Abstract
In this paper, we investigate the common approximate fixed points of monotone nonexpansive semigroups of nonlinear mappings $\{T(t)\}_{t \geq0}$ , i.e., a family such that $T(0)x=x$ , $T(s+t)x=T(s)\circ T(t)x $ , where the domain is a Banach space. In particular we prove that under suitable conditions, the common approximate fixed points are the same as the common approximate fixed points set of two mappings from the family. Then we give an algorithm of how to construct an approximate fixed point sequence of the semigroup in the case of a uniformly convex Banach space.
Highlights
Nonexpansive mappings are those maps which have Lipschitz constant equal to
The purpose of this paper is to prove the existence of approximate fixed points for semigroups of nonlinear monotone mappings acting in a Banach vector space endowed with a partial order
Let us recall that a family {T(t)}t≥ of Bachar and Khamsi Fixed Point Theory and Applications (2015) 2015:160 mappings forms a semigroup if T( )x = x and T(s + t)x = T(s)T(t)
Summary
Nonexpansive mappings are those maps which have Lipschitz constant equal to. The fixed point theory for such mappings is rich and varied. The existence of fixed points for nonexpansive mappings in Banach and metric spaces has been investigated since the early s; see, e.g., Belluce and Kirk [ , ], Browder [ ], Bruck [ ], Lim [ ]. The purpose of this paper is to prove the existence of approximate fixed points for semigroups of nonlinear monotone mappings acting in a Banach vector space endowed with a partial order. Let us recall that a family {T(t)}t≥ of Bachar and Khamsi Fixed Point Theory and Applications (2015) 2015:160 mappings forms a semigroup if T( )x = x and T(s + t)x = T(s)T(t).
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