共振条件下二阶积分边值问题解的存在性(英文)

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. Moreover， boundary value problems with Riemann-Stieltjes integral conditions constitute a very interesting and important class of problems. They include two， three， multi-point and integral boundary-value problems as special cases， see [1，2，3，4]. The existence and multiplicity of solutions for such problems have received a great deal of attentions. We refer the reader to [5，6，7，8] for some recent results at nonresonance. To the best of our knowledge， there are only few published papers that deal with the existence of solutions for local， nonlocal and， particularly， integral nonlocal boundary value problems at resonance（see [9-18]）. In [16]， Ma gave an existence result for solutions for the three-point boundary value problem

In this paper， we shall establish a theorem of existence of solution for the problem（1） and （2） at resonance by employing the methods of lower and upper solutions. Clearly， we generalize the main results of [10，16].

The proofs of the methods of lower and upper solution are based on the connectivity properties of the solution sets of parameterized families of compact vector fields； they are a direct consequence of Mawhin[19，20].

A contradiction！ This implies that t0= 0 or t0= 1 and there are only three cases to consider：

Case 1： There are 0

Case 2： There isξ∈（0，1） such that v（t）≤0 for all t∈[0，ξ] and v（t） > 0 for all t∈（ξ，1]；

Case 3： There isη∈（0，1） such that v（t）≤0 for all t∈[η，1] and v（t） > 0 for all t∈[0，η）.

We only prove Case 1， the others are similar

By the same argument， we see that y（t）≤u（t）， for t∈[0，1]. Since y（t）≤u（t）≤x（t） for t∈[0，1]， it follows that f = f？， and so u is a solution of problems （1） and（2）. The proof is complete.

Theorem 2Suppose f ： [0，1]×RR is continuous. If there exist strict upper solution x and strict lower solution y of problems （1） and （2） with y（t）≤x（t） for t∈[0，1]， then problems （1） and （2） has a solution u∈D.

Therefore， by the connectivity ofΣ， there must exist some c0∈（c2，c1） and w（c0）∈W（c0） such that （c0，w（c0））∈Σand （17） holds. Thus c0（1 + （ρ？1）t） + w（t） is a solution of （9） and （10）.

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

Example 1Consider the following boundary value problem

References

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[2] Karakostas G L， Tsamatos P C. Existence of multiple positive solutions for a nonlocal boundary value problem. Topol Methods Nonlinear Anal， 2002， 19： 109-121.

[3] Webb J R L， Infante G. Positive solutions of nonlocal boundary value problems involving integral conditions. Nonl DiffEqua Appl， 2008， 15： 5-67.

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[17] Zhang X， Feng M， Ge W. Existence result of second-order differential equations with integral boundary conditions at resonance. J Math Anal Appl， 2009， 353： 311-319.

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