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Keywords:
ordinary differential equations; singular boundary value problem; positive solutions
ordinary differential equations; singular boundary value problem; positive solutions
Summary:
In this note we consider the boundary value problem $y''=f(x,y,y')$ $\,(x\in [0,X];X>0)$, $y(0)=0$, $y(X)=a>0$; where $f$ is a real function which may be singular at $y=0$. We prove an existence theorem of positive solutions to the previous problem, under different hypotheses of Theorem 2 of L.E. Bobisud [J. Math. Anal. Appl. 173 (1993), 69–83], that extends and improves Theorem 3.2 of D. O'Regan [J. Differential Equations 84 (1990), 228–251].
References:
[1] Arino O., Gautier S., Penot J.P.: A fixed point theorem for sequentially continuous mappings with application to ordinary differential equations. Funkcial. Ekv. 27 (1984), 273-279. MR 0794756 | Zbl 0599.34008
[2] Bobisud L.E.: Existence of positive solutions to some nonlinear singular boundary value problems on finite and infinite intervals. J. Math. Anal. Appl. 173 (1993), 69-83. MR 1205910 | Zbl 0777.34017
[3] Diestel J., Uhl J.J.: Vector Measures. Math. Survey, no. 15, Amer. Soc., 1977. MR 0453964 | Zbl 0521.46035
[4] O'Regan D.: Existence of positive solutions to some singular and nonsingular second order boundary value problems. J. Differential Equations 84 (1990), 228-251. MR 1047568 | Zbl 0706.34030
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