Kamenev-type Oscillation Criteria Research Articles

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.

New Kamenev-type theorems for super-linear matrix differential equations

By using Riccati transformation and the integral averaging technique, some new Kamenev-type oscillation criteria are established for the super-linear matrix differential systems X ″ ( t ) + ( X n ( t ) Q ( t ) X ∗ n ( t ) ) X ( t ) = 0 and X ″ ( t ) + ( X ∗ n ( t ) Q ( t ) X n ( t ) ) X ( t ) = 0 , t ⩾ t 0 > 0 , n ≥ 1 , where Q ( t ) is an m × m continuous symmetric and positive definite matrix for t ∈ [ t 0 , ∞ ) . The results improve and complement those given by Tomastik [E.C. Tomastik, Oscillation of nonlinear matrix differential equations of second-order, Proc. Amer. Math. Soc. 19 (1968) 1427–1431], Ahlbrandt et al. [C.D. Ahlbrandt, J. Ridenhour, R.C. Thompson, Oscillation of super-linear matrix differential equation, Proc. Amer. Math. Soc. 105 (1989) 141–148] and Ou [L.M. Ou, Atkinson’s super-linear oscillation theorem for matrix dynamic equations on a time scale, J. Math. Anal. Appl. 299 (2004) 615–629], which is illustrated by an example at the end of the paper.

Oscillation of even order partial differential equations with distributed deviating arguments

Some Kamenev-type oscillation criteria are established for a class of boundary value problems associated with even-order partial differential equations with distributed deviating arguments. Our approach is to reduce the high-dimensional oscillation problem to a one-dimensional oscillation one, and the general means developed by Philos and Wong is used as the main tool. The results obtained here extend and improve some known results in the literature.

Kamenev-type oscillation criteria for delay difference equations

Some oscillation criteria are established by Raccati transformation techniques for the following second-order nonlinear neutral difference equation Δ( p n (Δ( x n + c n x n - τ )) γ) + q n x n - σ β = 0,n = 0,1,2 … which extend and include several oscillation criteria in [11], and also correct a theorem and its proof in [10].

Oscillations of solutions of second order quasilinear differential equations with impulses

Some Kamenev-type oscillation criteria are obtained for a second order quasilinear damped differential equation with impulses. These criteria generalize and improve some well-known results for second order differential equations with /and without impulses. In addition, new oscillation criteria are also obtained to generalize and improve known results. Two examples of applications are given to illustrate the theory.

Kamenev-type oscillation criteria for even order neutral differential equations with deviating arguments

Sufficient conditions are established for the oscillation of even order neutral differential equations of the form x ( t ) + ∑ i = 1 m p i ( t ) x ( τ i ( t ) ) ( n ) + ∑ j = 1 l q j ( t ) f j ( x ( σ j ( t ) ) ) = 0 , t ⩾ t 0 , where n ⩾ 2 is even integer.

New Kamenev-type oscillation criteria for second order neutral nonlinear differential equations

In this paper, some new oscillation criteria for second order neutral nonlinear differential equation [ r ( t ) ( y ( t ) + p ( t ) y ( σ ( t ) ) ) ′ ] ′ + ∑ i = 1 n q i ( t ) f i ( y ( τ i ( t ) ) ) = 0 , t ⩾ t 0 are established. New oscillation criteria are different from most known ones in the sense that they are based on a class of new functions Φ( t, s, l) defined in the sequel. Our results are sharper than some previous results which can be seen by the example at the end of this paper.

Fite and Kamenev type oscillation criteria for second order elliptic equations

Fite and Kamenev type oscillation criteria for the second order nonlinear damped elliptic differential equation $$ \sum_{i,j=1}^N D_i[a_{ij}(x)D_j y] + \sum_{i=1}^Nb_i(x)D_iy+p(x)f(y)=0 $$ are obtained. Our results are extensions of those for ordinary di

Oscillation of second order functional differential equations with damping

Using a class of new test functions Φ( t, s, r) defined in our recent work [Y.G. Sun, New Kamenev-type oscillation criteria for second order nonlinear differential equations with damping, J. Math. Anal. Appl. 291 (2004) 341–351], we establish some new oscillation criteria for the second order functional differential equation with damping x ″ ( t ) + p ( t ) x ′ ( t ) + ∫ a b q ( t , ξ ) f x [ g 1 ( t , ξ ) ] , … , x [ g m ( t , ξ ) ] d σ ( ξ ) = 0 . Our results are different from most known ones and can be applied to many cases which are not covered by existing results.

New kamenev-type oscillation criteria for second-order differential equations on a measure chain

We establish new Kamenev-type criteria for oscillation of the second-order linear differential equation, ( p( t) x Δ( t)) Δ + q( t) x( σ( t)) = 0, on a measure chain. Our results are extensions of those for differential equations and provide new oscillation criteria for difference equations. Several examples are given to show the significance of the results.

Kamenev type oscillation criteria for second order matrix differential systems with damping

Kamenev type oscillation criteria for second order matrix differential systems with damping

New Kamenev type theorems for second-order linear matrix differential systems

Some new Kamenev type oscillation criteria are established for the second-order matrix differential system ( P(t)Y′)′ + Q(t)Y = 0, where Y, P, and Q are n × n real continuous matrix functions with P(t), Q(t) symmetric and P(t) > 0 for t ε t 0, ∞) . Our results are sharper than some previous results, and can be applied to the cases which are not covered by known criteria.

New Kamenev-type oscillation criteria for second-order nonlinear differential equations with damping

Some new oscillation criteria are established for the nonlinear damped differential equation ( r( t) y′)′+ p( t) y′+ q( t) f( y)=0 that are different from most known ones in the sense that they are based on a class of new functions Φ( t, s, r) defined in the sequel. Our results are sharper than some previous results which can be seen by the examples at the end of this paper.

New Kamenev type oscillation criteria for linear matrix Hamiltonian systems

Some new Kamenev type criteria have been obtained for the oscillation of the linear matrix Hamiltonian system X ′= A( t) X+ B( t) Y, Y ′= C( t) X− A *( t) Y under the hypothesis: A( t), B( t)= B *( t)>0 and C( t)= C *( t) are n× n real continuous matrix functions on the interval [ t 0,∞) ( t 0>−∞). Our results are different from most known ones in the sense that they are given in the form of lim t→∞ sup g[·]>const. rather than in the form of lim t→∞ sup λ 1[·]=∞, where g is a positive linear functional on the linear space of n× n matrices with real entries. Consequently, our results improve some previous results to some extent, which can be seen by the examples given at the end of this paper.

Kamenev-type oscillation criteria for second-order matrix differential systems

By employing the generalized Riccati technique and the integral averaging technique, new Kamenev-type oscillation criteria are established for a class of second-order matrix differential systems. These criteria extend, improve, and unify a number of existing results and handle the cases which are not covered by known criteria. In particular, two interesting examples that illustrate the importance of our results are included.