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

  • Bashir Ahmad1Email author and

    Affiliated with

    • Juan J Nieto2

      Affiliated with

      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 http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ1_HTML.gif
      (11)
      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq2_HTML.gif is the Caputo fractional derivative, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq3_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq4_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ2_HTML.gif
      (12)

      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq5_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq6_HTML.gif are real numbers. Here, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq7_HTML.gif is a Banach space and http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq11_HTML.gif the Caputo derivative of fractional order http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq12_HTML.gif is defined as
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ3_HTML.gif
      (21)

      where http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq15_HTML.gif is defined as
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq16_HTML.gif for a function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq17_HTML.gif is defined by
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq19_HTML.gif th derivative of the function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq20_HTML.gif as http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq23_HTML.gif is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ6_HTML.gif
      (24)

      where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq24_HTML.gif ( http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ7_HTML.gif
      (25)

      for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq26_HTML.gif ( http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq29_HTML.gif Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq30_HTML.gif be the operators such that (i) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq31_HTML.gif whenever http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq32_HTML.gif , (ii) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq33_HTML.gif is compact and continuous, (iii) http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq35_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq36_HTML.gif

      Lemma 2.6.

      For any http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ8_HTML.gif
      (26)
      is given by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ9_HTML.gif
      (27)
      where http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq39_HTML.gif we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ11_HTML.gif
      (29)
      In view of the relations http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq40_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq41_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq42_HTML.gif we obtain
      http://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
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ14_HTML.gif
      (212)

      where http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq45_HTML.gif into relatively compact subsets of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq46_HTML.gif is continuous with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq47_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq49_HTML.gif such that

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

      (A2) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq53_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq54_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq55_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq56_HTML.gif http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ15_HTML.gif
      (31)
      with
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ16_HTML.gif
      (32)

      Proof.

      Define http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq58_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ17_HTML.gif
      (33)
      Setting http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq59_HTML.gif (by the assumption on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq60_HTML.gif ) and Choosing
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ18_HTML.gif
      (34)
      we show that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq61_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq62_HTML.gif For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq63_HTML.gif we have
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ19_HTML.gif
      (35)
      Now, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq64_HTML.gif and for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq65_HTML.gif we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ20_HTML.gif
      (36)
      where
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq66_HTML.gif therefore http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq68_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq69_HTML.gif and
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq70_HTML.gif

      Proof.

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

      which is independent of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq84_HTML.gif So http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq85_HTML.gif is relatively compact on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq86_HTML.gif Hence, By Arzela Ascoli Theorem, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq87_HTML.gif is compact on http://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 http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_Equ29_HTML.gif
      (315)
      Here, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq90_HTML.gif As http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F708576/MediaObjects/13661_2008_Article_875_IEq92_HTML.gif Further,
      http://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 http://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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