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The concept of resolvent operator associated with a P-η-accretive mapping is used in constructing of a new iterative algorithm for solving a new system of generalized multi-valued resolvent equations in the framework of Banach spaces. Some definitions along with some new concrete examples are provided. The main result of this paper is to prove the Lipschitz continuity of the resolvent operator associated with a P-η-accretive mapping and to compute an estimate of its Lipschitz constant under some new appropriate conditions imposed on the parameters and mappings involved in it. In part II, the convergence analysis of the sequences generated by our proposed iterative algorithm under some appropriate conditions is studied. The results presented in this paper are new, and improve and generalize many known corresponding results. [ABSTRACT FROM AUTHOR]

This paper focuses on investigating the problem of finding a common element of the set of solutions of a generalized nonlinear implicit variational-like inclusion problem involving an $ (A,\eta) $ (A , η) -maximal m-relaxed monotone mapping in the sense of L.-G.-s.i.p. and the set of fixed points of a total asymptotically nonexpansive mapping. To achieve such a purpose, a new iterative algorithm is constructed. Applying the concepts of graph convergence and generalized resolvent operator associated with an $ (A,\eta) $ (A , η) -maximal m-relaxed monotone mapping in the sense of L.-G.-s.i.p. As an application of the obtained equivalence relationship, the strong convergence of the sequence generated by our proposed iterative algorithm to a point belonging to the intersection of the two sets mentioned above is proved. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the investigation problem of finding a common element of the set of fixed points of a total uniformly L-Lipschitzian mapping and the set of solutions of a system of generalized nonlinear variational-like inequalities. For this end, based on P-η-proximal mapping, an iterative algorithm is constructed and under some suitable conditions, the strong convergence and stability of the sequence generated by our proposed iterative algorithm to a common element of the above two sets are proved. [ABSTRACT FROM AUTHOR]

The main contribution of this work is the construction of a new iterative algorithm with the help of the notion of resolvent operator associated with an $ \widehat{A} $-maximal $ m $-relaxed $ \eta $-accretive mapping for solving a new class of set-valued variational-like inclusion problems in the setting of Banach spaces. The convergence analysis of the iterative sequences generated by our proposed iterative algorithm is studied under some appropriate conditions. The main attention of the final section is paid to the investigating and analysis of the notion of $ (H(.,.), \eta) $-accretive operator introduced and studied by Wang and Ding (2010). In the light of the considered conditions for an $ (H(.,.), \eta) $-accretive operator, we point out that every $ (H(.,.), \eta) $-accretive operator is actually an $ \widehat{A} $-maximal $ m $-relaxed $ \eta $-accretive mapping and is not a new one. Finally, our paper is closed with some important comments regarding $ (H(.,.), \eta) $-accretive operator and the derived results related to it in the literature. [ABSTRACT FROM AUTHOR]

In this paper, under some new appropriate conditions imposed on the parameter and mappings involved in the resolvent operator associated with an (H; η)-monotone operator, its Lipschitz continuity is proved and an estimate of its Lipschitz constant is computed. This paper is also concerned with the establishment of a new equivalence relationship between the graph convergence of a sequence of (H; η)-monotone operators and their associated resolvent operators, respectively, to a given (H; η)-monotone operator and its associated resolvent operator. A new iterative scheme for approximating a common element of the set of solutions of a variational inclusion problem and the set of fixed points of a given total asymptotically nonexpansive mapping is constructed. As an application of the obtained equivalence conclusion concerning graph convergence, under some suitable conditions, the strong convergence of the sequence generated by our suggested iterative algorithm to a common element of the above-mentioned two sets is proved. Our results improve and generalize the corresponding results of recent works. [ABSTRACT FROM AUTHOR]

The present paper is in two folds. In the first fold, we prove the Lipschitz continuity of the proximal mapping associated with a general strongly H-monotone mapping and compute an estimate of its Lipschitz constant under some mild assumptions imposed on the mapping H involved in the proximal mapping. We provide two examples to show that a maximal monotone mapping need not be a general H-monotone for a single-valued mapping H from a Banach space to its dual space. A class of multi-valued nonlinear variational inclusion problems is considered, and by using the notion of proximal mapping and Nadler's technique, an iterative algorithm with mixed errors is suggested to compute its solutions. Under some appropriate hypotheses imposed on the mappings and parameters involved in the multi-valued nonlinear variational inclusion problem, the strong convergence of the sequences generated by the proposed algorithm to a solution of the aforesaid problem is verified. The second fold of this paper investigates and analyzes the notion of Cn-monotone mappings defined and studied in [S.Z. Nazemi, A new class of monotone mappings and a new class of variational inclusions in Banach spaces, J. Optim. Theory Appl. 155(3)(2012) 785-795]. Several comments related to the results and algorithm appeared in the above mentioned paper are given. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a regularized nonconvex mixed equilibrium problem and suggest iterative algorithms for solving such a problem by using the auxiliary principle technique. The convergence analysis of the proposed iterative algorithms is discussed either under pseudomono-tonicity or partially mixed relaxed and strong monotonicity of type (I) property of the bifunctions involved in the formulation. We also point out some fatal errors in [M.A. Noor et al.: On non-convex bifunction variational inequalities. Optim. Lett. 6, 1477-1488 (2012)] and [M.A. Noor et al.: Some iterative methods for solving nonconvex bifunction equilibrium variational inequalities. J. Appl. Math. Volume 2012, Article ID 280451]. Finally, we present the correct version of the results presented in these references. [ABSTRACT FROM AUTHOR]