Existence and Uniqueness of Positive and Nondecreasing Solutions for a Class of Singular Fractional Boundary Value Problems

  • J Caballero Mena1,

    Affiliated with

    • J Harjani1 and

      Affiliated with

      • K Sadarangani1Email author

        Affiliated with

        Boundary Value Problems20092009:421310

        DOI: 10.1155/2009/421310

        Received: 24 April 2009

        Accepted: 14 June 2009

        Published: 19 July 2009

        Abstract

        We establish the existence and uniqueness of a positive and nondecreasing solution to a singular boundary value problem of a class of nonlinear fractional differential equation. Our analysis relies on a fixed point theorem in partially ordered sets.

        1. Introduction

        Many papers and books on fractional differential equations have appeared recently. Most of them are devoted to the solvability of the linear fractional equation in terms of a special function (see, e.g., [1, 2]) and to problems of analyticity in the complex domain [3]. Moreover, Delbosco and Rodino [4] considered the existence of a solution for the nonlinear fractional differential equation http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq1_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq2_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq4_HTML.gif is a given continuous function in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq5_HTML.gif . They obtained results for solutions by using the Schauder fixed point theorem and the Banach contraction principle. Recently, Zhang [5] considered the existence of positive solution for equation http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq6_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq7_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq8_HTML.gif is a given continuous function by using the sub- and super-solution methods.

        In this paper, we discuss the existence and uniqueness of a positive and nondecreasing solution to boundary-value problem of the nonlinear fractional differential equation

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ1_HTML.gif
        (1.1)

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq9_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq10_HTML.gif is the Caputo's differentiation and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq11_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq12_HTML.gif (i.e., http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq13_HTML.gif is singular at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq14_HTML.gif ).

        Note that this problem was considered in [6] where the authors proved the existence of one positive solution for (1.1) by using Krasnoselskii's fixed point theorem and nonlinear alternative of Leray-Schauder type in a cone and assuming certain hypotheses on the function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq15_HTML.gif . In [6] the uniqueness of the solution is not treated.

        In this paper we will prove the existence and uniqueness of a positive and nondecreasing solution for the problem (1.1) by using a fixed point theorem in partially ordered sets.

        Existence of fixed point in partially ordered sets has been considered recently in [712]. This work is inspired in the papers [6, 8].

        For existence theorems for fractional differential equation and applications, we refer to the survey [13]. Concerning the definitions and basic properties we refer the reader to [14].

        Recently, some existence results for fractional boundary value problem have appeared in the literature (see, e.g., [1517]).

        2. Preliminaries and Previous Results

        For the convenience of the reader, we present here some notations and lemmas that will be used in the proofs of our main results.

        Definition 2.1.

        The Riemman-Liouville fractional integral of order http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq16_HTML.gif of a function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq17_HTML.gif is given by
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ2_HTML.gif
        (2.1)

        provided that the right-hand side is pointwise defined on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq18_HTML.gif .

        Definition 2.2.

        The Caputo fractional derivative of order http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq19_HTML.gif of a continuous function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq20_HTML.gif is given by
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ3_HTML.gif
        (2.2)

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq21_HTML.gif , provided that the right-hand side is pointwise defined on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq22_HTML.gif .

        The following lemmas appear in [14].

        Lemma 2.3.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq23_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq24_HTML.gif . Then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ4_HTML.gif
        (2.3)

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq25_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq26_HTML.gif

        Lemma 2.4.

        The relation
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ5_HTML.gif
        (2.4)

        is valid when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq27_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq28_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq29_HTML.gif .

        The following lemmas appear in [6].

        Lemma 2.5.

        Given http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq30_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq31_HTML.gif , the unique solution of
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ6_HTML.gif
        (2.5)
        is given by
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ7_HTML.gif
        (2.6)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ8_HTML.gif
        (2.7)

        Remark 2.6.

        Note that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq32_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq33_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq34_HTML.gif (see [6]).

        Lemma 2.7.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq35_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq36_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq37_HTML.gif is a continuous function with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq38_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq39_HTML.gif is a continuous function on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq40_HTML.gif . Then the function defined by
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ9_HTML.gif
        (2.8)

        is continuous on [0,1], where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq41_HTML.gif is the Green function defined in Lemma 2.5.

        Now, we present some results about the fixed point theorems which we will use later. These results appear in [8].

        Theorem 2.8.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq42_HTML.gif be a partially ordered set and suppose that there exists a metric http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq43_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq44_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq45_HTML.gif is a complete metric space. Assume that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq46_HTML.gif satisfies the following condition: if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq47_HTML.gif is a non decreasing sequence in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq48_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq49_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq50_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq51_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq52_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq53_HTML.gif be a nondecreasing mapping such that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ10_HTML.gif
        (2.9)

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq54_HTML.gif is continuous and nondecreasing function such that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq55_HTML.gif is positive in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq56_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq57_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq58_HTML.gif . If there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq59_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq60_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq61_HTML.gif has a fixed point.

        If we consider that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq62_HTML.gif satisfies the following condition:

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ11_HTML.gif
        (2.10)

        then we have the following theorem [8].

        Theorem 2.9.

        Adding condition (2.10) to the hypotheses of Theorem 2.8 one obtains uniqueness of the fixed point of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq63_HTML.gif .

        In our considerations, we will work in the Banach space http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq64_HTML.gif with the standard norm http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq65_HTML.gif .

        Note that this space can be equipped with a partial order given by

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ12_HTML.gif
        (2.11)

        In [10] it is proved that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq66_HTML.gif with the classic metric given by

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ13_HTML.gif
        (2.12)

        satisfies condition (2) of Theorem 2.8. Moreover, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq67_HTML.gif , as the function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq68_HTML.gif is continuous in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq69_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq70_HTML.gif satisfies condition (2.10).

        3. Main Result

        Theorem 3.1.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq71_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq72_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq73_HTML.gif is continuous and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq74_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq75_HTML.gif is a continuous function on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq76_HTML.gif . Assume that there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq77_HTML.gif such that for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq78_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq79_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq80_HTML.gif
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ14_HTML.gif
        (3.1)

        Then one's problem (1.1) has an unique nonnegative solution.

        Proof.

        Consider the cone
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ15_HTML.gif
        (3.2)

        Note that, as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq81_HTML.gif is a closed set of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq82_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq83_HTML.gif is a complete metric space.

        Now, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq84_HTML.gif we define the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq85_HTML.gif by

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ16_HTML.gif
        (3.3)
        By Lemma 2.7, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq86_HTML.gif . Moreover, taking into account Remark 2.6 and as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq87_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq88_HTML.gif by hypothesis, we get
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ17_HTML.gif
        (3.4)

        Hence, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq89_HTML.gif .

        In what follows we check that hypotheses in Theorems 2.8 and 2.9 are satisfied.

        Firstly, the operator http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq90_HTML.gif is nondecreasing since, by hypothesis, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq91_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ18_HTML.gif
        (3.5)
        Besides, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq92_HTML.gif
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ19_HTML.gif
        (3.6)
        As the function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq93_HTML.gif is nondecreasing then, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq94_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ20_HTML.gif
        (3.7)
        and from last inequality we get
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ21_HTML.gif
        (3.8)

        Put http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq95_HTML.gif . Obviously, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq96_HTML.gif is continuous, nondecreasing, positive in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq97_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq98_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq99_HTML.gif .

        Thus, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq100_HTML.gif

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ22_HTML.gif
        (3.9)

        Finally, take into account that for the zero function, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq101_HTML.gif , by Theorem 2.8 our problem (1.1) has at least one nonnegative solution. Moreover, this solution is unique since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq102_HTML.gif satisfies condition (2.10) (see comments at the beginning of this section) and Theorem 2.9.

        Remark 3.2.

        In [6, lemma  3.2] it is proved that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq103_HTML.gif is completely continuous and Schauder fixed point theorem gives us the existence of a solution to our problem (1.1).

        In the sequel we present an example which illustrates Theorem 3.1.

        Example 3.3.

        Consider the fractional differential equation (this example is inspired in [6])
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ23_HTML.gif
        (3.10)
        In this case, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq104_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq105_HTML.gif . Note that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq106_HTML.gif is continuous in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq107_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq108_HTML.gif . Moreover, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq109_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq110_HTML.gif we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ24_HTML.gif
        (3.11)
        because http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq111_HTML.gif is nondecreasing on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq112_HTML.gif , and
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ25_HTML.gif
        (3.12)

        Note that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq113_HTML.gif .

        Theorem 3.1 give us that our fractional differential (3.10) has an unique nonnegative solution.

        This example give us uniqueness of the solution for the fractional differential equation appearing in [6] in the particular case http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq114_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq115_HTML.gif

        Remark 3.4.

        Note that our Theorem 3.1 works if the condition (3.1) is changed by, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq116_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq117_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq118_HTML.gif
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ26_HTML.gif
        (3.13)

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq119_HTML.gif is continuous and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq120_HTML.gif satisfies

        (a) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq121_HTML.gif and nondecreasing;

        (b) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq122_HTML.gif ;

        (c) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq123_HTML.gif is positive in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq124_HTML.gif ;

        (d) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq125_HTML.gif .

        Examples of such functions are http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq126_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq127_HTML.gif .

        Remark 3.5.

        Note that the Green function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq128_HTML.gif is strictly increasing in the first variable in the interval http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq129_HTML.gif . In fact, for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq130_HTML.gif fixed we have the following cases

        Case 1.

        For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq131_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq132_HTML.gif as, in this case,
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ27_HTML.gif
        (3.14)
        It is trivial that
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ28_HTML.gif
        (3.15)

        Case 2.

        For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq133_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq134_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ29_HTML.gif
        (3.16)
        Now, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq135_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq136_HTML.gif then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ30_HTML.gif
        (3.17)

        Hence, taking into account the last inequality and (3.16), we obtain http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq137_HTML.gif .

        Case 3.

        For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq138_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq139_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ31_HTML.gif
        (3.18)

        and, as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq140_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq141_HTML.gif , it can be deduced that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq142_HTML.gif and consequently, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq143_HTML.gif .

        This completes the proof.

        Remark 3.5 gives us the following theorem which is a better result than that [6, Theorem  3.3] because the solution of our problem (1.1) is positive in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq144_HTML.gif and strictly increasing.

        Theorem 3.6.

        Under assumptions of Theorem 3.1, our problem (1.1) has a unique nonnegative and strictly increasing solution.

        Proof.

        By Theorem 3.1 we obtain that the problem (1.1) has an unique solution http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq145_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq146_HTML.gif . Now, we will prove that this solution is a strictly increasing function. Let us take http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq147_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq148_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ32_HTML.gif
        (3.19)

        Taking into account Remark 3.4 and the fact that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq149_HTML.gif , we get http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq150_HTML.gif .

        Now, if we suppose that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq151_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq152_HTML.gif and as, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq153_HTML.gif we deduce that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq154_HTML.gif a.e.

        On the other hand, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq155_HTML.gif a.e. then

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ33_HTML.gif
        (3.20)
        Now, as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq156_HTML.gif , then for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq157_HTML.gif there exists http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq158_HTML.gif such that for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq159_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq160_HTML.gif we get http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq161_HTML.gif . Observe that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq162_HTML.gif , consequently,
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_Equ34_HTML.gif
        (3.21)

        and this contradicts that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq163_HTML.gif a.e.

        Thus, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq164_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq165_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq166_HTML.gif . Finally, as http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq167_HTML.gif we have that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq168_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F421310/MediaObjects/13661_2009_Article_847_IEq169_HTML.gif .

        Declarations

        Acknowledgment

        This research was partially supported by "Ministerio de Educación y Ciencia" Project MTM 2007/65706.

        Authors’ Affiliations

        (1)
        Departamento de Matemáticas, Universidad de Las Palmas de Gran Canaria

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        © J. Caballero Mena 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.