This paper deals with the positive solutions of a fourth-order boundary value problem in Banach spaces. By using the fixed-point theorem of strict-set-contractions, some sufficient conditions for the existence of at least one or two positive solutions to a fourth-order boundary value problem in Banach spaces are obtained. An example illustrating the main results is given.
Keywords:-positive operator; boundary value problem; positive solution; fixed-point theorem; measure of noncompactness
In this paper, we consider the existence of multiple positive solutions for the fourth-order ordinary differential equation boundary value problem in a Banach space E
where is continuous, , θ is the zero element of E. This problem models deformations of an elastic beam in equilibrium state, whose two ends are simply supported. Owing to its importance in physics, the existence of this problem in a scalar space has been studied by many authors using Schauder’s fixed-point theorem and the Leray-Schauder degree theory (see [1-5] and references therein). On the other hand, the theory of ordinary differential equations (ODE) in abstract spaces has become an important branch of mathematics in last thirty years because of its application in partial differential equations and ODEs in appropriately infinite dimensional spaces (see, for example, [6-8]). For an abstract space, it is here worth mentioning that Guo and Lakshmikantham  discussed the multiple solutions of two-point boundary value problems of ordinary differential equations in a Banach space. Recently, Liu  obtained the sufficient condition for multiple positive solutions to fourth-order singular boundary value problems in an abstract space. In , by using the fixed-point index theory in a cone for a strict-set-contraction operator, the authors have studied the existence of multiple positive solutions for the singular boundary value problems with an integral boundary condition.
However, the above works in a Banach space were carried out under the assumption that the second-order derivative is not involved explicitly in the nonlinear term f. This is because the presence of second-order derivatives in the nonlinear function f will make the study extremely difficult. As a result, the goal of this paper is to fill up the gap, that is, to investigate the existence of solutions for fourth-order boundary value problems of (1.1) in which the nonlinear function f contains second-order derivatives, i.e., f depends on .
The main features of this paper are as follows. First, we discuss the existence results in an abstract space E, not . Secondly, we will consider the nonlinear term which is more extensive than the nonlinear term of [10,11]. Finally, the technique for dealing with fourth-order BVP is completely different from [10,11]. Hence, we improve and generalize the results of [10,11] to some degree, and so, it is interesting and important to study the existence of positive solutions of BVP (1.1). The arguments are based upon the -positive operator and the fixed-point theorem in a cone for a strict-set-contraction operator.
The paper is organized as follows. In Section 2, we present some preliminaries and lemmas that will be used to prove our main results. In Section 3, various conditions on the existence of positive solutions to BVP (1.1) are discussed. In Section 4, we give an example to demonstrate our result.
Let the real Banach space E with norm be partially ordered by a cone P of E, i.e., if and only if . P is said to be normal if there exists a positive constant N such that implies . We consider problem (1.1) in . Evidently, is a Banach space with norm and is a cone of the Banach space . In the following, is called a solution of problem (1.1) if it satisfies (1.1). x is a positive solution of (1.1) if, in addition, x is nonnegative and nontrivial, i.e., and for .
For a bounded set V in a Banach space, we denote by the Kuratowski measure of noncompactness (see [6-8] for further understanding). In this paper, we denote by the Kuratowski measure of noncompactness of a bounded set in E and in .
The key tool in our approach is the following fixed-point theorem of strict-set-contractions:
Lemma 2.2LetTbe-positive on a coneK. IfTis completely continuous, then, the spectral radius ofT, is the unique positive eigenvalue ofTwith its eigenfunction in K. Moreover, ifholds with, then for an arbitrary non-zero () the elementsuandare incomparable
For convenience, let us list the following assumptions:
Using the above transformation and (2.2), BVP (1.1) becomes
From (2.3) and (2.4), we have
Now, define an operator A on Q by
The following Lemma 2.3 can be easily obtained.
Since f is uniformly continuous and bounded on , we see from (2.5) that A is continuous and bounded on . Let , according to (2.5), it is easy to show that the functions are uniformly bounded and equicontinuous, and so in  we have
Using the obvious formula
From the fact of , we know
It follows from (2.6) and (2.7) that
3 Main results
In the following, we prove that W is bounded.
Since is the first eigenvalue of T, by (), the first eigenvalue of , . Therefore, by , the inverse operator exists and
It is easy to see that
By , we have . Let , by Gelfand’s formula, we have . Let as .
Now, we assert that
It is easy to see that
In fact, implies for , and consequently, in contradiction to . Now, notice that is a -positive operator with . Then by Lemma 2.2, we have for some , where is the positive eigenfunction of corresponding to . This together with implies that
By Lemma 2.4, A is a strict set contraction on . Observing (3.5) and (3.8) and using Theorem 2.1, we see that A has a fixed point on . This together with Lemma 2.3 implies that BVP (1.1) has at least one positive solution. □
On the other hand, it is easy to see that
It follows from (3.12) and (3.13) that
a contradiction. Thus (3.11) is true.
By Lemma 2.4, A is a strict set contraction on , and also on . Observing (3.9), (3.10), (3.11) and applying, respectively, Theorem 2.1 to A, and , we assert that there exist and such that and and, by Lemma 2.3 and (3.11), , are positive solutions of BVP (1.1). □
On the other hand, it is easy to see that
which is a contradiction. Hence, (3.16) holds.
By Lemma 2.4, A is a strict set contraction on and also on . Observing (3.14), (3.15), (3.16) and applying, respectively, Theorem 2.1 to A, and , we assert that there exist and such that and and, by Lemma 2.3 and (3.16), , are positive solutions of BVP (1.1). □
4 One example
Now, we consider an example to illustrate our results.
Example 4.1 Consider the following boundary value problem of the finite system of scalar differential equations:
The authors declare that they have no competing interests.
All authors typed, read and approved the final manuscript.
The authors would like to thank the referees for carefully reading this article and making valuable comments and suggestions. This work is supported by the Foundation items: NSFC (10971179), NSF (BS2010SF023, BS2012SF022) of Shandong Province.
Aftabizadeh, AR: Existence and uniqueness theorems for fourth-order boundary value problems. J. Math. Anal. Appl.. 116, 415–426 (1986). Publisher Full Text
Yang, Y: Fourth-order two-point boundary value problems. Proc. Am. Math. Soc.. 104, 175–180 (1988). Publisher Full Text
Cui, Y, Zou, Y: Existence and uniqueness theorems for fourth-order singular boundary value problems. Comput. Math. Appl.. 58, 1449–1456 (2009). Publisher Full Text
Guo, D, Lakshmikantham, V: Multiple solutions of two-point boundary value problem of ordinary differential equations in Banach space. J. Math. Anal. Appl.. 129, 211–222 (1988). Publisher Full Text
Zhang, X, Feng, M, Ge, W: Existence results for nonlinear boundary value problems with integral boundary conditions in Banach spaces. Nonlinear Anal.. 69, 3310–3321 (2008). Publisher Full Text