Approximating fixed points in modular spaces

Abstract

A generic two theorems for the two step iterative sequence of multivalued mappings are proved in a complete convex real modular space, and then cite some corollaries that are special cases of these theorems.

Some Results on Normalized Duality Mappings and Approximating Fixed Points in Convex Real Modular Spaces

“…The collection of all -balls in a modular space generates a locally convex Hausdorff topological linear space (7). Definition 4 (8) (1) A sequence { } ⊂ is said to be is ‫ــ‬ convergent to a point in then is called ‫ــ‬ closed.…”

“…(5) The distance between ∈ and ⊂ is ( − ) =inf { ( − ); ∈ }. Definition 5 (8) Let , be two nonempty subsets in 2 then ( , ) = { Є γ( -), Є ( -)} is Hausdorff distance of and .…”

“…If is single-valued mapping, is a fixed point⇔ = ( ). Definition 7 (8) is called uniformly convex if for any > 0, there exists a () > 0, ∋ if ( ) = ( ) = 1 and ( − )  , then (…”

On Invariant Approximations in Modular Spaces

“…Many works in this space can be found in [3][4][5][6]. Recently, Abed and Abdul Jabbar [7][8][9][10] introduced the concept of normalized duality mappings in the real convex modular spaces proved some of its properties which related to uniformly smooth of these spaces. Also, they presented some results concerning the convergence and equivalence iterative sequences for multivalued strongly pseudo-contractions mappings.…”

Results on a faster iterative scheme for a generalized monotone asymptotically

“…for any 11 , ( Later, many researchers introduced results on convergence of the Mann scheme and its modified versions for differe classes of mappings such as nonexpansine, pseudo-contractions, total asymptotically nonexpansive mappings ... etc. For example, see [1][2][3], [5][6], [12], [14], [18] and see [22][23], [24], [26], [31], [34], [35]. Recently, there are some convergence theorems of such a scheme in an POB (A,≾).…”