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

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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.

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:CC 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,yC.

We use F(T) to denote the set of all fixed points of T; F(T)={xC: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

TxTyxy

for all x,yC (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 CH is called proximal if for each xH, there exists an element yC such that

xy=dist(x,C)=inf { x z : z C } .

We denote by CB(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 CB(H) is defined by

H(A,B):=max { sup x A dist ( x , B ) , sup y B dist ( y , A ) }

for all A,BCB(H).

Let T:H 2 H be a multivalued mapping. An element xH is said to be a fixed point of T if xTx. A multivalued mapping T:HCB(H) is called nonexpansive if

H(Tx,Ty)xy,x,yH.

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 xC. Consider the classical Ky Fan inequality. Find a point x C such that

f ( x , y ) 0,yC,

where f(x,) is convex and subdifferentiable on C for every xC. 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)=Fx,yx for every x,yC, 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,yC.

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×CR 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,yC;
  2. (ii)

    monotone on C if

    f(x,y)+f(y,x)0,x,yC;
  3. (iii)

    pseudomonotone on C if

    f(x,y)0f(y,x)0,x,yC;
  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,zC.

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×CR be a monotone, continuous, and Lipschitz-type continuous bifunction, and let f(x,) be convex and subdifferentiable on C for every xC. 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 k2 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 qF(S)Sol(f,C) which solves the variational inequality

( I h ) q , x q 0,xF(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 xH. 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 xH the unique point Proj C xC 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 xH,

x Proj C x,y Proj C x0,yC.

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 vC, 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×CR be a bifunction. For each zC, 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 +2y,x+y,x,yH.

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 ,n0.

Then lim n a n =0.

Lemma 2.5 [38]

Let H be a real Hilbert space. Then for all x,y,zH 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 αβ x y 2 αγ x z 2 βγ 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 iN. Then there exists a nondecreasing sequence {τ(n)}N such that τ(n), and the following properties are satisfied by all (sufficiently large) numbers nN:

t τ ( n ) t τ ( n ) + 1 , t n t τ ( n ) + 1 .

In fact,

τ(n)=max{kn: 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×CR be a pseudomonotone and Lipschitz-type continuous bifunction. For each xC, 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 (12 λ n c 1 ) x n w n 2 (12 λ n c 2 ) w n z n 2 ,n0.

Lemma 2.8 [5]

Let C be nonempty closed convex subset of a real Hilbert space H, and let T:CC be β-pseudo-contraction mapping. Then IT 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=Tx.

Lemma 2.9 [5]

Let C be a closed convex subset of a Hilbert space H, and let T:CC 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:CCB(C) be a nonexpansive multivalued mapping. Assume that T(p)={p} for all pF(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:CK(C) be a nonexpansive multivalued mapping. If x n v and lim n dist( x n ,T x n )=0, then vTv.

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×CR be a monotone, continuous, and Lipschitz-type continuous bifunction. Suppose that f(x,) is convex and subdifferentiable on C for all xC. Let, T:CCB(C) be a multivalued nonexpansive mapping, and let S:CC be a β-strict pseudo-contraction mapping. Assume that F=F(T)F(S)Sol(f,C) and T(p)={p} for each pF. 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 qF, which solves the variational inequality

qhq,xq0,xF.
(2)

Proof Let Q= Proj F . It easy to see that Qh is a contraction. By the Banach contraction principle, there exists a qF such that q=(Qh)(q). Applying Lemma 2.7, we have

z n q 2 x n q 2 (12 λ n c 1 ) x n w n 2 (12 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 Tq={q}, by (4) we have

u n q=dist( u n ,Tq)H(T z n ,Tq) 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 qmax { x 0 q , h q q 1 k }

for all nN. 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 qhq,q x n 0,

where q=(Qh)(q). To show this inequality, we choose a subsequence { x n i } of { x n } such that

lim i qhq,q x n i = lim sup n qhq,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 0as 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 xC, 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 xC. 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,yC.
(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 ,yC

(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,yC.

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,yC.

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,yC.

This implies that x Sol(f,C), and hence x F. Since q=(Qh)(q) and x F, it follows that

lim sup n qhq,q x n = lim i qhq,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 jN. 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 qmax { x τ ( n ) q , x n q } x τ ( n ) + 1 q.

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

Now, let T:CP(C) be a multivalued mapping, and let

P T (x)= { y T x : x y = dist ( x , T x ) } ,xC.

Then, we have F(T)=F( P T ). Indeed, if pF(T), then P T (p)={p}, hence pF( P T ). On the other hand, if pF( P T ), since P T (p)Tp, we have pF(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 pF(T).

Theorem 3.2 Let C be a nonempty closed convex subset of a real Hilbert space H, and let f:C×CR be a monotone, continuous, and Lipschitz-type continuous bifunction. Suppose that f(x,) is convex and subdifferentiable on C for all xC. Let T:CP(C) be a multivalued mapping such that P T is nonexpansive, and let S:CC 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 qF, which solves the variational inequality

qhq,xq0,xF.
(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×CR be a monotone, continuous, and Lipschitz-type continuous bifunction. Suppose that f(x,) is convex and subdifferentiable on C for all xC. Let T:CC be a nonexpansive mapping, and let S:CC 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 qF, which solves the variational inequality

qhq,xq0,xF.
(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),yx for every x,yC with F:CH. Then, we obtain the classical variational inequality

find zC such that  F ( z ) , y z 0,yC.

The set of solutions of this problem is denoted by VI(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,zC.

Therefore,

| F ( x ) F ( y ) , y z | Lxyyz 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:CCB(C) be a multivalued nonexpansive mapping, and let S:CC be a β-strict pseudo-contraction mapping. Assume that F=F(T)F(S)VI(F,C) and T(p)={p} for each pF. 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 qF, which solves the variational inequality

qhq,xq0,xF.
(16)

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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.

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Correspondence to A Latif.

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The authors declare that they have no competing interests.

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All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

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Vahidi, J., Latif, A. & Eslamian, M. New iterative scheme with strict pseudo-contractions and multivalued nonexpansive mappings for fixed point problems and variational inequality problems. Fixed Point Theory Appl 2013, 213 (2013). https://doi.org/10.1186/1687-1812-2013-213

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Keywords

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