## 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

### Keywords:

Fixed points; Stability; Neutral equations; Perron theorem
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### References:

 [1] Bellman, R., Stability theory of differential equations, (1953), 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. · 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. [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. [11] Smart, D.R., Fixed point theorems, (1974), Cambridge University Press Cambridge · Zbl 0297.47042
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