# Time scale extensions of a theorem of Wintner on systems with asymptotic equilibrium

2011-01-01
Mert, R.
Zafer, Ağacık
Abstract We consider quasilinear dynamic systems of the form[image omitted]where is a time scale, and provide extensions of a theorem of Wintner on systems with asymptotic equilibrium to arbitrary time scales. More specifically, we give sufficient conditions for the asymptotic equilibrium of the above system in the sense that for any given constant vector c, there is a solution satisfying[image omitted] Our results are new for difference equations, q-difference equations and many other time scale systems even though their analogous for differential equations have been known for some time.
JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS

# Suggestions

 Interval criteria for second-order super-half-linear functional dynamic equations with delay and advance arguments Anderson, Douglas R.; Zafer, Ağacık (Informa UK Limited, 2010-01-01) Interval oscillation criteria are established for second-order forced super half-linear dynamic equations on time scales containing both delay and advance arguments, where the potentials and forcing term are allowed to change sign. Four discrete examples are provided to illustrate the relevance of the results. The theory can be applied to second-order dynamic equations regardless of the choice of delta or nabla derivatives.
 Oscillation Criteria for Second-Order Forced Dynamic Equations with Mixed Nonlinearities Agarwal, Ravi P.; Zafer, A. (Springer Science and Business Media LLC, 2009) We obtain new oscillation criteria for second-order forced dynamic equations on time scales containing mixed nonlinearities of the form (r(t)Phi(alpha)(x(Delta)))(Delta) + f(t,x(sigma)) = e(t), t is an element of [t(0), infinity)(T) with f (t, x) = q(t) Phi(alpha)(x) + Sigma(n)(i=1)q(i)(t)Phi(beta i)(x), Phi(*)(u) = vertical bar u vertical bar*(-1) u, where [t(0), infinity)(T) is a time scale interval with t(0) is an element of T, the functions r, q, q(i), e : [t(0), infinity)(T) -> R are right-dense contin...
 Oscillation for a nonlinear dynamic system on time scales Erbe, Lynn; Mert, Raziye (Informa UK Limited, 2011-01-01) We study the oscillation properties of a system of two first-order nonlinear equations on time scales. This form includes the classical Emden-Fowler differential and difference equations and many of its extensions. We generalize some well-known results of Atkinson, Belohorec, Waltman, Hooker, Patula and others and also describe the relation to solutions of a delay-dynamic system.
 Oscillation of Second-Order Mixed-Nonlinear Delay Dynamic Equations Unal, M.; Zafer, Ağacık (Springer Science and Business Media LLC, 2010-01-01) New oscillation criteria are established for second-order mixed-nonlinear delay dynamic equations on time scales by utilizing an interval averaging technique. No restriction is imposed on the coefficient functions and the forcing term to be nonnegative.
 Value sets of Lattes maps over finite fields Küçüksakallı, Ömer (Elsevier BV, 2014-10-01) We give an alternative computation of the value sets of Dickson polynomials over finite fields by using a singular cubic curve. Our method is not only simpler but also it can be generalized to the non-singular elliptic case. We determine the value sets of Lattes maps over finite fields which are rational functions induced by isogenies of elliptic curves with complex multiplication.
Citation Formats
R. Mert and A. Zafer, “Time scale extensions of a theorem of Wintner on systems with asymptotic equilibrium,” JOURNAL OF DIFFERENCE EQUATIONS AND APPLICATIONS, pp. 841–857, 2011, Accessed: 00, 2020. [Online]. Available: https://hdl.handle.net/11511/52148.