In this paper, we study and modify the algorithm of Kraikaew and Saejung for the class of total quasi-asymptotically nonexpansive case so that the strong convergence is guaranteed for the solution of split common fixed-point problems in Hilbert space. Moreover, we justify our result through an example. The results presented in this paper not only extend the result of Kraikaew and Saejung but also extend, improve, and generalize some existing results in the literature.

Abstract. 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.

In this paper, we study synchronal and cyclic algorithms for finding a common fixed point x * of a finite family of strictly pseudocontractive mappings, which solve the variational inequalitywhere f is a contraction mapping, G is an η-strongly accretive and L-Lipschitzian operator, N ≥ 1 is a positive integer, γ , μ > 0 are arbitrary fixed constants, andare N-strict pseudocontractions. Furthermore, we prove strong convergence theorems of such iterative algorithms in a real q-uniformly smooth Banach space. The results presented extend, generalize and improve the corresponding results recently announced by many authors. MSC: 47H06; 47H09; 47H10; 47J05; 47J20; 47J25

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 quasinonexpansive 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 quasinonexpansive 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 multipleset split feasibility problems.

In the present paper, we investigate a differential game of pursuit for an infinite system of simple motion in a plane. The control functions of the players satisfies both geometric and integral constraints respectively. In the plane, the game is assume to be completed if the state of the pursuer xk, k = 1, 2, ... is directly coincide with that of the evader yk, k = 1, 2, ..., i.e; xk(x ) = yk(x ), k = 1,2, ..., at some time x and the evader is tries to stop the incident. In addition to that the strategy of the pursuer with respect to geometric and integral constraints will be constructed. Moreover, a numerical example will be given to illustrate the result.

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In this paper, we introduced new algorithms for solving split equality fixed point problems for the class of demicontractive mappings in Hilbert spaces and proved the convergence results of the proposed algorithms. The results presented in this paper generalized a number of well-known results announced.