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Positive solutions of a second-order Neumann boundary value problem with a parameter. (English) Zbl 1256.34019

The authors are concerned with the following second-order Neumann boundary value problem: \[ \begin{cases} -(p(t)x(t))'+q(t)x(t)=\lambda g(t)f(x(t)), &0<t<1,\\ x'(0)=x'(1)=0,&\\ \end{cases} \] where \(\lambda>0\) is a parameter, \(p\in C^1[0,1]\) (\(p(t)>0)\), \(q\in C[0,1]\) (\(q(t)\geq0)\), \(f, g: [0,\infty)\to[0,\infty)\) are continuous functions with \(f\not\equiv0\) and \(\int_0^1g(s)ds>0\). Under some assumptions regarding the limits \(\lim\limits_{x\to+\infty,0^+}\sup\frac{f(x)}{x}\), the authors first prove the existence of single and twin positive solutions. They use Krasnosel’skii’s fixed point theorem of cone compression and expansion both with a fixed point theorem for strongly completely continuous mappings; a discussion upon the parameter \(\lambda\) is given and a nonexistence result presented. When the nonlinearity \(f\) further satisfies a monotonicity condition, a uniqueness result is obtained together with the continuous dependence of the solutions on \(\lambda\).

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47H10 Fixed-point theorems
34B08 Parameter dependent boundary value problems for ordinary differential equations
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