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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.

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