APPROXIMATION OF SOLUTIONS OF THE SPLIT MINIMIZATION PROBLEM WITH MULTIPLE OUTPUT SETS AND COMMON FIXED POINT PROBLEMS IN REAL BANACH SPACES

In this paper, we introduce and study a split minimization problem with multiple output sets. We propose a new iterative method, which employs the inertial Halpern approximation technique, for a common solution of the split minimization problem and the fixed point problem with a finite family of Bregman relatively nonexpansive mappings in the framework of p-uniformly convex and uniformly smooth Banach spaces. Our iterative method uses the step sizes which do not require prior knowledge of the operators norm, and we prove a strong convergence result under some mild conditions. Moreover, we present some applications of our result and further demonstrate the efficiency and applicability of our algorithm with some numerical examples. The results presented in this paper unify and complement several existing results in the literature.

Automated summary

In this paper, a split minimization problem with multiple output sets is introduced and studied. A new iterative method is proposed for solving the problem, using the inertial Halpern approximation technique in p-uniformly convex and uniformly smooth Banach spaces. The method does not require prior knowledge of the operators norm, and a strong convergence result is proven under mild conditions. Applications and numerical examples are presented to demonstrate the efficiency and applicability of the algorithm. The results in this paper unify and complement existing research in the literature.