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Existence of a positive solution to a nonlocal semipositone boundary value problem on a time scale

Christopher S. Goodrich (2013)

Commentationes Mathematicae Universitatis Carolinae

We consider the existence of at least one positive solution to the dynamic boundary value problem - y Δ Δ ( t ) = λ f ( t , y ( t ) ) , t [ 0 , T ] 𝕋 y ( 0 ) = τ 1 τ 2 F 1 ( s , y ( s ) ) Δ s y σ 2 ( T ) = τ 3 τ 4 F 2 ( s , y ( s ) ) Δ s , where 𝕋 is an arbitrary time scale with 0 < τ 1 < τ 2 < σ 2 ( T ) and 0 < τ 3 < τ 4 < σ 2 ( T ) satisfying τ 1 , τ 2 , τ 3 , τ 4 𝕋 , and where the boundary conditions at t = 0 and t = σ 2 ( T ) can be both nonlinear and nonlocal. This extends some recent results on second-order semipositone dynamic boundary value problems, and we illustrate these extensions with some examples.

Opial's type inequalities on time scales and some applications

S. H. Saker (2012)

Annales Polonici Mathematici

We prove some new Opial type inequalities on time scales and employ them to prove several results related to the spacing between consecutive zeros of a solution or between a zero of a solution and a zero of its derivative for second order dynamic equations on time scales. We also apply these inequalities to obtain a lower bound for the smallest eigenvalue of a Sturm-Liouville eigenvalue problem on time scales. The results contain as special cases some results obtained for second order differential...

Periodicity, almost periodicity for time scales and related functions

Chao Wang, Ravi P. Agarwal, Donal O’Regan (2016)

Nonautonomous Dynamical Systems

In this paper, we study almost periodic and changing-periodic time scales considered byWang and Agarwal in 2015. Some improvements of almost periodic time scales are made. Furthermore, we introduce a new concept of periodic time scales in which the invariance for a time scale is dependent on an translation direction. Also some new results on periodic and changing-periodic time scales are presented.

Theory of rapid variation on time scales with applications to dynamic equations

Jiří Vítovec (2010)

Archivum Mathematicum

In the first part of this paper we establish the theory of rapid variation on time scales, which corresponds to existing theory from continuous and discrete cases. We introduce two definitions of rapid variation on time scales. We will study their properties and then show the relation between them. In the second part of this paper, we establish necessary and sufficient conditions for all positive solutions of the second order half-linear dynamic equations on time scales to be rapidly varying. Note...

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