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

where satisfies the local Carathéodory function on () with

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

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

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

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).

() and there exists such that

for a.e. and each .

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

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

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.

Let be from (1.1). For , define , , and by the formulas

Let . Chose and put

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

for , and

Then, under condition (), and

Condition () gives

We investigate the regular differential equation

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

Let and denote by the Green function of the problem

Then,

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

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

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

For and , the inequalities

hold.

Let and let be a solution of the differential equation

satisfying the Lidstone boundary conditions

It follows from the definition of the Green function that

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

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

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

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

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

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

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))

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

Hence, , and so

Next, we deduce from the relation

and from (2.7) that

Therefore,

where . Since for , we have

The last inequality together with (3.21) gives

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

hold. By (), there exists such that

and therefore,

Let

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

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

for ,

(iv)the sequence is bounded in .

Proof.

Let us choose an arbitrary . By (2.5),

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

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),

Since

and, by [13, Lemma 6.2],

we have

Furthermore,

and (cf. (3.32) for )

since on by assertion (ii). Let

where

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

and have

In particular,

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

where

Notice that by (). Consequently,

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

where is from (). Due to the condition

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

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

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

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

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

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

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

Proof.

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

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 )

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

Then, by the Vitali theorem, and

Letting in the equality

we get

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

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).