Multistep Hybrid Extragradient Method for Triple Hierarchical Variational Inequalities

and Applied Analysis 3 where A n = α n I + A for all n ≥ 0. In particular, if V ≡ 0, then (11) reduces to the following iterative scheme: x 0 = x ∈ C chosen arbitrarily, y n = P C (x n − ] n A n x n ) , z n = β n x n + γ n P C (x n − ] n A n y n ) + σ n TP C (x n − ] n A n y n ) , x n+1 = P C [λ n (1 − δ n ) γSx n + (I − λ n μF) z n ] , ∀n ≥ 0. (12) Further, if S = V, then (11) reduces to the following iterative scheme: x 0 = x ∈ C chosen arbitrarily, y n = P C (x n − ] n A n x n ) , z n = β n x n + γ n P C (x n − ] n A n y n ) + σ n TP C (x n − ] n A n y n ) , x n+1 = P C [λ n γVx n + (I − λ n μF) z n ] , ∀n ≥ 0; (13) moreover, if S = V ≡ 0, then (12) reduces to the following iterative scheme: x 0 = x ∈ C chosen arbitrarily, y n = P C (x n − ] n A n x n ) , z n = β n x n + γ n P C (x n − ] n A n y n ) + σ n TP C (x n − ] n A n y n ) , x n+1 = P C [(I − λ n μF) z n ] , ∀n ≥ 0. (14) We prove that under appropriate conditions the sequence {x n } generated by Algorithm I converges strongly to a unique solution of Problem II. Our result improves and extends Theorem 4.1 in [22] in the following aspects. (a) Problem II generalizes Problem I from the fixed point set Fix(T) of a nonexpansive mapping T to the intersection Fix(T)∩Γ of the fixed point set of a strictly pseudocontractive mapping T and the solution set Γ of VIP (2). (b) The Korpelevich extragradient algorithm is extended to develop the multistep hybrid extragradient algorithm (i.e., Algorithm I) for solving Problem II by virtue of the iterative schemes inTheorem 4.1 in [22]. (c) The strong convergence of the sequence {x n } generated by Algorithm I holds under the lack of the same restrictions as those inTheorem 4.1 in [22]. (d) The boundedness requirement of the sequence {x n } in Theorem 4.1 in [22] is replaced by the boundedness requirement of the sequence {Sx n }. We also consider and study themultistep hybrid extragradient algorithm (i.e., Algorithm I) for solving the following system of hierarchical variational inequalities (SHVI). Problem III. Let F : C → H be κ-Lipschitzian and ηstrongly monotone on the nonempty, closed, and convex subset C of H, where κ and η are positive constants. Let A : C → H be a monotone and L-Lipschitzian mapping, V : C → H be a ρ-contraction with coefficient ρ ∈ [0, 1), S : C → C be a nonexpansive mapping, and T : C → C be a ζ-strictly pseudocontractive mapping with Fix(T)∩Γ ̸ = 0. Let 0 < μ < 2η/κ 2 and 0 < γ ≤ τ, where τ = 1−√1 − μ(2η − μκ2). Then the objective is to find x∗ ∈ Fix(T) ∩ Γ such that ⟨(μF − γV) x ∗ , x − x ∗ ⟩ ≥ 0, ∀x ∈ Fix (T) ∩ Γ, ⟨(μF − γS) x ∗ , y − x ∗ ⟩ ≥ 0, ∀y ∈ Fix (T) ∩ Γ. (15) In particular, if T = T 1 and A = I − T 2 where T i : C → C is ζ i -strictly pseudocontractive for i = 1, 2, Problem III reduces to the following Problem IV. Problem IV. Let F : C → H be κ-Lipschitzian and ηstrongly monotone on the nonempty, closed, and convex subset C of H, where κ and η are positive constants. Let V : C → H be a ρ-contraction with coefficient ρ ∈ [0, 1), S : C → C be a nonexpansive mapping, and, for i = 1, 2, T i : C → C be ζ i -strictly pseudocontractive mapping with Fix(T 1 ) ∩ Fix(T 2 ) ̸ = 0. Let 0 < μ < 2η/κ2 and 0 < γ ≤ τ, where τ = 1 −√1 − μ(2η − μκ2). Then the objective is to find x ∗ ∈ Fix(T 1 ) ∩ Fix(T 2 ) such that ⟨(μF − γV) x ∗ , x − x ∗ ⟩ ≥ 0, ∀x ∈ Fix (T 1 ) ∩ Fix (T 2 ) , ⟨(μF − γS) x ∗ , y − x ∗ ⟩ ≥ 0, ∀y ∈ Fix (T 1 ) ∩ Fix (T 2 ) . (16) We prove that under very mild conditions the sequence {x n } generated by Algorithm I converges strongly to a unique solution of Problem III. 2. Preliminaries Let C be a nonempty, closed, and convex subset of H and V : C → H be a (possibly nonself) ρ-contraction mapping with coefficient ρ ∈ [0, 1); that is, there exists a constant ρ ∈ [0, 1) such that ‖Vx − Vy‖ ≤ ρ‖x − y‖, for all x, y ∈ C. Now we present some known results and definitions which will be used in the sequel. The metric (or nearest point) projection from H onto C is the mapping P C : H → C which assigns to each point x ∈ H the unique point P C x ∈ C satisfying the property 󵄩󵄩󵄩x − PCx 󵄩󵄩󵄩 = inf y∈C 󵄩󵄩󵄩x − y 󵄩󵄩󵄩 =: d (x, C) . (17) The following properties of projections are useful and pertinent to our purpose. Proposition 1 (see [21]). Given any x ∈ H and z ∈ C. One has (i) z = P C x ⇔ ⟨x − z, y − z⟩ ≤ 0, for all y ∈ C, 4 Abstract and Applied Analysis (ii) z = P C x ⇔ ‖x − z‖ 2 ≤ ‖x − y‖ 2 − ‖y − z‖ 2, for all y ∈ C; (iii) ⟨P C x − P C y, x − y⟩ ≥ ‖P C x − P C y‖ 2, for all x, y ∈ H, which hence implies that P C is nonexpansive and monotone. Definition 2. A mapping T : H → H is said to be (a) nonexpansive if 󵄩󵄩󵄩Tx − Ty 󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩x − y 󵄩󵄩󵄩 , ∀x, y ∈ H; (18) (b) firmly nonexpansive if 2T − I is nonexpansive, or, equivalently, ⟨x − y, Tx − Ty⟩ ≥ 󵄩󵄩󵄩Tx − Ty 󵄩󵄩󵄩 2 , ∀x, y ∈ H; (19) alternatively, T is firmly nonexpansive if and only if T can be expressed as T = 1 2 (I + S) , (20) where S : H → H is nonexpansive; projections are firmly nonexpansive. Definition 3. Let T be a nonlinear operator whose domain is D(T) ⊆ H and whose range is R(T) ⊆ H. (a) T is said to be monotone if ⟨x − y, Tx − Ty⟩ ≥ 0, ∀x, y ∈ D (T) . (21) (b) Given a number β > 0, T is said to be β-strongly monotone if ⟨x − y, Tx − Ty⟩ ≥ β 󵄩󵄩󵄩x − y 󵄩󵄩󵄩 2 , ∀x, y ∈ D (T) . (22) (c) Given a number ] > 0, T is said to be ]-inverse strongly monotone (]-ism) if ⟨x − y, Tx − Ty⟩ ≥ ] 󵄩󵄩󵄩Tx − Ty 󵄩󵄩󵄩 2 , ∀x, y ∈ D (T) . (23) It can be easily seen that ifT is nonexpansive, then I−T is monotone. It is also easy to see that a projection P C is 1-ism. Inverse strongly monotone (also referred to as cocoercive) operators have been applied widely in solving practical problems in various fields, for instance, in traffic assignment problems; see [24, 25]. Definition 4. A mapping T : H → H is said to be an averaged mapping if it can be written as the average of the identity I and a nonexpansive mapping, that is, T ≡ (1 − α) I + αS, (24) where α ∈ (0, 1) and S : H → H are nonexpansive. More precisely, when the last equality holds, we say that T is αaveraged. Thus firmly nonexpansive mappings (in particular, projections) are 1/2-averaged maps. Proposition 5 (see [26]). Let T : H → H be a given mapping. (i) T is nonexpansive if and only if the complement I − T is 1/2-ism. (ii) If T is ]-ism, then, for γ > 0, γT is ]/γ-ism. (iii) T is averaged if and only if the complement I − T is ]-ism for some ] > 1/2. Indeed, for α ∈ (0, 1), T is α-averaged if and only if I − T is 1/2α-ism. Proposition 6 (see [26, 27]). Let S, T, V : H → H be given operators. (i) If T = (1 − α)S + αV for some α ∈ (0, 1) and if S is averaged and V is nonexpansive, then T is averaged. (ii) T is firmly nonexpansive if and only if the complement I − T is firmly nonexpansive. (iii) If T = (1 − α)S + αV for some α ∈ (0, 1) and if S is firmly nonexpansive and V is nonexpansive, then T is averaged. (iv) The composite of finitely many averaged mappings is averaged. That is, if each of the mappings {T i } N i=1 is averaged, then so is the composite T 1 ∘ ⋅ ⋅ ⋅ ∘ T N . In particular, if T 1 is α 1 -averaged and T 2 is α 2 -averaged, where α 1 , α 2 ∈ (0, 1), then the composite T 1 ∘ T 2 is αaveraged, where α = α 1 + α 2 − α 1 α 2 . On the other hand, it is clear that, in a real Hilbert space H, T : C → C is ζ-strictly pseudocontractive if and only if there holds the following inequality: ⟨Tx − Ty, x − y⟩ ≤ 󵄩󵄩󵄩x − y 󵄩󵄩󵄩 2 − 1 − ζ 2 󵄩󵄩󵄩(I − T) x − (I − T) y 󵄩󵄩󵄩 2 ,