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Solvability of \(m\)-point boundary value problems with nonlinear growth. (English) Zbl 0883.34020
Let \(f:[0,1] \times \mathbb{R}^2\to \mathbb{R}\) be a continuous function and \(e(.) \in L^1 (0,1)\). Let \(\xi_i\in (0,1)\), \(a_i\in \mathbb{R}\) with all the \(a_i\)’s having the same sign, \(i=1,2, \dots, m-2\), and \(0<\xi_1< \xi_2< \cdots <\xi_{m-2} <1\).
This paper is concerned with the existence of solutions of the second order ordinary differential equation \[ x''(t)= f\bigl(t,x(t),x'(t) \bigr)+e(t) \quad t\in (0,1) \tag{1} \] subject to one of the following nonlocal boundary conditions \[ x'(0)=0, \quad x(1) =\sum^{m-2}_{i=1} a_ix(\xi_i) \tag{2} \] \[ x(0)=0, \quad x(1)= \sum^{m-2}_{i=1} a_i x(\xi_i). \tag{3} \] Conditions for the existence of a solution for the above \(m\)-point boundary value problems are given.
The authors assume that \(f(t,x,p)= g(t,x,p)+ h(t,x,p)\) and deal with both the nonresonant and the resonant case, using the coincidence degree theory of Mawhin.

MSC:
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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