Existence of pseudo-symmetric solutions to a \(p\)-Laplacian four-point BVPs involving derivatives on time scales. (English) Zbl 1265.34340

Su, You-hui; Feng, Zhaosheng

Differ. Integral Equ. 25, No. 5-6, 441-466 (2012).

Summary: We are concerned with a four-point boundary-value problem of the \(p\)-Laplacian dynamic equation on time scales, where the nonlinear term contains the first-order derivatives of the dependent variable. By using Krasnosel’skii’s fixed-point theorem, some new sufficient conditions are obtained for the existence of at least one or a pair of positive pseudo-symmetric solutions to this problem. We also establish the existence of at least three or arbitrarily many odd positive pseudo-symmetric solutions to this problem by using the Avery-Peterson fixed-point theorem. As applications, two examples are given to illustrate and explain our main results.

MSC:

34N05	Dynamic equations on time scales or measure chains
34B15	Nonlinear boundary value problems for ordinary differential equations
47N20	Applications of operator theory to differential and integral equations
34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B10	Nonlocal and multipoint boundary value problems for ordinary differential equations

Keywords:

\(p\)-Laplacian; positive pseudo-symmetric solution; four-point boundary-value problem