Murakami, Satoru; Ngoc, Pham Huu Anh On stability and robust stability of positive linear Volterra equations in Banach lattices. (English) Zbl 1218.45013 Cent. Eur. J. Math. 8, No. 5, 966-984 (2010). The authors first introduce the notion of positive linear Volterra integro-differential equations in Banach lattices. Then they give a characterization of positive linear Volterra equations in terms of positivity of the co-semigroup generated by \(A\), where \(A\) is the infinitesimal generator of a co-semigroup, and positivity of the kernel function. Furthermore, an explicit spectral criterion for uniformly asymptotic stability of positive equations is presented. Finally, they deal with problems of robust stability of positive systems under structured perturbations. Some explicit stability bounds with respect to these perturbations are given. An example is given to illustrate the obtained results. Their analysis is based on the theory of positive co-semigroups on Banach lattices. Reviewer: Kun Soo Chang (Seoul) MSC: 45N05 Abstract integral equations, integral equations in abstract spaces 45M10 Stability theory for integral equations 45J05 Integro-ordinary differential equations 47D06 One-parameter semigroups and linear evolution equations 45D05 Volterra integral equations 45M20 Positive solutions of integral equations 46B42 Banach lattices Keywords:Banach lattice; Volterra integro-differential equation; positive system; stability; robust stability; positive co-semigroups; asymptotic stability PDFBibTeX XMLCite \textit{S. Murakami} and \textit{P. H. A. Ngoc}, Cent. Eur. J. Math. 8, No. 5, 966--984 (2010; Zbl 1218.45013) Full Text: DOI References: [1] Engel K.-J., Nagel R., One-Parameter Semigroups for Linear Evolution Equations, Grad. Texts in Math., 194, Springer, Berlin, 2000; · Zbl 0952.47036 [2] Gripenberg G., Londen L.O., Staffans O.J., Volterra Integral and Functional Equations, Encyclopedia Math. Appl., 34, Cambridge University Press, Cambridge, 1990; · Zbl 0695.45002 [3] Henríquez H. R., Periodic solutions of quasi-linear partial functional differential equations with unbounded delay, Funkcial. Ekvac., 1994, 37(2), 329-343; · Zbl 0814.35141 [4] Hille E., Phillips R.S., Functional Analysis and Semigroups, Amer. Math. Soc. Colloq. Publ., 31, AMS, Providence, 1957; · Zbl 0078.10004 [5] Hino Y., Murakami S., Stability properties of linear Volterra integrodifferential equations in a Banach space, Funkcial. Ekvac., 2005, 48(3), 367-392 http://dx.doi.org/10.1619/fesi.48.367; · Zbl 1113.45013 [6] Kantorovich L.V., Akilov G.P., Functional Analysis, Pergamon Press, 1982; · Zbl 0484.46003 [7] Meyer-Nieberg P., Banach Lattices, Universitext, Springer, Berlin, 1991; [8] Nagel R. (ed.), One-parameter Semigroups of Positive Operators, Lecture Notes in Math., 1184, Springer, Berlin, 1986; [9] Ngoc P.H.A., Son N.K., Stability radii of linear systems under multi-perturbations, Numer. Funct. Anal. Optim., 2004, 25(3-4), 221-238 http://dx.doi.org/10.1081/NFA-120039610; · Zbl 1071.34050 [10] Ngoc P.H.A., Son N.K., Stability radii of positive linear functional differential equations under multi-perturbations, SIAM J. Control Optim., 2005, 43(6), 2278-2295 http://dx.doi.org/10.1137/S0363012903434789; · Zbl 1090.34061 [11] Ngoc P.H.A., Minh N.V., Naito T., Stability radii of positive linear functional differential systems in Banach spaces, Int. J. Evol. Equ., 2007, 2(1), 75-97; · Zbl 1127.34044 [12] Ngoc P.H.A., Naito T., Shin J.S., Murakami S., On stability and robust stability of positive linear Volterra equations, SIAM J. Control Optim., 2008, 47(2), 975-996 http://dx.doi.org/10.1137/070679740; · Zbl 1206.45009 [13] Protter M.H., Weinberger H.F., Maximum Principles in Differential Equations, Springer, New York, 1984; · Zbl 0549.35002 [14] Prüss J., Evolutionary Integral Equations and Applications, Monogr. Math., 87, Birkhäuser, Basel, 1993; · Zbl 0784.45006 [15] Schaefer H.H., Banach Lattices and Positive Operators, Grundlehren Math. Wiss., 215, Springer, Berlin, 1974; · Zbl 0296.47023 [16] Stewart H.B., Generation of analytic semigroups by strongly elliptic operators under general boundary conditions, Trans. Amer. Math. Soc., 1980, 259(1), 299-310; · Zbl 0451.35033 [17] Zaanen A.C., Introduction to Operator Theory in Riesz Spaces, Springer, Berlin, 1997; · Zbl 0878.47022 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.