Monotonic solutions of a quadratic integral equation of Volterra type. (English) Zbl 1059.45002

Banaś, J.; Martinon, A.

Comput. Math. Appl. 47, No. 2-3, 271-279 (2004).

The paper concerns the existence of nondecreasing solutions of the nonlinear quadratic Volterra integral equation of the form \[ x(t)=a(t)+x(t)\int_0^tv(t,\tau,x(\tau))d\tau, \quad t\in I=[0,T] \] in the space \(C(I)\) consisting of all real functions defined and continuous on the interval \(I\), where \(a(t)\) and \(v(t,\tau,x)\) are given while \(x=x(t)\) is an unknown function. The proof of the main result is based on the Darbo type fixed point theorem formulated in terms of axiomatic measures of noncompactness introduced by J. Banás and K. Goebel [Measures of noncompactness in Banach spaces, Lect. Notes Pure Appl. Math. 60 (1980; Zbl 0441.47056)]. The authors illustrate the main result and compare its assumptions by giving suitable interesting examples.

Reviewer: Dariusz Bugajewski (Poznań)

MSC:

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

Keywords:

measure of noncompactness; monotonic solutions; Darbo type fixed point theorem; nondecreasing solutions; nonlinear quadratic Volterra integral equation

Citations:

Zbl 0441.47056

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