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Solvability of a boundary value problem with the nonlinearity satisfying a sign condition. (English) Zbl 0638.34015
Let f: [0,$$\pi$$ ]$$\times R\to R$$ be a given function satisfying Caratheodory’s conditions and the sign condition f(x,u)u$$\geq 0$$ for $$x\in [0,\pi]$$, $$u\in R$$. Let h: [0,$$\pi$$ ]$$\to R$$ be a given function in $$L^ 1[0,\pi]$$ such that $$\int^{\pi}_{0}h(x) \sin x dx=0.$$ This paper is concerned with the existence of a solution for the boundary value problem $(1)\quad -u''(x)-u(x)+f(x,u(x))=h(x),\quad u(0)=u(\pi)=0,$ and the boundary value problem $(2)\quad u''(x)+u(x)+f(x,u(x))=h(x),\quad u(0)=u(\pi)=0,$ when f additionally satisfies $$\limsup_{| u| \to \infty}(f(x,u)/u)=\beta <3$$. The boundary value problem (2) was earlier studied by J. Mawhin, J. R. Ward and M. Willem [Proc. Am. Math. Soc. 93, 667-674 (1985; Zbl 0559.34014)] using variational methods when f(x,u) is non-decreasing in u and $$h(x)=0$$. The methods of this paper use Leray-Schauder continuation theorem and estimates involving the use of Wirtinger type inequalities.
Reviewer: C.P.Gupta

##### MSC:
 34B15 Nonlinear boundary value problems for ordinary differential equations
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##### References:
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