Multivalued Quasi-nonexpansive Mappings Research Articles

Projection methods for quasi-nonexpansive multivalued mappings in Hilbert spaces

<abstract><p>This paper proposes a modified D-iteration to approximate the solutions of three quasi-nonexpansive multivalued mappings in a real Hilbert space. Due to the incorporation of an inertial step in the iteration, the sequence generated by the modified method converges faster to the common fixed point of the mappings. Furthermore, the generated sequence strongly converges to the required solution using a shrinking technique. Numerical results obtained indicate that the proposed iteration is computationally efficient and outperforms the standard forward-backward with inertial step.</p></abstract>

Some fixed point theorems for multi-valued mappings via Ishikawa-type iterations in CAT(0) spaces

In this manuscript, we establish strong convergence theorems for the Ishikawa iteration scheme involving quasi-nonexpansive multi-valued maps in the setting of CAT(0) spaces. Our results extend and improve some related results in the literature.

Viscosity Iterative Algorithm] {Accelerated extragradient algorithm for equilibrium and fixed point problems for countable family of certain multi-valued mappings

In this paper, we introduce a viscosity-type extragradient algorithm for finding a common point of the solution of a pseudomonotone equilibrium problem and a fixed point problem of an infinite family of multi-valued quasi-nonexpansive mappings in real Hilbert space. Using our algorithm, we state and prove a strong convergence result of our iteration sequences. An application to variational inequality problem was considered. Lastly, we give a numerical example of our main result. The result presented in this paper extends and complements some recent results in literature.

Approximating Common Solution of Minimization Problems Involving Asymptotically Quasi-Nonexpansive Multivalued Mappings

In this paper, an iterative scheme for finding common solutions of the set of fixed points for a pair of asymptotically quasi-nonexpansive mapping and the set of minimizers for the minimization problem is constructed. Using the idea of the jointly demicloseness principle, strong convergence results are achieved without imposing any compactness condition on the space or the operator. Our results improve, extend and generalize many important results in the literature.

A Modified Proximal Point Algorithm for Finite Families of Minimization Problems and Fixed Point Problems of Asymptotically Quasi-nonexpansive Multivalued Mappings

ln this paper, a new iterative algorithm for finding common ele-ments of the set of fixed points for a finite family of asymptotically quasi-nonexpansive multivalued mappings and the set of minimizers for a finite family of minimization problem is constructed. Under mild conditions on the control sequences, strong convergence of our algorithm was achieved without necessarily imposing any compactness condition on the space or the operator by using an independent approach. Our results improve, ex-tend and generalize many important results recently announced in current literature.

On multi-valued version of M-iteration process

We give a multi-valued version of M iterative process [K. Ullah and M. Arshad, Numerical reckoning fixed points Suzuki’s generalized nonexpansive mappings via new iteration process, Filomat 32 (2018) 187–196] in the setting of Banach spaces. After this, we prove some weak and strong convergence results for the class of multi-valued quasi-nonexpansive mappings (which contains the class of multi-valued nonexpansive mappings) with proper numerical examples using multi-valued version of M-iteration process. These results generalize several well-known comparable results in the literature.

A new fixed point technique for equilibrium problem with a finite family of multivalued quasi-nonexpansive mappings

A new fixed point technique for equilibrium problem with a finite family of multivalued quasi-nonexpansive mappings

Iterative algorithm for approximating fixed points of multivalued quasinonexpansive mappings in Banach spaces

Let E be a strictly convex real Banach space and let Dsubseteq E be a nonempty closed convex subset of E. Let T_{i}: {D}longrightarrow mathcal{P}({D}), i=1,2,3,ldots be a countable family of quasinonexpansive multivalued maps that are continuous with respect to the Hausdorff metric, mathcal{P}(D) is the family of proximinal and bounded subsets of D. Supposing that the family has at least one common fixed point, we show that a Krasnoselskii–Mann-type sequence converges strongly to a common fixed point. Our result generalizes and complements some important results for single-valued and multivalued quasinonexpansive maps.

On split generalized mixed equilibrium and fixed point problems of an infinite family of quasi-nonexpansive multi-valued mappings in real Hilbert spaces

In this paper, we study split generalized mixed equilibrium problem and fixed point problem in real Hilbert spaces with a view to analyze an iterative method for approximating a common solution of split generalized mixed equilibrium problem and fixed point problem of an infinite family of a quasi-nonexpansive multi-valued mappings. The iterative algorithm introduced in this paper is designed in such a way that it does not require the knowledge of the operator norm. We state and prove a strong convergence result of the aforementioned problems and also give application of our main result to split variational inequality problem. Our result complements and extends some related results in literature.

Convergence theorems for total asymptotically nonexpansive single-valued and quasi nonexpansive multi-valued mappings in hyperbolic spaces

Abstract In this paper, we discuss the Δ-convergence and strong convergence for the iterative sequence generated by the proposed scheme to approximate a common fixed point of a total asymptotically nonexpansive single-valued mapping and a quasi nonexpansive multi-valued mapping in a complete uniformly convex hyperbolic space. Finally, by giving an example, we illustrate our result.

Modified Hybrid Projection Methods with SP Iterations for Quasi-Nonexpansive Multivalued Mappings in Hilbert Spaces

In this paper, we present a modified SP iteration with the inertial technical term for three quasi-nonexpansive multivalued mappings in a Hilbert space. We then obtain weak convergence theorem under suitable conditions. The strong convergence theorems are given using CQ and shrinking projection methods with our modified iteration. Finally, we test some numerical experiments to illustrate that our inertial forward–backward method with the inertial technique term has a more effective convergence than that of the standard forward–backward method and Halpern algorithm.

Approximation of the Fixed Point of Multivalued Quasi-Nonexpansive Mappings via a Faster Iterative Process with Applications

In this paper, we approximate the fixed points of multivalued quasi-nonexpansive mappings via a faster iterative process and propose a faster fixed-point iterative method for finding the solution of two-point boundary value problems. We prove analytically and with series of numerical experiments that the Picard–Ishikawa hybrid iterative process has the same rate of convergence as the CR iterative process.

Weak and strong convergence results for the modified Noor iteration of three quasi-nonexpansive multivalued mappings in Hilbert spaces

The paper aims to present an advanced algorithm by taking help of the Noor-iteration scheme along with the inertial technical term for three quasi-nonexpansive multivalued in Hilbert spaces. A weak convergence theorem under certain conditions has been given and added the CQ and shrinking projection methods to our algorithm to obtain certain strong convergence results. Furthermore, numerical experiments are provided by constructing an example and comparison results have also been incorporated.

The Modified Inertial Iterative Algorithm for Solving Split Variational Inclusion Problem for Multi-Valued Quasi Nonexpansive Mappings with Some Applications

Based on the very recent work by Shehu and Agbebaku in Comput. Appl. Math. 2017, we introduce an extension of their iterative algorithm by combining it with inertial extrapolation for solving split inclusion problems and fixed point problems. Under suitable conditions, we prove that the proposed algorithm converges strongly to common elements of the solution set of the split inclusion problems and fixed point problems.

A modified Krasnosellkii-Mann algorithms for computing fixed points of multivalued quasi-nonexpansive mappings in Banach spaces

It is well known that Krasnoselskii-Mann iteration of nonexpansive mappings find application in many areas of mathematics and know to be weakly convergent in the infinite dimensional setting. In this paper, we introduce and study an explicit iterative scheme by a modified Krasnoselskii-Mann algorithm for approximating fixed points of multivalued quasi-nonexpansive mappings in Banach spaces. Strong convergence of the sequence generated by this algorithm is established. There is no compactness assumption. The results obtained in this paper are significant improvement on important recent results.

An iterative algorithm for solving equilibrium problems, variational inequalities and fixed point problems of multivalued quasi-nonexpansive mappings

An iterative algorithm for solving equilibrium problems, variational inequalities and fixed point problems of multivalued quasi-nonexpansive mappings

Proximal point algorithms for finding common fixed points of a finite family of quasi-nonexpansive multi-valued mappings in real Hilbert spaces

In this paper, we propose a modified proximal point algorithm for finding a common element of the set of common fixed points of a finite family of quasi-nonexpansive multi-valued mappings and the set of minimizers of convex and lower semi-continuous functions. We obtain weak and strong convergence of the proposed algorithm to a common element of the two sets in real Hilbert spaces. A numerical example to support our main results is also given.

A modified inertial shrinking projection method for solving inclusion problems and quasi-nonexpansive multivalued mappings

In this work, we propose a modified inertial and forward–backward splitting method for solving the fixed point problem of a quasi-nonexpansive multivalued mapping and the inclusion problem. Then, we establish the weak convergence theorem of the proposed method. The strongly convergent theorem is also established under suitable assumptions in Hilbert spaces using the shrinking projection method. Some preliminary numerical experiments are tested to illustrate the advantage performance of our methods.

Strong Convergence Theorems for a Countable Family of Multi-Valued Bregman Quasi-Nonexpansive Mappings in Reflexive Banach Spaces

ABSTRACTThis article uses the shrinking projection method introduced by Takahashi, Kubota and Takeuchi to propose an iteration algorithm for a countable family of Bregman multi-valued quasi-nonexpansive mappings in order to have the strong convergence under a limit condition in the framework of reflexive Banach spaces. We apply our results to a zero point problem of maximal monotone mappings and equilibrium problems in reflexive Banach spaces. The results presented in the article improve and extend the corresponding results of that found in the literature.

On split inclusion problem and fixed point problem for multi-valued mappings

Many of the iterative schemes for solving split inclusion and fixed point problems involve step-sizes that depend on the norm of a bounded linear operator. The implementation of such algorithms are usually difficult to handle. This is because they require the computation of the operator norm. In this paper, we propose an algorithm involving a step-size selected in such a way that its implementation does not require the computation or an estimate of some spectral radius. Using our algorithm we proved strong convergence theorem for split inclusion problem and fixed point problem for multi-valued quasi-nonexpansive mappings in real Hilbert spaces. Our result generalizes some important and recent results in the literature. Some applications of our main result to game theory and variational inequality problem are also presented.