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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.

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
Full Text: DOI
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