Abstract
In this work, we introduce Fibonacci– Halpern iterative scheme ( FH scheme) in partial ordered Banach space (POB space) for monotone total asymptotically non-expansive mapping (, MTAN mapping) that defined on weakly compact convex subset. We also discuss the results of weak and strong convergence for this scheme.
 Throughout this work, compactness condition of m-th iterate of the mapping for some natural m is necessary to ensure strong convergence, while Opial's condition has been employed to show weak convergence. Stability of FH scheme is also studied. A numerical comparison is provided by an example to show that FH scheme is faster than Mann and Halpern iterative schemes.
Highlights
As more general classes of asymptotically non-expansive mapping
Convergence Theorems Throughout this results, we assume that ‖ ‖ )is POB space such that order intervals are closed and convex and TAN mapping whenever
G is said to be compact if G is continuous and G(D) is relatively compact for any subset D of Theorem (2.6): Let D be a nonempty weakly compact convex subset of a uniformly convex
Summary
As more general classes of asymptotically non-expansive mapping. In [1], Alber introduced total asymptotically non-expansive mappings ( TAN). He studied the iterative method to determine their fixed points. Let be a Banach space with norm ‖ ‖ a mapping is called TAN if for. Where is increasing continuous function with ). Equation (1) reduces to the non-expansive mapping. See [2,3] for more details. Where D is a closed convex subset of Hilbert space. We consider a modification of Halpern iteration to be suitable for MTAN mappings. That equation(3) will be modified by employing Fibonacci sequence of numbers
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