Positive Solutions of Singular Complementary Lidstone Boundary Value Problems

  • RaviP Agarwal1Email author,

    Affiliated with

    • Donal O'Regan2 and

      Affiliated with

      • Svatoslav Staněk3

        Affiliated with

        Boundary Value Problems20102010:368169

        DOI: 10.1155/2010/368169

        Received: 7 October 2010

        Accepted: 21 November 2010

        Published: 2 December 2010

        Abstract

        We investigate the existence of positive solutions of singular problem http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq1_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq2_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq3_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq4_HTML.gif . Here, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq5_HTML.gif and the Carathéodory function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq6_HTML.gif may be singular in all its space variables http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq7_HTML.gif . The results are proved by regularization and sequential techniques. In limit processes, the Vitali convergence theorem is used.

        1. Introduction

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq8_HTML.gif be a positive constant, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq9_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq10_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq11_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq12_HTML.gif . We consider the singular complementary Lidstone boundary value problem
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ1_HTML.gif
        (1.1)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ2_HTML.gif
        (1.2)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq13_HTML.gif satisfies the local Carathéodory function on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq14_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq15_HTML.gif ) with
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ3_HTML.gif
        (1.3)

        The function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq16_HTML.gif is positive and may be singular at the value zero of all its space variables http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq17_HTML.gif .

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq18_HTML.gif . We say that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq19_HTML.gif is singular at the value zero of its space variable http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq20_HTML.gif if for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq21_HTML.gif and all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq22_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq23_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq24_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq25_HTML.gif , the relation
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ4_HTML.gif
        (1.4)

        holds.

        A function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq26_HTML.gif (i.e., http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq27_HTML.gif has absolutely continuous http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq28_HTML.gif th derivative on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq29_HTML.gif ) is a positive solution of problem (1.1), (1.2) if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq30_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq31_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq32_HTML.gif satisfies the boundary conditions (1.2) and (1.1) holds a.e. on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq33_HTML.gif .

        The regular complementary Lidstone problem
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ5_HTML.gif
        (1.5)

        was discussed in [1]. Here, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq34_HTML.gif is continuous at least in the interior of the domain of interest. Existence and uniqueness criteria for problem (1.5) are proved by the complementary Lidstone interpolating polynomial of degree http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq35_HTML.gif . No contributions exist, as far as we know, concerning the existence of positive solutions of singular complementary Lidstone problems.

        We observe that differential equations in complementary Lidstone problems as well as derivatives in boundary conditions are odd orders, in contrast to the Lidstone problem
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ6_HTML.gif
        (1.6)

        where the differential equation and derivatives in the boundary conditions are even orders. For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq36_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq37_HTML.gif ), regular Lidstone problems were discussed in [29], while singular ones in [1015].

        The aim of this paper is to give the conditions on the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq38_HTML.gif in (1.1) which guarantee that the singular problem (1.1), (1.2) has a solution. The existence results are proved by regularization and sequential techniques, and in limit processes, the Vitali convergence theorem [16, 17] is applied.

        Throughout the paper, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq39_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq40_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq41_HTML.gif stands for the norm in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq42_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq43_HTML.gif , respectively. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq44_HTML.gif denotes the set of functions (Lebesgue) integrable on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq45_HTML.gif and meas http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq46_HTML.gif the Lebesgue measure of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq47_HTML.gif .

        We work with the following conditions on the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq48_HTML.gif in (1.1).

        (H1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq50_HTML.gif and there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq51_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ7_HTML.gif
        (1.7)

        for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq52_HTML.gif and each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq53_HTML.gif .

        (H2) For a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq55_HTML.gif and for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq56_HTML.gif , the inequality
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ8_HTML.gif
        (1.8)
        is fulfilled, where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq57_HTML.gif is positive and nondecreasing in the second variable, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq58_HTML.gif is nonincreasing, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq59_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ9_HTML.gif
        (1.9)

        The paper is organized as follows. In Section 2, we construct a sequence of auxiliary regular differential equations associated with (1.1). Section 3 is devoted to the study of auxiliary regular complementary Lidstone problems. We show that the solvability of these problems is reduced to the existence of a fixed point of an operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq60_HTML.gif . The existence of a fixed point of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq61_HTML.gif is proved by a fixed point theorem of cone compression type according to Guo-Krasnosel'skii [18, 19]. The properties of solutions to auxiliary problems are also investigated here. In Section 4, applying the results of Section 3, the existence of a positive solution of the singular problem (1.1), (1.2) is proved.

        2. Regularization

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq62_HTML.gif be from (1.1). For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq63_HTML.gif , define http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq64_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq65_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq66_HTML.gif by the formulas
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ10_HTML.gif
        (2.1)
        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq67_HTML.gif . Chose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq68_HTML.gif and put
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ11_HTML.gif
        (2.2)
        for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq69_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq70_HTML.gif . Now, define an auxiliary function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq71_HTML.gif by means of the following recurrence formulas:
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ12_HTML.gif
        (2.3)
        for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq72_HTML.gif , and
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ13_HTML.gif
        (2.4)
        Then, under condition ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq73_HTML.gif ), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq74_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ14_HTML.gif
        (2.5)
        Condition ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq75_HTML.gif ) gives
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ15_HTML.gif
        (2.6)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ16_HTML.gif
        (2.7)
        We investigate the regular differential equation
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ17_HTML.gif
        (2.8)

        If a function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq76_HTML.gif satisfies (2.8) for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq77_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq78_HTML.gif is called a solution of (2.8).

        3. Auxiliary Regular Problems

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq79_HTML.gif and denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq80_HTML.gif the Green function of the problem
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ18_HTML.gif
        (3.1)
        Then,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ19_HTML.gif
        (3.2)
        By [2, 3, 20], the Green function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq81_HTML.gif can be expressed as
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ20_HTML.gif
        (3.3)
        and it is known that (see, e.g., [3, 20])
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ21_HTML.gif
        (3.4)

        Lemma 3.1 (see [10, Lemmas 2.1 and 2.3]).

        For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq82_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq83_HTML.gif , the inequalities
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ22_HTML.gif
        (3.5)
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ23_HTML.gif
        (3.6)

        hold.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq84_HTML.gif and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq85_HTML.gif be a solution of the differential equation
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ24_HTML.gif
        (3.7)
        satisfying the Lidstone boundary conditions
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ25_HTML.gif
        (3.8)
        It follows from the definition of the Green function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq86_HTML.gif that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ26_HTML.gif
        (3.9)
        It is easy to check that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq87_HTML.gif is a solution of problem (2.8), (1.2) if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq88_HTML.gif , and its derivative http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq89_HTML.gif is a solution of a problem involving the functional differential equation
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ27_HTML.gif
        (3.10)
        and the Lidstone boundary conditions (3.8). From (3.9) (for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq90_HTML.gif ), we see that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq91_HTML.gif is a solution of problem (3.10), (3.8) exactly if it is a solution of the equation
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ28_HTML.gif
        (3.11)
        in the set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq92_HTML.gif . Consequently, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq93_HTML.gif is a solution of problem (2.8), (1.2) if and only if it is a solution of the equation
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ29_HTML.gif
        (3.12)
        in the set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq94_HTML.gif . It means that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq95_HTML.gif is a solution of problem (2.8), (1.2) if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq96_HTML.gif is a fixed point of the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq97_HTML.gif defined as
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ30_HTML.gif
        (3.13)

        We prove the existence of a fixed point of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq98_HTML.gif by the following fixed point result of cone compression type according to Guo-Krasnosel'skii (see, e.g., [18, 19]).

        Lemma 3.2.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq99_HTML.gif be a Banach space, and let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq100_HTML.gif be a cone in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq101_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq102_HTML.gif be bounded open balls of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq103_HTML.gif centered at the origin with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq104_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq105_HTML.gif is completely continuous operator such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ31_HTML.gif
        (3.14)

        holds. Then, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq106_HTML.gif has a fixed point in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq107_HTML.gif .

        We are now in the position to prove that problem (2.8), (1.2) has a solution.

        Lemma 3.3.

        Let ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq108_HTML.gif ) and ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq109_HTML.gif ) hold. Then, problem (2.8), (1.2) has a solution.

        Proof.

        Let the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq110_HTML.gif be given in (3.13), and let
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ32_HTML.gif
        (3.15)

        Then, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq111_HTML.gif is a cone in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq112_HTML.gif and since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq113_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq114_HTML.gif by (3.4) and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq115_HTML.gif satisfies (2.5), we see that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq116_HTML.gif . The fact that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq117_HTML.gif is a completely continuous operator follows from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq118_HTML.gif , from Lebesgue dominated convergence theorem, and from the Arzelà-Ascoli theorem.

        Choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq119_HTML.gif and put http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq120_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq121_HTML.gif . Then, (cf. (2.5))
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ33_HTML.gif
        (3.16)
        Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq122_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq123_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq124_HTML.gif , the equality http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq125_HTML.gif holds with some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq126_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq127_HTML.gif . We now use the equality http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq128_HTML.gif and have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ34_HTML.gif
        (3.17)
        Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq129_HTML.gif , and so
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ35_HTML.gif
        (3.18)
        Next, we deduce from the relation
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ36_HTML.gif
        (3.19)
        and from (2.7) that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ37_HTML.gif
        (3.20)
        Therefore,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ38_HTML.gif
        (3.21)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq130_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq131_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq132_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ39_HTML.gif
        (3.22)
        The last inequality together with (3.21) gives
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ40_HTML.gif
        (3.23)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq133_HTML.gif is from ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq134_HTML.gif ). Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq135_HTML.gif is arbitrary, relations (3.18) and (3.21) imply that for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq136_HTML.gif , inequalities (3.18) and
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ41_HTML.gif
        (3.24)
        hold. By ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq137_HTML.gif ), there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq138_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ42_HTML.gif
        (3.25)
        and therefore,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ43_HTML.gif
        (3.26)
        Let
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ44_HTML.gif
        (3.27)
        Then, it follows from (3.18), (3.24), and (3.26) that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ45_HTML.gif
        (3.28)

        The conclusion now follows from Lemma 3.2 (for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq139_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq140_HTML.gif ).

        The properties of solutions to problem (2.8), (1.2) are collected in the following lemma.

        Lemma 3.4.

        Let ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq141_HTML.gif ) and ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq142_HTML.gif ) be satisfied. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq143_HTML.gif be a solution of problem (2.8), (1.2). Then, for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq144_HTML.gif , the following assertions hold:

        (i) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq145_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq146_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq147_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq148_HTML.gif for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq149_HTML.gif ,

        (ii) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq150_HTML.gif is increasing on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq151_HTML.gif , and for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq152_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq153_HTML.gif is decreasing on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq154_HTML.gif , and there is a unique http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq155_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq156_HTML.gif ,

        (iii) there exists a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq157_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ46_HTML.gif
        (3.29)

        for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq158_HTML.gif ,

        (iv) the sequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq159_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq160_HTML.gif .

        Proof.

        Let us choose an arbitrary http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq161_HTML.gif . By (2.5),
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ47_HTML.gif
        (3.30)
        and it follows from the definition of the Green function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq162_HTML.gif that the equality
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ48_HTML.gif
        (3.31)

        holds for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq163_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq164_HTML.gif . Now, using (1.2), (3.4), (3.30), and (3.31), we see that assertion (i) is true. Hence, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq165_HTML.gif is decreasing on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq166_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq167_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq168_HTML.gif is increasing on this interval. Due to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq169_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq170_HTML.gif , there exists a unique http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq171_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq172_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq173_HTML.gif . Consequently, assertion (ii) holds.

        Next, in view of (2.5), (3.6), and (3.31),
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ49_HTML.gif
        (3.32)
        Since
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ50_HTML.gif
        (3.33)
        and, by [13, Lemma 6.2],
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ51_HTML.gif
        (3.34)
        we have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ52_HTML.gif
        (3.35)
        Furthermore,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ53_HTML.gif
        (3.36)
        and (cf. (3.32) for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq174_HTML.gif )
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ54_HTML.gif
        (3.37)
        since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq175_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq176_HTML.gif by assertion (ii). Let
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ55_HTML.gif
        (3.38)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ56_HTML.gif
        (3.39)

        Then estimate (3.29) follows from relations (3.32)–(3.37).

        It remains to prove the boundedness of the sequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq177_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq178_HTML.gif . We use estimate (3.29), the properties of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq179_HTML.gif given in ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq180_HTML.gif ), and the inequality
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ57_HTML.gif
        (3.40)
        and have
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ58_HTML.gif
        (3.41)
        In particular,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ59_HTML.gif
        (3.42)
        for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq181_HTML.gif . Now, from the above estimates, from (2.6) and from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq182_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq183_HTML.gif , which is proved in (ii), we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ60_HTML.gif
        (3.43)
        where
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ61_HTML.gif
        (3.44)
        Notice that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq184_HTML.gif by ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq185_HTML.gif ). Consequently,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ62_HTML.gif
        (3.45)
        Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq186_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq187_HTML.gif , which follows from the fact that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq188_HTML.gif vanishes in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq189_HTML.gif by (1.2) and assertion (ii), inequality (3.45) yields
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ63_HTML.gif
        (3.46)
        where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq190_HTML.gif is from ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq191_HTML.gif ). Due to the condition
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ64_HTML.gif
        (3.47)
        in ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq192_HTML.gif ), there exists a positive constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq193_HTML.gif such that for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq194_HTML.gif the inequality
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ65_HTML.gif
        (3.48)

        is fulfilled. The last inequality together with estimate (3.46) gives http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq195_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq196_HTML.gif . Consequently, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq197_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq198_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq199_HTML.gif , and assertion (iv) follows.

        The following result gives the important property of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq200_HTML.gif for applying the Vitali convergent theorem in the proof of Theorem 4.1.

        Lemma 3.5.

        Let ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq201_HTML.gif ) and ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq202_HTML.gif ) hold. Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq203_HTML.gif be a solution of problem (2.8), (1.2). Then, the sequence
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ66_HTML.gif
        (3.49)
        is uniformly integrable on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq204_HTML.gif , that is, for each http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq205_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq206_HTML.gif such that if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq207_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq208_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq209_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ67_HTML.gif
        (3.50)

        Proof.

        By Lemma 3.4 (iv), there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq210_HTML.gif such that for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq211_HTML.gif , the inequality http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq212_HTML.gif holds. Now, we conclude from (2.5) and (2.6), from the properties of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq213_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq214_HTML.gif given in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq215_HTML.gif , and finally from (3.29) that for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq216_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq217_HTML.gif , the estimate
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ68_HTML.gif
        (3.51)
        is fulfilled, where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq218_HTML.gif is a positive constant. Since the functions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq219_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq220_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq221_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq222_HTML.gif ) belong to the set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq223_HTML.gif by assumption ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq224_HTML.gif ), in order to prove that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq225_HTML.gif is uniformly integrable on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq226_HTML.gif , it suffices to show that the sequences
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ69_HTML.gif
        (3.52)

        are uniformly integrable on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq227_HTML.gif . Due to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq228_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq229_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq230_HTML.gif by ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq231_HTML.gif ), this fact follows from [13, Criterion 11.10 (with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq232_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq233_HTML.gif )].

        4. The Main Result

        The following theorem is the existence result for the singular problem (1.1), (1.2).

        Theorem 4.1.

        Let ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq234_HTML.gif ) and ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq235_HTML.gif ) hold. Then, problem (1.1), (1.2) has a positive solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq236_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ70_HTML.gif
        (4.1)

        Proof.

        Lemma 3.3 guarantees that problem (2.8), (1.2) has a solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq237_HTML.gif . Consider the sequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq238_HTML.gif . By Lemma 3.4, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq239_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq240_HTML.gif ,
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ71_HTML.gif
        (4.2)
        and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq241_HTML.gif fulfils estimate (3.29), where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq242_HTML.gif is a positive constant and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq243_HTML.gif . Furthermore, the sequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq244_HTML.gif is uniformly integrable on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq245_HTML.gif by Lemma 3.5, and therefore, we deduce from the equality http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq246_HTML.gif for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq247_HTML.gif that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq248_HTML.gif is equicontinuous on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq249_HTML.gif . Now, by the Arzelà-Ascoli theorem and the Bolzano-Weierstrass theorem, we may assume without loss of generality that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq250_HTML.gif is convergent in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq251_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq252_HTML.gif is convergent in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq253_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq254_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq255_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq256_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq257_HTML.gif ). Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq258_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq259_HTML.gif satisfies the boundary conditions (1.2). Letting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq260_HTML.gif in (3.29) and (4.2), we get (for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq261_HTML.gif )
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ72_HTML.gif
        (4.3)
        Keeping in mind the definition of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq262_HTML.gif , we conclude from (4.3) that
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ73_HTML.gif
        (4.4)
        Then, by the Vitali theorem, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq263_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ74_HTML.gif
        (4.5)
        Letting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq264_HTML.gif in the equality
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ75_HTML.gif
        (4.6)
        we get
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ76_HTML.gif
        (4.7)

        As a result, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq265_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq266_HTML.gif is a solution of (1.1). Consequently, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq267_HTML.gif is a positive solution of problem (1.1), (1.2) and inequality (4.1) follows from (4.3).

        Example 4.2.

        Consider problem (1.1), (1.2) with
        http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ77_HTML.gif
        (4.8)

        on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq268_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq269_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq270_HTML.gif (that is, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq271_HTML.gif is essentially bounded and measurable on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq272_HTML.gif ) are nonnegative, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq273_HTML.gif for a.e. http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq274_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq275_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq276_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq277_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq278_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq279_HTML.gif , then, by Theorem 4.1, the problem has a positive solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq280_HTML.gif satisfying inequality (4.1).

        Declarations

        Acknowledgment

        This work was supported by the Council of Czech Government MSM no. 6198959214.

        Authors’ Affiliations

        (1)
        Department of Mathematical Sciences, Florida Institute of Technology
        (2)
        Department of Mathematics, National University of Ireland
        (3)
        Department of Mathematical Analysis, Faculty of Science, Palacký University

        References

        1. Agarwal RP, Pinelas S, Wong PJY: Complementary Lidstone interpolation and boundary value problems. Journal of Inequalities and Applications 2009, 2009:-30.
        2. Agarwal RP: Boundary Value Problems for Higher Order Differential Equations. World Scientific, Teaneck, NJ, USA; 1986:xii+307.View Article
        3. Agarwal RP, Wong PJY: Lidstone polynomials and boundary value problems. Computers & Mathematics with Applications 1989,17(10):1397-1421. 10.1016/0898-1221(89)90023-0MathSciNetView Article
        4. Davis JM, Henderson J, Wong PJY: General Lidstone problems: multiplicity and symmetry of solutions. Journal of Mathematical Analysis and Applications 2000,251(2):527-548. 10.1006/jmaa.2000.7028MathSciNetView Article
        5. Guo Y, Gao Y: The method of upper and lower solutions for a Lidstone boundary value problem. Czechoslovak Mathematical Journal 2005,55(130)(3):639-652.MathSciNetView Article
        6. Ma Y: Existence of positive solutions of Lidstone boundary value problems. Journal of Mathematical Analysis and Applications 2006,314(1):97-108.MathSciNetView Article
        7. Wong PJY, Agarwal RP: Results and estimates on multiple solutions of Lidstone boundary value problems. Acta Mathematica Hungarica 2000,86(1-2):137-168.MathSciNetView Article
        8. Yang Y-R, Cheng SS: Positive solutions of a Lidstone boundary value problem with variable coefficient function. Journal of Applied Mathematics and Computing 2008,27(1-2):411-419. 10.1007/s12190-008-0066-zMathSciNetView Article
        9. Zhang B, Liu X: Existence of multiple symmetric positive solutions of higher order Lidstone problems. Journal of Mathematical Analysis and Applications 2003,284(2):672-689. 10.1016/S0022-247X(03)00386-XMathSciNetView Article
        10. Agarwal RP, O'Regan D, Rachůnková I, Staněk S: Two-point higher-order BVPs with singularities in phase variables. Computers & Mathematics with Applications 2003,46(12):1799-1826. 10.1016/S0898-1221(03)90238-0MathSciNetView Article
        11. Agarwal RP, O'Regan D, Staněk S: Singular Lidstone boundary value problem with given maximal values for solutions. Nonlinear Analysis: Theory, Methods & Applications 2003,55(7-8):859-881. 10.1016/j.na.2003.06.001View Article
        12. Rachůnková I, Staněk S, Tvrdý M: Singularities and Laplacians in boundary value problems for nonlinear ordinary differential equations. In Handbook of Differential Equations: Ordinary Differential Equations. Vol. III, Handb. Differ. Equ.. Edited by: Cañada A, Drábek P, Fonda A. Elsevier/North-Holland, Amsterdam, The Netherlands; 2006:607-722.View Article
        13. Rachůnková I, Staněk S, Tvrdý M: Solvability of Nonlinear Singular Problems for Ordinary Differential Equations, Contemporary Mathematics and Its Applications. Volume 5. Hindawi Publishing Corporation, New York, NY, USA; 2008:x+268.
        14. Wei Z: Existence of positive solutions for n th-order singular sublinear boundary value problems. Journal of Mathematical Analysis and Applications 2005,306(2):619-636. 10.1016/j.jmaa.2004.10.037MathSciNetView Article
        15. Zhao Z: On the existence of positive solutions for n -order singular boundary value problems. Nonlinear Analysis: Theory, Methods & Applications 2006,64(11):2553-2561. 10.1016/j.na.2005.09.003MathSciNetView Article
        16. Bartle RG: A Modern Theory of Integration, Graduate Studies in Mathematics. Volume 32. American Mathematical Society, Providence, RI, USA; 2001:xiv+458.
        17. Natanson IP: Theorie der Funktionen einer reellen Veränderlichen, Mathematische Lehrbücher und Monographien. Akademie, Berlin, USA; 1969:xii+590.
        18. Guo DJ, Lakshmikantham V: Nonlinear Problems in Abstract Cones, Notes and Reports in Mathematics in Science and Engineering. Volume 5. Academic Press, Boston, Mass, USA; 1988:viii+275.
        19. Krasnosel'skii MA: Positive Solutions of Operator Equations. P. Noordhoff, Groningen, The Netherlands; 1964:381.
        20. Agarwal RP, Wong PJY: Error Inequalities in Polynomial Interpolation and Their Applications, Mathematics and Its Applications. Volume 262. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1993:x+365.View Article

        Copyright

        © Ravi P. Agarwal et al. 2010

        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.