## 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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### References:

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