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Keywords:
BVP on infinite intervals; $\phi $-Laplacian; fixed point theory in cones
BVP on infinite intervals; $\phi $-Laplacian; fixed point theory in cones
Summary:
We are concerned in this paper with the existence of positive solutions to the $\phi $-Laplacian third-order boundary value problem $$ \begin {cases} -(\phi (u''))'(t)=f(t,u(t),u'(t))\text { for a.e. }t\in J, \\ u(0)=0,\ u'(0)=a\text {, }\lim \limits _{t\rightarrow \infty }u''(t)=0, \\ \Delta u(t_{k})=I_{1,k}(u(t_{k}),u'(t_{k})),\ k=1,2,\ldots ,\\ \Delta u'(t_{k})=I_{2,k}(u(t_{k}),u'(t_{k})),\ k=1,2,\ldots , \\ -\Delta \phi (u'')(t_{k})=I_{3,k}(u(t_{k}),u'(t_{k})),\ k=1,2,\ldots ,\end {cases}$$ where $a\geq 0,$ $J=(0,\infty )$, $0<t_{1}<t_{2}<\ldots <t_{k}\ldots $, $t_{k}\rightarrow \infty $ as $k\rightarrow \infty $, $\Delta u(t_{k})=u(t_{k}^{+})-u(t_{k}^{-})$ and $J^{\ast }=J\backslash \{t_{k}\colon k\geq 1\}$. The function $\phi \colon \mathbb {R}\rightarrow \mathbb {R}$ is an increasing homeomorphism such that $\phi (0)=0$, $I_{i,k}\in C(I^{2},[0,\infty ))$ for $i=1,2,3$ and $k\geq 1$, and the nonlinearity $f\colon J^{3}\rightarrow \mathbb {R}^{+}$ is a Caratheodory function.\\ By means of a Guo-Krasnoselskii type fixed point theorem, we prove an existence result for at least one positive solution.
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