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Positive solutions of Neumann problems with singularities. (English) Zbl 1142.34315

The authors establish the existence of positive solutions to the boundary value problem
\[ x''+m^2x=f(t,x)+{\mathit{e}}(t),\;x'(0)=x'(1)=0, \]
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 \(x\in(0,r]\), \(\phi_r(t)+{\mathit{e}}(t)> 0\) 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.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
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References:

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