Modified Mann Iterations for Nonexpansive Semigroups in Banach Space

• Published in: Acta mathematica Sinica. English series Vol. 26; no. 1; pp. 193 - 202
• Main Authors: Chen, Ru Dong, He, Hui Min, Noor, Muhammad Aslam
• Format: Journal Article
• Language: English
• Published: Heidelberg Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society 2010; Springer Nature B.V
• Subjects: Applied mathematics; Banach spaces; Hilbert space; Mann迭代; Mathematics; Mathematics and Statistics; 变分不等式; 对偶映射; 田纳西州; 联合国; 自反Banach空间; 闭凸子集; 非扩张半群; 58E35; 47H10; fixed point; 47H09; reflexive Banach space; nonexpansive semigroups; strong convergence
• DOI: 10.1007/s10114-010-7446-7
• ISSN: 1439-8516

Let E be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E^＊, and C be a nonempty closed convex subset of E. Let {T（t） ： t ≥ 0} be a nonexpansive semigroup on C such that F ：=∩t≥0 Fix（T（t）） ≠ 0, and f ： C → C be a fixed contractive mapping. If {αn}, {βn}, {an}, {bn}, {tn} satisfy certain appropriate conditions, then we suggest and analyze the two modified iterative processes as：{yn=αnxn＋（1-αn）T（tn）xn,xn=βnf（xn）＋（1-βn）yn {u0∈C,vn=anun＋（1-an）T（tn）un,un＋1=bnf（un）＋（1-bn）vn We prove that the approximate solutions obtained from these methods converge strongly to q ∈∩t≥0 Fix（T（t））, which is a unique solution in F to the following variational inequality： 〈（I-f）q,j（q-u）〉≤0 u∈F Our results extend and improve the corresponding ones of Suzuki [Proc. Amer. Math. Soc., 131, 2133-2136 （2002）], and Kim and XU [Nonlear Analysis, 61, 51-60 （2005）] and Chen and He [Appl. Math. Lett., 20, 751-757 （2007）].