Aydın Tiryaki

Oscillation criteria for third-order nonlinear functional differential equations

Abstract In this work, we are concerned with oscillation of third-order nonlinear functional differential equations of the form ( r 2 ( t ) ( r 1 ( t ) y ′ ) ′ ) ′ + p ( t ) y ′ + q ( t ) f ( y ( g ( t ) ) ) = 0 . By using a Riccati type transformation and integral averaging technique, we establish some new sufficient conditions under which every solution y ( t ) either oscillates or converges to zero as t → ∞ . Unlike ones from the known works in the literature, our results are applicable to nonlinear functional differential equations of the above form when f ( u ) = | u | α − 1 u , α > 0 .

On Lyapunov-type inequality for quasilinear systems

In this paper, we sketch some recent developments in the theory of Lyapunov-type inequalities and present some new results relating to a quasilinear system, special cases of which contain some well-known differential equations such as half-linear and linear equations. Our result generalize the Lyapunov-type inequality given in [18].

Lyapunov-type inequalities for a certain class of nonlinear systems

In this paper, we state and prove some new Lyapunov-type inequalities for a class of nonlinear systems, special cases of which contain the well-known Hamiltonian system, Emden-Fowler, half-linear and linear differential equations of second order. Our results improve and generalize these types of inequalities related to all existing ones.

A note on Tang and He’s paper

Abstract In this paper, we establish new Lyapunov-type inequalities for two classes of one-dimensional quasilinear elliptic systems of resonant type, which improve the recent results of Tang and He [X.H. Tang, X. He, Lower bounds for generalized eigenvalues of the quasilinear systems, J. Math. Anal. Appl. 385 (2012) 72–85] when 1xa0 p i i xa0=xa01,xa02,xa0…xa0,xa0 n .

Oscillation criteria for a certain second-order nonlinear perturbed differential equations

In this paper, a class of second-order nonlinear perturbed differential equation and its special cases are studied. By using the generalized Riccati transformation and well-known techniques, some new oscillation criteria are established. The results obtained essentially generalize and improve some known results and can be applied to the well-known half-linear and damped half-linear-type equations.

On the Lyapunov-type inequalities of a three-point boundary value problem for third order linear differential equations

Abstract In this paper, by using Green’s function for second order differential equations with Dirichlet boundary condition, we establish new Lyapunov-type inequalities for third order linear differential equation which improve all existing results in the literature. As an application, we obtain sharp lower bound for the eigenvalues of corresponding equations and for the distance between the end points in the three consecutive zeros of the solution of the equations.

On the qualitative behaviors of solutions of third order nonlinear differential equations

Abstract We present some new results about oscillation and asymptotic behavior of solutions of third order nonlinear differential equations of the form ( r 2 ( t ) ( r 1 ( t ) y ′ ) ′ ) ′ + p ( t ) y ′ + q ( t ) f ( y ( g ( t ) ) ) = 0 . Our work is significantly different from the previous works and the results of this paper improve, extend, and complement a number of existing results in the literature. Several examples are also given to illustrate the importance of our results.

On the qualitative behaviors of solutions of third-order nonlinear functional differential equations

Abstract The third-order nonlinear functional differential equations of the form ( r 2 ( t ) ( r 1 ( t ) y ′ ) ′ ) ′ + p ( t ) y ′ + q ( t ) f ( y ( g ( t ) ) ) = 0 are considered. We present some new oscillatory and asymptotic behavior of solutions of this equation by modifying a method given for second-order differential equations. Our results are applicable to nonlinear functional differential equations of the above form. Several examples are also given to illustrate the importance of our results.

Sahoo- and Wayment-Type Integral Mean Value Theorems.

In this article, by using Rolles theorem, we establish some results related to the mean value theorem for integrals. Our results are different from the set of integral mean value theorems which are given by Wayment [An integral mean value theorem, Math. Gazette 54 (1970), pp. 300–301] and Sahoo [Some results related to the integral mean value theorem, Int. J. Math. Ed. Sci. Tech. 38(6) (2007), pp. 818–822]. The importance of our results are illustrated by interesting examples.

Reducibility and Stability Results for Linear System of Difference Equations

We first give a theorem on the reducibility of linear system of difference equations of the form . Next, by the means of Floquet theory, we obtain some stability results. Moreover, some examples are given to illustrate the importance of the results.