Open Access

Positive Solutions of Singular Complementary Lidstone Boundary Value Problems

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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq4_HTML.gif . Here, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq5_HTML.gif and the Carathéodory function https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq8_HTML.gif be a positive constant, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq9_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq10_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq11_HTML.gif , https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ1_HTML.gif
(1.1)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ2_HTML.gif
(1.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq13_HTML.gif satisfies the local Carathéodory function on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq14_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq15_HTML.gif ) with
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ3_HTML.gif
(1.3)

The function https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq17_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq18_HTML.gif . We say that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq20_HTML.gif if for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq21_HTML.gif and all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq22_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq23_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq24_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq25_HTML.gif , the relation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ4_HTML.gif
(1.4)

holds.

A function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq26_HTML.gif (i.e., https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq27_HTML.gif has absolutely continuous https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq28_HTML.gif th derivative on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq30_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq31_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq33_HTML.gif .

The regular complementary Lidstone problem
https://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, https://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 https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq36_HTML.gif ( https://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 https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq39_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq40_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq41_HTML.gif stands for the norm in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq42_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq43_HTML.gif , respectively. https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq45_HTML.gif and meas https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq46_HTML.gif the Lebesgue measure of https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq48_HTML.gif in (1.1).

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

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

(H2) For a.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq55_HTML.gif and for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq56_HTML.gif , the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ8_HTML.gif
(1.8)
is fulfilled, where https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq58_HTML.gif is nonincreasing, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq59_HTML.gif ,
https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq62_HTML.gif be from (1.1). For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq63_HTML.gif , define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq64_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq65_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq66_HTML.gif by the formulas
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ10_HTML.gif
(2.1)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq67_HTML.gif . Chose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq68_HTML.gif and put
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ11_HTML.gif
(2.2)
for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq69_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq70_HTML.gif . Now, define an auxiliary function https://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:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ12_HTML.gif
(2.3)
for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq72_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ13_HTML.gif
(2.4)
Then, under condition ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq73_HTML.gif ), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq74_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ14_HTML.gif
(2.5)
Condition ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq75_HTML.gif ) gives
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ15_HTML.gif
(2.6)
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ17_HTML.gif
(2.8)

If a function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq76_HTML.gif satisfies (2.8) for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq77_HTML.gif , then https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq79_HTML.gif and denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq80_HTML.gif the Green function of the problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ18_HTML.gif
(3.1)
Then,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq81_HTML.gif can be expressed as
https://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])
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq82_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq83_HTML.gif , the inequalities
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ22_HTML.gif
(3.5)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ23_HTML.gif
(3.6)

hold.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq84_HTML.gif and let https://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
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq86_HTML.gif that
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq88_HTML.gif , and its derivative https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq90_HTML.gif ), we see that https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ28_HTML.gif
(3.11)
in the set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq92_HTML.gif . Consequently, https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ29_HTML.gif
(3.12)
in the set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq94_HTML.gif . It means that https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq97_HTML.gif defined as
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq99_HTML.gif be a Banach space, and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq100_HTML.gif be a cone in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq101_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq102_HTML.gif be bounded open balls of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq103_HTML.gif centered at the origin with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq104_HTML.gif . Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq105_HTML.gif is completely continuous operator such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ31_HTML.gif
(3.14)

holds. Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq106_HTML.gif has a fixed point in https://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 ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq108_HTML.gif ) and ( https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ32_HTML.gif
(3.15)

Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq111_HTML.gif is a cone in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq112_HTML.gif and since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq113_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq114_HTML.gif by (3.4) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq115_HTML.gif satisfies (2.5), we see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq116_HTML.gif . The fact that https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq119_HTML.gif and put https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq120_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq121_HTML.gif . Then, (cf. (2.5))
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ33_HTML.gif
(3.16)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq122_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq123_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq124_HTML.gif , the equality https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq125_HTML.gif holds with some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq126_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq127_HTML.gif . We now use the equality https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq128_HTML.gif and have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ34_HTML.gif
(3.17)
Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq129_HTML.gif , and so
https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ37_HTML.gif
(3.20)
Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ38_HTML.gif
(3.21)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq130_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq131_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq132_HTML.gif , we have
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ40_HTML.gif
(3.23)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq133_HTML.gif is from ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq134_HTML.gif ). Since https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq136_HTML.gif , inequalities (3.18) and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ41_HTML.gif
(3.24)
hold. By ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq137_HTML.gif ), there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq138_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ42_HTML.gif
(3.25)
and therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ43_HTML.gif
(3.26)
Let
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq139_HTML.gif and https://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 ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq141_HTML.gif ) and ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq142_HTML.gif ) be satisfied. Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq144_HTML.gif , the following assertions hold:

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

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

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

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

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

Proof.

Let us choose an arbitrary https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq161_HTML.gif . By (2.5),
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq162_HTML.gif that the equality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ48_HTML.gif
(3.31)

holds for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq163_HTML.gif and https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq165_HTML.gif is decreasing on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq166_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq167_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq169_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq170_HTML.gif , there exists a unique https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq171_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq172_HTML.gif for https://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),
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ49_HTML.gif
(3.32)
Since
https://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],
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ51_HTML.gif
(3.34)
we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ52_HTML.gif
(3.35)
Furthermore,
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq174_HTML.gif )
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ54_HTML.gif
(3.37)
since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq175_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq176_HTML.gif by assertion (ii). Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ55_HTML.gif
(3.38)
where
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq177_HTML.gif in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq179_HTML.gif given in ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq180_HTML.gif ), and the inequality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ57_HTML.gif
(3.40)
and have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ58_HTML.gif
(3.41)
In particular,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ59_HTML.gif
(3.42)
for all https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq182_HTML.gif for some https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ60_HTML.gif
(3.43)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ61_HTML.gif
(3.44)
Notice that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq184_HTML.gif by ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq185_HTML.gif ). Consequently,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ62_HTML.gif
(3.45)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq186_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq187_HTML.gif , which follows from the fact that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq188_HTML.gif vanishes in https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ63_HTML.gif
(3.46)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq190_HTML.gif is from ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq191_HTML.gif ). Due to the condition
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ64_HTML.gif
(3.47)
in ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq192_HTML.gif ), there exists a positive constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq193_HTML.gif such that for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq194_HTML.gif the inequality
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq195_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq196_HTML.gif . Consequently, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq197_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq198_HTML.gif , https://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 https://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 ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq201_HTML.gif ) and ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq202_HTML.gif ) hold. Let https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ66_HTML.gif
(3.49)
is uniformly integrable on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq204_HTML.gif , that is, for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq205_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq206_HTML.gif such that if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq207_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq208_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq209_HTML.gif , then
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq210_HTML.gif such that for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq211_HTML.gif , the inequality https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq213_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq214_HTML.gif given in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq216_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq217_HTML.gif , the estimate
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ68_HTML.gif
(3.51)
is fulfilled, where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq219_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq220_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq221_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq222_HTML.gif ) belong to the set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq223_HTML.gif by assumption ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq224_HTML.gif ), in order to prove that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq225_HTML.gif is uniformly integrable on https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ69_HTML.gif
(3.52)

are uniformly integrable on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq227_HTML.gif . Due to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq228_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq229_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq230_HTML.gif by ( https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq232_HTML.gif and https://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 ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq234_HTML.gif ) and ( https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq236_HTML.gif and
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq237_HTML.gif . Consider the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq238_HTML.gif . By Lemma 3.4, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq239_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq240_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ71_HTML.gif
(4.2)
and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq241_HTML.gif fulfils estimate (3.29), where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq242_HTML.gif is a positive constant and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq243_HTML.gif . Furthermore, the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq244_HTML.gif is uniformly integrable on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq246_HTML.gif for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq247_HTML.gif that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq248_HTML.gif is equicontinuous on https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq250_HTML.gif is convergent in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq251_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq252_HTML.gif is convergent in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq253_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq254_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq255_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq256_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq257_HTML.gif ). Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq258_HTML.gif and https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq261_HTML.gif )
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq262_HTML.gif , we conclude from (4.3) that
https://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, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq263_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ74_HTML.gif
(4.5)
Letting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq264_HTML.gif in the equality
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ75_HTML.gif
(4.6)
we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ76_HTML.gif
(4.7)

As a result, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq265_HTML.gif and https://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, https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_Equ77_HTML.gif
(4.8)

on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq268_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq269_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq270_HTML.gif (that is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq271_HTML.gif is essentially bounded and measurable on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq272_HTML.gif ) are nonnegative, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq273_HTML.gif for a.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq274_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq275_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq276_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq277_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F368169/MediaObjects/13661_2010_Article_919_IEq278_HTML.gif for https://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 https://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

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© Ravi P. Agarwal et al. 2010

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