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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):

$\left\{\begin{array}{c}{x}^{\text{'}\text{'}\text{'}}\left(t\right)+f\left(t,x\left(t\right),{x}^{\text{'}}\left(t\right),{x}^{\text{'}\text{'}}\left(t\right)\right)=0,\phantom{\rule{3.33333pt}{0ex}}0

where $0<{\xi }_{i},{\eta }_{j}<1,\phantom{\rule{3.33333pt}{0ex}}i=1,2,\cdots ,m-2,\phantom{\rule{3.33333pt}{0ex}}j=1,2,\cdots ,n-2,\phantom{\rule{3.33333pt}{0ex}}A,B\in ℝ,$ and $f:\left[0,1\right]×{ℝ}^{3}\to ℝ,\phantom{\rule{3.33333pt}{0ex}}g:{ℝ}^{m}\to ℝ,\phantom{\rule{3.33333pt}{0ex}}h:{ℝ}^{n}\to ℝ$ 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:

$\left\{\begin{array}{c}{x}^{\text{'}\text{'}\text{'}}\left(t\right)+a\left(t\right){x}^{\text{'}\text{'}}\left(t\right)+b\left(t\right){x}^{\text{'}}\left(t\right)+c\left(t\right)x\left(t\right)=0,\phantom{\rule{3.33333pt}{0ex}}0

where $a\left(t\right),b\left(t\right),c\left(t\right)\in C\left[0,1\right],\phantom{\rule{3.33333pt}{0ex}}c\left(t\right)\ge 0,t\in \left[0,1\right],\phantom{\rule{3.33333pt}{0ex}}{p}_{1},{p}_{2},{q}_{1},{q}_{2},{r}_{i},{R}_{j}\in ℝ$ with ${q}_{1}\le 0,{q}_{2}\ge 0,{r}_{i}\le 0,{R}_{j}\le 0·$ Some illustrative examples are also presented.

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ODE 34B15 Nonlinear boundary value problems for ODE 47N20 Applications of operator theory to differential and integral equations