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Existence to singular boundary value problems with sign changing nonlinearities using an approximation method approach. (English) Zbl 1164.34351
Summary: This paper studies the existence of solutions to the singular boundary value problem \[ \begin{cases} -u''=g(t,u)+h(t,u),\quad t\in (0,1),\\ u(0)=0=u(1), \end{cases} \] where \(g\:(0,1)\times (0,\infty )\to \mathbb R\) and \(h\:(0,1)\times [0,\infty )\to [0,\infty )\) are continuous. So our nonlinearity may be singular at \(t=0,1\) and \(u=0\) and, moreover, may change sign. The approach is based on an approximation method together with the theory of upper and lower solutions.

MSC:
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
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References:
[1] R. P. Agarwal, D. O’Regan: Singular Differential and Integral Equations with Applications. Kluwer Academic Publishers, Dordrecht, 2003.
[2] P. Habets, F. Zanolin: Upper and lower solutions for a generalized Emden-Fower equation. J. Math. Anal. Appl. 181 (1994), 684–700. · Zbl 0801.34029
[3] D. O’Regan: Theory of Singular Boundary Value Problems. World Scientific, Singapore, 1994. · Zbl 0793.92020
[4] H. Lü, D. O’Regan, and R. P. Agarwal: An Approximation Approach to Eigenvalue Intervals for Singular Boundary Value Problems with Sign Changing Nonlinearities. To appear.
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