Multiple positive solutions of time scale boundary value problems on infinite intervals. II. (English) Zbl 1247.34141
The authors show the existence of three positive solutions of the nonlinear boundary value problem
\[
|p(t)\phi(x^\Delta(t))|^\nabla+f(t,x(t),x^\Delta)=0, \;\;t \in \mathbb {T}_\kappa,
\]
\[ x(0)-\int^\infty_\sigma(0)\alpha(t)x(t)\Delta t=0 \lim_{t\to \infty}\phi^{-1}(p(t))x^\Delta(t)-\int^\infty_0\beta(t)x^\Delta(t)\Delta t=0, \] where \(\mathbb {T}\) is a times scale which may be unbounded, \(\phi(x)=|x|^{p-2}x\) with \(p>1\), \(\alpha, \beta:\mathbb {T}\to [0, \infty)\) are re-continuous with \(\alpha, \beta\in L^1[0, \infty)\), \(p\) is continuous, \(f:\mathbb {T}_\kappa\times [0, \infty)^2\to [0, \infty)\) is a Carathéodory function. The main tool is the five functionals fixed point theorem.
\[ x(0)-\int^\infty_\sigma(0)\alpha(t)x(t)\Delta t=0 \lim_{t\to \infty}\phi^{-1}(p(t))x^\Delta(t)-\int^\infty_0\beta(t)x^\Delta(t)\Delta t=0, \] where \(\mathbb {T}\) is a times scale which may be unbounded, \(\phi(x)=|x|^{p-2}x\) with \(p>1\), \(\alpha, \beta:\mathbb {T}\to [0, \infty)\) are re-continuous with \(\alpha, \beta\in L^1[0, \infty)\), \(p\) is continuous, \(f:\mathbb {T}_\kappa\times [0, \infty)^2\to [0, \infty)\) is a Carathéodory function. The main tool is the five functionals fixed point theorem.
Reviewer: Ruyun Ma (Lanzhou)
MSC:
34N05 | Dynamic equations on time scales or measure chains |
34B15 | Nonlinear boundary value problems for ordinary differential equations |
34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |
34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |
47N20 | Applications of operator theory to differential and integral equations |