Graph convergence with an application for system of variational inclusions and fixed-point problems

This paper aims at proposing an iterative algorithm for finding an element in the intersection of the solutions set of a system of variational inclusions and the fixed-points set of a total uniformly L-Lipschitzian mapping. Applying the concepts of graph convergence and the resolvent operator associated with an Ĥ-accretive mapping, a new equivalence relationship between graph convergence and resolvent-operator convergence of a sequence of Ĥ-accretive mappings is established. As an application of the obtained equivalence relationship, the strong convergence of the sequence generated by our proposed iterative algorithm to a common point of the above two sets is proved under some suitable hypotheses imposed on the parameters and mappings. At the same time, the notion of H(⋅,⋅)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H(\cdot,\cdot)$\end{document}-accretive mapping that appeared in the literature, where H(⋅,⋅)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H(\cdot,\cdot)$\end{document} is an α, β-generalized accretive mapping, is also investigated and analyzed. We show that the notions H(⋅,⋅)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$H(\cdot,\cdot)$\end{document}-accretive and Ĥ-accretive operators are actually the same, and point out some comments on the results concerning them that are available in the literature.


Introduction
During the last six decades, variational inequality theory, originally introduced for the study of partial differential equations by Hartman and Stampacchia [1], has been recognized as a strong tool in the mathematical study of many nonlinear problems of physics and mechanics, as the complexity of the boundary conditions and the diversity of the constitutive equations lead to variational formulations of inequality type. As many nonlinear problems arising in optimization, operations research, structural analysis, and engineering sciences can be transformed into variational inequality problems (see, e.g., [2,3]), since the appearance of this theory, there has been an increasing interest in extending and generalizing variational inequalities in many different directions using novel and innovative techniques, see, for example, [4,5] and the references therein. Without doubt, one of the most important and well-known generalizations of variational inequalities is variational inclusions and thanks to their wide applications in the optimization and control, economics and transportation equilibrium, engineering science, etc., the study of different classes of variational inclusion problems continues to attract the interest of many researchers. For more related details, we refer the readers to [6][7][8][9][10][11][12][13][14][15][16][17][18][19] and the references therein.
It is important to emphasize that two important problems in the theory of the variational inequalities/icnlusions are the existence of solutions and approximation of solutions by the iterative algorithms. This has been one of the main motivations of researchers to develop alternative methods to study iterative algorithms for approximating solutions of various kinds of variational inequality/inclusion problems in the setting of Hilbert and Banach spaces. Among the methods that have appeared in the literature, the resolventoperator technique is interesting and important, and plays a crucial role in computing approximate solutions of different classes of variational inequality/inclusion problems and their generalizations. More information along with relevant commentaries can be found in [6, 11-14, 17, 18, 20-29] and the references therein.
Monotone operators and accretive mappings have gained impetus, due to their wide range of applicability, to resolve diverse problems emanating from the theory of nonlinear differential equations, integral equations, mathematical economics, optimal control, and so forth. Due to the importance and their many diverse applications in a huge variety of scientific fields, considerable attention has been paid to the development and generalization of monotone and accretive operators in the framework of different spaces. By the same token, the introduction of the notion of generalized m-accretive mapping as an extension of maximal monotone operators and m-accretive mapping along with a definition of its resolvent operator in a Banach space setting was first made by Huang and Fang [16] in 2001. Afterwards, many authors have shown interest in extending maximal monotone operators and generalized m-accretive mappings, and further generalizations of them have appeared in the literature. For instance, Huang and Fang [15], Ding and Lou [30] and Lee et al. [21], Fang and Huang [12], Xia and Huang [31], Fang and Huang [11], Fang et al. [14], Kazmi and Khan [24] and Peng and Zhu [25], Verma [27,32], Verma [33], and Lan et al. [17] introduced and studied the notions of η-monotone operators, η-subdifferential operators, H-monotone operators, general H-monotone operators, H-accretive (to avoid confusion, throughout the paper we call it H-accretive) mappings, (H, η)-monotone operators, P-η-accretive (also referred to as (H, η)-accretive) mappings, A-monotone operators, (A, η)-monotone operators, and (A, η)-accretive (also referred to as A-maximal m-relaxed η-accretive) mappings, respectively. Motivated by these advances, in 2008, Sun et al. [34] introduced the class of M-monotone operators as a generalization of maximal monotone and H-monotone operators. With inspiration and motivation from the work of Sun et al. [34], in the same year, Zou and Huang [35] succeeded in introducing the notion of H(·, ·)-accretive mappings in a Banach-space setting as a generalization of generalized m-accretive, H-monotone, H-accretive, and M-monotone operators.
The notion of graph convergence has attracted many researchers since 1984 after the pioneering work of Attouch [36]. It is worthwhile to stress that the attention of the author in [36] was limited to maximal monotone operators. In later years, considerable research efforts have been made to generalize and study the concept of graph convergence for generalized monotone operators and generalized accretive mappings available in the literature. For instance, Li and Huang [22] introduced the notion of graph convergence concerned with H(·, ·)-accretive operators in Banach spaces and proved some equivalence theorems between graph convergence and resolvent-operator convergence of a sequence of H(·, ·)accretive mappings. For a detailed description of the concept of graph convergence for other generalizations of generalized monotone (accretive) operators existing in the literature, we refer the interested reader to [6,22,23,28,36] and the references therein. Using the properties of graph convergence of H(·, ·)-accretive operators introduced by Li and Huang [22], recently, Tang and Wang [26] constructed a perturbed iterative algorithm for solving a system of ill-posed variational inclusions involving H(·, ·)-accretive operators. At the same time, they proved that under some suitable conditions, the sequence generated by their proposed iterative algorithm is strongly convergent to the unique solution of the system of variational inclusions considered in [26].
On the other hand, the theory of fixed points that the starting point of its study dates back to the beginning of the 1920s with the pioneering work of Polish mathematician Stefan Banach [37], is a very attractive subject, which has recently drawn much attention from the communities of physics, engineering, mathematics, etc. The existence of a strong connection between the variational inequality problems and the fixed-point problems motivated many investigators to study the problem of finding common elements of the set of solutions of variational inequalities/inclusions and the set of fixed points of given operators. For more details and information, the reader is referred to [4,[38][39][40][41][42][43][44][45][46] and the references therein.
In addition, after the emergence of the notion of nonexpansive mapping in the 1960s, the number of works dedicated to study fixed-point theory for nonexpansive mappings in the setting of different spaces has grown rapidly and has influenced several branches of mathematics. This is mainly because there is a very close relation between the classes of monotone and accretive operators, which arise naturally in the theory of differential equations, and the class of nonexpansive mappings. Due to its many diverse applications in the theory of fixed points, the interest in extending and generalizing the notion of nonexpansive mapping has increased rapidly over the past forty years. One of the first attempts in this direction was carried out by Goebel and Kirk [47] in 1972 who introduced a class of generalized nonexpansive mappings, the so-called asymptotically nonexpansive mappings. In 2005, Sahu [48] succeeded in introducing the concept of nearly asymptotically nonexpansive mapping as a generalization of the notion of asymptotically nonexpansive mapping. One year later, another class of generalized nonexpansive mappings, the so-called total asymptotically nonexpansive mappings, which is essentially more general than the classes of nearly asymptotically nonexpansive mappings and asymptotically nonexpansive mappings, was introduced and studied by Alber et al. [49]. The efforts in this direction have continued and in a successfully attempt by Kiziltunc and Purtas [50], the class of total uniformly L-Lipschitzian mappings was introduced as a unifying framework for the classes of generalized nonexpansive mappings existing in the literature. To find more information about different classes of generalized nonexpansive mappings and relevant commentaries, we refer the reader to [20,[47][48][49][50][51] and the references therein.
Motivated and inspired by the excellent work mentioned above, this paper pursues two purposes. The first objective is to prove the existence of a unique solution for a system of variational inclusions (SVI) involving H-accretive mappings under some suitable mild conditions. With the goal of finding a common point lying in the solutions set of the SVI and the set of fixed points of a total uniformly L-Lipschitzian mapping, an iterative algo-rithm is constructed. Employing the notions of graph convergence and the resolvent operator associated with an H-accretive mapping, a new equivalence relationship between the graph convergence of a sequence of H-accretive mappings and their associated resolvent operators, respectively, to a given H-accretive mapping and its associated resolvent operator under some appropriate conditions is established. As an application of the obtained equivalence relationship, we prove the strong convergence of the sequence generated by our proposed iterative algorithm to a common point of the set of fixed points of the total uniformly L-Lipschitzian mapping and the set of solutions of the SVI. The second goal of this paper is to investigate and analyze the notion of H(·, ·)-accretive mapping that appeared in [26], where H(·, ·) is an α, β-generalized accretive mapping, and to point out some comments concerning it. We prove that under the assumptions imposed on the H(·, ·)-accretive mapping considered in [26], every H(·, ·)-accretive mapping is actually an H-accretive mapping and is not a new one. All the results derived by the authors in [26] are reviewed and some remarks regarding them are stated. We show that our results improve and generalize the corresponding results of [26] and recent related works.

Notation and preliminaries
In this section, we briefly present the notation and some preliminary material to be used later in this paper. First, we make clear that all linear spaces used in this paper are assumed to be real. Unless it is stated otherwise, in this paper we denote by X a real Banach space with norm · , we denote by X * its topological dual, and ·, · will represent the duality pairing of X and X * . We denote by S X and S X * , respectively, the unit sphere in X and X * . For a given set-valued mapping M : X ⇒ X, the set Graph(M) defined by Let us recall that a normed space X is called strictly convex if S X is strictly convex, that is, the inequality x + y < 2 holds for all x, y ∈ S X with x = y. It is said to be smooth if for every x ∈ S X there is exactly one x * ∈ S X * such that x * (x) = 1. Equivalently, a normed space X is said to be smooth provided lim t→0 x+tyx t exists for all x, y ∈ S X . It is known that if a Banach space X is reflexive, then X is strictly convex if and only if X * is smooth and X is smooth if and only if X * is strictly convex.
With each x ∈ X, we associate the set The operator J : X ⇒ X * is called the normalized duality mapping of X. We observe immediately that if X = H, a Hilbert space, then J is the identity mapping on H. At the same time, from the Hahn-Banach theorem, it follows that J(x) is nonempty for each x ∈ X. In general, the normalized duality mapping is set-valued. However, it is single-valued in a smooth Banach space.

Definition 2.1
For a given real smooth Banach space X, an operator T : X → X is said to be (ii) strictly accretive if T is accretive and equality holds if and only if x = y; (iii) r-strongly accretive if there exists a constant r > 0 such that (iv) α-relaxed accretive if there exists a constant α > 0 such that (v) γ -Lipschitz continuous if there exists a constant γ > 0 such that Definition 2.2 For a given real smooth Banach space X, a set-valued operator M : (ii) m-accretive if M is accretive and (I + λM)(X) = X, for all λ > 0, where I denotes the identity mapping on X.
Let be a bounded domain in R n with smooth boundary ∂ , and let for 1 < p < ∞, L p ( ) be the space of all the Lebesgue measurable functions f : → R such that |f | p dx < ∞. Suppose further that T is a maximal monotone graph in R. With appropriate domains, (i) the operator A 1 u :=u + T(u) with homogeneous Neuman boundary condition, and the operator A 2 u :=u, -∂u ∂n ∈ T(u) on ∂ , where denotes the Laplacian, are accretive on L p ( ). Meanwhile, The operator -, where denotes the Laplacian, is an m-accretive operator.
Remark 2.5 As was pointed out in [52], the interest and importance of the concept of accretive mappings, which was introduced and studied independently by Browder [53] and Kato [54], stems from the fact that many physically significant problems can be modeled in terms of an initial-valued problem of the form where A is either an accretive or strongly accretive mapping on an appropriate Banach space. Typical examples of such evolution equations are found in models involving the heat, wave or Schrödinger equation (see, e.g., Browder [53]). An early fundamental result in the theory of accretive mappings, due to Browder [55], states that the initial-value problem (2.1) is solvable if A is locally Lipschitzian and accretive on X. Utilizing the existence result for the equation (2.1), Browder [53] proved that if A is locally Lipschitzian and accretive on X, then A is m-accretive. Obviously, a consequence of this is that the equation x + Tx = f , for a given f ∈ X, where T := I -A, has a solution. Martin [56,57] proved that the equation (2.1) is solvable if A is continuous and accretive on X, and using this result, he further established that if A is continuous and accretive, then A is m-accretive. In [52], the author verified that if A : X → X is a Lipschitz and strongly accretive mapping, then A is surjective. Consequently, for each f ∈ X, the equation Ax = f has a solution in X.
We note that M is an m-accretive mapping if and only if M is accretive and there is no other accretive mapping whose graph contains strictly Graph(M). The m-accretivity is to be understood in terms of inclusion of graphs. If M : X ⇒ X is an m-accretive mapping, then adding anything to its graph, so as to obtain the graph of a new set-valued mapping, destroys the accretivity. In fact, the extended mapping is no longer accretive. In other words, for every pair (x, u) ∈ X × X\ Graph(M) there exists (y, v) ∈ Graph(M) such that uv, J(xy) < 0. Thanks to the above-mentioned arguments, a necessary and sufficient condition for set-valued mapping M : X ⇒ X to be m-accretive is that the property is equivalent to u ∈ M(x). The above characterization of m-accretive mappings provides us with a useful and manageable way for recognizing that an element u belongs to M(x). Definition 2.6 Given a smooth Banach space X and a mapping H : X → X, the set-valued mapping M : X ⇒ X is said to be H-accretive if M is accretive and ( H + λM)(X) = X holds for all λ > 0.
It should be remarked that Fang and Huang [11] were the first to introduce the class of H-accretive mappings on q-uniformly smooth Banach spaces for some real constant q > 1. We recall that for a given real constant q > 1, X is called q-uniformly smooth if there exits a constant C > 0 such that ρ X (τ ) ≤ Cτ q , for all τ ∈ R + , where the function ρ X : R + → R + is given by formula At the same time, it should be pointed out that if H = I, then Definition 2.6 reduces to the definition of an m-accretive operator, and if X = H and H = I, then Definition 2.6 becomes just the definition of a maximal monotone operator.
The following example shows that for a given mapping H : X → X, an m-accretive mapping may not be H-accretive. Example 2.7 Let M 2 (C) be the space of all 2 × 2 matrices with complex entries. Then, is a Hilbert space together with the inner product A, B := tr(AB * ), for all A, B ∈ M 2 (C), where tr denotes the trace, that is, the sum of the diagonal entries, and B * denotes the Hermitian conjugate (or adjoint) of the matrix B, that is, B * = B t , the complex conjugate of the transpose B. The inner product defined above induces a norm on M 2 (C) as follows: Thereby, the Hilbert space (M 2 (C), · ) is a 2-uniformly smooth Banach space. For any Therefore, the set {μ k : k = 1, 2, 3, 4} spans the Hilbert space M 2 (C). Letting θ k := 1 √ 2 μ k for k = 1, 2, 3, 4, it is easy to see that the set B consisting of the rescaled 2 × 2 matrices θ k (k = 1, 2, 3, 4) is also a spanner of M 2 (C). At the same time, θ k = 1, for k = 1, 2, 3, 4 and θ k , θ j = 0 for 1 ≤ k, j ≤ 4, that is, the set B is orthonormal. Accordingly, the set B = {θ k : k = 1, 2, 3, 4} is an orthonormal basis for the Banach space i.e., M is an accretive mapping. By virtue of the fact that for any A ∈ M 2 (C) and λ > 0, we have where I is the identity mapping on M 2 (C), we conclude that (I + λM)(M 2 (C)) = M 2 (C), for every real constant λ > 0, that is, the mapping I + λM is surjective for every positive real constant λ. Hence, M is an m-accretive mapping. Since for any A ∈ M 2 (C), we have This implies that 0 / ∈ ( H + M)(H 2 (C)), i.e., H + M is not surjective. Thus, the mapping M is not H-accretive.
Example 2.8 Let H 2 (C) be the set of all Hermitian matrices with complex entries. We recall that a square matrix A is said to be Hermitian (or self-adjoint) if it is equal to its own Hermitian conjugate, i.e., A * = A t = A. In the light of the definition of a Hermitian 2 × 2 matrix, the condition A * = A implies that the 2 × 2 matrix A = a b c d is Hermitian if and only if a, d ∈ R and b =c. Therefore, Then, H 2 (C) is a subspace of M 2 (C), the space of all 2 × 2 matrices with complex entries, with respect to the operations of addition and scalar multiplication defined on M 2 (C), when M 2 (C) is considered as a real vector space. In other words, H 2 (C) together with the mentioned operators is a vector space over R. We introduce the scalar product on H 2 (C) as A, B := 1 2 tr(AB), for all A, B ∈ H 2 (C). By an easy check, we observe that (H 2 (C), ·, · ) is an inner product space. The inner product defined above induces a norm on H 2 (C) as follows: Taking into account that every finite-dimensional normed space is a Banach space, it follows that (H 2 (C), · ) is a Hilbert space and so it is a 2-uniformly smooth Banach space.
Let the mappings H 1 , H 2 , M : H 2 (C) → H 2 (C) be defined, respectively, by 1+y 2 e x e x +1 + 2i for all A = z x-iy x+iy w ∈ H 2 (C), where α and β are arbitrary positive real constants, γ and ς are arbitrary nonzero real constants, θ and ξ are arbitrary real constants, and k, l, and p are arbitrary but fixed odd natural numbers.
Then, for any A = , Taking into account the fact that k, l, and p are odd natural numbers, it can be easily Then, for any A = z x-iy x+iy w ∈ H 2 (C), it yields It can be easily seen that These facts imply that ( H 1 + M)(H 2 (C)) = H 2 (C), i.e., H 1 + M is not surjective and so M is not an H 1 -accretive mapping. Now, let λ > 0 be an arbitrary real constant and assume that the functions f , g, h, ϕ : R → R are defined, respectively, by Then, for any A = z x-iy x+iy w ∈ H 2 (C), we obtain Relying on the fact that k, l, and p are odd natural numbers, it is easy to see that Taking into account the arbitrariness in the choice of λ > 0, we conclude that M is an H 2 -accretive mapping.
As was pointed out, if H = I, then the definition of I-accretive mappings is that of maccretive mappings. In fact, the class of H-accretive mappings has a close relation with that of m-accretive mappings. In the same way as the proofs of Theorems 2.1 and 2.2 in [11], we obtain the following assertions in a smooth Banach space setting.

Lemma 2.9
Let X be a real smooth Banach space, H : X → X be a strictly accretive mapping, M : X ⇒ X be an H-accretive mapping, and let x, u ∈ X be given points. If uv, J(xy) ≥ 0 holds, for all (y, v) ∈ Graph(M), then u ∈ M(x), that is, M is an maccretive mapping. Lemma 2.10 Let X be a real smooth Banach space, H : X → X be a strictly accretive mapping and M : X ⇒ X be an H-accretive mapping. Then, the mapping ( H + λM) -1 is singlevalued for every constant λ > 0.
It is worth noting that Lemma 2.10 allows us to define the resolvent operator R H M,λ associated with H, M and an arbitrary real constant λ > 0 as follows.

Definition 2.11
Let X be a real smooth Banach space, H : X → X be a strictly accretive mapping and M : X ⇒ X be an H-accretive mapping. The resolvent operator R H M,λ : X → X associated with H, M and an arbitrary positive real constant λ is defined by By a similar proof as in Theorem 2.3 of [11], we conclude the Lipschitz continuity of the resolvent operator R H M,λ associated with H, M and λ > 0 and calculate its Lipschitz constant under some appropriate conditions as follows.

Lemma 2.12
Let X be a real smooth Banach space, H : X → X be an r-strongly accretive mapping and M : X ⇒ X be an H-accretive mapping. Then, the resolvent operator R H M,λ : X → X is Lipschitz continuous with constant 1 r , i.e., Remark 2.13 (i) It should be pointed out that Lemmas 2.1-2.3 improve, respectively, Theorems 2.1-2.3 in [11]. In fact, Theorems 2.1-2.3 in [11] have been presented in the framework of a q-uniformly smooth Banach space, whereas our results are given in a smooth Banach space setting.
(ii) There is a small mistake in the context of Theorem 2.3 of [11]. In fact, in the context of [11, Theorem 2.3], the inequality must be replaced by (2.2), as we have done in the context of Lemma 2.12.

System of variational inclusions: existence and uniqueness of solution and iterative algorithm
For given real Banach spaces X 1 and X 2 , and the mappings F : which is called a system of variational inclusions (SVI) involving H-accretive mappings.
It is important to emphasize that by taking different choices of the operators F, G, H i , M, N and the underlying spaces X i (i = 1, 2) in the SVI (3.1), one can easily obtain the problems studied in [12-14, 22, 29, 58] and the references therein.
The following conclusion, which tells us that SVI (3.1) is equivalent to a fixed-point problem, provides us with a characterization of the solution of the SVI (3.1).
Proof "(i) ⇒ (ii)" Let us first assume that (a, b) ∈ X 1 × X 2 is a solution of the SVI (3.1). Then, using Definition 2.11, it yields Then, in the light of Definition 2.11, we obtain which implies that i.e., (a, b) ∈ X 1 × X 2 is a solution of the SVI (3.1). The proof is completed.
Before proceeding to the main result of this section, we need to recall the following notion that will be used efficiently in its proof. Definition 3.2 A mapping F : X × X → X is said to be (i) ς -Lipschitz continuous with respect to its first argument if there exists a constant ς > 0 such that (ii) ξ -Lipschitz continuous with respect to its second argument if there exists a constant ξ > 0 such that Theorem 3.3 Let X 1 and X 2 be two real smooth Banach spaces with norms · 1 and · 2 , respectively, H 1 : X 1 → X 1 be a 1 -strongly accretive and r-Lipschitz continuous mapping, H 2 : X 2 → X 2 be a 2 -strongly accretive and k-Lipschitz continuous mapping, M : X 1 ⇒ X 1 be an H 1 -accretive set-valued mapping, and N : X 2 ⇒ X 2 be an H 2 -accretive set-valued mapping. Suppose further that the mapping F : X 1 × X 2 → X 1 is τ 1 -Lipschitz continuous with respect to its first argument and τ 2 -Lipschitz continuous with respect to its second argument, and the mapping G : X 1 × X 2 → X 2 is θ 1 -Lipschitz continuous with respect to its first argument and θ 2 -Lipschitz continuous with respect to its second argument. If r < 1 and k < 2 then, the SVI (3.1) admits a unique solution.
Given a real normed space X with a norm · , we recall that a nonlinear mapping It is well known that the class of nonexpansive mappings has a deep and close relation with the classes of monotone and accretive operators that arise naturally in the theory of differential equations. On the other hand, the fixed-point theory is an attractive and interesting subject with a large number of applications in various fields of mathematics and other branches of science. At the same time, the study of nonexpansive mappings is a very interesting research area in fixed-point theory. These facts have motivated many researchers to extend the notion of nonexpansive mapping and several interesting generalized nonexpansive mappings in the framework of different spaces have appeared in the literature. For example, two classes of generalized nonexpansive mappings are recalled in the next definition.

Definition 3.4 A nonlinear mapping
(ii) uniformly L-Lipschitzian if there exists a constant L > 0 such that for each n ∈ N, It is significant to emphasize that every uniformly L-Lipschitzian mapping is L-Lipschitzian but the converse need not be true. The following example illustrates that the class of L-Lipschitzian mappings contains properly the class of uniformly L-Lipschitzian mappings.
Example 3.5 Consider X = R with the Euclidean norm · = | · | and let the self-mapping T of X be defined by T(x) = kx for all x ∈ X, where k > 1 is an arbitrary real constant. Taking into account that for all x, y ∈ X, |T(x) -T(y)| = k|x -y|, it follows that T is a k-Lipschitzian mapping. However, thanks to the fact that k > 1, for any x, y ∈ X and n ∈ N\{1}, we obtain |T n (x) -T n (y)| = k n |x -y| > k|x -y|. This fact ensures that T is not a uniformly k-Lipschitzian mapping.
The introduction and study of the notion of asymptotically nonexpansive mapping as a generalization of the concept of nonexpansive mapping was first made by Goebel and Kirk [47]. Definition 3.6 ([47]) A nonlinear mapping T : X → X is said to be asymptotically nonexpansive if, there exists a sequence {a n } ⊂ (0, +∞) with lim n→∞ a n = 0 such that for each n ∈ N, Equivalently, we say that the mapping T is asymptotically nonexpansive if there exists a sequence {k n } ⊂ [1, +∞) with lim n→∞ k n = 1 such that for each n ∈ N, In recent decades, successful attempts in this direction have continued and several other interesting generalizations of nonexpansive mappings and asymptotically nonexpansive mappings are presented. For instance, in 2006, Alber et al. [49] succeeded in introducing a class of generalized nonexpansive mappings, the so-called total asymptotically nonexpansive mappings, which are more general than the classes of asymptotically nonexpansive mappings and nearly asymptotically nonexpansive mappings.

Definition 3.7 ([49])
A nonlinear mapping T : X → X is said to be total asymptotically nonexpansive (also referred to as ({a n }, {b n }, φ)-total asymptotically nonexpansive) if there exist nonnegative real sequences {a n } and {b n } with a n , b n → 0 as n → ∞ and a strictly increasing continuous function φ : R + → R + with φ(0) = 0 such that for all x, y ∈ X, T n (x) -T n (y) ≤ xy + a n φ xy + b n , ∀n ∈ N.
Using a modified Mann iteration process, they also studied the iterative approximation of the fixed point of total asymptotically nonexpansive mappings under some appropriate conditions. Note, in particular, that every asymptotically nonexpansive mapping is total asymptotically nonexpansive with b n = 0 (or equivalently b n = 0 and a n = k n -1) for all n ∈ N and φ(t) = t for all t ≥ 0, but the converse need not be true. In other words, the class of total asymptotically nonexpansive mappings is more general than the class of asymptotically nonexpansive mappings. This fact is shown in the next example.

Definition 3.9 ([50])
A nonlinear mapping T : X → X is said to be total uniformly L-Lipschitzian (or ({a n }, {b n }, φ)-total uniformly L-Lipschitzian) if there exist a constant L > 0, nonnegative real sequences {a n } and {b n } with a n , b n → 0 as n → ∞ and a strictly increasing continuous function φ : R + → R + with φ(0) = 0 such that for each n ∈ N, It is essential to note that, for given nonnegative real sequences {a n } and {b n } and a strictly increasing continuous function φ : R + → R + , an ({a n }, {b n }, φ)-total asymptotically nonexpansive mapping is ({a n }, {b n }, φ)-total uniformly L-Lipschitzian with L = 1, but the converse may not be true. In the following example, the fact that the class of total uniformly L-Lipschitzian mappings contains properly the class of total asymptotically nonexpansive mappings is illustrated.
Example 3.10 Let X = R endowed with the Euclidean norm · = | · | and let the selfmapping T of X be defined by where α > 0 and β > α+ √ α 2 +4 2 are arbitrary real constants such that αβ > 1. Since the mapping T is discontinuous at the points x = 0, α, 1 β , it follows that T is not Lipschitzian and so it is not an asymptotically nonexpansive mapping. Take a n = γ n and b n = α k n for each n ∈ N, where γ > 0 and k > 1 are arbitrary constants such that k = αβ. Let us now define the function φ : R + → R + by φ(t) = θ t m for all t ∈ R + , where m ∈ N and θ ∈ (0, k m (β 2 -αβ-1) βγ (k-1) m α m ) are arbitrary constants. Selecting x = α and y = α k , we have T(x) = 1 β and T(y) = β. With the help of the fact that 0 < θ < k m (β 2 -αβ-1) βγ (k-1) m α m , it follows that which implies that T is not a ({ γ n }, { α k n }, φ)-total asymptotically nonexpansive mapping. However, for all x, y ∈ X, we obtain and for all n ≥ 2, due to the fact that T n (z) = 1 β for all z ∈ X and n ≥ 2. Making use of (3.19) and (3.20), we deduce that T is a ({ γ n }, { α k n }, φ)-total uniformly kβ α -Lipschitzian mapping.

Then, Q is an ({a
Proof In view of the fact that for each j ∈ {1, 2}, φ j : R + → R + is a strictly increasing function, for all (x 1 , x 2 ), (y 1 , y 2 ) ∈ X 1 × X 2 and i ∈ N, yields where · * is a norm on X 1 × X 2 defined by (3.10). This fact ensures that Q is an , φ)-total uniformly max{L 1 , L 2 }-Lipschitzian mapping. The proof is completed.
Assume that X 1 and X 2 are two real smooth Banach spaces with norms · 1 and · 2 , respectively, S 1 : , φ 1 )-total uniformly L 1 -Lipschitzian mapping and S 2 : , φ 2 )-total uniformly L 2 -Lipschitzian mapping. Furthermore, let Q be a self-mapping of X 1 × X 2 defined as (3.21). Denote by Fix(S j ) (j = 1, 2) and Fix(Q) the sets of all the fixed points of S j (j = 1, 2) and Q, respectively. At the same time, denote by SVI(X j , H j , M, N, F, G : j = 1, 2) the set of all the solutions of the SVI (3.1), where for j = 1, 2, the nonlinear mappings H j : X j → X j are strictly accretive, and the set-valued mappings M : X ⇒ X 1 and N : X 2 ⇒ X 2 are H 1 -accretive and H 2accretive, respectively. Using (3.21), we infer that for any (x 1 , x 2 ) ∈ X 1 ×X 2 , (x 1 , x 2 ) ∈ Fix(Q) if and only if for j = 1, 2, x j ∈ Fix(S j ), that is, Fix(Q) = Fix(S 1 , S 2 ) = Fix(S 1 ) × Fix(S 2 ). If (a, b) ∈ Fix(Q) ∩ SVI(X j , A j , M, N, F, G : j = 1, 2), then with the help of Lemma 3.1 it can be easily observed that for each i ∈ N, ρG(a, b)]. (3.23) Using the fixed-point formulation (3.23) we now are able to construct the following iterative algorithm for finding a common element of the two sets of SVI(X j , H j , M, N, F, G : j = 1, 2) and Fix(Q) = Fix(S 1 , S 2 ).

Algorithm 3.12
Assume that X j (j = 1, 2), F and G are the same as in the SVI (3.1). Let for i ≥ 0 and j = 1, 2, H i,j : X j → X j be strictly accretive, M i : X 1 ⇒ X 1 be an H i,1 -accretive set-valued mapping and N i : X 2 ⇒ X 2 be an H i,2 -accretive set-valued mapping. Suppose further that for j = 1, 2, S j : where i ∈ N ∪ {0}; λ i , ρ i > 0 are real constants; and {α i } ∞ i=0 is a sequence in the interval [0, 1] such that lim sup i α i < 1.
If S j ≡ I j (j = 1, 2), the identity mapping on X j , then Algorithm 3.12 reduces to the following algorithm.

Algorithm 3.13 Let
where i ∈ N ∪ {0}; the constants λ i , ρ i > 0 and the sequence {α i } ∞ i=0 are the same as in Algorithm 3.12.
If H i,j = H j , λ i = λ and ρ i = ρ for each i ≥ 0 and j ∈ {1, 2}, then Algorithm 3.13 collapses to the following algorithm. Algorithm 3.14 Suppose that X j (j = 1, 2), F and G are the same as in Algorithm 3.12. Let for j = 1, 2, H j : X j → X j be strictly accretive mappings and let for each i ≥ 0, M i : X 1 ⇒ X 1 be an H 1 -accretive set-valued mapping and N i : X 2 ⇒ X 2 be an H 2 -accretive set-valued mapping. For any given where i ∈ N ∪ {0}; λ, ρ > 0 are two constants; and the sequence {α i } ∞ i=0 is the same as in Algorithm 3.12.

Graph convergence and an application
Before turning to the main results of this paper, we need to recall the following definition. We now establish a new equivalence relationship between the graph convergence of a sequence of H-accretive mappings and their associated resolvent operators, respectively, to a given H-accretive mapping and its associated resolvent operator under some appropriate conditions.
of real positive constants convergent to some positive real constant λ 0 , Proof "(i) ⇒ (ii)" Suppose that M i G − → M and let {λ i } ∞ i=0 be a sequence of real positive constants convergent to a constant λ > 0. Choose z ∈ X arbitrarily but fixed. The fact that M is an H-accretive mapping implies that ( H + λM)(X) = X, which guarantees the existence of a point (x, u) ∈ Graph(M) such that z = H(x) + λu. Then, thanks to Definition 4.1 there Picking z i = H i (x i ) + λ i u i for each i ≥ 0 and making use of Lemma 2.12, (4.1), and the assumptions, we derive that for all i ≥ 0, Since λ i → λ as i → ∞ and the sequences are also bounded. By virtue of the facts that x i → x, u i → u and λ i → λ as i → ∞, we conclude that the right-hand side of (4.2) tends to zero as i → ∞, which implies that R Taking into account that λ i,0 → λ 0 and λ i,0 u i → λ 0 u, as i → ∞, we deduce that the righthand side of (4.3) approaches zero, as i → ∞, which ensures that u i → u as i → ∞. The proof is finished.
We obtain the following corollary as a direct consequence of the above theorem immediately.

Corollary 4.3
Suppose that X is a real smooth Banach space, and H : X → X is astrongly accretive and γ -Lipschitz continuous mapping. Furthermore, let M i , M : X ⇒ X be H-accretive mappings for i = 1, 2, . . . . Then, the following statements are equivalent: Then, The following lemma plays a prominent role in studying the convergence analysis of our iterative algorithms proposed in the previous section.
are three real sequences of nonnegative numbers that satisfy the following conditions: Proof Let > 0 be chosen arbitrarily but fixed. Taking into account that lim sup i γ i < 1 and lim i→∞ t i = 0, one can choose i 0 ∈ N such that we have lim sup i γ i < 1 -and t i < 2 for all i ≥ i 0 . In the light of (ii), we deduce that Then, by taking θ = 1 -and ξ = 2 , from Lemma 4.4, it follows that which implies that lim sup i σ i ≤ . This completes the proof. Remark 4.6 (i) It should be pointed out that the condition lim sup i σ i < 1 imposed on the sequence {σ i } in Lemma 4.5 is essential and cannot be dropped. To illustrate this fact, let us take σ i = β, t i = β i , and γ i = 1 -1 i for all i ∈ N, where β > 0 is an arbitrary but fixed real number. Then, we have σ i+1 ≤ γ i σ i + t i for all i ∈ N, lim i→∞ t i = 0 and lim sup i γ i = 1, but lim i→∞ σ i = β = 0.
We are now ready, as an application of the notion of graph convergence for H-accretive mappings, to present the most important result of this paper in which the strong convergence of the iterative sequence generated by Algorithm 3.12 to a common element of the two sets SVI (X j , H j , M, N, F, G : j = 1, 2) and Fix(Q), where Q is a self-mapping of X 1 × X 2 defined by (3.23), is proved.  , H j , M, N, F, G : j = 1, 2 where for each i ≥ 0, λF(a, b) .
Taking into account that for each i ≥ 0, the mapping H i,1 is r i -Lipschitz continuous, and the mapping F is τ 1 -Lipschitz continuous and τ 2 -Lipschitz continuous with respect to its first and second arguments, respectively, it follows that and (4.7) Substituting (4.6) and (4.7) into (4.5), for all i ≥ 0, we obtain By following similar arguments as in the proofs of (4.5)-(4.8) with suitable changes, from (3.24), (4.5), Lemma 2.12, and the assumptions, one can deduce that where for each i ≥ 0, ρG(a, b) .
Taking S j ≡ I j , the identity mapping on X j , the following corollary follows from Theorem 4.7 immediately. that X j , H j , H i,j , M i , N i , λ i , ρ i , F, G, M, N (i ≥ 0 and j = 1, 2)

H(·, ·)-Accretive operators and some comments
In this section, our attention is turned to investigate and analyze the notion of an H(·, ·)accretive operator and the related results available in [26]. Some remarks together with relevant commentaries are also pointed out.
Let us first remark that throughout [26], X is assumed to be a real Banach space such that J is single-valued. As we know, J is single-valued if and only if X is smooth. Hence, throughout the rest of paper, unless otherwise stated, we assume that X is a real smooth Banach space. (iv) H(·, ·) is said to be ς -Lipschitz continuous with respect to B if there exists a constant ς > 0 such that Here, it is to be noted that, as was pointed out in [26], in a similar way to cases (iv) and (v) of Definition 2.1 in [26], one can define the generalized accretivity of the mapping H(·, ·) with respect to B and the Lipschitz continuity of the mapping H(·, ·) with respect to B, as we have done, respectively, in parts (ii) and (iv) of Definition 5.1. (i) If H(·, ·) is α, β-generalized accretive with respect to A, B, respectively, then H is (α + β)-strongly accretive and hence it is strictly accretive (resp., accretive and -(α + β)-relaxed accretive) provided that α + β > 0 (resp., α + β = 0 and α + β < 0); (ii) If H(·, ·) is r 1 -Lipschitz continuous with respect to A and r 2 -Lipschitz continuous with respect to B, then H is (r 1 + r 2 )-Lipschitz continuous. ≥ α xy 2 + β xy 2 = (α + β) xy 2 .
It is significant to emphasize that every bifunction H : X × X → X that is α, βgeneralized accretive with respect to A, B, respectively, is actually a univariate (α + β)strongly accretive (resp., accretive and -(α + β)-relaxed accretive) mapping provided that α + β > 0 (resp., α + β = 0 and α + β < 0) and is not a new one. At the same time, thanks to Proposition 5.2(ii), the notion of Lipschitz continuity of the bifunction H : X × X → X with respect to the mappings A, B : X → X presented in parts (iii) and (iv) of Definition 5.1 is exactly the same concept of Lipschitz continuity of a univariate mapping H = H(A, B) : X → X that appeared in Definition 2.1(v) and is not a new one. Remark 5.4 It is worth mentioning that the concept of an H(·, ·)-accretive operator was initially introduced by Zou and Huang [35] in 2008, and was studied for the case when H(·, ·) is α-strongly accretive with respect to A, β-relaxed accretive with respect to B, and α > β. Afterwards, several generalizations of this notion appeared in the literature. Recently, this notion has been considered by Tang and Wang [26] and has been studied in a more general case when H(·, ·) is α-generalized accretive with respect to A, βgeneralized accretive with respect to B. It should be pointed out that by defining the mapping H : X → X as H(x) := H(Ax, Bx) for all x ∈ X, Definition 5.3 coincides exactly with Definition 2.6. In other words, the concept of an H(·, ·)-accretive operator is actually the same notion of the H-accretive operator introduced and studied by Fang and Huang [11] and is not a new one.
According to the following conclusion, the authors [26] deduced that every H(·, ·)accretive operator is maximal under some appropriate conditions.  Bx) for all x ∈ X, from the assumptions and using Proposition 5.2(i) it follows that H is (α + β)-strongly accretive and so it is a strictly accretive mapping. At the same time, invoking Remark 5.4, M is an Haccretive mapping. We now note that all the conditions of Lemma 2.9 are satisfied and so the conclusion follows from Lemma 2.9 immediately.
It should be noted that the conclusion of Theorem 2.1 in [26] has been derived based on Theorem 3.1 in [35] without presenting any proof. In fact, in [35,Theorem 3.1], the authors proved that every H(·, ·)-accretive operator with respect to mappings A and B satisfying the appropriate conditions, where H(·, ·) is α-strongly accretive with respect to A, β-relaxed accretive with respect to B and α > β is maximal. Tang and Wang [26] concluded the same assertion for H(·, ·)-accretive mappings for the case when H(·, ·) is αβ-generalized accretive with respect to A, B, respectively, and α + β = 0. However, by following a similar argument as in the proof of [35,Theorem 3.1] with suitable modifications, we found that the condition α + β = 0 in the context of [26, Theorem 2.1] must be replaced by the condition α + β > 0, as has been done in the context of Lemma 5.5. Remark 5.7 It should be pointed out that the assertion of Theorem 2.2 in [26] is derived similarly to that of assertion of Theorem 3.3 in [35]. In fact, in Theorem 3.3 of [35], the authors proved that for a given H(·, ·)-accretive operator M : X ⇒ X with respect to the mappings A and B, where H(·, ·) is α-strongly accretive with respect to A, β-relaxed accretive with respect to B, and α > β, the operator (H(A, B) + λM) -1 is single-valued for every constant λ > 0. Without giving any proof, Tang and Wang [26] claimed that the same assertion holds for the case when H(·, ·) is α, β-generalized accretive with respect to A, B, respectively, and α + β = 0. However, by following a similar argument as in the proof of Theorem 3.3 presented in [35] with suitable changes, we inferred that the condition α + β = 0 in the context of [26, Theorem 2.2] must be replaced by the condition α + β > 0, as we have done in the context of Lemma 5.6.
Based on Theorem 2.2 in [26], the authors defined the resolvent operator associated with an H(·, ·)-accretive operator M : X ⇒ X as follows. Indeed, in view of the discussion mentioned above, the notion of the resolvent operator R H(·,·) M,λ associated with an H(·, ·)-accretive operator M : X ⇒ X and an arbitrary constant λ > 0, where H(·, ·) is α, β-generalized accretive with respect to A, B, respectively, and α + β > 0 is actually the same notion of the resolvent operator R H M,λ associated with the H-accretive operator M and the real constant λ > 0 given in Definition 2.11 and is not a new one. At the same time, it should be remarked that in Definition 2.3 of [26], the notion of a resolvent operator associated with an H(·, ·)-accretive operator M : X ⇒ X is defined based on Theorem 2.2 in [26]. However, as was pointed out in Remark 5.7, the condition α + β = 0 in the context of Theorem 2.2 of [26] must be replaced by the condition α + β > 0. Hence, this correction must be done in the context of Definition 2.3 of [26], as has done in the context of Definition 5.8.
With the aim of proving the Lipschitz continuity of the resolvent operator R H(·,·) M,λ and computing an estimate of its Lipschitz constant, Tang and Wang [26] presented one of the most important results of Sect. 2 of [26] without any proof as follows.
for all u, v ∈ X. This gives the desired result. = 2x -2y, xy Tang and Wang [26] deduced that H(·, ·) is -1, 2-generalized accretive with respective to A, B, respectively. By virtue of the facts that  Bx) for all x ∈ X. Then, for all x ∈ X, we have H(x) = Ax + Bx = -x + 2x = x. Now, taking α = -1, β = 2, r 1 = 1 and r 2 = 2, in light of the fact that α + β = -1 + 2 > 0, from parts (i) and (ii) of Proposition 5.2, it is expected that the mapping H is (α + β) = 1-strongly accretive and (r 1 + r 2 ) = 3-Lipschitz continuous. Since for all x, y ∈ X, these facts confirm our expectations, i.e., our observations are compatible with our derived assertions in Proposition 5.2. As was pointed out, by defining the mapping H : X → X by H(x) := H(Ax, Bx) for all x ∈ X, the notion of H(·, ·)-accretive mapping is exactly the same concept of H-accretive mapping and is not a new one. Thanks to this fact we found that if in Definition 5.11, the mappings M i (i ≥ 0) and M are assumed to be H-accretive, then Definition 5.11 becomes actually the same Definition 4.1. In fact, the notion of graph convergence for H(·, ·)accretive mappings introduced in [26,60] is exactly the same concept of graph convergence for H-accretive mappings as a special case of Definition 4.1 and is not a new one.
Taking into account the above-mentioned argument, it is significant to emphasize that contrary to the claim of the authors in [26], Lemma 5.13 (that is, [26, Lemma 3.1]) gives actually a characterization of the solution of the SVI (3.1) involving an H 1 -accretive mapping M and an H 2 -accretive mapping N not the SVI (3.1) involving H 1 (·, ·)-accretive and H 2 (·, ·)-accretive mappings M and N (that is, [26, the problem (3.1)]). Meanwhile, it should be remarked that throughout Sect. 3 of [26], the spaces X i (i = 1, 2) are assumed to be real Banach spaces such that for each i ∈ {1, 2}, the normalized duality mapping J i : X i ⇒ X * i is single-valued. It is known that, in general, J i (i = 1, 2) is single-valued if and only if X i is smooth. Hence, in the following, we may assume that X i (i = 1, 2) are real smooth Banach spaces, as we have assumed in the context of Lemma 5. 13.
Based on Lemma 5.13 and by assuming that for all i ≥ 0, M i is an H 1 (·, ·)-accretive mapping with respect to A 1 and B 1 , and N i is an H 2 (·, ·)-accretive mapping with respect to A 2 and B 2 , Tang and Wang [26] constructed the following iterative algorithm for finding an approximate solution of the SVI (3.1) involving an H 1 (·, ·)-accretive mapping M and an H 2 (·, ·)-accretive mapping N (that is, [26, the problem (3.1)]).
It is also remarkable that by defining the mapping H i : X i → X i for i = 1, 2 by H i (x i ) := H i (A i x i , B i x i ) for all x i ∈ X i , with the help of the assumptions and utilizing Proposition 5.2(i) we infer that the operators H i (i = 1, 2) are strictly accretive. In the light of