## Journal of Applied Mathematics

• J. Appl. Math.
• Volume 2014, Special Issue (2014), Article ID 867018, 10 pages.

### Solutions of Second-Order $m$-Point Boundary Value Problems for Impulsive Dynamic Equations on Time Scales

#### Abstract

We study a general second-order $m$-point boundary value problems for nonlinear singular impulsive dynamic equations on time scales ${u}^{\Delta \nabla }(t)+a(t){u}^{\Delta }(t)+b(t)u(t)+q(t)f(t,u(t))=\mathrm{0},t\in (\mathrm{0,1}),t\ne {t}_{k},{u}^{\Delta }({t}_{k}^{+})={u}^{\Delta }({t}_{k})-{I}_{k}(u({t}_{k}))$, and $k=\mathrm{1,2},\dots ,n,\mathrm{}u(\rho (\mathrm{0}))=\mathrm{0},u(\sigma (\mathrm{1}))={\sum }_{i=\mathrm{1}}^{m-\mathrm{2}}{\alpha }_{i}u({\eta }_{i})\mathrm{‍}.$ The existence and uniqueness of positive solutions are established by using the mixed monotone fixed point theorem on cone and Krasnosel’skii fixed point theorem. In this paper, the function items may be singular in its dependent variable. We present examples to illustrate our results.

#### Article information

Source
J. Appl. Math., Volume 2014, Special Issue (2014), Article ID 867018, 10 pages.

Dates
First available in Project Euclid: 1 October 2014

https://projecteuclid.org/euclid.jam/1412177574

Digital Object Identifier
doi:10.1155/2014/867018

Mathematical Reviews number (MathSciNet)
MR3198411

#### Citation

Xu, Xue; Wang, Yong. Solutions of Second-Order $m$ -Point Boundary Value Problems for Impulsive Dynamic Equations on Time Scales. J. Appl. Math. 2014, Special Issue (2014), Article ID 867018, 10 pages. doi:10.1155/2014/867018. https://projecteuclid.org/euclid.jam/1412177574

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