Let E be a real Banach space with dual space E *. A new class of relatively weak J-nonexpansive maps, T : E → E * , is introduced and studied. An algorithm to approximate a common element of J-fixed points for a countable family of relatively weak J-nonexpansive maps and zeros of a countable family of inverse strongly monotone maps in a 2-uniformly convex and uniformly smooth real Banach space is constructed. Furthermore, assuming existence, the sequence of the algorithm is proved to converge strongly. Finally, a numerical example is given to illustrate the convergence of the sequence generated by the algorithm.
Using a dynamical step size technique, a new self-adaptive CQ-algorithm is proposed in the presence of an inertial term to find the solution of convex feasibility problem and monotone inclusion problem involving a finite number of maximal monotone set valued operators. To do this, in certain Banach spaces, we construct an algorithm which converges to the fixed point of right Bregman strongly nonexpansive mappings and coincidentally solves the convex feasibility and monotone inclusion problems. Strong convergence of the algorithm is achieved without computation of the associated operator norms. Interesting numerical examples which illustrate the implementation and efficiency of our scheme are also given. Results obtained via this work improve and extend on previous results of its kind, in the literature.
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