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Zbl 1142.34315
Chu, Jifeng; Sun, Yigang; Chen, Hao
Positive solutions of Neumann problems with singularities.
(English)
[J] J. Math. Anal. Appl. 337, No. 2, 1267-1272 (2008). ISSN 0022-247X

The authors establish the existence of positive solutions to the boundary value problem $$x''+m^2x=f(t,x)+{\text{\it e}}(t),\;x'(0)=x'(1)=0,$$ where $m\in(0,{\pi\over2})$, $e\in C[0,1]$ and $f(t,x)$ may be singular at $x=0$. Moreover, they suppose that there exist a constant $r>0$, a nonincreasing function $g(x)\in C((0,r],(0,\infty))$, functions $h\in C((0,r],[0,\infty))$ and $k\in C([0,1],[0,\infty))$ and a continuous function $\phi_r(t)> 0$ such that $h(x)/g(x)$ is nondecreasing for $x\in(0,r]$, $\phi_r(t)+{\text{\it e}}(t)> 0$ for all $t\in[0,1]$ and $$\phi_r(t)\leq f(t,x)\leq k(t)\{f(x)+h(x)\}\text{ for all }(t,x)\in[0,1]\times(0,r].$$ The proof relies on a nonlinear alternative principle of Leray-Schauder and a truncation technique.
[Petio S. Kelevedjiev (Sliven)]
MSC 2000:
*34B18 Positive solutions of nonlinear boundary value problems
34B16 Singular nonlinear boundary value problems

Keywords: Singular boundary value problem; Neumann boundary conditions; existence; positive solutions; Leray-Schauder principle; truncation technique

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