Adeolu Taiwo

Bio: Adeolu Taiwo is an academic researcher from University of KwaZulu-Natal. The author has contributed to research in topics: Fixed point & Hilbert space. The author has an hindex of 14, co-authored 21 publications receiving 479 citations.

A Strong Convergence Theorem for Solving Pseudo-monotone Variational Inequalities Using Projection Methods

Abstract: Several iterative methods have been proposed in the literature for solving the variational inequalities in Hilbert or Banach spaces, where the underlying operator A is monotone and Lipschitz continuous. However, there are very few methods known for solving the variational inequalities, when the Lipschitz continuity of A is dispensed with. In this article, we introduce a projection-type algorithm for finding a common solution of the variational inequalities and fixed point problem in a reflexive Banach space, where A is pseudo-monotone and not necessarily Lipschitz continuous. Also, we present an application of our result to approximating solution of pseudo-monotone equilibrium problem in a reflexive Banach space. Finally, we present some numerical examples to illustrate the performance of our method as well as comparing it with related method in the literature.

Halpern-type iterative process for solving split common fixed point and monotone variational inclusion problem between Banach spaces

Abstract: In this paper, we study the split common fixed point and monotone variational inclusion problem in uniformly convex and 2-uniformly smooth Banach spaces. We propose a Halpern-type algorithm with two self-adaptive stepsizes for obtaining solution of the problem and prove strong convergence theorem for the algorithm. Many existing results in literature are derived as corollary to our main result. In addition, we apply our main result to split common minimization problem and fixed point problem and illustrate the efficiency and performance of our algorithm with a numerical example. The main result in this paper extends and generalizes many recent related results in the literature in this direction.

Inertial extragradient method via viscosity approximation approach for solving equilibrium problem in Hilbert space

Abstract: In this paper, we propose a new viscosity type inertial extragradient method with Armijo line-search technique for approximating a common solution of equilibrium problem with pseudo-monotone bifunc...

A unified algorithm for solving variational inequality and fixed point problems with application to the split equality problem

Abstract: In this paper, we propose a new extragradient method consisting of the hybrid steepest descent method, a single projection method and an Armijo line searching the technique for approximating a solution of variational inequality problem and finding the fixed point of demicontractive mapping in a real Hilbert space. The essence of this algorithm is that a single projection is required in each iteration and the step size for the next iterate is determined in such a way that there is no need for a prior estimate of the Lipschitz constant of the underlying operator. We state and prove a strong convergence theorem for approximating common solutions of variational inequality and fixed points problem under some mild conditions on the control sequences. By casting the problem into an equivalent problem in a suitable product space, we are able to present a simultaneous algorithm for solving the split equality problem without prior knowledge of the operator norm. Finally, we give some numerical examples to show the efficiency of our algorithm over some other algorithms in the literature.

A parallel combination extragradient method with Armijo line searching for finding common solutions of finite families of equilibrium and fixed point problems

Abstract: In this paper, we introduce a new parallel combination extragradient method for solving a finite family of pseudo-monotone equilibrium problems and finding a common fixed point of a finite family of demicontractive mappings in Hilbert space. The algorithm is designed such that at each iteration a single strongly convex program is solved and the stepsize is determined via an Armijo line searching technique. Also, the algorithm make a single projection onto a sub-level set which is constructed by the convex combination of finite convex functions. Under certain mild-conditions, we state and prove a strong convergence theorem for approximating a common solution of a finite family of equilibrium problems with pseudo-monotone bifunctions and a finite family of demicontractive mappings. Finally, we present numerical examples to illustrate the applicability of the algorithm proposed. This method improves many of the existing methods in the literature.

Variational inequalities

Abstract: If $- \infty < \alpha < \beta < \infty $ and $f \in C^{3} \left( [ \alpha , \beta ] \times {\bf R}^{2} , {\bf R} \right) $ is bounded, while $y \in C^{2} \left( [ \alpha , \beta ] , {\bf R} \right) $ solves the typical one-dimensional problem of the calculus of variations to minimize the function $$F \left( y \right) = \int_{ \alpha }^{ \beta }f \left( x, y(x), y'(x) \right) dx,$$ then for any ${\phi } \in C^{2} \left( [ \alpha , \beta ] , {\bf R} \right) $ for which ${\phi }^{(k)}( \alpha ) = {\phi }^{(k)}( \beta ) = 0$ for every $k \in \{ 0, 1, 2 \} $, we prove that $\int_{\alpha }^{\beta } \left( \frac{ {\partial }^{2}f }{ \partial y^{2} } {\phi }^{2} - \frac{ {\partial }^{3}f }{ \partial y^{2} \partial y' } 2 {\phi }^{3} \right) dx$ $\geq \int_{\alpha }^{\beta } \left( \frac{ {\partial }^{2}f }{ \partial y \partial y' } 2 \phi \phi ' + \frac{ {\partial }^{3}f }{ \partial y {\partial y'}^{2} } 2 {\phi }^{2} \phi ' + \frac{ {\partial }^{2}f }{ {\partial y'}^{2} } \phi \phi " + \frac{ {\partial }^{3}f }{ \partial y {\partial y'}^{2} } \phi ' {\phi }^{2} + \frac{ {\partial }^{3}f }{ {\partial y'}^{3} } \phi {\phi '}^{2} \right) dx$, so either the above are variational inequalities of motion or the Lagrangian of motion is not $C^{3}$

A Strong Convergence Theorem for Solving Pseudo-monotone Variational Inequalities Using Projection Methods

Abstract: Several iterative methods have been proposed in the literature for solving the variational inequalities in Hilbert or Banach spaces, where the underlying operator A is monotone and Lipschitz continuous. However, there are very few methods known for solving the variational inequalities, when the Lipschitz continuity of A is dispensed with. In this article, we introduce a projection-type algorithm for finding a common solution of the variational inequalities and fixed point problem in a reflexive Banach space, where A is pseudo-monotone and not necessarily Lipschitz continuous. Also, we present an application of our result to approximating solution of pseudo-monotone equilibrium problem in a reflexive Banach space. Finally, we present some numerical examples to illustrate the performance of our method as well as comparing it with related method in the literature.

Modified inertial subgradient extragradient method with self adaptive stepsize for solving monotone variational inequality and fixed point problems

Abstract: In this paper, we study a classical monotone and Lipschitz continuous variational inequality and fixed point problems defined on a level set of a convex function in the setting of Hilbert space. We...

Halpern-type iterative process for solving split common fixed point and monotone variational inclusion problem between Banach spaces

Abstract: In this paper, we study the split common fixed point and monotone variational inclusion problem in uniformly convex and 2-uniformly smooth Banach spaces. We propose a Halpern-type algorithm with two self-adaptive stepsizes for obtaining solution of the problem and prove strong convergence theorem for the algorithm. Many existing results in literature are derived as corollary to our main result. In addition, we apply our main result to split common minimization problem and fixed point problem and illustrate the efficiency and performance of our algorithm with a numerical example. The main result in this paper extends and generalizes many recent related results in the literature in this direction.

Inertial extragradient method via viscosity approximation approach for solving equilibrium problem in Hilbert space

Abstract: In this paper, we propose a new viscosity type inertial extragradient method with Armijo line-search technique for approximating a common solution of equilibrium problem with pseudo-monotone bifunc...