On generalized extragradient implicit method for systems of variational inequalities with constraints of variational inclusion and fixed point problems

1 Introduction

Let H be a real Hilbert space, in which the inner product and induced norm are denoted by the notations ⟨ ⋅ , ⋅ ⟩ and ‖ ⋅ ‖ , respectively. Given a nonempty, closed, and convex subset C ⊂ H . Let P C be the metric projection of H onto C . Given a mapping A : C → H . Consider the classical variational inequality problem (VIP) of finding a point u ∗ ∈ C s.t. ⟨ Au ∗ , v − u ∗ ⟩ ≥ 0 , ∀ v ∈ C . The solution set of the VIP is denoted by VI( C , A ). In 1976, Korpelevich [1] first designed an extragradient method, i.e., for any initial u 0 ∈ C , the sequence { u m } is generated by

with ℓ ∈ 0 , 1 L , which has been one of the most popular approaches for solving the VIP till now. In the case of VI ( C , A ) ≠ ∅ , the sequence { u m } has only weak convergence. Indeed, the convergence of { u m } only requires that the mapping A is monotone and Lipschitz continuous. To the best of our knowledge, Korpelevich’s extragradient method has received great attention from many authors, who have improved and modified it in various ways, see e.g., [2,3,4, 5,6,7, 8,9,10, 11,12] and references therein.

Recently, to solve the variational inclusion (VI) of finding v ∗ ∈ H s.t. 0 ∈ ( A + B ) v ∗ , Takahashi et al. [13] suggested a Halpern-type iterative method, i.e., for any given v 0 , u ∈ H , { v m } is the sequence generated by

where A is an α -inverse-strongly monotone operator on H and B is a maximal monotone operator on H . They proved the strong convergence of { v m } to a solution v ∗ ∈ ( A + B ) − 1 0 of the VI. Subsequently, Pholasa et al. [14] extended the result in [13] to the setting of Banach spaces, and proved the strong convergence of { v m } to a point of ( A + B ) − 1 0 .

In 2010, Takahashi et al. [15] invented a Mann-type Halpern iterative scheme for solving the fixed point problem (FPP) of a nonexpansive mapping S : C → C and the VI for an α -inverse-strongly monotone mapping A : C → H and a maximal monotone operator B : D ( B ) ⊂ C → H , i.e., for any given y 1 = y ∈ C , { y m } is the sequence generated by

where { λ m } ⊂ ( 0 , 2 α ) and { α m } , { β m } ⊂ ( 0 , 1 ) are such that (i) lim m → ∞ α m = 0 and ∑ m = 1 ∞ α m = ∞ ; (ii) 0 < a ≤ λ m ≤ b < 2 α and lim m → ∞ ( λ m − λ m + 1 ) = 0 and (iii) 0 < c ≤ β m ≤ d < 1 . They proved the strong convergence of { y m } to a point of Fix ( S ) ∩ ( A + B ) − 1 0 .

Since the VI is important and interesting, many researchers have presented and developed a great number of iterative methods for solving the VI in several approaches, see e.g., [7,13,14, 15,16,17, 18,19,20, 21,22,23, 24,25] and references therein. Meanwhile, we consider the FPP of finding a point u ∗ ∈ C such that u ∗ = S u ∗ , where S : C → C is a nonlinear mapping. The solution set of the FPP is denoted by Fix ( S ) . In the practical life, many mathematical models have been formulated to solve this problem. At present, many mathematicians are interested in finding a common solution to the VI and FPP, i.e., find a point u ∗ s.t. u ∗ ∈ Fix ( S ) ∩ ( A + B ) − 1 0 .

Assume that A : C → H is an inverse-strongly monotone mapping, B : D ( B ) ⊂ C → 2 H is a maximal monotone operator, and S : C → C is a nonexpansive mapping. In 2011, Manaka and Takahashi [22] suggested an iterative process, i.e., for any given u 0 ∈ C , { u m } is the sequence generated by

where { α m } ⊂ ( 0 , 1 ) and { λ m } ⊂ ( 0 , ∞ ) . They proved the weak convergence of { u m } to a point of Fix ( S ) ∩ ( A + B ) − 1 0 under some appropriate conditions.

Furthermore, suppose that q ∈ ( 1 , 2 ] and E is a real Banach space. Let f : E → E be a ρ -contraction and S : E → E be a nonexpansive mapping. Let A : E → E be an α -inverse-strongly accretive mapping of order q and B : E → 2 E be an m -accretive operator. Recently, to solve the FPP of S and the VI of finding u ∗ ∈ E s.t. 0 ∈ ( A + B ) u ∗ , Sunthrayuth and Cholamjiak [7] suggested a modified viscosity-type extragradient method in the setting of uniformly convex and q -uniformly smooth Banach space E with q -uniform smoothness coefficient κ q , i.e., for any given u 0 ∈ E , { u m } is the sequence generated by

where J λ m B = ( I + λ m B ) − 1 , { α m } , { β m } , { γ m } , { r m } ⊂ ( 0 , 1 ) , and { λ m } ⊂ ( 0 , ∞ ) are such that: (i) α m + β m + γ m = 1 ; (ii) lim m → ∞ α m = 0 and ∑ m = 1 ∞ α m = ∞ ; (iii) { β m } ⊂ [ a , b ] ⊂ ( 0 , 1 ) ; and (iv) 0 < λ ≤ λ m < λ m ∕ r m ≤ μ < ( α q ∕ κ q ) 1 ∕ ( q − 1 ) and 0 < r ≤ r m < 1 . They proved strong convergence of { u m } to a point of Fix ( S ) ∩ ( A + B ) − 1 0 , which solves a certain hierarchical variational inequality (HVI).

On the other hand, let J : E → 2 E ∗ be the normalized duality mapping from E into 2 E ∗ defined by J ( x ) = { ϕ ∈ E ∗ : ⟨ x , ϕ ⟩ = ‖ x ‖ 2 = ‖ ϕ ‖ 2 } , ∀ x ∈ E , where ⟨ ⋅ , ⋅ ⟩ denotes the generalized duality pairing between E and E ∗ . Recall that if E is smooth, then J is single-valued. Let B 1 , B 2 : C → E be two nonlinear mappings in a smooth Banach space E . The general system of variational inequalities (GSVI) is to find ( x ∗ , y ∗ ) ∈ C × C such that

where μ i is a positive constant for i = 1 , 2 . In particular, if E = H a real Hilbert space, it is easy to see that the GSVI (1.6) reduces to the GSVI considered in [3] as follows:

In [3], problem (1.7) is transformed into a fixed point problem in the following way.

Recently, using Lemma 1.1, Cai et al. [2] proposed a viscosity implicit rule for solving the GSVI (1.7) with the FPP constraint of an asymptotically nonexpansive mapping T with a sequence { θ n } , i.e., for any given x 0 ∈ C , the sequence { x n } is generated as follows:

where { α n } , { s n } ⊂ ( 0 , 1 ] are such that (i) lim n → ∞ α n = 0 , ∑ n = 0 ∞ α n = ∞ , and ∑ n = 0 ∞ ∣ α n + 1 − α n ∣ < ∞ ; (ii) lim n → ∞ θ n ∕ α n = 0 ; (iii) 0 < ε ≤ s n ≤ 1 , ∑ n = 0 ∞ ∣ s n + 1 − s n ∣ < ∞ ; and (iv) ∑ n = 0 ∞ ‖ T n + 1 y n − T n y n ‖ < ∞ . They proved that the sequence constructed by (1.8) converges strongly to a point of GSVI ( C , A 1 , A 2 ) ∩ Fix ( T ) , which solves a certain HVI.

In a real Banach space E , let the VI indicate a variational inclusion for two accretive operators and let the CFPP denote a common fixed point problem of countably many nonexpansive mappings. In this article, we introduce a generalized extragradient implicit method for solving the GSVI (1.6) with the VI and CFPP constraints. We then prove the strong convergence of the suggested method to a solution of the GSVI (1.6) with the VI and CFPP constraints under some suitable assumptions. Our results improve and extend the corresponding results in Manaka and Takahashi [22], Sunthrayuth and Cholamjiak [7], and Cai et al. [2] to a certain extent.

2 Preliminaries

Let C be a nonempty, closed, and convex subset of a real Banach space E with the dual E ∗ . For simplicity, we shall use the following notations: x n → x indicates the strong convergence of the sequence { x n } to x and x n ⇀ x denotes the weak convergence of the sequence { x n } to x . Given a self-mapping T on C . We use the notations R and Fix ( T ) to stand for the set of all real numbers and the fixed point set of T , respectively. Recall that T is said to be nonexpansive if ‖ T u − T v ‖ ≤ ‖ u − v ‖ , ∀ u , v ∈ C . A mapping f : C → C is called a contraction if ∃ δ ∈ [ 0 , 1 ) s.t. ‖ f ( u ) − f ( v ) ‖ ≤ δ ‖ u − v ‖ , ∀ u , v ∈ C . In addition, recall that the normalized duality mapping J defined by

is the one from E into the family of nonempty (by Hahn-Banach’s theorem), weak ∗ , and compact subsets of E ∗ , satisfying J ( τ u ) = τ J ( u ) and J ( − u ) = − J ( u ) for all τ > 0 and u ∈ E .

The modulus of convexity of E is the function δ E : ( 0 , 2 ] → [ 0 , 1 ] defined as follows:

The modulus of smoothness of E is the function ρ E : R + ≔ [ 0 , ∞ ) → R + defined as follows:

A Banach space E is said to be uniformly convex if δ E ( ε ) > 0 , ∀ ε ∈ ( 0 , 2 ] . It is said to be uniformly smooth if lim τ → 0 + ρ E ( τ ) ∕ τ = 0 . In addition, it is said to be q -uniformly smooth with q > 1 if ∃ c > 0 s.t. ρ E ( t ) ≤ c t q , ∀ t > 0 . If E is q -uniformly smooth, then q ≤ 2 and E is also uniformly smooth, and if E is uniformly convex, then E is also reflexive and strictly convex. It is known that Hilbert space H is 2-uniformly smooth. Furthermore, sequence space ℓ p and Lebesgue space L p are min { p , 2 } -uniformly smooth for every p > 1 [26].

Let q > 1 . The generalized duality mapping J q : E → 2 E ∗ is defined as follows:

where ⟨ ⋅ , ⋅ ⟩ denotes the generalized duality pairing between E and E ∗ . In particular, if q = 2 , then J 2 = J is the normalized duality mapping of E . It is known that J q ( x ) = ‖ x ‖ q − 2 J ( x ) , ∀ x ≠ 0 , and that J q is the subdifferential of the functional 1 q ‖ ⋅ ‖ q . If E is uniformly smooth, the generalized duality mapping J q is one-to-one and single-valued. Furthermore, J q satisfies J q = J p − 1 , where J p is the generalized duality mapping of E ∗ with 1 p + 1 q = 1 . Note that no Banach space is q -uniformly smooth for q > 2 , see [8] for more details.

The following lemma is an immediate consequence of the subdifferential inequality of the functional 1 q ‖ ⋅ ‖ q .

The following lemma can be obtained from the result in [26].

The following lemma is an analogue of Lemma 2.2(a).

Let D be a subset of C and let Π be a mapping of C into D . Then Π is said to be sunny if Π [ Π ( x ) + t ( x − Π ( x ) ) ] = Π ( x ) , whenever Π ( x ) + t ( x − Π ( x ) ) ∈ C for x ∈ C and t ≥ 0 . A mapping Π of C into itself is called a retraction if Π 2 = Π . If a mapping Π of C into itself is a retraction, then Π ( z ) = z for each z ∈ R ( Π ) , where R ( Π ) is the range of Π . A subset D of C is called a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction from C onto D . In terms of [28], we know that if E is smooth and Π is a retraction of C onto D , then the following statements are equivalent:

1. Π is sunny and nonexpansive;

2. ‖ Π ( x ) − Π ( y ) ‖ 2 ≤ ⟨ x − y , J ( Π ( x ) − Π ( y ) ) ⟩ , ∀ x , y ∈ C ;

3. ⟨ x − Π ( x ) , J ( y − Π ( x ) ) ⟩ ≤ 0 , ∀ x ∈ C , y ∈ D .

Let A : C → E be an α -inverse-strongly accretive mapping of order q and B : C → 2 E be an m -accretive operator. In the sequel, we will use the notation T λ ≔ J λ B ( I − λ A ) = ( I + λ B ) − 1 ( I − λ A ) , ∀ λ > 0 .

Recall that if E = H a Hilbert space, then the sunny nonexpansive retraction Π C from E onto C coincides with the metric projection P C from H onto C . Moreover, if E is uniformly smooth and T is a nonexpansive self-mapping on C with Fix ( T ) ≠ ∅ , then Fix ( T ) is a sunny nonexpansive retract from E onto C [32]. By Lemma 2.6, we know that x ∗ ∈ Fix ( T ) solves the HVI in Proposition 2.4 if and only if x ∗ solves the fixed point equation x ∗ = Π Fix ( T ) f ( x ∗ ) .

3 Main results

Throughout this article, we assume that E is a q -uniformly smooth and uniformly convex Banach space with q ∈ ( 1 , 2 ] . Let C be a nonempty, closed, and convex subset of E and Π C be a sunny nonexpansive retraction from E onto C . Let f : C → C be a δ -contraction with constant δ ∈ [ 0 , 1 ) and { S n } n = 0 ∞ be a countable family of nonexpansive self-mappings on C . Let A : C → E and B : C → 2 E be a σ -inverse-strongly accretive mapping of order q and an m -accretive operator, respectively. Suppose that B 1 and B 2 : C → E are α -inverse-strongly accretive mapping of order q and β -inverse-strongly accretive mapping of order q , respectively. Assume that Ω ≔ ⋂ n = 0 ∞ Fix ( S n ) ∩ GSVI ( C , B 1 , B 2 ) ∩ ( A + B ) − 1 0 ≠ ∅ .