E. Thandapani

On the oscillation of certain third order nonlinear functional differential equations

Abstract We offer some sufficient conditions for the oscillation of all solutions of third order nonlinear functional differential equations of the form d d t a ( t ) d 2 d t 2 x ( t ) α + q ( t ) f ( x [ g ( t ) ] ) = 0 and d d t a ( t ) d 2 d t 2 x ( t ) α = q ( t ) f ( x [ g ( t ) ] ) + p ( t ) h ( x [ σ ( t ) ] ) , when ∫ ∞ a - 1 / α ( s ) d s ∞ . The case when ∫ ∞ a - 1 / α ( s ) d s = ∞ is also included.

On the oscillation of higher-order half-linear delay differential equations

Abstract In this note, we study the oscillatory behavior of the following higher-order half-linear delay differential equation ( r ( t ) ( x ( n − 1 ) ( t ) ) α ) ′ + q ( t ) x β ( τ ( t ) ) = 0 , t ≥ t 0 , where we assume ∫ t 0 ∞ 1 r 1 / α ( t ) d t ∞ . An example is given to illustrate the main results.

Fourt-order nonlinear oscillations of difference equations

Abstract The authors consider the fourth-order difference equation where f ( n, u ) may be classified as superlinear, sublinear, strongly superlinear, and strongly sublinear and Σ ∞ n=n 0 ( n / r n )

Oscillation of two-dimensional difference systems

Abstract The authors consider the nonlinear two-dimensional difference system Δx n = b n g ( y n ), Δy n −1 =− a n f ( x n , ϵN ( n o ={ no , no +1…}, where n 0 ϵ N = 1, 2, …, a n and b n , n ϵ N ( n 0 ), are real sequences, and f , g : R → R are continuous with uf ( u ) > 0 and ug ( u ) > 0 for u ≠ 0. A solution ( x n , y n ) of the system is oscillatory if both components are oscillatory. The authors obtain sufficient conditions for all solutions of the system to be oscillatory. Some of their results allow a n to oscillate. Examples to illustrate the results are included.

Monotone properties of certain classes of solutions of second-order difference equations

The authors consider the difference equations (*)Δ(anΔxn)=qnxn+1 and (**)Δ(anΔxn)=qnf(xn+1) where an > 0, qn > 0, and f: R→R is continuous with uf(u) > 0 for u ≠ 0. They obtain necessary and sufficient conditions for the asymptotic behavior of certain types of nonoscillatory solutions of (*) and sufficient conditions for the asymptotic behavior of certain types of nonoscillatory solutions of (**). Sufficient conditions for the existence of these types of nonoscillatory solutions are also presented. Some examples illustrating the results and suggestions for further research are included.

Oscillation and comparison theorems for half-linear second-order difference equations

Abstract The authors consider second-order difference equations of the type where α > 0 is the ratio of odd positive integers, { q n } is a positive sequence, and {σ( n )} is a positive increasing sequence of integers with σ( n ) → ∞ as n → ∞. They give some oscillation and comparison results for equation (E).

Bounded and monotone properties of solutions of second-order quasilinear forced difference equations

Abstract The authors consider the quasilinear difference equation δ(p n−1 (δy n−1 )α)=q n y β n +r n and obtain results on the asymptotic behavior of solutions of (∗) including sufficient conditions for all solutions to be bounded or unbounded. Some results on the existence and behavior of nonincreasing solutions of (∗) are also obtained. Examples are inserted to illustrate the results.

Oscillatory and asymptotic behavior of solutions of nonlinear neutral-type difference equations

The authors consider the higher-order nonlinear neutral delay difference equation and obtain results on the asymptotic behavior of solutions when ( p n ) is allowed to oscillate about the bifurcation value –1. We also consider the case where the sequence { p n } has arbitrarily large zeros. Examples illustrating the results are included, and suggestions for further research are indicated.

On some new discrete inequalities

Some new discrete inequalities involving higher order differences have been obtained here. These inequalities can be used in the analysis of a class of summary difference equations as handy tools. Some applications are also given.

Asymptotic results for a class of fourth order quasilinear difference equations

The authors consider the fourth order quasilinear difference equation where α and β are positive constants and {p n } and {q n } are positive real sequences. They classify the nonoscillatory solutions according to their asymptotic behavior for large n and then give necessary and sufficient conditions for existence of solutions of these various types. The results are illustrated with examples.