In this section, we introduce our hybrid implicit viscosity scheme for solving the GSVI (13) with hierarchical fixed point problem constraint for a nonexpansive mapping and show the strong convergence theorem. First, we list several useful and helpful lemmas.
We now state and prove our first result.
Proof.
First, let us show that the net {xt} is defined well. As a matter of fact, define the mapping St:C→C as follows:
(37)Stx=tf(x)+(I-tA)[G(Tx)-θtFG(Tx)], ∀x∈C.
We may assume, without loss of generality, that t≤∥A∥-1. Utilizing Lemmas 7, 13, and 15, we have
(38)〈Stx-Sty,J(x-y)〉 =t〈f(x)-f(y),J(x-y)〉 +〈(I-tA)[(I-θtF)G(Tx)-(I-θtF)G(Ty)],hhhhhhJ(x-y)〉 ≤tβ∥x-y∥2+(1-tγ-) ×∥(I-θtF)G(Tx)-(I-θtF)G(Ty)∥∥x-y∥ ≤tβ∥x-y∥2+(1-tγ-)(1-θt(1-1-αλ)) ×∥G(Tx)-G(Ty)∥∥x-y∥ ≤tβ∥x-y∥2+(1-tγ-)∥Tx-Ty∥∥x-y∥ ≤tβ∥x-y∥2+(1-tγ-)∥x-y∥2 =(1-t(γ--β))∥x-y∥2.
Hence, it is known that St:C→C is a continuous and strongly pseudocontractive mapping with pseudocontractive coefficient 1-t(γ--β)∈(0,1) Thus, by Lemma 14, we deduce that there exists a unique fixed point in C, denoted by xt, which uniquely solves the fixed point equation
(39)xt=tf(xt)+(I-tA)[G(Txt)-θtFG(Txt)].
Let us show the uniqueness of the solution of VIP (36). Suppose that both p1∈Λ and p2∈Λ are solutions to VIP (36). Then, we have
(40)〈(A-f)p1,J(p1-p2)〉≤0,〈(A-f)p2,J(p2-p1)〉≤0.
Adding up the above two inequalities, we obtain
(41)〈(A-f)p1-(A-f)p2,J(p1-p2)〉≤0.
Note that
(42)〈(A-f)p1-(A-f)p2,J(p1-p2)〉 =〈A(p1-p2),J(p1-p2)〉 -〈f(p1)-f(p2),J(p1-p2)〉 ≥γ-∥p1-p2∥2-β∥p1-p2∥2 =(γ--β)∥p1-p2∥2≥0.
Consequently, we have p1=p2, and the uniqueness is proved.
Next, let us show that, for some a∈(0,1), {xt:t∈(0,a]} is bounded. Indeed, since {θt:t∈(0,1)}⊂[0,1) with limt→0(θt/t)=0, there exists some a∈(0,1) such that 0≤θt/t<1 for all t∈(0,a]. Take a fixed p∈Fix(Λ) arbitrarily. Utilizing Lemma 7, we have
(43)∥xt-p∥2=〈t(f(xt)-f(p))+(I-tA)[G(Txt)-θtFG(Txt)-p] -t(Ap-f(p)),J(xt-p)〉=t〈f(xt)-f(p),J(xt-p)〉 +〈(I-tA)[G(Txt)-θtFG(Txt)-p],J(xt-p)〉 -t〈(A-f)p,J(xt-p)〉≤tβ∥xt-p∥2+(1-tγ-)∥G(Txt)-θtFG(Txt)-p∥ ×∥xt-p∥+t∥(A-f)p∥∥xt-p∥≤tβ∥xt-p∥2+(1-tγ-) ×[∥(I-θtF)G(Txt)-(I-θtF)G(Tp)∥ +∥(I-θtF)G(Tp)-p∥]∥xt-p∥ +t∥(A-f)p∥∥xt-p∥≤tβ∥xt-p∥2+(1-tγ-)(1-θt(1-1-αλ)) ×∥G(Txt)-G(Tp)∥∥xt-p∥ +(1-tγ-)θt∥Fp∥∥xt-p∥+t∥(A-f)p∥∥xt-p∥≤tβ∥xt-p∥2+(1-tγ-)(1-θt(1-1-αλ)) ×∥Txt-Tp∥∥xt-p∥ +θt∥Fp∥∥xt-p∥+t∥(A-f)p∥∥xt-p∥≤tβ∥xt-p∥2+(1-tγ-)∥xt-p∥2 +θt∥Fp∥∥xt-p∥+t∥(A-f)p∥∥xt-p∥=(1-t(γ--β))∥xt-p∥2+θt∥Fp∥∥xt-p∥ +t∥(A-f)p∥∥xt-p∥,
and, hence, for all t∈(0,a],
(44)∥xt-p∥≤1γ--β(∥(A-f)p∥+θtt∥Fp∥)≤1γ--β(∥(A-f)p∥+∥Fp∥).
Thus, this implies that {xt:t∈(0,a]} is bounded and so are {f(xt):t∈(0,a]}, {Txt:t∈(0,a]}, and {G(Txt):t∈(0,a]}.
Let us show that ∥Txt-G(Txt)∥→0 as t→0.
Indeed, for simplicity, we put q=ΠC(I-μ2B2)p, x^t=Txt, ut=ΠC(I-μ2B2)x^t, and vt=ΠC(I-μ1B1)ut. Then, it is clear that p=ΠC(I-μ1B1)q and vt=G(x^t)=G(Txt). Hence, from (43), it follows that
(45)∥xt-p∥2 ≤tβ∥xt-p∥2+(1-tγ-)(1-θt(1-1-αλ)) ×∥G(Txt)-G(Tp)∥∥xt-p∥ +(1-tγ-)θt∥Fp∥∥xt-p∥+t∥(A-f)p∥∥xt-p∥ ≤tβ∥xt-p∥2+(1-tγ-)∥vt-p∥∥xt-p∥ +θt∥Fp∥∥xt-p∥+t∥(A-f)p∥∥xt-p∥.
From Lemma 12, we have
(46)∥ut-q∥2=∥ΠC(x^t-μ2B2x^t)-ΠC(p-μ2B2p)∥2≤∥x^t-p-μ2(B2x^t-B2p)∥2≤∥x^t-p∥2-2μ2(α2-κ2μ2)∥B2x^t-B2p∥2,∥vt-p∥2=∥ΠC(ut-μ1B1ut)-ΠC(q-μ1B1q)∥2≤∥ut-q-μ1(B1ut-B1q)∥2≤∥ut-q∥2-2μ1(α1-κ2μ1)∥B1ut-B1q∥2.
From the last two inequalities, we obtain
(47)∥vt-p∥2≤∥x^t-p∥2-2μ2(α2-κ2μ2) ×∥B2x^t-B2p∥2-2μ1(α1-κ2μ1) ×∥B1ut-B1q∥2≤∥xt-p∥2-2μ2(α2-κ2μ2)∥B2x^t-B2p∥2 -2μ1(α1-κ2μ1)∥B1ut-B1q∥2,
which together with (45) implies that
(48)∥xt-p∥2 ≤tβ∥xt-p∥2+(1-tγ-)∥vt-p∥∥xt-p∥+θt∥Fp∥ ×∥xt-p∥+t∥(A-f)p∥∥xt-p∥ ≤tβ∥xt-p∥2+t∥(A-f)p∥∥xt-p∥+(1-tγ-) ×12(∥xt-p∥2+∥vt-p∥2)+θt∥Fp∥∥xt-p∥ ≤tβ∥xt-p∥2+t∥(A-f)p∥∥xt-p∥ +(1-tγ-)12{∥xt-p∥2+∥xt-p∥2hhhhhhhhhhhhh-2μ2(α2-κ2μ2)∥B2x^t-B2p∥2hhhhhhhhhhhhh-2μ1(α1-κ2μ1)hhhhhhhhhhhhh×∥B1ut-B1q∥2}+θt∥Fp∥∥xt-p∥ =(1-t(γ--β))∥xt-p∥2+t∥(A-f)p∥∥xt-p∥ -(1-tγ-)[μ2(α2-κ2μ2)∥B2x^t-B2p∥2hhhhhhhhhhhh+ μ1(α1-κ2μ1)∥B1ut-B1q∥2] +θt∥Fp∥∥xt-p∥ ≤∥xt-p∥2+t∥(A-f)p∥∥xt-p∥-(1-tγ-) ×[μ2(α2-κ2μ2)∥B2x^t-B2p∥2hhhhhh+μ1(α1-κ2μ1)∥B1ut-B1q∥2]+θt∥Fp∥∥xt-p∥.
So, it immediately follows that
(49)(1-tγ-)[μ2(α2-κ2μ2)∥B2x^t-B2p∥2+μ1(α1-κ2μ1)hhhhhhh×∥B1ut-B1q∥2] ≤t∥(A-f)p∥∥xt-p∥+θt∥Fp∥∥xt-p∥.
Since 0<μi<αi/κ2, for i=1,2, we have
(50)limt→0∥B2x^t-B2p∥=0, limt→0∥B1ut-B1q∥=0.
Utilizing Proposition 2 and Lemma 8, we have that there exists g1 such that
(51)∥ut-q∥2=∥ΠC(x^t-μ2B2x^t)-ΠC(p-μ2B2p)∥2≤〈x^t-μ2B2x^t-(p-μ2B2p),J(ut-q)〉=〈x^t-p,J(ut-q)〉+μ2〈B2p-B2x^t,J(ut-q)〉≤12[∥x^t-p∥2+∥ut-q∥2hihh-g1(∥x^t-ut-(p-q)∥)∥x^t-p∥2] +μ2∥B2p-B2x^t∥∥ut-q∥,
which implies that
(52)∥ut-q∥2≤∥x^t-p∥2-g1(∥x^t-ut-(p-q)∥) +2μ2∥B2p-B2x^t∥∥ut-q∥.
In the same way, we derive that there exists g2:
(53)∥vt-p∥2=∥ΠC(ut-μ1B1ut)-ΠC(q-μ1B1q)∥2≤〈ut-μ1B1ut-(q-μ1B1q),J(vt-p)〉=〈ut-q,J(vt-p)〉+μ1〈B1q-B1ut,J(vt-p)〉≤12[∥ut-q∥2+∥vt-p∥2hhhh-g2(∥ut-vt+(p-q)∥)∥vt-p∥2] +μ1∥B1q-B1ut∥∥vt-p∥,
which implies that
(54)∥vt-p∥2≤∥ut-q∥2-g2(∥ut-vt+(p-q)∥) +2μ1∥B1q-B1ut∥∥vt-p∥.
Substituting (52) for (54), we get
(55)∥vt-p∥2≤∥x^t-p∥2-g1(∥x^t-ut-(p-q)∥) -g2(∥ut-vt+(p-q)∥) +2μ2∥B2p-B2x^t∥∥ut-q∥ +2μ1∥B1q-B1ut∥∥vt-p∥≤∥xt-p∥2-g1(∥x^t-ut-(p-q)∥) -g2(∥ut-vt+(p-q)∥) +2μ2∥B2p-B2x^t∥∥ut-q∥ +2μ1∥B1q-B1ut∥∥vt-p∥,
which together with (45) implies that
(56)∥xt-p∥2 ≤tβ∥xt-p∥2+(1-tγ-)∥vt-p∥∥xt-p∥+θt∥Fp∥ ×∥xt-p∥+t∥(A-f)p∥∥xt-p∥ ≤tβ∥xt-p∥2+t∥(A-f)p∥∥xt-p∥+(1-tγ-) ×12(∥xt-p∥2+∥vt-p∥2)+θt∥Fp∥∥xt-p∥ ≤tβ∥xt-p∥2+t∥(A-f)p∥∥xt-p∥+(1-tγ-) ×12{∥xt-p∥2+∥xt-p∥2hhhhhhh-g1(∥x^t-ut-(p-q)∥)hhhhhhh-g2(∥ut-vt+(p-q)∥)hhhhhhh+2μ2∥B2p-B2x^t∥∥ut-q∥hhhhhhh+2μ1∥B1q-B1ut∥∥vt-p∥∥xt-p∥2} +θt∥Fp∥∥xt-p∥ =(1-t(γ--β))∥xt-p∥2+t∥(A-f)p∥∥xt-p∥ -(1-tγ-) ×12[g1(∥x^t-ut-(p-q)∥)hhhhhh+g2(∥ut-vt+(p-q)∥)]+(1-tγ-) ×[μ2∥B2p-B2x^t∥∥ut-q∥hhhhhh+μ1∥B1q-B1ut∥∥vt-p∥] +θt∥Fp∥∥xt-p∥ ≤∥xt-p∥2+t∥(A-f)p∥∥xt-p∥-(1-tγ-) ×12[g1(∥x^t-ut-(p-q)∥)hhhhhhh+g2(∥ut-vt+(p-q)∥)] +μ2∥B2p-B2x^t∥∥ut-q∥ +μ1∥B1q-B1ut∥∥vt-p∥+θt∥Fp∥∥xt-p∥.
So, it immediately follows that
(57)(1-tγ-)12[g1(∥x^t-ut-(p-q)∥)hhhhhhhh+g2(∥ut-vt+(p-q)∥)] ≤t∥(A-f)p∥∥xt-p∥+μ2∥B2p-B2x^t∥∥ut-q∥ +μ1∥B1q-B1ut∥∥vt-p∥+θt∥Fp∥∥xt-p∥.
Hence, from (50), we conclude that
(58)limt→0g1(∥x^t-ut-(p-q)∥)=0,limt→0g2(∥ut-vt+(p-q)∥)=0.
Utilizing the properties of g1 and g2, we get
(59)limt→0∥x^t-ut-(p-q)∥=0,limt→0∥ut-vt+(p-q)∥=0,
which leads to
(60)∥x^t-vt∥≤∥x^t-ut-(p-q)∥ +∥ut-vt+(p-q)∥⟶0 as t⟶0.
That is,
(61)limt→0∥Txt-G(Txt)∥=limt→0∥x^t-vt∥=0.
Note that {xt:t∈(0,a]} is bounded and so are {f(xt):t∈(0,a]}, {Txt:t∈(0,a]}, and {G(Txt):t∈(0,a]}. Hence, we have
(62)∥xt-G(Txt)∥ =t∥f(xt)-AG(Txt)-θtt(I-tA)FG(Txt)∥⟶0,
as t→0. Also, observe that
(63)∥xt-Txt∥≤∥xt-G(Txt)∥+∥G(Txt)-Txt∥.
This together with (61) and (62) implies that
(64)limt→0∥xt-Txt∥=0.
Utilizing the nonexpansivity of G, we obtain
(65)∥xt-Gxt∥≤∥xt-G(Txt)∥+∥G(Txt)-Gxt∥≤∥xt-G(Txt)∥+∥Txt-xt∥,
which together with (62) and (64) implies that
(66)limt→0∥xt-Gxt∥=0.
Now, let {tk} be a sequence in (0,a] that converges to 0 as k→∞, and define a function g on C by
(67)g(x)=μk12∥xtk-x∥2, ∀x∈C,
where μ is a Banach limit. Define the set
(68)K:={w∈C:g(w)=min{g(y):y∈C}}
and the mapping
(69)Wx=(1-θ)Tx+θGx, ∀x∈C,
where θ is a constant in (0,1). Then, by Lemma 10, we know that Fix(W)=Fix(T)∩Fix(G)=Λ. We observe that
(70)∥xt-Wxt∥=∥(1-θ)(xt-Txt)+θ(xt-Gxt)∥≤(1-θ)∥xt-Txt∥+θ∥xt-Gxt∥.
So, from (64) and (66), we obtain
(71)limn→∞∥xt-Wxt∥=0.
Since X is a uniformly smooth Banach space, K is a nonempty bounded closed convex subset of C; for more details, see [14]. We claim that K is also invariant under the nonexpansive mapping W. Indeed, noticing (71), we have, for w∈K,
(72)g(Ww)=μk12∥xtk-Ww∥2=μk12∥Wxtk-Ww∥2≤μk12∥xtk-w∥2=g(w).
Since every nonempty closed bounded convex subset of a uniformly smooth Banach space X has the fixed point property for nonexpansive mappings and W is a nonexpansive mapping of K, W has a fixed point in K, say p. Utilizing Lemma 5, we get
(73)μk〈x-p,J(xtk-p)〉≤0, ∀x∈C.
Putting x=(f-A)p+p∈C, we have
(74)μk〈(f-A)p,J(xtk-p)〉≤0, ∀x∈C.
Since xtk-p=tk(f(xtk)-f(p))+(I-tkA)[G(Txtk)-θtkFG(Txtk)-p]-tk(A-f)p, we get
(75)∥xtk-p∥2 =tk〈f(xtk)-f(p),J(xtk-p)〉 +〈(I-tkA)(G(Txtk)-p),J(xtk-p)〉 -θtk〈(I-tkA)FG(Txtk),J(xtk-p)〉 -tk〈(A-f)p,J(xtk-p)〉 ≤tkβ∥xtk-p∥2+tk〈(f-A)p,J(xtk-p)〉 +(1-tkγ-)∥G(Txtk)-p∥∥xtk-p∥ +(1-tkγ-)θtk∥FG(Txtk)∥∥xtk-p∥ ≤tkβ∥xtk-p∥2+tk〈(f-A)p,J(xtk-p)〉 +(1-tkγ-)∥xtk-p∥2+θtk∥FG(Txtk)∥∥xtk-p∥.
It follows that
(76)∥xtk-p∥2≤1γ--β[+θtktk∥FG(Txtk)∥∥xtk-p∥]〈(f-A)p,J(xtk-p)〉hhhhhhh+θtktk∥FG(Txtk)∥∥xtk-p∥].
Since limk→∞(θtk/tk)=0, from (74) and the boundedness of sequences {FG(Txtk)}, {xtk}, it follows that
(77)μk∥xtk-p∥2≤1γ--βμk[+θtktk∥FG(Txtk)∥∥xtk-p∥]〈(f-A)p,J(xtk-p)〉hhhhhhhhh+θtktk∥FG(Txtk)∥∥xtk-p∥]=1γ--β[(θtktk∥FG(Txtk)∥∥xtk-p∥)μk〈(f-A)p,J(xtk-p)〉hhhhhhh+μk(θtktk∥FG(Txtk)∥∥xtk-p∥)]≤0.
Therefore, for the sequence {xtk} in {xt:t∈(0,a]}, there exists a subsequence which is still denoted by {xtk} that converges strongly to some fixed point p of W.
Now, we claim that such a p is the unique solution in Λ to the VIP (36).
Indeed, from (35), it follows that for all u∈Λ=Fix(T)∩Ω(78)∥xt-u∥2=t〈f(xt)-f(u),J(xt-u)〉 +〈(I-tA)[G(Txt)-θtFG(Txt)-u],J(xt-u)〉 -t〈(A-f)u,J(xt-u)〉=〈(I-tA)[(I-θtF)G(Txt)-(I-θtF)uhhhhhihhhh+(I-θtF)u-u],J(xt-u)〉 +t〈f(xt)-f(u),J(xt-u)〉-t〈(A-f)u,J(xt-u)〉≤(1-tγ-)[∥(I-θtF)G(Txt)-(I-θtF)u∥hhhihhhhh+∥(I-θtF)u-u∥] ×∥xt-u∥+tβ∥xt-u∥2-t〈(A-f)u,J(xt-u)〉≤(1-tγ-)[(1-θt(1-1-αλ))∥xt-u∥+θt∥Fu∥] ×∥xt-u∥+tβ∥xt-u∥2-t〈(A-f)u,J(xt-u)〉≤(1-tγ-)[∥xt-u∥+θt∥Fu∥]∥xt-u∥ +tβ∥xt-u∥2-t〈(A-f)u,J(xt-u)〉≤(1-t(γ--β))∥xt-u∥2+θt∥Fu∥ ×∥xt-u∥-t〈(A-f)u,J(xt-u)〉≤∥xt-u∥2+θt∥Fu∥∥xt-u∥-t〈(A-f)u,J(xt-u)〉,
which hence implies that
(79)〈(A-f)u,J(xt-u)〉≤θtt∥Fu∥∥xt-u∥, ∀u∈Λ.
Since xtk→p as tk→0 and limt→0(θt/t)=0, we obtain from the last inequality that
(80)〈(A-f)u,J(p-u)〉≤0, ∀u∈Λ.
Utilizing the well-known Minty-type Lemma, we get
(81)〈(A-f)p,J(p-u)〉≤0, ∀u∈Λ.
So, p is a solution in Λ to the VIP (36).
In order to prove that the net {xt:t∈(0,a]} converges strongly to p as t→0, suppose that there exists another subsequence {xsk}⊂{xt} such that xsk→q as sk→0; then we also have q∈Fix(W)=Fix(T)∩Ω=:Λ due to (71). Repeating the same argument as above, we know that q is another solution in Λ to the VIP (36). In terms of the uniqueness of solutions in Λ to the VIP (36), we immediately get p=q. This completes the proof.