Let C be a nonempty closed convex subset of a real Hilbert space H. The fixed point problem for the mapping T:C→C is to find x ∈ C such that

x=Tx.

We denote the fixed point set of a mapping T by Fix(T).

Definition 1.1. Let T:C→C be a mapping. Then

(i) a mapping T is called nonexpansive if

Tx−Ty≤x−y,∀x,y∈C;

(ii) T is said to be κ-strictly pseudo-contractive if there exists a constant κ∈[0,1)such that

Tx−Ty2≤x−y2+κ(I−T)x−(I−T)y2,∀x,y∈C.

Note that the class of κ-strict pseudo-contractions strictly includes the class of nonexpansive mappings, that is, a nonexpansive mapping is a 0-strictly pseudo-contractive mapping.

In 2008, Kohsaka and Takahashi [6] introduced a nonspreading mapping T in Hilbert space H as follows:

(1.1)2‖Tx−Ty‖2≤‖Tx−y‖2+‖x−Ty‖2,∀x,y∈C.

In 2009, it is shown by Iemoto and Takahashi [2] that (1.1) is equivalent to the following equation.

‖Tx−Ty‖2≤‖x−y‖2+2⟨x−Tx,y−Ty⟩, for all x,y∈C.

Later, in 2011, Osilike and Isiogugu [13] proposed a κ-strictly pseudononspreading mapping, that is, a mapping T:C→C is said to be a κ-strictly pseudononspreading mapping if there exists κ∈[0,1) such that

‖Tx−Ty‖2≤‖x−y‖2+κ‖(I−T)x−(I−T)y‖2+2⟨x−Tx,y−Ty⟩, for all x,y∈C.

Obviously, every nonspreading mapping is a κ-strictly pseudononspreading mapping, that is, a nonspreading mapping is a 0-stricly pseudononspreading mapping.

Many mathematicians tried to proposed iterative algorithms and proved the strong convergence theorems for a nonspreading mapping and a strictly pseudononspreading mapping in Hilbert space to find their fixed points, see, for instance, [7, 13, 8, 1].

Over the past decades, many others have constructed various types of iterative methods to approximate fixed points. The first one is the Mann iteration introduced by Mann [9] in 1953 and is defined as follows:

(1.2)x0∈H arbitrary chosen,xn+1=1−αnxn+αnTxn,∀n≥0,

where C is a nonempty closed convex subset of a normed space, T:C→C is a mapping and the sequence αn is in the interval (0,1). But this algorithm has only weak convergence. Thus, many mathematicians have been trying to modify Mann's iteration (1.1) and construct new iterative method to obtain the strong convergence theorem.

By modification of Mann's iteration (1.1), the next iteration process is referred to as Ishikawa's iteration process [3] which is defined recursively as follows:

(1.3)x0∈H arbitrary chosen,yn=βnxn+1−βnTxn,xn+1=αnxn+1−αnTyn,∀n≥0,

where αn and βn are real sequences in [0,1]. He also obtain the strong convergence theorem for the iterative method (1.3) converging to a fixed point of mapping T. Observe that if β_n=1, then the Ishikawa's iteration (1.3) reduces to the Mann's iteration (1.2).

In 2000, Moudafi [11] introduced the viscosity approximation method for nonexpansive mapping S as follows:

Let C be a closed convex subset of a real Hilbert space H and let S:C→C be a nonexpansive mapping such that Fix(S) is nonempty. Let f:C→C be a contraction, that is, there exists α∈(0,1) such that fx−fy≤αx−y,∀x,y∈C, and let xn be a sequence defined by

(1.4)x1∈C arbitrary chosen,xn+1=11+ϵnSxn+ϵn1+ϵnfxn,∀n∈ℕ,

where εn⊂(0,1) satisfies certain conditions. Then the sequence xn converges strongly to z∈Fix(S), where z=PFix(S)f(z) and PFix(S) is the metric projection of H onto Fix(S).

In 2006, using the concept of the viscosity approximation method (1.4), Marino and Xu [10] introduced the general iterative method and obtained the strong convergence theorem.

Let T:H→H be a nonexpansive mapping with Fix(T)≠∅. Let f:H→H be a contractive mapping on H and let xn be generated by

(1.5)x0∈H arbitrary chosen,xn+1=I−αnATxn+αnγfxn,n≥0,

where αn is a sequence in (0,1) satisfying the appropriate conditions. Then xn converges strongly to a fixed point x˜ of T which solves the variational inequality:

(A−γf)x˜,x˜−z≤0,z∈Fix(T).

In 2015, Yao et al. [18] proposed the intermixed algorithm for two strict pseudocontractions S and T as follows:

Algorithm 1.2. For arbitrarily given x0∈C,y0∈C, let the sequences xn and yn be generated iteratively by

(1.6)xn+1=1−βnxn+βnPCαnfyn+1−k−αnxn+kTxn,n≥0,yn+1=1−βnyn+βnPCαngxn+1−k−αnyn+kSyn,n≥0,

where T:C→C is a λ-strict pseudo-contraction, f:C→H is a ρ₁-contraction and g:C→H is a ρ2-contraction, k∈(0,1−λ) is a constant and αn, βn are two real number sequences in (0,1).

Furthermore, under some control conditions, they proved that the iterative sequences xn and yn defined by (1.6) converges independently to PFix(T)fy* and PFix(S)gx*, respectively, where x*∈Fix(T) and y*∈Fix(S).

Motivated by Yao et al. [18], in 2018, Suwannaut [15] introduce the S-intermixed iteration for two finite families of nonlinear mappings without considering the constant k as in the following algorithm:

Algorithm 1.3. Starting with x1,y1,z1∈C, let the sequences xn, yn and zn be defined by

(1.7)xn+1=1−βnxn+βnαnf1yn+1−αnSxn,yn+1=1−βnyn+βnαnf2xn+1−αnTyn,n≥1,

where S,T:C→C, is a nonlinear mapping with Fix(S)∩FixT≠∅, fi:C→C is a contractive mapping with coefficients αi;i=1,2 and βn,αn are real sequences in (0,1), ∀n≥1.

Under appropriate conditions, they prove a strong convergence theorem for finding a common solution of two finite families of equilibrium problems.

Inspired by the previous work, we introduce the new iterative method called the three-step intermixed iteration for two finite families of nonlinear mappings as the following algorithm:

Algorithm 1.4. Starting with x1,y1,z1∈C, let the sequences xn, yn and zn be defined by

(1.8)xn+1=δnxn+ηnS1xn+μnPCαnγ1f1(yn)+(I−αnA1)T1xn,yn+1=δnyn+ηnS2yn+μnPCαnγ2f2(zn)+(I−αnA2)T2yn,zn+1=δnzn+ηnS3zn+μnPCαnγ3f3(xn)+(I−αnA3)T3zn,n≥1,

where Si,Ti:C→C, where i=1,2,3, is nonlinear mappings with Fix(Si)∩FixTi≠∅,∀i=1,2,3, f_i is a contractive mapping with coefficients ξi,Ai:C→C is a strongly positive linear bounded operator with coefficient βi>0 and 0<γ<βξ, where γ=maxi∈{1,2,3}γi, ξ=maxi∈{1,2,3}ξi and β=mini∈{1,2,3}βi, δn, ηn, μn and αn are real sequences in (0,1) and δn+ηn+μn=1,∀n≥1.

Remark 1.5. From Algorithm 1.2 and 1.4, we observe that Algorithm 1.4 can be seen as a modification and extension of Algorithm 1.2 in senses that we choose to consider the three-step intermixed algorithm for approximating fixed points of two finite families of nonlinear mappings and we study the general iterative method without a constant k.

Remark 1.6. If we take Si≡I, γi=1 and Ai≡I for i=1,2,3, then the iterative method (1.8) reduces to

(1.9)xn+1=1−μnxn+μnαnf1(yn)+(1−αn)T1xn,yn+1=1−μnyn+μnαnf2(zn)+(1−αn)T2yn,zn+1=1−μnzn+μnαnf3(xn)+(1−αn)T3zn.

The iteration (1.9) is a modification and improvement of iteration (1.7) in sense that it extends to three-step iteration for three nonlinear mappings.

Inspired by the previous research, we introduce the three-steps intermixed iteration for two finite families of nonlinear mappings. Under appropriate conditions, we prove a strong convergence theorem for finding a common fixed point of a strictly pseudo-contractive mapping and a strictly pseudononspreading mapping. Finally, we give a numerical example for the main theorem in a space of real numbers.

We denote weak convergence and strong convergence by notations ‵‵⇀′′ and ‵‵→′′, respectively. For every x ∈ H, there is a unique nearest point P_Cx in C such that

‖x−PCx‖≤‖x−y‖,∀y∈C.

Such an operator P_C is called the metric projection of H onto C.

We now recall the following definition and well-known lemmas.

Lemma 2.1. ([16]) For a given z ∈ H and u ∈ C,

u=PCz⇔⟨u−z,v−u⟩≥0,∀v∈C.

Furthermore, P_C is a firmly nonexpansive mapping of H onto C and satisfies

PCx−PCy2≤PCx−PCy,x−y,∀x,y∈H.

Lemma 2.2. ([12]) Each Hilbert space H satisfies Opial's condition, i.e., for any sequence xn⊂H with xn⇀x, the inequality

liminfn→∞xn−x<liminfn→∞xn−y

holds for every y ∈ H with y ≠ x.

Lemma 2.3. ([13]) Let H be a real Hilbert space. Then the following results hold:

(i) For all x,y ∈ H and α∈[0,1],

αx+(1−α)y2=αx2+(1−α)y2−α(1−α)x−y2,

(ii) ‖x+y‖2≤‖x‖2+2⟨y,x+y⟩, for each x,y∈H.

Lemma 2.4. ([17]) Let {sn} be a sequence of nonnegative real numbers satisfying

sn+1≤(1−αn)sn+δn,∀n≥0,

where α_n is a sequence in (0,1) and {δn} is a sequence such that

(1) ∑ n=1∞αn=∞,

(2) limsupn→∞δnαn≤0 or ∑ n=1∞|δn|<∞.

Then, limn→∞sn=0.

Lemma 2.5. ([10]) Assume A is a strongly positive linear bounded operator on a Hilbert space H with coefficient β>0 and 0<δ<‖A‖−1. Then ‖I−δA‖≤1−δβ.

Lemma 2.6. ([4, 14]) Let C be a nonempty closed convex subset of a real Hilbert space H and let T:C→C be a κ-strictly pseudo-contractive mapping with Fix(T)≠∅. Then, we there hold the following statement:

(i) Fix(T)=VI(C,I−T);

(ii) For every u ∈ C and v∈Fix(T),

‖PC(I−λ(I−T))u−v‖≤‖u−v‖, for u∈C,v∈Fix(T) and λ∈(0,1−κ).

By applying Remark 2.10 in [5], we easily obtain the following result:

Lemma 2.7. Let S:C→C be a κ-strictly pseudo nonspreading mapping with Fix(S)≠∅. Define T:C→C by Tx:=(1−λ)x+λSx, where λ∈(0,1−κ). Then there hold the following statement:

(i) Fix(S) = Fix(T)

(ii) T is a quasi-nonexpansive mapping, that is,

‖Tx−y‖≤‖x−y‖, for every x∈C and y∈Fix(S).

Proof. It is clear to prove that (i) holds.

(ii) Let x ∈ H and y∈Fix(S). Then we derive

‖Tx−y‖2=‖(1−λ)(x−y)+λ(Sx−y)‖2    =(1−λ)‖x−y‖2+λ‖Sx−y‖2−λ(1−λ)‖Sx−x‖2    ≤(1−λ)‖x−y‖2+λ‖x−y‖2+κ‖x−Sx‖2−λ(1−λ)‖Sx−x‖2    =‖x−y‖2+κλ‖x−Sx‖2−λ(1−λ)‖Sx−x‖2    =‖x−y‖2−λ((1−κ)−λ)‖x−Sx‖2    ≤‖x−y‖2.

This implies that T is a quasi-nonexpansive mapping.

Let C be a nonempty closed convex subset of a real Hilbert space H. For i=1,2,3, let fi:C→C be a contractive mappings with a coefficient ξ_i and ξ=maxi∈{1,2,3}ξi, let Si:C→C be a κ_i-strictly pseudo-contractive mapping and Wi:C→C be ρ_i-strictly pseudo-nonspreading mapping with Ωi=Fix(Si)∩Fix(Wi)≠∅. For each i=1,2,3, define a mapping Tni:C→C by Tnix=1−ωnx+ωnWix, for all x ∈ C, and let Ai:C→C be a strongly positive linear bounded operator with a coefficient βi>0 and 0<γ<βξ, where γ=maxi∈{1,2,3,}γi and β=mini∈{1,2,3}βi. Let xn, yn and zn be sequences generated by x1,y1,z1∈C and

xn+1=δnxn+ηnPCI−λnI−S1xn  +μnPCαnγ1f1(yn)+(I−αnA1)Tn1xn,yn+1=δnyn+ηnPCI−λnI−S2yn  +μnPCαnγ2f2(zn)+(I−αnA2)Tn2yn,zn+1=δnzn+ηnPCI−λnI−S3zn  +μnPCαnγ3f3(xn)+(I−αnA3)Tn3zn,

for n ≥ 1, where αn, δn, ηn, μn⊂(0,1), λn⊂(0,1−κ), κ=mini∈{1,2,3}, ωn⊂(0,1−ρ), where ρ=mini∈{1,2,3} and δn+μn+ηn=1 satisfying the following conditions:

(i) limn→∞αn=0 and ∑ n=1∞αn=∞;

(ii) 0<τ≤δn,ηn,μn,≤υ<1, for some τ,υ>0;

(iii) ∑ n=1∞λn<∞, ∑ n=1∞ωn<∞;

(iv) ∑ n=1∞α n+1−αn<∞, ∑ n=1∞δ n+1−δn<∞, ∑ n=1∞μ n+1−μn<∞,

∑ n=1∞η n+1−ηn<∞, ∑ n=1∞λ n+1−λn<∞, ∑ n=1∞ω n+1−ωn<∞.

Then the sequences xn, yn and zn converge strongly to x˜=PΩ1(I−A1)x˜+γf1 y ˜ , y˜=PΩ2(I−A2)y˜+γf2 z ˜ and z˜=PΩ3(I−A3)z˜+γf3 x ˜ , respectively.

Proof. The proof of this theorem will be divided into five steps.

Step 1. We show that xn is bounded.

Since αn→0 as n→∞, without loss of generality, we may assume that αn<1Ai, for all i=1,2,3 and n∈ℕ.

Let x*∈Ω1, y*∈Ω2, z*∈Ω3, β=mini∈{1,2,3}βi, ξ=maxi∈{1,2,3}ξi and γ=maxi∈{1,2,3}γi. Then we have

(3.1) xn+1−x*≤‖δnxn−x*+ηnPCI−λnI−S1xn−x* +μnPCαnγ1f1(yn)+(I−αnA1)Tn1xn−x*‖≤δnxn−x*+ηnPCI−λnI−S1xn−x* +μnPCαnγ1f1(yn)+(I−αnA1)Tn1xn−x*≤1−μnxn−x*+μnαnγ1f1yn−A1x*+I−αnA1Tn1xn−x*≤1−μnxn−x* +μnαnγ1ξ1yn−y*+αnγ1f1y*−A1x*+1−αnβxn−x*≤1−μnαnβxn−x*+μnαnγξyn−y*+μnαnγ1f1y*−A1x*.

Similarly, we get

(3.2) yn+1−y*≤1−μnαnβyn−y*+μnαnγξzn−z*+μnαnγ2f2z*−A2y*

and

(3.3) zn+1−z*≤1−μnαnβzn−z*+μnαnγξxn−x*+μnαnγ3f3x*−A3z*.

Combining (3.1), (3.2) and (3.3), we have

 xn+1−x*+yn+1−y*+zn+1−z*≤1−μnαnβ−γξxn−x*+yn−y*+zn−z* +μnαnγ1f1y*−A1x*+γ2f2z*−A2y*+γ3f3x*−A3z*.

By induction, we can derive that

 xn−x*+yn−y*+zn−z*≤max{x1−x*+y1−y*+z1−z*, γ1f1y*−A1x*+γ2f2z*−A2y*+γ3f3x*−A3z*β−γξ},

for every n∈ℕ. This implies that xn,yn and zn are bounded.

Step 2. Claim that limn→∞xn+1−xn=0.

First, we let

un=PCαnγ1f1(yn)+(I−αnA1)Tn1xn, vn=PCαnγ2f2(zn)+(I−αnA2)Tn2yn

and

wn=PCαnγ3f3(xn)+(I−αnA3)Tn3zn.

Then, observe that

(3.4) un−un−1=∥PCαnγ1f1yn+I−αnA1Tn1xn −PCαn−1γ1f1yn−1+I−αn−1A1Tn−11xn−1∥≤αnγ1f1yn−f1yn−1+γ1αn−αn−1f1yn−1 +I−αnA1Tn1xn−Tn−11xn−1 +I−αnA1Tn−11xn−1−I−αn−1A1Tn−11xn−1≤αnγ1ξ1yn−yn−1+γ1αn−αn−1f1yn−1 +1−αnβ1(1−ωnxn−xn−1 +ωn−ωn−1xn−1+ωnW1xn−W1xn−1+ωn−ωn−1W1xn−1) +αn−αn−1A1Tn1xn−1≤αnγξyn−yn−1+αn−αn−1γf1yn−1+A1Tn1xn−1 +1−αnβ(1−ωnxn−xn−1+ωn−ωn−1xn−1+W1xn−1 +ωnW1xn−W1xn−1).

By the definition of x_n, we obtain

 xn+1−xn≤δnxn−xn−1+δn−δn−1xn−1 +ηnPCI−λn(I−S1)xn−PCI−λn−1(I−S1)xn−1 +ηn−ηn−1PCI−λn−1(I−S1)xn−1+μnun−un−1 +μn−μn−1un−1≤δnxn−xn−1+δn−δn−1xn−1+ηnxn−xn−1 +ηnλn(I−S1)xn−λn−1(I−S1)xn−1 +ηn−ηn−1PCI−λn−1(I−S1)xn−1 +μn[αnγξyn−yn−1+αn−αn−1γf1yn−1+A1Tn1xn−1 +1−αnβ(1−ωnxn−xn−1+ωn−ωn−1xn−1+W1xn−1 +ωnW1xn−W1xn−1)]+μn−μn−1un−1 ≤1−μnxn−xn−1+δn−δn−1xn−1 +ηnλn(I−S1)xn−(I−S1)xn−1 +ηnλn−λn−1(I−S1)xn−1 +ηn−ηn−1PCI−λn−1(I−S1)xn−1 +μnαnγξyn−yn−1+αn−αn−1γf1yn−1+A1Tn1xn−1 +μn1−αnβxn−xn−1+ωn−ωn−1xn−1+W1xn−1 +ωnW1xn−W1xn−1+μn−μn−1un−1 (3.5)≤1−μnαnβxn−xn−1+μnαnγξyn−yn−1+δn−δn−1xn−1 +λn(I−S1)xn−(I−S1)xn−1+λn−λn−1(I−S1)xn−1 +ηn−ηn−1PCI−λn−1(I−S1)xn−1 +αn−αn−1γf1yn−1+A1Tn1xn−1 +ωn−ωn−1xn−1+W1xn−1+ωnW1xn−W1xn−1 +μn−μn−1un−1.

Using the same method as derived in (3.5), we have

(3.6) yn+1−yn≤1−μnαnβyn−yn−1+μnαnγξzn−zn−1+δn−δn−1yn−1 +λn(I−S2)yn−(I−S2)yn−1+λn−λn−1(I−S2)yn−1 +ηn−ηn−1PCI−λn−1(I−S2)yn−1 +αn−αn−1γf2zn−1+A2Tn2yn−1 +ωn−ωn−1yn−1+W2yn−1+ωnW2yn−W2yn−1 +μn−μn−1vn−1

and

(3.7) zn+1−zn≤1−μnαnβzn−zn−1+μnαnγξxn−xn−1+δn−δn−1zn−1 +λn(I−S3)zn−(I−S3)zn−1+λn−λn−1(I−S3)zn−1 +ηn−ηn−1PCI−λn−1(I−S3)zn−1 +αn−αn−1γf3xn−1+A3Tn3zn−1 +ωn−ωn−1zn−1+W3zn−1+ωnW3zn−W3zn−1 +μn−μn−1wn−1.

From (3.5), (3.6) and (3.7), then we get

 xn+1−xn+yn+1−yn+zn+1−zn≤1−μnαnβ−γξxn−xn−1+yn−yn−1+zn−zn−1 +δn−δn−1xn−1+yn−1+zn−1+λn((I−S1)xn−(I−S1)xn−1 +(I−S2)yn−(I−S2)yn−1+(I−S3)zn−(I−S3)zn−1) +λn−λn−1(I−S1)xn−1+(I−S2)yn−1+(I−S3)zn−1 +ηn−ηn−1(PCI−λn−1(I−S1)xn−1+PCI−λn−1(I−S2)yn−1 +PCI−λn−1(I−S3)zn−1)+αn−αn−1(γ(f1xn−1+f2yn−1+f3zn−1)+A1Tn1xn−1+A2Tn2yn−1+A3Tn3zn−1)+ωn−ωn−1(xn−1+yn−1+zn−1+W1xn−1+W2yn−1+W3zn−1)+ωnW1xn−W1xn−1+W2yn−W2yn−1+W3zn−W3zn−1+μn−μn−1un−1+vn−1+wn−1.

Applying Lemma 2.4 and the condition(iii), (iv), we can conclude that

(3.8)xn+1−xn→0,yn+1−yn→0 and zn+1−zn→0 as n→∞.

Step 3. Prove that limn→∞un−PCI−λn(I−S1)un=limn→∞un−Tn1un=0.

To show this, take u˜n=αnγ1f1(yn)+(I−αnA1)Tn1xn. Then we derive that

 xn+1−x*2=δnxn−x*+ηnPCI−λn(I−S1)xn−x*+μnun−x*2≤δnxn−x*2+ηnPCI−λn(I−S1)xn−x*2+μnun−x*2 −δnηnxn−PCI−λn(I−S1)xn2≤1−μnxn−x*2+μnαnγ1f1yn−A1Tn1xn+Tn1xn−x*2 −δnηnxn−PCI−λn(I−S1)xn2≤1−μnxn−x*2 +μn[Tn1xn−x*2+2αnγ1f1 y n −A1Tn1xn,u˜ n−x*] −δnηnxn−PCI−λn(I−S1)xn2≤xn−x*2+2μnαnγ1f1 y n −A1Tn1xnu˜ n−x* −δnηnxn−PCI−λn(I−S1)xn2,

which implies that

 δnηn x n −PC I−λ n(I−S1)x n 2≤ x n −x* 2− xn+1 −x* 2+2μnαnγ1f1 yn −A1Tn1xnu˜ n−x*≤xn−x n+1 xn − x * + x n+1 − x *  +2μnαnγ1f1 yn −A1Tn1xnu˜ n−x*.

By (3.8), the condition (i) and (ii), thus we get

(3.9)xn−PCI−λn(I−S1)xn→0 as n→∞.

Observe that

xn+1−xn=ηnPCI−λn (I−S1 )xn−xn+μnun−xn.

This follows that

μnun−xn≤ηnPCI−λn (I−S1 )xn−xn+xn+1−xn.

From (3.8) and (3.9), we obtain

(3.10)un−xn→0 as n→∞.

Observe that

un−PCI−λn(I−S1)un≤un−xn+xn−PCI−λn(I−S1)xn+PCI−λn(I−S1)xn−PCI−λn(I−S1)un≤2un−xn+xn−PCI−λn(I−S1)xn+λnI−S1xn−I−S1un.

Hence, by (3.9), (3.10) and the condition (iii), we obtain

(3.11)un−PCI−λn(I−S1)un→0 as n→∞.

Applying the same argument as (3.11), we also obtain

(3.12)vn−PCI−λn(I−S2)vn→0 and wn−PCI−λn(I−S3)wn→0 as n→∞.

Consider

xn+1−un≤xn+1−xn+xn−un,

by (3.8) and (3.10), we have

(3.13)xn+1−un→0 as n→∞.

Since

xn−Tn1xn≤xn−xn+1+xn+1−un+un−Tn1xn    ≤xn−xn+1+xn+1−un+ u˜n−Tn1xn    =xn−xn+1+xn+1−un+αnγ1f1yn−A1Tn1xn,

from (3.8), (3.13) and the condition (i), we get

(3.14)xn−Tn1xn→0 as n→∞.

Consider

un−Tn1un≤un−xn+xn−Tn1xn+Tn1xn−Tn1un    ≤2un−xn+xn−Tn1xn+ωnW1xn−W1un.

Therefore, by (3.10), (3.14) and the condition (iii), we have

(3.15)un−Tn1un→0 as n→∞.

Applying the same method as (3.15), we also have

(3.16)vn−Tn2vn→0 and wn−Tn3wn→0 as n→∞.

Step 4. Claim that

limsupn→∞γ1f1y˜−A1x˜,un−x˜≤0, where     x˜=PΩ1(I−A1)x˜+γ1f1y˜.

First, take a subsequence unk of un such that

(3.17)limsupn→∞γ1f1y˜−A1x˜,un−x˜=limk→∞γ1f1y˜−A1x˜,unk−x˜.

Since xn is bounded, then we can assume that xnk⇀x^ as k→∞. From (3.10), we obtain unk⇀x^ as k→∞.

Next, assume x^∉FixS1. Since FixS1=FixPCI−λnk(I−S1), then we get x^≠PCI−λnk(I−S1)x^.

By nonexpansiveness of P_C, (3.11), the condition (iii) and the Opial's condition, we obtain

liminfk→∞unk−x^<liminfk→∞unk−PCI−λnk(I−S1)x^   ≤liminfk→∞[unk−PCI−λnk(I−S1)unk   +PCI−λnk(I−S1)unk−PCI−λnk(I−S1)x^]   ≤liminfk→∞[unk−PCI−λnk(I−S1)unk+unk−x^   +λnk(I−S1)unk−(I−S1)x^]   =liminfk→∞unk−x^.

This is a contradiction. Therefore

(3.18)x^∈FixS1.

Assume that x^∉FixW1. Because FixW1=FixTnk1, then we have x^≠Tnk1x^.

From (3.15) and the Opial's condition, we deduce that

liminfk→∞unk−x^<liminfk→∞unk−Tnk1x^      ≤liminfk→∞unk−Tnk1unk+Tnk1unk−Tnk1x^      ≤liminfk→∞[unk−Tnk1unk+unk−x^      +ωnkI−W1unk−I−W1x^]      =liminfk→∞unk−x^.

This is a contradiction. Thus we obtain

(3.19)x^∈FixW1.

By (3.18) and (3.19), this yields that

(3.20)x^∈Ω1=FixS1∩FixW1.

Since xnk⇀x^ as k→∞, (3.20) and Lemma 2.1, we can derive that

(3.21)limsupn→∞γ1f1y˜−A1x˜,un−x˜=limk→∞γ1f1y˜−A1x˜,unk−x˜            =γ1f1y˜−A1x˜,x^−x˜            =γ1f1y˜−A1x˜+x˜−x˜,x^−x˜            ≤0.

Following the same method as (3.21), we easily obtain that

(3.22)limsupn→∞γ2f2z˜−A2y˜,vn−y˜≤0 and limsupn→∞γ3f3x˜−A3z˜,wn−z˜≤0.

Step 5. Finally, Prove that the sequence xn, yn and zn converge strongly to x˜=PΩ1(I−A1)x˜+γ1f1 y ˜ , y˜=PΩ2(I−A2)y˜+γ2f2 z ˜ and z˜=PΩ3(I−A3)z˜+γ3f3 x ˜ , respectively.

By firmly-nonexpansiveness of P_C, we derive that

un−x˜2=PCun˜−x˜2   ≤ u n ˜−x˜,un−x˜   =αn γ1 f1 y n −A1 x ˜ + I−α n A1 T n1 x n − x ˜ ,un−x˜   =αnγ1f1 y n −A1x˜,un−x˜+ I−α n A1 T n1 x n − x ˜ ,un−x˜   ≤αnγ1f1 y n −f1y˜ ,un−x˜+αnγ1f1y˜ −A1x˜,un−x˜   + I−α n A1 T n1 x n − x ˜ un−x˜   ≤αnγ1ξ1yn−y˜un−x˜+αnγ1f1y˜ −A1x˜,un−x˜   +1−αnβ1xn−x˜un−x˜   ≤αnγξ2 y n −y ˜ 2+ u n − x ˜ 2+αnγ1f1y˜ −A1x˜,un−x˜   +1−αnβ2 x n − x ˜ 2+ u n − x ˜ 2   =αnγξ2yn−y˜2+1−αnβ2xn−x˜2+1−αn(β−γξ)2un−x˜2   +αnγ1f1y˜ −A1x˜,un−x˜,

which yields that

(3.23)un−x˜2≤αnγξ1+αn(β−γξ)yn−y˜2+1−αnβ1+αn(β−γξ)xn−x˜2    +2αn1+αn(β−γξ)γ1f1y˜ −A1x˜,un−x˜.

From the definition of x_n and (3.23), we get

(3.24) xn+1−x˜2≤δnx n−x˜2+ηnPCI−λ n(I−S1)x n−x˜2+μnu n−x˜2≤1−μnx n−x˜2+μn[αnγξ1+αn(β−γξ)y n−y˜2+1−αnβ1+αn(β−γξ)x n−x˜2 +2αn1+αn(β−γξ)γ1f1y˜ −A1x˜,un−x˜]=1−μn+ μn 1−αn β 1+αn (β−γξ)x n−x˜2+μnαnγξ1+αn(β−γξ)y n−y˜2 +2μnαn1+αn(β−γξ)γ1f1y˜ −A1x˜,un−x˜=1− μn 1+αn (β−γξ)−μn 1−αn β 1+αn (β−γξ)x n−x˜2+μnαnγξ1+αn(β−γξ)y n−y˜2=1− μn αn (2β−γξ) 1+αn (β−γξ)x n−x˜2+μnαnγξ1+αn(β−γξ)y n−y˜2 +2μnαn1+αn(β−γξ)γ1f1y˜ −A1x˜,un−x˜.

Similarly, as derived above, we also have

(3.25)yn+1−y˜2≤1− μn αn (2β−γξ) 1+αn (β−γξ)yn−y˜2+μnαnγξ1+αn(β−γξ)zn−z˜2     +2μnαn1+αn(β−γξ)γ2f2z˜ −A2y˜,vn−y˜

and

(3.26)zn+1−z˜2≤1− μn αn (2β−γξ) 1+αn (β−γξ)zn−z˜2+μnαnγξ1+αn(β−γξ)xn−x˜2     +2μnαn1+αn(β−γξ)γ3f3x˜ −A3z˜,wn−z˜.

From (3.24), (3.25) and (3.26), we deduces that

(3.27)xn+1−x˜2+yn+1−y˜2+zn+1−z˜2≤1− 2μn αn (β−γξ) 1+αn (β−γξ) xn − x ˜ 2+ yn − y ˜ 2+ zn − z ˜ 2+2μnαn1+αn(β−γξ)(γ1f1y˜ −A1x˜,un−x˜+γ2f2z˜ −A2y˜,vn−y˜+γ3f3x˜ −A3z˜,wn−z˜).

By (3.21), (3.22), the condition (i) and Lemma 2.4, this implies by (3.27) that the sequences xn, yn and zn converge strongly to x˜=PΩ1(I−A1)x˜+γ1f1 y ˜ , y˜=PΩ2(I−A2)y˜+γ2f2 z ˜ and z˜=PΩ3(I−A3)z˜+γ3f3 x ˜ , respectively. This completes the proof.

The following Corollary is a direct consequence of Theorem 3.1.

Corollary 3.2. Let C be a nonempty closed convex subset of a real Hilbert space H. For i=1,2,3, let fi:C→C be a contractive mappings with a coefficient ξ_i and ξ=maxi∈1,2,3ξi and let Wi:C→C be ρ_i-strictly pseudo-nonspreading mapping with Fix(Wi)≠∅. Define a mapping Tni:C→C by Tnix=1−ωnx+ωnWix, for all x ∈ C and i=1,2,3. Let xn, yn and zn be sequences generated by x1,y1,z1∈C and

(3.28)xn+1=1−μnxn+μnαnf1(yn)+(1−αn)Tn1xn,yn+1=1−μnyn+μnαnf2(zn)+(1−αn)Tn2yn,zn+1=1−μnzn+μnαnf3(xn)+(1−αn)Tn3zn,

for n≥1, where αn, μn⊂(0,1) and ωn⊂(0,1−ρ), where ρ=mini∈{1,2,3}ρi, satisfying the following conditions:

• (i) limn→∞αn=0 and ∑ n=1∞αn=∞;

• (ii) 0<τ≤μn,≤υ<1, for some τ,υ>0;

• (iii) ∑ n=1∞ωn<∞;

• (iv) ∑ n=1∞α n+1−αn<∞, ∑ n=1∞μ n+1−μn<∞, ∑ n=1∞λ n+1−λn<∞,∑ n=1∞ω n+1−ωn<∞

Then the sequences xn, yn and zn converge strongly to x˜=PFix(W1)f1 y ˜ , y˜=PFix(W2)f2 z ˜ and z˜=PFix(W3)f3 x ˜ , respectively.

Proof. For each i=1,2,3, put Si≡I, γi=1 and Ai≡I. Then, by Theorem 3.1, we obtain the desired result.