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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:
 [1] Ahmad, S., A resonance problem in which the nonlinearity may grow linearly, (), 381-384 · Zbl 0562.34011 [2] Cesari, L.; Kannan, R., Existence of solutions of nonlinear differential equations, (), 705 [3] Fucik, S., Surjectivity of operators involving linear noninvertible part and nonlinear compact perturbation, Funkcial. ekvac., 17, 73-83, (1974) · Zbl 0294.47041 [4] Gupta, C.P., On functional equations of Fredholm and Hammerstein type with applications to existence of periodic solutions of certain ordinary differential equations, J. integral equations, 3, 21-41, (1981) · Zbl 0457.34040 [5] Kazdan, J.L.; Warner, F., Remarks on some quasilinear elliptic equations, Comm. pure appl. math., 28, 587-597, (1975) · Zbl 0325.35038 [6] Mawhin, J., Landesman-lazer type problems for nonlinear equations, () [7] Mawhin, J., Topological degree methods in nonlinear boundary value problems, () · Zbl 0414.34025 [8] Mawhin, J., Compacité, monotonie et convexité dans l’étude de problèmes aux limites semi-linéaires, () · Zbl 0497.47033 [9] Mawhin, J.; Ward, J.R.; Willem, M., Necessary and sufficient conditions for the solvability of a nonlinear two point boundary value problem, (), 667-674 · Zbl 0559.34014 [10] Schechter, M.; Shapiro, J.; Snow, M., Solutions of the nonlinear problem A (u) = N (u) in a Banach space, Trans. amer. math. soc., 241, 168-178, (1978)
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