Abstract
We introduce and study a new notion of relativelyA-maximalm-relaxed monotonicity framework and discuss some properties of a new class of generalized relatively resolvent operator associated with the relativelyA-maximalm-relaxed monotone operator and the new generalized Yosida approximations based on relativelyA-maximalm-relaxed monotonicity framework. Furthermore, we give some remarks to show that the theory of the new generalized relatively resolvent operator and Yosida approximations associated with relativelyA-maximalm-relaxed monotone operators generalizes most of the existing notions on (relatively) maximal monotone mappings in Hilbert as well as Banach space and can be applied to study variational inclusion problems and first-order evolution equations as well as evolution inclusions.
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