Common Solution Of Finite Family Research Articles

Inertial iterative method with self-adaptive step size for finite family of split monotone variational inclusion and fixed point problems in Banach spaces

Abstract In this paper, we propose and study a new inertial iterative algorithm with self-adaptive step size for approximating a common solution of finite family of split monotone variational inclusion problems and fixed point problem of a nonexpansive mapping between a Banach space and a Hilbert space. This method combines the inertial technique with viscosity method and self-adaptive step size for solving the common solution problem. We prove a strong convergence result for the proposed method under some mild conditions. Moreover, we apply our result to study the split feasibility problem and split minimization problem. Finally, we provide some numerical experiments to demonstrate the efficiency of our method in comparison with some well-known methods in the literature. Our method does not require prior knowledge or estimate of the operator norm, which makes it easily implementable unlike so many other methods in the literature, which require prior knowledge of the operator norm for their implementation.

Strong Convergence Theorem for Finding a Common Solution of Convex Minimization and Fixed Point Problems in CAT(0) Spaces

In this paper, we introduce a proximal point algorithm for approximating a common solution of finite family of convex minimization problems and fixed point problems fork-demicontractive mappings in complete CAT(0) spaces. We prove a strong convergence result and obtain other consequence results which generalize and extend some recent results in the literature. We further provide a numerical example to illustrate the convergence behaviour of the sequence generated by our algorithm.

Strong convergence theorem for family of minimization and monotone inclusion problems in Hadamard spaces

In this paper, we introduce a modified Ishikawa-type proximal point algorithm for approximating a common solution of minimization problem, monotone inclusion problem and fixed point problem. We obtain a strong convergence of the proposed algorithm to a common solution of finite family of minimization problem, finite family of monotone inclusion problem and fixed point problem for asymptotically demicontractive mapping in Hadamard spaces. Numerical example is given to illustrate the applicability of our main result. Our results complement and extend some recent results in literature.

Parallel Hybrid Algorithm for Solving Pseudomonotone Equilibrium and Split Common Fixed Point Problems

Using the concept of Bregman W-mapping, we propose a parallel hybrid extragradient algorithm for approximating a common element of the set of solutions of pseudomonotone equilibrium problems and split common fixed point problems of Bregman weak relatively nonexpansive mappings. With the algorithm, we state and prove a strong convergence result for finding a common solution of finite family of equilibrium problem and split common fixed point problem in a real Banach space. The stepsize for the split common fixed point problem is chosen in such a way that the algorithm does not require a prior estimation of the operator norm. Finally, we present an application of our results to variational inequality problems. Our result improves and extends some existing results in the literature in these directions.

Approximation of common solution of finite family of monotone inclusion and fixed point problems for demicontractive multivalued mappings in CAT(0) spaces

In this paper, we use the gate condition on two multivalued k-demicontractive mappings to approximate a common solution of a finite family of monotone inclusion problem and fixed point problem in CAT(0) space. Furthermore, we propose a Halpern-type proximal point algorithm and prove its strong convergence to a common solution of a finite family of monotone inclusion problems and fixed point problem for two multivalued k-demicontractive mappings in a complete CAT(0) space. We also applied our result to the problem of finding a common solution of a finite family of minimization problem and fixed point problem in CAT(0) space. Finally, numerical experiments of our result are presented to further show its applicability.

Iterative algorithm for a family of monotone inclusion problems in cat(0) spaces

In this paper, we introduce a new mapping given by a finite family of multivalued monotone mappings in a CAT(0) space. We further propose a modified Halpern-type algorithm for the mapping and prove a strong convergence theorem for approximating a common solution of finite family of monotone inclusion problems in a complete CAT(0) space. We also applied our results to solve a finite family of minimization problems in a complete CAT(0) space. Some non-trivial examples of monotone mappings in complete CAT(0) space setting were also studied and a numerical example of our algorithm to further show its applicability is also presented. Our numerical experiment shows that our algorithm converges faster than that proposed in [42] by Takahashi and Shimoji, Mathematical and Computer Modelling 32 (2000), 1463–1471.

Strong convergence theorem for monotone inclusion problem in CAT(0) spaces

In this paper, we introduce Halpern-type proximal point algorithm for approximating a common solution of monotone inclusion problem and fixed point problem. We obtain a strong convergence of the proposed algorithm to a common solution of finite family of monotone inclusion problem and fixed point problem for nonexpansive mappings in complete CAT(0) spaces. Nontrivial application and numerical example were given. Our results complement and extend some recent results in literature.

Strong convergence theorems for a solution of finite families of equilibrium and variational inequality problems

We introduce an iterative process for finding an element of common solutions of finite family of equilibrium problems and finite family of variational inequality problems for continuous monotone mappings in Banach spaces. Our theorems improve and unify most of the results that have been proved for this important class of non-linear operators.

Strong convergence theorems for variational inequality problems and quasi- $${\phi}$$ -asymptotically nonexpansive mappings

In this paper, we introduce an iterative process which converges strongly to a common solution of finite family of variational inequality problems for ?-inverse strongly monotone mappings and fixed point of two continuous quasi- $${\phi}$$ -asymptotically nonexpansive mappings in Banach spaces. Our theorems extend and unify most of the results that have been proved for the class of monotone mappings.

A hybrid scheme for finite families of equilibrium, variational inequality and fixed point problems

We introduce an iterative process for finding an element in the common fixed point set of finite family of closed relatively quasi-nonexpansive mappings, common solutions of finite family of equilibrium problems and common solutions of finite family of variational inequality problems for monotone mappings in Banach spaces. Our theorem extends and unifies most of the results that have been proved for this important class of nonlinear operators.