Vahidi et al. Fixed Point Theory and Applications 2013, 2013:213http://www.fixedpointtheoryandapplications.com/content/2013/1/213

R E S E A R C H Open Access

New iterative scheme with strictpseudo-contractions and multivaluednonexpansive mappings for fixed pointproblems and variational inequality problemsJ Vahidi1, A Latif2* and M Eslamian1

*Correspondence: alatif@kau.edu.sa2Department of Mathematics, KingAbdulaziz University, P.O. Box 80203,Jeddah, 21589, Saudi ArabiaFull list of author information isavailable at the end of the article

AbstractIn this paper, we introduce an iterative scheme for finding a common element of thesets of fixed points for multivalued nonexpansive mappings, strictpseudo-contractive mappings and the set of solutions of an equilibrium problem fora pseudomonotone, Lipschitz-type continuous bifunctions. We prove the strongconvergence of the sequence, generated by the proposed scheme, to the solution ofthe variational inequality. Our results generalize and improve some known results.MSC: 47H10; 65K10; 65K15; 90C25

Keywords: Ky Fan inequality; strict pseudo-contractive mapping; multivaluednonexpansive mapping; common fixed point

1 IntroductionIn , Browder and Petryshyn [] introduced a concept of strict pseudo-contractive in areal Hilbert space. Let C be a nonempty subset of a real Hilbert space H , and let T : C → Cbe a single-valued mapping. A mapping T is called a β-strict pseudo-contractive on C []if there exists a constant β ∈ [, ) such that

‖Tx – Ty‖ ≤ ‖x – y‖ + β∥∥(x – Tx) – (y – Ty)

∥∥, ∀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 thatthe class of strictly pseudo-contractive mappings strictly includes the class of nonexpan-sive mappings, which are the mappings T on C such that

‖Tx – Ty‖ ≤ ‖x – y‖

for all x, y ∈ C (see []). Strictly pseudocontractive mappings have more powerful appli-cations than nonexpansive mappings in solving inverse problems, see Scherzer []. In theliterature, many interesting and important results have been appeared to approximate thefixed points of pseudo-contractive mappings. For example, see [–] and the referencestherein.

© 2013 Vahidi et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly cited.

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A subset C ⊂ H is called proximal if for each x ∈ H , there exists an element y ∈ C suchthat

‖x – y‖ = 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 sub-sets, nonempty compact subsets, and nonempty proximal bounded subsets of C, respec-tively. The Hausdorff metric H on CB(H) is defined by

H(A, B) := max{

supx∈A

dist(x, B), supy∈B

dist(y, A)}

for all A, B ∈ CB(H).Let T : H → H be a multivalued mapping. An element x ∈ H is said to be a fixed point

of T if x ∈ Tx. A multivalued mapping T : H → CB(H) is called nonexpansive if

H(Tx, Ty) ≤ ‖x – y‖, x, y ∈ H .

Much work has been done on the existence of common fixed points for a pair consistingof a single-valued and a multivalued mapping, see, for instance [–]. Let f be a bifunc-tion from C × C into R, such that f (x, x) = for all x ∈ C. Consider the classical Ky Faninequality. Find a point x∗ ∈ C such that

f(x∗, y

) ≥ , ∀y ∈ C,

where f (x, ·) is convex and subdifferentiable on C for every x ∈ C. The set of solutions forthis problem is denoted by Sol(f , C). In fact, the Ky Fan inequality can be formulated as anequilibrium problem. Further, if f (x, y) = 〈Fx, y – x〉 for every x, y ∈ C, where F is a mappingfrom C into H , then the Ky Fan inequality problem (equilibrium problem) becomes theclassical variational inequality problem, which is formulated as finding a point x∗ ∈ C suchthat

⟨Fx∗, y – x∗⟩ ≥ , ∀y ∈ C.

Such problems arise frequently in mathematics, physics, engineering, game theory, trans-portation, 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 solutionsof such problems, see, e.g., [–]. Further, in the recent years, iterative algorithms forfinding a common element of the set of solutions of equilibrium problem and the set offixed points of nonexpansive mappings in a real Hilbert space have been studied by manyauthors (see, e.g., [–]).

Definition . Let C be a nonempty closed convex subset of a Hilbert space H . The bi-function f : C × C → R is said to be

(i) strongly monotone on C with α > if

f (x, y) + f (y, x) ≤ –α‖x – y‖, ∀x, y ∈ C;

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(ii) monotone on C if

f (x, y) + f (y, x) ≤ , ∀x, y ∈ C;

(iii) pseudomonotone on C if

f (x, y) ≥ ⇒ f (y, x) ≤ , ∀x, y ∈ C;

(iv) Lipschitz-type continuous on C with constants c > and c > (in the sense ofMastroeni []) if

f (x, y) + f (y, z) ≥ f (x, z) – c‖x – y‖ – c‖y – z‖, ∀x, y, z ∈ C.

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

Theorem . Let C be a nonempty, closed, and convex subset of a real Hilbert space H . Letf : C × C → R be a monotone, continuous, and Lipschitz-type continuous bifunction, andlet f (x, ·) be convex and subdifferentiable on C for every x ∈ C. Let h be a contraction of Cinto itself with constant k ∈ (, ), let S be a nonexpansive mapping of C into itself, and letF(S) ∩ Sol(f , C) �= ∅. Let {xn}, {wn} and {zn} be sequences generated by x ∈ C and by

⎧⎪⎪⎨⎪⎪⎩

wn = arg min{λnf (xn, w) + ‖w – xn‖ : w ∈ C},

zn = arg min{λnf (wn, z) + ‖z – xn‖ : z ∈ C},

xn+ = αnh(xn) + βnxn + γn(μS(xn) + ( – μ)zn), ∀n ≥ ,

where μ ∈ (, ), and {αn}, {βn}, {γn}, and {λn} satisfy the following conditions:(i) limn→∞ αn = ,

∑∞n= αn = ∞,

(ii) limn→∞ |λn+ – λn| = , {λn} ⊂ [a, b] ⊂ (, L ), where L = max{c, c},

(iii) αn + βn + γn = and αn( – αn – βnk – γn) ∈ (, ),(iv) < lim infn→∞ βn ≤ lim supn→∞ βn < .

Then, the sequences {xn}, {wn} and {zn} converge strongly to q ∈ F(S)∩Sol(f , C) which solvesthe variational inequality

⟨(I – h)q, x – q

⟩ ≥ , ∀x ∈ F(S) ∩ Sol(f , C).

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

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2 PreliminariesLet H be a real Hilbert space with inner product 〈·, ·〉 and the norm ‖ · ‖. Let {xn} bea sequence in H , and let x ∈ H . Weak convergence of {xn} to x is denoted by xn ⇀ x,and strong convergence by xn → x. Let C be a nonempty closed convex subset of H . Thenearest point projection from H to C, denoted by ProjC , assigns to each x ∈ H the uniquepoint ProjC x ∈ C with the property

‖x – ProjC x‖ := inf{‖x – y‖,∀y ∈ C

}.

It is known that ProjC is a nonexpansive mapping, and for each x ∈ H ,

〈x – ProjC x, y – ProjC x〉 ≤ , ∀y ∈ C.

Definition . Let C be a nonempty, closed and convex subset of a Hilbert space H . De-note by NC(v) the normal cone of C at v ∈ C, i.e.,

NC(v) :={

z ∈ H : 〈z, y – v〉 ≤ ,∀y ∈ C}

.

Definition . Let C be a nonempty, closed and convex subset of a Hilbert space H , andlet f : C × C → R be a bifunction. For each z ∈ C, by ∂f (z, u) we denote the subgradientof the function f (z, ·) at u, i.e.,

∂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 . In a Hilbert space H , the following inequality holds:

‖x + y‖ ≤ ‖x‖ + 〈y, x + y〉, ∀x, y ∈ H .

Lemma . [] Let {an} be a sequence of nonnegative real numbers, let {αn} be a se-quence in (, ) with

∑∞n= αn = ∞, let {γn} be a sequence of nonnegative real numbers with∑∞

n= γn < ∞, and let {βn} be a sequence of real numbers with lim supn→∞ βn ≤ . Supposethat the following inequality holds:

an+ ≤ ( – αn)an + αnβn + γn, n ≥ .

Then limn→∞ an = .

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

‖αx + βy + γ z‖ = α‖x‖ + β‖y‖ + γ ‖z‖ – αβ‖x – y‖ – αγ ‖x – z‖ – βγ ‖z – y‖.

Lemma . [] Let {tn} be a sequence of real numbers such that there exists a subsequence{ni} of {n} such that tni < tni+ for all i ∈ N. Then there exists a nondecreasing sequence

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{τ (n)} ⊂N such that τ (n) → ∞, and the following properties are satisfied by all (sufficientlylarge) numbers n ∈N:

tτ (n) ≤ tτ (n)+, tn ≤ tτ (n)+.

In fact,

τ (n) = max{k ≤ n : tk < tk+}.

Lemma . [] Let C be a nonempty closed convex subset of a real Hilbert space H , andlet f : C × C → R be a pseudomonotone and Lipschitz-type continuous bifunction. Foreach x ∈ C, let f (x, ·) be convex and subdifferentiable on C. Let {xn}, {zn} and {wn} be thesequences, generated by x ∈ C and by

⎧⎨⎩

wn = arg min{λnf (xn, w) + ‖w – xn‖ : w ∈ C},

zn = arg min{λnf (wn, z) + ‖z – xn‖ : z ∈ C}.

Then for each x∗ ∈ Sol(f , C),

∥∥zn – x∗∥∥ ≤ ∥∥xn – x∗∥∥ – ( – λnc)‖xn – wn‖ – ( – λnc)‖wn – zn‖, ∀n ≥ .

Lemma . [] Let C be nonempty closed convex subset of a real Hilbert space H , and letT : C → C be β-pseudo-contraction mapping. Then I – T is demiclosed at . That is, if {xn}is a sequence in C such that xn ⇀ x and limn→∞ ‖xn – Txn‖ = , then x = Tx.

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

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

Lemma . [] 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 xn ⇀ v and limn→∞ dist(xn,Txn) = , then v ∈ Tv.

3 Main resultsNow, we are in a position to give our main results.

Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H , andlet 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 → CB(C) bea multivalued nonexpansive mapping, and let S : C → C be a β-strict pseudo-contractionmapping. Assume that F = F(T) ∩ F(S) ∩ Sol(f , C) �= ∅ and T(p) = {p} for each p ∈F . Let h

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be a k-contraction of C into itself. Let {xn}, {wn} and {zn} be sequences generated by x ∈ Cand by

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

wn = arg min{λnf (xn, w) + ‖w – xn‖ : w ∈ C},

zn = arg min{λnf (wn, z) + ‖z – xn‖ : z ∈ C},

yn = αnzn + βnun + γnSzn,

xn+ = ϑnh(xn) + ( – ϑn)yn, ∀n ≥ ,

()

where un ∈ Tzn. Let {αn}, {βn}, {γn}, {λn} and {ϑn} satisfy the following conditions:(i) {ϑn} ⊂ (, ), limn→∞ ϑn = ,

∑∞n= ϑn = ∞,

(ii) {λn} ⊂ [a, b] ⊂ (, L ), where L = max{c, c},

(iii) {αn}, {γn} ⊂ [a, ) ⊂ (, ), αn > β and αn + βn + γn = .Then, the sequence {xn} converges strongly to q ∈F , which solves the variational inequality

〈q – hq, x – q〉 ≥ , ∀x ∈F . ()

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

‖zn – q‖ ≤ ‖xn – q‖ – ( – λnc)‖xn – wn‖ – ( – λnc)‖wn – zn‖. ()

This implies that

‖zn – q‖ ≤ ‖xn – q‖. ()

Since T is nonexpansive and Tq = {q}, by () we have

‖un – q‖ = dist(un, Tq) ≤ H(Tzn, Tq) ≤ ‖zn – q‖ ≤ ‖xn – q‖. ()

We show that {xn} is bounded. Indeed, using inequality (), () and Lemma ., we have

‖yn – q‖ = ‖αnzn + βnun + γnSzn – q‖

≤ αn‖zn – q‖ + βn‖un – q‖ + γn‖Szn – q‖

– αnβn‖un – zn‖ – αnγn‖zn – Szn‖

≤ αn‖xn – q‖ + βn‖xn – q‖ + γn(‖zn – q‖ + β‖zn – Szn‖)

– αnβn‖un – zn‖ – αnγn‖zn – Szn‖

– αn( – λnc)‖xn – wn‖ – αn( – λnc)‖wn – zn‖

≤ ‖xn – q‖ – αnβn‖un – zn‖ – γn(αn – β)‖zn – Szn‖

– αn( – λnc)‖xn – wn‖ – αn( – λnc)‖wn – zn‖. ()

It follows that

‖yn – q‖ ≤ ‖xn – q‖ – γn(αn – β)‖zn – Szn‖.

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Since αn > β , we get that ‖yn – q‖ ≤ ‖xn – q‖. This implies that

‖xn+ – q‖ =∥∥ϑnhxn + ( – ϑn)yn – q

∥∥≤ ϑn‖hxn – q‖ + ( – ϑn)‖yn – q‖≤ ϑn

(‖hxn – hq‖ + ‖hq – q‖) + ( – ϑn)‖xn – q‖≤ ϑnk‖xn – q‖ + ϑn‖hq – q‖ + ( – ϑn)‖xn – q‖=

( – ϑn( – k)

)‖xn – q‖ + ϑn‖hq – q‖

≤ max

{‖xn – q‖,

‖hq – q‖ – k

}.

By induction, we get

‖xn – q‖ ≤ max

{‖x – q‖,

‖hq – q‖ – k

}

for all n ∈ N. This implies that {xn} is bounded, and we also obtain that {un}, {zn}, {hxn}and {Szn} are bounded. Next, we show that

limn→∞‖zn – Szn‖ = lim

n→∞‖zn – un‖ = limn→∞‖zn – xn‖ = .

Indeed, using inequality (), we have

‖xn+ – q‖ =∥∥ϑnhxn + ( – ϑn)yn – q

∥∥

≤ ϑn‖hxn – q‖ + ( – ϑn)‖yn – q‖

≤ ϑn‖hxn – q‖ + ( – ϑn)‖xn – p‖

– ( – ϑn)αnβn‖un – zn‖ – ( – ϑn)γn(αn – β)‖zn – Szn‖

– ( – ϑn)αn( – λnc)‖xn – wn‖ – ( – ϑn)αn( – λnc)‖wn – zn‖.

Therefore, we have

( – ϑn)γn(αn – β)‖zn – Szn‖ ≤ ‖xn – q‖ – ‖xn+ – q‖ + ϑn‖hxn – q‖. ()

In order to prove that xn → q as n → ∞, we consider the following two cases.Case . Suppose that there exists n such that {‖xn – q‖} is nonincreasing, for all n ≥ n.

Boundedness of {‖xn – q‖} implies that ‖xn – q‖ is convergent. Since {hxn} is bounded andlimn→∞ ϑn = , from () and our assumption that αn > β , we obtain that

limn→∞‖zn – Szn‖ = .

By similar argument we can obtain that

limn→∞‖un – zn‖ = lim

n→∞‖xn – wn‖ = limn→∞‖wn – zn‖ = . ()

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From this with inequality ‖xn – zn‖ ≤ ‖xn – wn‖ + ‖wn – zn‖, it follows that

limn→∞‖xn – zn‖ = . ()

Next, we show that

lim supn→∞

〈q – hq, q – xn〉 ≤ ,

where q = (Qh)(q). To show this inequality, we choose a subsequence {xni} of {xn} suchthat

limi→∞〈q – hq, q – xni〉 = lim sup

n→∞〈q – hq, q – xn〉.

Since {xni} is bounded, there exists a subsequence {xnij} of {xni}, which converges weakly

to x∗. Without loss of generality, we can assume that xni ⇀ x∗. From inequality (), we havezni ⇀ x∗. Now, since limn→∞ ‖zn –Szn‖ = , from Lemma ., we have x∗ ∈ F(S). Also from(), we have

dist(zn, Tzn) ≤ ‖un – zn‖ → as n → ∞.

It follows from Lemma . 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

wn = arg min

{λnf (xn, y) +

‖y – xn‖ : y ∈ C

}

if and only if

o ∈ ∂

(f (xn, y) +

‖y – xn‖

)(wn) + NC(wn),

where NC(x) is the (outward) normal cone of C at x ∈ C. This follows that

= λnv + wn – xn + un,

where v ∈ ∂f (xn, wn) and un ∈ NC(wn). By the definition of the normal cone NC , we have

〈wn – xn, y – wn〉 ≥ λn〈v, wn – y〉, ∀y ∈ C. ()

Since f (xn, ·) is subdifferentiable on C, there exists v ∈ ∂f (xn, wn) such that

f (xn, y) – f (xn, wn) ≥ 〈v, y – wn〉, ∀y ∈ C

(see, [, ]). Combining this with (), we have

λn(f (xn, y) – f (xn, wn)

) ≥ 〈wn – xn, wn – y〉, ∀y ∈ C.

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Hence

f (xni , y) – f (xni , wni ) ≥ λni

〈wni – xni , wni – y〉, ∀y ∈ C.

From (), we have that wni ⇀ x∗. Now by continuity of f and assumption that {λn} ⊂[a, b] ⊂ ],

L [, we have

f(x∗, y

) ≥ , ∀y ∈ C.

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

lim supn→∞

〈q – hq, q – xn〉 = limi→∞〈q – hq, q – xni〉 =

⟨q – hq, q – x∗⟩ ≤ .

By using Lemma . and inequality (), we have

‖xn+ – q‖ ≤ ∥∥( – ϑn)(yn – q)∥∥ + ϑn〈hxn – q, xn+ – q〉

≤ ( – ϑn)‖yn – q‖ + ϑn〈hxn – hq, xn+ – q〉 + ϑn〈hq – q, xn+ – q〉≤ ( – ϑn)‖xn – q‖ + ϑnk‖xn – q‖‖xn+ – q‖ + ϑn〈hq – q, xn+ – q〉≤ ( – ϑn)‖xn – q‖ + ϑnk

(‖xn – q‖ + ‖xn+ – q‖)+ ϑn〈hq – q, xn+ – q〉

≤ (( – ϑn) + ϑnk

)‖xn – q‖ + ϑnk‖xn+ – q‖ + ϑn〈hq – q, xn+ – q〉.

This implies that

‖xn+ – q‖ ≤(

–( – k)ϑn

– ϑnk

)‖xn – q‖ +

ϑn

– ϑnk‖xn – q‖

+ϑn

– ϑnk〈hq – q, xn+ – q〉.

From Lemma ., we conclude that the sequence {xn} converges strongly to q.Case . Assume that there exists a subsequence {xnj} of {xn} such that

‖xnj – q‖ < ‖xnj+ – q‖,

for all j ∈ N. In this case from Lemma ., there exists a nondecreasing sequence {τ (n)}of N for all n ≥ n (for some n large enough) such that τ (n) → ∞ as n → ∞, and thefollowing inequalities hold for all n ≥ n,

‖xτ (n) – q‖ < ‖xτ (n)+ – q‖, ‖xn – q‖ < ‖xτ (n)+ – q‖.

From (), we obtain limn→∞ ‖zτ (n) – Szτ (n)‖ = , and similarly we obtain

limn→∞‖xτ (n) – zτ (n)‖ = lim

n→∞‖uτ (n) – zτ (n)‖ = .

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Following an argument similar to that in Case , we have

limn→∞‖xτ (n) – q‖ = , lim

n→∞‖xτ (n)+ – q‖ = .

Thus, by Lemma ., we have

≤ ‖xn – q‖ ≤ max{‖xτ (n) – q‖,‖xn – q‖} ≤ ‖xτ (n)+ – q‖.

Therefore, {xn} converges strongly to q ∈F . This completes the proof. �

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

PT (x) ={

y ∈ Tx : ‖x – y‖ = dist(x, Tx)}

, x ∈ C.

Then, we have F(T) = F(PT ). Indeed, if p ∈ F(T), then PT (p) = {p}, hence p ∈ F(PT ). Onthe other hand, if p ∈ F(PT ), since PT (p) ⊂ Tp, we have p ∈ F(T). Now, using the similararguments as in the proof of Theorem ., we obtain the following result by replacing Tby PT , and removing the strict condition T(p) = {p} for all p ∈ F(T).

Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H , andlet 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 PT is nonexpansive, and let S : C → C be a β-strictpseudo-contraction mapping. Assume that F = F(T) ∩ F(S) ∩ Sol(f , C) �= ∅. Let h be a k-contraction of C into itself. Let {xn}, {wn} and {zn} be sequences generated by x ∈ C andby

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

wn = arg min{λnf (xn, w) + ‖w – xn‖ : w ∈ C},

zn = arg min{λnf (wn, z) + ‖z – xn‖ : z ∈ C},

yn = αnzn + βnun + γnSzn,

xn+ = ϑnh(xn) + ( – ϑn)yn, ∀n ≥ ,

()

where un ∈ PT (zn). Let {αn}, {βn}, {γn}, {λn} and {ϑn} satisfy the following conditions:(i) {ϑn} ⊂ (, ), limn→∞ ϑn = ,

∑∞n= ϑn = ∞,

(ii) {λn} ⊂ [a, b] ⊂ (, L ), where L = max{c, c},

(iii) {αn}, {γn} ⊂ [a, ) ⊂ (, ), αn > β and αn + βn + γn = .Then, the sequence {xn} converges strongly to q ∈F , which solves the variational inequality

〈q – hq, x – q〉 ≥ , ∀x ∈F . ()

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

Corollary . Let C be a nonempty closed convex subset of a real Hilbert space H , andlet 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 bea nonexpansive mapping, and let S : C → C be a β-strict pseudo-contraction mapping.

Vahidi et al. Fixed Point Theory and Applications 2013, 2013:213 Page 11 of 13http://www.fixedpointtheoryandapplications.com/content/2013/1/213

Assume that F = F(T) ∩ F(S) ∩ Sol(f , C) �= ∅. Let h be a k-contraction of C into itself. Let{xn}, {wn} and {zn} be sequences generated by x ∈ C and by

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

wn = arg min{λnf (xn, w) + ‖w – xn‖ : w ∈ C},

zn = arg min{λnf (wn, z) + ‖z – xn‖ : z ∈ C},

yn = αnzn + βnTzn + γnSzn,

xn+ = ϑnh(xn) + ( – ϑn)yn, ∀n ≥ .

()

Let {αn}, {βn}, {γn}, {λn} and {ϑn} satisfy the following conditions:(i) {ϑn} ⊂ (, ), limn→∞ ϑn = ,

∑∞n= ϑn = ∞,

(ii) {λn} ⊂ [a, b] ⊂ (, L ), where L = max{c, c},

(iii) {αn}, {γn} ⊂ [a, ) ⊂ (, ), αn > β and αn + βn + γn = .Then, the sequence {xn} converges strongly to q ∈F , which solves the variational inequality

〈q – hq, x – q〉 ≥ , ∀x ∈F . ()

4 Application to variational inequalitiesIn this section, we consider the particular Ky Fan inequality, corresponding to the functionf , defined by f (x, y) = 〈F(x), y – x〉 for every x, y ∈ C with F : C → H . Then, we obtain theclassical variational inequality

find z ∈ C such that⟨F(z), y – z

⟩ ≥ ,∀y ∈ C.

The set of solutions of this problem is denoted by VI(F , C). In that particular case, thesolution yn of the minimization problem

arg min

{λnf (xn, y) +

‖y – xn‖ : y ∈ C

}

can be expressed as

yn = ProjC(xn – λnF(xn)

).

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

(‖x – y‖ + ‖y – z‖),

and, hence, f satisfies Lipschitz-type continuous condition with c = c = L . Now, using

Theorem ., we obtain the following convergence theorem for finding a common elementof the set of common fixed points of a strict pseudo-contractive mapping and a multival-ued nonexpansive mapping and the solution set of the variational inequality problem.

Vahidi et al. Fixed Point Theory and Applications 2013, 2013:213 Page 12 of 13http://www.fixedpointtheoryandapplications.com/content/2013/1/213

Theorem . Let C be a nonempty closed convex subset of a real Hilbert space H , and letF be a function from C to H such that F is monotone and L-Lipschitz continuous on C. Let,T : C → CB(C) be a multivalued nonexpansive mapping, and let S : C → C be a β-strictpseudo-contraction mapping. Assume that F = F(T) ∩ F(S) ∩ VI(F , C) �= ∅ and T(p) = {p}for each p ∈ F . Let h be a k-contraction of C into itself. Let {xn}, {wn}, and let {zn} besequences generated by x ∈ C and by

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

wn = ProjC(xn – λnF(xn)),

zn = ProjC(xn – λnF(wn)),

yn = αnzn + βnun + γnSzn,

xn+ = ϑnh(xn) + ( – ϑn)yn, ∀n ≥ ,

()

where un ∈ Tzn. Let {αn}, {βn}, {γn}, {λn} and {ϑn} satisfy the following conditions:(i) {ϑn} ⊂ (, ), limn→∞ ϑn = ,

∑∞n= ϑn = ∞,

(ii) {λn} ⊂ [a, b] ⊂ (, L ),

(iii) {αn}, {γn} ⊂ [a, ) ⊂ (, ), αn > β and αn + βn + γn = .Then, the sequence {xn} converges strongly to q ∈F , which solves the variational inequality

〈q – hq, x – q〉 ≥ , ∀x ∈F . ()

Competing interestsThe authors declare that they have no competing interests.

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

Author details1Department of Applied Mathematics, Mazandaran University of Science and Technology, Behshahr, Iran. 2Departmentof Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia.

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

Received: 7 May 2013 Accepted: 25 July 2013 Published: 9 August 2013

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doi:10.1186/1687-1812-2013-213Cite this article as: Vahidi et al.: New iterative scheme with strict pseudo-contractions and multivaluednonexpansive mappings for fixed point problems and variational inequality problems. Fixed Point Theory andApplications 2013 2013:213.