Mappings In Real Banach Spaces Research Articles

Fast hybrid method for solving variational inequalities beyond monotonicity in real Banach spaces

In this work, we introduce a new inertial hybrid algorithm that combines the subgradient extragradient, Halpern, and S-iteration methods. We provide an approximate solution to variational inequality and common fixed point problem of relatively non-expansive mappings in real Banach spaces, where our cost operator is Lipschitz continuous and quasimonotone, and our step size is self-adaptive, non-monotone, and non-reliant on any line-search principle. We provide two strong convergence results with (and without) recourse to the monotonicity property, and further conduct several relevant numerical experiments to reveal the efficiency of our method over others in the literature.

(A(., .), η)-MONOTONE MAPPINGS AND A SYSTEM OF GENERALIZED SET-VALUED VARIATIONAL-LIKE INCLUSION PROBLEM

In this paper, we introduce and study a system of variational inclusions called system of generalized set-valued variational-like inclusion problem involving (A(., .), η)-monotone mappings in real Banach spaces. By means of resolvent operator technique, we prove the existence of solutions for this system of variational inclusions. Further, we suggest an iterative algorithm for finding the approximate solution of this system and discuss the convergence criteria of the sequences generated by the iterative algorithm under some suitable conditions.

Approximation of common fixed points of finite family of mixed-type total asymptotically quasi pseudocontractive-type mappings in Banach spaces

We propose an m-step hybrid-type iteration scheme for finite family of total asymptotically quasi pseudocontractive-type self mappings and finite family of total asymptotically quasi pseudocontractive-type nonself mappings. Under suitable conditions on the iteration parameters, we established strong convergence theorems of the scheme to the common fixed point of the mappings in real Banach spaces. Our results modify, improve and generalise numerous results currently in literature.

Approximation of solutions of Hammerstein equations with monotone mappings in real Banach spaces

Let E be a uniformly convex and uniformly smooth real Banach space with dual space, E∗. Let F : E → E∗, K : E∗ → E be maximal monotone mappings. An iterative algorithm is constructed and the sequence of the algorithm is proved to converge strongly to a solution of the Hammerstein equation u+KF u = 0. This theorem is a significant improvement of some important recent results which were proved in Lp spaces, 1 < p ≤ 2 under the assumption that F and K are bounded. This restriction on K and F have been dispensed with even in the more general setting considered here. Finally, a numerical experiment is presented to illustrate the convergence of the sequence of the algorithm which is found to be much faster, in terms of the number of iterations and the computational time than the convergence obtained with existing algorithms.

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.

STRONG CONVERGENCE BY PSEUDOCONTRACTIVE MAPPINGS FOR THE NOOR ITERATION SCHEME

In this paper, we establish a strong convergence for the Noor iterative scheme associated with Lipschitz strongly pseudocontractive mappings in real Banach spaces. It's proof-method is very simple by comparing with the previous proofs known.

CONVERGENCE OF MULTI-STEP ITERATIVE SEQUENCE FOR NONLINEAR UNIFORMLY L-LIPSCHITZIAN MAPPINGS

In this paper, by using the proof method of Xue, Rafiq and Zhou[19]
 some strong convergence results of multi-step iterative sequence are proved for
 nearly uniformly L− Lipschitzian mappings in real Banach spaces. Our results
 generalise and improve some recent results in this area of research.

Strong convergence of an implicit iteration process for a finite family of Lipschitz $\phi -$uniformly pseudocontractive mappings in Banach spaces

The purpose of this paper is to establish a strong convergence of an implicit iteration process to a common fixed point for a finite family of Lipschitz $\phi-$uniformly pseudocontractive mappings in real Banach spaces. The results presented here improve and extend the corresponding results in [2, 4, 6] and the consecutive remark explains in details about the facts.

Generalized strongly vector equilibrium problem for set-valued mappings

In this paper, we introduce and study a generalized strongly vector equilibrium problem for set-valued mappings in real Banach spaces. Using Minty type lemma and KKM-Fan theorem as basic tools, we prove existence theorems for generalized strongly vector equilibrium problem with and without monotonicity, respectively. Some examples are given.

Fixed point iterations of three asymptotically pseudocontractive mappings

Abstract In this paper, we establish the strong convergence for a modified three-step iterative scheme with errors associated with three mappings in real Banach spaces. Moreover, our technique of proofs is of independent interest. Remark at the end simplifies many known results.

Picard iterations for nonexpansive and Lipschitz strongly accretive mappings in a real Banach space

Abstract We study the convergence of a more general Picard iterative sequence for nonexpansive and Lipschitz strongly accretive mappings in an arbitrary real Banach space. Our results improve the results of Ćirić et al. (Nonlinear Anal. 70(12):4332-4337, 2009). MSC:47H06, 47H09, 47J05, 47J25.

The convergence of implicit Mann and Ishikawa iterations for weak generalized φ-hemicontractive mappings in real Banach spaces

Let E be a real Banach space and let D be a nonempty closed convex subset of E ,l et T : D → D be a continuous weak generalized ϕ-hemicontractive mapping. The existence theorem of a fixed point of a weak generalized ϕ-pseudocontractive mapping is obtained. And we also prove that implicit Mann and Ishikawa iterations converge strongly to the unique fixed point of T. Our results extend the corresponding results of Xiang (Nonlinear Anal. 70(6): 2277-2279, 2009). MSC: Primary 47H09; 47J25

The convergence of the modified Mann and Ishikawa iterations in Banach spaces

In this paper, under the new condition we show that the convergence of the modified Mann and Ishikawa iterations is equivalent for uniformly L-Lipschitz asymptotically pseudocontractive mappings in real Banach spaces. Our results extend and improve the corresponding results of Zeng (Acta Math. Sin. 47:219-228, 2004).

Strong convergence theorems for uniformly L-Lipschitzian asymptotically pseudocontractive mappings in Banach spaces

In this paper, we establish some strong convergence theorems of the modified Ishikawa and Mann iterations with errors of uniformly L-Lipschitzian asymptotically pseudocontractive mappings in real Banach spaces. Our results not only provide the new proof method, but also extend the known corresponding results given in (Chang in Proc. Am. Math. Soc. 129:845-853, 2001; Chang et al. in Appl. Math. Lett. 22:121-125, 2009; Goebel and Kirk in Proc. Am. Math. Soc. 35:171-174, 1972; Ofoedu in J. Math. Anal. Appl. 321:722-728, 2006; Schu in J. Math. Anal. Appl. 158:407-413, 1991). In order to get some applications of our results, we also provide specific examples.

Approximating Common Fixed Points for a Finite Family of Asymptotically Nonexpansive Mappings Using Iteration Process with Errors Terms

LetXbe a real Banach space andKa nonempty closed convex subset ofX. LetTi:K→K (i=1,2,…,m)bemasymptotically nonexpansive mappings with sequence{kn}⊂[1, ∞),∑n=1∞(kn-1)<∞, andℱ=⋂i=1m‍F(Ti)≠∅, whereFis the set of fixed points ofTi. Suppose that{ain}n=1∞, {bin}n=1∞, i=1,2,…,mare appropriate sequences in[0,1]and{uin}n=1∞, i=1,2,…,mare bounded sequences inKsuch that∑n=1∞bin<∞fori=1,2,…,m. We give{xn}defined byx1∈K,xn+1=(1-a1n-b1n)yn+m-2+a1nT1nyn+m-2+b1nu1n,yn+m-2=(1-a2n-b2n)yn+m-3+a2nT2nyn+m-3 +b2nu2n,…,yn+2=(1-a(m-2)n-b(m-2)n)yn+1+a(m-2)nTm-2nyn+1+b(m-2)nu(m-2)n,yn+1=(1-a(m-1)n -b(m-1)n)yn+a(m-1)nTm-1nyn+b(m-1)nu(m-1)n,yn=(1-amn-bmn)xn+amnTmnxn+bmnumn,m≥2,n≥1.The purpose of this paper is to study the above iteration scheme for approximating common fixed points of a finite family of asymptotically nonexpansive mappings and to prove weak and some strong convergence theorems for such mappings in real Banach spaces. The results obtained in this paper extend and improve some results in the existing literature.

Strong Convergence for HybridS-Iteration Scheme

We establish a strong convergence for the hybridS-iterative scheme associated with nonexpansive and Lipschitz strongly pseudocontractive mappings in real Banach spaces.

ON CONVERGENCE THEOREMS OF ITERATIVE SCHEMES FOR PSEUDOCONTRACTIVE MAPPINGS

Abstract. In this paper, we establish the strong convergence for theMann iterative scheme associated with uniformly continuous pseudocon-tractive mappings in real Banach spaces. Moreover, our technique ofproofs is of independent interest. 1. Introduction and preliminariesLet Ebe a real Banach space and Kbe a nonempty convex subset of E.Let Jdenote the normalized duality mapping from Eto 2 E de ned byJ(x) = ff 2E : hx;fi= jjxjj 2 and jjfjj= jjxjjg;where E denotes the dual space of Eand h;idenotes the generalized dualitypairing. We shall denote the single-valued duality map by j.Let T: D(T) ˆE!Ebe a mapping with domain D(T) in E:De nition 1.1. Tis said to be Lipschitzian if there exists L>1 such that forall x;y2D(T)kTx TykLkx yk:De nition 1.2. T is said to be nonexpansive if for all x, y 2D(T), thefollowing inequality holds:kTx Tykkx ykfor all x;y2D(T):De nition 1.3. T is said to be pseudocontractive if there exists j(x y) 2J(x y) such thathTx Ty;j(x y)ikx yk 2 for all x;y2D(T); n1:

On the Convergence of Multistep Iteration for Uniformly Continuous Φ‐Hemicontractive Mappings

It is shown that the convergence of the multistep iterative process with errors is obtained for uniformly continuous Φ‐hemicontractive mappings in real Banach spaces. We also revise the problems of C. E. Chidume and C. O. Chidume (2005).

On Convergence Results for Lipschitz Pseudocontractive Mappings

We establish the strong convergence for the Ishikawa iteration scheme associated with Lipschitz pseudocontractive mappings in real Banach spaces. Moreover, our technique of proofs is of independent interest.