Periodic solutions of non-linear discrete Volterra equations with finite memory

✍ Scribed by Christopher T.H. Baker; Yihong Song

• Publisher: Elsevier Science
• Year: 2010
• Tongue: English
• Weight: 525 KB
• Volume: 234
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/j.cam.2010.01.019

✦ Synopsis

In this paper we discuss the existence of periodic solutions of discrete (and discretized) non-linear Volterra equations with finite memory. The literature contains a number of results on periodic solutions of non-linear Volterra integral equations with finite memory, of a type that arises in biomathematics. The ''summation'' equations studied here can arise as discrete models in their own right but are (as we demonstrate) of a type that arise from the discretization of such integral equations. Our main results are in two parts: (i) results for discrete equations and (ii) consequences for quadrature methods applied to integral equations. The first set of results are obtained using a variety of fixed-point theorems. The second set of results address the preservation of properties of integral equations on discretizing them. The effect of weak singularities is addressed in a final section. The detail that is presented, which is supplemented using appendices, reflects the differing prerequisites of functional analysis and numerical analysis that contribute to the outcomes.