Local asymptotic attractivity for nonlinear quadratic functional integral equations. (English) Zbl 1173.47056

Dhage, Bapurao C.

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 70, No. 5, 1912-1922 (2009).

The paper deals with the quadratic functional-integral equation of mixed type \[ x(t)=[f(t,x(\alpha(t)))]\big(q(t)+\int_0^{\beta(t)}g(t,s,x(\gamma(s)))\,ds\big). \tag{1} \] Using a result about operator equations in Banach algebras, the author proves the existence of a solution of (1) in the space \(BC(\mathbb{R}_+,\mathbb{R})\). It is also proved that the solutions of (1) are uniformly locally asymptotically attractive on \(\mathbb{R}_+\).

Reviewer: Mirosława Zima (Rzeszow)

MSC:

47N20	Applications of operator theory to differential and integral equations
47H10	Fixed-point theorems
34K99	Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
45G10	Other nonlinear integral equations

Keywords:

quadratic functional integral equation; fixed point theorem; uniformly locally asymptotically attractive solution

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