×

Perron-type stability theorems for neutral equations. (English) Zbl 1044.34028

This article deals with asymptotic stability results for a neutral functional-differential equation of the form \[ x'(t)= Sx(t)+ Px(t-r)+ \frac{d}{dt} Q(t,x_t)+ G(t,x_t), \tag{1} \] with an asymptotically stable linear part \(x'(t)= Sx(t)+ Px(t-r)\). As is well known, the problem of the asymptotical stability of the zero solution to (1) is reduced to the problem of finding a fixed point for some nonlinear integral operator \(A\). The author describes conditions under which the operator \(A\) satisfies conditions of the following Krasnosel’skij fixed-point principle: if \(A= A_1+ A_2\), \(A_1\) is a contraction of a closed convex nonempty subset \(M\) in a Banach space \(S\), \(A_2\) is continuous and \(A_2M\) compact, and \(A_1M+ M_2M\subseteq M\), then there exists a fixed point of \(A\). The use of this theorem allows the author to consider the case when the functions \(Q\) and \(G\) are unbounded with respect to \(t\) and \(G\) is not differentiable or Lipschitzian.

MSC:

34K20 Stability theory of functional-differential equations
34K40 Neutral functional-differential equations
47H10 Fixed-point theorems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Bellman, R., Stability Theory of Differential Equations (1953), McGraw-Hill: McGraw-Hill New York · Zbl 0052.31505
[2] Burton, T. A., Liapunov functionals, fixed points, and stability by Krasnoselskii’s theorem, Nonlinear Stud., 9, 181-190 (2002) · Zbl 1084.47522
[3] Burton, T. A.; Furumochi, Tetsuo, Fixed points and problems in stability theory, Dyn. Systems Appl., 10, 89-116 (2001) · Zbl 1021.34042
[4] Burton, T. A.; Tetsuo, Furumochi, Krasnoselskii’s fixed point theorem and stability, Nonlinear Anal., 49, 445-454 (2002) · Zbl 1015.34046
[5] Krasnoselskii, M. A., Some problems of nonlinear analysis, Amer. Math. Soc. Transl. (2), 10, 345-409 (1958)
[6] V. Lakshmikantham, S. Leela, Differential and Integral Inequalities, Vol. I, Academic Press, New York, 1969.; V. Lakshmikantham, S. Leela, Differential and Integral Inequalities, Vol. I, Academic Press, New York, 1969. · Zbl 0177.12403
[7] Perron, O., Die stabilitatsfrage bei differentialgleichungen, Math. Z., 32, 703-728 (1930) · JFM 56.1040.01
[8] I.A. Rus, Picard Operators and Applications, Babes-Bolyai University, Cluj-Napoca, Romania, 1996.; I.A. Rus, Picard Operators and Applications, Babes-Bolyai University, Cluj-Napoca, Romania, 1996.
[9] Saito, Seiji, Global stability of solutions for quasilinear ordinary differential systems, Math. Japonica, 34, 821-929 (1989) · Zbl 0681.34048
[10] M.-A. Serban, Global asymptotic stability for some difference equations via fixed point technique, Seminar on Fixed Point Theory, Vol. 2, Cluj-Napoca, 2002, pp. 87-96.; M.-A. Serban, Global asymptotic stability for some difference equations via fixed point technique, Seminar on Fixed Point Theory, Vol. 2, Cluj-Napoca, 2002, pp. 87-96.
[11] Smart, D. R., Fixed Point Theorems (1974), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0297.47042
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.