Multiple positive solutions of nonlinear singular m-point boundary value problem for second-order dynamic equations with sign changing coefficients on time scales

Fuyi Xu, Shandong University of Technology, Zibo, Shandong, P. R. China

E. J. Qualitative Theory of Diff. Equ., No. 20. (2010), pp. 1-13.

Abstract: Let $\mathbb{T}$ be a time scale. In this paper, we study the existence of multiple positive solutions for the following nonlinear singular $m$-point boundary value problem dynamic equations with sign changing coefficients on time scales

$$\left\{\begin{array}{lll}
u^{\triangle\nabla}(t)+ a(t)f(u(t))=0, (0,T)_{\mathbb{T}},
\cr\
u^{\triangle}(0)=\sum_{i=1}^{m-2}a_{i}u^{\triangle}(\xi_i),
\cr
u(T)=\sum_{i=1}^{k}b_{i}u(\xi_i)-\sum_{i=k+1}^{s}b_{i}u(\xi_i)-\sum_{i=s+1}^{m-2}b_{i}u^{\triangle}(\xi_i),
\end{array}\right.$$

where $1\leq k\leq s\leq m-2, a_i, b_i\in(0,+\infty)$ with $0<\sum_{i=1}^{k}b_{i}-\sum_{i=k+1}^{s}b_{i}<1,
0<\sum_{i=1}^{m-2}a_{i}<1, 0<\xi_1<\xi_2<\cdots<\xi_{m-2}<\rho(T)$, $f\in C( [0,+\infty),[0,+\infty))$, $a(t)$ may be singular at $t=0$. We show that there exist two positive solutions by using two different fixed point theorems respectively. As an application, some examples are included to illustrate the main results. In particular, our criteria extend and improve some known results.

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