# Convergence theorems on generalized equilibrium problems and fixed point problems with applications/Uldistatud tasakaalu ja pusipunkti ulesannete koonduvusteoreemid rakendustega.

1. INTRODUCTION AND PRELIMINARIES

Throughout this paper, we always assume that C is a nonempty, closed and convex subset of a real Hilbert space H. Let A : C [right arrow] H be a nonlinear mapping. Recall the following definitions.

(a) A is said to be monotone if

<Ax - Ay, x - yi> 0; [for all] x, y [member of] C:

(b) A is said to be strongly monotone if there exists a constant [alpha] > 0 such that

<Ax - Ay, x - y> [greater than or equal to] [alpha] [parallel]x- y[[parallel].sup.2], [for all]x, y [member of] C.

For such a case, T is said to be a-strongly-monotone.

(c) A is said to be inverse-strongly monotone if there exists a constant [alpha] > 0 such that

<Ax - Ay, x - y> [greater than or equal to] [alpha] [parallel]Ax- Ay[[parallel].sup.2], [for all]x, y [member of] C.

For such a case, A is said to be a-inverse-strongly monotone.

Recall that the classical variational inequality problem, denoted by VI(C,A), is to find u [member of] C such that

<Au, v - u> [greater than or equal to] 0, [for all]v [member of] C. (1.1)

Given z [member of] H and u [member of] C, we see that the following inequality holds

<u - z, v - u> [greater than or equal to] 0, [for all]v [member of] C,

if and only if u = [P.sub.C]z, where [P.sub.C] denotes the metric projection from H onto C. From the above we see that u [member of] C is a solution to problem (1.1) if and only if u satisfies the following equation:

u = [P.sub.C](u - [rho]Tu), (1.2)

where [rho] > 0 is a constant. This implies that problem (1.1) and problem (1.2) are equivalent. This alternative formula is very important form the numerical analysis point of view.

Let T:C [right arrow] C be a nonlinear mapping. In this paper, we use F(T) to denote the set of fixed points of T. Recall the following definitions.

(d) The mapping T is said to be contractive if there exists a constant [alpha] [member of] (0,1) such that

[parallel]Tx - Ty[parallel] [less than or equal to] [parallel]x - y[parallel], [for all]x, y [member of] C.

(e) The mapping T is said to be nonexpansive if

[parallel]Tx - Ty[parallel] [less than or equal to] [parallel]x - y[parallel], [for all]x, y [member of] C.

(f) T is said to be strictly pseudo-contractive with the coefficient k 2 [0,1) if

[parallel]Tx - Ty[[parallel].sup.2] [less than or equal to] [parallel]x - y[[parallel].sup.2] + k[parallel](I - T)x - (I - T)y[[parallel].sup.2], [for all]x, y [member of] C.

For such a case, T is also said to be a k-strict pseudo-contraction.

(g) T is said to be pseudo-contractive if

<Tx - Ty,x - y> [less than or equal to] [parallel]x - y[[parallel].sup.2], [for all]x, y [member of] C.

Clearly, the class of strict pseudo-contractions falls into the one between the classes of non-expansive mappings and pseudo-contractions.

Let A:C [right arrow] H be an [alpha]-inverse-strongly monotone mapping, F a bi- function of C x C into R, where R denotes the set of real numbers. We consider the following generalized equilibrium problem.

Find x [member of] C such that F(x,y) + <Ax,y - x> [greater than or equal to] 0, [for all]y [member of] C. (1.3)

In this paper, the set of such an x [member of] C is denoted by EP(F,A), i.e.,

EP(F,A) = {x [member of] C:F(x,y) + <Ax,y - x> 0, 8y [member of] Cg.

Next, we give some special cases of problem (1.3).

(i) If A [equivalent to] 0, the zero mapping, then problem (1.3) is reduced to the the following equilibrium problem.

Find x [member of] C such that F(x,y) [greater than or equal to] 0, [for all]y [member of] C. (1.4)

In this paper, the set of such an x [member of] C is denoted by EP(F), i.e.,

EP(F) = {x [member of] C:F(x,y)> [greater than or equal to] 0, [for all]y [member of] C}.

(ii) If F [equivalent to] 0, then problem (1.3) is reduced to the classical variational inequality problem (1.1). Problem (1.3) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, mini-max problems, the Nash equilibrium problem in noncooperative games, and others, see, for instance, [1,12].

To study the equilibrium problems (1.3) and (1.4), we may assume that F satisfies the following conditions.

(A1) F(x,x) = 0 for all x [member of] C,

(A2) F is monotone, i.e., F(x,y) + F(y,x) [less than or equal to] 0 for all x,y [member of] C,

(A3) for each x,y, z [member of] C, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(A4) for each x [member of] C, y [??] F(x,y) is convex and lower semi- continuous.

Recently, Takahashi and Takahashi [23] considered problem (1.4) by an iterative method. To be more precise, they proved the following theorem.

Theorem TT1. Let C be a nonempty closed convex subset of H. Let F be a bi- function from C x C to R satisfying (A1) - (A4) and let S be a nonexpansive mapping of C into H such that F(S) [intersection] EP(f) [not equal to] 0. Let f be a contraction of H into itself and let {[x.sub.n]} and {[u.sub.n]} be sequences generated by [x.sub.1] [member of] H and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where {[[alpha].sub.n]} [member of] [0,1] and {[r.sub.n]} [subset] (0, [infinity]) satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Then {[x.sub.n]} and {[y.sub.n]} converge strongly to z [member of] F(S) [intersection] EP(F), where z = [P.sub.F(S) [intersection] EP(F) f (z).

Very recently, Takahashi and Takahashi [24] further considered the generalized equilibrium problem (1.3). They obtained the following result in a real Hilbert space.

Theorem TT2. Let C be a closed convex subset of a real Hilbert space H and let F:C x C [right arrow] R be a bi-function satisfying (A1), (A2), (A3), and (A4). Let A be an a-inverse- strongly monotone mapping of C into H and let S be a nonexpansive mapping of C into itself such that F(S) [intersection] EP(F,A) [not equal to] 0. Let u [member of] C and [x.sub.1] [member of] C and let {[z.sub.n]} [subsection] C and {[x.sub.n]} [subsection] C be sequences generated by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where {[[alpha].sub.n]} [subset] [0,1], {[[beta].sub.n]} [subset] [0,1], and {[r.sub.n]} [subset] [0,2 [alpha]], satisfy

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Then, {[z.sub.n]} converges strongly to z = [P.sub.F(S) [intersection] EP(F,A)]u.

In this paper, motivated by the research going on in this direction [4,5,7,9,10,13-17,19,20,22-25,27], we introduce a general iterative algorithm for the problem of finding a common element in the set of solutions to problem (1.3) and the set of fixed points of a strict pseudo-contraction. Strong convergence theorems are established in the framework of Hilbert spaces. The results presented in this paper improve and extend the corresponding results reported by many others.

In order to prove our main results, we need the following lemmas.

The following lemma can be found in [1] and [9].

Lemma 1.1. LetC be a nonempty closed convex subset of H and let F:C x C [right arrow] R be a bi-function satisfying (A1) - (A4). Then, for any r > 0 and x [member of] H there exists z [member of] C such that

F(z,y) + 1/r <y - z, z - x> [less than or equal to] 0, [for all]y [member of] C.

Further, define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

for all r > 0 and x [member of] H. Then, the following hold.

(1) [T.sub.r] is single-valued,

(2) [T.sub.r] is firmly nonexpansive, i.e., for any x,y [member of] H,

[parallel][T.sub.r]x - [parallel][T.sub.r]y[[parallel].sup.2] [less than or equal to] <[T.sub.r]x - [T.sub.r]y,x - y>;

(3) F([T.sub.r]) = EP(F),

(4) EP(F) is closed and convex.

Lemma 1.2 ([21]). Let {[x.sub.n]} and {[y.sub.n]} be bounded sequences in a Banach space E and let {[[beta].sub.n]} be a sequence in [0,1] with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Suppose that [x.sub.n+1] = (1 - [[beta].sub.n])[y.sub.n] + [[beta].sub.n][x.sub.n] for all integers n [greater than or equal to] 0 and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Then [lim.sub.n [right arrow] [infinity]] [parallel][y.sub.n] - [x.sub.n][parallel] = 0.

Lemma 1.3 ([3]). Let C be a closed convex subset of a strictly convex Banach space E. Let {Tn:n [member of] N} be a sequence of nonexpansive mappings on C. Suppose that [[intersection].sup.[infinity].sub.n = 1] F([T.sub.n]) is nonempty. Let {[[lambda].sub.n]} be a sequence of positive numbers with [[SIGMA].sup.[infinity].sub.n = 1] = 1. Then a mapping S on C defined by

Sx = [[infinity].summation over (n = 1)] [[lambda].sub.n] [T.sub.n]x

for x [member of] C is well defined, nonexpansive and F(S) = [[intersection].sup.[infinity].sub.n = 1] F([T.sub.n]) holds.

Lemma 1.4 ([2]). Let E be a uniformly convex Banach space, C be a nonempty closed convex subset of E, and S:C [right arrow] C be a nonexpansive mapping. Then I - S is demi-closed at zero.

Lemma 1.5 ([26]). Assume that {[[alpha].sub.n]} is a sequence of nonnegative real numbers such that

[[alpha].sub.n + 1] [less than or equal to] (1 - [[gamma].sub.n]) [[alpha].sub.n] + [[delta].sub.n],

where {[[gamma].sub.n]} is a sequence in (0,1) and {[[delta].sub.n]} is a sequence such that

(i) [[infinity].summation over (n = 1)] [[gamma].sub.n] = [infinity];

(ii) lim [sup.sub.n [right arrow] [infinity] [[delta].sub.n]/[[gamma].sub.n] [less than or equal to] 0 or [[infinity].summation over (n = 1)] [absolute value of [[delta].sub.n]] < [infinity].

Then [lim.sub.n [right arrow] [infinity]] [[alpha].sub.n] = 0.

Lemma 1.6 ([28]). Let C be a nonempty closed convex subset of a real Hilbert space H and T:C [right arrow] H a k-strict pseudo-contraction with a fixed point. Then F(T) is closed and convex. Define S:C [right arrow] H by Sx = kx + (1 - k)Tx for each x [member of] C. Then S is nonexpansive such that F(S) = F(T).

2. MAIN RESULTS

Theorem 2.1. Let C be a nonempty closed convex subset of a Hilbert space H. Let [F.sub.1] and [F.sub.2] be two bi-functions from C x C to R satisfying (A1)-(A4), respectively. Let A:C [right arrow] H be an inverse-strongly monotone mapping and B:C [right arrow] H a [beta]-inverse-strongly monotone mapping. Let T:C [right arrow] C be a k-strict pseudo-contraction with a fixed point. Define a mapping S:C [right arrow] C by Sx = kx + (1 - k)Tx, [for all] x [member of] C. Assume that F = EP([F.sub.1],A) [intersection] EP([F.sub.2],B) [intersection] F(T) [not equal to] 0. Let u [member of] C, [x.sub.1] [member of] C, and {[x.sub.n]} be a sequence generated by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] ([GAMMA])

where {[[alpha].sub.n]}, {[[beta].sub.n]}, and {[[gamma].sub.n]} are sequences in (0,1), r [member of] (0,2[alpha]), and s [member of] (0,2[beta]). If the above control sequences satisfy the following restrictions

(a) [lim.sub.n [right arrow] [infinity]] [[alpha].sub.n] = 0 and [[infinity].summation over (n = 1)] [[alpha].sub.n] = [infinity];

(b) 0 < [liminf.sub.n [right arrow] [infinity]] [[beta].sub.n] [less than or equal to] [limsup.sub.n [right arrow] [infinity]] [[beta].sub.n] < 1;

(c) [lim.sub.n [right arrow] [infinity]] [[gamma].sub.n] = [gamma] [member of] (0,1), then the sequence {[x.sub.n]} defined by the iterative algorithm ([GAMMA]) will converge strongly to z [member of] F, where z = [P.sub.F]u.

Proof. The proof is divided into five steps.

Step 1. Show that the sequence {[x.sub.n]} is bounded.

First, we claim that the mappings I - rA and I - sB are nonexpansive. Indeed, for each x,y [member of] C, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

It follows from the condition r [member of] (0,2[alpha]) that the mapping I - rA is nonexpansive, so is I - sB. Note that [u.sub.n] can be rewritten as [u.sub.n] = [T.sub.r](I - rA)[x.sub.n] and [[upsilon].sub.n] can be rewritten as [[upsilon].sub.n] = [T.sub.s](I - sB)[x.sub.n] for each n [greater than or equal to] 1. Let p [member of] F.

It follows from Lemma 1.1 that

p = [T.sub.r](I - rA)p = [T.sub.s](I - sB)p = Tp.

Notice that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

From Lemma 1.6, we see that S is a nonexpansive mapping with F(T) = F(S). It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Putting [M.sub.1] = max{[parallel][x.sub.1] - p[parallel],[parallel]u - p[parallel]}, we have that [parallel][x.sub.n] - p[parallel] [less than or equal to] [M.sub.1] for all n [greater than or equal to] 1. Indeed, we can easily see that [parallel][x.sub.1] - p[parallel] [less than or equal to] [M.sub.1]. Suppose that [parallel][x.sub.k] - p[parallel] [less than or equal to] [M.sub.1] for some k. Then, we have that

[parallel][x.sub.k+1] - p[parallel] [less than or equal to] [1 - [[alpha].sub.k](1 - [[beta]k)][M.sub.1] + (1 - [[beta]k) [[alpha].sub.k][M.sub.1] = [M.sub.1].

This shows that {[x.sub.n]} is bounded, so are {[y.sub.n]}, {[u.sub.n]}, and {[v.sub.n]}.

Step 2. Show that [x.sub.n+1] - [x.sub.n] [less than or equal to] 0 as n [right arrow] [infinity].

Note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.1)

where [M.sub.2] is an appropriate constant such that [M.sub.2] [greater than or equal to] [sup.sub.n[greater than or equal to] 1] {[parallel][u.sub.n] - [v.sub.n][parallel]}. Put

[[rho].sub.n] = [[alpha].sub.n] u + (1 - [[alpha].sub.n])[y.sub.n], [for all]n [greater than or equal to] 1.

Notice that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

It follows from (2.1) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

This implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [M.sub.3] is an appropriate constant such that [M.sub.3]> [sup.sub.n [greater than or equal to] 1]{[parallel]u - [y.sub.n][parallel]}. This implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

From the conditions (a) and (c), we arrive at

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Thanks to Lemma 1.2, we obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.2)

Notice that

[parallel][x.sub.n+1] - [x.sub.n][parallel] = (1 - [[beta].sub.n])[parallel]S[[rho].sub.n] - [x.sub.n][parallel].

From the condition (b), we see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.3)

Step 3. Show that [x.sub.n] - [Sx.sub.n] [right arrow] 0 as n [right arrow] [infinity].

For each p [member of] F, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.4)

It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.5)

This implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

From the conditions (a)-(c) and (2.3), we see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.6)

It also follows from (2.5) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

From the conditions (a)-(c) and (2.3), we obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.7)

On the other hand, we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.8)

Similarly, we can obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.9)

Note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.10)

Substituting (2.8) and (2.9) into (2.10), we see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.11)

It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

From the conditions (a), (c), (2.3), (2.6), and (2.7), we obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.12)

It also follows from (2.11) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Thanks to the conditions (a), (c), (2.3), (2.6), and (2.7), we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.13)

On the other hand, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In view of the condition (a), (2.12), and (2.13), we obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.14)

It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

From (2.2) and (2.13), we see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.15)

Step 4. Show that [lim.sub.n [right arrow] [infinity]]<u - z, [[rho].sub.n] - z> [less than or equal to] 0, where z = [P.sub.F]u.

First, we show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.16)

To show (2.16), we may choose a subsequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] of {[x.sub.n]} such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.17)

Since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is bounded, we can choose a subsequence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] that converges weakly to q. We may assume without loss of generality that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Noticing (2.15) and apply>ng Lemma 1.4, we obtain that q [member of] F(S) = F(T). Next, we define a mapping R:C [right arrow] C by

Rx = [delta][T.sub.r](I - rA)x + (1 - [delta]) [T.sub.s] (I - sB)x, [for all]x [member of] C,

where (0,1) [??] [delta] = [lim.sub.n [right arrow] [infinity]] [[delta].sub.n]. From Lemma 1.3, we see that R is a nonexpansive mapping with

F(R) = F([T.sub.r](I - rA)) [intersection] F([T.sub.s](I - sB)) = EP([F.sub.1],A) [intersection] FP([F.sub.2],B).

Note that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

From (2.12) and (2.13), we see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

On the other hand, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [M.sub.4] is an appropriate constant such that [M.sub.4]> [sup.sub.n [greater than or equal to] 1] {[parallel][T.sub.r](I - rA)[x.sub.n][parallel] + {[parallel][T.sub.s](I - sB)[x.sub.n][parallel]}. It follows that [lim.sub.n [right arrow] [infinity]] [parallel][Rx.sub.n] - [x.sub.n][parallel] = 0. This implies that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In view of Lemma 1.4, we obtain that q [member of] F(R). That is,

q [member of] EP([F.sub.1],A) [intersection] FP([F.sub.2],B) [intersection] F(T).

It follows from (2.17) that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Notice that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

From (2.14), we conclude that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (2.18)

Step 5. Show that [x.sub.n] [right arrow] z as n [right arrow] [infinity].

Notice that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Since [[alpha].sub.n](1 - [[beta].sub.n]) [right arrow] 0, [[infinity].summation over (n = 1)] [[alpha].sub.n](1 - [[beta].sub.n]) = [infinity] and [lim.sub.n [right arrow] [infinity]] 2<u - z, [[rho].sub.n] - z> [less than or equal to] 0, we get the desired conclusion by Lemma 1.5. This completes the proof.

3. APPLICATIONS

First, we consider the following convex feasibility problem (CFP).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where N [greater than or equal to] 1 is an integer and each [C.sub.i] is assumed to be the set of solutions of an equilibrium problem with the bi-functions [F.sub.i], i = 1,2, ..., N. There is a considerable investigation on CFP in the setting of Hilbert spaces which captures applications in various disciplines such as image restoration [8,11], computer tomography [18], and radiation therapy treatment planning [6]. The following result can be obtained from Theorem 2.1. We, therefore, omit the proof.

Theorem 3.1. Let C be a nonempty closed convex subset of a Hilbert space H. Let [F.sub.1],[F.sub.2], ..., [F.sub.r] be r bi-functions from C x C to R satisfying (A1) - (A4). Let [A.sub.i]:C [right arrow] H be a [k.sub.i]-inverse-strongly monotone mapping for each i [member of] {1,2, ..., r}. Assume that F = [[intersection].sup.r.sub.i = 1] EP([F.sub.i], [A.sub.i]) [not equal to]. Let u [member of] C, [x.sub.1] [member of] C, and {[x.sub.n]} be a sequence generated by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where {[[alpha].sub.n]}, {[[beta].sub.n]}, and {[[gamma].sub.n,i]} are sequences in (0,1), [s.sub.i] [member of] (0,[2k.sub.i]) for each i [member of] {1,2, ...,r}. If the above control sequences satisfy the following restrictions

(a) [lim.sub.n [right arrow] [infinity]] [[alpha].sub.n] = 0 and [[infinity].summation over (n = 1)] [[alpha].sub.n] = [infinity],

(b) 0 < [liminf.sub.n [right arrow] [infinity]] [[beta].sub.n] [less than or equal to] [limsup.sub.n [right arrow] [infinity]] [[beta].sub.n] < 1,

(c) [r.summation over (i = 1) [[gamma].sub.n, i] = 1, [lim.sub.n [right arrow] [infinity]] [[gamma].sub.n,i] = [[gamma].sub.i] [member of] (0,1) for each i [member of] {1,2, ..., r}, then the sequence {[z.sub.n]} converges strongly to z [member of] F, where z = [P.sub.F]u.

Theorem 3.2. Let C be a nonempty closed convex subset of a Hilbert space H. Let [F.sub.1] and [F.sub.2] be two bi-functions from C x C to R satisfying (A1) - (A4), respectively. Let T:C [right arrow] C be a k-strict pseudocontraction with a fixed point. Define a mapping S:C ][right arrow] C by Sx = kx + (1 - k) Tx, [for all] x [member of] C. Assume that F = EP([F.sub.1]) [intersection] EP([F.sub.2]) [intersection] F(T) [not equal to] 0. Let u [member of] C, [x.sub.1] [member of] C, and {[x.sub.n]} be a sequence generated by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where {[[alpha].sub.n]}, {[[beta].sub.n]}, and {[[gamma].sub.n]} are sequences in (0,1), r > 0, and s > 0. If the above control sequences satisfy the following restrictions

(a) [lim.sub.n [right arrow] [infinity]] [[alpha].sub.n] = 0 and [[infinity].summation over (n = 1)] [[alpha].sub.n] = [infinity],

(b) 0 < [liminf.sub.n [right arrow] [infinity]] [[beta].sub.n] [less than or equal to] [limsup.sub.n [right arrow] [infinity]] [[beta].sub.n] < 1,

(c) [lim.sub.n [right arrow] [infinity]] [[gamma].sub.n] = [gamma] [member of] (0,1), then the sequence {[x.sub.n]} converges strongly to z [member of] F, where z = [P.sub.F]u.

Proof. Putting A = B = 0, we see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

From Theorem 2.1, we can draw the desired conclusion immediately.

Theorem 3.3. Let C be a nonempty closed convex subset of a Hilbert space H. Let A:C [right arrow] H be an [[alpha]-inverse-strongly monotone mapping and B:C [right arrow] H a inverse-strongly monotone mapping. Let T:C [right arrow] C be a k-strict pseudo-contraction with a fixed point. Define a mapping S:C ][right arrow] C by Sx = kx + (1 - k)Tx, [for all] x [member of] C. Assume that F = VI(C,A) [intersection] VI(C,B) [intersection] F(T) [not equal to] 0. Let u [member of] C, [x.sub.1] [member of] C, and {[x.sub.n]} be a sequence generated by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where {[[alpha].sub.n]}, {[[alpha].sub.n]}, and {[[alpha].sub.n]} are sequences in (0,1), r [member of] (0,2[alpha]), and s [member of] (0,2[beta]). If the above control sequences satisfy the following restrictions

(a) [lim.sub.n [right arrow] [infinity]] n = 0 and [[infinity].summation over (n = 1)] [[alpha].sub.n] = [infinity],

(b) 0 < [liminf.sub.n [right arrow] [infinity]] [[beta].sub.n] [less than or equal to] [limsup.sub.n [right arrow] [infinity]] [[beta].sub.n] < 1,

(c) [lim.sub.n [right arrow] [infinity]] [[gamma].sub.n] = [gamma] [member of] (0,1), then the sequence {[x.sub.n]} converges strongly to z [member of] F, where z = [P.sub.F]u.

Proof. Putting [F.sub.1] [equivalent to] 0, we see that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

is equivalent to

<[x.sub.n] - [rAx.sub.n] - [u.sub.n],[u.sub.n] - [x.sub.n]> [greater than or equal to] 0, [for all]u [member of] C.

That is, [u.sub.n] = [P.sub.C]([x.sub.n] - [rAx.sub.n]). Similarly, putting [F.sub.2] [equivalent to] 0, we can obtain that [v.sub.n] = [P.sub.C]([x.sub.n] - [sBx.sub.n]). From the proof of Theorem 2.1, we can draw the desired conclusion easily.

Next, we consider another class of nonlinear mappings. strict pseudo- contractions.

Theorem 3.4. Let C be a nonempty closed convex subset of a Hilbert space H. Let [F.sub.1] and [F.sub.2] be two bi-functions from C x C to R satisfying (A1) - (A4), respectively. Let [T.sub.A]:C [right arrow] C be a [k.sub.[alpha]-strict pseudo-contraction and [T.sub.B]:C [right arrow] H a [k.sub.[beta]-strict pseudo-contraction. Let T:C [right arrow] C be a k-strict pseudocontraction with a fixed point. Define a mapping S:C [right arrow] C by Sx = kx + (1 - k) Tx, [for all]x [member of] C. Assume F =EP([F.sub.1], (I - [T.sub.B])) [intersection] EP([F.sub.2], (I - [T.sub.B])) [intersection] F(T) [not equal to] 0. Let u [member of] C, [x.sub.1] [member of] C, and {[x.sub.n]} be a sequence generated by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where {[[alpha].sub.n]}, {[[alpha].sub.n]}, and {[[alpha].sub.n]} are sequences in (0,1), r [member of] (0, (1 - [k.sub.[alpha]])), and s [member of] (0, (1 - [k.sub.[beta]])). If the above control sequences satisfy the following restrictions

(a) [lim.sub.n [right arrow] [infinity]] n = 0 and [[infinity].summation over (n = 1)] [[alpha].sub.n] = [infinity],

(b) 0 < [liminf.sub.n [right arrow] [infinity]] [[beta].sub.n] [less than or equal to] [limsup.sub.n [right arrow] [infinity]] [[beta].sub.n] < 1,

(c) [lim.sub.n [right arrow] [infinity]] [[gamma].sub.n] = [gamma] [member of] (0,1), then the sequence {[x.sub.n]} will converge strongly to z [member of] F, where z = [P.sub.F]u.

Proof. Putting A = I - [T.sub.A] and B = I - [T.sub.B], respectively, we see that A is 1 - [k.sub.[alpha]]/2-inverse-strongly monotone and B is 1 - [k.sub.[[beta]]/2-inverse-strongly monotone. The desired result is not hard to derive from the proof of Theorem 2.1.

doi: 10.3176/proc.2009.3.04

ACKNOWLEDGEMENT

This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).

Received 17 December 2008, revised 6 February 2009, accepted 18 March 2009

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* Corresponding author, smkang@gnu.ac.kr

Xiaolong Qin (a), Shin Min Kangb (b), and Yeol Je Cho (c)

(a) Department of Mathematics, Gyeongsang National University, Jinju 660-701, Korea, ljjhqxl@yahoo.com.cn

(b) Department of Mathematics and the RINS, Gyeongsang National University, Jinju 660-701, Korea

(c) Department of Mathematics Education and the RINS, Gyeongsang National University, Jinju 660-701, Korea, fyjcho@gsnu.ac.kr