Multiple Positive Solutions of Fourth-Order Impulsive Differential Equations with Integral Boundary Conditions and One-Dimensional -Laplacian

Abstract

By using the fixed point theory for completely continuous operator, this paper investigates the existence of positive solutions for a class of fourth-order impulsive boundary value problems with integral boundary conditions and one-dimensional -Laplacian. Moreover, we offer some interesting discussion of the associated boundary value problems. Upper and lower bounds for these positive solutions also are given, so our work is new.

1. Introduction

The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments. Processes with such a character arise naturally and often, especially in phenomena studied in physics, chemical technology, population dynamics, biotechnology, and economics. For an introduction of the basic theory of impulsive differential equations, see Lakshmikantham et al. [1]; for an overview of existing results and of recent research areas of impulsive differential equations, see Benchohra et al. [2]. The theory of impulsive differential equations has become an important area of investigation in recent years and is much richer than the corresponding theory of differential equations (see, e.g., [3–18] and references cited therein).

Moreover, the theory of boundary-value problems with integral boundary conditions for ordinary differential equations arises in different areas of applied mathematics and physics. For example, heat conduction, chemical engineering, underground water flow, thermo-elasticity, and plasma physics can be reduced to the nonlocal problems with integral boundary conditions. For boundary-value problems with integral boundary conditions and comments on their importance, we refer the reader to the papers by Gallardo [19], Karakostas and Tsamatos [20], Lomtatidze and Malaguti [21], and the references therein. For more information about the general theory of integral equations and their relation with boundary-value problems, we refer to the book of Corduneanu [22] and Agarwal and O'Regan [23].

On the other hand, boundary-value problems with integral boundary conditions constitute a very interesting and important class of problems. They include two, three, multipoint and nonlocal boundary-value problems as special cases. The existence and multiplicity of positive solutions for such problems have received a great deal of attention in the literature. To identify a few, we refer the reader to [24–46] and references therein. In particular, we would like to mention some results of Zhang et al. [34], Kang et al. [44], and Webb et al. [45]. In [34], Zhang et al. studied the following fourth-order boundary value problem with integral boundary conditions

(1.1)

where is a positive parameter, , is the zero element of , and . The authors investigated the multiplicity of positive solutions to problem (1.1) by using the fixed point index theory in cone for strict set contraction operator.

In [44], Kang et al. have improved and generalized the work of [34] by applying the fixed point theory in cone for a strict set contraction operator; they proved that there exist various results on the existence of positive solutions to a class of fourth-order singular boundary value problems with integral boundary conditions

(1.2)

where and may be singular at or ; are continuous and may be singular at , and ; , and , and , and are nonnegative, .

More recently, by using a unified approach, Webb et al. [45] considered the widely studied boundary conditions corresponding to clamped and hinged ends and many nonlocal boundary conditions and established excellent existence results for multiple positive solutions of fourth-order nonlinear equations which model deflections of an elastic beam

(1.3)

subject to various boundary conditions

(1.4)

where denotes a linear functional on given by involving a Stieltjes integral, and is a function of bounded variation.

At the same time, we notice that there has been a considerable attention on -Laplacian BVPs [18, 32, 35, 36, 38, 42] as -Laplacian appears in the study of flow through porous media (), nonlinear elasticity (), glaciology (), and so forth. Here, it is worth mentioning that Liu et al. [43] considered the following fourth-order four-point boundary value problem:

(1.5)

where , , and . By using upper and lower solution method, fixed-point theorems, and the properties of Green's function and , the authors give sufficient conditions for the existence of one positive solution.

Motivated by works mentioned above, in this paper, we consider the existence of positive solutions for a class of boundary value problems with integral boundary conditions of fourth-order impulsive differential equations:

(1.6)

Here is -Laplace operator, that is, , , (where is fixed positive integer) are fixed points with , where and represent the right-hand limit and left-hand limit of at , respectively, and is nonnegative.

For the case of , problem (1.6) reduces to the problem studied by Zhang et al. in [33]. By using the fixed point theorem in cone, the authors obtained some sufficient conditions for the existence and multiplicity of symmetric positive solutions for a class of -Laplacian fourth-order differential equations with integral boundary conditions.

For the case of , and , problem (1.6) is related to fourth-order two-points boundary value problem of ODE. Under this case, problem (1.6) has received considerable attention (see, e.g., [40–42] and references cited therein). Aftabizadeh [40] showed the existence of a solution to problem (1.6) under the restriction that is a bounded function. Bai and Wang [41] have applied the fixed point theorem and degree theory to establish existence, uniqueness, and multiplicity of positive solutions to problem (1.6). Ma and Wang [42] have proved that there exist at least two positive solutions by applying the existence of positive solutions under the fact that is either superlinear or sublinear on by employing the fixed point theorem of cone extension or compression.

Being directly inspired by [18, 34, 43], in the present paper, we consider some existence results for problem (1.6) in a specially constructed cone by using the fixed point theorem. The main features of this paper are as follows. Firstly, comparing with [39–43], we discuss the impulsive boundary value problem with integral boundary conditions, that is, problem (1.6) includes fourth-order two-, three-, multipoint, and nonlocal boundary value problems as special cases. Secondly, the conditions are weaker than those of [33, 34, 46], and we consider the case . Finally, comparing with [33, 34, 39–43, 46], upper and lower bounds for these positive solutions also are given. Hence, we improve and generalize the results of [33, 34, 39–43, 46] to some degree, and so, it is interesting and important to study the existence of positive solutions of problem (1.6).

The organization of this paper is as follows. We shall introduce some lemmas in the rest of this section. In Section 2, we provide some necessary background. In particular, we state some properties of the Green's function associated with problem (1.6). In Section 3, the main results will be stated and proved. Finally, in Section 4, we offer some interesting discussion of the associated problem (1.6).

To obtain positive solutions of problem (1.6), the following fixed point theorem in cones is fundamental, which can be found in [47, page 94].

Lemma 1.1.

Let and be two bounded open sets in Banach space , such that and . Let be a cone in and let operator be completely continuous. Suppose that one of the following two conditions is satisfied:

(a), and ;

(b), and .

Then, has at least one fixed point in .

2. Preliminaries

In order to define the solution of problem (1.6), we shall consider the following space.

Let , and

(2.1)

Then is a real Banach space with norm

(2.2)

where .

A function is called a solution of problem (1.6) if it satisfies (1.6).

To establish the existence of multiple positive solutions in of problem (1.6), let us list the following assumptions:

;

with

(2.3)

Write

(2.4)

From , it is clear that .

We shall reduce problem (1.6) to an integral equation. To this goal, firstly by means of the transformation

(2.5)

we convert problem (1.6) into

(2.6)

(2.7)

Lemma 2.1.

Assume that and hold. Then problem (2.6) has a unique solution given by

(2.8)

where

(2.9)

(2.10)

Proof.

The proof follows by routine calculations.

Write . Then from (2.9) and (2.10), we can prove that have the following properties.

Proposition 2.2.

If holds, then we have

(2.11)

Proposition 2.3.

For , we have

(2.12)

Proposition 2.4.

If holds, then for , we have

(2.13)

where

(2.14)

Proof.

By (2.6) and (2.12), we have

(2.15)

On the other hand, noticing , we obtain

(2.16)

The proof of Proposition 2.4 is complete.

Remark 2.5.

From (2.9) and (2.13), we can obtain that

(2.17)

Lemma 2.6.

If and hold, then problem (2.7) has a unique solution and can be expressed in the following form:

(2.18)

where

(2.19)

and is defined in (2.10).

Proof.

First suppose that is a solution of problem (2.7).

If it is easy to see by integration of problem (2.7) that

(2.20)

If then integrate from to ,

(2.21)

Similarly, if we have

(2.22)

Integrating again, we can get

(2.23)

Letting in (2.23), we find

(2.24)

Substituting and (2.24) into (2.23), we obtain

(2.25)

where

(2.26)

Therefore, we have

(2.27)

Let

(2.28)

Then,

(2.29)

and the proof of sufficient is complete.

Conversely, if is a solution of (2.18).

Direct differentiation of (2.18) implies, for ,

(2.30)

Evidently,

(2.31)

The Lemma is proved.

Remark 2.7.

From (2.19), we can prove that the properties of are similar to that of .

Suppose that is a solution of problem (1.6). Then from Lemmas 2.6 and 2.1, we have

(2.32)

For the sake of applying Lemma 1.1, we construct a cone in via

(2.33)

where

(2.34)

It is easy to see that is a closed convex cone of .

Define an operator by

(2.35)

From (2.35), we know that is a solution of problem (1.6) if and only if is a fixed point of operator .

Definition 2.8 (see [1]).

The set is said to be quasi-equicontinuous in if for any there exist such that if then

(2.36)

We present the following result about relatively compact sets in which is a consequence of the Arzela-Ascoli Theorem. The reader can find its proof partially in [1].

Lemma 2.9.

is relatively compact if and only if is bounded and quasi-equicontinuous on .

Write

(2.37)

where .

Lemma 2.10.

Suppose that and hold. Then and is completely continuous.

Proof.

For , it is clear that , and

(2.38)

From (2.35) and Remark 2.5, we obtain the following cases.

Case 1.

if , noticing , then we have

(2.39)

Case 2.

if , noticing and , then we have

(2.40)

Therefore, , that is, . Also, we have since . Hence we have .

Next, we prove that is completely continuous.

It is obvious that is continuous. Now we prove is relatively compact.

Let be a bounded set. Then, for all , we have

(2.41)

Therefore is uniformly bounded.

On the other hand, for all with , we have

(2.42)

and by the continuity of , we have

(2.43)

and then is quasi-equicontinuous. It follows that is relatively compact on by Lemma 2.9. So is completely continuous.

3. Main Results

In this section, we apply Lemma 1.1 to establish the existence of positive solutions of problem (1.6). We begin by introducing the following conditions on and .

There exist numbers such that

(3.1)

(3.2)

where is defined in (2.4), are defined in (2.14), respectively, and write

(3.3)

where denotes or

Theorem 3.1.

Assume that hold. Then problem (1.6) has at least one positive solution with

(3.4)

Proof.

Let be the cone preserving, completely continuous operator that was defined by (2.35). For with , (2.13), (2.19), and (3.1) imply

(3.5)

where

(3.6)

Now if we let , then (3.5) shows that

(3.7)

Further, let

(3.8)

Then, and implies

(3.9)

that is,

(3.10)

Hence, for all . Therefore, for all , (3.2) implies

(3.11)

that is, implies

(3.12)

Applying (b) of Lemma 1.1 to (3.7) and (3.12) yields that has a fixed point with . Hence, since for we have , it follows that (3.4) holds. This and Lemma 2.9 complete the proof.

As a special case of Theorem 3.1, we can prove the following results.

Corollary 3.2.

Assume that and hold. If , and , then, for being sufficiently small and being sufficiently large, BVP (1.6) has at least one positive solution with property (3.4).

Proof.

The proof is similar to that of Theorem of [6].

In Theorem 3.3, we assume the following condition on and .

There exist numbers such that

(3.13)

(3.14)

(3.15)

where are defined in (2.14), and write

(3.16)

Theorem 3.3.

Assume that , and hold. Then problem (1.6) has at least one positive solution with

(3.17)

Proof.

For with , (3.13) implies

(3.18)

that is, implies

(3.19)

Next, we turn to (3.14) and (3.15). From (3.14), (3.15), and (3.16), we have

(3.20)

Let

(3.21)

Thus, for , we have

(3.22)

(3.23)

Then, (3.22) and (3.23) imply

(3.24)

Applying (a) of Lemma 1.1 to (3.19) and (3.24) yields that has a fixed point with . Hence, since for we have , it follows that (3.17) holds. This and Lemma 2.9 complete the proof.

As a special case of Theorem 3.3, we can prove the following results.

Corollary 3.4.

Assume that and hold. If and ; then, for being sufficiently small and being sufficiently large, BVP (1.6) has at least one positive solution with property (3.17).

Proof.

The proof is similar to that of Theorem of [6].

Theorem 3.5.

Assume that (3.1) of and (3.14) and (3.15) of hold. In addition, letting and satisfy the following condition:

There is a such that and implies

(3.25)

Then, problem (1.6) has at least two positive solutions and with

(3.26)

where and satisfy

(3.27)

Proof.

If (3.1) of holds, similar to the proof of (3.7), we can prove that

(3.28)

If (3.14) and (3.15) of hold, similar to the proof of (3.23), we have

(3.29)

Finally, we show that

(3.30)

In fact, for with then by (2.18), we have

(3.31)

and it follows from that

(3.32)

which implies that (3.30) holds.

Applying Lemma 1.1 to (3.28), (3.29), and (3.30) yields that has two fixed point with , and. Hence, since for we have , it follows that (3.26) holds. This and Lemma 2.9 complete the proof.

Remark 3.6.

Similar to the proof of that of [5], we can prove that problem (1.6) can be generalized to obtain many positive solutions.

4. Discussion

In this section, we offer some interesting discussions associated with problem (1.6).

Discussion.

Generally, it is difficult to obtain the upper and lower bounds of positive solutions for nonlinear higher-order boundary value problems (see, e.g., [33, 34, 39–43, 46, 48, 49] and their references).

For example, we consider the following problems:

(4.1)

Here is -Laplace operator, that is, , , (where is fixed positive integer) are fixed points with , where and represent the right-hand limit and left-hand limit of at , respectively, and is nonnegative.

By means of the transformation (2.5), we can convert problem (4.1) into

(4.2)

(4.3)

Using the similar proof of that of Lemmas 2.1 and 2.6, we can obtain the following results. In addition, if we replace by in , respectively, then we obtain , where

(4.4)

Lemma 4.1.

If and hold, then BVP (4.2) has a unique solution and can be expressed in the form

(4.5)

where

(4.6)

is defined in (2.10).

Lemma 4.2.

If and hold, then BVP (4.3) has a unique solution and can be expressed in the form

(4.7)

where

(4.8)

It is not difficult to prove that and have the similar properties to that of and . But for , and have no property (2.13). In fact, if , then we can prove that and have the following properties.

Proposition 4.3.

If holds, then for , we have

(4.9)

(4.10)

where

(4.11)

Proof.

We only consider (4.9). By (4.3), we have

(4.12)

On the other hand, noticing , we obtain

(4.13)

Similarly, we can prove that (4.10) holds, too.

Remark 4.4.

Since , .

From (4.9) and (4.10), we can only define a cone by

(4.14)

which implies that we cannot obtain the lower bounds for the positive solutions of problem (4.1).

5. Example

To illustrate how our main results can be used in practice, we present an example.

Example 5.1.

Consider the following boundary value problem:

(5.1)

where , and

(5.2)

Conclusion.

Equation(5.1) has at least one positive solution for with

(5.3)

Proof.

By simple computation, we have . Select then for , we have

(5.4)

By Theorem 3.1, (5.1) has a positive solution with .