The Existence of Countably Many Positive Solutions for Nonlinear th-Order Three-Point Boundary Value Problems

The existence of positive solutions for nonlinear second-order and higher-order multipoint boundary value problems has been studied by several authors, for example, see [1–12] and the references therein. However, there are a few papers dealing with the existence of positive solutions for the th-order multipoint boundary value problems with infinitely many singularities. Hao et al. [13] discussed the existence and multiplicity of positive solutions for the following th-order nonlinear singular boundary value problems:

(1.1)

where , , may be singular at and/or . Hao et al. established the existence of at least two positive solution for the boundary value problems if is either superlinear or sublinear by applying the Krasnosel'skii-Guo theorem on cone expansion and compression.

In [14], Kaufmann and Kosmatov showed that there exist countably many positive solutions for the two-point boundary value problems with infinitely many singularities of following form:

(1.2)

where for some and has countably many singularities in .

In [15], Ji and Guo proved the existence of countably many positive solutions for the th-order ordinary differential equation

(1.3)

with one of the following -point boundary conditions:

(1.4)

where , ( ), , ( , ), , for some and has countably many singularities in .

Motivated by the result of [13–15], in this paper we are interested in the existence of countably many positive solutions for nonlinear th-order three-point boundary value problem

(1.5)

where , , , , , ( , ), , for some and has countably many singularities in . We show that the problem (1.5) has countably many solutions if and satisfy some suitable conditions. Our approach is based on the Krasnosel'skii fixed point theorem and Leggett-Williams fixed point theorem in cones.

Suppose that the following conditions are satisfied.

There exists a sequence such that , , , and for all

There exists such that for all .

Assuming that satisfies the conditions - (we cite [15, Example ] to verify existence of ) and imposing growth conditions on the nonlinearity , it will be shown that problem (1.5) has infinitely many solutions.

The paper is organized as follows. In Section 2, we provide some necessary background material such as the Krasnosel'skii fixed-point theorem and Leggett-Williams fixed point theorem in cones. In Section 3, the associated Green's function for the th-order three-point boundary value problem is first given and we also look at some properties of the Green's function associated with problem (1.5). In Section 4, we prove the existence of countably many positive solutions for problem (1.5) under suitable conditions on and . In Section 5, we give two simple examples to illustrate the applications of obtained results.

Definition 2.1.

Let be a Banach space over . A nonempty convex closed set is said to be a cone provided that

(i) for all and for all ;

(ii) implies .

Definition 2.2.

The map is said to be a nonnegative continuous concave functional on provided that is continuous and

(2.1)

for all and Similarly, we say that the map is a nonnegative continuous convex functional on provided that is continuous and

(2.2)

for all and

Definition 2.3.

Let be given and let be a nonnegative continuous concave functional on . Define the convex sets and by

(2.3)

The following Krasnosel'skii fixed point theorem and Leggett-Williams fixed point theorem play an important role in this paper.

Theorem 2.4 ([16], Krasnosel'skii fixed point theorem).

Let be a Banach space and let be a cone. Assume that , are bounded open subsets of such that . Suppose that

(2.4)

is a completely continuous operator such that, either

(i) , and , or

(ii) , and .

Then has a fixed point in .

Theorem 2.5 ([17], Leggett-Williams fixed point theorem).

Let be a completely continuous operator and let be a nonnegative continuous concave functional on such that for all . Suppose there exist such that

, and  for ,

 for ,

 for , with .

Then has at least three fixed points , and such that

(2.5)

In order to establish some of the norm inequalities in Theorems 2.4 and 2.5 we will need Holder's inequality. We use standard notation of for the space of measurable functions such that

(2.6)

where the integral is understood in the Lebesgue sense. The norm on , is defined by

(2.7)

Theorem 2.6 ([18], Holder's inequality).

Let and , where and . Then and, moreover

(2.8)

Let and . Then and

(2.9)

To prove the main results, we need the following lemmas.

Lemma 3.1 (see [15]).

For the boundary value problem

(3.1)

has a unique solution

(3.2)

Lemma 3.2 (see [15]).

The Green's function for the boundary value problem

(3.3)

is given by

(3.4)

Lemma 3.3 (see [15]).

The Green's function defined by (3.4) satisfies that

(i) is continuous on ;

(ii) for all and there exists a constant for any such that

(3.5)

where

(3.6)

Lemma 3.4.

Suppose then for the boundary value problem

(3.7)

has a unique solution

(3.8)

Proof.

The general solution of can be written as

(3.9)

Since for , we get for . Now we solve for by and , it follows that

(3.10)

By solving the above equations, we get

(3.11)

Therefore, (3.7) has a unique solution

(3.12)

Lemma 3.5.

Suppose , the Green's function for the boundary value problem

(3.13)

is given by

(3.14)

where is defined by (3.4).

We omit the proof as it is immediate from Lemma 3.4 and (3.4).

Lemma 3.6.

Suppose , the Green's function defined by (3.14) satisfies that

(i) is continuous on ;

(ii) for all and there exists a constant for any such that

(3.15)

where

(3.16)

Proof.

From Lemma 3.3 and (3.14), we get

(3.17)

From Lemma 3.3 and (3.14), we have

(3.18)

Next, we prove that (3.15) holds.

From Lemma 3.3 and (3.14), for , we have

(3.19)

for all , where , .

We use inequality (3.15) to define our cones. Let , then is a Banach space with the norm . For a fixed , define the cone by

(3.20)

Define the operator by

(3.21)

Obviously, is a solution of (1.5) if and only if is a fixed point of operator .

Theorems 2.4 and 2.5 require the operator to be completely continuous and cone preserving. If is continuous and compact, then it is completely continuous. The next lemma shows that for and that is continuous and compact.

Lemma 3.7.

The operator is completely continuous and for each .

Proof.

Fix . Since for all , and since for all , then for all .

Let , by (3.15) and (3.21) we have

(3.22)

for all . Thus

(3.23)

Clearly operator (3.21) is continuous. By the Arzela-Ascoli theorem is compact. Hence, the operator is completely continuous and the proof is complete.

In this section we present that problem (1.5) has countably many solutions if and satisfy some suitable conditions.

For convenience, we denote

(4.1)

Theorem 4.1.

Suppose conditions and hold, let be such that Let and be such that

(4.2)

where , . Furthermore, for each natural number , assume that satisfies the following two growth conditions:

 for all ,

 for all .

Then problem (1.5) has countably many positive solutions such that for each

Proof.

Consider the sequences and of open subsets of defined by

(4.3)

Let be as in the hypothesis and note that for all . For each , define the cone by

(4.4)

Fixed and let . For , we have

(4.5)

By condition , we get

(4.6)

Now let , then for all . By condition , we get

(4.7)

It is obvious that . Therefore, by Theorem 2.4, the operator has at least one fixed point such that . Since was arbitrary, Theorem 4.1 is completed.

Let is defined by Theorem 4.1. We define the nonnegative continuous concave functionals on by

(4.8)

We observe here that, for each , .

For convenience, we denote

(4.9)

Theorem 4.2.

Suppose conditions and hold, let be such that Let , , and be such that

(4.10)

where , . Furthermore,  for each natural number , assume that satisfies the following growth conditions:

 for all ,

 for all ,

 for all .

Then problem (1.5) has three infinite families of solutions , and such that

(4.11)

for each

Proof.

We note first that is completely continuous operator. If , then from properties of , , and by Lemma 3.7, . Consequently, .

If , then , and by condition , we have

(4.12)

Therefore, . Standard applications of Arzela-Ascoli theorem imply that is completely continuous operator.

In a completely analogous argument, condition implies that condition of Theorem 2.5 is satisfied.

We now show that condition of Theorem 2.5 is satisfied. Clearly,

(4.13)

If , then , for . By condition , we get

(4.14)

Therefore, condition of Theorem 2.5 is satisfied.

Finally, we show that condition of Theorem 2.5 is also satisfied.

If and , then

(4.15)

Therefore, condition is also satisfied. By Theorem 2.5, There exist three infinite families of solutions , , and for problem (1.5) such that

(4.16)

for each Thus, Theorem 4.2 is completed.

In this section, we cite an example (see [15]) to verify existence of , and two simple examples are presented to illustrate the applications for obtained conclusion of Theorems 4.1 and 4.2.

Example 5.1.

As an example of problem (1.5), we mention the boundary value problem

(5.1)

where is defined by [15, Example ] and ,

(5.2)

We notice that , , , .

If we take , , , then , and = = min , =

It follows from a direct calculation that

(5.3)

so

(5.4)

In addition, if we take , , , , , then

(5.5)

and satisfies the following growth conditions:

(5.6)

Then all the conditions of Theorem 4.1 are satisfied. Therefore, by Theorem 4.1 we know that problem (5.1) has countably many positive solutions such that for each

Example 5.2.

As another example of problem (1.5), we mention the boundary value problem

(5.7)

where is defined by [15, Example ] and ,

(5.8)

We notice that , , , .

If we take = , = , = , = then , and = , = , =

It follows from a direct calculation that

(5.9)

In addition, if we take , , , , , , then

(5.10)

and satisfies the following growth conditions:

(5.11)

Then all the conditions of Theorem 4.2 are satisfied. Therefore, by Theorem 4.2 we know that problem (5.7) has countably many positive solutions such that

(5.12)

for each

Remark 5.3.

In [8–12], the existence of solutions for local or nonlocal boundary value problems of higher-order nonlinear ordinary (fractional) differential equations that has been treated did not discuss problems with singularities. In [13], the singularity only allowed to appear at and/or , the existence and multiplicity of positive solutions were asserted under suitable conditions on . Although, [14, 15] seem to have considered the existence of countably many positive solutions for the second-order and higher-order boundary value problems with infinitely many singularities in . However, in [15], only the boundary conditions or have been considered. It is clear that the boundary conditions of Examples 5.1 and 5.2 are and . Hence, we generalize second-order and higher-order multipoint boundary value problem.