Abstract
In this study, we establish some results for strong convergence of a sequence to a common fixed point of a subfamily of a nonexpansive and periodic evolution family of bounded linear operators acting on a closed and bounded subset J of a strictly convex Banach space X . In fact, we generalized the results from semigroups of the operator to an evolution family of operators.
Highlights
Consider the autonomous Cauchy problem⎧⎨ Ω_ (a) Z(Ω(a)), a ≥ 0, ⎩ Ω(0) Ω0, (1)consider the following nonautonomous system:⎧⎨ Ω_ (r) Z(a)Ω(a) + eiρaI, a ≥ 0, (2)where Z(a) is a matrix of order α. e solution of such families leads to the idea of an evolution family
Many researchers have studied the convergence of nonexpansive semigroups to a fixed point [4, 6,7,8,9]
All the above results are related to the iteration method for strong convergence of a sequence to a fixed point
Summary
Many researchers have studied the convergence of nonexpansive semigroups to a fixed point [4, 6,7,8,9]. Assume that ξ∞ > 0 and V(r,0): r ∈ [0, ξ∞) is a family of operators on J, such that the following conditions hold: (i) For every r ∈ [0, ξ∞), V(r,0) is nonexpansive (ii) ere is ξm, a strictly increasing sequence in [0, ξ∞), such that ξ1 0, ξm converges to ξ∞ and weakly continuous mappings r↦V(r,0)τ on [ξm, ξm+1) for all τ ∈ J and m ∈ N. Define a family V(a,0); a ∈ [0, 1) of nonexpansive mappings as follows: if ξm ≤ r < ξm+1,
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