Let H be a real Hilbert space with inner product ⟨·, ·⟩ and norm ||·||, and let C be a nonempty closed convex subset of H. Let f : H × H → ℝ be a bifunction such that f(x, x) = 0 for all x ∈ C. The classical Ky Fan inequality [7] consists of finding a point x^* in C such that

(1.1)f(x*,y)≥0,   ∀y∈C.

The set of solutions of problem (1.1) is denoted by Sol(f,C). In fact, the Ky Fan inequality can be formulated as an equilibrium problem. If f(x, y) = ⟨Ax, y − x⟩, where A : C → H is a operator, then problem (1.1) become the following variational inequality problem (shortly, VI(A,C)): find x^* ∈ C such that

(1.2)⟨Ax*,y-x*⟩≥0,   ∀y∈C.

The equilibrium problem which was considered as the Ky Fan inequality is very general in the sense that it includes, as special cases, the optimization problem, the variational inequality problem, the complementarity problem, the saddle point problem, the Nash equilibrium problem in noncooperative games and the Kakutani fixed point problem, etc., see [1, 4, 5, 9, 10, 18] and the references therein. Recently, algorithms for solving the Ky Fan inequality have been studied extensively.

In 2001, Yamada [27] proved that the sequence {x_n} generated by the projected gradient algorithm

(1.3){x1∈C,xn+1=PC(xn-λAxn),   ∀n∈ℕ,

converges to the unique solution x^* of VI(A,C) under the assumption that A is strongly monotone and Lipschitz continuous, the mapping P_C(I − λA) is strictly contractive over C. If A is monotone and Lipschitz, the projected gradient algorithm (1.3) may not be convergent. In order to deal with this situation, Korpelevich [15] introduced an extragradient algorithm:

(1.4){x1∈C,yn=PC(xn-λAxn),xn+1=PC(xn-λAyn),   ∀n∈ℕ.

He also proved that the sequences {x_n} and {y_n} converge to the same solution x^* of VI(A,C) under the assumptions that A is L-Lipschitz and monotone, λ∈(0,1L).

In 2008, the extragradient algorithm (1.4) has been extended to Ky Fan inequality problem by Muu et al. [17] as follows:

(1.5){x1∈C,yn=argminw∈C [λf(xn,w)+12‖w-xn‖2],xn+1=argminz∈C [λf(yn,z)+12‖z-xn‖2],   ∀n∈ℕ.

Under assumptions that f is pseudomonotone and Lipschitz-type continuous, the authors showed that the sequence {x_n} converges to an element of Sol(f,C).

For obtaining a common element of set of solutions of Ky Fan inequality (1.1) and the set of fixed points of a nonexpansive mapping T in a real Hilbert space H, Anh [3] introduced an iterative algorithm by the modified viscosity approximation method. The sequence {x_n} is defined by

(1.6){x1∈C,yn=argminw∈C [λnf(xn,w)+12‖w-xn‖2],zn=argminz∈C [λnf(yn,z)+12‖z-xn‖2],xn+1=αnh(xn)+βnxn+γn(μTxn+(1-μ)zn),   ∀n∈ℕ,

where C is a nonempty closed convex subset of H and h is a contractive mapping of C into itself. The author showed that under certain conditions, the sequence {x_n} converges strongly to an element of Sol(f,C) ∩ F(T).

Later in 2013, Vahidi et al. [24] introduced an iterative algorithm for finding a common element of the sets of fixed points for nonexpansive multi-valued mappings, strict pseudo-contractive single-valued mappings and the set of solutions of Ky Fan inequality for pseudomonotone and Lipschitz-type continuous bifunctions in Hilbert spaces.

In this paper, motivated by the research described above, we propose a new iterative algorithm for finding a common element of the sets of fixed points for demicontractive single-valued mappings, quasi-nonexpansive multi-valued mappings, the set of solutions of Ky Fan inequality for pseudomonotone and Lipschitz-type continuous bifunctions, and the set of solutions of variational inequality for φ-inverse strongly monotone mappings in real Hilbert spaces. We obtain strong convergence theorems for the sequence generated by the proposed algorithm in a real Hilbert space. Our results generalize and improve a number of known results including the results of Anh [3] and Vahidi et al. [24].

In this section, we recall some definitions and results for further use. Let C be a nonempty closed convex subset of a real Hilbert space H. We denote the strong convergence and the weak convergence of the sequence {x_n} to a point x ∈ H by x_n → x and x_n ⇀ x, respectively. It is also known in [19] that a Hilbert space H satisfies Opial’s condition, that is, for any sequence {x_n} with x_n ⇀ x, the inequality

lim supn→∞‖xn-x‖ <lim supn→∞‖xn-y‖

holds for every y ∈ H with y ≠ x. Let P_C be the metric projection of H onto C i.e., for x ∈ H, P_Cx satisfies the property

‖x-PCx‖ =miny∈C‖x-y‖.

Since C is nonempty closed and convex, P_Cx exists and is unique. It is also known that P_C has the following characteristic properties, see [11, 23] for more details.

Lemma 2.1

Let C be a nonempty closed convex subset of a real Hilbert space H and let P_C : H → C be the metric projection. Then

(i) for all x ∈ C, y ∈ H, ‖x-PCy‖2+‖PCy-y‖2≤ ‖x-y‖2;

(ii) P_Cx = y if and only if there holds the inequality⟨x-y,y-z⟩≥0,   ∀z∈C.

Lemma 2.2.([23])

Let C be a nonempty closed convex subset of a Hilbert space H and let A be a mapping of C into H. Let u ∈ C. Then for η > 0,

u=PC(I-ηA)u⇔u∈VI(A,C).

Definition 2.3.([13])

A mapping A : C → H is called δ-inverse strongly monotone if there exists a positive real number δ such that

⟨x-y,Ax-Ay⟩≥δ‖Ax-Ay‖2,   ∀x,y∈C.

We now give some concepts of the monotonicity of a bifunction.

Definition 2.4

Let H be a real Hilbert space, C be a nonempty closed convex subset of H, and let f : H × H → ℝ be a bifunction. A bifunction f is said to be:

(i) strongly monotone on C if there exists a constant α > 0 such that f(x,y)+f(y,x)≤-α‖x-y‖2,   ∀x,y∈C;

(ii) monotone on C if f(x,y)+f(y,x)≤0,   ∀x,y∈C;

(iii) pseudomonotone on C if f(x,y)≥0⇒f(y,x)≤0,   ∀x,y∈C;

(iv) Lipschitz-type continuous on C if there exist two positive constants c₁ and c₂ such that f(x,y)+f(y,z)≥f(x,z)-c1‖x-y‖2-c2‖y-z‖2,   ∀x,y∈C.

From the definition above we obviously have the following implications: (1) It is clear that (i) ⇒ (ii) ⇒ (iii), (2) If f(x, y) = ⟨Φ(x), y − x⟩ for a mapping Φ : H → H. Then the notions of monotonicity of bifunction f collapse to the notions of monotonicity of mapping Φ, respectively. In addition, if mapping Φ is L-Lipschitz on C, i.e., ||Φ(x) − Φ(y)|| ≤ L||x − y|| for all x, y ∈ C. Then, f is also Lipschitz-type continuous on C, for example, with constants L1=L2ɛ,L2=Lɛ2, for any ε > 0.

Definition 2.5

Let H be a real Hilbert space, and let f : H × H → ℝ be a bifunction. For each z ∈ H, by ∂f(z, u) we denote the subdifferential of the function f(z, ·) at u, i.e.,

∂f(z,u)={ξ∈H:f(z,t)-f(z,u)≥⟨ξ,t-u⟩,∀t∈H}.

Definition 2.6

Let H be a real Hilbert space and C be a nonempty closed convex subset of H. The normal cone of C at v ∈ C is defined by

NC(v)={z∈H:⟨z,y-v⟩≤0,∀y∈C}.

Lemma 2.7.([6])

Let H be a real Hilbert space, C be a nonempty closed convex subset of H, and f : H × H → ℝ be a bifunction. For each z ∈ H, suppose that f(z, ·) is subdifferentiable on C. Then x^*is a solution to the following convex problem:

min{f(z,x):x∈C}

if and only if 0 ∈ ∂f(z, x^*) + N_C(x^*), where f(z, ·) denotes the subdifferential of f(z, ·) and N_C(x^*) is the normal cone of C at x^* ∈ C.

Lemma 2.8.([2, 17])

Let H be a real Hilbert space, C be a nonempty closed convex subset of H, and let f : H × H → 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 x₁ ∈ C and by

wn=argminw∈C [λnf(xn,w)+12‖w-xn‖2],zn=argminz∈C [λnf(wn,z)+12‖z-xn‖2].

Then for each x^* ∈ Sol(f,C),

(2.1)‖zn-x*‖2≤ ‖xn-x*‖2-(1-2λnc1)‖xn-wn‖2-(1-2λnc2)‖wn-zn‖2, ∀n∈ℕ.

A mapping h : C → C is a contraction if there exists a constant η ∈ (0, 1) such that ||h(x) − h(y)||≤ η||x − y|| for all x, y ∈ C. Let T : C → C be a single-valued mapping. An element x ∈ C is said to be a fixed point of T if x = Tx. The fixed point set of T is denoted by F(T) = {x ∈ C : x = Tx}. A single-valued mapping T is called strictly pseudononspreading [20] if there exists k ∈ [0, 1) such that, for all x, y ∈ C,

‖Tx-TY‖2≤ ‖x-y‖2+ k‖(I-T)x-(I-T)y‖2+ 2⟨x-Tx,y-Ty⟩,

where I denotes the identity mapping. Note that if k = 0, a mapping T is called nonspreading [14]. As a generalization of the class of strictly pseudononspreading mappings, the class of demicontractive mappings was introduced by Hicks and Kubicek [12] in 1977.

Recall that a single-valued mapping T is said to be demicontractive if F(T) ≠ ∅︀ and there exists κ ∈ [0, 1) such that, for all x ∈ C and for all z ∈ F(T),

‖Tx-z‖2≤ ‖x-z‖2+ κ‖x-Tx‖2.

We call κ the contraction coefficient. Clearly, strictly pseudononspreading mapping with a nonempty fixed point set is demicontractive.

We now give two examples for the class of demicontractive mappings.

Example 2.9

Let H be the real line and C = [0, 1]. Define a mapping T : C → C by

Tx={47x sin (1x),x≠0,0,x=0.

Obviously, F(T) = {0}. Also, for all x ∈ C, we have ∣Tx-T0∣2= ∣Tx∣2 = ∣47x sin (1x)∣2≤ ∣4x7∣2≤ ∣x∣2 ≤ ∣x-0∣2+ k∣x-Tx∣2 for all k ∈ [0, 1). Therefore, T is demicontractive.

Example 2.10

Let H be the real line and C = [−1, 1]. Define a mapping T : C → C by

Tx={9-x10,x∈[-1,0),x+910,x∈[0,1].

Obviously, F(T) = {1} and T is demicontractive.

The following lemma obtained by Suantai and Phuengrattana [22] is useful for our results.

Lemma 2.11

Let H be a Hilbert space and C be a nonempty closed convex subset of H. Let T : C → C be a demicontractive mapping with contraction coefficient κ. Then, the following hold:

(i) F(T) = F(P_C(I − μ(I − T))) for all μ > 0;

(ii) P_C(I − μ(I − T)) is quasi-nonexpansive, for all μ ∈ (0, 1 − κ].

The set C of H is called proximinal if for each x ∈ H there exists z ∈ C such that

‖x-z‖ =inf{‖x-y‖:y∈C}=dist(x,C).

It is clear that every nonempty closed convex subset of a real Hilbert space is proximinal. We denote by CB(C) and KC(C) the families of all nonempty closed bounded subsets, and nonempty compact convex subsets of C, respectively. The Pompeiu-Hausdorff metric ℋ on CB(C) is defined by

H(A,B):=max{supx∈A dist(x,B),supy∈B dist(y,A)},   ∀A,B∈CB(C).

Let S : C → CB(C) be a multi-valued mapping. An element x ∈ C is said to be a fixed point of S if x ∈ Sx. The fixed point set of S is denoted by F(S) = {x ∈ C : x ∈ Sx}.

Definition 2.12

A multi-valued mapping S : C → CB(C) is said to

(i) be nonexpansive if ℋ(Sx,Sy) ≤ ||x − y|| for all x, y ∈ C;

(ii) be quasi-nonexpansive if F(S) ≠ ∅︀ and ℋ(Sx,Sz) ≤ ||x − z|| for all x ∈ C and z ∈ F(S);

(iii) satisfy condition (E_μ) if there exists μ ≥ 1 such that for each x, y ∈ C, dist(x,Sy)≤μdist(x,Sx)+‖x-y‖.

We say that S satisfies condition (E) whenever S satisfies (E_μ) for some μ ≥ 1.

From the above definitions, it is clear that:

(i) if S is nonexpansive, then T satisfies the condition (E₁);

(ii) if C is compact, then S is hemicompact.

We now give an example for the class of quasi-nonexpansiveness multi-valued mapping satisfying the condition (E).

Example 2.13

Let C = [0,∞) and S : C → CB(C) be defined by

Sx=[x4,x2]   for all x∈C.

Then S is quasi-nonexpansive and satisfies condition (E).

Although the condition (E) implies the quasi-nonexpansiveness for single-valued mappings, but it is not true for multi-valued mappings as the following example.

Example 2.14.([25])

Let C = [0,∞) and S : C → CB(C) be defined by

Sx=[x,2x]   for   all x∈C.

Then S satisfies condition (E) and is not quasi-nonexpansive.

Notice also that the classes of (multi-valued) quasi-nonexpansive mappings and mappings satisfying condition (E) are different (see Examples 2.15).

Example 2.15.([8])

Let C = [−1, 1] and S : C → CB(C) be defined by

Sx={{x1+∣x∣sin(1x)}ifx≠0;{0}ifx=0.

Then S is quasi-nonexpansive and does not satisfy condition (E).

Lemma 2.16.([16])

Let {t_n} be a sequence of real numbers such that there exists a subsequence {n_i} of {n} such that t_ni < t_ni+1for 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:

tτ(n)≤tτ(n)+1,         tn≤tτ(n)+1.

In fact,

τ(n)=max{k≤n:tk<tk+1}.

Lemma 2.17.([23])

In Hilbert space H, the following inequality holds:

‖x+y‖2≤ ‖x‖2+ 2⟨y,x+y⟩,   ∀x,y∈H.

Lemma 2.18.([28])

Let H be a Hilbert space. Let x₁, x₂,..., x_N ∈ H and α₁, α₂,..., α_N be real numbers in [0, 1] such that ∑i=1Nαi=1. Then,

‖∑i=1Nαixi‖2=∑i=1Nαi‖xi‖2-∑1≤i,j≤Nαiαj‖xi-xj‖2.

Lemma 2.19.([26])

Let {a_n} be a sequence of nonnegative real numbers, let {b_n} be a sequence in (0, 1) with ∑n=1∞bn=∞, let {d_n} be a sequence of nonnegative real numbers with ∑n=1∞dn<∞, and let {c_n} be a sequence of real numbers with lim sup_n→∞c_n ≤ 0. Suppose that the following inequality holds:

an+1≤(1-bn)an+bncn+dn,   ∀n∈ℕ.

Then lim_n→∞a_n = 0.

In this section, we show strong convergence theorems for the sequence generated by the hybrid algorithm (3.1) based on extragradient algorithm which solve the problem of finding of four sets, i.e., F(T), F(S), Sol(f,C), and VI(B,C).

Now, let C be a nonempty, closed and convex subset of a real Hilbert space H and f : H × H → ℝ be a bifunction such that f(x, x) = 0, for all x ∈ C. In order to find a point in F(T) ∩ F(S) ∩ Sol(f,C) ∩ VI(B,C) ≠ ∅︀, we make use of the following blanket assumptions:

Assumptions

(A1) f is monotone on C;

(A2) F is Lipschitz-type continuous on C with constants c₁ > 0 and c₂ > 0;

(A3) f(x, ·) is convex and subdifferentiable on C, for all x ∈ C;

(A4) f is jointly weakly continuous on C × C in the sense that, if x, y ∈ C and {x_n}, {y_n} ⊂ C converge weakly to x and y, respectively, then f(x_n, y_n) → f(x, y) as n→∞.

We are now in a position to prove our main results.

Theorem 3.1

Let H be a real Hilbert space and C be a nonempty closed convex subset of H. Let f be a bifunction satisfying assumptionson C, T : C → C be a demicontractive single-valued mapping with contraction coefficient κ, S : C → KC(C) be a quasi-nonexpansive multi-valued mapping satisfying the condition (E), and B : C → H be a δ-inverse strongly monotone mapping. Assume that ℱ = F(T)∩F(S)∩Sol(f,C)∩VI(B,C) ≠ ∅︀ and Sp = {p} for all p ∈ ℱ. Let h : C → C be a k-contraction. For x₁ ∈ C, let {x_n}, {y_n}, {z_n}, and {w_n} be sequences generated by

(3.1){wn=argminw∈C [λnf(xn,w)+12‖w-xn‖2],zn=argminz∈C [λnf(wn,z)+12‖z-xn‖2],yn=αnzn+βnun+γnPC(I-μn(I-T))zn+ζnPC(I-ηnB)zn,xn+1=σnh(xn)+(1-σn)yn,   ∀n∈ℕ,

where u_n ∈ Sz_n and {α_n}, {β_n}, {γ_n}, {ζ_n}, {σ_n}, {μ_n}, {η_n}, and {λ_n} satisfy the following conditions:

(C1) {σ_n} ⊂ (0, 1), lim_n→∞σ_n = 0, ∑n=1∞σn=∞;

(C2) {λn}⊂[a,b]⊂(0,1L), where L = max{2c₁, 2c₂};

(C3) μ_n ∈ (0, 1 − κ] with lim_n→∞μ_n = 0;

(C4) η_n ∈ [d, e] for some d, e ∈ (0, 2δ) and for all n ∈ ℕ;

(C5) 0 < a ≤ α_n, β_n, γ_n, ζ_n ≤ b < 1 and α_n + β_n + γ_n + ζ_n = 1 for all n ∈ ℕ.

Then the sequence {x_n} converges strongly to q ∈ ℱ, which solves the variational inequality

⟨q-h(q),x-q⟩≥0,   ∀x∈ℱ.