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Upper and lower solutions for \(\Phi\)-Laplacian third-order BVPs on the half line. (English) Zbl 1319.34038
The authors study the existence of positive solutions for third-order boundary value problems of the type \[ \begin{aligned} (\Phi(-x''(t)))' + f (t, x(t)) = 0, \quad t \in (0,+\infty), \\ x(0)=\mu x'(0), \;x'(+\infty)=x''(+\infty)=0, \end{aligned} \] where \(\mu\geq0\), \(f\) is a continuous map and \(\Phi:\mathbb{R}\to\mathbb{R}\) is a continuous, increasing homeomorphism such that \(\Phi(0)=0\). The map \(\Phi\), the so-called \(\Phi\)-Laplacian, is a generalization of the one-dimensional \(p\)-Laplacian. The map \(f\) verifies some monotonicity assumption and may be singular at \(0\).
Using the method of upper and lower solutions and fixed point techniques, the existence of at least one positive solution is proved. The result is illustrated by a concrete example.
MSC:
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B40 Boundary value problems on infinite intervals for ordinary differential equations
47H10 Fixed-point theorems
34B16 Singular nonlinear boundary value problems for ordinary differential equations
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