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In this paper, we introduce two modified Tseng's extragradient algorithms with a new generalized adaptive stepsize for solving monotone variational inequalities (VI) in reflexive Banach spaces. The advantage of our methods is that stepsizes do not require prior knowledge of the Lipschitz constant of the cost mapping. Based on Bregman projection-type methods, we prove weak and strong convergence of the proposed algorithms to a solution of VI. Some numerical experiments to show the efficiency of our methods including a comparison with related methods are provided. [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]

A novel iterative algorithm is introduced for solving the split common fixed point problem with demicontractive operators. Weak and strong convergence of the algorithm is discussed. The step-sizes of the algorithm are chosen in such a way that the implementation of the algorithm does not depend on operator norms. The main results proven in this paper extend and improve some results of Yao et al. [Algorithms with strong convergence for the split common solution of the feasibility problem and fixed point problem. Fixed Point Theor Appl. 2014;2014:183], Boikanyo [A strongly convergent algorithm for the split common fixed point problem. Appl Math Comput. 2015;265:844–853] and Cui and Wang [Iterative methods for the split common fixed point problem in Hilbert spaces. Fixed Point Theor Appl. 2014;2014:78]. [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 present two penalty-splitting inspired iteration schemes (PFFSA) and (PFFSA) for hierarchical equilibrium problems in Hilbert space. Based on the Opial-Passty lemma, we propose weak ergodic convergence and weak convergence of the iterative sequences generated by the Forward-Forward algorithm (PFFSA) and the Forward-Backward algorithm (PFFSA), which are proved under quite mild conditions: the bifunction of the two level equilibrium problems are supposed pseudomonotone. For strong convergence, we first add a strong monotonicity condition on the objective bifunction. We present after, a strong convergence result of algorithm (PFFSA) by adding a topological assumption, i.e. the objective bifunction is of class . Some examples are given to illustrate our results. The first example deals with pseudomonotone variational inequalities and convex minimization problem in the upper level problem. In the second one, we propose a convex minimization in the lower-level problem, where strong convergence of (PFFSA) to a minimum point is insured under infcompactness condition for objective function. These convergence results are new and generalize some recent results in this field. [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]

Letbe a real Hilbert space,a closed convex subset of,andcontinuous convex and bounded from below on. We study the asymptotic behaviour of the non-autonomous differential inclusion with project operator as followswhereis an absolutely continuous decreasing control function with strictly positive value and. On the one hand, when, each solution of PS with initial point(not necessarily in) converges to an element ofin the weak topology. On the other hand, whenand, the weak and strong convergence of PS to the minimizer ofovercan be guaranteed under some conditions. Strong convergence can be obtained whenoris strongly convex on. Another two sufficient conditions are given to ensure the weak convergence. Moreover, asymptotic analysis for the special case thatis also considered. Finally, we present some discussion on the global existence of the solutions of PS. [ABSTRACT FROM PUBLISHER]