The global attractivity and asymptotic stability of solution of a nonlinear integral equation. (English) Zbl 1108.45006

Hu, Xiaoling; Yan, Jurang

J. Math. Anal. Appl. 321, No. 1, 147-156 (2006).

The paper is devoted to the study of the nonlinear Volterra integral equations \[ x(t)=f\left(t,x(t),\int^t_{0}u(t,s,x(s))ds\right),\;\;t\geq 0,\tag{1} \] and \[ x(t)=g(t,x(t))+x(t)\int^t_{0}u(t,s,x(s))ds,\;\;t\geq 0,\tag{2} \] in the Banach space consisting of all real functions defined, bounded and continuous on \([0,+\infty)\). Using the measures of noncompactness and the fixed point theorem, conditions are given when equation (1) has at least one globally attractive solution and when equation (2) has at least one asymptotically stable solution. The last statement for the special case of equation (2) with \(g(t,u)\equiv 0\) improves the corresponding result of J. Banas and B. Rzepka [J. Math. Anal. Appl. 284, No. 1, 165–173 (2003; Zbl 1029.45003)]. Three examples are given.

Reviewer: Anatoliy Aleksandrovich Kilbas (Minsk)

MSC:

45M10	Stability theory for integral equations
45G10	Other nonlinear integral equations
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

Keywords:

nonlinear Volterra integral equation; fixed point theorem; asymptotic stability; global attractivity; measures of noncompactness

Citations:

Zbl 1029.45003

References:

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