The purpose of this paper is to establish some strong convergence theorems for a common fixed point of two total quasi-ϕ-asymptotically nonexpansive mappings in Banach space by means of the hybrid method in mathematical programming. The results presented in this paper extend and improve on the corresponding ones announced by Martinez-Yanes and Xu (2006), Plubtieng and Ungchittrakool (2007), Qin et al. (2009), and many others.

1. Introduction

Let E be a smooth Banach space and let E* be the topological dual of E. Let the function ϕ:E×E→R+ be defined by
(1)ϕ(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2,∀x,y∈E,
where J is the normalized duality mapping from E to E*. Let K be a nonempty subset of E and T:K→K a nonlinear mapping. Recall that a mapping T is called nonexpansive if ∥Tx-Ty∥≤∥x-y∥,∀x,y∈K. A point x∈K is said to be a fixed point of T [1] if Tx=x. Let F(T):={x∈K:Tx=x} be the set of fixed points of T. A point p∈K is said to be an asymptotic fixed point of T [1] if K contains a sequence {xn} which converges weakly to p such that the strong limn→∞(xn-Txn)=0. The set of asymptotic fixed points of T will be denoted by F^(T).

Recall that a mapping T:K→K is called (i) relatively nonexpansive [2] if F^(T)=F(T)≠ϕ and ϕ(p,Tx)≤ϕ(p,x),∀x∈K,∀p∈F(T); (ii) closed if for any sequence {xn}⊂K with xn→x and Txn→y, Tx=y; (iii) quasi-ϕ-nonexpansive [3] if F(T)≠ϕ and ϕ(p,Tx)≤ϕ(p,x),∀x∈K,∀p∈F(T); (iv) quasi-ϕ-asymptotically nonexpansive [3] if F(T)≠ϕ and there exists a sequence {μn}⊂[0,∞) with limn→∞μn=0 such that ϕ(p,Tnx)≤(1+μn)ϕ(p,x),∀x∈K,∀p∈F(T),∀n∈N; (v) total quasi-ϕ-asymptotically nonexpansive [4–6] if F(T)≠ϕ and there exist nonnegative real sequences {μn}{νn} with limn→∞μn=limn→∞νn=0 and a strictly increasing continuous function ζ:R+→R+ such that ϕ(p,Tnx)≤ϕ(p,x)+μnζ(ϕ(p,x))+νn,∀x∈K,∀p∈F(T),∀n∈N; (vi) uniformly L-Lipschitz continuous [4] if there exists a constant L>0 such that ∥Tnx-Tny∥≤L∥x-y∥,∀x,y∈K,∀n≥1.

It is well known that total quasi-ϕ-asymptotically nonexpansive mappings contain relatively nonexpansive mappings, quasi-ϕ-nonexpansive mappings, and quasi-ϕ-asymptotically nonexpansive mappings as its special cases; see [4, 6, 7] for more details.

Iterative approximation of fixed points for nonexpansive mappings has been considered by many papers for either the Mann iteration [8] or the Ishkawa iteration [9]; see, for example, [9–12] and the references therein.

In 1967, Halpern [11] considered the following explicit iteration which was referred to as Halpern iteration:
(2)x0∈Karbitrarilychosen,xn+1=αnx0+(1-αn)Txn,∀n≥0,
where T is nonexpansive. He proves the strong convergence of {xn} to a point of T provided that αn=n-θ, where θ∈(0,1).

Recently, many authors improved and generalized the results of Halpern [11] by means of different methods; see, for example, [2, 4, 10, 12–17] and the references therein. In general, there are two ways as follows.

One of the methods is to combine Halpern iteration with Mann iteration; see, for example, C.E. Chidume and C.O. Chidume [10] and Hu [12].

Another method is to use the hybrid projection algorithm method; see, for example, Martinez-Yanes and Xu [15], Plubtieng and Ungchittrakool [16], and Qin et al. [17].

Motivated and inspired by [2, 4, 10, 12–17], the purpose of this paper is to modify Halpern iteration (2) by means of both methods above for two total quasi-ϕ-asymptotically nonexpansive mappings and then to prove the strong convergence in the framework of Banach spaces. The results presented in this paper extend and improve the corresponding results of Martinez-Yanes and Xu [15], Plubtieng and Ungchittrakool [16], Qin et al. [17], and others.

Throughout the paper, we denote ⇀ for weak convergence and → for strong convergence.

2. Preliminaries

Let E be a real Banach space endowed with the norm ∥·∥ and let E* be the topological dual of E. For all x∈E and x*∈E*, the value of f at x is denoted by 〈x,f〉 and is called the duality pairing. Then, the normalized duality mapping J:E→2E* is defined by
(3)J(x)={f∈E*:〈x,f〉=∥x∥2=∥f∥2},∀x∈E.
It is well known that the operator J is well defined and J is the identity mapping if and only if E is a Hilbert space. But in general, J is nonlinear and multiple-valued.

The following basic properties for a Banach space E can be found in [18].

If E is uniformly smooth, then J is unformly continuous on each bounded subset of E.

If E is reflexive and strictly convex, then J-1 is norm-weak-continuous.

If E is a smooth, strictly convex, and reflexive Banach space, then J is single-valued, one-to-one, and onto.

A Banach space E is uniformly smooth if and only if E* is uniformly convex.

Each uniformly convex Banach space E has the Kadec-Klee property; that is, for any sequence {xn}∈E, if xn⇀x∈E and ∥xn∥→∥x∥, then xn→x.

Every uniformly smooth Banach space is reflexive.

Moreover, we have the Lyapunov functional ϕ defined by (1). It is obvious from the definition of the function ϕ that we have the following property.

Property 1 (see [<xref ref-type="bibr" rid="B13">19</xref>]).

It follows Albert [20] that the generalized projection ΠK:E→K is defined by ΠK(x)=arginfy∈Kϕ(y,x),∀x∈E. So we have the following lemmas.

Lemma 1 (see [<xref ref-type="bibr" rid="B2">20</xref>]).

Let E be a smooth, strictly convex, and reflexive Banach space and let K be a nonempty closed convex subset of E. Then the following conclusions hold:

ϕ(x,ΠKy)+ϕ(ΠKy,y)≤ϕ(x,y),∀x∈K,∀y∈E;

if x∈K and z∈K, then z=ΠKx⇔〈z-y,Jx-Jz〉≥0,∀y∈K;

for x,y∈E, ϕ(x,y)=0 if and only if x=y.

Remark 2.

If H is a real Hilbert space, then ϕ(x,y)=∥x-y∥2 and ΠK=PK (the metric projection of H onto a closed convex subset K).

Lemma 3 (see [<xref ref-type="bibr" rid="B2">20</xref>]).

Let E be a uniformly convex and smooth Banach space and let {xn}, {yn} be sequences of E. If ϕ(xn,yn)→0 (as n→∞) and either {xn} or {yn} is bounded, then xn-yn→0 (as n→∞).

Lemma 4 (see [<xref ref-type="bibr" rid="B4">4</xref>]).

Let E be a smooth, strictly convex, and reflexive Banach space with Kadec-Klee property, and let K be a nonempty closed convex subset of E. Let T:K→K be a closed and totally quasi-ϕ-asymptotically nonexpansive mapping with nonnegative real sequences {μn},{νn} and a strictly increasing continuous function ζ:R+→R+ such that limn→∞μn=limn→∞νn=0 and ζ(0)=0. If ν1=0, then the fixed point set F(T) of T is a closed and convex subset of K.

Lemma 5 (see [<xref ref-type="bibr" rid="B9">21</xref>]).

Let E be a uniformly convex Banach space and let Br(0)={x∈E:∥x∥≤r} be a closed ball of E. Then there exists a continuous strictly increasing convex function g:[0,∞)→[0,∞) with g(0)=0 such that
(4)∥λx+μy+γz∥2≤λ∥x∥2+μ∥y∥2+γ∥z∥2-λμg(∥x-y∥),
for all x,y,z∈Br(0) and λ,μ,γ∈[0,1] with λ+μ+γ=1.

3. Main Results

To prove the main results, we need the following hypotheses for the sequences {αn} and {βn(j)}(j=1,2,3):

0<αn<1 for all n∈N∪{0} and limn→∞αn=0,

0≤βn(1),βn(2),βn(3)≤1, βn(1)+βn(2)+βn(2)=1 for all n∈N∪{0}, limn→∞βn(1)=0 and liminfn→∞βn(2)βn(3)>0.

The main results of this paper are stated as follows.

Theorem 6.

Let E be a real strictly convex and uniformly smooth Banach space with Kadec-Klee property, and let K be a nonempty closed convex subset of E. Let S,T:K→K be two closed, uniformly L-Lipschitz continuous and totally quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences {μn(j)},{νn(j)},j=1,2 and a strictly increasing continuous function ζ:R+→R+ such that limn→∞μn(j)=limn→∞νn(j)=0,j=1,2 and ζ(0)=0. Let {xn} be a sequence defined by
(5)x0∈C1=Karbitrarilychosen,yn=J-1(αnJx0+(1-αn)Jzn),zn=J-1(βn(1)Jxn+βn(2)JTnxn+βn(3)JSnxn),Cn+1={z∈Cn:ϕ(z,yn)≤αnϕ(z,x0)+(1-αn)ϕ(z,xn)+(1-αn)ξn},xn+1=ΠCn+1x0,∀n≥0,
where ξn=(βn(2)μn(2)+βn(3)μn(1))supp∈Fζ(ϕ(p,xn))+(βn(2)νn(2)+βn(3)νn(1)). Assume that the sequences {αn} and {βn(j)}(j=1,2,3) satisfy hypotheses (i) and (ii). If F:=F(S)∩F(T)≠ϕ is bounded and ν1=0, then the sequence {xn} converges strongly to ΠFx0.

Proof.

We divide the proof of Theorem 6 into six steps.

We first show that F is a closed and convex subset in K.

By Lemma 4, it is trivial to show that F(S) and F(T) are two closed and convex subsets of K. Therefore F is closed and convex in K.

Next we prove that Cn is a closed and convex subset in K for all n≥1.

As a matter of fact, by hypothesis, C1=K is closed and convex. Suppose that Ck is closed and convex for some k≥1. From the definition of ϕ, we may know that
(6)Ck+1={z∈Ck:ϕ(z,yk)≤αkϕ(z,x0)+(1-αk)ϕ(z,xk)+(1-αk)ξk},={∥yk∥2+(1-αk)ξkz∈:2〈z,Jxk-Jyk〉+2αk〈z,Jx0〉≤αk∥x0∥2+(1-αk)∥xk∥2-∥yk∥2+(1-αk)ξk}∩Ck
and thus Ck+1 is closed and convex. Therefore, by induction principle, Cn are closed and convex for all n≥1. This also shows that ΠFx0 is well defined.

Now we show that F⊂Cn,∀n≥1.

Indeed, it is obvious that F⊂C1=K. Suppose that F⊂Cn for some n≥1. Hence for any u∈F⊂Cn, by Lyapunov functional (1) and Lemma 5, we have
(7)ϕ(u,zn)=ϕ(u,J-1(βn(1)Jxn+βn(2)JTnxn+βn(3)JSnxn))=∥u∥2-2〈u,βn(1)Jxn+βn(2)JTnxn+βn(3)JSnxn〉+∥βn(1)Jxn+βn(2)JTnxn+βn(3)JSnxn∥2≤∥u∥2-2βn(1)〈u,Jxn〉-2βn(2)〈u,JTnxn〉-2βn(3)〈u,JSnxn〉+βn(1)∥xn∥2+βn(2)∥Tnxn∥2+βn(3)∥Snxn∥2-βn(2)βn(3)g(∥Tnxn-Snxn∥)≤βn(1)ϕ(u,xn)+βn(2)ϕ(u,Tnxn)+βn(3)ϕ(u,Snxn)-βn(2)βn(3)g(∥Tnxn-Snxn∥)≤βn(1)ϕ(u,xn)+βn(2)[ϕ(u,xn)+μn(2)ζ(ϕ(u,xn))+νn(2)]+βn(3)[ϕ(u,xn)+μn(1)ζ(ϕ(u,xn))+νn(1)]-βn(2)βn(3)g(∥JTnxn-JSnxn∥)≤ϕ(u,xn)+ξn-βn(2)βn(3)g(∥JTnxn-JSnxn∥)≤ϕ(u,xn)+ξn.
Moreover, it follows from Property 1(ii) that we have
(8)ϕ(u,yn)=ϕ(u,J-1(αnJx0+(1-αn)Jzn))≤αnϕ(u,x0)+(1-αn)ϕ(u,zn).
Combining (7) with (8), we have
(9)ϕ(u,yn)≤αnϕ(u,x0)+(1-αn)(ϕ(u,xn)+ξn).
This shows that u∈Cn+1. Therefore, by induction principle, we have F⊂Cn for all n≥1.

Next we prove that {xn} converges strongly to some point p*∈K.

In fact, since xn=ΠCnx0, it follows from Lemma 1(ii) that we have
(10)〈xn-y,Jx0-Jxn〉≥0,∀y∈Cn.
Again since F⊂Cn for all n≥1, we have
(11)〈xn-u,Jx0-Jxn〉≥0,∀u∈F.
It follows from Lemma 1(i) that for each u∈F and for each n≥1 we have
(12)ϕ(xn,x0)=ϕ(ΠCnx0,x0)≤ϕ(u,x0)-ϕ(u,xn)≤ϕ(u,x0).
Therefore {ϕ(xn,x0)} is bounded, and hence {xn} is bounded.

On the other hand, since xn=ΠCnx0 and xn+1=ΠCn+1x0∈Cn+1⊂Cn, we have ϕ(xn,x0)≤ϕ(xn+1,x0) for all n≥1. This implies that {ϕ(xn,x0)} is nondecreasing, and so limn→∞ϕ(xn,x0) exists. By the construction of {ϕ(xn,x0)}, we have Cm⊂Cn and xm=ΠCmx0∈Cn for any positive integer m>n. Therefore, using Lemma 1(i), we have
(13)ϕ(xn+m,xn)=ϕ(xn+m,ΠCnx0)≤ϕ(xn+m,x0)-ϕ(xn,x0)
for all n≥1. Since limn→∞ϕ(xn,x0) exists, we obtain that
(14)ϕ(xn+m,xn)⟶0(asn⟶∞),∀m≥1.
Thus, by Lemma 3, we have
(15)limn→∞∥xn+m-xn∥=0,∀m≥1.
This implies that the sequence {xn} is a Cauchy sequence in K. Since K is a nonempty closed subset of Banach space E, this implies that it is complete. Hence there exists an p*∈K such that
(16)limn→∞xn=p*.
By the way, it is easy from (16) to see that
(17)limn→∞ξn=limn→∞[(βn(2)μn(2)+βn(3)μn(1))supp∈Fζ(ϕ(p,xn))+(βn(2)νn(2)+βn(3)νn(1))(βn(2)μn(2)+βn(3)μn(1))supp∈Fζ(ϕ(p,xn))]=0.

Now we prove that p*∈F.

Since xn+1∈Cn+1 and by the structure of Cn+1, we have
(18)ϕ(xn+1,yn)≤αnϕ(xn+1,x0)+(1-αn)(ϕ(xn+1,xn)+ξn).
Hence, by means of limits (14), (17) and limn→∞αn=0 and using Lemma 3, we get that limn→∞∥xn+1-yn∥=0. But
(19)∥xn-yn∥≤∥xn-xn+1∥+∥xn+1-yn∥.
Thus limn→∞∥xn-yn∥=0. This implies that
(20)limn→∞yn=p*.

Meanwhile, it follows from (7) and (8) that we have
(21)(1-αn)βn(2)βn(3)g(∥JTnxn-JSnxn∥)≤αnϕ(u,x0)+(1-αn)(ϕ(u,xn)+ξn)-ϕ(u,yn)≤αn(ϕ(u,x0)-ϕ(u,yn))+(1-αn)ξn+(1-αn)(ϕ(u,xn)-ϕ(u,yn)),
where g:[0,∞)→[0,∞) is a continuous strictly increasing convex function with g(0)=0. Thus, by virtue of (16), (17), (20), (21), limn→∞αn=0, and liminfn→∞βn(2)βn(3)>0, we have limn→∞g(∥JTnxn-JSnxn∥)=0, and hence limn→∞∥JTnxn-JSnxn∥=0. Since J-1 is also uniformly norm-to-norm continuous on bounded sets, we get
(22)limn→∞∥Tnxn-Snxn∥=limn→∞∥J-1(JTnxn-JSnxn)∥=0.
Next, using the convexity of ∥·∥2, Property 1(iv), and hypothesis (ii), we obtain that
(23)ϕ(Tnxn,zn)=ϕ(Tnxn,J-1(βn(1)Jxn+βn(2)JTnxn+βn(3)JSnxn))=∥Tnxn∥2-2〈Tnxn,βn(1)Jxn+βn(2)JTnxn+βn(3)JSnxn〉+∥βn(1)Jxn+βn(2)JTnxn+βn(3)JSnxn∥2≤∥Tnxn∥2-2βn(1)〈Tnxn,Jxn〉-2βn(2)〈Tnxn,JTnxn〉-2βn(3)〈Tnxn,JSnxn〉+βn(1)∥xn∥2+βn(2)∥Tnxn∥2+βn(3)∥Snxn∥2=βn(1)ϕ(Tnxn,xn)+βn(3)ϕ(Tnxn,Snxn)⟶0(asn⟶∞).
Hence, from Property 1(ii), hypothesis (i), and (23), we have
(24)ϕ(Tnxn,yn)=ϕ(Tnxn,J-1(αnJx0+(1-αn)Jzn))≤αnϕ(Tnxn,x0)+(1-αn)ϕ(Tnxn,zn)⟶0hhhhhhhhhhhhhihhhhhhhh(asn⟶∞).
Thus, by Lemma 3, we have limn→∞∥Tnxn-yn∥=0 and hence
(25)∥Tnxn-xn∥≤∥Tnxn-yn∥+∥yn-xn∥⟶0mmmmmmmmmiiiiiiiuuukkkkkki(asn⟶∞).
Moreover, we observe that
(26)∥Snxn-xn∥≤∥Snxn-Tnxn∥+∥Tnxn-xn∥⟶0kokokokokokkkkkkkkkkiiiiuuuyuyuyu(asn⟶∞).
Since limn→∞xn=p* and E is uniformly smooth, it follows from (25) and (26) that we have
(27)limn→∞Tnxn=p*,(28)limn→∞Snxn=p*.
Next, by the assumption that T is uniformly L-Lipschitz continuous, we have
(29)∥Tn+1xn-Tnxn∥≤∥Tn+1xn-Tn+1xn+1∥+∥Tn+1xn+1-xn+1∥+∥xn+1-xn∥+∥xn-Tnxn∥≤(L+1)∥xn+1-xn∥+∥Tn+1xn+1-xn+1∥+∥xn-Tnxn∥.
Combining (15) and (25) with (29), we have
(30)limn→∞∥Tn+1xn-Tnxn∥=0.
Hence, it follows from (27) that we have
(31)limn→∞Tn+1xn=p*,
that is,
(32)limn→∞TTnxn=p*.
Furthermore, in view of (27) and the closeness of T, it yields that Tp*=p*. Similarly, we have Sp*=p*. Therefore, this implies that p*∈F.

Finally we prove that p*=ΠFx0 and so xn→ΠFx0.

Let w=ΠFx0. Since w∈F⊂Cn and xn→ΠFx0, we have ϕ(xn,x0)≤ϕ(w,x0),∀n≥1. This implies that
(33)ϕ(p*,x0)=limn→∞ϕ(xn,x0)≤ϕ(w,x0).
Since w=ΠFx0, this implies that p*=w. Therefore, xn→ΠFx0. This completes the proof.

If S=T,βn(1)=0 or βn(1)=βn(3)=0 for all n≥1 in Theorem 6, then we have the following corollary.

Corollary 7.

Let E,K be the same as Theorem 6. Let T:K→K be closed, uniformly L-Lipschitz continuous, and totally quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences {μn},{νn} and a strictly increasing continuous function ζ:R+→R+ such that limn→∞μn=limn→∞νn=0 and ζ(0)=0. Let {xn} be a sequence defined by
(34)x0∈C1=Karbitrarilychosen,yn=J-1(αnJx0+(1-αn)JTnxn),Cn+1={z∈Cn:ϕ(z,yn)≤αnϕ(z,x0)+(1-αn)ϕ(z,xn)+ξn},xn+1=ΠCn+1x0,∀n≥0,
where ξn=(1-αn)(μnsupp∈Fζ(ϕ(p,xn))+νn) and ΠCn+1 is the generalized projection from E onto Cn+1. Assume that the sequence {αn} satisfies hypothesis (i). If F:=F(T)≠ϕ is bounded and ν1=0, then the sequence {xn} converges strongly to ΠFx0.

It is well known that the quasi-ϕ-asymptotically nonexpansive mappings are a special case of the total quasi-ϕ-asymptotically nonexpansive mappings. So we obtain the following theorem.

Theorem 8.

Let E,K be the same as Theorem 6. Let S,T:K→K be two closed, uniformly L-Lipschitz continuous, and quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences {μn(1)},{μn(2)} such that limn→∞μn(1)=limn→∞μn(2)=0. Let {xn} be a sequence defined by
(35)x0∈C1=Karbitrarilychosen,yn=J-1(αnJx0+(1-αn)Jzn),zn=J-1(βn(1)Jxn+βn(2)JTnxn+βn(3)JSnxn),Cn+1={z∈Cn:ϕ(z,yn)≤αnϕ(z,x0)+(1-αn)ϕ(z,xn)+(1-αn)ξn},xn+1=ΠCn+1x0,∀n≥0,
where ξn=(βn(2)μn(2)+βn(3)μn(1))supp∈Fϕ(p,xn). Assume that the sequences {αn} and {βn(j)}(j=1,2,3) satisfy hypotheses (i) and (ii). If F:=F(S)∩F(T)≠ϕ is bounded and ν1=0, then the sequence {xn} converges strongly to ΠFx0.

Proof.

The proof is similar to that of Theorem 6 and hence we omit it. This completes the proof.

If S=T,βn(1)=0 or βn(1)=βn(3)=0 for all n≥1 in Theorem 8, then we have the following corollary.

Corollary 9.

Let E,K be the same as Theorem 6. Let T:K→K be closed, uniformly L-Lipschitz continuous, and quasi-ϕ-asymptotically nonexpansive mappings with nonnegative real sequences {μn} such that limn→∞μn=0. Let {xn} be a sequence defined by
(36)x0∈C1=Karbitrarilychosen,yn=J-1(αnJx0+(1-αn)JTnxn),Cn+1={z∈Cn:ϕ(z,yn)≤αnϕ(z,x0)+(1-αn)ϕ(z,xn)+ξn},xn+1=ΠCn+1x0,∀n≥0,
where ξn=(1-αn)μnsupp∈Fϕ(p,xn) and ΠCn+1 is the generalized projection from E onto Cn+1. Assume that the sequence {αn} satisfies hypothesis (i). If F:=F(T)≠ϕ is bounded, then the sequence {xn} converges strongly to ΠFx0.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors would like to thank the referees for reading this paper carefully, providing valuable suggestions and comments, and pointing out a major error in the original version of this paper.

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