** 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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