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We introduce a novel class of asymptotically α-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. J. Convex Anal. 16 (2009), no. 2, 587-600.], Moudafi [Moudafi, A. The split common fixed-point problem for demicontractive mappings. Inverse Problems 26 (2010), no. 5:055007.; Moudafi, A. A note on the split common fixed-point problem for quasi-nonexpansive operators. Nonlinear Anal. 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. J. Nigerian Math. Soc. 38 (2019), no. 3, 363-390.], Fan et al [Fan, Q.; Peng, J.; He, H. Weak and strong convergence theorems for the split common fixed point problem with demicontractive operators. Optimization 70 (2021), no. 5-6, 1409-1423.] and host of other related results in literature. [ABSTRACT FROM AUTHOR]

This paper aims to present two inertial iterative algorithms for estimating the solution of split variational inclusion (S p VI s P) and its extended version for estimating the common solution of (S p VI s P) and fixed point problem (FPP) of a nonexpansive mapping in the setting of real Hilbert spaces. We establish the weak convergence of the proposed algorithms and strong convergence of the extended version without using the pre-estimated norm of a bounded linear operator. We also exhibit the reliability and behavior of the proposed algorithms using appropriate assumptions in a numerical example. [ABSTRACT FROM AUTHOR]

In this article, we enquire for some weak and strong convergence results for a class of mappings satisfying Condition (E) via Picard Normal S-iterative algorithm (PNSA) in the setting of uniformly convex Banach space (UCBS). Eventually, we furnish an example to substantial our attained findings and numerically compare PNSA with that of some other well-known iterative algorithms. Additionally, we discuss the existence of the solution of a nonlinear functional integral equation as an application of our result. [ABSTRACT FROM AUTHOR]

In this paper, we present self-adaptive inertial iterative algorithms involving Yosida approximation to investigate a split variational inclusion problem (SVIP) and common solutions of a fixed point problem (FPP) and SVIP in Hilbert spaces. We analyze the weak convergence of the proposed iterative algorithm to explore the approximate solution of the SVIP and strong convergence to estimate the common solution of the SVIP and FPP under some mild suppositions. A numerical example is demonstrated to validate the theoretical findings, and comparison of our iterative methods with some known schemes is outlined.

In this paper, we present self-adaptive inertial iterative algorithms involving Yosida approximation to investigate a split variational inclusion problem (SVIP) and common solutions of a fixed point problem (FPP) and SVIP in Hilbert spaces. We analyze the weak convergence of the proposed iterative algorithm to explore the approximate solution of the SVIP and strong convergence to estimate the common solution of the SVIP and FPP under some mild suppositions. A numerical example is demonstrated to validate the theoretical findings, and comparison of our iterative methods with some known schemes is outlined. [ABSTRACT FROM AUTHOR]

In this work, we propose and study a framework of generalized proximal point algorithms associated with a maximally monotone operator. We indicate sufficient conditions on the regularization and relaxation parameters of generalized proximal point algorithms for the equivalence of the boundedness of the sequence of iterations generated by this algorithm and the non-emptiness of the zero set of the maximally monotone operator, and for the weak and strong convergence of the algorithm. Our results cover or improve many results on generalized proximal point algorithms in our references. Improvements of our results are illustrated by comparing our results with related known ones. [ABSTRACT FROM AUTHOR]

In this work, we introduce and study the modified inertial Mann’s algorithm and inertial hybrid algorithm for approximating some fixed points of a k-strict pseudo-contractive mapping in Hilbert spaces. Weak convergence to a solution of fixed-point problems for a k-strict pseudo-contractive mapping is obtained by using the modified inertial Mann’s algorithm. In order to obtain strong convergence, we introduce an inertial hybrid algorithm by using the inertial extrapolation method mixed with the convex combination of three iterated vectors and forcing for strong convergence by the hybrid projection method for a k-strict pseudo-contractive mapping in Hilbert spaces. The strong convergence theorem of the proposed method is proved under mild assumptions on the scalars. For illustrating the performance of the proposed algorithms, we provide some new nonlinear k-strict pseudo-contractive mappings which are not nonexpansive to create some numerical experiments to show the advantage of the two new inertial algorithms for a k-strict pseudo-contractive mapping. [ABSTRACT FROM AUTHOR]

In this work, we establish two new self-adaptive parallel algorithms to solve the generalized split common fixed point problem which is to find a point which belongs to the intersection of finite family of fixed point sets of demimetric mappings such that its image under a finite number of linear transformations belongs to the intersection of another finite family of fixed point sets of demimetric mappings in the image space. Under suitable assumptions, the weak and strong convergence theorems are analysed. The obtained results generalize and improve the recent results announced by many other authors in the framework of split inverse problem. As a direct consequence of our two main algorithms, we obtain several new algorithms. Preliminary numerical experiments are provided to illustrate the efficiency and implementation of our new methods and also to compare with others. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new splitting algorithm to find the zero of a monotone inclusion problem that features the sum of three maximal monotone operators and a Lipschitz continuous monotone operator in Hilbert spaces. We prove that the sequence of iterates generated by our proposed splitting algorithm converges weakly to the zero of the considered inclusion problem under mild conditions on the iterative parameters. Several splitting algorithms in the literature are recovered as special cases of our proposed algorithm. Another interesting feature of our algorithm is that one forward evaluation of the Lipschitz continuous monotone operator is utilized at each iteration. Numerical results are given to support the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we alter Wang's new iterative method as well as apply it to find the common solution of fixed point problem (FPP) and split variational inclusion problem ( S p VIP) in Hilbert space. We discuss the weak convergence for ( S p VIP) and strong convergence for the common solution of ( S p VIP) and (FPP) using appropriate assumptions. Some consequences of the proposed methods are studied. We compare our iterative schemes with other existing related schemes. [ABSTRACT FROM AUTHOR]

The aim of this paper is to propose two modified forward-backward splitting algorithms for zeros of the sum of a maximal monotone operator and a Bregman inverse strongly monotone operator in reflexive Banach spaces. We prove weak and strong convergence theorems of the generated sequences by the proposed methods under some suitable conditions. We apply our results to study the variational inequality problem and the equilibrium problem. Finally, a numerical example is given to illustrate the proposed methods. The results presented in this paper improve and generalize many known results in recent literature. [ABSTRACT FROM AUTHOR]

In this paper, we study a new hybrid iterative method for solving the generalized mixed equilibrium problems and the fixed point problems of quasi- φ-asymptotically nonexpansive multivalued mapping in uniformly smooth and uniformly convex Banach space. We prove weak and strong convergence theorem. Our results generalize and improve many recent announced results in the literature. [ABSTRACT FROM AUTHOR]

We propose and study new projection-type algorithms for solving pseudomonotone variational inequality problems in real Hilbert spaces without assuming Lipschitz continuity of the cost operators. We prove weak and strong convergence theorems for the sequences generated by these new methods. The numerical behavior of the proposed algorithms when applied to several test problems is compared with that of several previously known algorithms. [ABSTRACT FROM AUTHOR]

Two new iterative algorithms are introduced for solving the split common null point problem (SCNPP). The first algorithm is shown to converge weakly and the second is shown to converge strongly to a solution of SCNPP in a Hilbert space. The algorithms are forward methods (i.e, involving no computation of the resolvent of a monotone operator). Another feature of the algorithms is that the selection of the step-sizes does not need prior knowledge of operator norms. Numerical experiments are presented to illustrate the performance of the algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we study the iterations of strongly (asymptotic) quasi ø-nonexpansive mappings in Banach spaces. First, we prove weak convergence of the generated sequence to a common fixed point of an infinite family of strongly asymptotic quasi ø-nonexpansive mappings. Next we prove strong convergence of the generated sequence by an additional assumption. In the sequel, invoke of Halpern regularization method, we prove strong convergence of the generated sequence to a common fixed point of the family of mappings without any extra conditions. Finally, we give some applications of our main results in convex minimization and equilibrium problems and present numerical examples to illustrate and support them. [ABSTRACT FROM AUTHOR]

In this paper, we alter Wang’s new iterative method as well as apply it to find the common solution of fixed point problem (FPP) and split variational inclusion problem (SpVIP) in Hilbert space. We discuss the weak convergence for (SpVIP) and strong convergence for the common solution of (SpVIP) and (FPP) using appropriate assumptions. Some consequences of the proposed methods are studied. We compare our iterative schemes with other existing related schemes.

We present two projection extragradient algorithms for solving equilibrium problems without monotonicity and Lipschitz-type property in Hilbert spaces. Our strategy consists in embedding a subgradient projection step in the extragradient algorithm and employing an Armijo-linesearch. The strategy guarantees that the sequences generated by the presented algorithms converge weakly and strongly to a solution of the equilibrium problem, respectively. The convergence does not require any monotonicity and Lipschitz-type property of the bifunction but the nonemptyness of the solution set of the associated Minty equilibrium problem. Some numerical experiments illustrate the efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we propose a CQ-type algorithm for solving the split feasibility problem (SFP) in real Hilbert spaces. The algorithm is designed such that the step-sizes are directly computed at each iteration. We will show that the sequence generated by the proposed algorithm converges in norm to the minimum-norm solution of the SFP under appropriate conditions. In addition, we give some numerical examples to verify the implementation of our method. Our result improves and complements many known related results in the literature. [ABSTRACT FROM AUTHOR]

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Inspired by the works of López et al. [21] and the recent paper of Dang et al. [15], we devise a new inertial relaxation of the CQ algorithm for solving Split Feasibility Problems (SFP) in real Hilbert spaces. Under mild and standard conditions we establish weak convergence of the proposed algorithm. We also propose a Mann-type variant which converges strongly. The performances and comparisons with some existing methods are presented through numerical examples in Compressed Sensing and Sparse Binary Tomography by solving the LASSO problem. [ABSTRACT FROM AUTHOR]

Combining inertial-type methods with projection extragradient methods, we propose three inertial extragradient algorithms for solving nonmonotone and non-Lipschitzian equilibrium problems in Hilbert spaces. Under the assumption that the solution set of the associated Minty equilibrium problem is nonempty, we establish the weak and strong convergence of the proposed algorithms, respectively. The convergence is guaranteed without any monotonicity and Lipschitz-type continuity of the equilibrium bifunction. Some numerical experiments illustrate that the presented algorithms are faster than the existing projection extragradient ones. [ABSTRACT FROM AUTHOR]

In this article, we consider a split monotone variational inclusions constraints over the set of fixed points of finite families of firmly non-expansive mappings, named split hierarchical monotone variational inclusions problem (in short: SHMVIP). We propose an iterative algorithm for solving SHMVIP. Further, we prove that the sequence generated by proposed algorithm converges weakly to the solution of SHMVIP. Strong convergence of the iterative algorithm is also discussed under some suitable assumptions. We give an example in support of our problem. [ABSTRACT FROM AUTHOR]

In this paper, first we study the weak convergence of the proximal point algorithm for an infinite family of equilibrium problems of pseudo-monotone type in Hilbert spaces. Then with additional conditions on the bifunctions, we prove the strong convergence for the family to a common equilibrium point. We also study a regularization of Halpern type and prove the strong convergence of the generated sequence to an equilibrium point of the family of infinite pseudo-monotone bifunctions without any additional assumptions on the bifunctions. A concrete example of a family of pseudo-monotone bifunctions is also presented. [ABSTRACT FROM PUBLISHER]

A new class of extragradient-type methods is introduced for solving an equilibrium problem in a real Hilbert space without any monotonicity assumption on the equilibrium function. The strategy is to replace the second projection step in the classical extragradient method by a projection onto shrinking convex subsets of the feasible set. Furthermore, to ensure a sufficient decrease on the equilibrium function, a general Armijo-type condition is imposed. This condition is shown to be satisfied for four different linesearches used in the literature. Then, the weak and strong convergence of the resulting algorithms is obtained under non-monotonicity assumptions. Finally, some numerical experiments are reported. [ABSTRACT FROM AUTHOR]

In this paper, we study the weak and strong convergence of the proximal point algorithm for equilibrium problems of pseudo-monotone type in Hilbert spaces. We prove the weak convergence of the generated sequence to a common solution of two equilibrium problems and some strong convergence results with additional assumptions on pseudo-monotone bifunctions. Then we study a regularization of Halpern-type and prove the strong convergence of the generated sequence to an equilibrium point of two pseudo-monotone bifunctions without any additional assumption on bifunctions. Finally, some examples of pseudo-monotone bifunctions from pseudo-monotone operators and Nash-Cournot oligopolistic equilibrium models are also presented. Our results extend some similar results in the literature for monotone and pseudo-monotone equilibrium problems and also the related results for variational inequalities associated with monotone and pseudo-monotone operators. [ABSTRACT FROM AUTHOR]

Based on the results proposed by Li and Song [Modified proximal-point algorithm for maximal monotone operators in Banach spaces], J. Optim. Theory appl. 138 (2008) 45-64.], we modify and generate our new iterative scheme for solving generalized mixed equilibrium problems and finding zeroes of maximal monotone operators in a Banach space under the appropriate conditions. We also prove strong and weak convergence theorems of this proximal point algorithm and give an example with numerical test which corresponding to our main results. Furthermore, we also consider the convex minimization problem and the problem of finding a zero point of an α-inverse strongly monotone operator as its appplications. [ABSTRACT FROM AUTHOR]

Abstract: Several strong convergence results involving two distinct four parameter proximal point algorithms are proved under different sets of assumptions on these parameters and the general condition that the error sequence converges to zero in norm. Thus our results address the two important problems related to the proximal point algorithm — one being that of strong convergence (instead of weak convergence) and the other one being that of acceptable errors. One of the algorithms discussed was introduced by Yao and Noor (2008) while the other one is new and it is a generalization of the regularization method initiated by Lehdili and Moudafi (1996) and later developed by Xu (2006) . The new algorithm is also ideal for estimating the convergence rate of a sequence that approximates minimum values of certain functionals. Although these algorithms are distinct, it turns out that for a particular case, they are equivalent. The results of this paper extend and generalize several existing ones in the literature. [Copyright &y& Elsevier]

This article is concerned with a class of nonsmooth constrained convex optimization in a real Hilbert space. Coupling with the penalty method, we propose an automatic system (AS) and a nonautomatic system (NS) modeled by differential inclusions. Under a suitable assumption on the feasible region and a proper condition on the objective and constrained functions, some valuable convergence properties of (AS) are obtained. In order to obtain strong convergence result in general cases, based on evolution differential inclusion, we propose a nonautomatic system (NS). When the control item ε(t) of (NS) satisfies some basal conditions, global and unique existence of the solution, finite time convergence to the feasible region and slow solution choice are obtained. Moreover, under different conditions of ε(t), we give some strong convergence results of (NS). Furthermore, we end the article by numerical experiments to illustrate the efficiency and good performance of the proposed systems in this article. [ABSTRACT FROM AUTHOR]

In this paper, we propose two Ishikawa iterative algorithms for finding a common element of the set of solutions of a generalized equilibrium problem and the set of fixed points of a Lipschitz continuous pseudo-contraction mapping. We obtain both strong convergence theorems and weak convergence theorems in a Hilbert space. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce and study two hybrid proximal-point algorithms for finding a common element of the set of solutions of an equilibrium problem and the set of solutions to the equation 0∈ Tx for a maximal monotone operator T in a uniformly smooth and uniformly convex Banach space X. Strong and weak convergence results of these two hybrid proximal-point algorithms are established, respectively. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we introduce general iterative methods for finding zeros of a maximal monotone operator in a Hilbert space which unify two previously studied iterative methods: relaxed proximal point algorithm [H.K. Xu, Iterative algorithms for nonlinear operators, J. London Math Soc. 66 (2002) 240–256] and inexact hybrid extragradient proximal point algorithm [R.S. Burachik, S. Scheimberg, B.F. Svaiter, Robustness of the hybrid extragradient proximal-point algorithm, J. Optim. Theory Appl. 111 (2001) 117–136]. The paper establishes both weak convergence and strong convergence of the methods under suitable assumptions on the algorithm parameters. [Copyright &y& Elsevier]