By using the fixed point theory for completely continuous operator, this paper investigates the existence of positive solutions for a class of fourthorder impulsive boundary value problems with integral boundary conditions and onedimensional 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 boundaryvalue 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, thermoelasticity, and plasma physics can be reduced to the nonlocal problems with integral boundary conditions. For boundaryvalue 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 boundaryvalue problems, we refer to the book of Corduneanu [22] and Agarwal and O'Regan [23].
On the other hand, boundaryvalue problems with integral boundary conditions constitute a very interesting and important class of problems. They include two, three, multipoint and nonlocal boundaryvalue 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 fourthorder boundary value problem with integral boundary conditions
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 fourthorder singular boundary value problems with integral boundary conditions
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 fourthorder nonlinear equations which model deflections of an elastic beam
subject to various boundary conditions
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 fourthorder fourpoint boundary value problem:
where , , and . By using upper and lower solution method, fixedpoint 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 fourthorder impulsive differential equations:
Here is Laplace operator, that is, , , (where is fixed positive integer) are fixed points with , where and represent the righthand limit and lefthand 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 fourthorder differential equations with integral boundary conditions.
For the case of , and , problem (1.6) is related to fourthorder twopoints 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 fourthorder 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
Then is a real Banach space with norm
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
Write
From , it is clear that .
We shall reduce problem (1.6) to an integral equation. To this goal, firstly by means of the transformation
we convert problem (1.6) into
Lemma 2.1.
Assume that and hold. Then problem (2.6) has a unique solution given by
where
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
Proposition 2.3.
For , we have
Proposition 2.4.
If holds, then for , we have
where
Proof.
By (2.6) and (2.12), we have
On the other hand, noticing , we obtain
The proof of Proposition 2.4 is complete.
Remark 2.5.
From (2.9) and (2.13), we can obtain that
Lemma 2.6.
If and hold, then problem (2.7) has a unique solution and can be expressed in the following form:
where
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
If then integrate from to ,
Similarly, if we have
Integrating again, we can get
Letting in (2.23), we find
Substituting and (2.24) into (2.23), we obtain
where
Therefore, we have
Let
Then,
and the proof of sufficient is complete.
Conversely, if is a solution of (2.18).
Direct differentiation of (2.18) implies, for ,
Evidently,
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
For the sake of applying Lemma 1.1, we construct a cone in via
where
It is easy to see that is a closed convex cone of .
Define an operator by
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 quasiequicontinuous in if for any there exist such that if then
We present the following result about relatively compact sets in which is a consequence of the ArzelaAscoli Theorem. The reader can find its proof partially in [1].
Lemma 2.9.
is relatively compact if and only if is bounded and quasiequicontinuous on .
Write
where .
Lemma 2.10.
Suppose that and hold. Then and is completely continuous.
Proof.
For , it is clear that , and
From (2.35) and Remark 2.5, we obtain the following cases.
Case 1.
if , noticing , then we have
Case 2.
if , noticing and , then we have
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
Therefore is uniformly bounded.
On the other hand, for all with , we have
and by the continuity of , we have
and then is quasiequicontinuous. 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
where is defined in (2.4), are defined in (2.14), respectively, and write
where denotes or
Theorem 3.1.
Assume that hold. Then problem (1.6) has at least one positive solution with
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
where
Now if we let , then (3.5) shows that
Further, let
Then, and implies
that is,
Hence, for all . Therefore, for all , (3.2) implies
that is, implies
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
where are defined in (2.14), and write
Theorem 3.3.
Assume that , and hold. Then problem (1.6) has at least one positive solution with
Proof.
For with , (3.13) implies
that is, implies
Next, we turn to (3.14) and (3.15). From (3.14), (3.15), and (3.16), we have
Let
Thus, for , we have
Then, (3.22) and (3.23) imply
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
Then, problem (1.6) has at least two positive solutions and with
where and satisfy
Proof.
If (3.1) of holds, similar to the proof of (3.7), we can prove that
If (3.14) and (3.15) of hold, similar to the proof of (3.23), we have
Finally, we show that
In fact, for with then by (2.18), we have
and it follows from that
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 higherorder boundary value problems (see, e.g., [33, 34, 39–43, 46, 48, 49] and their references).
For example, we consider the following problems:
Here is Laplace operator, that is, , , (where is fixed positive integer) are fixed points with , where and represent the righthand limit and lefthand limit of at , respectively, and is nonnegative.
By means of the transformation (2.5), we can convert problem (4.1) into
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
Lemma 4.1.
If and hold, then BVP (4.2) has a unique solution and can be expressed in the form
where
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
where
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
where
Proof.
We only consider (4.9). By (4.3), we have
On the other hand, noticing , we obtain
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
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:
where , and
Conclusion.
Equation(5.1) has at least one positive solution for with
Proof.
By simple computation, we have . Select then for , we have
By Theorem 3.1, (5.1) has a positive solution with .
Acknowledgments
The authors thank the referee for his/her careful reading of the manuscript and useful suggestions. This work is sponsored by the Funding Project for Academic Human Resources Development in Institutions of Higher Learning under the Jurisdiction of Beijing Municipality (PHR201008430), the Scientific Research Common Program of Beijing Municipal Commission of Education (KM201010772018), and Beijing Municipal Education Commission (71D0911003).
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