Kamenev-type Oscillation Criteria Research Articles

Oscillation and asymptotic behavior of third-order nonlinear delay differential equations with positive and negative terms

We study the oscillation and nonoscillation of third order delay differential equations with positive and negative terms. We establish Kiguradze Lemma, and offer the novel and useful estimate x(τ1(t))x(t)≥τ1(t)t which plays an important role in our main results. First of all, we give Leighton–Wintner type criteria. Then, by using Riccati transformation, we establish new oscillation criteria including Kamenev-type oscillation criteria. Finally, we present a sufficient and necessary condition which guarantees that the nonoscillatory solution x(t) has an upper bound and tends to zero. Examples illustrate the validity and practicability of our results.

Kamenev type oscillation criteria for second order impulsive differential equations

The oscillation of second order impulsive differential equations is discussed using Riccati transformations technique. New oscillation criteria are established, to improve and extend some recent results in the literature. Two illustrative examples are given.

Kamenev and Philos-types oscillation criteria for fourth-order neutral differential equations

This work is concerned with the oscillatory behavior of solutions of fourth-order neutral differential equations. By using the Riccati transformation and integral averaging techniques we obtain some new Kamenev-type and Philos-type oscillation criteria. Our results extend and improve some known results in the literature. An example is given to illustrate our main results.

An Approach for Studying Asymptotic Properties of Solutions of Neutral Differential Equations

The paper is devoted to the study of oscillation of even-order neutral differential equations. New Kamenev-type oscillation criteria are established, and they essentially improve and complement some the well-known results reported in the literature. Ideas of symmetry help us determine the correct ways to study these topics and show us the correct direction, because they are often invisible. To illustrate the main results, some examples are mentioned.

Kamenev Type Oscillatory Criteria for Linear Conformable Fractional Differential Equations

Using integral average method and properties of conformable fractional derivative, new Kamenev type oscillation criteria are given firstly for conformable fractional differential equations, which improve known results in oscillation theory. Examples are also given to illustrate the effectiveness of the main results.

Kamenev-Type Oscillation Criteria for Generalized Second Order Sublinear -Difference Equations

By means of Riccati transformation techniques, authors establish some new oscillation criteria for generalized second order nonlinear -difference equation when and are quotient of odd positive integers.

New oscillation criteria of special type second-order non-linear dynamic equations on time scales

By the Riccati transformation technique, we study some new oscillatory properties for the second-order dynamic equation on an arbitrary time scale mathbb {T}. We also establish the Kamenev-type and Philos-type oscillation criteria. At the end, we give examples which illustrate our main results.

Kamenev-Type oscillation criteria for second order matrix differential systems with damping

By using the positive linear functional, including the generalized averaging technique, some new Kamenev-type oscillation criteria are established for the second order matrix differential system \[ (r(t)P(t)\psi(X(t))K(X'(t)))'+p(t)R(t)\psi(X(t))K(X'(t))+Q(t)F(X'(t))G(X(t))=0. \] The results improve and generalize those given in some previous papers.

The Kamenev type oscillation criteria of nonlinear differential equations under impulse effects

In this paper, a general class of second-order nonlinear damped differential equations under impulse effects is studied. By introducing an auxiliary function of two variables and using Riccati transformation, several Kamenev type interval oscillation criteria are established, which generalize and extend some known ones, such as those of Huang and Feng. Moreover, examples are given to illustrate the effectiveness and non-emptiness of our results.

Generalized Philos-type oscillation criteria for second order nonlinear neutral delay dynamic equations on time scales

By employing the generalized Riccati transformation w(s) and showing that [H(σ(t),s)w(s)]Δs≤0, we establish new Philos-type oscillation criteria for second order nonlinear dynamic equations on time scales. The obtained results essentially generalize and improve the well-known oscillation results for half-linear dynamic equations such as Philos-type and Kamenev-type oscillation criteria. We illustrate the versatility of our results by means of examples.

Oscillation of certain third-order quasilinear neutral differential equations

Abstract In this paper, new oscillation criteria for the third-order quasilinear neutral differential equation $$\left( {a\left( t \right)\left( {z''\left( t \right)} \right)^\gamma } \right)^\prime + q\left( t \right)x^\gamma \left( {\tau \left( t \right)} \right) = 0, t \geqslant t_0 ,$$ are established, where z(t) = x(t) + p(t)x(δ(t)), and γ is a ratio of odd positive integers. Those results extend the oscillation criteria due to Sun [SUN, Y. G.: New Kamenev-type oscillation criteria for second-order nonlinear differential equations with damping, J. Math. Anal. Appl. 291 (2004) 341–351] to the equation, and complement the existing results in literature. Two examples are provided to illustrate the relevance of our main theorems.

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

Using functions in some function classes and a generalized Riccati technique, we establish Kamenev-type oscillation criteria for second-order dynamic equations with damping on time scales of the form(r(t)(xΔ(t))γ)Δ+p(t)(xΔ(t)γ)+f(t,x(g(t)))=0. Two examples are included to show the significance of the results.

New Kamenev-type oscillation criteria for second-order nonlinear advanced dynamic equations

This paper presents a new Kamenev-type oscillation criterion for a second-order advanced dynamic equation on a time scale. Two examples are included to show that the result obtained complements and improves those reported in the literature.

Kamenev-type oscillation criteria for higher-order nonlinear dynamic equations on time scales

Abstract In this paper, we investigate the oscillation of the following higher-order dynamic equation: { r n ( t ) [ ( r n − 1 ( t ) ( ⋯ ( r 1 ( t ) x △ ( t ) ) △ ⋯ ) △ ) △ ] γ } △ + F ( t , x ( τ ( t ) ) ) = 0 on an arbitrary time scale T, where n ≥ 2 , 1 r k ( t ) ( 1 ≤ k ≤ n ) are positive rd-continuous functions on T, and γ is the quotient of two odd positive integers, τ : T → T with τ ( t ) > t and F ∈ C ( T × R , R ) . We give sufficient conditions under which every solution of this equation is either oscillatory or tends to zero. MSC:34K11, 39A10, 39A99.

Kamenev-Type Oscillation Criteria of Second-Order Nonlinear Dynamic Equations on Time Scales

Using functions from some function classes and a generalized Riccati technique, we establish Kamenev-type oscillation criteria for second-order nonlinear dynamic equations on time scales of the form(p(t)ψ(x(t))k∘xΔ(t))Δ+f(t,x(σ(t)))=0. Two examples are included to show the significance of the results.

Oscillation of impulsive functional differential equations with oscillatory potentials and Riemann-Stieltjes integrals

This paper addresses the oscillation problem of a class of impulsive differential equations with delays and Riemann-Stieltjes integrals that cover many equations in the literature. In the case of oscillatory potentials, both El-Sayed type and Kamenev type oscillation criteria are established by overcoming the difficulty caused by impulses and oscillatory potentials in the estimation of the delayed argument. The main results not only generalize some existing results but also drop a restrictive condition imposed on impulse constants. Finally, two examples are presented to illustrate the theoretical results. MSC:34K11.

New Kamenev-type oscillation criteria for half-linear partial differential equations

We establish new Kamenev-type oscillation criteria for the half-linear partial differential equation with damping $$div(A(x)\left\| {\nabla u} \right\|^{p - 2} \nabla u) + \left\langle {b(x),\left\| {\nabla u} \right\|^{p - 2} \nabla u} \right\rangle + c(x)\left| u \right|^{p - 2} u = 0$$ (E) under quite general conditions. These results are extensions of the recent results developed by Sun [Y.G. Sun, New Kamenev-type oscillation criteria of second order nonlinear differential equations with damping, J. Math. Anal. Appl. 291 (2004) 341–351] for second order ordinary differential equations in a natural way, and improve some existing results in the literature. As applications, we illustrate our main results using two different types of half-linear partial differential equations.

Interval criteria for oscillation of a forced impulsive differential equation with Riemann–Stieltjes integral

In this paper, both El-Sayed type and Kamenev type oscillation criteria for a forced impulsive differential equation with Riemann–Stieltjes integral are established. By using a generalized El-Sayed type function and Kong’s technique in terms of the number of impulse moments on a series of intervals, we not only drop the restriction on the impulse constants ck and dk that dk≥ck in the literature, but also extend some existing results to the case of Riemann–Stieltjes integral and improve some results in the case of Riemann integral. Two examples are also considered to illustrate the main results.

Kamenev-type oscillation criteria for second order nonlinear dynamic equations on time scales

The purpose of this paper to establish oscillation criteria for second order nonlinear dynamic equation(r(t)(xΔ(t))γ)Δ+f(t,x(g(t)))=0,on an arbitrary time scale T, where γ is a quotient of odd positive integers and r is a positive rd-continuous function on T. The function g:T→T satisfies g(t)⩾t and limt→∞g(t)=∞ and f∈C(T×R,R). We establish some new sufficient conditions such that the above equation is oscillatory by using generalized Riccati transformation. Our results in the special cases when T=R and T=N involve and improve some oscillation results for second-order differential and difference equations; and when T=hN,T=qN0 and T=N2 our oscillation results are essentially new. Some examples illustrating the importance of our results are included.

Oscillation of second order semilinear elliptic equations with damping

We obtain some new Kamenev-type oscillation theorems for the second order semilinear elliptic differential equation with damping $$ \sum\limits_{i,j = 1}^N {D\left[ {a_{ij} \left( x \right)D_j y} \right] + } \sum\limits_{i = 1}^N {b_i \left( x \right)D_i y + c\left( x \right)f\left( y \right) = 0} $$ under quite general assumptions. These results are extensions of the recent results of Sun [Sun, Y. G.: New Kamenev-type oscillation criteria of second order nonlinear differential equations with damping. J. Math. Anal. Appl., 291, 341–351 (2004)] in a natural way. In particular, we do not impose any additional conditions on the damped functions bi(x) except the continuity. Several examples are given to illustrate the main results.