disciplines as the mathematical modelling of systems and processes in the fields of physics, chemistry, aerodynamics, electrodynamics of complex medium, polymer rheology, and so forth, involves derivatives of fractional order. In consequence, the subject of fractional differential equations is gaining much importance and attention. For examples and details, see [122] and the references therein. However, the theory of boundary value problems for nonlinear fractional differential equations is still in the initial stages and many aspects of this theory need to be explored.

Integrodifferential equations arise in many engineering and scientific disciplines, often as approximation to partial differential equations, which represent much of the continuum phenomena. Many forms of these equations are possible. Some of the applications are unsteady aerodynamics and aero elastic phenomena, visco elasticity, visco elastic panel in super sonic gas flow, fluid dynamics, electrodynamics of complex medium, many models of population growth, polymer rheology, neural network modeling, sandwich system identification, materials with fading memory, mathematical modeling of the diffusion of discrete particles in a turbulent fluid, heat conduction in materials with memory, theory of lossless transmission lines, theory of population dynamics, compartmental systems, nuclear reactors, and mathematical modeling of a hereditary phenomena. For details, see [2329] and the references therein.

Integral boundary conditions have various applications in applied fields such as blood flow problems, chemical engineering, thermoelasticity, underground water flow, population dynamics, and so forth. For a detailed description of the integral boundary conditions, we refer the reader to a recent paper [30]. For more details of nonlocal and integral boundary conditions, see [3137] and references therein.

In this paper, we consider the following boundary value problem for a nonlinear fractional integrodifferential equation with integral boundary conditions
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ1_HTML.gif
(11)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq2_HTML.gif is the Caputo fractional derivative, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq3_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq4_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ2_HTML.gif
(12)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq5_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq6_HTML.gif are real numbers. Here, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq7_HTML.gif is a Banach space and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq8_HTML.gif denotes the Banach space of all continuous functions from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq9_HTML.gif endowed with a topology of uniform convergence with the norm denoted by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq10_HTML.gif

2. Preliminaries

First of all, we recall some basic definitions [15, 18, 20].

Definition 2.1.

For a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq11_HTML.gif the Caputo derivative of fractional order https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq12_HTML.gif is defined as
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ3_HTML.gif
(21)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq13_HTML.gif denotes the integer part of the real number https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq14_HTML.gif

Definition 2.2.

The Riemann-Liouville fractional integral of order https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq15_HTML.gif is defined as
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ4_HTML.gif
(22)

provided the integral exists.

Definition 2.3.

The Riemann-Liouville fractional derivative of order https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq16_HTML.gif for a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq17_HTML.gif is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ5_HTML.gif
(23)

provided the right hand side is pointwise defined on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq18_HTML.gif

In passing, we remark that the definition of Riemann-Liouville fractional derivative, which did certainly play an important role in the development of theory of fractional derivatives and integrals, could hardly produce the physical interpretation of the initial conditions required for the initial value problems involving fractional differential equations. The same applies to the boundary value problems of fractional differential equations. It was Caputo definition of fractional derivative which solved this problem. In fact, the Caputo derivative becomes the conventional https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq19_HTML.gif th derivative of the function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq20_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq21_HTML.gif and the initial conditions for fractional differential equations retain the same form as that of ordinary differential equations with integer derivatives. Another difference is that the Caputo derivative for a constant is zero while the Riemann-Liouville fractional derivative of a constant is nonzero. For more details, see [20].

Lemma 2.4 (see [22]).

For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq22_HTML.gif the general solution of the fractional differential equation https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq23_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ6_HTML.gif
(24)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq24_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq25_HTML.gif ).

In view of Lemma 2.4, it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ7_HTML.gif
(25)

for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq26_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq27_HTML.gif ).

Now, we state a known result due to Krasnosel'skiĭ [38] which is needed to prove the existence of at least one solution of (1.1).

Theorem 2.5.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq28_HTML.gif be a closed convex and nonempty subset of a Banach space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq29_HTML.gif Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq30_HTML.gif be the operators such that (i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq31_HTML.gif whenever https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq32_HTML.gif , (ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq33_HTML.gif is compact and continuous, (iii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq34_HTML.gif is a contraction mapping. Then there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq35_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq36_HTML.gif

Lemma 2.6.

For any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq37_HTML.gif the unique solution of the boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ8_HTML.gif
(26)
is given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ9_HTML.gif
(27)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq38_HTML.gif is the Green's function given by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ10_HTML.gif
(28)

Proof.

Using (2.5), for some constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq39_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ11_HTML.gif
(29)
In view of the relations https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq40_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq41_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq42_HTML.gif we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ12_HTML.gif
(210)
Applying the boundary conditions for (2.6), we find that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ13_HTML.gif
(211)
Thus, the unique solution of (2.6) is
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ14_HTML.gif
(212)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq43_HTML.gif is given by (2.8). This completes the proof.

3. Main Results

Theorem 3.1.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq44_HTML.gif is jointly continuous and maps bounded subsets of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq45_HTML.gif into relatively compact subsets of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq46_HTML.gif is continuous with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq47_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq48_HTML.gif are continuous functions. Further, there exist positive constants https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq49_HTML.gif such that

(A1) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq50_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq51_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq52_HTML.gif

(A2) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq53_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq54_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq55_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq56_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq57_HTML.gif

Then the boundary value problem (1.1) has a unique solution provided
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ15_HTML.gif
(31)
with
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ16_HTML.gif
(32)

Proof.

Define https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq58_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ17_HTML.gif
(33)
Setting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq59_HTML.gif (by the assumption on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq60_HTML.gif ) and Choosing
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ18_HTML.gif
(34)
we show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq61_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq62_HTML.gif For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq63_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ19_HTML.gif
(35)
Now, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq64_HTML.gif and for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq65_HTML.gif we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ20_HTML.gif
(36)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ21_HTML.gif
(37)

which depends only on the parameters involved in the problem. As https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq66_HTML.gif therefore https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq67_HTML.gif is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle.

Theorem 3.2.

Assume that (A1)-(A2) hold with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq68_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq69_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ22_HTML.gif
(38)

Then the boundary value problem (1.1) has at least one solution on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq70_HTML.gif

Proof.

Let us fix
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ23_HTML.gif
(39)
and consider https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq71_HTML.gif We define the operators https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq72_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq73_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq74_HTML.gif as
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ24_HTML.gif
(310)
For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq75_HTML.gif we find that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ25_HTML.gif
(311)
Thus, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq76_HTML.gif It follows from the assumption (A1), (A2) that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq77_HTML.gif is a contraction mapping for
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ26_HTML.gif
(312)
Continuity of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq78_HTML.gif implies that the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq79_HTML.gif is continuous. Also, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq80_HTML.gif is uniformly bounded on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq81_HTML.gif as
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ27_HTML.gif
(313)
Now we prove the compactness of the operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq82_HTML.gif In view of (A1), we define https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq83_HTML.gif and consequently we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ28_HTML.gif
(314)

which is independent of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq84_HTML.gif So https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq85_HTML.gif is relatively compact on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq86_HTML.gif Hence, By Arzela Ascoli Theorem, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq87_HTML.gif is compact on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq88_HTML.gif Thus all the assumptions of Theorem 2.5 are satisfied and the conclusion of Theorem 2.5 implies that the boundary value problem (1.1) has at least one solution on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq89_HTML.gif

Example 3.3.

Consider the following boundary value problem:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ29_HTML.gif
(315)
Here, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq90_HTML.gif As https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq91_HTML.gif therefore, (A1) and (A2) are satisfied with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq92_HTML.gif Further,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ30_HTML.gif
(316)

Thus, by Theorem 3.1, the boundary value problem (3.15) has a unique solution on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq93_HTML.gif

Declarations

Acknowledgments

The authors are grateful to the anonymous referee for his/her valuable suggestions that led to the improvement of the original manuscript. The research of J. J. Nieto has been partially supported by Ministerio de Educacion y Ciencia and FEDER, Project MTM2007-61724, and by Xunta de Galicia and FEDER, project PGIDIT05PXIC20702PN.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, King Abdulaziz University
(2)
Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela

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© B. Ahmad and J.J. Nieto 2009

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