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Stability in neutral nonlinear differential equations with functional delays using fixed-point theory. (English) Zbl 1083.34536

The paper deals with the stability of the zero solution of the scalar neutral differential equation \[ x'(t)=-a(t) x(t)+c(t) x'(t-g(t))+q(t,x(t),x(t-g(t))), \] where \(a\), \(b\), \(g\) and \(q\) are continuous functions of their arguments. Noting that the construction of a Lyapunov functional solving this problem is an open problem (the difficulties that arise are illustrated by the case \(q\equiv 0\)), the author gets sufficient conditions for the stability of the zero solution on the base of the contraction mapping principle applied to the equivalent Volterra-type integral equation. Both bounded and unbounded delays are considered and the obtained results are illustrated by examples.

MSC:

34K20 Stability theory of functional-differential equations
34K40 Neutral functional-differential equations
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[1] Hatvani, L., Annulus arguments in the stability theory for functional differential equations, Differential and integral equations, 10, 975-1002, (1997) · Zbl 0897.34060
[2] Raffoul, Y.N., Periodic solutions in neutral nonlinear differential equations with functional delay, Electron. J. differential equations, 102, 7, 1-7, (2003) · Zbl 1054.34115
[3] Burton, T.A.; Furumochi, T., Fixed points and problems in stability theory, Dynamical systems and appl., 10, 89-116, (2001) · Zbl 1021.34042
[4] Hale, J., Theory of functional differential equations, (1977), Springer New York
[5] Burton, T.A., Volterra integral and differential equations, (1983), Academic Press New York · Zbl 0515.45001
[6] Burton, T.A., Stability and periodic solutions of ordinary and functional differential equations, (1985), Academic Press New York · Zbl 0635.34001
[7] Yohizawa, T., Stability theory by Liapunov’s second method, Tokyo math. soc., Japan, (1966)
[8] Burton, T.A.; Hatvani, L., Stability theorems for non-autonomous functional differential equations by Liapunov functionals, Tohoku math. J., 41, 295-306, (1989)
[9] Hino, Y.; Murakami, S., (), 31-46
[10] Raffoul, Y.N., Uniform asymptotic stability in linear Volterra systems with nonlinear perturbation, Int. J. differ. equ. appl., 6, 1, 19-28, (2002) · Zbl 1046.45006
[11] Zhang, B., Asymptotic criteria and integrability properties of the resolvent of Volterra and functional equations, Funkcialaj ekvacioj, 40, 335-351, (1997) · Zbl 0897.45009
[12] Becker, L.C., Stability consideration for Volterra integro-differential equations, ()
[13] Eloe, P.; Islam, M., Stability properties and integrability of the resolvent of linear Volterra equations, Tohoku math. J., 47, 263-269, (1995) · Zbl 0826.45004
[14] Eloe, P.; Islam, M.; Zhang, B., Uniform asymptotic stability in linear Volterra integrodifferential equations with applications to delay systems, Dynamic systems and applications, 9, 331-334, (2000) · Zbl 0970.45004
[15] Hino, Y.; Murakami, S., Total stability and uniform asymptotic stability for linear Volterra equations, J. London math. soc., 43, 305-312, (1991) · Zbl 0728.45007
[16] Islam, M.; Raffoul, Y., Stability properties of linear Volterra integrodifferential equations with nonlinear perturbation, Communication of applied analysis, 7, 2-3, 405-416, (2003) · Zbl 1085.45501
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