Oscillation for a nonlinear neutral dynamic equations on time‐scales with variable exponents

• Published in: Mathematical methods in the applied sciences Vol. 42; no. 12; pp. 4146 - 4169
• Main Authors: Negi, Shekhar Singh, Abbas, Syed, Malik, Muslim
• Format: Journal Article
• Language: English
• Published: Freiburg Wiley Subscription Services, Inc 01.08.2019
• Subjects: dynamic equation; Economic models; Exponents; Mathematical analysis; neutral equations; Nonlinear equations; oscillation; Quotients; Time; timescale calculus
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.5636

In the present manuscript, we study the oscillation theory for a first‐order nonlinear neutral dynamic equations on timescales with variable exponents of the form (x(σ(t))−R(t)xξ(t−η))Δ+T(t)∏m=1n|fm(x(t−τm))|αm(t)sign(x(t−τm))=0,∀t∈[t∗,∞)T, where ξ is a quotient of odd positive integers; t∗∈T be a fixed number; [t∗,∞)T is a timescale interval; η,τm > 0; fm∈C(R,R) for m = 1,2,…,n such that xfm(x)>0∀x∈R\{0};R,T∈Crd([t∗,∞)T,R), and the variable exponents αm(t) satisfy ∑m=1nαm(t)=1. The principal goal of this paper is to establish some new succinct sufficient conditions for oscillation. Furthermore, we introduce a forcing term Ξ(·,x(·))∈Crd(T×R,R) and then study the oscillation. Afterward, some interesting special cases are also studied to obtain similar sufficient conditions of oscillation under certain conditions. Moreover, the oscillatory behaviour of the solutions of a first‐order neutral dynamic equation on timescale with a nonlocal condition and a forced nonlinear neutral dynamic equation on time scale are studied. But the proofs are based on the prior estimates obtained in this paper. Some enthralling examples are constructed to show the effectiveness of our analytic results. These counterparts are quite different in the literature even when T=R. Finally, the Kamenev‐type and Philos‐type oscillation criterions are established.