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The hybrid algorithm for the system of mixed equilibrium problems, the general system of finite variational inequalities and common fixed points for nonexpansive semigroups and strictly pseudo-contractive mappings

Fixed Point Theory and Applications20122012:84

https://doi.org/10.1186/1687-1812-2012-84

Received: 31 October 2011

Accepted: 18 May 2012

Published: 18 May 2012

Abstract

In this article, we introduce a new iterative algorithm by the shrinking projection method for finding a common element of the set of solutions of generalized mixed equilibrium problems, the set of common solutions of general system of finite variational inequalities, the set of solutions of fixed points for nonexpansive semigroups and the set of common fixed points for an infinite family of strictly pseudo-contractive mappings in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above four sets under some mind conditions. Our results improve and extend the corresponding recent results in literature work.

1 Introduction

Throughout this article, we assume that C is a closed convex subset of a real Hilbert space H with inner product and norm are denoted by 〈.,.〉 and ., respectively.

Let A, B: CH be two mappings. We consider the following problem of finding (x*, y*) C × C such that
λ A y * + x * - y * , x - x * 0 , x C , μ B x * + y * - x * , x - y * 0 , x C ,
(1)
which is called a general system of variational inequalities, where λ ≥ 0 and μ ≥ 0 are two constants. The set of solution of (1) is denoted by SVI(C, A, B). In particular, if A = B, then problem (1) reduces to finding (x*, y*) C × C such that
λ A y * + x * - y * , x - x * 0 , x C , μ A x * + y * - x * , x - y * 0 , x C ,
(2)

which is defined by Verma [1] (see also Verma [2]), and is called the new general system of variational inequalities. Further, if we set B = 0, then problem (1) reduces to the classical variational inequality VI(C, A) which was originally introduced and studied by Stampacchia [3] in 1964.

By the system of variational inequality problems above, we extend into the general system of finite variational inequalities is to find ( x 1 * , x 2 * , , x M * ) C × C × × C and is defined by
λ M A M x M * + x 1 * - x M * , x - x 1 * 0 , x C , λ M - 1 A M - 1 x M - 1 * + x M * - x M - 1 * , x - x M * 0 , x C , λ 2 A 2 x 2 * + x 3 * - x 2 * , x - x 3 * 0 , x C , λ 1 A 1 x 1 * + x 2 * - x 1 * , x - x 2 * 0 , x C ,
(3)

where { A l } l = 1 M : C H is a family of mappings, λ l ≥ 0, l {1,2, ..., M}. The set of solution of (3) is denoted by GSVI(C, A l ). In particular, if M = 2 , A 1 = B , A 2 = A , λ 1 = μ , λ 2 = λ , x 1 * = x * , and x 2 * = y * , then the problem (3) is reduced to the problem (1).

Recall that a mapping T : CC is said to be a k-strict pseudo-contraction (see also [4]) if there exists 0 ≤ k < 1 such that
T x - T y 2 x - y 2 + k ( I - T ) x - ( I - T ) y 2 , x , y C ,
where I denotes the identity operator on C (see also [5]). If k = 0, a mapping T : CC is said to be nonexpansive[6], that is,
T x - T y x - y , x , y C .
If k = 1, a mapping T : CC is said to be pseudo-contraction, that is,
T x - T y 2 x - y 2 + ( I - T ) x - ( I - T ) y 2 , x , y C .

Clearly, the class of k-strict pseudo-contraction falls into the one between classes of nonexpansive mappings and pseudo-contraction mappings. We denote the set of fixed points of T by F(T).

Let = {F k }k Γbe a countable family of bifunctions from C × C to , where is the set of real numbers and Γ is an arbitrary index set. Let φ : C {+∞} be a proper extended real-valued function. The system of mixed equilibrium problems is to find x C such that
F k ( x , y ) + φ ( y ) φ ( x ) , k Γ , k C .
(4)
The set of solutions of (4) is denoted by SMEP(F k , φ), that is
S M E P ( F k , φ ) = { x C : F k ( x , y ) + φ ( y ) φ ( x ) , k Γ , y C } .
(5)
If Γ is a singleton, the problem (4) reduces to find the following mixed equilibrium problem (see also Flores-Bazán [7]). For finding x C such that
F ( x , y ) + φ ( y ) φ ( x ) , y C .
(6)
The set of solutions of (6) is denoted by MEP(F, φ). Combettes and Hirstoaga [8] introduced the following system of equilibrium problems. For finding x C such that,
F k ( x , y ) 0 , k Γ , y C .
(7)
The set of solutions of (7) is denoted by SEP(), that is,
S E P ( ) = { x C : F k ( x , y ) 0 , k Γ , y C } .
(8)
If Γ is a singleton, the problem (7) becomes the following equilibrium problem. For finding x C such that
F ( x , y ) 0 , y C .
(9)

The set of solution of (9) is denoted by EP(F). The mixed equilibrium problems include fixed point problems, variational inequality problems, optimization problems, Nash equilibrium problems, noncooperative games, economics and the equilibrium problem as special cases (see [919]). In the last two decades, many articles have appeared in the literature on the existence of solutions of equilibrium problems; see, for example [13] and references therein. Some solution methods have been proposed to solve the mixed equilibrium problems; see, for example, (see [1114, 1624]) and references therein.

A family S = { S ( s ) : 0 s < } of mappings of C into itself is called a nonexpansive semigroup on C if it satisfies the following conditions:
  1. (i)

    S(0)x = x for all x C;

     
  2. (ii)

    S(s + t) = S(s) S(t) for all s, t ≥ 0;

     
  3. (iii)

    S(s)x - S(s)yx - y for all x, y C and s ≥ 0;

     
  4. (iv)

    for all x C, s S(s)x is continuous.

     

We denote by F ( S ) the set of all common fixed points of S = { S ( s ) : s 0 } , i.e., F ( S ) = s 0 F ( S ( s ) ) . It is well known that F ( S ) is closed and convex (see also [25, 26]).

In 2011, Shehu [21] introduced a new iterative scheme by hybrid method for finding a common element of the set of common fixed points of an infinite family of k-strictly pseudocontractive mappings and the set of common solutions to a system of generalized mixed equilibrium problems and the set of solution of variational inequality problems in Hilbert spaces. Starting with an arbitrary x 0 C , C 1 , i = C , C 1 = i = 1 C 1 , i , x 1 = P C 1 x 0 define sequences {x n }, {w n }, {u n }, {z n }, and {yn,i} as follows:
z n = T r n ( F 1 , φ 1 ) ( x n - r n A x n ) y n = T λ n ( F 2 , φ 2 ) ( z n - λ n B z n ) w n = P C ( u n - s n D u n ) y n , i = α n , i w n + ( 1 - α n , i ) T i w n , n 1 , C n + 1 , i = { z C n , i : y n , i - z x n - z } , n 1 , C n + 1 = i = 1 C n + 1 , i , x n + 1 = P C n + 1 x 0 , n 1 ,
(10)

where T i be a k i -strictly pseudocontractive mapping and for some 0 ≤ k i < 1, A, B are α, β-inverse-strongly monotone mappings of C into H, respectively. He proved that if the sequences {α n,i }, {r n }, {s n }, and {λ n } of parameters satisfies appropriate conditions, then {x n } is generated by (10) converges strongly to PΩx0, where PΩ is metric projection on H in to Ω : = M E P ( F 1 , φ 1 ) M E P ( F 2 , φ 2 ) V I ( C , A ) ( i = 1 F ( T i ) ) . For using the hybrid method, we can see [2729].

In this article, motivated by the above results, we present a new iterative algorithm for finding a common element of the set of solutions for a system of mixed equilibrium problems, the set of common solutions of general system of finite variational inequality problems, the set of solutions of fixed points for nonexpansive semigroup mappings and the set of common fixed points for an infinite family of strictly pseudo-contractive mappings in a real Hilbert space. Then, we prove strong convergence theorem under some mind conditions. The results presented in this article extend and improve the results of Shehu [21] and many authors.

2 Preliminaries

Let H be a real Hilbert space with norm and inner product 〈,〉, respectively. Let C be a closed convex subset of H. The sequence {x n } is a sequence in H, x n x means {x n } converges weakly to x and x n x means {x n } converges strongly to x. In a real Hilbert space H, we have
x - y 2 = x 2 - y 2 - 2 x - y , y
(11)
and
λ x + ( 1 - λ ) y 2 = λ x 2 + ( 1 - λ ) y 2 - λ ( 1 - λ ) x - y 2 , x , y H ,
(12)
and λ . For every point x H, there exists a unique nearest point in C, denoted by P C x, such that
x - P C x x - y , y C .
(13)
P C is called the metric projection of H onto C. It is well known that P C is a nonexpansive mapping of H onto C and satisfies
x - y , P C x - P C y P C x - P C y 2 , x , y H .
(14)
Moreover, P C x is characterized by the following properties: P C x C and
x - P C x , y - P C x 0 ,
(15)
x - y 2 x - P C x 2 + y - P C x 2 , x H , y C .
(16)
Recall that a mapping A of C into H is called α-inverse-strongly monotone if there exists a positive real number α such that
A x - A y , x - y α A x - A y 2 , x , y C .
(17)

It is obvious that any α-inverse-strongly monotone mappings A is 1 α -Lipschitz monotone and continuous mappings.

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

Lemma 2.1. [30]Let V : CH be a k-strict pseudo-contraction, then

(1) the fixed point set F(V) of V is closed convex so that the projection PF(V)is well defined;

(2) define a mapping T : CH by
T x = t x + ( 1 - t ) V x , x C .
(18)

If t [k, 1), then T is a nonexpansive mapping such that F(V) = F(T).

A family of mappings { V i : C H } i = 1 is called a family of uniformly k-strict pseudo-contractions, if there exists a constant k [0, 1) such that
V i x - V i y 2 x - y 2 + k ( 1 - V i ) x - ( I - V i ) y 2 , x , y C , i 1 .
Let { V i : C C } i = 1 be a countable family of uniformly k-strict pseudo-contractions. Let { T i : C C } i = 1 be the sequence of nonexpansive mappings defined by (18), i.e.,
T i x = t x + ( 1 - t ) V i x , x C , i 1 , t [ k , 1 ) .
(19)
Let {T i } be a sequence of nonexpansive mappings of C into itself defined by (19) and let {μ i } be a sequence of nonnegative numbers in [0,1]. For each n ≥ 1, define a mapping W n of C into itself as follows:
U n , n + 1 = I , U n , n = μ n T n U n , n + 1 + ( 1 - μ n ) I , U n , n - 1 = μ n - 1 T n - 1 U n , n + ( 1 - μ n - 1 ) I , U n , k = μ k T k U n , k + 1 + ( 1 - μ k ) I , U n , k - 1 = μ k - 1 T k - 1 U n , k + ( 1 - μ k - 1 ) I , U n , 2 = μ 2 T 2 U n , 3 + ( 1 - μ 2 ) I , W n = U n , 1 = μ 1 T 1 U n , 2 + ( 1 - μ 1 ) I .
(20)

Such a mapping W n is nonexpansive from C to C and it is called the W-mapping generated by T1, T2, ..., T n and μ1, μ2, ..., μ n .

For each n, k , let the mapping Un,kbe defined by (20). Then we can have the following crucial conclusions concerning W n . You can find them in [31]. Now we only need the following similar version in Hilbert spaces.

Lemma 2.2. [31]Let C be a nonempty closed convex subset of a real Hilbert space H. Let T1, T2, ... be nonexpansive mappings of C into itself such that n = 1 F ( T n ) is nonempty, let μ1, μ2, ... be real numbers such that 0 ≤ μ n b < 1 for every n ≥ 1. Then,

(1) W n is nonexpansive and F ( W n ) = i = 1 n F ( T i ) , n 1 ;

(2) for every x C and k , the limit limn→∞Un,kx exists;

(3) a mapping W : CC defined by
W x : = lim n W n x = lim n U n , 1 x , x C
(21)

is a nonexpansive mapping satisfying F ( W ) = i = 1 F ( T i ) and it is called the W-mapping generated by T1,T2, ... and μ1, μ2, ....

Lemma 2.3. [32]Let C be a nonempty closed convex subset of a Hilbert space H, {T i : CC} be a countable family of nonexpansive mappings with i = 1 F ( T i ) , { μ i } be a real sequence such that 0 < μ i b < 1, i ≥ 1. If D is any bounded subset of C, then
lim n sup x D W x - W n x = 0 .
Lemma 2.4. [33]Each Hilbert space H satisfies Opial's condition, i.e., for any sequence {x n } H with x n x, the inequality
lim inf n x n - x < lim inf n x n - y ,

hold for each y H with yx.

Lemma 2.5. [22]Let C be a nonempty bounded closed convex subset of a Hilbert space H and let S = { S ( s ) : 0 s < } be a nonexpansive semigroup on C, then for any h ≥ 0,
lim t sup x C 1 t 0 t S ( s ) x d s - S ( h ) ( 1 t 0 t S ( s ) x d s ) = 0 .

Lemma 2.6. [34]Let C be a nonempty bounded closed convex subset of H, {x n } be a sequence in C and S = { S ( s ) : 0 s < } be a nonexpansive semigroup on C. If the following conditions are satisfied:

(i) x n z;

(ii) lim sups→∞lim supn→∞S(s)x n - x n = 0, then z F ( S ) .

Lemma 2.7. Let C be a nonempty closed convex subset of Hilbert space H, A l : CH be a β l -inverse-strongly monotone and λ l (0, 2β l ) where l {1, 2, ..., M}. If P : C C is defined by
P ( x ) = P C ( I - λ M A M ) P C ( I - λ M - 1 A M - 1 ) P C ( I - λ 2 A 2 ) P C ( I - λ 1 A 1 ) x , x C ,

then is nonexpansive.

Proof. Taking P C l = P C ( I - λ l A l ) P C ( I - λ l - 1 A l - 1 ) P C ( I - λ 2 A 2 ) P C ( I - λ 1 A 1 ) , l { 1 , 2 , 3 , , M } and P C 0 = I , where I is the identity mapping on H. Then we have P = P C M .

For any x, y C, we have
P ( x ) - P ( y ) = P C M x - P C M y = P C ( I - λ M A M ) P C M - 1 x - P C ( I - λ M A M ) P C M - 1 y ( I - λ M A M ) P C M - 1 x - ( I - λ M A M ) P C M - 1 y P C M - 1 x - P C M - 1 y P C 0 x - P C 0 y = x - y .

This show that is nonexpansive on C.

Lemma 2.8. Let C be a nonempty closed and convex subset of a real Hilbert space H, A l :CH be nonlinear mappings, where l {1, 2, ..., M}. For x l * C , l { 1 , 2 , , M } , then ( x 1 * , x 2 * , , x M * ) is a solution of problem (3) if and only if
x 1 * = P C ( I - λ M A M ) x M * x 2 * = P C ( I - λ 1 A 1 ) x 1 * x 3 * = P C ( I - λ 2 A 2 ) x 2 * x M * = P C ( I - λ M - 1 A M - 1 ) x M - 1 * ,
(22)
that is
x 1 * = P C ( I - λ M A M ) P C ( I - λ M - 1 A M - 1 ) P C ( I - λ 2 A 2 ) P C ( I - λ 1 A 1 ) x 1 * .
Proof. From the problem (3), we can rewrite as
x 1 * - ( x M * - λ M A M x M * ) , x - x 1 * 0 , x C , x M * - ( x M - 1 * - λ M - 1 A M - 1 x M - 1 * ) , x - x M * 0 , x C , x 3 * - ( x 2 * - λ 2 A 2 x 2 * ) , x - x 3 * 0 , x C , x 2 * - ( x 1 * - λ 1 A 1 x 1 * ) , x - x 2 * 0 , x C .
(23)

From (15), we conclude that (23) is equivalent to (22).

Lemma 2.9. (Demi-closedness Principle[6]) Assume that S is a nonexpansive self-mapping of a nonempty closed convex subset C of a real Hilbert space H. If S has a fixed point, the I - S is demi-closed: that is, whenever {x n } is a sequence in C converging weakly to some x C (for short, x n x), and the sequence {(I - S)x n } converges strongly to some y (for short, (I - S)x n y), it follows that (I - S)x = y. Here I is the identity operator of H.

For solving the system of mixed equilibrium problems, let us assume that bifunction F k : C × C, k = 1,2, ..., N satisfies the following conditions:

(H1) F k is monotone, i.e., F k (x, y) + F k (y, x) ≤ 0, x, y C;

(H2) for each fixed y C, x F k (x,y) is convex and upper semicontinuous;

(H3) for each fixed x C, y F k (x, y) is convex.

Lemma 2.10. [35]Let C be a nonempty closed convex subset of a real Hilbert space H and let φ be a lower semicontinuous and convex functional from C to . Let F be a bifunction from C × C to satisfying (H1)-(H3). Assume that

(i) η : C × CH is k Lipschitz continuous with constant k > 0 such that;

(a) η(x, y) + η(y, x) = 0, x, y C,

(b) η(,) is affine in the first variable,

(c) for each fixed x C, y η(x, y) is sequentially continuous from the weak topology to the weak topology,

(ii) K : C R is η-strongly convex with constant σ > 0 and its derivative K is sequentially continuous from the weak topology to the strong topology;

(iii) for each x C, there exist a bounded subset D x C and z x C such that for any y C\D x ,
F ( y , z x ) + φ ( z x ) - φ ( y ) + 1 r K ( y ) - K ( x ) , η ( z x , y ) < 0 .
For given r > 0, Let K r F : C C be the mapping defined by:
K r F ( x ) = y C : F ( y , z ) + φ ( z ) - φ ( y ) + 1 r K ( y ) - K ( x ) , η ( z , y ) 0 , z C
(24)

for all x C. Then the following hold

(1) K r F is single-valued;

(2) K r F is nonexpansive if K is Lipschitz continuous with constant ν > 0 such that σkν;

(3) F ( K r F ) = M E P ( F , φ ) ;

(4) MEP(F, φ) is closed and convex.

3 Main result

In this section, we prove a strong convergence theorem in a real Hilbert space.

Theorem 3.1. Let C be a nonempty closed convex subset of a real Hilbert Space H. Let {F k : C × C, k = 1, 2, ..., N} be a finite family of bifunctions satisfying conditions (H1)-(H3). Let A l be β l -inverse-strongly monotone mappings of C into H, where l {1, 2, ..., M}. Let S = { S ( s ) : 0 s < } be a nonexpansive semigroup on C and let {t n } be a positive real divergent sequence. Let { V i : C C } i = 1 be a countable family of uniformly k-strict pseudo-contractions, { T i : C C } i = 1 be a countable family of nonexpansive mappings defined by T i x = tx + (1 - t)V i x, x C, i ≥ 1, t [k, 1). For l {1,2, ..., M}, suppose Θ : = F ( S ) ( i = 1 F ( T i ) ) ( k = 1 N S M E P ( F k ) ) G S V I ( C , A l ) . Let {x n } be a sequence generated by x 0 C , C 1 , i = C , C 1 = i = 1 C 1 , i , x 1 = P C 1 x 0 and
u n = K r N , n F N K r N - 1 , n F N - 1 K r 2 , n F 2 K r 1 , n F 1 x n , w n = P C ( I - λ M A M ) P C ( I - λ M - 1 A M - 1 ) P C ( I - λ 2 A 2 ) P C ( I - λ 1 A 1 ) u n , y n , i = α n , i x 0 + ( 1 - α n , i ) 1 t n 0 t n S ( s ) W n w n d s , C n + 1 , i = { z C n , i : y n , i - z 2 x n - z 2 + α n , i ( x 0 2 + 2 x n - x 0 , z ) } , C n + 1 = i = 1 C n + 1 , i , x n + 1 = P C n + 1 x 0
(25)

for every n ≥ 0, where K r k F k : C C , is the mapping defined by (24), r k > 0, k = 1, 2, ..., N are constants and { α n , i } n = 1 ( 0 , 1 ) satisfy the following conditions:

(i) η k : C × CH is L k -Lipschitz continuous with constant k = 1,2, ..., N such that

(1) η k (x, y) + η k (y, x) = 0, x, y C,

(2) x η k (x, y) is affine,

(3) for each fixed y C, y η k (x, y) is sequentially continuous from the weak topology to the weak topology;

(ii) K k : C R is η k -strongly convex with constant σ k > 0 and its derivative K k is not only sequentially continuous from the weak topology to the strong topology but also Lipschitz continuous with a Lipschitz constant ν k > 0 such that σ k > L k ν k ;

(iii) For each k {1, 2, ..., N} and for all x C, there exist a bounded subset D x C and z x C such that for any y C\D x ,
F k ( y , z x ) + φ ( z x ) - φ ( y ) + 1 r k K ( y ) - K ( x ) , η ( z x , y ) < 0 ;

(iv) limn→∞α n,i = 0,i ≥ 1;

(v) {λ l } (0,2β l ), l = 1, 2, ..., M;

(vi) lim infn→∞r k,n > 0, k = 1, 2, 3, ..., N.

Then, {x n } converges strongly to PΘx0.

Proof. First, we show that I - λ l A l for all l {1, 2, ..., M} is nonexpansive mappings.

Indeed, for all x, y C and λ l (0, 2β l ), we observe that
( I - λ 1 A l ) x - ( I - λ 1 A l ) y 2 = x - y - λ l ( A l x - A l y ) 2 = x - y 2 - 2 λ l x - y , A l x - A l y + λ l 2 A l x - A l y 2 x - y 2 - 2 λ 1 β l A l x - A l y 2 + λ l 2 A l x - A l y 2 x - y 2 + λ l ( λ l - 2 β l ) A l x - A l y 2 x - y 2 ,
(26)

which implies that the mapping I - λ l A l is nonexpansive for all l {1, 2, ..., M}.

Let p Θ. Taking
n k = K r k , n F k K r k - 1 , n F k - 1 K r 2 , n F 2 K r 1 , n F 1 , k { 1 , 2 , 3 , , N } , n
and
P C l = P C ( I - λ l A l ) P C ( I - λ l - 1 A l - 1 ) P C ( I - λ 2 A 2 ) P C ( I - λ 1 A 1 ) for all l { 1 , 2 , 3 , , M } ,
n 0 = P C 0 = I , where I is the identity mapping on H. From the definition of K r k , n F k and P C are nonexpansive then n k , k { 1 , 2 , 3 , , N } and P C l , l { 1 , 2 , 3 , , M } also. We note that u n = n N x n and p = r k , n F k p , we have
u n - p = n N x n - n N p x n - p .
(27)
It follows that
w n - p = P C l u n - P C l p u n - p x n - p , l { 1 , 2 , 3 , , M } .
(28)

Next, we will divide the proof into five steps.

Step 1. We show that {x n } is well defined. Let n = 1, then C1,i= C is closed and convex for each i ≥ 1. Suppose that C n,i is closed convex for some n > 1. Then, from definition of Cn+1,i, we know that Cn+1,iis closed convex for the same n ≥ 1. Hence, C n,i is closed convex for n ≥ 1 and for each i ≥ 1. This implies that C n is closed convex for n ≥ 1. Furthermore, we show that Θ C n . For n = 1, Θ C = C1,i. For n ≥ 2, let p Θ. Then,
y n , i - p 2 = α n , i ( x 0 - p ) + ( 1 - α n , i ) ( 1 t n 0 t n S ( s ) W n w n d s - p ) 2 α n , i x 0 - p 2 + ( 1 - α n , i ) 1 t n 0 t n S ( s ) W n w n d s - p 2 α n , i x 0 - p 2 + ( 1 - α n , i ) w n - p 2 = w n - p 2 + α n , i ( x 0 - p 2 - w n - p 2 ) x n - p 2 + α n , i ( x 0 2 + 2 x n - x 0 , p ) ,
(29)

which shows that p C n,i , n ≥ 2, i ≥ 1. Thus, Θ C n,i , n ≥ 1, i ≥ 1. Hence, it follows that Θ C n , n 1 . This implies that {x n } is well-defined.

Step 2. We claim that limn→∞xn+1- x n = 0 and limn→∞yn,i- x n = 0, for i ≥ 1. Since x n = P C n x 0 and x n + 1 = P C n + 1 x 0 C n + 1 C n , n 1 , we have
x n - x 0 x n + 1 - x 0 .
(30)
Also, as Θ C n by (13), it follows that
x n - x 0 z - x 0 , z Θ , n 1 .
(31)
Form (30) and (31), we have that lim n→∞x n - x0 exists. Hence {x n } is bounded and so are {yn,i}, i ≥ 1, {w n }, {u n }, {W n w n }, and 1 t n 0 t n S ( s ) W n w n d s . For m > n ≥ 1, we have that x m = P C m x 0 C m C n . By (16), we obtain
x m - x n 2 x n - x 0 2 - x m - x 0 2 .
(32)
Letting m, n → ∞ and taking the limit in (32), we have x m - x n → 0, which shows that {x n } is a Cauchy sequence. In particular,
lim n x n + 1 - x n = 0 .
(33)
Since, {x n } is a Cauchy sequence, we assume that x n z C. Since x n + 1 = P C n + 1 x 0 C n + 1 , then for each i ≥ 1,
y n , i - x n + 1 2 x n - x n + 1 2 + α n , i ( x 0 2 + 2 x n - x 0 , x n + 1 ) 0 , n .
It follows that
y n , i - x n y n , i - x n + 1 + x n + 1 - x n .
Therefore
lim n y n , i - x n = 0 , i 1 .
(34)
Step 3. We claim that the following statements hold:
  1. (1)

    lim n n k x n - n k - 1 x n = 0 , k = 1 , 2 , , N ;

     
  2. (2)

    limn→∞u n - x n = 0;

     
  3. (3)

    limn→∞u n - w n = 0.

     
Indeed, for p Θ, note that K r k , n F k , k = 1 , 2 , , N is the firmly nonexpansive, so we have
n k x n - n k p 2 = K r k , n F k n k - 1 x n - K r k , n F k p 2 n k x n - p , n k - 1 x n - p = 1 2 n k x n - p 2 + n k - 1 x n - p 2 - n k x n - n k - 1 x n 2 .
Thus, we get
n k x n - n k p 2 n k - 1 x n - p 2 - n k x n - n k - 1 x n 2 .
It follows that
u n - p 2 n k x n - n k p 2 n k - 1 x n - p 2 - n k x n - n k - 1 x n 2 x n - p 2 - n k x n - n k - 1 x n 2 .
(35)
From (29) and (35), we have for i ≥ 1,
y n , i - p 2 α n , i x 0 - p 2 + ( 1 - α n , i ) w n - p 2 α n , i x 0 - p 2 + ( 1 - α n , i ) u n - p 2 α n , i x 0 - p 2 + ( 1 - α n , i ) [ x n - p 2 - n k x n - n k - 1 x n 2 ] ,
it follows that
( 1 - α n , i ) n k x n - n k - 1 x n 2 α n , i x 0 - p 2 + x n - p 2 - y n , i - p 2 α n , i x 0 - p 2 + x n - y n , i ( x n - p + y n , i - p ) .
By the condition (iv) and (34), we have
lim n n k x n - n k - 1 x n = 0 .
(36)
For p Θ and again since K r k , n F k , k = 1 , 2 , , N is the firmly nonexpansive, we obtain
u n - p 2 = n k x n - n k p 2 n k x n - n k p , x n - p = 1 2 n k x n - n k p 2 + x n - p 2 - n k x n - x n 2 = 1 2 u n - p 2 + x n - p 2 - u n - x n 2
and hence
u n - p 2 x n - p 2 - u n - x n 2 .
(37)
From (29) and (37), for i ≥ 1, we have
y n , i - p 2 α n , i x 0 - p 2 + ( 1 - α n , i ) w n - p 2 α n , i x 0 - p 2 + ( 1 - α n , i ) u n - p 2 α n , i x 0 - p 2 + ( 1 - α n , i ) [ x n - p 2 - u n - x n 2 ] ,
it follows that
( 1 - α n , i ) u n - x n 2 α n , i x 0 - p 2 + x n - p 2 - y n , i - p 2 α n , i x 0 - p 2 + x n - y n , i ( x n - p + y n , i - p ) .
By the condition (iv) and (34), we have
lim n u n - x n = 0 .
(38)
From (26), we note that
w n - p 2 = P C M u n - P C M p 2 = P C ( I - λ M A M ) P C M - 1 u n - P C ( I - λ M A M ) P C M - 1 p 2 ( I - λ M A M ) P C M - 1 u n - ( I - λ M A M ) P C M - 1 p 2 P C M - 1 u n - P C M - 1 p 2 + λ M ( λ M - 2 β M ) A M P C M - 1 u n - A M P C M - 1 p 2 u n - p 2 + l = 1 M λ l ( λ l - 2 β i ) A l P C l - 1 u n - A l P C l - 1 p 2 x n - p 2 + l = 1 M λ l ( λ l - 2 β l ) A l P C l - 1 u n - A l P C l - 1 p 2 .
(39)
It follows that, for i ≥ 1
y n , i - p 2 α n , i x 0 - p 2 + ( 1 - α n , i ) w n - p 2 = α n , i x 0 - p 2 + ( 1 - α n , i ) x n - p 2 + l = 1 M λ l ( λ l - 2 β l ) A l P C l - 1 u n - A l P C l - 1 p 2 α n , i x 0 - p 2 + x n - p 2 + ( 1 - α n , i ) l = 1 M λ l ( λ l - 2 β l ) A l P C l - 1 u n - A l P C l - 1 p 2 ,
(40)
which implies that
( 1 - α n , i ) l = 1 M λ l ( 2 β l - λ l ) A l P C l - 1 u n - A l P C l - 1 p 2 = α n , i x 0 - p 2 + x n - p 2 - y n , i - p 2 α n , i x 0 - p 2 + x n - y n , i ( x n - p + y n , i - p ) .
(41)
By the conditions (iv), (v) and (34), we obtain
lim n A l P C l - 1 u n - A l P C l - 1 p = 0 .
(42)
On the other hand, we note that
P C M u n - P C M p 2 = P C ( I - λ M A M ) P C M - 1 u n - P C ( I - λ M A M ) P C M - 1 p 2 ( I - λ M A M ) P C M - 1 u n - ( I - λ M A M ) P C M - 1 p , P C M u n - P C M p = 1 2 ( I - λ M A M ) P C M - 1 u n - ( I - λ M A M ) P C M - 1 p 2 + P C M u n - P C M p 2 - ( I - λ M A M ) P C M - 1 u n - ( I - λ M A M ) P C M - 1 p - ( P C M u n - P C M p ) 2 1 2 P C M - 1 u n - P C M - 1 p 2 + P C M u n - P C M p 2 - ( P C M - 1 u n - P C M - 1 p - P C M u n + P C M p ) - λ M ( A M P C M - 1 u n - A M P C M - 1 p ) 2 = 1 2 P C M - 1 u n - P C M - 1 p 2 + P C M u n - P C M p 2 - P C M - 1 u n - P C M - 1 p - P C M u n + P C M p 2 + 2 λ M P C M - 1 u n - P C M - 1 p - P C M u n + P C M p , A M P C M - 1 u n - A M P C M - 1 p - λ M 2 A M P C M - 1 u n - A M P C M - 1 p 2 1 2 P C M - 1 u n - P C M - 1 p 2 + P C M u n - P C M p 2 - P C M - 1 u n - P C M - 1 p - P C M u n + P C M p 2 + 2 λ M P C M - 1 u n - P C M - 1 p - P C M u n + P C M p A M P C M - 1 u n - A M P C M - 1 p ,
which implies that
P C M u n - P C M p 2 P C M - 1 u n - P C M - 1 p 2 - P C M - 1 u n - P C M - 1 p - P C M u n + P C M p 2 + 2 λ M P C M - 1 u n - P C M - 1 p - P C M u n + P C M p A M P C M - 1 u n - A M P C M - 1 p u n - p 2 - l = 1 M P C l - 1 u n - P C l - 1 p - P C l u n + P C l p 2 + l = 1 M 2 λ l P C l - 1 u n - P C l - 1 p - P C l u n + P C l p A l P C l - 1 u n - A l P C l - 1 p x n - p 2 - l = 1 M P C l - 1 u n - P C l - 1 p - P C l u n + P C l p 2 + l = 1 M 2 λ l P C l - 1 u n - P C l - 1 p - P C l u n + P C l p A l P C l - 1 u n - A l P C l - 1 p .
(43)
From (29) and (43), for i ≥ 1, we note that
y n , i - p 2 α n , i x 0 - p 2 + ( 1 - α n , i ) w n - p 2 = α n , i x 0 - p 2 + ( 1 - α n , i ) P C M u n - P C M p 2 α n , i x 0 - p 2 + ( 1 - α n , i ) x n - p 2 - l = 1 M P C l - 1 u n - P C l - 1 p - P C l u n + P C l p 2 + l = 1 M 2 λ l P C l - 1 u n - P C l - 1 p - P C l u n + P C l p A l P C l - 1 u n - A l P C l - 1 p α n , i x 0 - p 2 + x n - p 2 - ( 1 - α n , i ) l = 1 M P C l - 1 u n - P C l - 1 p - P C l u n + P C l p 2 + l = 1 M 2 λ l P C l - 1 u n - P C l - 1 p - P C l u n + P C l p A l P C l - 1 u n - A l P C l - 1 p .
(44)
This implies that
( 1 - α n , i ) l = 1 M P C l - 1 u n - P C l - 1 p - P C l u n + P C l p 2 α n , i x 0 - p 2 + x n - p 2 - y n , i - p 2 + l = 1 M 2 λ l P C l - 1 u n - P C l - 1 p - P C l u n + P C l p A l P C l - 1 u n - A l P C l - 1 p α n , i x 0 - p 2 + x n - y n , i ( x n - p + y n , i - p ) + l = 1 M 2 λ l P C l - 1 u n - P C l - 1 p - P C l u n + P C l p A l P C l - 1 u n - A l P C l - 1 p .
(45)
Then, by condition (iv), (34) and (42), we obtain that
lim n P C l - 1 u n - P C l - 1 p - P C l u n + P C l p = 0 .
(46)
Therefore, we have
u n - w n = P C 0 u n - P C l u n l = 1 M P C l - 1 u n - P C l - 1 p - P C l u n + P C l p 2 0 as n .
(47)
On the other hand, by condition (iv) implies that
y n , i - 1 t n 0 t n S ( s ) W n w n d s 2 = α n , i x 0 - 1 t n 0 t n S ( s ) W n w n d s 0 , i 1
(48)
it follows that
x n - 1 t n 0 t n S ( s ) W n w n d s x n - y n , i + y n , i + 1 t n 0 t n S ( s ) W n w n d s 0 .
(49)
Step 4. We show that z Θ : = F ( S ) ( i = 1 F ( T i ) ) ( k = 1 N S M E P ( F k ) ) G S V I ( C , A l ) , l { 1 , 2 , , M } . Since { w n i } is bounded, there exists a subsequence { w n i j } of { w n i } which converges weakly to z C. Without loss of generality, we can assume that w n i z .
  1. (1)
    First, we prove that z F ( S ) . From (38), (47), and (49), we get
    lim n w n - 1 t n 0 t n S ( s ) W n w n d s = 0 .
    (50)
     
Since {W n w n } is a bounded and from Lemma 2.5 for any h ≥ 0, we have
lim n 1 t n 0 t n S ( s ) W n w n d s - S ( h ) 1 t n 0 t n S ( s ) W n w n d s = 0 ,
(51)
and since
w n - S ( h ) w n w n - 1 t n 0 t n S ( s ) W n w n d s + 1 t n 0 t n S ( s ) W n w n d s - S ( h ) 1 t n 0 t n S ( s ) W n w n d s + S ( h ) 1 t n 0 t n S ( s ) W n w n d s - S ( h ) w n 2 w n - 1 t n 0 t n S ( s ) W n w n d s + 1 t n 0 t n S ( s ) W n w n d s - S ( h ) 1 t n 0 t n S ( s ) W n w n d s
for all 0 ≤ s < ∞. It follows from (50) and (51), we get
lim n w n - S ( h ) w n = 0 .
(52)
Indeed, from Lemma 2.6 and (52), we get z F ( S ) , i.e., z = S(s)z, s ≥ 0.
  1. (2)
    Next, we show that z F ( W ) = n = 1 F ( W n ) = i = 1 F ( T i ) , where F ( W n ) = i = 1 n F ( T i ) , n 1 and F(W n+ 1) F(W n ). Assume that z F(W), then there exists a positive integer m such that z F(T m ) and so z i = 1 m F ( T i ) . Hence for any n m , z i = 1 n F ( T i ) = F ( W n ) , i.e., zW n z. This together with z = S(s)z, s ≥ 0 shows z = S(s)zS(s)W n z, s ≥ 0, therefore we have z 1 t n 0 t n S ( s ) W n z d s , n m . It follows from the Opial's condition and (50) that
    lim inf i w n i - z < lim inf i w n i - 1 t n i 0 t n i S ( s ) W n i z d s lim inf i w n i - 1 t n i 0 t n i S ( s ) W n i w n i d s + 1 t n i 0 t n i S ( s ) W n i w n i d s - 1 t n i 0 t n i S ( s ) W n i z d s lim inf i w n i - z ,
     
which is a contradiction. Thus, we get z i = 1 F ( T i ) .
  1. (3)
    We prove that z k = 1 N S M E P ( F k