zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Strong convergence of an iterative algorithm for nonself multimaps in Banach spaces. (English) Zbl 1222.47092
Summary: Let $E$ be a uniformly convex Banach space having a uniformly Gâteaux differentiable norm, $D$ a nonempty closed convex subset of $E$, and $T:D\rightarrow K(E)$ a nonself multimap such that $F(T)\neq \varnothing$ and $P_T$ is nonexpansive, where $F(T)$ is the fixed point set of $T$, $K(E)$ is the family of nonempty compact subsets of $E$ and $P_T(x)=\{u_x\in T_X: \|x-u_x\| = d(x,Tx)\}$. Suppose that $D$ is a nonexpansive retract of $E$ and that, for each $v\in D$ and $t\in (0,1)$, the contraction $S_t$ defined by $S_tx=tP_Tx+(1 - t)v$ has a fixed point $x_t\in D$. Let $\{\alpha _n\},\{\beta _n\}$ and $\{\gamma _n\}$ be three real sequences in (0,1) satisfying approximate conditions. Then, for fixed $u\in D$ and arbitrary $x_{0}\in D$, the sequence $\{x_n\}$ generated by $$x_n\in \alpha _nu+\beta _nx_{n-1}+\gamma _nP_T(x_n), \quad n\ge 0,$$ converges strongly to a fixed point of $T$.

MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H05 Monotone operators (with respect to duality) and generalizations 47H10 Fixed-point theorems for nonlinear operators on topological linear spaces
Full Text:
References:
 [1] Ciorenescu, I.: Geometry of Banach spaces duality mapping and nonlinear problems. (1990) [2] Xu, H. K.; Yin, X. M.: Strong convergence theorems for nonexpansive non-self-mappings. Nonlinear anal. 24, 223-228 (1995) · Zbl 0826.47038 [3] Xu, H. K.: On weakly nonexpansive and \ast-nonexpansive multivalued mappings. Math. japonica 36, 441-445 (1991) · Zbl 0733.54010 [4] Jung, J. S.; Kim, S. S.: Strong convergence theorems for nonexpansive nonself-mappings in Banach spaces. Nonlinear anal. 33, 321-329 (1998) · Zbl 0988.47033 [5] Jung, J. S.; Kim, T. H.: Strong convergence of approximating fixed points for nonexpansive nonself-mappings in Banach spaces. Kodai math. J. 21, 259-272 (1998) · Zbl 0928.47040 [6] Ceng, L. C.; Cubiotti, P.; Yao, J. C.: Strong convergence theorems for finite many nonexpansive mappings and applications. Nonlinear anal. 67, 1464-1473 (2007) · Zbl 1123.47044 [7] Kim, T. H.; Jung, J. S.: Approximating fixed points of nonlinear mappings in Banach spaces, in: Proceedings of workshop on fixed point theory (Kazimierz dolny, 1997). Ann. univ. Mariae Curie-sklodowska sect. A 51, 149-165 (1997) [8] Zeng, L. C.; Yao, J. C.: Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings. Nonlinear anal. 64, 2507-2515 (2006) · Zbl 1105.47061 [9] Ceng, L. C.; Xu, H. K.: Strong convergence of a hybrid viscosity approximation method with perturbed mappings for nonexpansive and accretive operators. Taiwanese J. Math. 11, 661-682 (2007) · Zbl 1219.47102 [10] Shahzad, N.; Zegeye, H.: Strong convergence results for nonself multimaps in Banach spaces. Proc. amer. Math. soc. 136, 539-548 (2008) · Zbl 1135.47054 [11] Deimling, K.: Zeros of accretive operators. Manuscripta math. 13, 365-374 (1974) · Zbl 0288.47047 [12] Takahashi, W.; Jeong, D. H.: Fixed point theorem for nonexpansive semigroups on Banach spaces. Proc. amer. Math. soc. 122, 1175-1179 (1994) · Zbl 0818.47055 [13] Rafiq, A.: On Mann iteration in Hilbert spaces. Nonlinear anal. 66, 2230-2236 (2007) · Zbl 1136.47047 [14] Yao, Y.; Liou, Y. C.; Chen, R.: Strong convergence of an iterative algorithm for pseudocontractive mapping in Banach spaces. Nonlinear anal. 67, 3311-3317 (2007) · Zbl 1129.47059 [15] Ceng, L. C.; Xu, H. K.; Yao, J. C.: Strong convergence of an iterative method with perturbed mappings for nonexpansive and accretive operators. Numer. funct. Anal. optim. 29, 324-345 (2008) · Zbl 1140.47050 [16] Mann, W. R.: Mean value methods in iteration. Proc. amer. Math. soc. 4, 506-610 (1953) · Zbl 0050.11603 [17] Nadler, S.: Multivalued contraction mappings. Pacific J. Math. 30, 475-488 (1969) · Zbl 0187.45002 [18] Jung, J. S.: Strong convergence theorems for multivalued nonexpansive nonself-mappings in Banach spaces. Nonlinear anal. 66, 2345-2354 (2006) · Zbl 1123.47047 [19] Lim, T. C.: A fixed point theorem for weakly inward multivalued contractions. J. math. Anal. appl. 247, 323-327 (2000) · Zbl 0957.47040 [20] Sahu, D. R.: Strong convergence theorems for nonexpansive type and non-self multi-valued mappings. Nonlinear anal. 37, 401-407 (1999) · Zbl 0938.47039 [21] Deimling, K.: Multivalued differential equations. (1992) · Zbl 0760.34002 [22] Hussain, T.; Latif, A.: Fixed points of multivalued nonexpansive maps. Math. japonica 33, 385-391 (1988) · Zbl 0667.47028 [23] Ceng, L. C.; Ansari, Q. H.; Yao, J. C.: Mann-type steepest-descent and modified hybrid steepest-descent methods for variational inequalities in Banach spaces. Numer. funct. Anal. optim. 29, 987-1033 (2008) · Zbl 1163.49002 [24] Xu, H. K.: Multivalued nonexpansive mappings in Banach spaces. Nonlinear anal. 43, 693-706 (2001) · Zbl 0988.47034 [25] Shahzad, N.; Lone, A.: Fixed points of multimaps which are not necessarily nonexpansive. Fixed point theory appl 2005, No. 2, 169-176 (2005) · Zbl 1112.47050 [26] Barbu, V.; Precupanu, Th.: Convexity and optimization in Banach spaces. (1978) · Zbl 0379.49010 [27] Acedo, G. Lopez; Xu, H. K.: Remarks on multivalued nonexpansive mappings. Soochow J. Math. 21, 107-115 (1995) · Zbl 0826.47037