We introduce hybrid-iterative schemes for solving a system of the zero-finding problems of maximal monotone operators, the equilibrium problem, and the fixed point problem of weak relatively nonexpansive mappings. We then prove, in a uniformly smooth and uniformly convex Banach space, strong convergence theorems by using a shrinking projection method. We finally apply the obtained results to a system of convex minimization problems.

1. Introduction

Let E be a real Banach space and C a nonempty subset of E. Let E^* be the dual space of E. We denote the value of x^* ∈ E^* at x ∈ E by 〈x^*, x〉. Let T : C → C be a nonlinear mapping. We denote by F(T) the fixed points set of T, that is, F(T) = {x ∈ C : x = Tx}. Let be a set-valued mapping. We denote D(A) by the domain of A, that is, D(A) = {x ∈ E : Ax ≠ ∅} and also denote G(A) by the graph of A, that is, G(A) = {(x, x^*) ∈ E × E^* : x^* ∈ Ax}. A set-valued mapping A is said to be monotone if 〈x^* − y^*, x − y〉 ≥ 0 whenever (x, x^*), (y, y^*) ∈ G(A). It is said to be maximal monotone if its graph is not contained in the graph of any other monotone operators on E. It is known that if A is maximal monotone, then the set A⁻¹(0^*) = {z ∈ E : 0^* ∈ Az} is closed and convex.

The problem of finding a zero point of maximal monotone operators plays an important role in optimizations. This is because it can be reformulated to a convex minimization problem and a variational inequality problem. Many authors have studied the convergence of such problems in various spaces (see, e.g., [1–16]). Initiated by Martinet [17], in a real Hilbert space H, Rockafellar [18] introduced the following iterative scheme: x₁ ∈ H and

()

where {λ_n} ⊂ (0, ∞), J_λ is the resolvent of A defined by J_λ : = J_λA = (I+λA)⁻¹ for all λ > 0, and A is a maximal monotone operator on H. Such an algorithm is called the proximal point algorithm. It was proved that the sequence {x_n} generated by (1.1) converges weakly to an element in A⁻¹(0) provided that liminf _n→∞ λ_n > 0. Recently, Kamimura and Takahashi [19] introduced the following iteration in a real Hilbert space: x₁ ∈ H and

()

where {α_n} ⊂ [0,1] and {λ_n} ⊂ (0, ∞). The weak convergence theorems are also established in a real Hilbert space under suitable conditions imposed on {α_n} and {λ_n}.

In 2004, Kamimura et al. [20] extended the above iteration process to a much more general setting. In fact, they proposed the following algorithm: x₁ ∈ E and

()

where {α_n} ⊂ [0,1], {λ_n} ⊂ (0, ∞), and J_λ : = J_λA = (J+λA)⁻¹J for all λ > 0. They proved, in a uniformly smooth and uniformly convex Banach space, a weak convergence theorem.

Let F : C × C → ℝ, where ℝ is the set of real numbers, be a bifunction. The equilibrium problem is to find x ∈ C such that

()

The solutions set of (1.4) is denoted by EP (F).

For solving the equilibrium problem, we assume that

(A1)
F(x, x) = 0 for all x ∈ C,
(A2)
F is monotone, that is F(x, y) + F(y, x) ≤ 0 for all x, y ∈ C,
(A3)
for all x, y, z ∈ C, limsup _t↓0 F(tz + (1 − t)x, y) ≤ F(x, y),
(A4)
for all x ∈ C, F(x, ·) is convex and lower semi-continuous.

Recently, Takahashi and Zembayashi [21] introduced the following iterative scheme for a relatively nonexpansive mapping T : C → C in a uniformly smooth and uniformly convex Banach space: x₁ ∈ C and

()

where {α_n} ⊂ [0,1] and {r_n} ⊂ (0, ∞). Such an algorithm is called the shrinking projection method which was introduced by Takahashi et al. [22]. They proved that the sequence {x_n} converges strongly to an element in F(T)∩EP (F) under appropriate conditions. The equilibrium problem has been intensively studied by many authors (see, e.g., [23–31]).

Motivated by the previous results, we introduce a hybrid-iterative scheme for finding a zero point of maximal monotone operators (i = 1,2, …, N) which is also a common element in the solutions set of an equilibrium problem for F and in the fixed points set of weak relatively nonexpansive mappings T_i : C → C ( i = 1,2, …). Using the projection technique, we also prove that the sequence generated by a constructed algorithm converges strongly to an element in in a uniformly smooth and uniformly convex Banach space. Finally, we apply our results to a system of convex minimization problems.

2. Preliminaries and Lemmas

In this section, we give some useful preliminaries and lemmas which will be used in the sequel.

Let E be a real Banach space and let U = {x ∈ E : ∥x∥ = 1} be the unit sphere of E. A Banach space E is said to be strictly convex if for any x, y ∈ U,

()

A Banach space E is said to be uniformly convex if, for each ɛ ∈ (0,2], there exists δ > 0 such that for any x, y ∈ U,

()

It is known that a uniformly convex Banach space is reflexive and strictly convex. The function δ : [0,2] → [0,1] which is called the modulus of convexity of E is defined as follows:

()

Then E is uniformly convex if and only if δ(ɛ) > 0 for all ɛ ∈ (0,2]. A Banach space E is said to be smooth if the limit

()

exists for all x, y ∈ U. It is also said to be uniformly smooth if the limit (2.4) is attained uniformly for x, y ∈ U. The duality mapping

is defined by

()

for all x ∈ E. It is also known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on bounded subsets of E (see [32] for more details).

Let E be a smooth Banach space. The function ϕ : E × E → ℝ is defined by

()

for all x, y ∈ E. From the definition of ϕ, we see that

()

for all x, y, z ∈ E.

Let C be a closed and convex subset of E, and let T be a mapping from C into itself. A point p in C is said to be an asymptotic fixed point of T [33] 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 is said to be relatively nonexpansive [33, 34] if and ϕ(p, Tx) ≤ ϕ(p, x) for all p ∈ F(T) and x ∈ C. A point p in C is said to be a strong asymptotic fixed point of T if C contains a sequence {x_n} which converges strongly to p such that lim _n→∞ ∥x_n − Tx_n∥ = 0. The set of strong asymptotic fixed points of T will be denoted by . A mapping T is said to be weak relatively nonexpansive [35] if and ϕ(p, Tx) ≤ ϕ(p, x) for all p ∈ F(T) and x ∈ C. It is obvious by definition that the class of weak relatively nonexpansive mappings contains the class of relatively nonexpansive mappings. Indeed, for any mapping T : C → C, we see that . Therefore, if T is a relatively nonexpansive mapping, then .

Nontrivial examples of weak relatively nonexpansive mappings which are not relatively nonexpansive can be found in [36].

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 [37], is a mapping Π_C : E → C, that assigns to an arbitrary point x ∈ E the minimum point of the function ϕ(y, x), that is,

, where

is the solution to the minimization problem

()

In a Hilbert space, Π_C is coincident with the metric projection denoted by P_C.

Lemma 2.1 (see [38].)Let E be a uniformly convex and smooth Banach space and let {x_n}, {y_n} be two sequences in E. If lim _n→∞ ϕ(x_n, y_n) = 0 and either {x_n} or {y_n} is bounded, then lim _n→∞ ∥x_n − y_n∥ = 0.

Lemma 2.2 (see [37], [38].)Let C be a nonempty, closed, and convex subset of a smooth, strictly convex and reflexive Banach space E, let x ∈ E and let z ∈ C. Then z = Π_C(x) if and only if 〈y − z, Jx − Jz〉 ≤ 0 for all y ∈ C.

Lemma 2.3 (see [37], [38].)Let C be a nonempty, closed, and convex subset of a smooth, strictly convex, and reflexive Banach space E. Then

()

Lemma 2.4 (see [39].)Let E be a smooth and strictly convex Banach space, and let C be a nonempty, closed, and convex subset of E. Let T be a mapping from C into itself such that F(T) is nonempty and ϕ(u, Tx) ≤ ϕ(u, x) for all (u, x) ∈ F(T) × C. Then F(T) is closed and convex.

Let E be a reflexive, strictly convex, and smooth Banach space. It is known that is maximal monotone if and only if R(J + λA) = E^* for all λ > 0, where R(B) stands for the range of B.

Define the resolvent of A by J_λA = (J+λA)⁻¹J for all λ > 0. It is known that J_λA is a single-valued mapping from E to D(A) and A⁻¹(0^*) = F(J_λA) for all λ > 0. For each λ > 0, the Yosida approximation of A is defined by

()

for all x ∈ E. We know that A_λ(x) ∈ A(J_λA(x)) for all λ > 0 and x ∈ E.

Lemma 2.5 (see [5].)Let E be a smooth, strictly convex, and reflexive Banach space, let A ⊂ E × E^* be a maximal monotone operator with A⁻¹(0^*) ≠ ∅, and let J_λA = (J + λA) ⁻¹J for each λ > 0. Then

()

for all λ > 0, p ∈ A⁻¹(0^*), and x ∈ E.

Lemma 2.6 (see[40]). Let C be a closed and convex subset of a smooth, strictly convex, and reflexive Banach space E, let F be a bifunction from C × C to ℝ satisfying (A1)–(A4), and let r > 0 and x ∈ E. Then, there exists z ∈ C such that

()

Lemma 2.7 (see [41].)Let C be a closed and convex subset of a uniformly smooth, strictly convex, and reflexive Banach space E, and let F be a bifunction from C × C to ℝ satisfying (A1)–(A4). For all r > 0 and x ∈ E, define the mapping T_r : E → C as follows:

()

Then, the following holds:

(1)
T_r is single-valued;
(2)
T_r is a firmly nonexpansive-type mapping [42], that is, for all x, y ∈ E,
()
(3)
F(T_r) = EP (F);
(4)
EP (F) is closed and convex.

Lemma 2.8 (see [41].)Let C be a closed and convex subset of a smooth, strictly, and reflexive Banach space E, let F be a bifunction from C × C to ℝ satisfying (A1)–(A4), let r > 0. Then

()

for all x ∈ E and p ∈ F(T_r).

3. Strong Convergence Theorems

In this section, we are now ready to prove our main theorem.

Theorem 3.1. Let E be a uniformly smooth and uniformly convex Banach space, and let C be a nonempty, closed and convex subset of E. Let (i = 1,2, …, N) be maximal monotone operators, let F : C × C → ℝ be a bifunction, and let T_i : C → C ( i = 1,2, …) be weak relatively nonexpansive mappings such that . Let be the sequence such that lim _n→∞ e_n = 0. Define the sequence in C as follows:

()

for each i = 1,2, …, N and liminf _n→∞ r_n > 0, then the sequence {x_n} converges strongly to q = Π_ℱ(x₁).

Proof. We split the proof into several steps as follows.

Step 1. ℱ ⊂ C_n for all n ≥ 1.

From Lemma 2.4, we know that is closed and convex. From Lemma 2.7(4), we also know that EP (F) is closed and convex. On the other hand, since A_i (i = 1,2, …, N) are maximal monotone, are closed and convex for each i = 1,2, …, N; consequently, is closed and convex. Hence ℱ is a nonempty, closed, and convex subset of C.

We next show that C_n is closed and convex for all n ≥ 1. Obviously, C₁ = C is closed and convex. Now suppose that C_k is closed and convex for some k ∈ ℕ. Then, for each z ∈ C_k and i ≥ 1, we see that ϕ(z, T_iu_k) ≤ ϕ(z, x_k) is equivalent to

()

By the construction of the set C_k+1, we see that

()

Hence, C_k+1 is closed and convex. This shows, by induction, that C_n is closed and convex for all n ≥ 1. It is obvious that ℱ ⊂ C₁ = C. Now, suppose that ℱ ⊂ C_k for some k ∈ ℕ. For any p ∈ ℱ, by Lemmas 2.5 and 2.8, we have

()

This shows that ℱ ⊂ C_k+1. By induction, we can conclude that ℱ ⊂ C_n for all n ≥ 1.

Step 2. lim _n→∞ ϕ(x_n, x₁) exists.

From and , we have

()

From Lemma 2.3, for any p ∈ ℱ ⊂ C_n, we have

()

Combining (3.5) and (3.6), we conclude that lim _n→∞ ϕ(x_n, x₁) exists.

Step 3. lim _n→∞ ∥J(T_iy_n) − J(x_n + e_n)∥ = 0.

Since for m > n ≥ 1, by Lemma 2.3, it follows that

()

Letting m, n → ∞, we have ϕ(x_m, x_n) → 0. By Lemma 2.1, it follows that ∥x_m − x_n∥ → 0 as m, n → ∞. Therefore, {x_n} is a Cauchy sequence. By the completeness of the space E and the closedness of C, we can assume that x_n → q ∈ C as n → ∞. In particular, we obtain that

()

Since e_n → 0, we have

()

Since

, for each i ≥ 1,

()

Since E is uniformly smooth, J is uniformly norm-to-norm continuous on bounded sets. It follows from (3.9) and by the boundedness of {x_n} that

()

for all i = 1,2, …. So from Lemma 2.1, we have

()

and, since e_n → 0, therefore

()

for all i = 1,2, …. Since J is uniformly norm-to-norm continuous on bounded subsets of E,

()

for all i = 1,2, ….

Step 4. lim _n→∞ ∥T_iu_n − u_n∥ = 0 for all i = 1,2, ….

Denote that for each i ∈ {1,2, …, N} and for each n ≥ 1. We note that for each n ≥ 1.

To this end, we will show that

()

for all i = 1,2, …, N.

For any p ∈ ℱ, by (3.4), we see that

()

Since p ∈ ℱ, by Lemma 2.5 and (3.16), it follows that

()

From (3.13) and (3.14), we get that

. So we obtain that

()

Again, since p ∈ ℱ,

()

From (3.13) and (3.14), we get that

()

It also follows that

()

Continuing in this process, we can show that

()

So, we now conclude that

()

for each i = 1,2, …, N. By the uniform norm-to-norm continuity of J, we also have

()

for each i = 1,2, …, N. Using (3.23), it is easily seen that

()

From

, by Lemma 2.8, it follows that

()

This implies that lim _n→∞ ϕ(u_n, y_n) = 0 and hence

()

Combining (3.13), (3.25), and (3.27), we obtain that

()

for all i ≥ 1.

Step 5. .

Since x_n → q and e_n → 0, x_n + e_n → q. So from (3.25) and (3.27), we have u_n → q. Note that T_i (i = 1,2, …) are weak relatively nonexpansive. Using (3.28), we can conclude that for all i ≥ 1. Hence .

Step 6. .

Noting that for each i = 1,2, …, N, we obtain that

()

From (3.24) and

, we have

()

We note that

for each i = 1,2, …, N. If (w, w^*) ∈ G(A_i) for each i = 1,2, …, N, then it follows from the monotonicity of A_i that

()

We see that

for each i = 1,2, …, N. Thus, from (3.30) and (3.31), we have

()

By the maximality of A_i, it follows that

for each i = 1,2, …, N. Therefore,

Step 7. q ∈ EP (F).

From , we have

()

By (A2), we have

()

Note that ∥Ju_n − Jy_n∥/r_n → 0 since liminf _n→∞ r_n > 0. From (A4) and u_n → q, we get F(y, q) ≤ 0 for all y ∈ C. For 0 < t < 1 and y ∈ C, define that y_t = ty + (1 − t)q. Then y_t ∈ C, which implies that F(y_t, q) ≤ 0. From (A1), we obtain that 0 = F(y_t, y_t) ≤ tF(y_t, y) + (1 − t)F(y_t, q) ≤ tF(y_t, y). Thus, F(y_t, y) ≥ 0. From (A3), we have F(q, y) ≥ 0 for all y ∈ C. Hence, q ∈ EP (F). From Steps 5, 6, and 7, we now can conclude that q ∈ ℱ.

Step 8. q = Π_ℱ(x₁).

From , we have

()

Since ℱ ⊂ C_n, we also have

()

Letting n → ∞ in (3.36), we obtain that

()

This shows that q = Π_ℱ(x₁) by Lemma 2.2. We thus complete the proof.

As a direct consequence of Theorem 3.1, we can also apply to a system of convex minimization problems.

Theorem 3.2. Let E be a uniformly smooth and uniformly convex Banach space, and let C be a nonempty, closed, and convex subset of E. Let f_i : E → (−∞, ∞] (i = 1,2, …, N) be proper lower semicontinuous convex functions, let F : C × C → ℝ be a bifunction, and let T_i : C → C (i = 1,2, …) be weak relatively nonexpansive mappings such that . Let be the sequence such that lim _n→∞ e_n = 0. Define the sequence in C as follows:

()

for each i = 1,2, …, N and liminf _n→∞ r_n > 0, then the sequence {x_n} converges strongly to q = Π_ℱ(x₁).

Proof. By Rockafellar′s theorem [43, 44], ∂f_i are maximal monotone operators for each i = 1,2, …, N. Let λⁱ > 0 for each i = 1,2, …, N. Then, if and only if

()

which is equivalent to

()

Using Theorem 3.1, we thus complete the proof.

If E = H is a real Hilbert space, we then obtain the following results.

Corollary 3.3. Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let A_i : H → 2^H (i = 1,2, …, N) be maximal monotone operators, let F : C × C → ℝ be a bifunction, and let T_i : C → C (i = 1,2, …) be weak relatively nonexpansive mappings such that . Let be the sequence such that lim _n→∞ e_n = 0. Define the sequence in C as follows:

()

for each i = 1,2, …, N and liminf _n→∞ r_n > 0, then the sequence {x_n} converges strongly to q = P_ℱ(x₁).

Corollary 3.4. Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let f_i : H → (−∞, ∞] (i = 1,2, …, N) be proper lower semi-continuous convex functions, let F : C × C → ℝ be a bifunction, and let T_i : C → C (i = 1,2, …) be weak relatively nonexpansive mappings such that . Let be the sequence such that lim _n→∞ e_n = 0. Define the sequence in C as follows:

()

for each i = 1,2, …, N and liminf _n→∞ r_n > 0, then the sequence {x_n} converges strongly to q = P_ℱ(x₁).

Remark 3.5. Using the shrinking projection method, we can construct a hybrid-proximal point algorithm for solving a system of the zero-finding problems, the equilibrium problems, and the fixed point problems of weak relatively nonexpansive mappings.

Remark 3.6. Since every relatively nonexpansive mapping is weak relatively nonexpansive, our results also hold if T_i : C → C (i = 1,2, …) are relatively nonexpansive mappings.

Acknowledgments

The authors thank the editor and the referee(s) for valuable suggestions. The first author was supported by the Thailand Research Fund, the Commission on Higher Education, and the University of Phayao under Grant MRG5380202. The second and the third authors wish to thank the Thailand Research Fund and the Centre of Excellence in Mathematics, Thailand.

References

1 Cholamjiak P., Cho Y. J., and Suantai S., Composite iterative schemes for maximal monotone operators in reflexive Banach spaces, Fixed Point Theory and Applications. (2011) 2011, no. article 7, 10, 2820311.
10.1186/1687-1812-2011-7
Google Scholar
2 Cho Y. J., Kang S. M., and Zhou H., Approximate proximal point algorithms for finding zeroes of maximal monotone operators in Hilbert spaces, Journal of Inequalities and Applications. (2008) 2008, 10, 598191, https://doi.org/10.1155/2008/598191, 2395541, ZBL1153.47052.
10.1155/2008/598191
Google Scholar
3 Ceng L.-C., Liou Y.-C., and Naraghirad E., Iterative approaches to find zeros of maximal monotone operators by hybrid approximate proximal point methods, Fixed Point Theory and Applications. (2011) 2011, 18, 282171, 2739901, ZBL1206.47065.
10.1155/2011/282171
Google Scholar
4 Güler O., On the convergence of the proximal point algorithm for convex minimization, SIAM Journal on Control and Optimization. (1991) 29, no. 2, 403–419, 1092735, https://doi.org/10.1137/0329022.
10.1137/0329022
Web of Science® Google Scholar
5 Kohsaka F. and Takahashi W., Strong convergence of an iterative sequence for maximal monotone operators in a Banach space, Abstract and Applied Analysis. (2004) no. 3, 239–249, https://doi.org/10.1155/S1085337504309036, 2058504, ZBL1064.47068.
10.1155/S1085337504309036
Google Scholar
6 Marino G. and Xu H.-K., Convergence of generalized proximal point algorithms, Communications on Pure and Applied Analysis. (2004) 3, no. 4, 791–808, 2106300, https://doi.org/10.3934/cpaa.2004.3.791, ZBL1095.90115.
10.3934/cpaa.2004.3.791
Web of Science® Google Scholar
7 Onjai-uea N. and Kumam P., A new iterative scheme for equilibrium problems, fixed point problems for nonexpansive mappings and maximal monotone operators, Fixed Point Theory and Applications. In press.
Google Scholar
8 Qin X., Kang S. M., and Cho Y. J., Approximating zeros of monotone operators by proximal point algorithms, Journal of Global Optimization. (2010) 46, no. 1, 75–87, https://doi.org/10.1007/s10898%2D009%2D9410%2D6, 2566137, ZBL1181.47070.
10.1007/s10898-009-9410-6
Web of Science® Google Scholar
9 Saewan S. and Kumam P., A hybrid iterative scheme for a maximal monotone operator and two countable families of relatively quasi-nonexpansive mappings for generalized mixed equilibrium and variational inequality problems, Abstract and Applied Analysis. (2010) 2010, 31, 123027, https://doi.org/10.1155/2010/123027, 2735003, ZBL1204.65062.
10.1155/2010/123027
Google Scholar
10 Sunthrayuth P. and Kumam P., A system of generalized mixed equilibrium problems, maximal monotone operators, and fixed point problems with application to optimization problems, Abstract and Applied Analysis. (2012) 2012, 39, 316276, https://doi.org/10.1155/2012/316276.
10.1155/2012/316276
Google Scholar
11 Wattanawitoon K. and Kumam P., Hybrid proximal-point methods for zeros of maximal monotone operators, variational inequalities and mixed equilibrium problems, International Journal of Mathematics and Mathematical Sciences. (2011) 2011, 31, 174796, https://doi.org/10.1155/2011/174796, 2771219, ZBL1216.47101.
10.1155/2011/174796
Google Scholar
12 Wattanawitoon K. and Kumam P., Generalized mixed equilibrium problems for maximal monotone operators and two relatively quasi-nonexpansive mappings, Thai Journal of Mathematics. (2011) 9, no. 1, 171–195, 2833762.
Google Scholar
13 Wattanawitoon K. and Kumam P., A new iterative scheme for generalized mixed equilibrium, variational inequality problems and a zero point of maximal monotone operators, Journal of Applied Mathematics. (2012) 2012, 27, 692829, https://doi.org/10.1155/2012/692829.
10.1155/2012/692829
Google Scholar
14 Witthayarat U., Cho Y. J., and Kumam P., Convergence of an iterative algorithm for common solutions for zeros of maximal accretive operator with applications, Journal of Applied Mathematics. (2012) 2012, 17, 185104.
10.1155/2012/185104
Google Scholar
15 Yao Y. and Noor M. A., On convergence criteria of generalized proximal point algorithms, Journal of Computational and Applied Mathematics. (2008) 217, no. 1, 46–55, 2427430, https://doi.org/10.1016/j.cam.2007.06.013, ZBL1147.65049.
10.1016/j.cam.2007.06.013
Web of Science® Google Scholar
16 Yao Y. and Shahzad N., Strong convergence of a proximal point algorithm with general errors, Optimization Letters. (2012) 6, no. 4, 621–628.
10.1007/s11590-011-0286-2
Web of Science® Google Scholar
17 Martinet B., Régularisation d′inéquations variationnelles par approximations successives, 1970, 4, no. R-3, 154–158, 0298899, ZBL0215.21103.
Google Scholar
18 Rockafellar R. T., Monotone operators and the proximal point algorithm, SIAM Journal on Control and Optimization. (1976) 14, no. 5, 877–898, 0410483, https://doi.org/10.1137/0314056, ZBL0358.90053.
10.1137/0314056
Web of Science® Google Scholar
19 Kamimura S. and Takahashi W., Approximating solutions of maximal monotone operators in Hilbert spaces, Journal of Approximation Theory. (2000) 106, no. 2, 226–240, 1788273, https://doi.org/10.1006/jath.2000.3493, ZBL0992.47022.
10.1006/jath.2000.3493
Web of Science® Google Scholar
20 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
21 Takahashi W. and Zembayashi K., Strong convergence theorem by a new hybrid method for equilibrium problems and relatively nonexpansive mappings, Fixed Point Theory and Applications. (2008) 2008, 11, 528476, 2395311, ZBL1187.47054.
10.1155/2008/528476
Google Scholar
22 Takahashi W., Takeuchi Y., and Kubota R., Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, Journal of Mathematical Analysis and Applications. (2008) 341, no. 1, 276–286, 2394082, https://doi.org/10.1016/j.jmaa.2007.09.062, ZBL1134.47052.
10.1016/j.jmaa.2007.09.062
Web of Science® Google Scholar
23 Cholamjiak P., A hybrid iterative scheme for equilibrium problems, variational inequality problems, and fixed point problems in Banach spaces, Fixed Point Theory and Applications. (2009) 2009, 18, 719360, 2511308, ZBL1167.65379.
10.1155/2009/719360
Google Scholar
24 Cholamjiak P. and Suantai S., Convergence analysis for a system of equilibrium problems and a countable family of relatively quasi-nonexpansive mappings in Banach spaces, Abstract and Applied Analysis. (2010) 2010, 17, 141376, https://doi.org/10.1155/2010/141376, 2680415, ZBL1206.47069.
10.1155/2010/141376
Google Scholar
25 Cholamjiak W. and Suantai S., A hybrid method for a countable family of multivalued maps, equilibrium problems, and variational inequality problems, Discrete Dynamics in Nature and Society. (2010) 2010, 14, 349158, https://doi.org/10.1155/2010/349158, 2660692, ZBL1194.90104.
10.1155/2010/349158
Web of Science® Google Scholar
26 Cholamjiak W. and Suantai S., A new hybrid algorithm for a countable family of quasi-nonexpansive mappings and equilibrium problems, Journal of Nonlinear and Convex Analysis. (2011) 12, no. 2, 381–398, 2859035, ZBL1228.47059.
Web of Science® Google Scholar
27 Kumam P., A hybrid approximation method for equilibrium and fixed point problems for a monotone mapping and a nonexpansive mapping, Nonlinear Analysis. Hybrid Systems. (2008) 2, no. 4, 1245–1255, https://doi.org/10.1016/j.nahs.2008.09.017, 2479367, ZBL1163.49003.
10.1016/j.nahs.2008.09.017
Web of Science® Google Scholar
28 Kumam P., A new hybrid iterative method for solution of equilibrium problems and fixed point problems for an inverse strongly monotone operator and a nonexpansive mapping, Journal of Applied Mathematics and Computing. (2009) 29, no. 1-2, 263–280, 2472110, https://doi.org/10.1007/s12190%2D008%2D0129%2D1, ZBL1220.47102.
10.1007/s12190-008-0129-1
Google Scholar
29 Qin X., Cho Y. J., and Kang S. M., Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces, Journal of Computational and Applied Mathematics. (2009) 225, no. 1, 20–30, 2490167, https://doi.org/10.1016/j.cam.2008.06.011, ZBL1165.65027.
10.1016/j.cam.2008.06.011
Web of Science® Google Scholar
30 Shehu Y., Iterative approximation method for finite family of relatively quasi nonexpansive mappings and systems of equilibrium problems, Journal of Global Optimization. (2011) 51, no. 1, 69–78, https://doi.org/10.1007/s10898%2D010%2D9619%2D4, 2825486.
10.1007/s10898-010-9619-4
Web of Science® Google Scholar
31 Shehu Y., A new hybrid iterative scheme for countable families of relatively quasi-nonexpansive mappings and system of equilibrium problems, International Journal of Mathematics and Mathematical Sciences. (2011) 2011, 23, 131890, https://doi.org/10.1155/2011/131890, 2802444, ZBL1221.47124.
10.1155/2011/131890
Google Scholar
32 Takahashi W., Nonlinear Functional Analysis, Fixed Point Theory and Its Application, 2000, Yokohama Publishers, Yokohama, Japan, 1864294.
Google Scholar
33 Butnariu D., Reich S., and Zaslavski A. J., Asymptotic behavior of relatively nonexpansive operators in Banach spaces, Journal of Applied Analysis. (2001) 7, no. 2, 151–174, https://doi.org/10.1515/JAA.2001.151, 1875804, ZBL1010.47032.
10.1515/JAA.2001.151
Google Scholar
34 Butnariu D., Reich S., and Zaslavski A. J., Weak convergence of orbits of nonlinear operators in reflexive Banach spaces, Numerical Functional Analysis and Optimization. An International Journal. (2003) 24, no. 5-6, 489–508, 1995998, https://doi.org/10.1081/NFA%2D120023869, ZBL1071.47052.
10.1081/NFA-120023869
Web of Science® Google Scholar
35 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
36 Su Y., Xu H.-K., and Zhang X., Strong convergence theorems for two countable families of weak relatively nonexpansive mappings and applications, Nonlinear Analysis. Theory, Methods & Applications. (2010) 73, no. 12, 3890–3906, https://doi.org/10.1016/j.na.2010.08.021, 2728563, ZBL1215.47091.
10.1016/j.na.2010.08.021
Web of Science® Google Scholar
37 Alber Y. I., A. G. Kartsatos, Metric and generalized projection operators in Banach spaces: properties and applications, Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, 1996, 178, Marcel Dekker, New York, NY, USA, 15–50, 1386667, ZBL0883.47083.
Google Scholar
38 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
39 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, 2142300, https://doi.org/10.1016/j.jat.2005.02.007, ZBL1071.47063.
10.1016/j.jat.2005.02.007
Web of Science® Google Scholar
40 Blum E. and Oettli W., From optimization and variational inequalities to equilibrium problems, The Mathematics Student. (1994) 63, no. 1–4, 123–145, 1292380, ZBL0888.49007.
Google Scholar
41 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
42 Kohsaka F. and Takahashi W., Existence and approximation of fixed points of firmly nonexpansive-type mappings in Banach spaces, SIAM Journal on Optimization. (2008) 19, no. 2, 824–835, 2448915, https://doi.org/10.1137/070688717, ZBL1168.47047.
10.1137/070688717
Web of Science® Google Scholar
43 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
44 Rockafellar R. T., On the maximal monotonicity of subdifferential mappings, Pacific Journal of Mathematics. (1970) 33, 209–216, 0262827, ZBL0199.47101.
10.2140/pjm.1970.33.209
Web of Science® Google Scholar

Citing Literature

All articles

Convergence Theorems for Maximal Monotone Operators, Weak Relatively Nonexpansive Mappings and Equilibrium Problems

Abstract

1. Introduction

2. Preliminaries and Lemmas

3. Strong Convergence Theorems

Acknowledgments

References

Citing Literature

References

Information

About Wiley Online Library

Help & Support

Opportunities

Connect with Wiley

Convergence Theorems for Maximal Monotone Operators, Weak Relatively Nonexpansive Mappings and Equilibrium Problems

Abstract

1. Introduction

2. Preliminaries and Lemmas

3. Strong Convergence Theorems

Acknowledgments

References

Citing Literature

References

Related

Information