The purpose of this paper is to introduce a new hybrid projection method for finding a common
element of the set of common fixed points of two relatively quasi-nonexpansive mappings, the set of
the variational inequality for an α-inverse-strongly monotone, and the set of solutions of the generalized
equilibrium problem in the framework of a real Banach space. We obtain a strong convergence theorem
for the sequences generated by this process in a 2-uniformly convex and uniformly smooth Banach space.
Base on this result, we also get some new and interesting results. The results in this paper generalize,
extend, and unify some well-known strong convergence results in the literature.
1. Introduction
Let E be a real Banach space, E* the dual space of E. A Banach space E is said to be strictly convex if ∥(x+y)/2∥<1 for all x,y∈E, with ∥x∥=∥y∥=1 and x≠y. Let U={x∈E:∥x∥=1} be the unit sphere of E. Then a Banach space E is said to be smooth if the limit
limt→0∥x+ty∥-∥x∥t
exists for each x,y∈U. It is also said to be uniformly smooth if the limit is attained uniformly for x,y∈U. Let E be a Banach space. The modulus of convexity of E is the function δ:[0,2]→[0,1] defined by
δ(ε)=inf{1-∥x+y2∥:x,y∈E,∥x∥=∥y∥=1,∥x-y∥≥ε}.
A Banach space E is uniformly convex if and only if δ(ε)>0 for all ε∈(0,2]. Let p be a fixed real number with p≥2. A Banach space E is said to be p-uniformly convex if there exists a constant c>0 such that δ(ε)≥cεp for all ε∈[0,2]; see [1, 2] for more details. Observe that every p-uniform convex is uniformly convex. One should note that no Banach space is p-uniform convex for 1<p<2. It is well known that a Hilbert space is 2-uniformly convex, uniformly smooth. For each p>1, the generalized duality mapping Jp:E→2E* is defined by
Jp(x)={x*∈E*:〈x,x*〉=∥x∥p,∥x*∥=∥x∥p-1}
for all x∈E. In particular, J=J2 is called the normalized duality mapping. If E is a Hilbert space, then J=I, where I is the identity mapping. It is also known that if E is uniformly smooth, then J is uniformly norm-to-norm continuous on each bounded subset of E.
Let E be a real Banach space with norm ∥·∥ and E* denotes the dual space of E. Consider the functional defined by
ϕ(x,y)=∥x∥2-2〈x,Jy〉+∥y∥2∀x,y∈E.
Observe that, in a Hilbert space H, (1.4) reduces to ϕ(x,y)=∥x-y∥2, x,y∈H. The generalized projection ΠC:E→C is a map that assigns to an arbitrary point x∈E, the minimum point of the functional ϕ(x,y), that is, ΠCx=x̅, where x̅ is the solution to the minimization problem
ϕ(x̅,x)=infy∈Cϕ(y,x);
existence and uniqueness of the mapping ΠC follow from the properties of the functional ϕ(x,y) and strict monotonicity of the mapping J (see, e.g., [3–7]). In Hilbert spaces, ΠC=PC. It is obvious from the definition of function ϕ that
(∥y∥-∥x∥)2≤ϕ(y,x)≤(∥y∥+∥x∥)2,∀x,y∈E.
Remark 1.1.
If E is a reflexive, strictly convex, and smooth Banach space, then for x,y∈E, ϕ(x,y)=0 if and only if x=y. It is sufficient to show that if ϕ(x,y)=0, then x=y. From (2.13), we have ∥x∥=∥y∥. This implies that 〈x,Jy〉=∥x∥2=∥Jy∥2. From the definition of J, one has Jx=Jy. Therefore, we have x=y; see [5, 7] for more details.
Next, we give some examples which are closed relatively quasi-nonexpansive (see [8]).
Example 1.2.
Let ΠC be the generalized projection from a smooth, strictly convex and reflexive Banach space E onto a nonempty closed and convex subset C of E. Then, ΠC is a closed relatively quasi-nonexpansive mapping from E onto C with F(ΠC)=C.
Let E be a real Banach space and let C be a nonempty closed and convex subset of E, and A:C→E* be a mapping. The classical variational inequality problem for a mapping A is to find x*∈C such that
〈Ax*,y-x*〉≥0,∀y∈C.
The set of solutions of (1.4) is denoted by VI(A,C). Recall that A is called
monotone if
〈Ax-Ay,x-y〉≥0,∀x,y∈C,
an α-inverse-strongly monotone if there exists a constant α>0 such that
〈Ax-Ay,x-y〉≥α∥x-y∥2,∀x,y∈C.
Such a problem is connected with the convex minimization problem, the complementary problem, and the problem of finding a point x*∈E satisfying Ax*=0.
Let f be a bifunction from C×C to ℝ, where ℝ denotes the set of real numbers. The equilibrium problem (for short, EP) is to find x*∈C such that
f(x*,y)≥0,∀y∈C.
The set of solutions of (1.10) is denoted by EP(f). Given a mapping T:C→E*, let f(x,y)=〈Tx,y-x〉 for all x,y∈C. Then x*∈EP(f) if and only if 〈Tx*,y-x*〉≥0 for all y∈C; that is, x* is a solution of the variational inequality. Numerous problems in physics, optimization, and economics reduce to find a solution of (1.10). Some methods have been proposed to solve the equilibrium problem; see, for instance, [9–11].
Let C be a closed convex subset of E; a mapping T:C→C is said to be nonexpansive if
∥Tx-Ty∥≤∥x-y∥,∀x,y∈C.
A point x∈C is a fixed point of T provided that Tx=x. Denote by F(T) the set of fixed points of T; that is, F(T)={x∈C:Tx=x}. Recall that a point p in C is said to be an asymptotic fixed point of T [12] if C contains a sequence {xn} which converges weakly to p such that limn→∞∥xn-Txn∥=0. The set of asymptotic fixed points of T will be denoted by F(T)̂. A mapping T from C into itself is said to be relatively nonexpansive [13–15] if F(T)̂=F(T) and ϕ(p,Tx)≤ϕ(p,x) for all x∈C and p∈F(T). The asymptotic behavior of a relatively nonexpansive mapping was studied in [16–18]. T is said to be ϕ-nonexpansive, if ϕ(Tx,Ty)≤ϕ(x,y) for x,y∈C. T is said to be relatively quasi-nonexpansive if F(T)≠∅ and ϕ(p,Tx)≤ϕ(p,x) for x∈C and p∈F(T). A mapping T in a Banach space E is closed if xn→x and Txn→y, then Tx=y.
Remark 1.3.
The class of relatively quasi-nonexpansive mappings is more general than the class of relatively nonexpansive mappings [16–19] which requires the strong restriction F(T)=F(T)̂.
In Hilbert spaces H, Iiduka et al. [20] proved that the sequence {xn} defined by: x1=x∈C and
xn+1=PC(xn-λnAxn),
where PC is the metric projection of H onto C, and {λn} is a sequence of positive real numbers, and converges weakly to some element of VI(A,C).
It is well know that if C is a nonempty closed and convex subset of a Hilbert space H and PC:H→C is the metric projection of H onto C, then PC is nonexpansive. This fact actually characterizes Hilbert spaces and consequently, it is not available in more general Banach spaces. In this connection, Alber [4] recently introduced a generalized projection mapping ΠC in a Banach space E which is an analogue of the metric projection in Hilbert spaces.
In 2008, Iiduka and Takahashi [21] introduced the following iterative scheme for finding a solution of the variational inequality problem for inverse-strongly monotone A in a 2-uniformly convex and uniformly smooth Banach space E: x1=x∈C and
xn+1=ΠCJ-1(Jxn-λnAxn)
for every n=1,2,3,…, where ΠC is the generalized metric projection from E onto C, J is the duality mapping from E into E*, and {λn} is a sequence of positive real numbers. They proved that the sequence {xn} generated by (1.13) converges weakly to some element of VI(A,C).
Matsushita and Takahashi [22] introduced the following iteration: a sequence {xn} defined by
xn+1=ΠCJ-1(αnJxn+(1-αn)JTxn),
where the initial guess element x0∈C is arbitrary, {αn} is a real sequence in [0,1], T is a relatively nonexpansive mapping, and ΠC denotes the generalized projection from E onto a closed convex subset C of E. They proved that the sequence {xn} converges weakly to a fixed point of T.
In 2005, Matsushita and Takahashi [19] proposed the following hybrid iteration method (it is also called the CQ method) with generalized projection for relatively nonexpansive mapping T in a Banach space E:
They proved that {xn} converges strongly to ΠF(T)x0, where ΠF(T) is the generalized projection from C onto F(T).
Recently, Takahashi and Zembayashi [23] proposed the following modification of iteration (1.15) for a relatively nonexpansive mapping:
x0=x∈C,yn=J-1(αnJxn+(1-αn)JSxn),un∈Csuchthatf(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Hn={z∈C:ϕ(z,un)≤ϕ(z,xn)},Wn={z∈C:〈xn-z,Jx-Jxn〉≥0},xn+1=∏Hn∩Wnx,
where J is the duality mapping on E. Then, {xn} converges strongly to ΠF(S)∩EP(f)x, where ΠF(S)∩EP(f) is the generalized projection of E onto F(S)∩EP(f). Also, Takahashi and Zembayashi [24] proved the following iteration for a relatively nonexpansive mapping:
yn=J-1(αnJxn+(1-αn)JSxn),un∈Csuchthatf(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)},xn+1=∏Cn+1x,
where J is the duality mapping on E. Then, {xn} converges strongly to ΠF(S)∩EP(f)x, where ΠF(S)∩EP(f) is the generalized projection of E onto F(S)∩EP(f). Qin and Su [25] proved the following iteration for relatively nonexpansive mappings T in a Banach space E:
x0∈C,chosenarbitrarily,yn=J-1(αnJxn+(1-αn)JTzn),zn=J-1(βnJxn+(1-βn)JTxn),Cn={v∈C:ϕ(v,yn)≤αnϕ(v,xn)+(1-αn)ϕ(v,zn)},Qn={v∈C:〈Jx0-Jxn,xn-v〉≥0},xn+1=∏Cn∩Qnx0,
the sequence {xn} generated by (1.18) converges strongly to ΠF(T)x0.
In 2009, Wei et al. [26] proved the following iteration for two relatively nonexpansive mappings in a Banach space E:
x0∈C,Jzn=αnJxn+(1-αn)JTxn,Jun=(βnJxn+(1-βn)JSzn),Hn={v∈C:ϕ(v,un)≤βnϕ(v,xn)+(1-βn)ϕ(v,zn)≤ϕ(v,xn)},Wn={z∈C:〈z-xn,Jx0-Jxn〉≤0},xn+1=QHn∩Wnx0;
if {αn} and {βn} are sequences in [0,1) such that αn≤1-δ1 and βn≤1-δ2 for some δ1,δ2∈(0,1), then {xn} generated by (1.19) converges strongly to a point QF(T)∩F(S)x0, where the mapping QC of E onto C is the generalized projection. Very recently, Cholamjiak [27] proved the following iteration:
zn=ΠCJ-1(Jxn-λnAxn),yn=J-1(αnJxn+βnJTxn+γnJSzn),un∈Csuchthatf(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤ϕ(z,xn)},xn+1=∏Cn+1x0,
where J is the duality mapping on E. Assume that αn, βn, and γn are sequences in [0,1]. Then {xn} converges strongly to q=ΠFx0, where F:=F(T)∩F(S)∩EP(f)∩VI(A,C).
Motivated and inspired by Iiduka and Takahashi [21], Takahashi and Zembayashi [23, 24], Wei et al. [26], Cholamjiak [27], and Kumam and Wattanawitoon [28], we introduce a new hybrid projection iterative scheme which is difference from the algorithm (1.20) of Cholamjiak in [27, Theorem 3.1] for two relatively quasi-nonexpansive mappings in a Banach space. For an initial point x0∈E with x1=ΠC1x0 and C1=C, define a sequence {xn} as follows:
wn=ΠCJ-1(Jxn-λnAxn),zn=J-1(βnJxn+(1-βn)JTwn),yn=J-1(αnJxn+(1-αn)JSzn),un∈Csuchthatf(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤αnϕ(z,xn)+(1-αn)ϕ(z,zn)≤ϕ(z,xn)},xn+1=∏Cn+1x0,∀n≥1,
where J is the duality mapping on E. Then, we prove that under certain appropriate conditions on the parameters, the sequences {xn} and {un} generated by (1.21) converge strongly to ΠF(S)∩F(T)∩EP(f)∩VI(A,C).
The results presented in this paper improve and extend the corresponding results announced by Iiduka and Takahashi [21], Wei et al. [26], Kumam and Wattanawitoon [28], and many other authors in the literature.
2. Preliminaries
We also need the following lemmas for the proof of our main results.
Lemma 2.1 (Beauzamy [29] and Xu [30]).
If E is a 2-uniformly convex Banach space, then, for all x,y∈E we have
∥x-y∥≤2c2∥Jx-Jy∥,
where J is the normalized duality mapping of E and 0<c≤1.
The best constant 1/c in the Lemma is called the p-uniformly convex constant of E.
Lemma 2.2 (Beauzamy [29] and Zǎlinescu [31]).
If E is a p-uniformly convex Banach space and p is a given real number with p≥2, then for all x,y∈E,Jx∈Jp(x), and Jy∈Jp(y),〈x-y,Jx-Jy〉≥cp2p-2p∥x-y∥p,
where Jp is the generalized duality mapping of E and 1/c is the p-uniformly convexity constant of E.
Lemma 2.3 (Kamimura and Takahashi [6]).
Let E be a uniformly convex and smooth Banach space and let {xn} and {yn} be two sequences of E. If ϕ(xn,yn)→0 and either {xn} or {yn} is bounded, then ∥xn-yn∥→0.
Lemma 2.4 (Alber [4]).
Let C be a nonempty closed and convex subset of a smooth Banach space E and x∈E. Then, x0=ΠCx if and only if
〈x0-y,Jx-Jx0〉≥0,∀y∈C.
Lemma 2.5 (Alber [4]).
Let E be a reflexive, strictly convex, and smooth Banach space, let C be a nonempty closed and convex subset of E, and let x∈E. Then
ϕ(y,ΠCx)+ϕ(ΠCx,x)≤ϕ(y,x),∀y∈C.
Lemma 2.6 (Qin et al. [8]).
Let E be a uniformly convex and smooth Banach space, let C be a closed and convex subset of E, and let T be a closed relatively quasi-nonexpansive mapping from C into itself. Then F(T) is a closed and convex subset of C.
For solving the equilibrium problem for a bifunction f:C×C→ℝ, let us assume that f satisfies the following conditions:
f(x,x)=0 for all x∈C;
f is monotone, that is, f(x,y)+f(y,x)≤0 for all x,y∈C;
for each x,y,z∈C,
limt↓0f(tz+(1-t)x,y)≤f(x,y);
for each x∈C, y↦f(x,y) is convex and lower semi-continuous.
Lemma 2.7 (Blum and Oettli [9]).
Let C be a closed and convex subset of a smooth, strictly convex and reflexive Banach space E, let f be a bifunction from C×C to ℝ satisfying (A1)–(A4), and let r>0 and x∈E. Then, there exists z∈C such that
f(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C.
Lemma 2.8 (Combettes and Hirstoaga [10]).
Let C be a closed and convex subset of a uniformly smooth, strictly convex and reflexive Banach space E and let f be a bifunction from C×C to ℝ satisfying (A1)–(A4). For r>0 and x∈E, define a mapping Tr:E→C as follows:
Trx={z∈C:f(z,y)+1r〈y-z,Jz-Jx〉≥0,∀y∈C},
for all x∈C. Then the following holds:
Tr is single-valued;
Tr is a firmly nonexpansive-type mapping, for all x,y∈E,
〈Trx-Try,JTrx-JTry〉≤〈Trx-Try,Jx-Jy〉;
F(Tr)=EP(f);
EP(f) is closed and convex.
Lemma 2.9 (Takahashi and Zembayashi [24]).
Let C be a closed and convex subset of a smooth, strictly convex, and reflexive Banach space E, let f be a bifunction from C×C to ℝ satisfying (A1)–(A4), and let r>0. Then, for x∈E and q∈F(Tr),ϕ(q,Trx)+ϕ(Trx,x)≤ϕ(q,x).
Let E be a reflexive, strictly convex, and smooth Banach space and J the duality mapping from E into E*. Then J-1 is also single value, one-to-one, surjective, and it is the duality mapping from E* into E. We make use of the following mapping V studied in Alber [4]:
V(x,x*)=∥x∥2-2〈x,x*〉+∥x*∥2
for all x∈E and x*∈E*, that is, V(x,x*)=ϕ(x,J-1(x*)).
Lemma 2.10 (Alber [4]).
Let E be a reflexive, strictly convex, and smooth Banach space and let V be as in (2.10). Then
V(x,x*)+2〈J-1(x*)-x,y*〉≤V(x,x*+y*)
for all x∈E and x*,y*∈E*.
Let A be an inverse-strongly monotone of C into E* which is said to be hemicontinuous if for all x,y∈C, the mapping F of [0,1] into E*, defined by F(t)=A(tx+(1-t)y), is continuous with respect to the weak* topology of E*. We define by NC(v)the normal cone for C at a point v∈C; that is,
NC(v)={x*∈E*:〈v-y,x*〉≥0,∀y∈C}.
Theorem 2.11 (Rockafellar [32]).
Let C be a nonempty, closed and convex subset of a Banach space E, and A a monotone, hemicontinuous mapping of C into E*. Let T⊂E×E* be a mapping defined as follows:
Tv={Av+NC(v),v∈C;∅,otherwise.
Then T is maximal monotone and T-10=VI(A,C).
3. Main Results
In this section, we establish a new hybrid iterative scheme for finding a common element of the set of solutions of an equilibrium problems, the common fixed point set of two relatively quasi-nonexpansive mappings, and the solution set of variational inequalities for α-inverse strongly monotone in a 2-uniformly convex and uniformly smooth Banach space.
Theorem 3.1.
Let C be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space E. Let f be a bifunction from C×C to ℝ satisfying (A1)–(A4) and let A be an α-inverse-strongly monotone mapping of C into E* satisfying ∥Ay∥≤∥Ay-Au∥,forally∈C and u∈VI(A,C)≠∅. Let T,S:C→C be closed relatively quasi-nonexpansive mappings such that Ω:=F(T)∩F(S)∩EP(f)∩VI(A,C)≠∅. For an initial point x0∈E with x1=ΠC1x0 and C1=C, we define the sequence {xn} as follows:
wn=ΠCJ-1(Jxn-λnAxn),zn=J-1(βnJxn+(1-βn)JTwn),yn=J-1(αnJxn+(1-αn)JSzn),un∈Csuchthatf(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤αnϕ(z,xn)+(1-αn)ϕ(z,zn)≤ϕ(z,xn)},xn+1=∏Cn+1x0,∀n≥1,
where J is the duality mapping on E, {αn} and {βn} are sequences in [0,1] such that αn≤1-δ1 and βn≤1-δ2, for some δ1,δ2∈(0,1), {rn}⊆(0,2α) and {λn}⊂[a,b] for some a,b with 0<a<b<c2α/2, where 1/c is the 2-uniformly convexity constant of E. Then {xn} converges strongly to p∈Ω, where p=ΠΩx0.
Proof.
We have several steps to prove this theorem as follows:Step 1.
We show that Cn+1 is closed and convex.
Clearly C1=C is closed and convex. Suppose that Cn is closed and convex for each n∈ℕ. Since for any z∈Cn, we know that
ϕ(z,un)≤ϕ(z,xn)
is equivalent to
2〈z,Jxn-Jun〉≤∥xn∥2-∥un∥2.
So, Cn+1 is closed and convex. Then, by induction, Cn is closed and convex for all n≥1.
Step 2.
We show that {xn} is well defined.
Put un=Trnyn for all n≥0. On the other hand, from Lemma 2.8 one has Trn is relatively quasi-nonexpansive mappings and Ω⊂C1=C. Supposing Ω⊂Ck for k∈ℕ, by the convexity of ∥·∥2, for each q∈Ω⊂Ck, we have
ϕ(q,uk)=ϕ(q,Trkyk)≤ϕ(q,yk)=ϕ(q,J-1(αkJxk+(1-αk)JSzk))=∥q∥2-2〈q,αkJxk+(1-αk)JSzk〉+∥αkJxk+(1-αk)JSzk∥2≤∥q∥2-2αk〈q,Jxk〉-2(1-αk)〈q,JSzk〉+αk∥xk∥2+(1-αk)∥Szk∥2=αkϕ(q,xk)+(1-αk)ϕ(q,Szk)≤αkϕ(q,xk)+(1-αk)ϕ(q,zk),
and so
ϕ(q,zk)=ϕ(q,J-1(βkJxk+(1-βk)JTwk))=∥q∥2-2〈q,βkJxk+(1-βk)JTwk〉+∥βkJxk+(1-βk)JTwk∥2≤∥q∥2-2βk〈q,Jxk〉-2(1-βk)〈q,JTwk〉+βk∥Jxk∥2+(1-βk)∥JTwk∥2=βkϕ(q,xk)+(1-βk)ϕ(q,Twk)≤βkϕ(q,xk)+(1-βk)ϕ(q,wk).
For all q∈Ω, we know from Lemma 2.10, that
ϕ(q,wk)=ϕ(q,ΠCJ-1(Jxk-λkAxk))≤ϕ(q,J-1(Jxk-λkAxk))=V(q,Jxk-λkAxk)≤V(q,(Jxk-λkAxk)+λkAxk)-2〈J-1(Jxk-λkAxk)-q,λkAxk〉=V(q,Jxk)-2λk〈J-1(Jxk-λkAxk)-q,Axk〉=ϕ(q,xk)-2λk〈xk-q,Axk〉+2〈J-1(Jxk-λkAxk)-xk,-λkAxk〉.
Since q∈VI(A,C) and from A being an α-inverse-strongly monotone, we get
-2λk〈xk-q,Axk〉=-2λk〈xk-q,Axk-Aq〉-2λk〈xk-q,Aq〉≤-2λk〈xk-q,Axk-Aq〉=-2αλk∥Axk-Aq∥2.
From Lemma 2.1 and A being an α-inverse-strongly monotone, we obtain
2〈J-1(Jxk-λkAxk)-xk,-λkAxk〉=2〈J-1(Jxk-λkAxk)-J-1(Jxk),-λkAxk〉≤2∥J-1(Jxk-λkAxk)-J-1(Jxk)∥∥λkAxk∥≤4c2∥JJ-1(Jxk-λkAxk)-JJ-1(Jxk)∥∥λkAxk∥=4c2∥Jxk-λkAxk-Jxk∥∥λkAxk∥=4c2∥λkAxk∥2=4c2λk2∥Axk∥2≤4c2λk2∥Axk-Aq∥2.
Substituting (3.7) and (3.8) into (3.6), we have
ϕ(q,wk)≤ϕ(q,xk)-2αλk∥Axk-Aq∥2+4c2λk2∥Axk-Aq∥2=ϕ(q,xk)+2λk(2c2λk-α)∥Axk-Aq∥2≤ϕ(q,xk).
Replacing (3.9) into (3.5), we get
ϕ(q,zk)≤ϕ(q,xk).
Substituting (3.10) into (3.4), we also have
ϕ(q,uk)≤αkϕ(q,xk)+(1-αk)ϕ(q,xk),=ϕ(q,xk).
This shows that q∈Ck+1 and hence, Ω⊂Ck+1. Hence, Ω⊂Cn for all n≥1. This implies that the sequence {xn} is well defined.
Step 3.
We show that limn→∞ϕ(xn,x0) exists and {xn} is bounded.
From xn=ΠCnx0 and xn+1=ΠCn+1x0, we have
ϕ(xn,x0)≤ϕ(xn+1,x0),∀n≥1,
and from Lemma 2.5, we have
ϕ(xn,x0)=ϕ(ΠCn(x0),x0)≤ϕ(p,x0)-ϕ(p,xn)≤ϕ(p,x0),∀p∈Ω.
From (3.12) and (3.13), then {ϕ(xn,x0)} are nondecreasing and bounded. So, we obtain that limn→∞ϕ(xn,x0) exists. In particular, by (1.6), the sequence {(∥xn∥-∥x0∥)2} is bounded. This implies that {xn} is also bounded.
Step 4.
We show that {xn} is a Cauchy sequence in C.
Since xm=ΠCmx0∈Cm⊂Cn, for m>n, by Lemma 2.5, we have
Taking m,n→∞, we have ϕ(xm,xn)→0. We have limn→∞ϕ(xn+1,x0)=0. From Lemma 2.3, we get limn→∞∥xn+1-x0∥=0. Thus {xn} is a Cauchy sequence.
Step 5.
We cliam that ∥Jun-Jxn∥→0, as n→∞.
By the completeness of E, the closedness of C and {xn} is a Cauchy sequence (from Step 4); we can assume that there exists p∈C such that {xn}→p as n→∞.
By definition of ΠCnx0, we have
ϕ(xn+1,xn)=ϕ(xn+1,ΠCnx0)≤ϕ(xn+1,x0)-ϕ(ΠCnx0,x0)=ϕ(xn+1,x0)-ϕ(xn,x0).
Since limn→∞ϕ(xn,x0) exists, we get
limn→∞ϕ(xn+1,xn)=0.
It follow form Lemma 2.3, that
limn→∞∥xn+1-xn∥=0.
Since xn+1=ΠCn+1x0∈Cn+1⊂Cn and from the definition of Cn+1, we have
ϕ(xn+1,un)≤ϕ(xn+1,xn),∀n≥1
and so
limn→∞ϕ(xn+1,un)=0.
Hence
limn→∞∥xn+1-un∥=0.
By using the triangle inequality, we obtain
∥un-xn∥=∥un-xn+1+xn+1-xn∥≤∥un-xn+1∥+∥xn+1-xn∥.
By (3.17), (3.20), we get
limn→∞∥un-xn∥=0.
Since J is uniformly norm-to-norm continuous on bounded subsets of E, we have
limn→∞∥Jun-Jxn∥=0.
Step 6.
Show that xn→p∈EP(f).
Applying (3.4) and (3.11), we get ϕ(p,yn)≤ϕ(p,xn). From Lemma 2.9 and un=Trnyn, we observe that
ϕ(un,yn)=ϕ(Trnyn,yn)≤ϕ(p,yn)-ϕ(p,Trnyn)≤ϕ(p,xn)-ϕ(p,Trnyn)=ϕ(p,xn)-ϕ(p,un)=∥p∥2-2〈p,Jxn〉+∥xn∥2-(∥p∥2-2〈p,Jun〉+∥un∥2)=∥xn∥2-∥un∥2-2〈p,Jxn-Jun〉≤∥xn-un∥(∥xn+un∥)+2∥p∥∥Jxn-Jun∥.
From (3.22), (3.23) and Lemma 2.3, we get
limn→∞∥un-yn∥=0.
Since J is uniformly norm-to-norm continuous, we obtain
limn→∞∥Jun-Jyn∥=0.
From rn>0, we have ∥Jun-Jyn∥/rn→0 as n→∞ and
f(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C.
By (A2), that
∥y-un∥∥Jun-Jyn∥rn≥1rn〈y-un,Jun-Jyn〉≥-f(un,y)≥f(y,un),∀y∈C
and un→p, we get f(y,p)≤0 for all y∈C. For 0<t<1, define yt=ty+(1-t)p. Then yt∈C which implies that f(yt,p)≤0. From (A1), we obtain that
0=f(yt,yt)≤tf(yt,y)+(1-t)f(yt,p)≤tf(yt,y).
Thus f(yt,y)≥0. From (A3), we have f(p,y)≥0 for all y∈C. Hence p∈EP(f).
Step 7.
We show that xn→p∈F(T)∩F(S).
From definition of Cn, we have
αnϕ(z,xn)+(1-αn)ϕ(z,zn)≤ϕ(z,xn)⇔ϕ(z,zn)≤ϕ(z,xn).
Since xn+1=ΠCn+1x0∈Cn+1, we have
ϕ(xn+1,zn)≤ϕ(xn+1,xn).
It follows from (3.16) that
limn→∞ϕ(xn+1,zn)=0,
again from Lemma 2.3, we get
limn→∞∥xn+1-zn∥=0.
By using the triangle inequality, we get
∥zn-xn∥≤∥zn-xn+1∥+∥xn+1-xn∥.
Again by (3.17) and (3.33), we also have
limn→∞∥zn-xn∥=0.
Since J is uniformly norm-to-norm continuous, we obtain
limn→∞∥Jzn-Jxn∥=0.
Since
∥yn-zn∥≤∥yn-un∥+∥un-xn∥+∥xn-zn∥,
from (3.22), (3.25), and (3.35), we have
limn→∞∥yn-zn∥=0.
Since J is uniformly norm-to-norm continuous, we also have
limn→∞∥Jyn-Jzn∥=0.
From (3.1), we get
∥Jyn-Jzn∥=∥αn(Jxn-Jzn)+(1-αn)(JSzn-Jzn)∥=∥(1-αn)(JSzn-Jzn)-αn(Jzn-Jxn)∥≥(1-αn)∥JSzn-Jzn∥-αn∥Jzn-Jxn∥;
it follows that
(1-αn)∥JSzn-Jzn∥≤∥Jyn-Jzn∥+αn∥Jzn-Jxn∥,
and hence
∥JSzn-Jzn∥≤11-αn(∥Jyn-Jzn∥+αn∥Jzn-Jxn∥).
Since αn≤1-δ1 for some δ1∈(0,1), (3.36), and (3.39), one has limn→∞∥JSzn-Jzn∥=0. Since J-1 is uniformly norm-to-norm continuous, we get
limn→∞∥Szn-zn∥=0.
Since
∥Sxn-xn∥≤∥Sxn-Szn∥+∥Szn-zn∥+∥zn-xn∥≤∥xn-zn∥+∥Szn-zn∥+∥zn-xn∥,
from (3.35) and (3.43), we obtain
limn→∞∥Sxn-xn∥=0.
Since S is closed and xn→p, we have p∈F(S).
On the other hand, we note that
ϕ(q,xn)-ϕ(q,un)=∥xn∥2-∥un∥2-2〈q,Jxn-Jun〉≤∥xn-un∥(∥xn+un∥)+2∥q∥∥Jxn-Jun∥.
It follows from ∥xn-un∥→0 and ∥Jxn-Jun∥→0, that
ϕ(q,xn)-ϕ(q,un)→0.
Furthermore, from (3.4) and (3.5),
ϕ(q,un)≤ϕ(q,yn)≤αnϕ(q,xn)+(1-αn)ϕ(q,zn)≤αnϕ(q,xn)+(1-αn)[βnϕ(q,xn)+(1-βn)ϕ(q,wn)]=αnϕ(q,xn)+(1-αn)βnϕ(q,xn)+(1-αn)(1-βn)ϕ(q,wn)≤αnϕ(q,xn)+(1-αn)βnϕ(q,xn)+(1-αn)(1-βn)×[ϕ(q,xn)-2λn(α-2c2λn)∥Axn-Aq∥2]=αnϕ(q,xn)+(1-αn)βnϕ(q,xn)+(1-αn)(1-βn)ϕ(q,xn)-(1-αn)(1-βn)2λn(α-2c2λn)∥Axn-Aq∥2=ϕ(q,xn)-(1-αn)(1-βn)2λn(α-2c2λn)∥Axn-Aq∥2,
and hence
δ1δ22a(α-2ac2)∥Axn-Aq∥2≤(1-αn)(1-βn)2λn(α-2c2λn)∥Axn-Aq∥2≤ϕ(q,xn)-ϕ(q,un).
From (3.47) and (3.49), we have
∥Axn-Aq∥→0.
From Lemma 2.5, Lemma 2.10, and (3.8), we compute
ϕ(xn,wn)=ϕ(xn,ΠCJ-1(Jxn-λnAxn))≤ϕ(xn,J-1(Jxn-λnAxn))=V(xn,Jxn-λnAxn)≤V(xn,(Jxn-λnAxn)+λnAxn)-2〈J-1(Jxn-λnAxn)-xn,λnAxn〉=ϕ(xn,xn)+2〈J-1(Jxn-λnAxn)-xn,-λnAxn〉=2〈J-1(Jxn-λnAxn)-xn,-λnAxn〉≤4λn2c2∥Axn-Aq∥2≤4b2c2∥Axn-Aq∥2.
Applying Lemmas 2.3 and (3.50), we obtain that
∥xn-wn∥→0.
Again since J is uniformly norm-to-norm continuous on bounded set, we have
∥Jxn-Jwn∥→0.
Since
∥zn-wn∥≤∥zn-xn∥+∥xn-wn∥,
by (3.35) and (3.52), we have
limn→∞∥zn-wn∥=0,
and hence
limn→∞∥Jzn-Jwn∥=0.From (3.1) we obtain that∥Jzn-Jwn∥=∥βnJxn+(1-βn)JTwn-Jwn∥≥(1-βn)∥JTwn-Jwn∥-βn∥Jwn-Jxn∥,
and hence
(1-βn)∥JTwn-Jwn∥≤∥Jzn-Jwn∥+βn∥Jwn-Jxn∥,
so
∥JTwn-Jwn∥≤11-βn∥Jzn-Jwn∥+βn∥Jwn-Jxn∥.By (3.53), (3.56) andcondition βn≤1-δ2 for some δ2∈(0,1), we obtain
∥JTwn-Jwn∥→0.
Since J-1 is uniformly norm-to-norm continuous on bounded set, we obtain
∥Twn-wn∥→0.
Since xn→wn, then ∥Txn-xn∥→0. Thus by the closedness of T and xn→p, we get p∈F(T). Hence p∈F(T)∩F(S).
Step 8.
We show that xn→p∈VI(A,C).
Define T⊂E×E* by Theorem 2.11; T is maximal monotone and T-10=VI(A,C). Let (v,w)∈G(T). Since w∈Tv=Av+NC(v), we get w-Av∈NC(v).
From wn∈C, we have
〈v-wn,w-Av〉≥0.
On the other hand, since wn=ΠCJ-1(Jxn-λnAxn), then by Lemma 2.4, we have
〈v-wn,Jwn-(Jxn-λnAxn)〉≥0,
and hence
〈v-wn,Jxn-Jwnλn-Axn〉≤0.
It follows from (3.62) and (3.64), that
〈v-wn,w〉≥〈v-wn,Av〉≥〈v-wn,Av〉+〈v-wn,Jxn-Jwnλn-Axn〉=〈v-wn,Av-Axn〉+〈v-wn,Jxn-Jwnλn〉=〈v-wn,Av-Awn〉+〈v-wn,Awn-Axn〉+〈v-wn,Jxn-Jwnλn〉≥-∥v-wn∥∥wn-xn∥α-∥v-wn∥∥Jxn-Jwn∥a≥-M(∥wn-xn∥α+∥Jxn-Jwn∥a).
Where M=supn≥1∥v-wn∥. Taking the limit as n→∞ and (3.53), we obtain 〈v-p,w〉≥0. By the maximality of T, we have p∈T-10; that is, p∈VI(A,C).
Step 9.
We show that p=ΠΩx0.
From xn=ΠCnx0, we have 〈Jx0-Jxn,xn-z〉≥0, ∀z∈Cn. Since Ω⊂Cn, we also have
〈Jx0-Jxn,xn-y〉≥0,∀y∈Ω.
By taking limit n→∞, we obtain that
〈Jx0-Jp,p-y〉≥0,∀y∈Ω.
By Lemma 2.4, we can conclude that p=ΠΩx0 and xn→p as n→∞. This completes the proof.
Setting S≡T in Theorem 3.1., so, we obtain the following corollary.
Corollary 3.2.
Let C be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space E. Let f be a bifunction from C×C to ℝ satisfying (A1)–(A4) and let A be an α-inverse-strongly monotone mapping of C into E* satisfying ∥Ay∥≤∥Ay-Au∥,for ally∈C and u∈VI(A,C)≠∅. Let T:C→C be closed relatively quasi-nonexpansive mappings such that Ω:=F(T)∩EP(f)∩VI(A,C)≠∅. For an initial point x0∈E with x1=ΠC1x0 and C1=C, define a sequence {xn} as follows:
wn=ΠCJ-1(Jxn-λnAxn),zn=J-1(βnJxn+(1-βn)JTwn),yn=J-1(αnJxn+(1-αn)JTzn),un∈Csuchthatf(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤αnϕ(z,xn)+(1-αn)ϕ(z,zn)≤ϕ(z,xn)},xn+1=∏Cn+1x0,∀n≥1,
where J is the duality mapping on E. Assume that {αn} and {βn} are sequences in [0,1] such that αn≤1-δ1 and βn≤1-δ2, for some δ1,δ2∈(0,1), {rn}⊆(0,2α), and {λn}⊂[a,b] for some a,b with 0<a<b<c2α/2, where 1/c is the 2-uniformly convexity constant of E. Then {xn} converges strongly to p∈Ω, where p=ΠΩx0.
If A≡0 in Theorem 3.1, then we obtain the following corollary.
Corollary 3.3.
Let C be a nonempty closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space E. Let f be a bifunction from C×C to ℝ satisfying (A1)–(A4). Let T,S:C→C is closed relatively quasi-nonexpansive mappings such that Ω:=F(T)∩F(S)∩EP(f)≠∅. For an initial point x0∈E with x1=ΠC1x0 and C1=C, define a sequence {xn} as follows:
zn=J-1(βnJxn+(1-βn)JTwn),yn=J-1(αnJxn+(1-αn)JSzn),un∈Csuchthatf(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤αnϕ(z,xn)+(1-αn)ϕ(z,zn)≤ϕ(z,xn)},xn+1=∏Cn+1x0,∀n≥1,
where J is the duality mapping on E. Assume that {αn} and {βn} are sequences in [0,1] such that αn≤1-δ1 and βn≤1-δ2, for some δ1,δ2∈(0,1) and {rn}⊆(0,2α). Then {xn} converges strongly to p∈Ω, where p=ΠΩx0.
4. Application4.1. Complementarity Problem
Let K be a nonempty, closed and convex cone E, A a mapping of K into E*. We define its polar in E* to be the set
K*={y*∈E*:〈x,y*〉≥0,∀x∈K}.
Then the element u∈K is called a solution of the complementarity problem if
Au∈K*,〈u,Au〉=0.
The set of solutions of the complementarity problem is denoted by C(K,A).
Theorem 4.1.
Let K be a nonempty and closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E. Let f be a bifunction from K×K to ℝ satisfying (A1)–(A4) and let A be an α-inverse-strongly monotone of E into E* satisfying ∥Ay∥≤∥Ay-Au∥,forally∈K and u∈C(K,A)≠∅. Let T,S:K→K be closed relatively quasi-nonexpansive mappings and Ω:=F(T)∩F(S)∩EP(f)∩C(K,A)≠∅. For an initial point x0∈E with x1=ΠK1 and K1=K, we define the sequence {xn} as follows:
wn=ΠKJ-1(Jxn-λnAxn),zn=J-1(βnJxn+(1-βn)JTwn),yn=J-1(αnJxn+(1-αn)JSzn),un∈Csuchthatf(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈K,Cn+1={z∈Cn:ϕ(z,un)≤αnϕ(z,xn)+(1-αn)ϕ(z,zn)≤ϕ(z,xn)},xn+1=∏Cn+1x0,∀n≥1,
where J is the duality mapping on E, {αn} and {βn} are sequences in [0,1] such that αn≤1-δ1 and βn≤1-δ2, for some δ1,δ2∈(0,1), {rn}⊆(0,2α), and {λn}⊂[a,b] for some a,b with 0<a<b<c2α/2, where 1/c is the 2-uniformly convexity constant of E. Then {xn} converges strongly to p∈Ω, where p=ΠΩx0.
Proof.
As in the proof of Takahashi in [7, Lemma 7.11], we get that VI(K,A)=C(K,A). So, we obtain the result.
4.2. Approximation of a Zero of a Maximal Monotone Operator
Let B be a multivalued mapping from E to E* with domain D(B)={z∈E:Az≠∅} and range R(B)=∪{Bz:z∈D(B)}. A mapping B is said to be a monotone operator if 〈x1-x2,y1-y2〉≥0 for each xi∈D(B) and yi∈Axi,i=1,2. A monotone operator B is said to be maximal if its graph G(B)={(x,y):y∈Ax} is not property contained in the graph of any other monotone operator. We know that if B is a maximal monotone operator, then B-1(0) is closed and convex. Let E be a reflexive, strictly convex, and smooth Banach space, and let B be a monotone operator from E to E*, we know that B is maximal if and only if R(J+rB)=E* for all r>0. Let Jr:E→D(B)be defined by Jr=(J+rB)-1Jand such a Jr is called the resolvent of B. We know that Jr is a relatively nonexpansive (closed relatively quasi-nonexpansive for example; see [8]), and B-1(0)=F(Jr) for all r>0 (see [7, 33–35] for more details).
Theorem 4.2.
Let C be a nonempty and closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E. Let f be a bifunction from C×C to ℝ satisfying (A1)–(A4) and let A be α-inverse-strongly monotone of E into E* satisfying ∥Ay∥≤∥Ay-Au∥,for ally∈C and u∈VI(A,C)≠∅. Let B be a maximal monotone operator of E into E* and let Jr be a resolvent of B and a closed mapping such that Ω:=B-1(0)∩F(S)∩EP(f)∩VI(A,C)≠∅. For an initial point x0∈E with x1=ΠC1 and C1=C, we define the sequence {xn} as follows:
wn=ΠCJ-1(Jxn-λnAxn),zn=J-1(βnJxn+(1-βn)JJrwn),yn=J-1(αnJxn+(1-αn)JSzn),un∈Csuchthatf(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤αnϕ(z,xn)+(1-αn)ϕ(z,zn)≤ϕ(z,xn)},xn+1=∏Cn+1x0,∀n≥1,
where J is the duality mapping on E, {αn} and {βn} are sequences in [0,1] such that αn≤1-δ1 and βn≤1-δ2, for some δ1,δ2∈(0,1), {rn}⊆(0,2α) and {λn}⊂[a,b] for some a,b with 0<a<b<c2α/2, where 1/c is the 2-uniformly convexity constant of E. Then {xn} converges strongly to p∈Ω, where p=ΠΩx0.
Proof.
Since Jr is a closed relatively nonexpansive mapping and B-10=F(Jr). So, we obtain the result.
Corollary 4.3.
Let C be a nonempty and closed convex subset of a 2-uniformly convex and uniformly smooth Banach space E. Let f be a bifunction from C×C to ℝ satisfying (A1)–(A4) and let A be α-inverse-strongly monotone of E into E* satisfying ∥Ay∥≤∥Ay-Au∥,for ally∈C and u∈VI(A,C)≠∅. Let B be a maximal monotone operator of E into E* and let Jr be a resolvent of B and closed such that Ω:=B-1(0)∩EP(f)∩VI(A,C)≠∅. For an initial point x0∈E with x1=ΠC1 and C1=C, we define the sequence {xn} as follows:
wn=ΠCJ-1(Jxn-λnAxn),zn=J-1(βnJxn+(1-βn)JJrwn),yn=J-1(αnJxn+(1-αn)JJrzn),un∈Csuchthatf(un,y)+1rn〈y-un,Jun-Jyn〉≥0,∀y∈C,Cn+1={z∈Cn:ϕ(z,un)≤αnϕ(z,xn)+(1-αn)ϕ(z,zn)≤ϕ(z,xn)},xn+1=∏Cn+1x0,∀n≥1,
where J is the duality mapping on E, {αn} and {βn} are sequences in [0,1] such that αn≤1-δ1 and βn≤1-δ2, for some δ1,δ2∈(0,1), {rn}⊆(0,2α) and {λn}⊂[a,b] for some a,b with 0<a<b<c2α/2, where 1/c is the 2-uniformly convexity constant of E. Then {xn} converges strongly to p∈Ω, where p=ΠΩx0.
Acknowledgments
The authors would like to thank the referee for the valuable suggestions on the manuscript. Siwaporn Saewan would like to thank the Office of the Higher Education Commission, Thailand for supporting by grant fund under the program Strategic Scholarships for Frontier Research Network for the Join Ph.D. Program Thai Doctoral degree for this research. Moreover, Poom Kumam was supported by the Thailand Research Fund and the Commission on Higher Education (MRG5180034).
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