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A non-resonant multi-point boundary-value problem for a $$p$$-Laplacian type operator. (English) Zbl 1032.34012
Summary: Let $$\varphi$$ be an odd increasing homeomorphism from $$\mathbb{R}$$ onto $$\mathbb{R}$$ with $$\varphi(0)= 0$$, $$f: [0,1]\times \mathbb{R}^2\to \mathbb{R}$$ be a function satisfying Carathéodory conditions and $$e(t)\in L^1[0,1]$$. Let $$\xi_i\in (0,1)$$, $$a_i\in\mathbb{R}$$, $$i= 1,2,\dots, m-2$$, $$\sum_{i=1}^{m-2} a_i\neq 1$$, $$0< \xi_1< \xi_2<\cdots< \xi_{m-2}< 1$$ be given. This paper is concerned with the existence of a solution to the multipoint boundary value problem $(\varphi(x'(t)))'= f(t,x(t),x'(t))+ e(t), \quad 0< t< 1,$
$x(0)=0, \quad \varphi(x'(1))= \sum_{i=1}^{m-2} a_i\varphi (x'(\xi_i)).$ The author gives conditions for the existence of a solution to the above boundary value problem using some new Poincaré-type a priori estimates. In the case $$\varphi(t) \equiv t$$ for $$t\in \mathbb{R}$$, this problem was studied earlier by Gupta, Trofimchuk and by Gupta, Ntouyas and Tsamatos. We give a priori estimates needed for this problem that are similar to a priori estimates obtained by Gupta, Trofimchuk. We then obtain existence theorems for the above multipoint boundary value problem using the a priori estimates and Leray-Schauder continuation theorem.

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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