A Dirichlet type multi-point boundary value problem for second order ordinary differential equations.(English)Zbl 0847.34018

The author considers the second order ordinary differential equation (1) $$x'' (t)= f(t, x(t), x'(t))+ e(t)$$, $$0< t< 1$$, where $$f: [0, 1] \times \mathbb{R}^2\to \mathbb{R}$$ satisfies the Carathéodory conditions and $$e\in L^1 [0, 1]$$. He proves the existence of a solution to (1) which satisfies the multipoint boundary value condition (2) $$x(0)= 0$$, $$x(1)= \sum^{m- 1}_{i= 1} a_i x(\xi_i)$$, with $$a_i\in \mathbb{R}$$, $$\xi_i\in (0, 1)$$, $$i= 1, 2, \dots, m-2$$, $$0< \xi_1< \xi_2< \dots< \xi_{m-2}< 1$$. He directs his attention at the nonresonant case $$\sum^{m-2}_{i=1} a_i \xi_i\neq 1$$. The existence results are proved by means of Mawhin’s version of the Leray-Schauder continuation theorem and complete the earlier ones by the author, S. K. Ntouyas and P. Ch. Tsamatos in [Nonlinear Anal., Theory Methods Appl. 23, No. 11, 1427-1436 (1994; Zbl 0815.34012)].

MSC:

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

Zbl 0815.34012
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References:

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