Baoguo Jia

Kiguradze-type Oscillation Theorems for Second Order Superlinear Dynamic Equations on Time Scales

Consider the second order superlinear dynamic equation (�) x �� (t) + p(t)f(x(�(t))) = 0 where p 2 C(T;R), T is a time scale, f : R ! R is continuously differentiable and satisfies f 0 (x) > 0, and xf(x) > 0 for x 6 0. Furthermore, f(x) also satisfies a superlinear condition, which includes the nonlinear function f(x) = xwith � > 1, commonly known as the Emden-Fowler case. Here the coefficient function p(t) is allowed to be negative for arbitrarily large values of t. In addition to extending the result of Kiguradze for (�) in the real case T = R, we obtain analogues in the difference equation and q-difference equation cases.

Monotonicity results for delta fractional differences revisited

Abstract In this paper, by means of a recently obtained inequality, we study the delta fractional difference, and we obtain the following interrelated theorems, which improve recent results in the literature. Theorem A Assume that f : ℕa → ℝ and that Δaνf(t)

Monotonicity and convexity for nabla fractional (q, h)-differences

\Delta^\nu_af(t)

Asymptotic behavior of solutions of fractional nabla q-difference equations

≥ 0, for each t ∈ ℕa+2−ν, with 1 < ν < 2. If f(a+1)≥νk+2f(a),

Liapunov functional and stability of linear nabla (q, h)-fractional difference equations

f(a+1) \geq \frac{\nu}{k+2}f(a),

A Butler-type oscillation theorem for second-order dynamic equations on discrete timescales

for each k ∈ ℕ0, then Δ f(t) ≥ 0 for t ∈ ℕa+1. Theorem B Assume that f : ℕa → ℝ and that Δaνf(t)

Oscillation of Certain Emden-Fowler Dynamic Equations on Time Scales

\Delta^\nu_af(t)

Oscillation of Second-Order Sublinear Dynamic Equations with Damping on Isolated Time Scales

≥ 0, for each t ∈ ℕa+2−ν, with 1 < ν < 2. If f(a+2)≥νk+1f(a+1)+(k+1−ν)ν(k+2)(k+3)f(a)

Survey of the qualitative properties of fractional difference operators: monotonicity, convexity, and asymptotic behavior of solutions

f(a+2)\geq\displaystyle\frac{\nu}{k+1}f(a+1)+\frac{(k+1-\nu)\nu}{(k+2)(k+3)}f(a)