Positive Solutions for Third-Order p -Laplacian Functional Dynamic Equations on Time Scales

Abstract

We study the following third-order $p$-Laplacian functional dynamic equation on time scales: ${\left[{\mathrm{\Phi }}_p(u^{\Delta \nabla }(t))\right]}^{\nabla }+a(t)f(u(t),u(\mu (t)))=0$, $t\in {\left(0,T\right)}_{\mathbf{T}}$, $u(t)=\phi (t)$, $t\in {\left[-r,0\right]}_{\mathbf{T}}$, $u^{\Delta }(0)=u^{\Delta \nabla }(T)=0$, and $u(T)+B_0(u^{\Delta }(\eta ))=0$. By applying the Five-Functional Fixed Point Theorem, the existence criteria of three positive solutions are established.