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General nonlinear variational inclusion systems involving \(A\)-maximal \((m)\)-relaxed monotone, RMM, RMRM, PSM and cocoercive mappings. (English) Zbl 1168.49011

Summary: Based on the notions of \(A\)-maximal \((m)\)-relaxed monotonicity (also referred to as \(A\)-monotonicity in literature), RMM and RMRM, the solvability of a system of nonlinear variational inclusion problems based on the generalized resolvent operator technique is discussed. Furthermore, \(A\)-maximal \((m)\)-relaxed monotonicity has a wide range of applications, especially to the Yosida regularization corresponding to first-order evolution equations as well as to evolution inclusions in Hilbert space as well as in Banach space settings. Among the existing models for applications to nonlinear differential equations, RMM models uniquely do the job, even better than other more general models in literature, for stance RMRM models. Furthermore, a simple notion of partially strongly monotone (PSM) mappings is introduced and studied, while all strongly monotone mappings are PSM.

MSC:

49J40 Variational inequalities
65B05 Extrapolation to the limit, deferred corrections
47J20 Variational and other types of inequalities involving nonlinear operators (general)
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