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Register a new userCommon fixed point theorems for occasionally weakly compatible maps
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Hakima Bouhadjera
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In this paper, we establish a common fixed point theorem for two pairs of occasionally weakly compatible single and set-valued maps satisfying a strict contractive condition in a metric space. Our result extends many results existing in the literature as those of Aliouche and Popa [15-20]. Also we establish another common fixed point theorem for four owc single and set-valued maps of Gregu% v{s} type which generalizes the results of Djoudi and Nisse, Pathak, Cho, Kang and Madharia and we end our work by giving a third theorem which extends the results given by Elamrani & Mehdaoui and Mbarki.
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Let $X$ and $Y$ be completely regular spaces and $E$ and $F$ be Hausdorff topological vector spaces. We call a linear map $T$ from a subspace of $C(X,E)$ into $C(Y,F)$ a emph{Banach-Stone map} if it has the form $Tf(y) = S_{y}(f(h(y))$ for a family of linear operators $S_{y} : E to F$, $y in Y$, and a function $h: Y to X$. In this paper, we consider maps having the property: cap^{k}_{i=1}Z(f_{i}) eqemptysetiffcap^{k}_{i=1}Z(Tf_{i}) eq emptyset, where $Z(f) = {f = 0}$. We characterize linear bijections with property (Z) between spaces of continuous functions, respectively, spaces of differentiable functions (including $C^{infty}$), as Banach-Stone maps. In particular, we confirm a conjecture of Ercan and Onal: Suppose that $X$ and $Y$ are realcompact spaces and $E$ and $F$ are Hausdorff topological vector lattices (respectively, $C^{*}$-algebras). Let $T: C(X,E) to C(Y,F)$ be a vector lattice isomorphism (respectively, *-algebra isomorphism) such that Z(f) eqemptysetiff Z(Tf) eqemptyset. Then $X$ is homeomorphic to $Y$ and $E$ is lattice isomorphic (respectively, $C^{*}$-isomorphic) to $F$. Some results concerning the continuity of $T$ are also obtained.
The split common fixed point problems has found its applications in various branches of mathematics both pure and applied. It provides us a unified structure to study a large number of nonlinear mappings. Our interest here is to apply these mappings and propose some iterative methods for solving the split common fixed point problems and its variant forms, and we prove the convergence results of these algorithms. As a special case of the split common fixed problems, we consider the split common fixed point equality problems for the class of finite family of quasi-nonexpansive mappings. Furthermore, we consider another problem namely split feasibility and fixed point equality problems and suggest some new iterative methods and prove their convergence results for the class of quasi-nonexpansive mappings. Finally, as a special case of the split feasibility and fixed point equality problems, we consider the split feasibility and fixed point problems and propose Ishikawa-type extra-gradients algorithms for solving these split feasibility and fixed point problems for the class of quasi-nonexpansive mappings in Hilbert spaces. In the end, we prove the convergence results of the proposed algorithms. Results proved in this chapter continue to hold for different type of problems, such as; convex feasibility problem, split feasibility problem and multiple-set split feasibility problems.
We consider sequential iterative processes for the common fixed point problem of families of cutter operators on a Hilbert space. These are operators that have the property that, for any point xinH, the hyperplane through Tx whose normal is x-Tx always cuts the space into two half-spaces one of which contains the point x while the other contains the (assumed nonempty) fixed point set of T. We define and study generalized relaxations and extrapolation of cutter operators and construct extrapolated cyclic cutter operators. In this framework we investigate the Dos Santos local acceleration method in a unified manner and adopt it to a composition of cutters. For these we conduct convergence analysis of successive iteration algorithms.
In this paper, we prove some common coupled fixed point theorems for mappings satisfying different contractive conditions in the context of complete $C^*$-algebra-valued metric spaces. Moreover, the paper provides an application to prove the existence and uniqueness of a solution for Fredholm nonlinear integral equations.
String-averaging is an algorithmic structure used when handling a family of operators in situations where the algorithm at hand requires to employ the operators in a specific order. Sequential orderings are well-known and a simultaneous order means that all operators are used simultaneously (in parallel). String-averaging allows to use strings of indices, constructed by subsets of the index set of all operators, to apply the operators along these strings and then to combine their end-points in some agreed manner to yield the next iterate of the algorithm. String-averaging methods were discussed and used for solving the common fixed point problem or its important special case of the convex feasibility problem. In this paper we propose and investigate string-averaging methods for the problem of best approximation to the common fixed point set of a family of operators. This problem involves finding a point in the common fixed point set of a family of operators that is closest to a given point, called an anchor point. We construct string-averaging methods for solving the best approximation problem to the common fixed points set of either finite or infinite families of firmly nonexpansive operators in a real Hilbert space. We show that the simultaneous Halpern-Lions-Wittman-Bauschke algorithm, the Halpern-Wittman algorithm and the Combettes algorithm, which were not labeled as string-averaging methods, are actually special cases of these methods. Some of our string-averaging methods are labeled as static because they use a fixed pre-determined set of strings. Others are labeled as quasi-dynamic because they allow the choices of strings to vary, between iterations, in a specific manner and belong to a finite fixed pre-determined set of applicable strings.
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