## Global asymptotic stability of solutions of a functional integral equation.(English)Zbl 1154.45005

The authors consider the solvability in the space $$BC(\mathbb{R}_{+})$$ of the following functional integral equation
$x(t)=f(t,x(\alpha(t)))+\int_0^{\beta(t)} g(t,s,x(\gamma(s))) \,ds,$
where $$\alpha,\beta,\gamma : \mathbb{R}_{+}\to\mathbb{R}_{+}$$ are continuous and $$\alpha(t)\to\infty$$ as $$t\to\infty$$. Using the technique of measures of noncompactness, namely a fixed point theorem of Darbo type, they prove a theorem on the existence and global asymptotic stability of solutions of the above functional integral equation. A few realizations of the result obtained are indicated.

### MSC:

 45G10 Other nonlinear integral equations 45M10 Stability theory for integral equations 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H10 Fixed-point theorems 45M05 Asymptotics of solutions to integral equations
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### References:

 [1] Banaś, J.; Goebel, K., () [2] Banaś, J., Measures of noncompactness in the space of continuous tempered functions, Demonstratio math., 14, 127-133, (1981) · Zbl 0462.47035 [3] Banaś, J.; Rzepka, B., An application of a measure of noncompactness in the study of asymptotic stability, Appl. math. lett., 16, 1-6, (2003) · Zbl 1015.47034 [4] J. Banaś, D. O’Regan, On existence and local attractivity of solutions of a quadratic Volterra integral equation of fractional order, Preprint [5] Burton, T.A.; Zhang, B., Fixed points and stability of an integral equation: nonuniqueness, Appl. math. lett., 17, 839-846, (2004) · Zbl 1066.45002 [6] Deimling, K., Nonlinear functional analysis, (1985), Springer-Verlag Berlin · Zbl 0559.47040 [7] Dhage, B.C.; Ntouyas, S., Existence results for nonlinear functional integral equations via a fixed point theorem of krasnosel’skii – schaefer type, Nonlinear stud., 9, 307-317, (2002) · Zbl 1009.47054 [8] Hu, X.; Yan, J., The global attractivity and asymptotic stability of solution of a nonlinear integral equation, J. math. anal. appl., 321, 147-156, (2006) · Zbl 1108.45006 [9] O’Regan, D.; Meehan, M., Existence theory for nonlinear integral and integrodifferential equations, (1998), Kluwer Academic Dordrecht · Zbl 0932.45010 [10] Väth, M., ()
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