Devrim Çakmak

Lyapunov-type integral inequalities for certain higher order differential equations

In this paper, first we will give a short survey of the most basic results on Lyapunov inequality, and next we obtain this-type integral inequalities for certain higher order differential equations. Our results are sharper than some results of Yang (2003) [20].

Oscillation criteria for certain forced second-order nonlinear differential equations

Abstract In this paper, two new oscillation criteria for forced second-order nonlinear differential equations of the form r(t)ψ(y(t))|y′(t)| α−1 y′(t) ′ + q(t)f(y(t))=e(t), t≥t o , are established. Our results are based on the information on a sequence of subintervals of [t0, ∞) only, rather than on the whole half-line. Our methodology is somewhat different from that of previous authors. The results presented here are much more general than a recent result of Li and Cheng [1].

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].

On the oscillation of certain second-order nonlinear differential equations

Abstract In this paper, oscillation criteria for the nonlinear second-order ordinary differential equation r(t)Ψ x(t) x′(t) p−2 x′(t) ′+c(t)f x(t) =0 are given. The results extend the integral averaging technique and include earlier results. Our methodology is somewhat different from that of previous authors.

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.

Lyapunov-type inequalities for two classes of nonlinear systems with anti-periodic boundary conditions

In this paper, we establish new Lyapunov-type inequalities for nonlinear cycled and strongly coupled systems with anti-periodic boundary conditions.

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.

Oscillation criteria for nth-order forced functional differential equations

In this paper, by using the method of general means, some new oscillation criteria for forced functional differential equations of the form x(n)(t)+∑i=1n−1aix(i)(t)+q(t)fxg(t)=e(t),t0⩾0, are established, where ai are real constants, q(t), f(t), e(t), and g(t) are real continuous functions, xf(x)>0 whenever x≠0, and limt→∞g(t)=∞.

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.