Positive solutions of a nonlinear elastic beam equation rigidly fastened on the left and simply supported on the right

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Abstract

The purpose of this paper is to establish several local existence theorems concerned with n positive solutions for a fourth-order two-point boundary value problem, where n is an arbitrary positive integer and the nonlinear term may be singular. In mechanics, the problem describes deflection of an elastic beam 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: {u(4)(t)=f(t,u(t)),0t1,u(0)=u(0)=u(1)=u(1)=0.

Here the function uC[0,1] is called a positive solution of problem (P) if u is a solution of (P) and u(t)>0,0<t<1.

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 f:[0,1]×(,+)(,+) 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 f:[0,1]×(,+)(,+) is continuous and there exists a positive number d>0 such thatmax{|f(t,u)|:0t1,|u|2d}6553639+5533d.Then problem (P)has at least one solution uC3[0,1] .

The local existence theorem shows that the problem (P) has at least one solution provided the “height” of the nonlinear term f(t,u) is appropriate on the bounded set [0,1]×[2d,2d].

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 u(4)(t)=f(t,u(t)),0t1,u(0)=u(0)=u(1)=u(1)=0, which describes deflection of a beam simply supported at both ends (see [7], [8], [9], [10]), the problem u(4)(t)=f(t,u(t)),0t1,u(0)=u(0)=u(1)=u(1)=0, which describes deflection of a beam rigidly fastened at both ends (see [11], [12], [13]), and the problem u(4)(t)=f(t,u(t)),0t2π,u(i)(0)=u(i)(2π),i=0,1,2,3, which describes the equilibrium state of an elastic beam with periodic boundary condition (see [14]). In these studies the nonlinear terms f(t,u) 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 f(t,u) to be singular. In fact, we require only that the nonlinear term f(t,u) is locally continuous. To be more exact, the nonlinear term f(t,u) satisfies one of the following conditions with singularity:

(C1) f:D[a,b][0,+) is continuous, where D[a,b]={(t,u):0t1,aq(t)ub}, and q(t)=23t2(1t).

(C2) f:D(e×0)c[0,+) is continuous, where D(e×0)c={(t,u):0t1,0<u<+}{(t,0):t[0,1]e}, where e(0,1) is a closed subset.

The condition (C2) implies that the nonlinear term f(t,u) may be singular on the set e×{0}. The condition (C1) implies that f(t,u) may be singular on a certain closed subset of the set {(t,u):0<t<1,0u<aq(t)}.

A simple result is that, if there exists a closed subset e(0,1) such that f:D(e×0)c[0,+) is continuous, then f:D[a,b][0,+) is continuous for any 0<a<b<+; if f:[0,1]×[0,+)[0,+) is continuous, then f:D(e×0)c[0,+) is continuous for any closed subset e(0,1).

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 f(t,u) and considering heights of the nonlinear term f(t,u) 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 n positive solutions, where n 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 f(t,u) at u=0 and u=+ 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 f:[0,1]×[0,+)[0,+) is continuous.

Section snippets

Preliminaries

From [3], the Green function G(t,s) of the homogeneous linear problem u(4)(t)=0,0t1,u(0)=u(0)=u(1)=u(1)=0 has the form G(t,s)={112(1t)s2[3(1s)(1t)2(3s)],0st1,112t2(1s)[3(1t)(1s)2(3t)],0ts1. Obviously, G(t,s)0,0t,s1.

Let H(s)=14s2(1s) and q(t)=23t2(1t). It is easy to see that max0t1q(t)=q(23)=881.

Lemma 2.1

q(t)H(s)G(t,s)H(s),0t,s1.

Proof

Since G(t,0)=G(t,1)=0,0t1 and H(0)=H(1)=0, the inequalities hold for s=0 or s=1.

If 0<st1 and s<1, then G(t,s)112(1t)s23(1s)14s2(1s)=H(s),

Existence and multiplicity under condition (C1)

We obtain the following local existence theorems.

Theorem 3.1

Assume that

  • (a1)

    There exist two positive numbers 0<a<b such that f:D[a,b][0,+) is continuous.

  • (a2)

    One of the following conditions is satisfied:

    • (i)

      φ(a)aA,ψ(b)bB.

    • (ii)

      ψ(a)aB,φ(b)bA.

Then problem (P)has at least one positive solution uK satisfying aub.

Proof

We consider only the case (i). The proof for case (ii) is similar. Define Ωa={uK:u<a},Ωb={uK:u<b}.

If uΩa, then u=a and 0uK. Since aq(t)u(t)a,0t1, we have f(t,u(t))φ(a)aA,0t1. 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 f(t,u) at u=0 or u=+.

Theorem 4.1

Assume that

  • (f1)

    There exists a closed set e(0,1) such that f:D(e×0)c[0,+) is continuous.

  • (f2)

    There exist 0<t1<t2<1 such that lim infu0+mint1tt2f(t,u)>0 (in particular, lim infu0+mint1tt2f(t,u)=+ ).

  • (f3)

    There exists a>0 such that φ(a)aA.

Then problem (P)has at least one positive solution uK satisfying 0<ua.

Proof

By the condition (f2), there exist 0

An interesting example with a physical interpretation

Example 5.1

Let α=13,β=23. Consider the fourth-order boundary value problem {u(4)(t)=48100u2(t),0t1,u(0)=u(0)=u(1)=u(1)=0. Since limu0+max0t1f3(t,u)/u=0,limu+min0t1f3(t,u)/u=+, the nonlinear term f3(t,u)=48100u2 is superlinear. Since φ(100)=max{f3(t,u):0t1,100q(t)u100}=4800=100A,ψ(1000000)=min{f3(t,u):13t23,1000000q(t)u1000000}=48100(1000000σ)21170550000>10093800001000000B(13,23), we assert that the problem (P3) has a positive solution u satisfying 100u1000000 by Theorem 3.1.

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