It is worth emphasizing that the definition of the inverse strongly accretive mapping is based on that of the inverse strongly monotone mapping, which was studied by so many authors; see, e.g., [15, 30, 31].
This means that our results contain the study of finding a common element of (general) variational inequalities problems and fixed points set of nonexpansive mapping, which were studied in [4 8] as special cases.
A special case of problem (2.1) is the following: if is a Hilbert space, is the zero operator in, and, then problem (2.1) becomes the inclusion problem with a -maximal monotone mapping, which was studied by Verma [8].
(i If is a Hilbert space, is the zero operator in, is the identity operator in, and, then problem (2.5) becomes the parametric usual variational inclusion with a -maximal monotone mapping, which was studied by Verma [12].
(ii If is a real Banach space, is the identity operator in, and, then problem (2.5) becomes the parametric usual variational inclusion with a -accretive mapping, which was studied by Li [13].
If X is a real Banach space, Q = I is the identity operator in X, and u = 0, then problem (3.3) becomes the parametric usual variational inclusion u ∈ N ( x, g ( x ) ) + M ( x ) with an ( A, η ) -accretive mapping, which was studied by Li [54].
(iii) If X is a real Banach space, Q = I is the identity operator in X, and u = 0, then problem (3.3) becomes the parametric usual variational inclusion u ∈ N ( x, g ( x ) ) + M ( x ) with an ( A, η ) -accretive mapping, which was studied by Li [54]. .
Weak contractions are generalizations of the Banach contraction mapping, which have been studied by several authors.
Let us finish this section with such terminology for the modular space mapping which will be studied throughout.
Weak contractions are generalizations of Banach contraction mapping which have been studied by several authors, and in particular some types of weak contractions in complete metric spaces were introduced in [1].
