Periodic solutions of non-linear discrete Volterra equations with finite memory |
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Authors: | Christopher T.H. Baker Yihong Song |
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Affiliation: | a Department of Mathematics, University of Chester, CH1 4BJ, UK b School of Mathematics, University of Manchester, UK c Department of Mathematics, Suzhou University, Jiangsu 215006, PR China |
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Abstract: | 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. |
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Keywords: | 65Q05 37M05 39A12 47B34 47B60 |
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