Existence of solutions to second-order BVPs on time scales

Existence of Solutions to Second-Order BVPs on Time Scales

2 School of Mathematics The University of New South Wales, Sydney NSW 2052, Australia Abstract:We consider a boundary value problem (BVP) for systems of second-order dynamic equations on time scales. Using methods involving dynamic inequalities, we formulate conditions under which all solutions to a certain family of systems of dynamic equations satisfy certain a priori bounds. These results ar...

Existence of Positive Solution to Second-Order Three-Point BVPs on Time Scales

We are concerned with the following nonlinear second-order three-point boundary value problem on time scales −xΔΔ t f t, x t , t ∈ a, b T , x a 0, x σ2 b δx η , where a, b ∈ T with a < b, η ∈ a, b T and 0 < δ < σ2 b − a / η − a . A new representation of Green’s function for the corresponding linear boundary value problem is obtained and some existence criteria of at least one positive solution ...

Existence of positive solutions for a second-order p-Laplacian impulsive boundary value problem on time scales

In this paper, we investigate the existence of positive solutions for a second-order multipoint p-Laplacian impulsive boundary value problem on time scales. Using a new fixed point theorem in a cone, sufficient conditions for the existence of at least three positive solutions are established. An illustrative example is also presented.

Existence of Positive Solutions for Semipositone Higher-Order BVPS on Time Scales

We offer conditions on semipositone function f t, u0, u1, . . . , un−2 such that the boundary value problem, uΔ n t f t, u σn−1 t , uΔ σn−2 t , . . . , uΔ n−2 σ t 0, t ∈ 0, 1 ∩ T, n ≥ 2, uΔi 0 0, i 0, 1, . . . , n − 3, αuΔ 0 − βuΔ 0 0, γuΔ σ 1 δuΔ σ 1 0, has at least one positive solution, where T is a time scale and f t, u0, u1, . . . , un−2 ∈ C 0, 1 × R 0,∞ n−1,R −∞,∞ is continuous with f t, ...

Existence of Nonoscillatory Solutions to Second-Order Neutral Delay Dynamic Equations on Time Scales