Iterative Convergence For The Fixed Points Of Asymptotically Nonexpansive Mappings And The Solution Of A Family Of Differential Systems

We investigated the fixed point problems of asymptotically nonexpansive mappings in Hilbert spaces in this paper,and set up a new iterative algorithm.We also proved the strong convergence of the iterative process under some conditions.Meanwhile,we also established a new iterative algorithm in Lp(?)spaces to solve the existence of solutions of a family of differential systems,proved the strong convergence of this iteration process.Our results extended and improved some corresponding reports by other authors.The first result,let C(?)H be a nonempty bounded convex closed set of a Hilbert space H and ??C.Let T:C ?C be a asymptotically nonexpansive mapping with a sequence {kn} such that F(T)?(?).Suppose {?n?,{?n},??n?,{?n} are real number sequences in(0,1).Let {xn} be generated by Under some conditions,the sequence {xn} converges strongly to a fixed point x*?F(T)The second result,we investigate the follow differential systems;where ? is a bounded conical domain of a Euclidean space Rn(n ? 1),? is the boundary of? with ??C1[7]and v denotes the exterior normal derivative to ?,<·,·>and?·? denotes the Euclidean inner-product and Euclidean norm in Rn,respectively.?u(i)=((?)u(i)/(?)x1,···,(?)u(i)/(?)xn)and(x1,…?xn)??.?x is the subdifferential of ?x,where ?x=?(x,·):R?R for x??,? is a non-negative constant and K is a constant.For solved this differential systcms,we constructed a new iterative algorithm as fol-lows:Let E=Lq(?),C=Lq?(?),where q=sup{qi},q'=sup{qi?},q?=sup{qi?},i=1,2,...,n.Let f:E?E be a contraction with contractive constant k E(0,1),T:E?E be a,strongly positive linear bounded operator with coefficient(?)Suppose 0???(?)/2k.Let Ai:C?E be m-accretive mapping as that in Lemma 1.1,Si:C?E be ?i-inversely strongly accretive mapping as that in Lemma 1.2,where {?i}(?)[0,1],for i=1,2,...,n.Suppose{?n},{?n},{?n?,{?n},{?n},{?n},{?i},and{bi} are real number sequences in(0,1),where n? 0 and i = 1,2,...,n.Suppose{rn,i),{ui} and {ci} are real number sequences in(0,+?),where n ? 0 and i= 1,2,...,n.Let {zn} be generated by the following terative algorithm:Under some conditions,the sequence zn?q0?(?)N(Ai+Bi)and satisfies the following variational inequality:for(?)y?(?)N(Ai+Bi),?(T-?f)q0,J(q0-y)??0.Our results extended and improved some corresponding reports by other authors.The structure of this paper is that:we introduced some related research backgrounds,some relevant concepts and lemmas in the first chapter;in the second chapter,strong convergence of a new iterative algorithm for fixed points of asymptotically nonexpansive mappings;in the third chapter,the existence of solution of a family of systems.