Abstract
AbstractA generalization of Halpern's iteration is investigated on a compact convex subset of a smooth Banach space. The modified iteration process consists of a combination of a viscosity term, an external sequence, and a continuous nondecreasing function of a distance of points of an external sequence, which is not necessarily related to the solution of Halpern's iteration, a contractive mapping, and a nonexpansive one. The sum of the real coefficient sequences of four of the above terms is not required to be unity at each sample but it is assumed to converge asymptotically to unity. Halpern's iteration solution is proven to converge strongly to a unique fixed point of the asymptotically nonexpansive mapping.
Highlights
Fixed point theory is a powerful tool for investigating the convergence of the solutions of iterative discrete processes or that of the solutions of differential equations to fixed points in appropriate convex compact subsets of complete metric spaces or Banach spaces, in general, 1–12
The fixed point formalism is useful in stability theory to investigate the asymptotic convergence of the solution to stable attractors which are stable equilibrium points
The theory is useful for stability problems subject to multiple stable equilibrium points
Summary
Fixed point theory is a powerful tool for investigating the convergence of the solutions of iterative discrete processes or that of the solutions of differential equations to fixed points in appropriate convex compact subsets of complete metric spaces or Banach spaces, in general, 1–12. The objective of this paper is to investigate further generalizations for Halpern’s iteration process via fixed point theory by using two more driving terms, namely, an external one taking values on C plus a nonlinear term given by a continuous nondecreasing function, subject to an inequalitytype constraint as proposed in 2 , whose argument is the distance between pairs of points of sequences in certain complete metric space which are not necessarily directly related to the sequence solution taking values in the subset C of the Banach space X. Another generalization point is that the sample-by-sample sum of the scalar coefficient sequences of all the driving terms is not necessarily unity but it converges asymptotically to unity
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