A hybrid iterative method for common solutions of variational inequality problems and fixed point problems for single-valued and multi-valued mappings with applications

Revie

Strong Convergence Theorems for Maximal Monotone Operators with Nonspreading Mappings in a Hilbert Space

We prove the strong convergence theorems for finding a common element of the set of fixed points of a nonspreading mapping T and the solution sets of zero of a maximal monotone mapping and an α-inverse strongly monotone mapping in a Hilbert space. Manaka and Takahashi (2011) proved weak convergence theorems for maximal monotone operators with nonspreading mappings in a Hilbert space; there we introduced new iterative algorithms and got some strong convergence theorems for maximal monotone operators with nonspreading mappings in a Hilbert space

Strong Convergence Theorems under Hybrid Methods for Two Nonlinear Mappings in Banach Spaces (Study on Nonlinear Analysis and Convex Analysis)

In this article, using the hybrid method defined by Nakajo and Takahashi [17], we first obtain a strong convergence theorem for two noncommutative nonlinear mappings in a Banach space. Next, using the shrinking projection method defined by Takahashi, Takeuchi and Kubota [25], we prove another strong convergence theorem for the mappings in a Banach space. Using these results, we get well-known and new strong convergence theorems by the hybrid method and the shrinking projection method in a Hilbert space and a Banach space

A study of optimization and fixed point problems in certain geodesic metric spaces.

Doctoral Degree. University of KwaZulu-Natal, Durban.Abstract available in PDF

On the generalized asymptotically nonspreading mappings in convex metric spaces

[EN] In this article, we propose a new class of nonlinear mappings, namely, generalized asymptotically nonspreading mapping, and prove the existence of fixed points for such mapping in convex metric spaces. Furthermore, we also obtain the demiclosed principle and a delta-convergence theorem of Mann iteration for generalized asymptotically nonspreading mappings in CAT(0) spaces.Phuengrattana, W. (2017). On the generalized asymptotically nonspreading mappings in convex metric spaces. Applied General Topology. 18(1):117-129. doi:10.4995/agt.2017.6578.SWORD117129181Abkar, A., & Eslamian, M. (2012). Fixed point and convergence theorems for different classes of generalized nonexpansive mappings in CAT(0) spaces. Computers & Mathematics with Applications, 64(4), 643-650. doi:10.1016/j.camwa.2011.12.075Bruhat, F., & Tits, J. (1972). Groupes Réductifs Sur Un Corps Local. Publications mathématiques de l’IHÉS, 41(1), 5-251. doi:10.1007/bf02715544Dhompongsa, S., Kaewkhao, A., & Panyanak, B. (2012). On Kirk’s strong convergence theorem for multivalued nonexpansive mappings on CAT(0) spaces. Nonlinear Analysis: Theory, Methods & Applications, 75(2), 459-468. doi:10.1016/j.na.2011.08.046Dhompongsa, S., Kirk, W. A., & Sims, B. (2006). Fixed points of uniformly lipschitzian mappings. Nonlinear Analysis: Theory, Methods & Applications, 65(4), 762-772. doi:10.1016/j.na.2005.09.044Dhompongsa, S., & Panyanak, B. (2008). On △-convergence theorems in CAT(0) spaces. Computers & Mathematics with Applications, 56(10), 2572-2579. doi:10.1016/j.camwa.2008.05.036Iemoto, S., & Takahashi, W. (2009). Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space. Nonlinear Analysis: Theory, Methods & Applications, 71(12), e2082-e2089. doi:10.1016/j.na.2009.03.064Khan, S. H., & Abbas, M. (2011). Strong and △-convergence of some iterative schemes in CAT(0) spaces. Computers & Mathematics with Applications, 61(1), 109-116. doi:10.1016/j.camwa.2010.10.037Kirk, W. A., & Panyanak, B. (2008). A concept of convergence in geodesic spaces. Nonlinear Analysis: Theory, Methods & Applications, 68(12), 3689-3696. doi:10.1016/j.na.2007.04.011Kohsaka, F., & Takahashi, W. (2008). Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces. Archiv der Mathematik, 91(2), 166-177. doi:10.1007/s00013-008-2545-8Nanjaras, B., Panyanak, B., & Phuengrattana, W. (2010). Fixed point theorems and convergence theorems for Suzuki-generalized nonexpansive mappings in CAT(0) spaces. Nonlinear Analysis: Hybrid Systems, 4(1), 25-31. doi:10.1016/j.nahs.2009.07.003Phuengrattana, W. (2011). Approximating fixed points of Suzuki-generalized nonexpansive mappings. Nonlinear Analysis: Hybrid Systems, 5(3), 583-590. doi:10.1016/j.nahs.2010.12.006Shimizu, T., & Takahashi, W. (1996). Fixed points of multivalued mappings in certain convex metric spaces. Topological Methods in Nonlinear Analysis, 8(1), 197. doi:10.12775/tmna.1996.028Takahashi, W. (1970). A convexity in metric space and nonexpansive mappings. I. Kodai Mathematical Seminar Reports, 22(2), 142-149. doi:10.2996/kmj/113884611

On common fixed points approximation of countable families of certain multi-valued maps in hilbert spaces.

Master of Science in Mathematics, Statistics and Computer Science. University of KwaZulu-Natal, Durban 2017.Fixed point theory and its applications have been widely studied by many researchers. Di erent iterative algorithms have been used extensively to approximate solutions of xed point problems and other related problems such as equilibrium problems, variational in- equality problems, optimization problems and so on. In this dissertation, we rst introduce an iterative algorithm for nding a common solution of multiple-set split equality mixed equilibrium problem and xed point problem for in nite families of generalized ki-strictly pseudo-contractive multi-valued mappings in real Hilbert spaces. Using our iterative algo- rithm, we obtain weak and strong convergence results for approximating a common solution of multiple-set split equality mixed equilibrium problem and xed point problem. As ap- plication, we utilize our result to study the split equality mixed variational inequality and split equality convex minimization problems . Also, we present another iterative algorithm that does not require the knowledge of the oper- ator norm for approximating a common solution of split equilibrium problem and xed point problem for in nite family of multi-valued quasi-nonexpansive mappings in real Hilbert spaces. Using our iterative algorithm, we state and prove a strong convergence result for approximating a common solution of split equilibrium problem and xed point problem for in nite family of multi-valued quasi-nonexpansive mappings in real Hilbert spaces. We apply our result to convex minimization problem and also present a numerical example

A study of optimization problems and fixed point iterations in Banach spaces.

Doctoral Degree. University of KwaZulu-Natal, Durban.Abstract available in PDF

Iterative schemes for approximating common solutions of certain optimization and fixed point problems in Hilbert spaces.

Masters Degree. University of KwaZulu-Natal, Durban.In this dissertation, we introduce a shrinking projection method of an inertial type with self-adaptive step size for finding a common element of the set of solutions of Split Gen- eralized Equilibrium Problem (SGEP) and the set of common fixed points of a countable family of nonexpansive multivalued mappings in real Hilbert spaces. The self-adaptive step size incorporated helps to overcome the difficulty of having to compute the operator norm while the inertial term accelerates the rate of convergence of the propose algorithm. Under standard and mild conditions, we prove a strong convergence theorem for the sequence generated by the proposed algorithm and obtain some consequent results. We apply our result to solve Split Mixed Variational Inequality Problem (SMVIP) and Split Minimiza- tion Problem (SMP), and present numerical examples to illustrate the performance of our algorithm in comparison with other existing algorithms. Moreover, we investigate the problem of finding common solutions of Equilibrium Problem (EP), Variational Inclusion Problem (VIP)and Fixed Point Problem (FPP) for an infinite family of strict pseudo- contractive mappings. We propose an iterative scheme which combines inertial technique with viscosity method for approximating common solutions of these problems in Hilbert spaces. Under mild conditions, we prove a strong theorem for the proposed algorithm and apply our results to approximate the solutions of other optimization problems. Finally, we present a numerical example to demonstrate the efficiency of our algorithm in comparison with other existing methods in the literature. Our results improve and complement contemporary results in the literature in this direction