Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1135.47054
Shahzad, N.; Zegeye, H.
Strong convergence results for nonself multimaps in Banach spaces. (English)
[J] Proc. Am. Math. Soc. 136, No. 2, 539-548 (2008). ISSN 0002-9939; ISSN 1088-6826/e

Let $E$ be a uniformly convex Banach space, $D$ be a nonempty closed convex subset of $E$ and $T:D\rightarrow K(E)$ be a multimap, where $K(E)$ is the family of all nonempty compact subsets of $E$. If we denote $$ P_T(x)=\{u_x\in Tx: \left\Vert x-u_x\right\Vert =d(x,Tx)\}, $$ then $P_T:D\rightarrow K(E)$ is nonempty and compact for every $x\in D$. \par The first main result of the paper (Theorem 3.1) shows that, if $D$ is a nonexpansive retract of $E$ and if, for each $u\in D$ and $t\in (0,1)$, the multivalued contraction $S_t$ defined by $S_tx=tP_Tx+(1-t)u$ has a fixed point $x_t\in D$, then $T$ has a fixed point if and only if $\{x_t\}$ remains bounded as $t\rightarrow 1$. Moreover, in this case, $\{x_t\}$ converges strongly to a fixed point of $T$ as $t\rightarrow 1$. \par A similar result (Theorem 3.2) is then obtained for nonself-multimaps satisfying the inwardness condition in the case of reflexive Banach spaces having a uniformly Gâteaux differentiable norm. Several corollaries of these results are also presented.
[Vasile Berinde (Baia Mare)]

MSC 2000:: *47J25 Methods for solving nonlinear operator equations (general)
47H04 Set-valued operators
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47H09 Mappings defined by "shrinking" properties

Keywords: Banach space; nonexpansive multimap; fixed point; multi-valued contraction; strong convergence theorem; nonexpansive retract; Banach limit; inwardness

Citations: Zbl 1123.47047

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Scientific prize winners of the ICM 2010
	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.

Advanced Search

Zentralblatt MATH Berlin [Germany]