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Positive solutions to singular boundary value problems with sign changing nonlinearities on the half-line via upper and lower solutions. (English) Zbl 1133.34017
The authors study the following boundary value problem on the half line
\[ x''-m^2x+\Phi(t)f(t,x)=0,\quad t\in(0,+\infty), \]
\[ x(0)=0, x(\infty)=0, \] where \(f\in C(\mathbb{R}^+\times \mathbb{R}_0^+, \mathbb{R})\) and \(m>0\) is a constant, \(\Phi\in C(\mathbb{R}_0^+, \mathbb{R}^+)\); here, \(\mathbb{R}^+=[0,\infty), \mathbb{R}_0^+=(0,\infty)\). In this paper, \(f(t,x)\) can change sign and may be singular at \(x=0\). The lower and upper solution technique is presented for the above boundary value problem and some results on the existence of positive solutions are established.

MSC:
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B40 Boundary value problems on infinite intervals for ordinary differential equations
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