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New iterative scheme with strict pseudo-contractions and multivalued nonexpansive mappings for fixed point problems and variational inequality problems

Fixed Point Theory and Applications20132013:213

https://doi.org/10.1186/1687-1812-2013-213

  • Received: 7 May 2013
  • Accepted: 25 July 2013
  • Published:

Abstract

In this paper, we introduce an iterative scheme for finding a common element of the sets of fixed points for multivalued nonexpansive mappings, strict pseudo-contractive mappings and the set of solutions of an equilibrium problem for a pseudomonotone, Lipschitz-type continuous bifunctions. We prove the strong convergence of the sequence, generated by the proposed scheme, to the solution of the variational inequality. Our results generalize and improve some known results.

MSC:47H10, 65K10, 65K15, 90C25.

Keywords

  • Ky Fan inequality
  • strict pseudo-contractive mapping
  • multivalued nonexpansive mapping
  • common fixed point

1 Introduction

In 1967, Browder and Petryshyn [1] introduced a concept of strict pseudo-contractive in a real Hilbert space. Let C be a nonempty subset of a real Hilbert space H, and let T : C C be a single-valued mapping. A mapping T is called a β-strict pseudo-contractive on C [1] if there exists a constant β [ 0 , 1 ) such that
T x T y 2 x y 2 + β ( x T x ) ( y T y ) 2 , x , y C .
We use F ( T ) to denote the set of all fixed points of T; F ( T ) = { x C : x = T ( x ) } . Note that the class of strictly pseudo-contractive mappings strictly includes the class of nonexpansive mappings, which are the mappings T on C such that
T x T y x y

for all x , y C (see [2]). Strictly pseudocontractive mappings have more powerful applications than nonexpansive mappings in solving inverse problems, see Scherzer [3]. In the literature, many interesting and important results have been appeared to approximate the fixed points of pseudo-contractive mappings. For example, see [46] and the references therein.

A subset C H is called proximal if for each x H , there exists an element y C such that
x y = dist ( x , C ) = inf { x z : z C } .
We denote by C B ( C ) , K ( C ) and P ( C ) the collection of all nonempty closed bounded subsets, nonempty compact subsets, and nonempty proximal bounded subsets of C, respectively. The Hausdorff metric H on C B ( H ) is defined by
H ( A , B ) : = max { sup x A dist ( x , B ) , sup y B dist ( y , A ) }

for all A , B C B ( H ) .

Let T : H 2 H be a multivalued mapping. An element x H is said to be a fixed point of T if x T x . A multivalued mapping T : H C B ( H ) is called nonexpansive if
H ( T x , T y ) x y , x , y H .
Much work has been done on the existence of common fixed points for a pair consisting of a single-valued and a multivalued mapping, see, for instance [714]. Let f be a bifunction from C × C into , such that f ( x , x ) = 0 for all x C . Consider the classical Ky Fan inequality. Find a point x C such that
f ( x , y ) 0 , y C ,
where f ( x , ) is convex and subdifferentiable on C for every x C . The set of solutions for this problem is denoted by Sol ( f , C ) . In fact, the Ky Fan inequality can be formulated as an equilibrium problem. Further, if f ( x , y ) = F x , y x for every x , y C , where F is a mapping from C into H, then the Ky Fan inequality problem (equilibrium problem) becomes the classical variational inequality problem, which is formulated as finding a point x C such that
F x , y x 0 , y C .

Such problems arise frequently in mathematics, physics, engineering, game theory, transportation, economics and network. Due to importance of the solutions of such problems, many researchers are working in this area and studying on the existence of the solutions of such problems, see, e.g., [1520]. Further, in the recent years, iterative algorithms for finding a common element of the set of solutions of equilibrium problem and the set of fixed points of nonexpansive mappings in a real Hilbert space have been studied by many authors (see, e.g., [2133]).

Definition 1.1 Let C be a nonempty closed convex subset of a Hilbert space H. The bifunction f : C × C R is said to be
  1. (i)
    strongly monotone on C with α > 0 if
    f ( x , y ) + f ( y , x ) α x y 2 , x , y C ;
     
  2. (ii)
    monotone on C if
    f ( x , y ) + f ( y , x ) 0 , x , y C ;
     
  3. (iii)
    pseudomonotone on C if
    f ( x , y ) 0 f ( y , x ) 0 , x , y C ;
     
  4. (iv)
    Lipschitz-type continuous on C with constants c 1 > 0 and c 2 > 0 (in the sense of Mastroeni [34]) if
    f ( x , y ) + f ( y , z ) f ( x , z ) c 1 x y 2 c 2 y z 2 , x , y , z C .
     

Recently, Anh [35, 36] introduced some methods for finding a common element of the set of solutions of monotone Lipschitz-type continuous equilibrium problem and the set of fixed points of a nonexpansive mapping T in a Hilbert space H. In [35], he proved the following theorem.

Theorem 1.2 Let C be a nonempty, closed, and convex subset of a real Hilbert space H. Let f : C × C R be a monotone, continuous, and Lipschitz-type continuous bifunction, and let f ( x , ) be convex and subdifferentiable on C for every x C . Let h be a contraction of C into itself with constant k ( 0 , 1 ) , let S be a nonexpansive mapping of C into itself, and let F ( S ) Sol ( f , C ) . Let { x n } , { w n } and { z n } be sequences generated by x 0 C and by
{ w n = arg min { λ n f ( x n , w ) + 1 2 w x n 2 : w C } , z n = arg min { λ n f ( w n , z ) + 1 2 z x n 2 : z C } , x n + 1 = α n h ( x n ) + β n x n + γ n ( μ S ( x n ) + ( 1 μ ) z n ) , n 0 ,
where μ ( 0 , 1 ) , and { α n } , { β n } , { γ n } , and { λ n } satisfy the following conditions:
  1. (i)

    lim n α n = 0 , n = 1 α n = ,

     
  2. (ii)

    lim n | λ n + 1 λ n | = 0 , { λ n } [ a , b ] ( 0 , 1 L ) , where L = max { 2 c 1 , 2 c 2 } ,

     
  3. (iii)

    α n + β n + γ n = 1 and α n ( 2 α n 2 β n k 2 γ n ) ( 0 , 1 ) ,

     
  4. (iv)

    0 < lim inf n β n lim sup n β n < 1 .

     
Then, the sequences { x n } , { w n } and { z n } converge strongly to q F ( S ) Sol ( f , C ) which solves the variational inequality
( I h ) q , x q 0 , x F ( S ) Sol ( f , C ) .

In this paper, we introduce an iterative algorithm for finding a common element of the sets of fixed points for multivalued nonexpansive mappings, strict pseudo-contractive mappings and the set of solutions of an equilibrium problem for a pseudomonotone, Lipschitz-type continuous bifunctions. We prove the strong convergence of the sequence generated by the proposed algorithm to the solution of the variational inequality. Our results generalize and improve a number of known results including the results of Anh [35].

2 Preliminaries

Let H be a real Hilbert space with inner product , and the norm . Let { x n } be a sequence in H, and let x H . Weak convergence of { x n } to x is denoted by x n x , and strong convergence by x n x . Let C be a nonempty closed convex subset of H. The nearest point projection from H to C, denoted by Proj C , assigns to each x H the unique point Proj C x C with the property
x Proj C x : = inf { x y , y C } .
It is known that Proj C is a nonexpansive mapping, and for each x H ,
x Proj C x , y Proj C x 0 , y C .
Definition 2.1 Let C be a nonempty, closed and convex subset of a Hilbert space H. Denote by N C ( v ) the normal cone of C at v C , i.e.,
N C ( v ) : = { z H : z , y v 0 , y C } .
Definition 2.2 Let C be a nonempty, closed and convex subset of a Hilbert space H, and let f : C × C R be a bifunction. For each z C , by 2 f ( z , u ) we denote the subgradient of the function f ( z , ) at u, i.e.,
2 f ( z , u ) = { ξ H : f ( z , t ) f ( z , u ) ξ , t u , t C } .

The following lemmas are crucial for the proofs of our results.

Lemma 2.3 In a Hilbert space H, the following inequality holds:
x + y 2 x 2 + 2 y , x + y , x , y H .

Lemma 2.4 [37]

Let { a n } be a sequence of nonnegative real numbers, let { α n } be a sequence in ( 0 , 1 ) with n = 1 α n = , let { γ n } be a sequence of nonnegative real numbers with n = 1 γ n < , and let { β n } be a sequence of real numbers with lim sup n β n 0 . Suppose that the following inequality holds:
a n + 1 ( 1 α n ) a n + α n β n + γ n , n 0 .

Then lim n a n = 0 .

Lemma 2.5 [38]

Let H be a real Hilbert space. Then for all x , y , z H and α , β , γ [ 0 , 1 ] with α + β + γ = 1 , we have
α x + β y + γ z 2 = α x 2 + β y 2 + γ z 2 α β x y 2 α γ x z 2 β γ z y 2 .

Lemma 2.6 [39]

Let { t n } be a sequence of real numbers such that there exists a subsequence { n i } of { n } such that t n i < t n i + 1 for all i N . Then there exists a nondecreasing sequence { τ ( n ) } N such that τ ( n ) , and the following properties are satisfied by all (sufficiently large) numbers n N :
t τ ( n ) t τ ( n ) + 1 , t n t τ ( n ) + 1 .
In fact,
τ ( n ) = max { k n : t k < t k + 1 } .

Lemma 2.7 [36]

Let C be a nonempty closed convex subset of a real Hilbert space H, and let f : C × C R be a pseudomonotone and Lipschitz-type continuous bifunction. For each x C , let f ( x , ) be convex and subdifferentiable on C. Let { x n } , { z n } and { w n } be the sequences, generated by x 0 C and by
{ w n = arg min { λ n f ( x n , w ) + 1 2 w x n 2 : w C } , z n = arg min { λ n f ( w n , z ) + 1 2 z x n 2 : z C } .
Then for each x Sol ( f , C ) ,
z n x 2 x n x 2 ( 1 2 λ n c 1 ) x n w n 2 ( 1 2 λ n c 2 ) w n z n 2 , n 0 .

Lemma 2.8 [5]

Let C be nonempty closed convex subset of a real Hilbert space H, and let T : C C be β-pseudo-contraction mapping. Then I T is demiclosed at 0. That is, if { x n } is a sequence in C such that x n x and lim n x n T x n = 0 , then x = T x .

Lemma 2.9 [5]

Let C be a closed convex subset of a Hilbert space H, and let T : C C be a β-strict pseudo-contraction on C and the fixed-point set F ( T ) of T is nonempty, then F ( T ) is closed and convex.

Lemma 2.10 [40]

Let C be a closed convex subset of a real Hilbert space H. Let T : C C B ( C ) be a nonexpansive multivalued mapping. Assume that T ( p ) = { p } for all p F ( T ) . Then F ( T ) is closed and convex.

Lemma 2.11 [25]

Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C K ( C ) be a nonexpansive multivalued mapping. If x n v and lim n dist ( x n , T x n ) = 0 , then v T v .

3 Main results

Now, we are in a position to give our main results.

Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H, and let f : C × C R be a monotone, continuous, and Lipschitz-type continuous bifunction. Suppose that f ( x , ) is convex and subdifferentiable on C for all x C . Let, T : C C B ( C ) be a multivalued nonexpansive mapping, and let S : C C be a β-strict pseudo-contraction mapping. Assume that F = F ( T ) F ( S ) Sol ( f , C ) and T ( p ) = { p } for each p F . Let h be a k-contraction of C into itself. Let { x n } , { w n } and { z n } be sequences generated by x 0 C and by
{ w n = arg min { λ n f ( x n , w ) + 1 2 w x n 2 : w C } , z n = arg min { λ n f ( w n , z ) + 1 2 z x n 2 : z C } , y n = α n z n + β n u n + γ n S z n , x n + 1 = ϑ n h ( x n ) + ( 1 ϑ n ) y n , n 0 ,
(1)
where u n T z n . Let { α n } , { β n } , { γ n } , { λ n } and { ϑ n } satisfy the following conditions:
  1. (i)

    { ϑ n } ( 0 , 1 ) , lim n ϑ n = 0 , n = 1 ϑ n = ,

     
  2. (ii)

    { λ n } [ a , b ] ( 0 , 1 L ) , where L = max { 2 c 1 , 2 c 2 } ,

     
  3. (iii)

    { α n } , { γ n } [ a , 1 ) ( 0 , 1 ) , α n > β and α n + β n + γ n = 1 .

     
Then, the sequence { x n } converges strongly to q F , which solves the variational inequality
q h q , x q 0 , x F .
(2)
Proof Let Q = Proj F . It easy to see that Qh is a contraction. By the Banach contraction principle, there exists a q F such that q = ( Q h ) ( q ) . Applying Lemma 2.7, we have
z n q 2 x n q 2 ( 1 2 λ n c 1 ) x n w n 2 ( 1 2 λ n c 2 ) w n z n 2 .
(3)
This implies that
z n q x n q .
(4)
Since T is nonexpansive and T q = { q } , by (4) we have
u n q = dist ( u n , T q ) H ( T z n , T q ) z n q x n q .
(5)
We show that { x n } is bounded. Indeed, using inequality (4), (5) and Lemma 2.5, we have
y n q 2 = α n z n + β n u n + γ n S z n q 2 α n z n q 2 + β n u n q 2 + γ n S z n q 2 α n β n u n z n 2 α n γ n z n S z n 2 α n x n q 2 + β n x n q 2 + γ n ( z n q 2 + β z n S z n 2 ) α n β n u n z n 2 α n γ n z n S z n 2 α n ( 1 2 λ n c 1 ) x n w n 2 α n ( 1 2 λ n c 2 ) w n z n 2 x n q 2 α n β n u n z n 2 γ n ( α n β ) z n S z n 2 α n ( 1 2 λ n c 1 ) x n w n 2 α n ( 1 2 λ n c 2 ) w n z n 2 .
(6)
It follows that
y n q 2 x n q 2 γ n ( α n β ) z n S z n 2 .
Since α n > β , we get that y n q x n q . This implies that
x n + 1 q = ϑ n h x n + ( 1 ϑ n ) y n q ϑ n h x n q + ( 1 ϑ n ) y n q ϑ n ( h x n h q + h q q ) + ( 1 ϑ n ) x n q ϑ n k x n q + ϑ n h q q + ( 1 ϑ n ) x n q = ( 1 ϑ n ( 1 k ) ) x n q + ϑ n h q q max { x n q , h q q 1 k } .
By induction, we get
x n q max { x 0 q , h q q 1 k }
for all n N . This implies that { x n } is bounded, and we also obtain that { u n } , { z n } , { h x n } and { S z n } are bounded. Next, we show that
lim n z n S z n = lim n z n u n = lim n z n x n = 0 .
Indeed, using inequality (6), we have
x n + 1 q 2 = ϑ n h x n + ( 1 ϑ n ) y n q 2 ϑ n h x n q 2 + ( 1 ϑ n ) y n q 2 ϑ n h x n q 2 + ( 1 ϑ n ) x n p 2 ( 1 ϑ n ) α n β n u n z n 2 ( 1 ϑ n ) γ n ( α n β ) z n S z n 2 ( 1 ϑ n ) α n ( 1 2 λ n c 1 ) x n w n 2 ( 1 ϑ n ) α n ( 1 2 λ n c 2 ) w n z n 2 .
Therefore, we have
( 1 ϑ n ) γ n ( α n β ) z n S z n 2 x n q 2 x n + 1 q 2 + ϑ n h x n q .
(7)

In order to prove that x n q as n , we consider the following two cases.

Case 1. Suppose that there exists n 0 such that { x n q } is nonincreasing, for all n n 0 . Boundedness of { x n q } implies that x n q is convergent. Since { h x n } is bounded and lim n ϑ n = 0 , from (7) and our assumption that α n > β , we obtain that
lim n z n S z n = 0 .
By similar argument we can obtain that
lim n u n z n = lim n x n w n = lim n w n z n = 0 .
(8)
From this with inequality x n z n x n w n + w n z n , it follows that
lim n x n z n = 0 .
(9)
Next, we show that
lim sup n q h q , q x n 0 ,
where q = ( Q h ) ( q ) . To show this inequality, we choose a subsequence { x n i } of { x n } such that
lim i q h q , q x n i = lim sup n q h q , q x n .
Since { x n i } is bounded, there exists a subsequence { x n i j } of { x n i } , which converges weakly to  x . Without loss of generality, we can assume that x n i x . From inequality (9), we have z n i x . Now, since lim n z n S z n = 0 , from Lemma 2.8, we have x F ( S ) . Also from (8), we have
dist ( z n , T z n ) u n z n 0 as  n .
It follows from Lemma 2.9 that x F ( T ) . Now, we show that x Sol ( f , C ) . Since f ( x , ) is convex on C for each x C , we see that
w n = arg min { λ n f ( x n , y ) + 1 2 y x n 2 : y C }
if and only if
o 2 ( f ( x n , y ) + 1 2 y x n 2 ) ( w n ) + N C ( w n ) ,
where N C ( x ) is the (outward) normal cone of C at x C . This follows that
0 = λ n v + w n x n + u n ,
where v 2 f ( x n , w n ) and u n N C ( w n ) . By the definition of the normal cone N C , we have
w n x n , y w n λ n v , w n y , y C .
(10)
Since f ( x n , ) is subdifferentiable on C, there exists v 2 f ( x n , w n ) such that
f ( x n , y ) f ( x n , w n ) v , y w n , y C
(see, [41, 42]). Combining this with (10), we have
λ n ( f ( x n , y ) f ( x n , w n ) ) w n x n , w n y , y C .
Hence
f ( x n i , y ) f ( x n i , w n i ) 1 λ n i w n i x n i , w n i y , y C .
From (8), we have that w n i x . Now by continuity of f and assumption that { λ n } [ a , b ] ] 0 , 1 L [ , we have
f ( x , y ) 0 , y C .
This implies that x Sol ( f , C ) , and hence x F . Since q = ( Q h ) ( q ) and x F , it follows that
lim sup n q h q , q x n = lim i q h q , q x n i = q h q , q x 0 .
By using Lemma 2.3 and inequality (6), we have
x n + 1 q 2 ( 1 ϑ n ) ( y n q ) 2 + 2 ϑ n h x n q , x n + 1 q ( 1 ϑ n ) 2 y n q 2 + 2 ϑ n h x n h q , x n + 1 q + 2 ϑ n h q q , x n + 1 q ( 1 ϑ n ) 2 x n q 2 + 2 ϑ n k x n q x n + 1 q + 2 ϑ n h q q , x n + 1 q ( 1 ϑ n ) 2 x n q 2 + ϑ n k ( x n q 2 + x n + 1 q 2 ) + 2 ϑ n h q q , x n + 1 q ( ( 1 ϑ n ) 2 + ϑ n k ) x n q 2 + ϑ n k x n + 1 q 2 + 2 ϑ n h q q , x n + 1 q .
This implies that
x n + 1 q 2 ( 1 2 ( 1 k ) ϑ n 1 ϑ n k ) x n q 2 + ϑ n 2 1 ϑ n k x n q 2 + 2 ϑ n 1 ϑ n k h q q , x n + 1 q .

From Lemma 2.4, we conclude that the sequence { x n } converges strongly to q.

Case 2. Assume that there exists a subsequence { x n j } of { x n } such that
x n j q < x n j + 1 q ,
for all j N . In this case from Lemma 2.6, there exists a nondecreasing sequence { τ ( n ) } of for all n n 0 (for some n 0 large enough) such that τ ( n ) as n , and the following inequalities hold for all n n 0 ,
x τ ( n ) q < x τ ( n ) + 1 q , x n q < x τ ( n ) + 1 q .
From (3), we obtain lim n z τ ( n ) S z τ ( n ) = 0 , and similarly we obtain
lim n x τ ( n ) z τ ( n ) = lim n u τ ( n ) z τ ( n ) = 0 .
Following an argument similar to that in Case 1, we have
lim n x τ ( n ) q = 0 , lim n x τ ( n ) + 1 q = 0 .
Thus, by Lemma 2.6, we have
0 x n q max { x τ ( n ) q , x n q } x τ ( n ) + 1 q .

Therefore, { x n } converges strongly to q F . This completes the proof. □

Now, let T : C P ( C ) be a multivalued mapping, and let
P T ( x ) = { y T x : x y = dist ( x , T x ) } , x C .

Then, we have F ( T ) = F ( P T ) . Indeed, if p F ( T ) , then P T ( p ) = { p } , hence p F ( P T ) . On the other hand, if p F ( P T ) , since P T ( p ) T p , we have p F ( T ) . Now, using the similar arguments as in the proof of Theorem 3.1, we obtain the following result by replacing T by P T , and removing the strict condition T ( p ) = { p } for all p F ( T ) .

Theorem 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H, and let f : C × C R be a monotone, continuous, and Lipschitz-type continuous bifunction. Suppose that f ( x , ) is convex and subdifferentiable on C for all x C . Let T : C P ( C ) be a multivalued mapping such that P T is nonexpansive, and let S : C C be a β-strict pseudo-contraction mapping. Assume that F = F ( T ) F ( S ) Sol ( f , C ) . Let h be a k-contraction of C into itself. Let { x n } , { w n } and { z n } be sequences generated by x 0 C and by
{ w n = arg min { λ n f ( x n , w ) + 1 2 w x n 2 : w C } , z n = arg min { λ n f ( w n , z ) + 1 2 z x n 2 : z C } , y n = α n z n + β n u n + γ n S z n , x n + 1 = ϑ n h ( x n ) + ( 1 ϑ n ) y n , n 0 ,
(11)
where u n P T ( z n ) . Let { α n } , { β n } , { γ n } , { λ n } and { ϑ n } satisfy the following conditions:
  1. (i)

    { ϑ n } ( 0 , 1 ) , lim n ϑ n = 0 , n = 1 ϑ n = ,

     
  2. (ii)

    { λ n } [ a , b ] ( 0 , 1 L ) , where L = max { 2 c 1 , 2 c 2 } ,

     
  3. (iii)

    { α n } , { γ n } [ a , 1 ) ( 0 , 1 ) , α n > β and α n + β n + γ n = 1 .

     
Then, the sequence { x n } converges strongly to q F , which solves the variational inequality
q h q , x q 0 , x F .
(12)

As a consequence, we obtain the following result for single-valued mappings.

Corollary 3.3 Let C be a nonempty closed convex subset of a real Hilbert space H, and let f : C × C R be a monotone, continuous, and Lipschitz-type continuous bifunction. Suppose that f ( x , ) is convex and subdifferentiable on C for all x C . Let T : C C be a nonexpansive mapping, and let S : C C be a β-strict pseudo-contraction mapping. Assume that F = F ( T ) F ( S ) Sol ( f , C ) . Let h be a k-contraction of C into itself. Let { x n } , { w n } and { z n } be sequences generated by x 0 C and by
{ w n = arg min { λ n f ( x n , w ) + 1 2 w x n 2 : w C } , z n = arg min { λ n f ( w n , z ) + 1 2 z x n 2 : z C } , y n = α n z n + β n T z n + γ n S z n , x n + 1 = ϑ n h ( x n ) + ( 1 ϑ n ) y n , n 0 .
(13)
Let { α n } , { β n } , { γ n } , { λ n } and { ϑ n } satisfy the following conditions:
  1. (i)

    { ϑ n } ( 0 , 1 ) , lim n ϑ n = 0 , n = 1 ϑ n = ,

     
  2. (ii)

    { λ n } [ a , b ] ( 0 , 1 L ) , where L = max { 2 c 1 , 2 c 2 } ,

     
  3. (iii)

    { α n } , { γ n } [ a , 1 ) ( 0 , 1 ) , α n > β and α n + β n + γ n = 1 .

     
Then, the sequence { x n } converges strongly to q F , which solves the variational inequality
q h q , x q 0 , x F .
(14)

4 Application to variational inequalities

In this section, we consider the particular Ky Fan inequality, corresponding to the function f, defined by f ( x , y ) = F ( x ) , y x for every x , y C with F : C H . Then, we obtain the classical variational inequality
find  z C  such that  F ( z ) , y z 0 , y C .
The set of solutions of this problem is denoted by V I ( F , C ) . In that particular case, the solution y n of the minimization problem
arg min { λ n f ( x n , y ) + 1 2 y x n 2 : y C }
can be expressed as
y n = Proj C ( x n λ n F ( x n ) ) .
Let F be L-Lipschitz continuous on C. Then
f ( x , y ) + f ( y , z ) f ( x , z ) = F ( x ) F ( y ) , y z , x , y , z C .
Therefore,
| F ( x ) F ( y ) , y z | L x y y z L 2 ( x y 2 + y z 2 ) ,

and, hence, f satisfies Lipschitz-type continuous condition with c 1 = c 2 = L 2 . Now, using Theorem 3.1, we obtain the following convergence theorem for finding a common element of the set of common fixed points of a strict pseudo-contractive mapping and a multivalued nonexpansive mapping and the solution set of the variational inequality problem.

Theorem 4.1 Let C be a nonempty closed convex subset of a real Hilbert space H, and let F be a function from C to H such that F is monotone and L-Lipschitz continuous on C. Let, T : C C B ( C ) be a multivalued nonexpansive mapping, and let S : C C be a β-strict pseudo-contraction mapping. Assume that F = F ( T ) F ( S ) V I ( F , C ) and T ( p ) = { p } for each p F . Let h be a k-contraction of C into itself. Let { x n } , { w n } , and let { z n } be sequences generated by x 0 C and by
{ w n = Proj C ( x n λ n F ( x n ) ) , z n = Proj C ( x n λ n F ( w n ) ) , y n = α n z n + β n u n + γ n S z n , x n + 1 = ϑ n h ( x n ) + ( 1 ϑ n ) y n , n 0 ,
(15)
where u n T z n . Let { α n } , { β n } , { γ n } , { λ n } and { ϑ n } satisfy the following conditions:
  1. (i)

    { ϑ n } ( 0 , 1 ) , lim n ϑ n = 0 , n = 1 ϑ n = ,

     
  2. (ii)

    { λ n } [ a , b ] ( 0 , 1 L ) ,

     
  3. (iii)

    { α n } , { γ n } [ a , 1 ) ( 0 , 1 ) , α n > β and α n + β n + γ n = 1 .

     
Then, the sequence { x n } converges strongly to q F , which solves the variational inequality
q h q , x q 0 , x F .
(16)

Declarations

Acknowledgements

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the second author acknowledges with thanks DSR, KAU for financial support. The authors thank the referees for their valuable comments and suggestions.

Authors’ Affiliations

(1)
Department of Applied Mathematics, Mazandaran University of Science and Technology, Behshahr, Iran
(2)
Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia

References

  1. Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert spaces. J. Math. Anal. Appl. 1967, 20: 197–228. 10.1016/0022-247X(67)90085-6MathSciNetView ArticleGoogle Scholar
  2. Chidume CE, Mutangadura SA: An example on the Mann iteration method for Lipschitz pseudo-contractions. Proc. Am. Math. Soc. 2001, 129: 2359–2363. 10.1090/S0002-9939-01-06009-9MathSciNetView ArticleGoogle Scholar
  3. Scherzer O: Convergence criteria of iterative methods based on Landweber iteration for solving nonlinear problems. J. Math. Anal. Appl. 1991, 194: 911–933.MathSciNetView ArticleGoogle Scholar
  4. Acedo GL, Xu HK: Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 2007, 67: 2258–2271. 10.1016/j.na.2006.08.036MathSciNetView ArticleGoogle Scholar
  5. Marino G, Xu HK: Weak and strong convergence theorems for strictly pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 2007, 329: 336–349. 10.1016/j.jmaa.2006.06.055MathSciNetView ArticleGoogle Scholar
  6. Zhou HY: Convergence theorems of common fixed points for a finite family of Lipschitz pseudo-contractions in Banach spaces. Nonlinear Anal. 2008, 68: 2977–2983. 10.1016/j.na.2007.02.041MathSciNetView ArticleGoogle Scholar
  7. Dhompongsa S, Kaewkhao A, Panyanak B:Lim’s theorem for multivalued mappings in CAT ( 0 ) spaces. J. Math. Anal. Appl. 2005, 312: 478–487. 10.1016/j.jmaa.2005.03.055MathSciNetView ArticleGoogle Scholar
  8. Latif A, Tweddle I: Some results on coincidence points. Bull. Aust. Math. Soc. 1999, 59: 111–117. 10.1017/S0004972700032652MathSciNetView ArticleGoogle Scholar
  9. Hussain N, Khamsi MA: On asymptotic pointwise contractions in metric spaces. Nonlinear Anal. 2009, 71: 4423–4429. 10.1016/j.na.2009.02.126MathSciNetView ArticleGoogle Scholar
  10. Abkar A, Eslamian M:Common fixed point results in CAT ( 0 ) spaces. Nonlinear Anal. 2011, 74: 1835–1840. 10.1016/j.na.2010.10.056MathSciNetView ArticleGoogle Scholar
  11. Espinola R, Nicolae A: Geodesic Ptolemy spaces and fixed points. Nonlinear Anal. 2011, 74: 27–34. 10.1016/j.na.2010.08.009MathSciNetView ArticleGoogle Scholar
  12. Garcia-Falset J, Llorens-Fuster E, Moreno-Galvez E: Fixed point theory for multivalued generalized nonexpansive mappings. Appl. Anal. Discrete Math. 2012, 6: 265–286. 10.2298/AADM120712017GMathSciNetView ArticleGoogle Scholar
  13. Markin JT: Fixed points for generalized nonexpansive mappings in R-trees. Comput. Math. Appl. 2011, 62: 4614–4618. 10.1016/j.camwa.2011.10.045MathSciNetView ArticleGoogle Scholar
  14. Abkar A, Eslamian M: Fixed point theorems for Suzuki generalized nonexpansive multivalued mappings in Banach spaces. Fixed Point Theory Appl. 2010., 2010: Article ID 457935 10.1155/2010/457935Google Scholar
  15. Blum E, Oettli W: From optimization and variational inequalities to equilibrium problems. Math. Stud. 1994, 63: 123–145.MathSciNetGoogle Scholar
  16. Flam SD, Antipin AS: Equilibrium programming using proximal-link algorithms. Math. Program. 1997, 78: 29–41.MathSciNetView ArticleGoogle Scholar
  17. Combettes PL, Hirstoaga SA: Equilibrium programming in Hilbert spaces. J. Nonlinear Convex Anal. 2005, 6: 117–136.MathSciNetGoogle Scholar
  18. Iiduka H, Yamada I: A use of conjugate gradient direction for the convex optimization problem over the fixed point set of a nonexpansive mapping. SIAM J. Optim. 2009, 19: 1881–1893. 10.1137/070702497MathSciNetView ArticleGoogle Scholar
  19. Iiduka H, Yamada I: A subgradient-type method for the equilibrium problem over the fixed point set and its applications. Optimization 2009, 58: 251–261. 10.1080/02331930701762829MathSciNetView ArticleGoogle Scholar
  20. Mainge PE: A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J. Control Optim. 2008, 47: 1499–1515. 10.1137/060675319MathSciNetView ArticleGoogle Scholar
  21. Tada A, Takahashi W: Weak and strong convergence theorems for a nonexpansive mapping and an equilibrium problem. J. Optim. Theory Appl. 2007, 133: 359–370. 10.1007/s10957-007-9187-zMathSciNetView ArticleGoogle Scholar
  22. Takahashi S, Takahashi W: Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 2007, 331: 506–515. 10.1016/j.jmaa.2006.08.036MathSciNetView ArticleGoogle Scholar
  23. Ansari QH, Khan Z: Densely relative pseudomonotone variational inequalities over product of sets. J. Nonlinear Convex Anal. 2006, 7(2):179–188.MathSciNetGoogle Scholar
  24. Ceng LC, Ansari QH, Yao JC: Viscosity approximation methods for generalized equilibrium problems and fixed point problems. J. Glob. Optim. 2009, 43: 487–502. 10.1007/s10898-008-9342-6MathSciNetView ArticleGoogle Scholar
  25. Eslamian M: Convergence theorems for nonspreading mappings and nonexpansive multivalued mappings and equilibrium problems. Optim. Lett. 2011. 10.1007/s11590-011-0438-4Google Scholar
  26. Ceng LC, Yao JC: Hybrid viscosity approximation schemes for equilibrium problems and fixed point problems of infinitely many nonexpansive mappings. Appl. Math. Comput. 2008, 198: 729–741. 10.1016/j.amc.2007.09.011MathSciNetView ArticleGoogle Scholar
  27. Ceng LC, Yao JC: A hybrid iterative scheme for mixed equilibrium problems and fixed point problems. J. Comput. Appl. Math. 2008, 214: 186–201. 10.1016/j.cam.2007.02.022MathSciNetView ArticleGoogle Scholar
  28. Ceng LC, Schaible S, Yao JC: Implicit iteration scheme with perturbed mapping for equilibrium problems and fixed point problems of finitely many nonexpansive mappings. J. Optim. Theory Appl. 2008, 139: 403–418. 10.1007/s10957-008-9361-yMathSciNetView ArticleGoogle Scholar
  29. Ceng LC, Al-Homidan S, Ansari QH, Yao JC: An iterative scheme for equilibrium problems and fixed point problems of strict pseudo-contraction mappings. J. Comput. Appl. Math. 2009, 223: 967–974. 10.1016/j.cam.2008.03.032MathSciNetView ArticleGoogle Scholar
  30. Ceng LC, Guu SM, Yao JC: Hybrid iterative method for finding common solutions of generalized mixed equilibrium and fixed point problems. Fixed Point Theory Appl. 2012., 2012: Article ID 92Google Scholar
  31. Ceng LC, Ansari QH, Schaible S: Hybrid extragradient-like methods for generalized mixed equilibrium problems, systems of generalized equilibrium problems and optimization problems. J. Glob. Optim. 2012, 53: 69–96. 10.1007/s10898-011-9703-4MathSciNetView ArticleGoogle Scholar
  32. Ceng LC, Yao JC: A relaxed extragradient-like method for a generalized mixed equilibrium problem, a general system of generalized equilibria and a fixed point problem. Nonlinear Anal. 2010, 72: 1922–1937. 10.1016/j.na.2009.09.033MathSciNetView ArticleGoogle Scholar
  33. Ceng LC, Petrusel A, Yao JC: Iterative approaches to solving equilibrium problems and fixed point problems of infinitely many nonexpansive mappings. J. Optim. Theory Appl. 2009, 143: 37–58. 10.1007/s10957-009-9549-9MathSciNetView ArticleGoogle Scholar
  34. Mastroeni G: On auxiliary principle for equilibrium problems. In Equilibrium Problems and Variational Models. Edited by: Daniele P, Giannessi F, Maugeri A. Kluwer Academic, Dordrecht; 2003.Google Scholar
  35. Anh PN: A hybrid extragradient method extended to fixed point problems and equilibrium problems. Optimization 2013, 62: 271–283. 10.1080/02331934.2011.607497MathSciNetView ArticleGoogle Scholar
  36. Anh PN: Strong convergence theorems for nonexpansive mappings and Ky Fan inequalities. J. Optim. Theory Appl. 2012. 10.1007/s10957-012-0005-xGoogle Scholar
  37. Xu HK: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 2003, 116: 659–678. 10.1023/A:1023073621589MathSciNetView ArticleGoogle Scholar
  38. Osilike MO, Igbokwe DI: Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations. Comput. Math. Appl. 2000, 40: 559–567. 10.1016/S0898-1221(00)00179-6MathSciNetView ArticleGoogle Scholar
  39. Mainge PE: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 2008, 16: 899–912. 10.1007/s11228-008-0102-zMathSciNetView ArticleGoogle Scholar
  40. Dhompongsa S, Kaewkhao A, Panyanak B:On Kirk strong convergence theorem for multivalued nonexpansive mappings on CAT ( 0 ) spaces. Nonlinear Anal. 2012, 75: 459–468. 10.1016/j.na.2011.08.046MathSciNetView ArticleGoogle Scholar
  41. Rockafellar RT: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 1976, 14: 877–898. 10.1137/0314056MathSciNetView ArticleGoogle Scholar
  42. Daniele P, Giannessi F, Maugeri A: Equilibrium Problems and Variational Models. Kluwer, Norwell; 2003.View ArticleMATHGoogle Scholar

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