An algorithmic approach for a system of extended multi-valued variational-like inclusions in Banach spaces

Abstract

In this paper, we pursue two purposes. Our first goal is to study a new system of extended multi-valued nonlinear
variational inclusions in Banach spaces and to establish its equivalence with a system of fixed point problems with
the help of the concept of $(H,\eta)$-proximal mapping. The obtained alternative equivalent formulation is used and
a new iterative algorithm for finding its approximate solution is proposed. Under some appropriate assumptions imposed
on the mappings and parameters involved in the system of extended multi-valued nonlinear variational inclusions,
the existence of solution for the system mentioned above is proved and the convergence analysis of the sequences
generated by our suggested iterative algorithm is discussed. The second objective of this paper is to investigate
and analyze the notion of the $C_n$-$\eta$-monotone mapping, which is an extension of the concept of $C_n$-monotone
mapping, and pointing out some remarks relating to $C_n$-$\eta$-monotone mapping and the results concerning it
appeared in the literature. The results presented in this paper are new, and improve and generalize many known corresponding results.