Existence theory for positive solutions of p-laplacian multi-point BVPs on time scales

Authors: YOU-HUI SU

Abstract: This paper is concerned with the one-dimensional p-Laplacian multi-point boundary value problem on time scales T: (\varphi_p(u^{\Delta}))^{\nabla} + h(t)f(u) = 0, t \in [0,T]_T, subject to multi-point boundary conditions u(0) - B_0(\sum_{i=1}^{m-2}a_i u^{\Delta}(\xi_i)) = 0, u^{\Delta}(T) = 0, or u^{\Delta}(0) = 0, u(T) + B_1(\sum_{i=1}^{m-2}b_iu^{\Delta}(\xi'_i)) = 0, where \varphi_p(u) is p-Laplacian operator, i.e., \varphi_p(u = |u|^{p-2}u, p>1, \xi_i,\xi'_i\in [0,T]_T, m \geq 3 and satisfy 0 \leq \xi_1 < \xi_2 < ... < \xi_{m-2} < \rho(T), \sigma(0) < \xi'_1 < \xi'_2 < ... < \xi'_{m-2} \leq T, a_i, b_i\in [0,\infty) (i=1,2,..., m-2). Some new sufficient conditions are obtained for the existence of at least one positive solution by using Krasnosel'skii's fixed-point theorem and new sufficient conditions are obtained for the existence of twin, triple or arbitrary odd positive solutions by using generalized Avery and Henderson fixed-point theorem and Avery-Peterson fixed-point theorem. Our results include and extend some known results. As applications, two examples are given to illustrate the main results and their differences. These results are new even for the special cases of continuous and discrete equations, as well as in the general time scale setting.

Keywords: Time scales; boundary value problem; positive solutions; p-Laplacian; fixed-point theorem

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