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Existence of positive solutions in generalized boundary value problem for \(p\)-Laplacian dynamic equations on time scales. (English) Zbl 1195.34144

The authors study the one-dimensional \(p\)-Laplacian dynamic equation:
\[ \left(\phi_p\left(u^{\Delta}(t)\right)\right)^\nabla+h(t)\, f(t,u(t))=0, \quad t \in (0,T]_{\mathbf{T}}, \]
coupled with the \(m\)-point boundary value conditions
\[ u(0)-\beta\, B_0\left(u^\Delta(0)\right)=\sum_{i=1}^{m-2}{B\left(u^\Delta(\xi_i)\right)}, \quad \phi_p\left(u^\Delta(T)\right)=\sum_{i=1}^{m-2}{a_i\, \phi_p\left(u^\Delta(\xi_i)\right)}. \]
Here, \(\phi_p\) is the one-dimensional \(p\)-Laplacian operator, \(0 \leq \beta\), \(0 \leq a_i\) for \(i=1, \ldots,m-2\), \(0 < \xi_1 < \xi_2 < \cdots < \xi_{m-2} < T\) and \(\sum_{i=1}^{m-2}{a_i} < 1\).
Under suitable assumptions on the functions \(h\), \(f\), \(B_0\) and \(B\), they deduce the existence of at least two or three positive solutions. The results hold by the Avery-Henderson fixed point theorem and the five functional fixed point theorem.

MSC:

34N05 Dynamic equations on time scales or measure chains
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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