Approaching the split common solution problem for nonlinear demicontractive mappings by means of averaged iterative algorithms

In this paper we propose new averaged iterative algorithms designed for solving a split common fixed point problem in the class of demicontractive mappings. The algorithms are obtained by inserting an averaged term into the algorithms used in [Li, R. and He, Z., A new iterative algorithm for split solution problems of quasi-nonexpansive mappings J. Inequal. Appl.131 (2015), 1–12.] for solving the same problem but in the class of quasi-nonexpansive mappings, which is a subclass of demicontractive mappings. Basically, our investigation is based on the embedding of demicontractive operators in the class of quasi-nonexpansive operators by means of averaged mappings. For the considered algorithms we prove weak and strong convergence theorems in the setting of a real Hilbert space. A numerical example is given to illustrate the results.

We introduce a novel class of asymptotically $\alpha-$hemicontractive mappings and demonstrate its relationship with the existing related families of mappings. We establish certain interesting properties of the fixed point set of the new class of mappings. Furthermore, we propose and investigate a new iterative algorithm for solving split common fixed point problem for the new class of mappings. In particular, weak and strong convergence theorems for solving split common fixed point problem for our new class of mappings in Hilbert spaces are proved. Moreover, using our method, we require no prior knowledge of norm of the transfer operator. The results presented in the paper extend and improve the results of Censor and Segal [Censor, Y.; Segal, A. The split common fixed point problem for directed operators. {\it J. Convex Anal.} {\bf 16 } (2009), no. 2, 587–600.], Moudafi [Moudafi, A. The split common fixed-point problem for demicontractive mappings. {\it Inverse Problems} {\bf 26} (2010), no. 5:055007.; Moudafi, A. A note on the split common fixed-point problem for quasi-nonexpansive operators. {\it Nonlinear Anal.} {\bf 74} (2011), no. 12, 4083–4087.], Chima and Osilike [Chima, E. E.; Osilike, M. O. Split common fixed point problem for class of asymptotically hemicontractive mappings. {\it J. Nigerian Math. Soc.} {\bf 38} (2019), no. 3, 363--390.], Fan \textit{et al} [Fan, Q.; Peng, J.; He, H. Weak and strong convergence theorems for the split common fixed point problem with demicontractive operators. {\it Optimization} {\bf 70} (2021), no. 5-6, 1409--1423.] and host of other related results in literature.

Weak and strong convergence theorems for nonspreading-type mappings in Hilbert spaces

We consider mixed parallel and cyclic iterative algorithms in this paper to solve the multiple-set split equality common fixed-point problem which is a generalization of the split equality problem and the split feasibility problem for the demicontractive mappings without prior knowledge of operator norms in real Hilbert spaces. Some weak and strong convergence results are established. The results obtained in this paper generalize and improve the recent ones announced by many others.

An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings

In this paper, we introduce a new iterative algorithm for solving the split equality generalized mixed equilibrium problems. The weak and strong convergence theorems are proved for demi-contractive mappings in real Hilbert spaces. Several special cases are also discussed. As applications, we employ our results to get the convergence results for the split equality convex differentiable optimization problem, the split equality convex minimization problem, and the split equality mixed equilibrium problem. The results in this paper generalize, extend, and unify some recent results in the literature.

In this paper, first we introduce an iterative algorithm which does not require prior knowledge of operator norm and prove strong convergence theorem for approximating a solution of split common null point problem of demicontractive mappings in a real Hilbert space. Widely known the computation of algorithms involving the operator norm for solving split common null point problem may be difficult and for this reason, authors have recently started constructing iterative algorithms with a way of selecting the step-sizes such that the implementation of the algorithm does not require the calculation or estimation of the operator norm. We introduce a new algorithm for solving the split common null point problem for demicontractive mappings with a way of selecting the step-sizes such that the implementation of the algorithm does not require the calculation or estimation of the operator norm and then prove strong convergence of the sequence in real Hilbert spaces. Finally, we give some numerical examples to illustrate our main result.

The split common fixed problem (SCFPP) has been intensively studied by numerous author due to its various applications in many physical problem. However, to employ the algorithm for solving such a problem, one needs to know the prior information on the normed of bounded linear operator. Recently, Cui and Wang introduced the new algorithm for solving such a problem which does not needs any prior information on the normed on bounded linear operator and they established the weak convergence results under some mild conditions. It is well-known that in setting of infinite dimensional Hilbert space, the weak convergence does not implies strong convergence. It is the aims of this article to continue studying this problem (SCFPP) and establish the strong convergence result based on the result of Cui and Wang, this will be done for the class of demicontractive mappings. The results presented in this paper, not only extend and improve the result of Cui and Wang, but also extend, improve and generalize several well-known results announced.

We extend the notion of k-strictly pseudononspreading mappings introduced in Nonlinear Analysis 74 (2011) 1814-1822 to the notion of the more general pseudononspreading mappings. It is shown with example that the class of pseudononspreading mappings is more general than the class of k-strictly pseudonon-spreading mappings. Furthermore, it is shown with explicit examples that the class of pseudononspreading mappings and the important class of pseudocontractive mappings are independent. Some fundamental properties of the class of pseudononspreading mappings are proved. In particular, it is proved that the fixed point set of certain class of pseudononspsreading selfmappings of a nonempty closed and convex subset of a real Hilbert space is closed and convex. Demiclosedness property of such class of pseudonon-spreading mappings is proved. Certain weak and strong convergence theorems are then proved for the iterative approximation of fixed points of the class of pseudononspreading mappings.

Very recently, Plubtieng and Kumam [S. Plubtieng, P. Kumam, Weak convergence theorem for monotone mappings and a countable family of nonexpansive mappings, J. Comput. Appl. Math. 224 (2009) 614-621] proposed an iterative algorithm for finding a common solution of a variational inequality problem for an inverse-strongly monotone mapping and a fixed point problem of a countable family of nonexpansive mappings, and obtained a weak convergence theorem. In this paper, based on Plubtieng-Kumam's iterative algorithm we introduce a new iterative algorithm for finding a common solution of a generalized mixed equilibrium problem with perturbation and a fixed point problem of a countable family of nonexpansive mappings in a Hilbert space. We first derive a strong convergence theorem for this new algorithm under appropriate assumptions and then consider a special case of this new algorithm. Moreover, we establish a weak convergence theorem for this special case under some weaker assumptions. Such a weak convergence theorem unifies, improves and extends Plubtieng-Kumam's weak convergence theorem. It is worth pointing out that the proof method of strong convergence theorem is very different from the one of weak convergence theorem.

In this paper, a new class of mapping that unifies various classes of mappings associated with the class of asymptotically nonexpansive mappings is introduced. In addition, an iterative technique for approximation of fixed points of this class of mappings is introduced and studied in the setting of uniformly convex real Banach space. Moreover, Demiclosedness principle for the class of mapping under study is proved; in addition, weak and strong convergence theorems are obtained. The theorems obtained augment, generalize, improve and unify several results that are recently announced. The method of proof used is of independent interest.

Very recently, Plubtieng and Kumam [S. Plubtieng, P. Kumam, Weak convergence theorem for monotone mappings and a countable family of nonexpansive mappings, J. Comput. Appl. Math. 224 (2009) 614-621] proposed an iterative algorithm for finding a common solution of a variational inequality problem for an inverse-strongly monotone mapping and a fixed point problem of a countable family of nonexpansive mappings, and obtained a weak convergence theorem. In this paper, based on Plubtieng-Kumam's iterative algorithm we introduce a new iterative algorithm for finding a common solution of a generalized mixed equilibrium problem with perturbation and a fixed point problem of a countable family of nonexpansive mappings in a Hilbert space. We first derive a strong convergence theorem for this new algorithm under appropriate assumptions and then consider a special case of this new algorithm. Moreover, we establish a weak convergence theorem for this special case under some weaker assumptions. Such a weak convergence theorem unifies, improves and extends Plubtieng-Kumam's weak convergence theorem. It is worth pointing out that the proof method of strong convergence theorem is very different from the one of weak convergence theorem.

In this paper, the class of total asymptotically nonexpansive mappings is considered. A weak convergence theorem of Mann-type iterative algorithm is established. Hybrid projection methods are considered for the class of total asymptotically nonexpansive mappings. Strong convergence theorems are also established in the framework of Hilbert spaces.

In this paper, we introduce and study convergence analysis of a new two-step iteration process when applied to class of G-nonexpansive mappings. Weak and strong convergence theorems are established for the new two-step iterative scheme in a uniformly convex Banach space with a directed graph. Moreover, weak convergence theorem without making use of the Opial’s condition is proved. We also show the numerical experiment for supporting our main results and comparing rate of convergence of the proposed method with the Ishikawa iteration and the modified S-iteration.

In this paper, we introduce two iterative algorithms for the split feasibility problem in real Hilbert spaces by reformulating it as a fixed point equation. Under suitable conditions, weak and strong convergence theorems are established. As a consequence, we obtain weak and strong convergence iterative sequences for the split equality problem introduced by Moudafi. The efficiency of the proposed algorithms is illustrated by numerical experiments. Our results improve and extend the corresponding results announced by many others.