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Pure Mathematics

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Browsing Pure Mathematics by Author "Alakoya, Timilehin Opeyemi."

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    Fixed point approach for solving optimization problems in Hilbert, Banach and convex metric spaces.
    (2023) Ogwo, Grace Nnennaya.; Mewomo, Oluwatosin Temitope.; Alakoya, Timilehin Opeyemi.
    In this thesis, we study the fixed point approach for solving optimization problems in real Hilbert, Banach and Hadamard spaces. These optimization problems include the variational inequality problem, split variational inequality problem, generalized variational inequality problem, split equality problem, monotone inclusion problem, split monotone inclusion problem, minimization problem, split equilibrium problem, among others. We consider some interesting classes of mappings such as the nonexpansive semigroup in real Hilbert spaces, strict pseudo-contractive mapping in real Hilbert spaces and 2-uniformly convex real Banach spaces, nonexpansive mapping between a Hilbert space and a Banach space, and quasi-pseudocontractive mapping in Hilbert spaces and Hadamard spaces. We introduce several iterative schemes for approximating the solutions of the various aforementioned optimization problems and fixed point problems and prove their convergence results. We adopt and implement several inertial methods such as the inertial-viscosity-type algorithm, relaxed inertial subgradient extragradient, modified inertial forward-backward splitting algorithm viscosity method, among others. Furthermore, we present several novel and practical applications of our results to solve other optimization problems, image restoration problem, among others. Finally we present several numerical examples in comparison with some results in the literature to illustrate the applicability of our proposed methods.
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    Iterative approximations of certain nonlinear optimization, generalized eqilibrium and fixed point problems in Hilbert and Banach spaces.
    (2023) Godwin, Emeka Chigaemezu.; Mewomo, Oluwatosin Temitope.; Alakoya, Timilehin Opeyemi.
    In this thesis, in the framework of Banach spaces, we study several iterative methods for finding the solutions of many important problems in fixed point theory and optimization. Some of these include the equilibrium problem, monotone variational inclusion problem, variational inequality problem, split common fixed point problem and split minimization problem. In addition, we study fixed point problem for some important and interesting classes of mappings such as nonexpansive mappings, pseudocontractive mappings, asymptotically demicontractive mappings, quasi-pseudocontractive mappings, demimetric mappings and multivalued demicontractive mappings in real Hilbert spaces. Furthermore, we study some other classes of mappings which include the class of Bregman quasi-nonexpansive mappings and Bregman relatively nonexpansive mappings in real p-uniformly convex Banach spaces which are also uniformly smooth. Another important problem considered is the split equality problem. The split equality problem has gained attention from authors because of its vast applications to real life problems. This problem is known to contain several other optimization problems as special cases. Based on its numerous applications, we study a multiple set split equality equilibrium problem consisting of pseudomonotone bifunctions together with fixed point problem of certain nonlinear mappings in p-uniformly convex and uniformly smooth Banach spaces. In each case, we propose and study iterative algorithms for approximating the solutions of these problems and prove strong convergence theorems under suitable conditions on the control parameters. In most cases, we incorporate the inertial term which is known to speed up the convergence rate of iterative schemes. In addition, we employ several efficient iterative techniques which include the projection and contraction method, alternative regularization method, modified Halpern’s method, inertial Tseng’s extragradient method and viscosity approximation method. In all the cases, we design our algorithms in such a way that the step size does not depend on the knowledge of the Lipschitz constants of the cost operator or the norm of the bounded linear operators. We present some applications of our results to solve convex minimization problems, multiple set split variational inequality problem, image restoration problem, oligopolistic market equilibrium problem, among others. Also, we present several numerical experiments to demonstrate the efficiency, applicability and usefulness of our iterative schemes in comparison with several of the existing methods in the literature. The results obtained in this thesis extend and improve many existing results in the literature in a unified way.