Abstract
In this work, the existence criteria of extremal solutions of periodic boundary value problems for the first-order dynamic equations on time scales are given by using the method of lower and upper solutions coupled with the monotone iterative technique. Our results generalize and improve some existing results. Two examples are provided to show the effectiveness and feasibility of the obtained results.
Highlights
The theory of impulsive differential equations has been investigated extensively in simulating processes and phenomena subject to short-time perturbations during their evolution, such equations have a tremendous potential for applications in biology, physics, epidemic models, engineering, ect
The study of impulsive dynamic equations on time scales has attracted much attention since it provides an unifying structure for differential equations in the continuous cases and the finite difference equations in the discrete cases, see [5,6,7,8,9,10,11,12,13,14,15,16,17] and references therein
Most of them were devoted to the existence of solutions for periodic boundary value problems (PBVP) by means of some fixed point theorems [18,19,20,21,22]
Summary
The theory of impulsive differential equations has been investigated extensively in simulating processes and phenomena subject to short-time perturbations during their evolution, such equations have a tremendous potential for applications in biology, physics, epidemic models, engineering, ect. (see [1,2,3,4]). There are few papers to deal with the existence for the extremal solutions to periodic boundary value problems of first order dynamic equations on time scales based on the method of lower and upper solutions coupled with monotone iterative technique under these two cases. 3. Well-ordered lower and upper solutions we prove the existence theorem of extremal solutions for periodic boundary value problem of first-order dynamic equations on time scales under the case of a ≤ b, where a and b are lower and upper solutions of PBVP (1.3). Assumed that the following conditions are satisfied (H1) There exist two functions : a, b Î PC ∩ C1(J\{t1, t2, ..., tq}, R) a(t) ≤ b(t) such that α (t) + p(t)ασ (t) ≤ f (t, ασ (t)) − rα(t), t ∈ J, t = tk α(tk) ≤ Ik(α(tk−)) − dαk, k = 1, 2, .
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