Second-order Half-linear Dynamic Equations Research Articles

Fite–Hille–Wintner-type oscillation criteria for second-order half-linear dynamic equations with deviating arguments

We study oscillatory behavior of solutions to a class of second-order half-linear dynamic equations with deviating arguments under the assumptions that allow applications to dynamic equations with delayed and advanced arguments. Several improved Fite–Hille–Wintner-type criteria are obtained that do not need some restrictive assumptions required in related results. Illustrative examples and conclusions are presented to show that these criteria are sharp for differential equations and provide sharper estimates for oscillation of corresponding q-difference equations.

Oscillation of Second-Order Half-Linear Dynamic Equations on Time Scales

Oscillation of Second-Order Half-Linear Dynamic Equations on Time Scales

On the oscillation of second-order half-linear dynamic equations1

We obtain some oscillation criteria for solutions to the second-order half-linear dynamic equation when or . These criteria unify and extend known criteria for corresponding half-linear differential and difference equations. Some of our results are new even in the continuous and the discrete cases.

New comparison and oscillation theorems for second-order half-linear dynamic equations on time scales

Let T be a time scale (i.e., a closed nonempty subset of R ) with sup T = + ∞ . Consider the second-order half-linear dynamic equation ( r ( t ) ( x Δ ( t ) ) α ) Δ + p ( t ) x α ( σ ( t ) ) = 0 , where r ( t ) > 0 , p ( t ) are continuous, ∫ t 0 ∞ ( r ( t ) ) − 1 α Δ t = ∞ , α is a quotient of odd positive integers. In particular, no explicit sign assumptions are made with respect to the coefficient p ( t ) . We give conditions under which every positive solution of the equations is strictly increasing. For α = 1 , T = R , the result improves the original theorem [see: [Lynn Erbe, Oscillation theorems for second-order linear differential equation, Pacific J. Math. 35 (2) (1970) 337–343]]. As applications, we get two comparison theorems and an oscillation theorem for half-linear dynamic equations which improve and extend earlier results. Some examples are given to illustrate our theorems.

Oscillation criteria for half-linear dynamic equations on time scales

This paper is concerned with oscillation of the second-order half-linear dynamic equation ( r ( t ) ( x Δ ) γ ) Δ + p ( t ) x γ ( t ) = 0 , on a time scale T where γ is the quotient of odd positive integers, r ( t ) and p ( t ) are positive rd-continuous functions on T . Our results solve a problem posed by [R.P. Agarwal, D. O'Regan, S.H. Saker, Philos-type oscillation criteria for second-order half linear dynamic equations, Rocky Mountain J. Math. 37 (2007) 1085–1104; S.H. Saker, Oscillation criteria of second order half-linear dynamic equations on time scales, J. Comput. Appl. Math. 177 (2005) 375–387] and our results in the special cases when T = R and T = Z involve and improve some oscillation results for second-order differential and difference equations; and when T = h Z , T = q N 0 and T = N 0 2 , etc., our oscillation results are essentially new. Some examples illustrating the importance of our results are also included.

Oscillation criteria of second-order half-linear dynamic equations on time scales

In this paper, by using the Riccati transformation technique, chain rule and inequality A λ - λ AB λ - 1 + ( λ - 1 ) B λ ⩾ 0 , λ > 1 , where A and B are positive constants, we will establish some oscillation criteria for the second-order half-linear dynamic equation ( p ( t ) ( x Δ ( t ) ) γ ) Δ + q ( t ) x γ ( t ) = 0 , t ∈ [ a , b ] on time scales, where γ > 1 is an odd positive integer. Our results not only unify the oscillation of half-linear differential and half-linear difference equations but can be applied on different types of time scales and improve some well-known results in the difference equation case. Some examples are considered here to illustrate our main results.

Positive solutions of second-order half-linear dynamic equations on time scales

Necessary and sufficient conditions are obtained for the existence of positive solutions of a second-order half-linear dynamic equation on time scales. Relations between this equation and an advanced type dynamic equation are also discussed.