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August  2020, 25(8): 3033-3048. doi: 10.3934/dcdsb.2020050

The limits of solutions of a linear delay integral equation

Department of Mathematical Sciences, Osaka Prefecture University, Sakai 599-8531, Japan

* Corresponding author: Hideaki Matsunaga

Received  March 2019 Revised  October 2019 Published  February 2020

In this paper we classify the limits of solutions of a linear integral equation with finite delay. In particular, if the solution tends to a point or a periodic orbit, we establish the explicit expressions depending on given initial functions by using analysis of characteristic roots and the formal adjoint theory. Our results also present a necessary and sufficient condition for the exponential stability of the equation.

Citation: Kazuki Himoto, Hideaki Matsunaga. The limits of solutions of a linear delay integral equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3033-3048. doi: 10.3934/dcdsb.2020050
References:
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H. MatsunagaS. Murakami and V. M. Nguyen, Decomposition and variation-of-constants formula in the phase space for integral equations, Funkcial. Ekvac., 55 (2012), 479-520.  doi: 10.1619/fesi.55.479.  Google Scholar

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H. MatsunagaS. Murakami and Y. Nagabuchi, Formal adjoint operators and asymptotic formula for solutions of autonomous linear integral equations, J. Math. Anal. Appl., 410 (2014), 807-826.  doi: 10.1016/j.jmaa.2013.08.035.  Google Scholar

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D. Melchor-Aguilar, On stability of integral delay systems, Appl. Math. Comput., 217 (2010), 3578-3584.  doi: 10.1016/j.amc.2010.08.058.  Google Scholar

show all references

References:
[1]

T. AlarcónP. Getto and Y. Nakata, Stability analysis of a renewal equation for cell population dynamics with quiescence, SIAM J. Appl. Math., 74 (2014), 1266-1297.  doi: 10.1137/130940438.  Google Scholar

[2]

T. A. Burton, Volterra Integral and Differential Equations, Mathematics in Science and Engineering, 202, Elsevier B. V., Amsterdam, 2005.  Google Scholar

[3] C. Corduneanu, Integral Equations and Applications, Cambridge University Press, Cambridge, 1991.  doi: 10.1017/CBO9780511569395.  Google Scholar
[4]

O. Diekmann and M. Gyllenberg, Equations with infinite delay: Blending the abstract and the concrete, J. Differential Equations, 252 (2012), 819-851.  doi: 10.1016/j.jde.2011.09.038.  Google Scholar

[5] G. GripenbergS.-O. Londen and O. Staffans, Volterra Integral and Functional Equations, Encyclopedia of Mathematics and its Applications, 34, Cambridge University Press, Cambridge, 1990.  doi: 10.1017/CBO9780511662805.  Google Scholar
[6]

H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology, Springer, Singapore, 2017. doi: 10.1007/978-981-10-0188-8.  Google Scholar

[7]

H. MatsunagaS. Murakami and V. M. Nguyen, Decomposition and variation-of-constants formula in the phase space for integral equations, Funkcial. Ekvac., 55 (2012), 479-520.  doi: 10.1619/fesi.55.479.  Google Scholar

[8]

H. MatsunagaS. Murakami and Y. Nagabuchi, Formal adjoint operators and asymptotic formula for solutions of autonomous linear integral equations, J. Math. Anal. Appl., 410 (2014), 807-826.  doi: 10.1016/j.jmaa.2013.08.035.  Google Scholar

[9]

D. Melchor-Aguilar, On stability of integral delay systems, Appl. Math. Comput., 217 (2010), 3578-3584.  doi: 10.1016/j.amc.2010.08.058.  Google Scholar

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