Positive Solutions of Singular Complementary Lidstone Boundary Value Problems

Abstract

We investigate the existence of positive solutions of singular problem , , , . Here, and the Carathéodory function may be singular in all its space variables . The results are proved by regularization and sequential techniques. In limit processes, the Vitali convergence theorem is used.

1. Introduction

Let be a positive constant, and , , . We consider the singular complementary Lidstone boundary value problem

(1.1)

(1.2)

where satisfies the local Carathéodory function on () with

(1.3)

The function is positive and may be singular at the value zero of all its space variables .

Let . We say that is singular at the value zero of its space variable if for a.e. and all , , such that , the relation

(1.4)

holds.

A function (i.e., has absolutely continuous th derivative on ) is a positive solution of problem (1.1), (1.2) if for , satisfies the boundary conditions (1.2) and (1.1) holds a.e. on .

The regular complementary Lidstone problem

(1.5)

was discussed in [1]. Here, 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 . 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

(1.6)

where the differential equation and derivatives in the boundary conditions are even orders. For (), regular Lidstone problems were discussed in [2–9], while singular ones in [10–15].

The aim of this paper is to give the conditions on the function 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, and , stands for the norm in and , respectively. denotes the set of functions (Lebesgue) integrable on and meas the Lebesgue measure of .

We work with the following conditions on the function in (1.1).

(H₁) and there exists such that

(1.7)

for a.e. and each .

(H₂) For a.e. and for all , the inequality

(1.8)

is fulfilled, where is positive and nondecreasing in the second variable, is nonincreasing, ,

(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 . The existence of a fixed point of 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 be from (1.1). For , define , , and by the formulas

(2.1)

Let . Chose and put

(2.2)

for . Now, define an auxiliary function by means of the following recurrence formulas:

(2.3)

for , and

(2.4)

Then, under condition (), and

(2.5)

Condition () gives

(2.6)

(2.7)

We investigate the regular differential equation

(2.8)

If a function satisfies (2.8) for a.e. , then is called a solution of (2.8).

3. Auxiliary Regular Problems

Let and denote by the Green function of the problem

(3.1)

Then,

(3.2)

By [2, 3, 20], the Green function can be expressed as

(3.3)

and it is known that (see, e.g., [3, 20])

(3.4)

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

For and , the inequalities

(3.5)

(3.6)

hold.

Let and let be a solution of the differential equation

(3.7)

satisfying the Lidstone boundary conditions

(3.8)

It follows from the definition of the Green function that

(3.9)

It is easy to check that is a solution of problem (2.8), (1.2) if and only if , and its derivative is a solution of a problem involving the functional differential equation

(3.10)

and the Lidstone boundary conditions (3.8). From (3.9) (for ), we see that is a solution of problem (3.10), (3.8) exactly if it is a solution of the equation

(3.11)

in the set . Consequently, is a solution of problem (2.8), (1.2) if and only if it is a solution of the equation

(3.12)

in the set . It means that is a solution of problem (2.8), (1.2) if is a fixed point of the operator defined as

(3.13)

We prove the existence of a fixed point of by the following fixed point result of cone compression type according to Guo-Krasnosel'skii (see, e.g., [18, 19]).

Lemma 3.2.

Let be a Banach space, and let be a cone in . Let be bounded open balls of centered at the origin with . Suppose that is completely continuous operator such that

(3.14)

holds. Then, has a fixed point in .

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

Lemma 3.3.

Let () and () hold. Then, problem (2.8), (1.2) has a solution.

Proof.

Let the operator be given in (3.13), and let

(3.15)

Then, is a cone in and since for by (3.4) and satisfies (2.5), we see that . The fact that is a completely continuous operator follows from , from Lebesgue dominated convergence theorem, and from the Arzelà-Ascoli theorem.

Choose and put for . Then, (cf. (2.5))

(3.16)

Since and for , the equality holds with some for . We now use the equality and have

(3.17)

Hence, , and so

(3.18)

Next, we deduce from the relation

(3.19)

and from (2.7) that

(3.20)

Therefore,

(3.21)

where . Since for , we have

(3.22)

The last inequality together with (3.21) gives

(3.23)

where is from (). Since is arbitrary, relations (3.18) and (3.21) imply that for all , inequalities (3.18) and

(3.24)

hold. By (), there exists such that

(3.25)

and therefore,

(3.26)

Let

(3.27)

Then, it follows from (3.18), (3.24), and (3.26) that

(3.28)

The conclusion now follows from Lemma 3.2 (for and ).

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

Lemma 3.4.

Let () and () be satisfied. Let be a solution of problem (2.8), (1.2). Then, for all , the following assertions hold:

(i) for , , and for a.e. ,

(ii) is increasing on , and for , is decreasing on , and there is a unique such that ,

(iii) there exists a positive constant such that

(3.29)

for ,

(iv) the sequence is bounded in .

Proof.

Let us choose an arbitrary . By (2.5),

(3.30)

and it follows from the definition of the Green function that the equality

(3.31)

holds for and . Now, using (1.2), (3.4), (3.30), and (3.31), we see that assertion (i) is true. Hence, is decreasing on for and is increasing on this interval. Due to for , there exists a unique such that for . Consequently, assertion (ii) holds.

Next, in view of (2.5), (3.6), and (3.31),

(3.32)

Since

(3.33)

and, by [13, Lemma 6.2],

(3.34)

we have

(3.35)

Furthermore,

(3.36)

and (cf. (3.32) for )

(3.37)

since on by assertion (ii). Let

(3.38)

where

(3.39)

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

It remains to prove the boundedness of the sequence in . We use estimate (3.29), the properties of given in (), and the inequality

(3.40)

and have

(3.41)

In particular,

(3.42)

for all . Now, from the above estimates, from (2.6) and from for some , which is proved in (ii), we get

(3.43)

where

(3.44)

Notice that by (). Consequently,

(3.45)

Since for , which follows from the fact that vanishes in by (1.2) and assertion (ii), inequality (3.45) yields

(3.46)

where is from (). Due to the condition

(3.47)

in (), there exists a positive constant such that for all the inequality

(3.48)

is fulfilled. The last inequality together with estimate (3.46) gives for . Consequently, for , , and assertion (iv) follows.

The following result gives the important property of for applying the Vitali convergent theorem in the proof of Theorem 4.1.

Lemma 3.5.

Let () and () hold. Let be a solution of problem (2.8), (1.2). Then, the sequence

(3.49)

is uniformly integrable on , that is, for each , there exists such that if and , then

(3.50)

Proof.

By Lemma 3.4 (iv), there exists such that for , the inequality holds. Now, we conclude from (2.5) and (2.6), from the properties of and given in , and finally from (3.29) that for and , the estimate

(3.51)

is fulfilled, where is a positive constant. Since the functions , , and () belong to the set by assumption (), in order to prove that is uniformly integrable on , it suffices to show that the sequences

(3.52)

are uniformly integrable on . Due to and for by (), this fact follows from [13, Criterion 11.10 (with and )].

4. The Main Result

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

Theorem 4.1.

Let () and () hold. Then, problem (1.1), (1.2) has a positive solution and

(4.1)

Proof.

Lemma 3.3 guarantees that problem (2.8), (1.2) has a solution . Consider the sequence . By Lemma 3.4, is bounded in ,

(4.2)

and fulfils estimate (3.29), where is a positive constant and . Furthermore, the sequence is uniformly integrable on by Lemma 3.5, and therefore, we deduce from the equality for a.e. that is equicontinuous on . Now, by the Arzelà-Ascoli theorem and the Bolzano-Weierstrass theorem, we may assume without loss of generality that is convergent in and is convergent in for . Let and (). Then and satisfies the boundary conditions (1.2). Letting in (3.29) and (4.2), we get (for )

(4.3)

Keeping in mind the definition of , we conclude from (4.3) that

(4.4)

Then, by the Vitali theorem, and

(4.5)

Letting in the equality

(4.6)

we get

(4.7)

As a result, and is a solution of (1.1). Consequently, 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

(4.8)

on , where , (that is, is essentially bounded and measurable on ) are nonnegative, for a.e. . If for and , for , then, by Theorem 4.1, the problem has a positive solution satisfying inequality (4.1).