Three positive solutions for \(p\)-Laplacian functional dynamic equations on time scales. (English) Zbl 1183.34141

Guan, W.

Electron. J. Qual. Theory Differ. Equ. 2008, Paper No. 28, 7 p. (2008).

Summary: Existence criteria for three positive solutions to the following \(p\)-Laplacian functional dynamic equation on time scales \[ \begin{cases}[\Phi_p(u^\Delta(t))]^\nabla+a(t)f(u(t),u(\mu(t)))=0,\quad & t\in \left(0,T\right),\\ u_0(t)=\varphi(t),\quad & t\in \left[ -r,0\right], \\ u(0)-B_0(u^\nabla(\eta))=0, & u^\Delta(T)=0,\end{cases} \] are established by using the well-known Five Functionals Fixed Point Theorem.

MSC:

34N05	Dynamic equations on time scales or measure chains
34K10	Boundary value problems for functional-differential equations
47N20	Applications of operator theory to differential and integral equations

Keywords:

time scale; \(p\)-Laplacian functional dynamic equation; boundary value problem; positive solution; fixed point theorem