Oscillation Tests for Fractional Difference Equations

George E. Chatzarakis 1 , Palaniyappan Gokulraj 2  and Thirunavukarasu Kalaimani 2
  • 1 School of Pedagogical and Technological Education (ASPETE),, Athens, Greece
  • 2 Dhirajlal Gandhi College of Technology, Salem, India

Abstract

In this paper, we study the oscillatory behavior of solutions of the fractional difference equation of the form

Δ(r(t)g(Δαx(t)))+p(t)f(s=t0t1+α(ts1)(α)x(s))=0,t𝕅t0+1α,

where Δα denotes the Riemann-Liouville fractional difference operator of order α, 0 < α ≤ 1, ℕt0+1−α={t0+1−αt0+2−α…}, t0 > 0 and γ > 0 is a quotient of odd positive integers. We establish some oscillatory criteria for the above equation, using the Riccati transformation and Hardy type inequalities. Examples are provided to illustrate the theoretical results.

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