We introduce an iterative process which converges strongly to a common point of solution of variational inequality problem for a monotone mapping and fixed point of uniformly Lipschitzian relatively asymptotically nonexpansive mapping in Banach spaces. As a consequence, we provide a scheme that converges strongly to a common zero of finite family of monotone mappings under suitable conditions. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

1. Introduction

Let E be a smooth Banach space. Throughout this paper, we denote by ϕ : E × E → ℝ the function defined by

()

which was studied by Alber [1], Kamimura and Takahashi [2], and Reich [3], where J is the normalized duality mapping from E to

defined by

()

where 〈·, ·〉 denotes the duality pairing. It is well known that if E is smooth, then J is single-valued, and, if E has uniformly Gâteaux differentiable norm, then J is uniformly continuous on bounded subsets of E. Moreover, if E is a reflexive and strictly convex Banach space with a strictly convex dual, then J⁻¹ is single valued, one-to-one, surjective, and it is the duality mapping from E^* into E, and thus

and J⁻¹J = I_E (see [4]).

It is obvious from the definition of the function ϕ that

()

and, in a Hilbert space H, (1.1) reduces to ϕ(x, y) = ∥x−y∥², for x, y ∈ H.

Let E be a reflexive, strictly convex, and smooth Banach space, and let C be a nonempty closed and convex subset of E. The generalized projection mapping, introduced by Alber [1], is a mapping Π_C : E → C that assigns an arbitrary point x ∈ E to the minimizer,

, of ϕ(·, x) over C, that is,

, where

is the solution to the minimization problem

()

Let E be a real Banach space with dual E^*. A mapping A : D(A) ⊂ E → E^* is said to be monotone if, for each x, y ∈ D(A), the following inequality holds:

()

A is said to be γ-inverse strongly monotone if there exists positive real number γ such that

()

If A is γ-inverse strongly monotone, then it is Lipschitz continuous with constant 1/γ, that is, ∥Ax − Ay∥ ≤ (1/γ)∥x − y∥, for all x, y ∈ D(A), and it is called strongly monotone if there exists k > 0 such that, for all x, y ∈ D(A),

()

Clearly, the class of monotone mappings include the class of strongly monotone and γ-inverse strongly monotone mappings.

Suppose that A is monotone mapping from C into E^*. The variational inequality problem is formulated as finding a point u ∈ C such that 〈v − u, Au〉 ≥ 0, for all v ∈ C. The set of solutions of the variational inequality problems is denoted by VI (C, A).

The notion of monotone mappings was introduced by Zarantonello [5], Minty [6], and Kacurovskii [7] in Hilbert spaces. Monotonicity conditions in the context of variational methods for nonlinear operator equations were also used by Vainberg and Kachurovisky [8]. Variational inequalities were initially studied by Stampacchia [9, 10] and ever since have been widely studied in general Banach spaces (see, e.g., [2, 11–13]). Such a problem is connected with the convex minimization problem, the complementarity problem, the problem of finding point u ∈ C satisfying 0 ∈ Au.

If E = H, a Hilbert space, one method of solving a point u ∈ VI (C, A) is the projection algorithm which starts with any point x₁ = x ∈ C and updates iteratively as x_n+1 according to the formula

()

where P_C is the metric projection from H onto C and {α_n} is a sequence of positive real numbers. In the case that A is γ-inverse strongly monotone, Iiduka et al. [14] proved that the sequence {x_n} generated by (3.35) converges weakly to some element of VI (C, A).

In the case that E is a 2-uniformly convex and uniformly smooth Banach space, Iiduka and Takahashi [15] introduced the following iteration scheme for finding a solution of the variational inequality problem for an inverse strongly monotone operator A:

()

where Π_C is the generalized projection from E onto C, J is the normalized duality mapping from E into E^*, and {α_n} is a sequence of positive real numbers. They proved that the sequence {x_n} generated by (1.9) converges weakly to some element of VI (C, A) provided that A satisfies ∥Ax∥ ≤ ∥Ax − Ap∥, for x ∈ C and p ∈ VI (C, A).

It is worth to mention that the convergence is weak convergence.

To obtain strong convergence, when E = H, a Hilbert space and A is γ-inverse strongly monotone; Iiduka et al. [14] studied the following iterative scheme:

()

where {α_n} is a sequence in [0, 2γ]. They proved that the sequence {x_n} generated by (1.10) converges strongly to P_VI (C,A)(x₀), where P_VI (C,A) is the metric projection from H onto VI (C, A) provided that A satisfies ∥Ax∥ ≤ ∥Ax − Ap∥, for x ∈ C and p ∈ VI (C, A).

In the case that E is 2-uniformly convex and uniformly smooth Banach space, Iiduka and Takahashi [11] studied the following iterative scheme for a variational inequality problem for γ-inverse strongly monotone mapping:

()

where

is the generalized projection from E onto C_n∩Q_n, J is the normalized duality mapping from E into E^*, and {α_n} is a positive real sequence satisfying certain condition. Then, they proved that the sequence {x_n} converges strongly to an element of VI (C, A) provided that VI (C, A) ≠ ∅ and A satisfies ∥Ax∥ ≤ ∥Ax − Ap∥ for all x ∈ C and p ∈ VI (C, A).

Remark 1.1. We remark that the computation of x_n+1 in Algorithms (1.10) and (1.11) is not simple because of the involvement of computation of C_n+1 from C_n and Q_n for each n ≥ 1.

Let T be a mapping from C into itself. We denote by F(T) the fixed points set of T. A point p in C is said to be an asymptotic fixed point of T (see [3]) if C contains a sequence {x_n} which converges weakly to p such that lim _n→∞∥x_n − Tx_n∥ = 0. The set of asymptotic fixed points of T will be denoted by . A mapping T from C into itself is said to be nonexpansive if ∥Tx − Ty∥ ≤ ∥x − y∥ for each x, y ∈ C and is called relatively nonexpansive if (R1) F(T) ≠ ∅; (R2) ϕ(p, Tx) ≤ ϕ(p, x) for x ∈ C and (R3) . T is called relatively quasi-nonexpansive if F(T) ≠ ∅ and ϕ(p, Tx) ≤ ϕ(p, x) for all x ∈ C, and p ∈ F(T).

A mapping T from C into itself is said to be asymptotically nonexpansive if there exists {k_n} ⊂ [1, ∞) such that k_n → 1 and ∥Tⁿx − Tⁿy∥ ≤ k_n∥x − y∥ for each x, y ∈ C and is called relatively asymptotically nonexpansive if there exists {k_n} ⊂ [1, ∞) such that (N1) F(T) ≠ ∅; (N2) ϕ(p, Tⁿx) ≤ k_nϕ(p, x) for x ∈ C and p ∈ F(T), and (N3) , where k_n → 1 as n → ∞. A-self mapping on C is called uniformly L-Lipschitzian if there exists L > 0 such that ∥Tⁿx − Tⁿy∥ ≤ L∥x − y∥ for all x, y ∈ C. T is called closed if x_n → x and Tx_n → y, then Tx = y.

Clearly, we note that the class of relatively nonexpansive mappings is contained in a class of relatively asymptotically nonexpansive mappings but the converse is not true. Now, we give an example of relatively asymptotically nonexpansive mapping which is not relatively nonexpansive.

Example 1.2 (see [16].)Let X = l^p, where 1 < p < ∞, and C = {x = (x₁, x₂, …) ∈ X; x_n ≥ 0}. Then C is closed and convex subset of X. Note that C is not bounded. Obviously, X is uniformly convex and uniformly smooth. Let {λ_n} and be sequences of real numbers satisfying the following properties:

(i)
0 < λ_n < 1, , λ_n↑1, and ,
(ii)
and for all n and j (e.g., λ_n = 1 − 1/(n + 1), ). Then, the map T : C → C defined by
()

for all x = (x₁, x₂, …) ∈ C, is uniformly Lipschitzian which is relatively asymptotically nonexpansive but not relatively nonexpansive (see [16] for the details). Note also that F(T) = {0}.

In 2005, Matsushita and Takahashi [17] proposed the following hybrid iteration method with generalized projection for relatively nonexpansive mapping T in a Banach space E:

()

They proved that, if the sequence {α_n} is bounded above from one, then the sequence {x_n} generated by (1.13) converges strongly to Π_F(T)x₀.

Recently, many authors have considered the problem of finding a common element of the fixed-point set of relatively nonexpansive mapping and the solution set of variational inequality problem for γ-inverse monotone mapping (see, e.g., [12, 13, 18–20]).

In [21], Iiduka and Takahashi studied the following iterative scheme for a common point of solution of a variational inequality problem for γ-inverse strongly monotone mapping A and fixed point of nonexpansive mapping T in a Hilbert space H:

()

where {α_n} is sequences satisfying certain condition. They proved that the sequence {x_n} converges strongly to an element of F∶ = F(S)∩VI (C, A) provided that F ≠ ∅.

In the case that E is a Banach space more general than Hilbert spaces, Zegeye et al. [12] studied the following iterative scheme for a common point of solution of a variational inequality problem for γ-inverse strongly monotone mapping A and fixed point of a closed relatively quasi-nonexpansive mapping T in a 2-uniformly convex and uniformly smooth Banach space E:

()

where {λ_n} is sequences satisfying certain condition. They proved that the sequence {x_n} converges strongly to an element of F : = F(S)∩VI (C, A) ≠ ∅ provided that F ≠ ∅ and A satisfies ∥Ax∥ ≤ ∥Ax − Ap∥ for all x ∈ C and p ∈ F.

Furthermore, Zegeye and Shahzad [22] studied the following iterative scheme for common point of solution of a variational inequality problem for γ-inverse strongly monotone mapping A and fixed point of a relatively asymptotically nonexpansive mapping on a closed convex and bounded set C which is a subset of a real Hilbert space H:

()

where

is the metric projection from H into C_n and

and {α_n}, {λ_n} are sequences satisfying certain condition. Then, they proved that the sequence {x_n} converges strongly to an element of F∶ = F(S)∩VI (C, A) ≠ ∅ provided that F ≠ ∅ and A satisfies ∥Ax∥ ≤ ∥Ax − Ap∥ for all x ∈ C and p ∈ F.

Remark 1.3. We again remark that the computation of x_n+1 in Algorithms (1.13), (1.15), and (1.16) is not simple because of the involvement of computation of C_n+1 from C_n for each n ≥ 1.

It is our purpose in this paper to introduce an iterative scheme {x_n} which converges strongly to a common point of solution of variational inequality problem for a monotone operator A : C → E^* satisfying appropriate conditions, for some nonempty closed convex subset C of a Banach space E and fixed points of uniformly L-Lipschitzian relatively asymptotically nonexpansive mapping in Banach spaces. As a consequence, we provide a scheme which converges strongly to a common zero of finite family of monotone mappings. Our scheme does not involve computation of C_n+1 from C_n or Q_n, for each n ≥ 1. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

2. Preliminaries

Let E be a normed linear space with dim E ≥ 2. The modulus of smoothness of E is the function ρ_E : [0, ∞) → [0, ∞) defined by

()

The space E is said to be smooth if ρ_E(τ) > 0, for all τ > 0, and E is called uniformly smooth if and only if

The modulus of convexity of E is the function δ_E : (0,2] → [0,1] defined by

()

E is called uniformly convex if and only if δ_E(ϵ) > 0, for every ϵ ∈ (0,2].

In the sequel, we will need the following results.

Lemma 2.1 (see [23].)Let C be a nonempty closed and convex subset of a real reflexive, strictly convex, and smooth Banach space E. If A : C → E^* is continuous monotone mapping, then VI (C, A) is closed and convex.

Lemma 2.2. Let C be a closed convex subset of a uniformly convex and smooth Banach space E, and let S be continuous relatively asymptotically nonexpansive mapping from C into itself. Then, F(S) is closed and convex.

Lemma 2.3 (see [1].)Let K be a nonempty closed and convex subset of a real reflexive, strictly convex, and smooth Banach space E, and let x ∈ E. Then, for all y ∈ K,

()

Lemma 2.4 (see [2].)Let E be a real smooth and uniformly convex Banach space, and let {x_n} and {y_n} be two sequences of E. If either {x_n} or {y_n} is bounded and ϕ(x_n, y_n) → 0 as n → ∞, then x_n − y_n → 0, as n → ∞.

We make use of the function V : E × E^* → ℝ defined by

()

studied by Alber [1]. That is, V(x, y) = ϕ(x, J⁻¹x^*) for all x ∈ E and x^* ∈ E^*. We know the following lemma.

Lemma 2.5 (see [1].)Let E be reflexive strictly convex and smooth Banach space with E^* as its dual. Then,

()

for all x ∈ E and x^*, y^* ∈ E^*.

Lemma 2.6 (see [1].)Let C be a convex subset of a real smooth Banach space E. Let x ∈ E. Then x₀ = Π_Cx if and only if

()

Lemma 2.7 (see [12].)Let E be a uniformly convex Banach space and B_R(0) a closed ball of E. Then, there exists a continuous strictly increasing convex function g : [0, ∞) → [0, ∞) with g(0) = 0 such that

()

for α_i ∈ (0,1) such that α₁ + α₂ + α₃ = 1 and x_i ∈ B_R(0)∶ = {x ∈ E : ∥x∥ ≤ R}, for i = 1,2, 3.

Let E be a smooth and strictly convex Banach space, C a nonempty closed convex subset of E, and A : C → E^* a monotone operator satisfying

()

for r > 0. Then, we can define the resolvent Q_r : C → D(A) of A by

()

In other words, Q_rx = (J+rA)⁻¹Jx for x ∈ C. We know that Q_rx is single-valued mapping from C into D(A), for all x ∈ C and r > 0 and F(Q_r) = A⁻¹(0), where F(Q_r) is the set of fixed points of Q_r (see, [4]).

Lemma 2.8 (see [24].)Let E be a smooth and strictly convex Banach space, C a nonempty closed convex subset of E, and A ⊂ E × E^* a monotone operator satisfying (2.8) and A⁻¹(0) is nonempty. Let Q_r be the resolvent of A. Then, for each r > 0,

()

for all u ∈ A⁻¹(0) and x ∈ C, that is, Q_r is relatively nonexpansive.

Lemma 2.9 (see [25].)Let {a_n} be a sequence of nonnegative real numbers satisfying the following relation:

()

where {β_n} ⊂ (0,1) and {δ_n} ⊂ R satisfying the following conditions:

, and limsup _n→∞δ_n ≤ 0. Then, lim _n→∞a_n = 0.

Lemma 2.10 (see [26].)Let {a_n} be sequences of real numbers such that there exists a subsequence {n_i} of {n} such that for all i ∈ N. Then, there exists a nondecreasing sequence {m_k} ⊂ N such that m_k → ∞, and the following properties are satisfied by all (sufficiently large) numbers k ∈ N:

()

In fact, m_k = max {j ≤ k : a_j < a_j+1}.

3. Main Result

We note that, as it is mentioned in [27], if C is a subset of a real Banach space E and A : C → E^* is a mapping satisfying ∥Ax∥ ≤ ∥Ax − Ap∥, for all x ∈ C and p ∈ VI (C, A), then

()

In fact, clearly, A⁻¹(0)⊆VI (C, A). Now, we show that VI (C, A)⊆A⁻¹(0). Let p ∈ VI (C, A), then we have by hypothesis that ∥Ap∥ ≤ ∥Ap − Ap∥ = 0 which implies that p ∈ A⁻¹(0). Hence, VI (C, A)⊆A⁻¹(0). Therefore, VI (C, A) = A⁻¹(0). Now we prove the main theorem of our paper.

Theorem 3.1. Let C be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth real Banach space E. Let A : C → E^* be a monotone mapping satisfying (2.8) and ∥Ax∥ ≤ ∥Ax − Ap∥, for all x ∈ C and p ∈ VI (C, A). Let T : C → C be a uniformly L-Lipschitzian relatively asymptotically nonexpansive mapping with sequence {k_n}. Assume that F∶ = VI (C, A)∩F(S) is nonempty. Let Q_r be the resolvent of A and {x_n} a sequence generated by

()

where α_n ∈ (0,1) such that

, {β_n}, {γ_n}, {θ_n} ⊂ [c, d] ⊂ (0,1) such that β_n + γ_n + θ_n = 1. Then, {x_n} converges strongly to an element of F.

Proof. Let p∶ = Π_Fw. Then, from (3.2), Lemma 2.3, and property of ϕ, we get that

()

Now, from (3.2) and relatively asymptotically nonexpansiveness of T, relatively nonexpansiveness of Q_r, property of ϕ, and (3.3), we get that

()

where c_n = α_n(γ_nk_n + θ_n), since there exists N₀ > 0 such that γ_n(k_n − 1)/α_n ≤ ϵ(γ_nk_n + θ_n) for all n ≥ N₀ and for some ϵ > 0 satisfying (1 − ϵ)c_n ≤ 1. Thus, by induction,

()

which implies that {x_n}, and hence {y_n} is bounded. Now, let z_n = J⁻¹(α_nJw + (1 − α_n)Jx_n). Then we have that y_n = Π_Cz_n. Using Lemmas 2.3, 2.5, and property of ϕ, we obtain that

()

Furthermore, from (3.2), Lemma 2.7, relatively asymptotically nonexpansiveness of T, relatively nonexpansiveness of Q_r, and (3.6), we have that

()

for some M > 0, where δ_n = (γ_n + θ_n)α_n.

Similarly, from (3.7), we obtain that

()

for some M > 0. Note that {δ_n} satisfies that lim _n→∞δ_n = 0 and ∑δ_n = ∞.

Now, the rest of the proof is divided into two parts.

Case 1. Suppose that there exists n₀ ∈ N > N₀ such that {ϕ(p, x_n)} is nonincreasing for all n ≥ n₀. In this situation, {ϕ(p, x_n)} is then convergent. Then, from (3.8) and (*), we have that

()

which implies, by the property of g, that

()

and, hence, since J⁻¹ is uniformly continuous on bounded sets, we obtain that

()

Furthermore, Lemma 2.3, property of ϕ, and the fact that α_n → 0 as n → ∞ imply that

()

and hence

()

Therefore, from (3.12) and (3.14), we obtain that

()

But observe that from (3.2) and (3.11), we have

()

as n → ∞. Thus, as J⁻¹ is uniformly continuous on bounded sets, we have that x_n+1 − x_n → 0 which implies from (3.14) that x_n+1 − y_n → 0, as n → ∞, and that

()

Furthermore, since

()

we have from (3.17), (3.15), and uniform continuity of T that

()

Since {z_n} is bounded and E is reflexive, we choose a subsequence

of {z_n} such that

and

. Then, from (3.14) and (3.15) we get that

()

Thus, since T satisfies condition (N3), we obtain from (3.19) that z ∈ F(T) and the fact that Q_r is relatively nonexpansive and

implies that z ∈ F(Q_r) = A⁻¹(0), and, hence, using (3.1), we obtain that z ∈ VI (C, A).

Therefore, from the above discussions, we obtain that z ∈ F : = F(T)∩VI (C, A). Hence, by Lemma 2.6, we immediately obtain that . It follows from Lemma 2.9 and (3.9) that ϕ(p, x_n) → 0, as n → ∞. Consequently, x_n → p.

Case 2. Suppose that there exists a subsequence {n_i} of {n} such that

()

for all i ∈ N. Then, by Lemma 2.10, there exists a nondecreasing sequence {m_k} ⊂ N such that m_k → ∞,

and

for all k ∈ N. Then, from (3.8), (*) and the fact δ_n → 0, we have

()

Thus, using the same proof as in Case 1, we obtain that

, as k → ∞, and, hence, we obtain that

()

Then, from (3.9), we have that

()

Since

, (3.24) implies that

()

In particular, since

, we get

()

Then, from (3.23) and the fact that

, we obtain

, as k → ∞. This together with (3.24) gives

, as k → ∞. But

, for all k ∈ N, thus we obtain that x_k → p. Therefore, from the above two cases, we can conclude that {x_n} converges strongly to p and the proof is complete.

If, in Theorem 3.1, we assume that T is relatively nonexpansive, we get the following corollary.

Corollary 3.2. Let C be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth real Banach space E. Let A : C → E^* be a monotone mapping satisfying (2.8) and ∥Ax∥ ≤ ∥Ax − Ap∥, for all x ∈ C and p ∈ VI (C, A). Let T : C → C be a relatively nonexpansive mapping. Assume that F∶ = VI (C, A)∩F(S) is nonempty. Let {x_n} be a sequence generated by

()

where α_n ∈ (0,1) such that lim _n→∞α_n = 0,

, {β_n}, {γ_n}, {θ_n} ⊂ [c, d] ⊂ (0,1) such that β_n + γ_n + θ_n = 1. Then, {x_n} converges strongly to an element of F.

Proof. We note that the method of proof of Theorem 3.1 provides the required assertion.

If E = H, a real Hilbert space, then E is uniformly convex and uniformly smooth real Banach space. In this case, J = I, identity map on H and Π_C = P_C, projection mapping from H onto C. Thus, the following corollary holds.

Corollary 3.3. Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let A : C → H be a monotone mapping satisfying (2.8) and ∥Ax∥ ≤ ∥Ax − Ap∥, for all x ∈ C and p ∈ VI(C, A). Let T : C → C be a uniformly L-Lipschitzian relatively asymptotically nonexpansive mapping with sequence {k_n}. Assume that F∶ = VI(C, A)∩F(S) is nonempty. Let {x_n} be a sequence generated by

()

where α_n ∈ (0,1) such that lim _n→∞α_n = 0, lim _n→∞((k_n − 1)/α_n) = 0,

, {β_n}, {γ_n}, {θ_n} ⊂ [c, d] ⊂ (0,1) such that β_n + γ_n + θ_n = 1. Then, {x_n} converges strongly to an element of F.

Now, we state the second main theorem of our paper.

Theorem 3.4. Let C be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth real Banach space E. Let A : C → E^* be a monotone mapping satisfying (2.8). Let T : C → C be a uniformly L-Lipschitzian relatively asymptotically nonexpansive mapping with sequence {k_n}. Assume that F : = A⁻¹(0)∩F(S) is nonempty. Let Q_r be the resolvent of A and {x_n} a sequence generated by

()

where α_n ∈ (0,1) such that lim _n→∞α_n = 0, lim _n→∞((k_n − 1)/α_n) = 0,

, {β_n}, {γ_n}, {θ_n} ⊂ [c, d] ⊂ (0,1) such that β_n + γ_n + θ_n = 1. Then, {x_n} converges strongly to an element of F.

Proof. Similar method of proof of Theorem 3.1 provides the required assertion.

If, in Theorem 3.4, A = 0, then we have the following corollary. Similar proof of Theorem 3.1 provides the assertion.

Corollary 3.5. Let C be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth real Banach space E. Let T : C → C be a uniformly L-Lipschitzian relatively asymptotically nonexpansive mapping with sequence {k_n}. Assume that F∶ = F(S) is nonempty. Let {x_n} be a sequence generated by

()

where α_n ∈ (0,1) such that lim _n→∞α_n = 0, lim _n→∞((k_n − 1)/α_n) = 0,

, {β_n} ⊂ [c, d] ⊂ (0,1). Then, {x_n} converges strongly to an element of F.

If, in Theorem 3.4, T = I, identity mapping on C, then we have the following corollary.

Corollary 3.6. Let C be a nonempty, closed, and convex subset of a uniformly smooth and uniformly convex real Banach space E. Let A : C → E^* be a monotone mapping satisfying (2.8). Assume that F : = A⁻¹(0) is nonempty. Let Q_r be the resolvent of A and {x_n} a sequence generated by

()

where α_n ∈ (0,1) such that lim _n→∞α_n = 0,

, {β_n} ⊂ [c, d] ⊂ (0,1). Then, {x_n} converges strongly to an element of F.

If, in Theorem 3.4, we assume that T is relatively nonexpansive, we get the following corollary.

Corollary 3.7. Let C be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth real Banach space E. Let A : C → E^* be a monotone mapping satisfying (2.8). Let T : C → C be a relatively nonexpansive mapping. Assume that F∶ = A⁻¹(0)∩F(S) is nonempty. Let {x_n} be a sequence generated by

()

where α_n ∈ (0,1) such that lim _n→∞α_n = 0,

, {β_n}, {γ_n}, {θ_n} ⊂ [c, d] ⊂ (0,1) such that β_n + γ_n + θ_n = 1. Then, {x_n} converges strongly to an element of F.

We may also get the following corollary for a common zero of monotone mappings.

Corollary 3.8. Let C be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth real Banach space E. Let A, B : C → E^* be monotone mappings satisfying (2.8). Suppose that T₁ = (J+rA)⁻¹J and T₂ = (J+rB)⁻¹J. Assume that F : = A⁻¹(0)∩B⁻¹(0) is nonempty. Let {x_n} be a sequence generated by

()

where α_n ∈ (0,1) such that lim _n→∞α_n = 0,

, {β_n}, {γ_n}, {θ_n} ⊂ [c, d] ⊂ (0,1) such that β_n + γ_n + θ_n = 1. Then, {x_n} converges strongly to an element of F.

Proof. Clearly, from Lemma 2.8, we know that T₁ and T₂ are relatively nonexpansive mappings. We also have that F(T₁) = A⁻¹(0) and F(T₂) = B⁻¹(0). Thus, the conclusion follows from Corollary 3.7.

Remark 3.9. We remark that from Corollary 3.8 the scheme converges strongly to a common zero of two monotone operators. We may also have the following theorem for a common zero of finite family of monotone mappings.

Theorem 3.10. Let C be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth real Banach space E. Let A_i : C → E^*, i = 1,2, …, N be monotone mappings satisfying (2.8). Suppose that , and assume that is nonempty. Let {x_n} be a sequence generated by

()

where α_n ∈ (0,1) such that lim _n→∞α_n = 0,

, {β_n,i} ⊂ [c, d] ⊂ (0,1), for i = 0,1, 2, …, N, such that

. Then, {x_n} converges strongly to an element of F.

A monotone mapping A : C → E^* is said to be maximal monotone if its graph is not properly contained in the graph of any monotone operator. We know that if A is maximal monotone operator, then A⁻¹(0) is closed and convex: see [4] for more details. The following Lemma is well known.

Lemma 3.11 (see [28].)Let E be a smooth and strictly convex and reflexive Banach space, let C be a nonempty closed convex subset of E, and let A : C → E^* be a monotone operator. Then A is maximal if and only if R(J + rA) = E^* for all r > 0.

We note from the above lemma that if A is maximal then it satisfies condition (2.8) and hence we have the following corollary.

Corollary 3.12. Let C be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth real Banach space E. Let A : C → E^* be a maximal monotone mapping. Let T : C → C be a uniformly L-Lipschitzian relatively asymptotically nonexpansive mapping with sequence {k_n}. Assume that F : = A⁻¹(0)∩F(S) is nonempty. Let {x_n} be a sequence generated by

()

where α_n ∈ (0,1) such that lim _n→∞α_n = 0, lim _n→∞((k_n − 1)/α_n) = 0,

, {β_n}, {γ_n}, {θ_n} ⊂ [c, d] ⊂ (0,1) such that β_n + γ_n + θ_n = 1. Then, {x_n} converges strongly to an element of F.

4. Application

In this section, we study the problem of finding a minimizer of a lower semicontinuous continuously convex functional in Banach spaces.

Theorem 4.1. Let E be a uniformly convex and uniformly smooth real Banach space. Let f, g : E → (−∞, ∞) be a proper lower semicontinuous convex functions. Assume that F∶ = (∂f)⁻¹(0)∩(∂g)⁻¹(0) is nonempty. Let {x_n} be a sequence generated by

()

where α_n ∈ (0,1) such that lim _n→∞α_n = 0,

, {β_n}, {γ_n}, {θ_n} ⊂ [c, d] ⊂ (0,1) such that β_n + γ_n + θ_n = 1. Then, {x_n} converges strongly to an element of F.

Proof. Let A and B be operators defined by A = ∂f and B = ∂g and Q_r = (J+rA)⁻¹J, for all r > 0. Then, by Rockafellar [29], A and B are maximal monotone mappings. We also have that

()

for all y ∈ E and r > 0. Furthermore, we have that

and

. Thus, by Corollary 3.8, we obtain the desired result.

Remark 4.2. Consider the following.

(1)
Theorem 3.1 improves and extends the corresponding results of Zegeye et al. [12] and Zegeye and Shahzad [22] in the sense that either our scheme does not require computation of C_n+1 for each n ≥ 1 or the space considered is more general.
(2)
Corollary 3.5 improves the corresponding results of Nakajo and Takahashi [30] and Matsushita and Takahashi [17] in the sense that either our scheme does not require computation of C_n+1 for each n ≥ 1 or the class of mappings considered in our corollary is more general.
(3)
Corollary 3.6 improves the corresponding results of Iiduka and Takahashi [11], Iiduka et al. [14], and Alber [1] in the sense that our scheme does not require computation of C_n+1 for each n ≥ 1 or the class of mappings considered in our corollary is more general.

References

1 Alber Y. I., Metric and generalized projection operators in Banach spaces: properties and applications, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, 1996, 178, Dekker, New York, NY, USA, 15–50, Lecture Notes in Pure and Applied Mathematics, 1386667, ZBL0883.47083.
Google Scholar
2 Kamimura S. and Takahashi W., Strong convergence of a proximal-type algorithm in a Banach space, SIAM Journal on Optimization. (2002) 13, no. 3, 938–945, https://doi.org/10.1137/S105262340139611X, 1972223.
10.1137/S105262340139611X
Web of Science® Google Scholar
3 Reich S., A. G. Kartsatos, A weak convergence theorem for the alternating method with Bergman distance, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, 1996, 178, Dekker, New York, NY, USA, 313–318, Lecture Notes in Pure and Applied Mathematics, 1386686, ZBL0943.47040.
Google Scholar
4 Takahashi W., Nonlinear Functional Analysis, 1988, Kindikagaku, Tokyo, Japan.
Google Scholar
5 Zarantonello E. H., Solving functional equations by contractive averaging, Mathematics Research Center Report, 1960, no. 160, Mathematics Research Centre, Univesity of Wisconsin, Madison, Wis, USA.
Google Scholar
6 Minty G. J., Monotone (nonlinear) operators in Hilbert space, Duke Mathematical Journal. (1962) 29, 341–346, 0169064, https://doi.org/10.1215/S0012%2D7094%2D62%2D02933%2D2, ZBL0111.31202.
10.1215/S0012-7094-62-02933-2
Web of Science® Google Scholar
7 Kacurovskii R. I., Monotone operators and convex functionals, Uspekhi Matematicheskikh Nauk. (1960) 15, 213–215.
Google Scholar
8 Vaĭnberg M. M. and Kačurovskiĭ R. I., On the variational theory of non-linear operators and equations, Doklady Akademii Nauk SSSR. (1959) 129, 1199–1202, 0114103.
Google Scholar
9 Kinderlehrer D. and Stampaccia G., An Iteration to Variational Inequalities and Their Applications, 1990, Academic Press, New York, NY, USA.
Google Scholar
10 Lions J.-L. and Stampacchia G., Variational inequalities, Communications on Pure and Applied Mathematics. (1967) 20, 493–519, 0216344, https://doi.org/10.1002/cpa.3160200302, ZBL0152.34601.
10.1002/cpa.3160200302
Web of Science® Google Scholar
11 Iiduka H. and Takahashi W., Strong convergence studied by a hybrid type method for monotone operators in a Banach space, Nonlinear Analysis. Theory, Methods & Applications. (2008) 68, no. 12, 3679–3688, https://doi.org/10.1016/j.na.2007.04.010, 2416075, ZBL1220.47095.
10.1016/j.na.2007.04.010
Web of Science® Google Scholar
12 Zegeye H., Ofoedu E. U., and Shahzad N., Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings, Applied Mathematics and Computation. (2010) 216, no. 12, 3439–3449, https://doi.org/10.1016/j.amc.2010.02.054, 2661700, ZBL1198.65100.
10.1016/j.amc.2010.02.054
Web of Science® Google Scholar
13 Zegeye H. and Shahzad N., Strong convergence theorems for monotone mappings and relatively weak nonexpansive mappings, Nonlinear Analysis. Theory, Methods & Applications. (2009) 70, no. 7, 2707–2716, https://doi.org/10.1016/j.na.2008.03.058, 2499738, ZBL1223.47108.
10.1016/j.na.2008.03.058
Web of Science® Google Scholar
14 Iiduka H., Takahashi W., and Toyoda M., Approximation of solutions of variational inequalities for monotone mappings, Panamerican Mathematical Journal. (2004) 14, no. 2, 49–61, 2049049, ZBL1060.49006.
Google Scholar
15 Iiduka H. and Takahashi W., Weak convergence of projection algorithm for variational inequalities in Banach spaces, Journal of Mathematical Analysis and Applications. (2008) 339, no. 1, 668–679, https://doi.org/10.1016/j.jmaa.2007.07.019.
10.1016/j.jmaa.2007.07.019
Web of Science® Google Scholar
16 Kim T.-H. and Takahashi W., Strong convergence of modified iteration processes for relatively asymptotically nonexpansive mappings, Taiwanese Journal of Mathematics. (2010) 14, no. 6, 2163–2180, 2742357, ZBL1226.47079.
10.11650/twjm/1500406068
Web of Science® Google Scholar
17 Matsushita S.-Y. and Takahashi W., A strong convergence theorem for relatively nonexpansive mappings in a Banach space, Journal of Approximation Theory. (2005) 134, no. 2, 257–266, https://doi.org/10.1016/j.jat.2005.02.007, 2142300, ZBL1071.47063.
10.1016/j.jat.2005.02.007
Web of Science® Google Scholar
18 Tada A. and Takahashi W., Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem, Journal of Optimization Theory and Applications. (2007) 133, no. 3, 359–370, https://doi.org/10.1007/s10957%2D007%2D9187%2Dz, 2333820, ZBL1147.47052.
10.1007/s10957-007-9187-z
Web of Science® Google Scholar
19 Takahashi W. and Zembayashi K., Strong and weak convergence theorems for equilibrium problems and relatively nonexpansive mappings in Banach spaces, Nonlinear Analysis. Theory, Methods & Applications. (2009) 70, no. 1, 45–57, https://doi.org/10.1016/j.na.2007.11.031, 2468217, ZBL1170.47049.
10.1016/j.na.2007.11.031
Web of Science® Google Scholar
20 Zegeye H. and Shahzad N., Strong convergence theorems for variational inequality problems and quasi-ϕ-asymptotically nonexpansive mappings, Journal of Global Optimization. In press.
Google Scholar
21 Iiduka H. and Takahashi W., Strong convergence theorems for nonexpansive mappings and α inverse-strongly monotone mappings, Nonlinear Analysis. Theory, Methods & Applications. (2005) 61, no. 3, 341–350, https://doi.org/10.1016/j.na.2003.07.023, 2123081, ZBL1093.47058.
10.1016/j.na.2003.07.023
Web of Science® Google Scholar
22 Zegeye H. and Shahzad N., A hybrid approximation method for equilibrium, variational inequality and fixed point problems, Nonlinear Analysis. Hybrid Systems. (2010) 4, no. 4, 619–630, https://doi.org/10.1016/j.nahs.2010.03.005, 2680234.
10.1016/j.nahs.2010.03.005
Web of Science® Google Scholar
23 Zegeye H. and Shahzad N., A hybrid scheme for finite families of equilibrium, variational inequality and fixed point problems, Nonlinear Analysis. Theory, Methods & Applications. (2011) 74, no. 1, 263–272, https://doi.org/10.1016/j.na.2010.08.040, 2734995.
10.1016/j.na.2010.08.040
Web of Science® Google Scholar
24 Kamimura S., Kohsaka F., and Takahashi W., Weak and strong convergence theorems for maximal monotone operators in a Banach space, Set-Valued Analysis. (2004) 12, no. 4, 417–429, https://doi.org/10.1007/s11228%2D004%2D8196%2D4, 2112848, ZBL1078.47050.
10.1007/s11228-004-8196-4
Google Scholar
25 Xu H.-K., Another control condition in an iterative method for nonexpansive mappings, Bulletin of the Australian Mathematical Society. (2002) 65, no. 1, 109–113, https://doi.org/10.1017/S0004972700020116, 1889384, ZBL1030.47036.
10.1017/S0004972700020116
Web of Science® Google Scholar
26 Maingé P.-E., Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization, Set-Valued Analysis. (2008) 16, no. 7-8, 899–912, https://doi.org/10.1007/s11228%2D008%2D0102%2Dz, 2466027, ZBL1156.90426.
10.1007/s11228-008-0102-z
Web of Science® Google Scholar
27 Zegeye H. and Shahzad N., Approximating common solution of variational inequality problems for two monotone mappings in Banach spaces, Optimization Letters. (2011) 5, no. 4, 691–704, https://doi.org/10.1007/s11590%2D010%2D0235%2D5, 2836047.
10.1007/s11590-010-0235-5
Web of Science® Google Scholar
28 Rockafellar R. T., Monotone operators and proximal point algorithm, Transactions of the American Mathematical Society. (1970) 194, 75–88.
10.1090/S0002-9947-1970-0282272-5
Google Scholar
29 Rockafellar R. T., Characterization of the subdifferentials of convex functions, Pacific Journal of Mathematics. (1966) 17, 497–510, 0193549, ZBL0145.15901.
10.2140/pjm.1966.17.497
Web of Science® Google Scholar
30 Nakajo K. and Takahashi W., Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, Journal of Mathematical Analysis and Applications. (2003) 279, no. 2, 372–379, https://doi.org/10.1016/S0022%2D247X(02)00458%2D4, 1974031, ZBL1035.47048.
10.1016/S0022-247X(02)00458-4
Web of Science® Google Scholar

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An Iteration to a Common Point of Solution of Variational Inequality and Fixed Point-Problems in Banach Spaces

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An Iteration to a Common Point of Solution of Variational Inequality and Fixed Point-Problems in Banach Spaces

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2. Preliminaries

3. Main Result

4. Application

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