Existence of positive periodic solutions for nonlinear neutral dynamic equations with variable coefficients on a time scale

Abstract

Let T be a periodic time scale. The purpose of this paper is to use Krasnosel'skiȋ's fixed point theorem to prove the existence of positive periodic solutions for nonlinear neutral dynamic equations with variable coefficients on a time scale. We invert these equations to construct a sum of a contraction and a compact map which is suitable for applying the Krasnosel'skiȋ's theorem. The results obtained here extend the work of Candan [11].

Some existence results on positive solutions for an iterative second-order boundary-value problem with integral boundary conditions

“…The aim of this section is to convert our boundary-value problem (1.1)-(1.2) to a fixed point problem where the proof of our main result relies on Schauder's fixed point Theorem. By virtue of Lemma 2.1, we define an operator A : CB Int (L, M ) −→ CB Int as follows: [1] (s), ϕ [2] (s), ..., ϕ [n] (s) ds [1] (s), ϕ [2] (s), ..., ϕ [n] (s) ds − t 0 (t − s)f ϕ [0] (s), ϕ [1] (s), ϕ [2] (s), ..., ϕ [n] (s) ds.…”

“…( Proof: To show that A is well defined it suffices to show that (Aϕ) (0) = 0 and α η 0 [1] (s), ϕ [2] (s), ..., ϕ [n] (s) ds [1] (s), ϕ [2] (s), ..., ϕ [n] (s) ds [1] (s), ϕ [2] (s), ..., ϕ [n] (s) ds [1] (s), ϕ [2] (s), ..., ϕ [n] (s) ds [1] (s), ϕ [2] (s), ..., ϕ [n] (s) ds, and [1] (s), ϕ [2] (s), ..., ϕ [n] (s) ds dt [1] (s), ϕ [2] (s), ..., ϕ [n] (s) ds dt [1] (s), ϕ [2] (s), ..., ϕ [n] (s) ds dt, which implies [1] (s), ϕ [2] (s), ..., ϕ [n] (s) ds [1] (s), ϕ [2] (s), ..., ϕ [n] (s) ds…”

“…(s), ϕ [1] (s), ϕ [2] (s), ..., ϕ [n] (s) ds [1] (s), ϕ [2] (s), ..., ϕ [n] (s) ds [1] (s), ϕ [2] (s), ..., ϕ [n] Proof: For ϕ, ψ ∈ CB Int (L, M ) , we have [1] (s), ϕ [2] (s), ..., ϕ [n] (s) −f ψ [0] (s), ψ [1] (s), ψ [2] (s), ..., ψ [n] (s) ds…”

“…This work is a continuation of the obove montioned works, some other works on positivity and boundary value problems [1,3] and our recent papers on iterative problems (see [5,6]). Our main contribution to this important area is to show that the fixed point theory can be applied successfully to iterative problems with integral boundary conditions.…”

“…where x [0] (t) = t, x [1] (t) = x (t) , ..., x [n] (t) = x [n−1] (x (t)) and f : [0, b] × R n → [0, +∞) is a continuous function with respect to its arguments. Equation (1.1) describes diffusion phenomena with a source or a reaction term.…”