Existence and Iteration of Positive Solutions to Third-Order BVP for a Class of p-Laplacian Dynamic Equations on Time Scales

Abstract

We investigate the existence and iteration of positive solutions for the following third-order $p$-Laplacian dynamic equations on time scales: $({\varphi }_p(u^{\mathrm{\Delta }\mathrm{\Delta }}(t)){)}^{\nabla }+q(t)f(t,u(t),u^{\mathrm{\Delta }\mathrm{\Delta }}(t))=\mathrm{0},\hspace{0.17em}\hspace{0.17em}t\in [a,b],$$\alpha u(\rho (a))-\beta u^{\mathrm{\Delta }}(\rho (a))=\mathrm{0},\hspace{0.17em}\hspace{0.17em}\gamma u(b)+\delta u^{\mathrm{\Delta }}(b)=\mathrm{0},\hspace{0.17em}\hspace{0.17em}{u}^{\mathrm{\Delta }\mathrm{\Delta }}(\rho (a))=\mathrm{0},$ where ${\varphi }_p(s)$ is $p$-Laplacian operator; that is, ${\varphi }_p(s)={\left|s\right|}^{p-\mathrm{2}}s,\hspace{0.17em}\hspace{0.17em}p>\mathrm{1},\hspace{0.17em}\hspace{0.17em}{\varphi }_p^{-\mathrm{1}}={\varphi }_q$, and $\mathrm{1}/p+\mathrm{1}/q=\mathrm{1}.$ By applying the monotone iterative technique and without the assumption of the existence of lower and upper solutions, we not only obtain the existence of positive solutions for the problem, but also establish iterative schemes for approximating the solutions.