A new projection and contraction method for solving split monotone variational inclusion, pseudomonotone variational inequality, and common fixed point problems

Abstract

Several authors have studied and proposed different iterative methods for approximating a common solution of variational inequality problem and other optimization problems. In solving this common solution problem, authors often require that the variational inequality operator be co-coercive and very stringent conditions are often imposed on the control parameters for convergence. These restrictions may limit the usefulness of these existing methods in several applications. To remedy these drawbacks, we introduce a new projection and contraction method, which employs inertial technique and self-adaptive step size for approximating a common solution of split monotone variational inclusion problem (SMVIP), variational inequality problem (VIP) and common fixed point problem (CFPP) for an infinite family of strict pseudo-contractions. We establish strong convergence result for the proposed method when the variational inequality operator is pseudomonotone and Lipschitz continuous, but without the knowledge of the Lipschitz constant nor knowledge of the operator norm and without assuming the sequentially weakly continuity condition often assumed by authors. Finally, we apply our results to study other optimization problems and we present several numerical experiments with graphical illustrations to demonstrate the efficiency of our method in comparison with some of the existing methods. Our results in this study complement several existing ones in this direction in the current literature.

Appendix 7.1.

(Algorithm 3.1 in [31])

where $\gamma \in (0, \frac{1}{L}),$ where L is the spectral radius of the operator $A^*A,$ and $A^*$ is the adjoint of A and $T: H_1\rightarrow H_1$ is a nonexpansive mapping and

• (i) $\alpha _n\in (0,1),\lim _{n\rightarrow +\infty }\alpha _n=0, \sum _{n=1}^{\infty }\alpha _n=+\infty ;$       (ii) $\sum _{n=1}^{\infty }|\alpha _n-\alpha _{n-1}|<+\infty .$