STRONG CONVERGENCE THEOREM FOR UNIFORMLY L-LIPSCHITZIAN MAPPING OF GREGUS TYPE IN BANACH SPACES

In this paper, we introduced a new mapping called Uniformly L-Lipschitzian mapping of Gregus type, and used the Mann iterative scheme to approximate the fixed point. A Strong convergence result for the sequence generated by the scheme is shown in real Banach space. Our result generalized and unifybmany recent results in this area of research. In addition, using Java(jdk1.8.0_101), we give a numericalbexample to support our claim.

Convergence and stability of Fibonacci-Mann iteration for a monotone non-Lipschitzian mapping

In this paper, we prove strong convergence and Δ−convergence of Fibonacci-Mann iteration for a monotone non-Lipschitzian mapping (i.e. nearly asymptotically nonexpansive mapping) in partially ordered hyperbolic metric space. Moreover, we prove stability of Fibonacci-Mann iteration. Further, we construct a numerical example to illustrate results. Our results simultaneously generalize the results of Alfuraidan and Khamsi [Bull. Aust. Math. Soc., 2017, 96, 307–316] and Schu [J. Math. Anal. Appl., 1991, 58, 407–413].

Fixed points of nearly weak uniformly L-Lipschitzian mappings in real Banach spaces

Let K be a nonempty convex subset of a real Banach space X. Let T be a nearly weak uniformly L-Lipschitzian mapping. A modified Mann-type iteration scheme is proved to converge strongly to the unique fixed point of T. Our result is a significant improvement and generalization of several known results in this area of research. We give a specific example to support our result. Furthermore, an interesting equivalence of T-stability result between the convergence of modified Mann-type and modified Mann iterations is included.

Strong convergence results for nonlinear mappings in real Banach spaces

Let X be a real Banach space, K be a nonempty closed convex subset of X, T : K → K be a nearly uniformly L-Lipschitzian mapping with sequence {an}. Let kn ⊂ [1, ∞) and En be sequences with limn→∞ kn = 1, limn→∞ En = 0 and F(T) = {ρ ∈ K : T ρ = ρ} 6= ∅. Let {αn}n≥0 be real sequence in [0, 1] satisfying the following conditions: (i)P n≥0 αn = ∞ (ii) limn→∞ αn = 0. For arbitrary x0 ∈ K, let {xn}n≥0 be iteratively defined by xn+1 = (1 − αn)xn + αnT nxn, n ≥ 0. If there exists a strictly increasing function Φ : [0, ∞) → [0, ∞) with Φ(0) = 0 such that < T nx − T nρ, j(x − ρ) >≤ knkx − ρk 2 − Φ(kx − ρk) + En for all x ∈ K, then, {xn}n≥0 converges strongly to ρ ∈ F(T). It is also proved that the sequence of iteration {xn} defined by xn+1 = (1 − bn − dn)xn + bnT nxn + dnwn, n ≥ 0, where {wn}n≥0 is a bounded sequence in K and {bn}n≥0, {dn}n≥0 are sequences in [0,1] satisfying appropriate conditions, converges strongly to a fixed point of T.

A General Iterative Algorithm with Strongly Positive Operators for Strict Pseudo-Contractions

This paper deals with a new iterative algorithm{xn}with a strongly positive operatorAfor ak-strict pseudo-contractionTand a non-self-Lipschitzian mappingSin Hilbert spaces. Under certain appropriate conditions, the sequence{xn}converges strongly to a fixed point ofT, which solves some variational inequality. The results here improve and extend some recent related results.

Algorithms for Solving System of Extended General Variational Inclusions and Fixed Points Problems

We introduce a new system of extended general nonlinear variational inclusions with different nonlinear operators and establish the equivalence between the aforesaid system and the fixed point problem. By using this equivalent formulation, we prove the existence and uniqueness theorem for solution of the system of extended general nonlinear variational inclusions. We suggest and analyze a new resolvent iterative algorithm to approximate the unique solution of the system of extended general nonlinear variational inclusions which is a fixed point of a nearly uniformly Lipschitzian mapping. Subsequently, the convergence analysis of the proposed iterative algorithm under some suitable conditions is considered. Furthermore, some related works to our main problem are pointed out and discussed.

Hybrid Iteration Method for Common Fixed Points of a Finite Family of Nonexpansive Mappings in Banach Spaces

LetEbe a real uniformly convex Banach space, and let{Ti:i∈I}beNnonexpansive mappings fromEinto itself withF={x∈E:Tix=x, i∈I}≠ϕ, whereI={1,2,…,N}. From an arbitrary initial pointx1∈E, hybrid iteration scheme{xn}is defined as follows:xn+1=αnxn+(1−αn)(Tnxn−λn+1μA(Tnxn)),n≥1, whereA:E→Eis anL-Lipschitzian mapping,Tn=Ti,i=n(mod N),1≤i≤N,μ>0,{λn}⊂[0,1), and{αn}⊂[a,b]for somea,b∈(0,1). Under some suitable conditions, the strong and weak convergence theorems of{xn}to a common fixed point of the mappings{Ti:i∈I}are obtained. The results presented in this paper extend and improve the results of Wang (2007) and partially improve the results of Osilike, Isiogugu, and Nwokoro (2007).

Weak and strong convergence of an iterative method for nonexpansive mappings in Hilbert spaces

In a real Hilbert space H, starting from an arbitrary initial point x0 H, an iterative process is defined as follows: xn+1 = anxn +(1-an)T?n+1 f yn, yn = bnxn + (1 - bn)T?n g xn, n ? 0, where T ?n+1 f x = Tx - ?n+1?f f(Tx), T?n g x = Tx - ?n?gg(Tx), (8 x 2 H), T : H ? H a nonexpansive mapping with F(T) 6= ; and f (resp. g) : H ? H an ?f (resp. ?g)-strongly monotone and kf (resp. kg)-Lipschitzian mapping, {an} _ (0, 1), {bn} _ (0, 1) and {?n} _ [0, 1), {?n} _ [0, 1). Under some suitable conditions, several convergence results of the sequence {xn} are shown.