Abstract
Abstract Assuming ∫ t 0 ∞ Δ t r 1 / γ ( t ) < ∞ , a new oscillation criterion for a half-linear second-order neutral dynamic equation ( r ( t ) ( ( x ( t ) + p ( t ) x ( t − τ ) ) Δ ) γ ) Δ + q ( t ) | x ( t − δ ) | γ − 1 x ( t − δ ) = 0 is presented. An interesting example is provided to show that the delayed function δ ( t ) = t − δ plays an important role in the oscillatory behavior. MSC:34K11, 34N05, 39A10.
Highlights
1 Introduction This work is concerned with the oscillatory behavior of solutions to a second-order halflinear neutral delay dynamic equation r(t) x(t) + p(t)x(t – τ ) γ + q(t) x(t – δ) γ – x(t – δ) =
Our attention is restricted to those solutions of ( . ) which exist on some half-line [tx, ∞) T and satisfy sup{|x(t)| : t ∈ [t, ∞) T} >
3 Applications Due to Theorem . , we present the following results for oscillation of equations ( . )( . )
Summary
∞t t0 r1/γ (t) < ∞, a new oscillation criterion for a half-linear second-order neutral dynamic equation (r(t)((x(t) + p(t)x(t – τ )) )γ ) + q(t) x(t – δ) γ –1x(t – δ) = 0 is presented. 1 Introduction This work is concerned with the oscillatory behavior of solutions to a second-order halflinear neutral delay dynamic equation r(t) x(t) + p(t)x(t – τ ) γ + q(t) x(t – δ) γ – x(t – δ) = There has been much research activity concerning the oscillatory and nonoscillatory behavior of solutions to various classes of differential, difference, and dynamic equations.
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