Hybrid Extragradient Methods for Finding Zeros of Accretive Operators and Solving Variational Inequality and Fixed Point Problems in Banach Spaces

and Applied Analysis 3 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, for example, [7–9]. Very recently, Cai and Bu [10] considered the following general system of variational inequalities (GSVI) in a real smooth Banach space X, which involves finding (x∗, y∗) ∈ C × C such that ⟨μ 1 B 1 y ∗ + x ∗ − y ∗ , J (x − x ∗ )⟩ ≥ 0, ∀x ∈ C, ⟨μ 2 B 2 x ∗ + y ∗ − x ∗ , J (x − y ∗ )⟩ ≥ 0, ∀x ∈ C, (14) where C is a nonempty, closed, and convex subset of X, B 1 , B 2 : C → X are two nonlinear mappings, and μ 1 and μ 2 are two positive constants. Here the set of solutions of GSVI (14) is denoted by GSVI(C, B 1 , B 2 ). In particular, if X = H, a real Hilbert space, then GSVI (14) reduces to the following GSVI of finding (x∗, y∗) ∈ C × C such that ⟨μ 1 B 1 y ∗ + x ∗ − y ∗ , x − x ∗ ⟩ ≥ 0, ∀x ∈ C, ⟨μ 2 B 2 x ∗ + y ∗ − x ∗ , x − y ∗ ⟩ ≥ 0, ∀x ∈ C, (15) where μ 1 and μ 2 are two positive constants. The set of solutions of problem (15) is still denoted by GSVI(C, B 1 , B 2 ). In particular, if B 1 = B 2 = A, then problem (15) reduces to the new system of variational inequalities (NSVI), introduced and studied by Verma [11]. Furthermore, if x∗ = y∗ additionally, then the NSVI reduces to the classical variational inequality problem (VIP) of finding x∗ ∈ C such that ⟨Ax ∗ , x − x ∗ ⟩ ≥ 0, ∀x ∈ C. (16) The solution set of the VIP (16) is denoted by VI(C, A). Variational inequality theory has been studied quite extensively and has emerged as an important tool in the study of a wide class of obstacle, unilateral, free, moving, equilibrium problems. It is now well known that the variational inequalities are equivalent to the fixed point problems, the origin of which can be traced back to Lions and Stampacchia [12]. This alternative formulation has been used to suggest and analyze projection iterative method for solving variational inequalities under the conditions that the involved operator must be strongly monotone and Lipschitz continuous. Recently, Ceng et al. [13] transformed problem (15) into a fixed point problem in the following way. Lemma 4 (see [13]). For a given x, y ∈ C, (x, y) is a solution of problem (15) if and only if x is a fixed point of the mapping G : C → C defined by G (x) = P C [P C (x − μ 2 B 2 x) −μ 1 B 1 P C (x − μ 2 B 2 x)] , ∀x ∈ C, (17) where y = P C (x−μ 2 B 2 x) and P C is the projection ofH ontoC. In particular, if the mapping B i : C → H is β i -inverse strongly monotone for i = 1, 2, then the mapping G is nonexpansive provided μ i ∈ (0, 2β i ) for i = 1, 2. In 1976, Korpelevič [14] proposed an iterative algorithm for solving the VIP (16) in Euclidean space R as follows: y n = P C (x n − τAx n ) , x n+1 = P C (x n − τAy n ) , n ≥ 0, (18) with τ > 0 a given number, which is known as the extragradientmethod (see also [15]).The literature on theVIP is vast and Korpelevich’s extragradient method has received great attention given by many authors, who improved it in various ways; see, for example, [10, 13, 16–23] the and references therein, to name but a few. In particular, whenever X is still a real smooth Banach space, B 1 = B 2 = A, and x∗ = y, then GSVI (17) reduces to the variational inequality problem (VIP) of finding x∗ ∈ C such that ⟨Ax ∗ , J (x − x ∗ )⟩ ≥ 0, ∀x ∈ C. (19) which was considered by Aoyama et al. [24]. Note that VIP (19) is connected with the fixed point problem for nonlinear mapping (see, e.g., [15, 25]), the problem of finding a zero point of a nonlinear operator (see, e.g., [1, 26]), and so on. It is clear that VIP (19) extends VIP (16) from Hilbert spaces to Banach spaces. In order to find a solution of VIP (19), Aoyama et al. [24] introduced the following iterative scheme for an accretive operator A: x n+1 = α n x n + (1 − α n )Π C (x n − λ n Ax n ) , ∀n ≥ 1, (20) whereΠ C is a sunny nonexpansive retraction fromX ontoC. Then they proved a weak convergence theorem. Beyond doubt, it is an interesting and valuable problem of constructing some algorithms with strong convergence for solving GSVI (14) which contains VIP (19) as a special case. Very recently, Cai and Bu [10] constructed an iterative algorithm for solving GSVI (14) and a common fixed point problem of an infinite family of nonexpansive mappings in a uniformly convex and 2-uniformly smooth Banach space. They proved the strong convergence of the proposed algorithm by virtue of the following inequality in a 2-uniformly smooth Banach spaceX. Lemma 5 (see [27]). Let X be a 2-uniformly smooth Banach space. Then 󵄩󵄩󵄩󵄩x + y 󵄩󵄩󵄩󵄩 2 ≤ ‖x‖ 2 + 2 ⟨y, J (x)⟩ + 2 󵄩󵄩󵄩󵄩κy 󵄩󵄩󵄩󵄩 2 , ∀x, y ∈ X, (21) where κ is the 2-uniformly smooth constant of X and J is the normalized duality mapping from X intoX. Define the mapping G : C → C as follows: G (x) := Π C (I − μ 1 B 1 )Π C (I − μ 2 B 2 ) x, ∀x ∈ C. (22) The fixed point set of G is denoted by Ω. Then their strong convergence theorem on the proposed method is stated as follows. 4 Abstract and Applied Analysis Theorem 6 (see [10, Theorem 3.1]). Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space X. Let Π C be a sunny nonexpansive retraction from X onto C. Let the mapping B i : C → X be β i -inverse strongly accretive with 0 < μ i < β i /κ 2 for i = 1, 2. Let f be a contraction ofC into itself with coefficient δ ∈ (0, 1). Let {S n } ∞ n=1 be an infinite family of nonexpansive mappings of C into itself such that F = ⋂∞ i=1 Fix(S i ) ∩ Ω ̸ = 0, where Ω is the fixed point set of the mappingG defined by (22). For arbitrarily given x 1 ∈ C, let {x n } be the sequence generated by x n+1 = β n x n + (1 − β n ) S n y n , y n = α n f (x n ) + (1 − α n ) z n , z n = Π C (u n − μ 1 B 1 u n ) , u n = Π C (x n − μ 2 B 2 x n ) , ∀n ≥ 1. (23) Suppose that {α n } and {β n } are two sequences in (0, 1) satisfying the following conditions: (i) lim n→∞ α n = 0 and ∑∞ n=1 α n = ∞; (ii) 0 < lim inf n→∞ β n ≤ lim sup n→∞ β n < 1. Assume that∑∞ n=1 sup x∈D ‖S n+1 x − S n x‖ < ∞ for any bounded subset D of C and let S be a mapping of C into X defined by Sx = lim n→∞ S n x for all x ∈ C and suppose that Fix(S) = ⋂ ∞ n=1 Fix(S n ). Then {x n } converges strongly to q ∈ F, which solves the following VIP: ⟨q − f (q) , J (q − p)⟩ ≤ 0, ∀p ∈ F. (24) Corollary 7 (see [10, Corollary 3.2]). Let C be a nonempty closed convex subset of a uniformly convex and 2-uniformly smooth Banach space X. Let Π C be a sunny nonexpansive retraction from X onto C. Let the mapping B i : C → X be β i -inverse strongly accretive with 0 < μ i < β i /κ 2 for i = 1, 2. Let f be a contraction ofC into itself with coefficient δ ∈ (0, 1). Let S be a nonexpansive mapping of C into itself such that F = Fix(S) ∩ Ω ̸ = 0, where Ω is the fixed point set of the mapping G defined by (22). For arbitrarily given x 1 ∈ C, let {x n } be the sequence generated by x n+1 = β n x n + (1 − β n ) Sy n , y n = α n f (x n ) + (1 − α n ) z n , z n = Π C (u n − μ 1 B 1 u n ) , u n = Π C (x n − μ 2 B 2 x n ) , ∀n ≥ 1. (25) Suppose that {α n } and {β n } are two sequences in (0, 1) satisfying the following conditions: (i) lim n→∞ α n = 0 and ∑∞ n=1 α n = ∞; (ii) 0 < lim inf n→∞ β n ≤ lim sup n→∞ β n < 1. Then {x n } converges strongly to q ∈ F, which solves the following VIP: ⟨q − f (q) , J (q − p)⟩ ≤ 0, ∀p ∈ F. (26) We remark that in Theorem 6, the Banach space X is both uniformly convex and 2-uniformly smooth. According to Lemma 5, the 2-uniform smoothness of X guarantees the nonexpansivity of the mapping I − μ i B i for α i -inversestrongly accretive mapping B i : C → X with 0 ≤ μ i ≤ α i /κ 2 for i = 1, 2, and hence the composite mapping G : C → C is nonexpansive where G = Π C (I − μ 1 B 1 )Π C (I − μ 2 B 2 ). In the meantime, for the convenience of implementing the argument techniques in [13], they have applied the following inequality in a real smooth and uniform convex Banach space X. Proposition 8 (see [28]). LetX be a real smooth and uniform convex Banach space and let r > 0. Then there exists a strictly increasing, continuous, and convex function g : [0, 2r] → R, g(0) = 0 such that g ( 󵄩󵄩󵄩󵄩x − y 󵄩󵄩󵄩󵄩) ≤ ‖x‖ 2 − 2 ⟨x, J (y)⟩ + 󵄩󵄩󵄩󵄩y 󵄩󵄩󵄩󵄩 2 , ∀x, y ∈ B r , (27) where B r = {x ∈ X : ‖x‖ ≤ r}. Let C be a nonempty closed convex subset of a uniformly convex Banach space X which has a uniformly Gateaux differentiable norm. LetΠ C be a sunny nonexpansive retraction from X onto C. Motivated and inspired by the research going on this area, we introduce and analyze hybrid implicit and explicit extragradient methods for finding a zero of an accretive operator A ⊂ X × X such that D(A) ⊂ C ⊂ ⋂ r>0 R(I + rA) and solving GSVI (14) and a fixed point problem of an infinite family of nonexpansive self-mappings onC. We establish some strong convergence theorems for hybrid implicit and explicit extragradient algorithms under suitable assumptions. Furthermore, we derive the strong convergence of hybrid implicit and explicit extragradient algorithms for finding a common element of the set of zeros of an accretive operator and the common fixed point set of an infinite family of nonexpansive self-mappings on C and a self-mapping whose complement is strictly pseudocontractive and strongly accretive on C. The results presented in this paper improve, extend, supplement, and develop the corresponding results announced in the earlier and very recent literature; see, for example, [5, 10, 13, 16]. 2. Preliminaries Let X be a real Banach space. X is said to be smooth if the limit