Research On The Split Equation Equalization Problem And The Neighboring Point Algorithm

Since the split feasibility problem has been widely applied, it becomes a very important problem in nonlinear functional analysis, and attracted the attention of many scholars. In 1994, Censor and Elfving[l] firstly introduced split feasibility problem in finitely dimensional Hilbert spaces. It provided with crucial theoretical basis for solving problems in different spaces.In 2010, Moudafi[43] introduced split common fixed point problem, it gener-alized the split feasibility problem and convex feasilibility problem. Moudafi[15]introduced the split equality fixed point problem in 2013, which generaliezd the split common fixed point problem. For solving the split equality fixed point prob-lem, Moudafi introduced the alternating CQ algoritm in [15], and proved the weak convergence theorem. In 2013, Kazmi and Rizvi[30] propesed the split equilibrium problem, it also generalized the split common fixed point problem. Recently, be-cause of the widespread use of the split equilibrium problem, many scholar devote themselves to the split equilibrium problem[26,30,31]. For example, Witthayarat,Abdou and Cho in [26] propesed a new shrinking method for solving split equi-librium problems and fixed point problems in Hilbert spaces.In the thesis, we mainly research the split equality equilibrium problem. For solving the split equality equilibrium problem in Hilbert spaces, in 2015, Ma,Wang, Chang and Duan in [42] introduced a new iterative scheme, proved the weak convergence theorems, and under the condition of semi-compact, they got the strong convergence theorems. But the condition of semi-compact is very strong.So combining the iterative scheme of Witthayarat, Abdou and Cho involving the split equilibrium problem in [26] , we introduced a new shrinking method iterative scheme to prove the strong convergence theorems for split equility equilibrium problem without semi-compact.Because Hilbert spaces are complete inner spaces, in the next we generalized the results involving the split equility equilibrium problem in Hilbert spaces to Banach spaces. Up to now, the split equility equilibrium problem in Banach spaces is not yet resolved. Therefore, we investigate the split equality equilibrium problems in Banach spaces. By using generalized projection, we introduce an iterative scheme to solve this problem, and obtain strong theorems.Recently, many convergence results by the proximal point algorithm for solv-ing optimization problems have been extended from the classical linear spaces such as Euclidean spaces, Hilbert spaces and Banach spaces to many other spaces. The minimizers of the objective convex functions in the spaces with nonlinearity playa crucial role in the branch of analysis and geometry. Optimization problems can be applied in many axeas, such as in computer vision, machine learning, electronic structure computation, system balancing and robot manipulation[50-56].Very recently, Chang, Wu, Wang, Wang [59] introduced and studied the mod-ified proximal point algorithm involving fixed point for nonspreading-type multi-valued mappings in Hilbert spaces. Inspired and motivated by the above works,we extend the main results in [59] from onspreading-type multivalued mappings to asymptotically onexpansive multivalued mappings.The content of this thesis is organized by four sections.Firstly, we illustrate the background and current state of split equality equi-librium problems.Secondly, we introduce a new iterative algorithm to solve split equality equi-librium problems in Hilbert spaces, and obtain some strong convergence theorems without the condition of semi-compact.Thirdly, we investigate the split equality equilibrium problems in Banach spaces. By using generalized projection, we introduce an iterative scheme to solve this problem, and obtain strong theorems.Finally, we generalize the results of Chang, Wu, Wang, Wang in [59] to asymptotically nonexpansive multivalued mappings in Hilbert spaces.