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

We consider the existence of countably many positive solutions for nonlinear th-order three-point boundary value problem , , , , , where , for some and has countably many singularities in . The associated Green's function for the th-order three-point boundary value problem is first given, and growth conditions are imposed on nonlinearity which yield the existence of countably many positive solutions by using the Krasnosel'skii fixed point theorem and Leggett-Williams fixed point theorem for operators on a cone.

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:

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:

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

with one of the following -point boundary conditions:

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

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

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

for all and

Definition 2.3.

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

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

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

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

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

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

Let and , where and . Then and, moreover

Let and . Then and

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

Lemma 3.1 (see [15]).

For the boundary value problem

has a unique solution

Lemma 3.2 (see [15]).

The Green's function for the boundary value problem

is given by

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

where

Lemma 3.4.

Suppose then for the boundary value problem

has a unique solution

Proof.

The general solution of can be written as

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

By solving the above equations, we get

Therefore, (3.7) has a unique solution

Lemma 3.5.

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

is given by

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

where

Proof.

From Lemma 3.3 and (3.14), we get

From Lemma 3.3 and (3.14), we have

Next, we prove that (3.15) holds.

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

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

Define the operator by

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

for all . Thus

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

Theorem 4.1.

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

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

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

Fixed and let . For , we have

By condition , we get

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

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

We observe here that, for each , .

For convenience, we denote

Theorem 4.2.

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

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

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

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,

If , then , for . By condition , we get

Therefore, condition of Theorem 2.5 is satisfied.

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

If and , then

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

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

where is defined by [15, Example ] and ,

We notice that , , , .

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

It follows from a direct calculation that

so

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

and satisfies the following growth conditions:

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

where is defined by [15, Example ] and ,

We notice that , , , .

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

It follows from a direct calculation that

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

and satisfies the following growth conditions:

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

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.

The project is supported by the Natural Science Foundation of Hebei Province (A2009000664), the Foundation of Hebei Education Department (2008153), the Foundation of Hebei University of Science and Technology (XL2006040), and the National Natural Science Foundation of PR China (10971045).

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