Relaxed Proximal Point Algorithms Based on A−maximal Relaxed Monotonicity Frameworks with Applications

Based on the A−maximal (m)−relaxed monotonicity frameworks, the approximation solvability of a general class of variational inclusion problems using the relaxed proximal point algorithm is explored, while generalizing most of the investigations, especially of Xu (2002) on strong convergence of modified version of the relaxed proximal point algorithm, Eckstein and Bertsekas (1992) on weak convergence using the relaxed proximal point algorithm to the context of the Douglas-Rachford splitting method, and Rockafellar (1976) on weak as well as strong convergence results on proximal point algorithms in real Hilbert space settings. Furthermore, the main result has been applied to the context of the H−maximal monotonicity frameworks for solving a general class of variational inclusion problems. It seems the obtained results can be used to generalize the Yosida approximation that, in turn, can be applied to firstorder evolution inclusions, and can also be applied to Douglas-Rachford splitting methods for finding the zero of the sum of two A−maximal (m)relaxed monotone mappings.