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March  2015, 14(2): 397-406. doi: 10.3934/cpaa.2015.14.397

## On the asymptotic stability of Volterra functional equations with vanishing delays

 1 Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR 2 Department of Mathematics and Statistics, Memorial University of Newfoundland, St.Johns, Newfoundland, A1C 5S7

Received  November 2013 Revised  July 2014 Published  December 2014

We analyze the asymptotic stability of solutions of linear Volterra integral equations with general continuous convolution kernels and vanishing delays. The analysis is based on an extension of the variation-of-parameter formula for non-delay Volterra integral equations and on energy function techniques. The delay integral equations studied in this paper will be of interest in the (still open) stability analysis of numerical methods (e.g. collocation and Runge-Kutta-type methods) for Volterra integral equations with vanishing delays.
Citation: Hermann Brunner, Chunhua Ou. On the asymptotic stability of Volterra functional equations with vanishing delays. Communications on Pure & Applied Analysis, 2015, 14 (2) : 397-406. doi: 10.3934/cpaa.2015.14.397
##### References:
 [1] A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations,, Clarendon Press, (2003).  doi: 10.1093/acprof:oso/9780198506546.001.0001.  Google Scholar [2] H. Brunner, The numerical analysis of functional integral and integro-differential equations of Volterra type,, {\em Acta Numer.}, 13 (2004), 55.  doi: 10.1017/CBO9780511569975.002.  Google Scholar [3] H. Brunner, Current work and open problems in the numerical analysis of Volterra functional equations with vanishing delays,, {\em Front. Math. China}, 4 (2009), 3.  doi: 10.1007/s11464-009-0001-0.  Google Scholar [4] H. Brunner and H. Liang, Stability of collocation methods for delay differential equations with vanishing delays,, \emph{BIT Numer. Math.}, 50 (2010), 693.  doi: 10.1007/s10543-010-0285-1.  Google Scholar [5] A. Iserles, On the generalized pantograph functional differential equation,, \emph{Europ. J. Appl. Math.}, 4 (1993), 1.  doi: 10.1017/S0956792500000966.  Google Scholar [6] A. Iserles and J. Terjéki, Stability and asymptotic stability of functional-differential equations,, \emph{J. London Math. Soc.}, 51 (1995), 559.  doi: 10.1112/jlms/51.3.559.  Google Scholar [7] T. Kato and J. B. McLeod, The functional-differential equation $y'(x)=ay(\lambda x)+by(x)$,, \emph{Bull. Amer. Math. Soc.}, 77 (1970), 891.   Google Scholar

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##### References:
 [1] A. Bellen and M. Zennaro, Numerical Methods for Delay Differential Equations,, Clarendon Press, (2003).  doi: 10.1093/acprof:oso/9780198506546.001.0001.  Google Scholar [2] H. Brunner, The numerical analysis of functional integral and integro-differential equations of Volterra type,, {\em Acta Numer.}, 13 (2004), 55.  doi: 10.1017/CBO9780511569975.002.  Google Scholar [3] H. Brunner, Current work and open problems in the numerical analysis of Volterra functional equations with vanishing delays,, {\em Front. Math. China}, 4 (2009), 3.  doi: 10.1007/s11464-009-0001-0.  Google Scholar [4] H. Brunner and H. Liang, Stability of collocation methods for delay differential equations with vanishing delays,, \emph{BIT Numer. Math.}, 50 (2010), 693.  doi: 10.1007/s10543-010-0285-1.  Google Scholar [5] A. Iserles, On the generalized pantograph functional differential equation,, \emph{Europ. J. Appl. Math.}, 4 (1993), 1.  doi: 10.1017/S0956792500000966.  Google Scholar [6] A. Iserles and J. Terjéki, Stability and asymptotic stability of functional-differential equations,, \emph{J. London Math. Soc.}, 51 (1995), 559.  doi: 10.1112/jlms/51.3.559.  Google Scholar [7] T. Kato and J. B. McLeod, The functional-differential equation $y'(x)=ay(\lambda x)+by(x)$,, \emph{Bull. Amer. Math. Soc.}, 77 (1970), 891.   Google Scholar
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