Advanced Search

Query:

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

Format:

Display: entries per page entries

Zbl 1123.47047
Jung, Jong Soo
Strong convergence theorems for multivalued nonexpansive nonself-mappings in Banach spaces. (English)
[J] Nonlinear Anal., Theory Methods Appl. 66, No. 11, A, 2345-2354 (2007). ISSN 0362-546X

The paper gives convergence theorems for approximating fixed points of multivalued nonexpansive nonself mappings by means of viscosity type methods. The main result (Theorem~1) goes as follows. Let $E$ be a uniformly convex Banach space with uniformly Gâteaux differentiable norm, $C$ be a nonempty closed convex subset of $E$ and $T:C\rightarrow \mathcal{K}(E)$ be a nonself nonexpansive multivalued mapping (here, $\mathcal{K}(E)$ denotes the set of all nonempty compact subsets of $E$). Suppose that $C$ is a nonexpansive retract of $E$ and $T$ has only strict fixed points, that is, $T(y)=\{y\}$ for all fixed points $y$ of $T$. For each $u\in C$ and $t\in (0,1)$, consider the multivalued contraction $G_t:C\rightarrow \mathcal{K}(E)$ defined by $$G_t=tTx+(1-t)u,\ x\in C,$$ and assume that $G_t$ has a fixed point $x_t\in C$. Then $T$ has a fixed point if and only if $x_t$ remains bounded as $t\rightarrow 1$ and, in this case, $x_t$ converges strongly as $t\rightarrow 1$ to a fixed point of $T$. Several other related results are obtained in the same way or as corollaries.
[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; multivalued nonexpansive nonself mapping; fixed point; viscosity method; strong convergence

Cited in: Zbl 1245.54042 Zbl 1135.47054

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]