Nonlinear Analysis: Theory, Methods & Applications
Positive solutions of a nonlinear elastic beam equation rigidly fastened on the left and simply supported on the right
Introduction
The purpose of this paper is to consider the existence of multiple positive solutions for the following fourth-order two-point boundary value problem:
Here the function is called a positive solution of problem (P) if is a solution of (P) and .
Fourth-order two-point boundary value problems appear in beam analysis; see [1], [2] for more details. The deflection of an elastic beam rigidly fastened on the left and simply supported on the right leads to the problem (P). If is continuous, the existence of a single solution of (P) has been studied by several authors; for example, see [2], [3], [4], [5], [6]. In particular, Agarwal [3] established the following local existence theorem:
Theorem 1.1 Assume is continuous and there exists a positive number such thatThen problem (P)has at least one solution .
The local existence theorem shows that the problem (P) has at least one solution provided the “height” of the nonlinear term is appropriate on the bounded set .
Applying the localization idea and the Krasnosel’skii fixed point theorem of cone expansion–compression type, we have proved the existence of multiple positive solutions for some fourth-order boundary value problems with other boundary conditions, for example, the problem which describes deflection of a beam simply supported at both ends (see [7], [8], [9], [10]), the problem which describes deflection of a beam rigidly fastened at both ends (see [11], [12], [13]), and the problem which describes the equilibrium state of an elastic beam with periodic boundary condition (see [14]). In these studies the nonlinear terms are continuous functions.
In most real problems, only the positive solution is significant. To the best of our knowledge, the existence of multiple positive solutions for the problem (P) has not been considered by any authors. In this paper, we will study the question.
Moreover, we will allow that nonlinear term to be singular. In fact, we require only that the nonlinear term is locally continuous. To be more exact, the nonlinear term satisfies one of the following conditions with singularity:
(C1) is continuous, where and .
(C2) is continuous, where where is a closed subset.
The condition (C2) implies that the nonlinear term may be singular on the set . The condition (C1) implies that may be singular on a certain closed subset of the set .
A simple result is that, if there exists a closed subset such that is continuous, then is continuous for any ; if is continuous, then is continuous for any closed subset .
In this paper, we will improve the localization method developed in papers [7], [8], [9], [10], [11], [12], [13], [14] and apply the method to study the existence of multiple positive solutions of (P). The main ingredient is the Guo–Krasnosel’skii fixed point theorem of cone expansion–compression type. By constructing control functions concerned with the nonlinear term and considering heights of the nonlinear term on some bounded sets, we will prove a local existence theorem, that is Theorem 3.1, under the condition (C1). After that, we will establish the existence of positive solutions, where is an arbitrary positive integer. In Section 4, we will study the existence and multiplicity of positive solutions by considering the growth rates of the nonlinear term at and under the condition (C2). Finally, we will illustrate the physical interest of this work with an example in Section 5.
In this paper, all results are derived from Theorem 3.1. Therefore, Theorem 3.1 is a basic existence criterion of positive solution for the problem (P). On the other hand, all results are new even if is continuous.
Section snippets
Preliminaries
From [3], the Green function of the homogeneous linear problem has the form Obviously, .
Let and . It is easy to see that
Lemma 2.1 .
Proof Since and , the inequalities hold for or . If and , then
Existence and multiplicity under condition (C1)
We obtain the following local existence theorems.
Theorem 3.1 Assume that There exist two positive numbers such that is continuous. One of the following conditions is satisfied: . .
Then problem (P)has at least one positive solution satisfying .
Proof We consider only the case (i). The proof for case (ii) is similar. Define If , then and . Since , we have It follows
Existence and multiplicity under condition (C2)
In this section, we verify the existence and multiplicity of positive solutions by considering the growth rates of the nonlinear term at or .
Theorem 4.1 Assume that There exists a closed set such that is continuous. There exist such that (in particular, ). There exists such that .
Then problem (P)has at least one positive solution satisfying .
Proof By the condition (f2), there exist
An interesting example with a physical interpretation
Example 5.1 Let . Consider the fourth-order boundary value problem Since the nonlinear term is superlinear. Since we assert that the problem (P3) has a positive solution satisfying by Theorem 3.1.
References (15)
Existence of solutions and/or positive solutions to a semipositone elastic beam equation
Nonlinear Anal. TMA
(2007)Positive solutions for eigenvalue problems of fourth-order elastic beam equations
Appl. Math. Letters
(2004)Existence, multiplicity and infinite solvability of positive solutions to a nonlinear fourth-order periodic boundary value problem
Nonlinear Anal. TMA
(2005)Boundary Value Problems for Higher Order Differential Equations
(1986)Existence and uniqueness theorems for the bending of an elastic beam equation
Appl. Anal.
(1988)On fourth order boundary value problems arising in beam analysis
Differential and Integral Equations
(1989)- et al.
On the global solvability of a class of fourth-order nonlinear boundary value problems
Internat. J. Math. Math. Sci.
(1997)