Elvan Akin-Bohner

Oscillation Properties of an Emden-Fowler Type Equation on Discrete Time Scales

In this paper, we explore the oscillation properties of on a time scale T with only isolated points, where p(t) is defined on T and γ is a quotient of odd positive integers. We define oscillation in this setting, and generate conditions on the integral of p(t) which guarantee oscillation and find conditions which give the existence of a nonoscillatory solution. In addition, we consider the case when solutions of this equation has asymptotically positively bounded differences.

On the Asymptotic Integration of Nonlinear Dynamic Equations

The purpose of this paper is to study the existence and asymptotic behavior of solutions to a class of second-order nonlinear dynamic equations on unbounded time scales. Four different results are obtained by using the Banach fixed point theorem, the Boyd and Wong fixed point theorem, the Leray-Schauder nonlinear alternative, and the Schauder fixed point theorem. For each theorem, an illustrative example is presented. The results provide unification and some extensions in the time scale setup of the theory of asymptotic integration of nonlinear equations both in the continuous and discrete cases.

Almost oscillatory three-dimensional dynamical system

In this article, we investigate oscillation and asymptotic properties for 3D systems of dynamic equations. We show the role of nonlinearities and we apply our results to the adjoint dynamic systems.2010 Mathematics Subject Classification: 39A10

Positive decreasing solutions of quasilinear dynamic equations

We consider a quasilinear dynamic equation reducing to a half-linear equation, an Emden-Fowler equation or a Sturm-Liouville equation under some conditions. Any nontrivial solution of the quasilinear dynamic equation is eventually monotone. In other words, it can be either positive decreasing (negative increasing) or positive increasing (negative decreasing). In particular, we investigate the asymptotic behavior of all positive decreasing solutions which are classified according to certain integral conditions. The approach is based on the Tychonov fixed point theorem.

Some Dynamic Equations

In this chapter we consider several dynamic equations and present methods on how to solve these equations. Among them are linear equations of higher order, Euler-Cauchy equations of higher order, logistic equations (or Verhulst equations), Bernoulli equations, Riccati equations, and Clairaut equations.