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(Ω)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 θ∈C.Let T:C →C be a asymptotically nonexpansive mapping with a sequence {kn} such that F(T)≠(?).Suppose {αn｝,{βn},｛λn｝,{ξn} 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 Ω is a bounded conical domain of a Euclidean space Rn(n ≥ 1),Γ is the boundary ofΩ with Γ∈C1[7]and v denotes the exterior normal derivative to Γ,<·,·>and∣·∣ denotes the Euclidean inner-product and Euclidean norm in Rn,respectively.▽u(i)=((?)u(i)/(?)x1,···,(?)u(i)/(?)xn)and(x1,…，xn)∈Ω.βx is the subdifferential of Φx,where Φx=Φ(x,·):R→R for x∈Γ,ε 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(Ω),C=Lq′(Ω),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＜η＜(?)/2k.Let Ai:C→E be m-accretive mapping as that in Lemma 1.1,Si:C→E be μi-inversely strongly accretive mapping as that in Lemma 1.2,where {μi}(?)[0,1],for i=1,2,...,n.Suppose{αn},{βn},{γn｝,{τn},{δn},{ξn},{αi},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-ηf)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.