Abstract
We investigate the oscillation and boundedness of first and second order dynamic equations with mixed nonlinearities. Our results extend and improve known results for oscillation of first and second order dynamic equations that have been established by Agarwal and Bohner. Some examples are given to illustrate the main results.
Highlights
Does the new theory of so-called “dynamic equations” unify the theories of differential equations and difference equations, and extends these classical cases to cases “in between”, e.g., to so-called q-difference equations when T = qN0 and can be applied to different types of time scales like T = hZ, T = N20, and the space of harmonic numbers T = {Hn}
The purpose of this paper is to extend the oscillation and boundedness criteria to first order dynamic equations of the form x∆(t) + p(t)xγ (h(t)) + q1(t)xα(h(t)) = f (t) x∆(t) + p(t)xγ (h(t)) + q1(t)xα(h(t)) + q2(t)xβ (h(t)) = f (t), and we examine second order dynamic equations of the form
If (2.8) and (3.6) hold, every nonoscillatory solution of equation (1.4) is bounded
Summary
We assume that the function h : T → T satisfies lim h(t) = ∞, and our results will improve and extend results in [2]. We give oscillation criteria for solutions to first order dynamic equations of the form (1.1) and (1.2). This case leads to a contradiction as in the first case, provided t lim inf t→∞
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