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A sharper condition for the solvability of a three-point second order boundary value problem. (English) Zbl 0874.34014
Let $$f:[0,1] \times\mathbb{R}^2 \to\mathbb{R}$$ be a function satisfying Carathéodory’s conditions and $$e(t)\in L^1[0,1]$$. Let $$\eta\in(0,1)$$, $$\alpha\in\mathbb{R}$$, $$\alpha>1$$, $$\alpha\eta \neq 1$$ be given. This paper is concerned with the problem of existence of a solution for the three-point boundary value problem $x''(t)=f \bigl(t,x(t), x'(t)\bigr)+ e(t),\;0<t<1,\;x(0)=0,\;x(1)= \alpha x(\eta). \tag{*}$ In the general case this problem was studied by the first author as a multipoint boundary value problem. In the present paper, sharper existence conditions are obtained for the solvability of the above boundary value problem in the general case.

##### MSC:
 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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##### References:
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