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Uniqueness and existence results for a third-order nonlinear multi-point boundary value problem. (English) Zbl 1166.34008

This paper studies the existence and uniqueness of solutions for the following nonlinear multi-point third-order boundary value problem (BVP):

x ''' (t)+f(t,x(t),x ' (t),x '' (t))=0,0<t<1,x(0)=0,g(x ' (0),x '' (0),x(ξ 1 ),,x(ξ m-2 ))=A,h(x ' (1),x '' (1),x(η 1 ),,x(η n-2 ))=B,(1·1)

where 0<ξ i ,η j <1,i=1,2,,m-2,j=1,2,,n-2,A,B, and f:[0,1]× 3 ,g: m ,h: n are continuous functions. Assuming some monotonicity conditions on the functions f,g,h, the existence of a solution for the BVP (1.1) is proved by applying the method of lower and upper solutions, and Leray-Schauder degree theory.

In order to establish the uniqueness of the solution for BVP (1.1), the authors make use of the following auxiliary boundary value problem:

x ''' (t)+a(t)x '' (t)+b(t)x ' (t)+c(t)x(t)=0,0<t<1,x(0)=0,p 1 x ' (0)+q 1 x '' (0)+ i=1 m-2 r i x(ξ i )=0,p 2 x ' (1)+q 2 x '' (1)+ j=1 n-2 R j x(η j )=0,

where a(t),b(t),c(t)C[0,1],c(t)0,t[0,1],p 1 ,p 2 ,q 1 ,q 2 ,r i ,R j with q 1 0,q 2 0,r i 0,R j 0· Some illustrative examples are also presented.

34B10Nonlocal and multipoint boundary value problems for ODE
34B15Nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations