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Evolution equations in discrete and continuous time for nonexpansive operators in Banach spaces

Guillaume Vigeral (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider some discrete and continuous dynamics in a Banach space involving a non expansive operator J and a corresponding family of strictly contracting operators Φ (λ, x): = λ J( 1 - λ λ x) for λ ∈ ] 0,1] . Our motivation comes from the study of two-player zero-sum repeated games, where the value of the n-stage game (resp. the value of the λ-discounted game) satisfies the relation vn = Φ( 1 n , v n - 1 ) (resp.  v λ = Φ(λ, v λ )) where J is the Shapley operator of the game. We study the evolution equation u'(t) =...

Existence of a common solution for a system of nonlinear integral equations via fixed point methods inb-metric spaces

Oratai Yamaod, Wutiphol Sintunavarat, Yeol Je Cho (2016)

Open Mathematics

In this paper we introduce a property and use this property to prove some common fixed point theorems in b-metric space. We also give some fixed point results on b-metric spaces endowed with an arbitrary binary relation which can be regarded as consequences of our main results. As applications, we applying our result to prove the existence of a common solution for the following system of integral equations: x (t) =  ∫ a b K 1  (t,r,x(r)) dr, x (t) =  ∫ a b K 2  (t,r,x(r)) dr,       x ( t ) = ∫ a b K 1 ( t , r , x ( r ) ) d r , x ( t ) = ∫ a b K 2 ( t , r , x ( r ) ) d r , where a, b...

Existence theorem for the Hammerstein integral equation

Mieczysław Cichoń, Ireneusz Kubiaczyk (1996)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper we prove an existence theorem for the Hammerstein integral equation x ( t ) = p ( t ) + λ ∫ I K ( t , s ) f ( s , x ( s ) ) d s , where the integral is taken in the sense of Pettis. In this theorem continuity assumptions for f are replaced by weak sequential continuity and the compactness condition is expressed in terms of the measures of weak noncompactness. Our equation is considered in general Banach spaces.

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