Volterra integral equations on time scales: stability under constant perturbations via Liapunov direct method

Abstract

In this paper we consider Volterra integral equations on time scales and describe our study about the long time behavior of their solutions. We provide sufficient conditions for the stability under constant perturbations by using the direct Lyapunov method and we present some examples of application

Nonsmooth spin densities for continuous Heisenberg spin chains

“…Although we shall not pursue such an asymptotic analysis in this paper, it could in principle be used to analyze Heisenberg spin chains, where the spin density has jump discontinuities. We observe that recently the study of the long time behavior of the Volterra equations in a different, but significative, context has been performed in [6,15]. At the end of the paper we prove the representations (7) under the assumptions made in [7] instead of under the more general assumptions of Sect.…”

Regularization and uniqueness of solutions of Volterra linear integral equations of the third kind

“…In particular, Volterra integral equations were used to calculate dynamics of multi span uniform continuous beams subjected to a moving load [1], for population growth of a species within a closed system [2], and for duality theory solutions [3]. Recently the Volterra integral equations have been studied and reported for the oscillation of equations with positive and negative nonlinearities [4], Volterra integral equations and evolution equations with an integral condition [5], and on time scales -the long time behavior of solutions [6]. A noticeable moment to the development of this area was related to the research [7], which…”