On θ-generalized demimetric mappings and monotone operators in Hadamard spaces

Definition 2.1

[41] Let X be a CAT(0) space and denote the pair (u,v)∈X×X by uv→. Then, a mapping 〈.,.〉:(X×X)×(X×X)→ℝ defined by

is a quasilinearization mapping.

It is easy to verify that 〈uv→,uv→〉=d2(u,v),〈vu→,wx→〉=−〈uv→,wx→〉,〈uv→,wx→〉=〈uy→,wx→〉+〈yv→,wx→〉 and 〈uv→,wx→〉=〈wx→,uv→〉 for all u,v,w,x,y∈X.

Lim [42] introduced the notion of Δ-convergence in metric spaces, and Kirk and Panyanak [43] extended this concept to CAT(0) spaces. Recall that a bounded sequence {xn} in X is said to be Δ-convergent to a point x∈X if A({xnk})={x} for every subsequence {xnk} of {xn} (we write Δ−limn→∞xn=x), where A({xn})≔{x∈X:r(x,{xn})=r({xn})} is the asymptotic center of {xn}, r({xn})≔inf{x,{xn}):x∈X} is the asymptotic radius of {x_n} and r(⋅,{xn}):X→[0,∞) is a continuous functional defined by r(x,{xn})=limsupn→∞d(x,xn).

Kakavandi and Amini [44] introduced the concept of the dual space of a Hadamard space as follows: consider the map θ:ℝ×X×X→C(X,ℝ) defined by

where C(X,ℝ) is the space of all continuous real valued functions on a Hadamard space X. A pseudometric D on ℝ×X×X is defined by

The pseudometric space (ℝ×X×X,D) is the subspace of the pseudometric space of all real valued Lipschitz functions (Lip(X,ℝ),L). We know from [44] that D((t,u,v),(s,w,y))=0 if and only if t〈uv→,xe→〉=s〈wy→,xe→〉forallx,e∈X. Therefore, an equivalence relation on ℝ×X×X is induced by D, where the equivalence class of (t,u,v) is defined by:

Thus, X⁎={[tuv→]:(t,u,v)∈ℝ×X×X} endowed with D([tuv→],[swy→])≔D((t,u,v),(s,w,y)) is the dual space of X. In a real Hilbert space H, t(v−u)≡[tuv→] for all t∈ℝ, u,v∈H (see [44]). Note also that X⁎ acts on X×X by

Using the same notation as in [45, p. 1649], we denote the zero element of X⁎ by 0. That is, the (unique) element of X⁎ such that the evaluation 〈0,⋅〉 vanishes for all elements in X×X. In fact, 0≡[txx→] for any t∈ℝ and x∈X (see [31,36]).

Let X⁎ be the dual space of a Hadamard space X. An operator A:X→2X⁎ is monotone if and only if

A monotone operator A is called a maximal monotone operator if the graph G(A) of A defined by

is not properly contained in the graph of any other monotone operator. The resolvent of A of order λ>0 is the mapping JλA:X→2X defined by

The operator A satisfies the range condition if for every λ>0,D(JλA)=X.

Definition 2.2

Let C be a nonempty closed and convex subset of a Hadamard space X. A mapping T:C→C is said to be Δ-demiclosed, if for any bounded sequence {xn}inX such that Δ−limn→∞xn=x and limn→∞d(xn,Txn)=0, then x=Tx.

Definition 2.3

Let C be a nonempty closed and convex subset of a CAT(0) space X. The metric projection is a mapping PC:X→C which assigns to each x∈X, the unique point PCx∈C such that

Recall that a mapping T is firmly nonexpansive [36], if

It is well known that firmly nonexpansive mappings are nonexpansive (see [10]).

Lemma 2.4

Let X be aCAT(0)space,u,v,w∈X and t∈[0,1]. Then,

1. d(tu⊕(1−t)v,w)≤td(u,w)+(1−t)d(v,w)(see [23]).

2. d2(tu⊕(1−t)v,w)≤td2(u,w)+(1−t)d2(v,w)−t(1−t)d2(u,v)(see [23]).

3. d2(tu⊕(1−t)v,w)≤t2d2(u,w)+(1−t)2d2(v,w)+2t(1−t)〈uw→,vw→〉(see [46]).

Lemma 2.5

[23] Every bounded sequence in a Hadamard space always has a Δ-convergence subsequence.

Lemma 2.6

[47] Let X be a Hadamard space,{xn}be a sequence in X andu∈X. Then,{xn}Δ-converges to u if and only iflimsupn→∞〈uxn→,uw→〉≤0for allw∈C.

Lemma 2.7

[48] Let X be a Hadamard space andT:X→Xbe a nonexpansive mapping. Then, T isΔ-demiclosed.

Lemma 2.8

[49] Let{an}be a sequence of non-negative real numbers such that

Lemma 2.9

[36] Let X be a CAT(0) space andJλAbe the resolvent of A with orderλ.Then,

1. For anyλ>0,R(JλA)⊂D(A)andF(JλA)=A−1(0),whereR(JλA)is the range ofJλA.

2. If A is monotone, thenJλAis a single-valued and firmly nonexpansive mapping.

3. If A is monotone and0<λ≤μ,thend2(JλAx,JμAx)≤μ−λμ+λd2(x,JμAx), which implies thatd(x,JλAx)≤2d(x,JμAx).

Lemma 2.10

[34] Let X be a CAT(0) space andA:X→2Xbe monotone. Then,

Remark 2.11

We observe that inequality (12) is a property of any firmly nonexpansive mapping. That is, if T is a firmly nonexpansive mapping, then from (9) and (11), we obtain

Lemma 2.12

[34] LetX⁎be the dual space of a Hadamard space X andT:X→Xbe a nonexpansive mapping for eachi=1,2,…,N.LetJλibe the resolvent of monotone operatorsAiof orderλ>0.Then,

Lemma 2.13

[50] Let X be a Hadamard space, for anyt∈[0,1]andu,v∈X,letut=tu⊕(1−t)v.Then, for allx,y∈X,we have

Definition 3.1

Let C be a nonempty subset of a CAT(0) space X and T:C⊆X→X be a nonlinear mapping. Then, T is called θ-generalized demimetric, if F(T)≠∅ and there exists θ≠0 such that

for all u∈Candv∈F(T).

Example 3.2

Define T:[0,1]→ℝ by Tx=x+x2. Then, T is θ-generalized demimetric with θ=−1 but not a k-demimetric mapping.

Proof

Observe that F(T)={0}. Thus, for any x∈[0,1], we have

Remark 3.3

Other examples of θ-generalized demimetric mappings in CAT(0) spaces include the following.

1. If T:X→X is a k-strictly pseudocontractive mapping with k∈[0,1) and F(T)≠∅, then T is (21−k)-generalized demimetric mapping. The proof is similar to that in [2].

2. If F(T)≠∅ and T:X→X is a generalized hybrid mapping, then T is 2-generalized demimetric. Indeed, for u∈F(T) and v∈X, we obtain from (1) that

   (16)d2(u,Tv)≤d2(u,v).

   Also, from (9), we have that

   2〈vu→,vTv→〉=d2(v,Tv)+d2(u,v)−d2(u,Tv),

   which implies from (16) that

   2〈vu→,vTv→〉≥d2(v,Tv)+d2(u,v)−d2(u,v)=d2(v,Tv).

   Hence, T is a 2-generalized demimetric mapping. Therefore, nonexpansive, nonspreading and hybrid mappings are examples of θ-generalized demimetric mappings.

3. If T:X→X is a k-demicontractive mapping, then T is a (21−k)-generalized demimetric.

Proposition 3.4

Let X be a Hadamard space andT:X→Xbe aθ-generalized demimetric mapping withθ>0.Then, T is a(1−2θ)-demimetric.

Proof

It follows from the definition of demimetric mapping and θ-generalized demimetric mapping.□

Proposition 3.5

Let X be a Hadamard space andT:X→Xbe a θ-generalized demimetric mapping withθ≠0.Then,F(T)is closed and convex.

Proof

We first show that F(T) is closed. Let {xn} be a sequence in F(T) such that {xn} converges to x⁎. Then, from the definition of θ-generalized demimetric mappings, we have

which implies from the Cauchy Schwartz inequality that

Taking limits, we have that 0≥d2(x⁎,Tx⁎), which implies that x⁎=Tx⁎.

Thus, F(T) is closed. Next, we show that F(T) is convex. For this, let u,v∈F(T). Then, it suffices to show that (tu⊕(1−t)v)∈F(T), for t∈[0,1]. Let w=tu⊕(1−t)v,  t∈[0,1], then we obtain from Lemma 2.13 that

which implies that 1θd2(w,Tw)≤0. Since θ≠0, we obtain that w∈F(T) as required.□

Lemma 3.6

Let X be a CAT(0) space andT:X→Xbe a θ-generalized demimetric mapping withθ≠0.Suppose thatSλu=λu⊕(1−λ)Tuwithθ≤21−λandλ∈(0,1),thenSλis quasi-nonexpansive andF(Sλ)=F(T).

Proof

Let u∈X and w∈F(T), then since T is θ-generalized demimetric, we obtain by Lemma 2.13 that

Now, from (8), we obtain that d2(u,Sλu)=(1−λ)2d2(u,Tu). Thus, we obtain from (18) that

which implies that

Thus, we obtain that

which implies that

Hence, Sλ is quasi-nonexpansive.

Next, we show that F(Sλ)=F(T). From (8), we obtain that

This implies that Sλu=u if and only if Tu=u. Therefore, F(Sλ)=F(T).□

Theorem 3.7

Let X be a Hadamard space andX⁎be its dual space. LetAi:X→2X⁎,i=1,2,…,Nbe a finite family of multivalued monotone mappings satisfying the range condition andT:X→Xbe a θ-generalized demimetric mapping withθ≠0.Suppose thatΓ≔F(T)∩(⋂i=1NAi−1(0))≠∅and for arbitraryu,x1∈X,the sequence{xn}is defined by

(C1) limn→∞αn=0,∑n=1∞αn=∞,

(C2) 0<a≤βn, γn≤b<1.

Then,{xn}converges strongly to an element ofΓ.

Proof

We first show that {xn} is bounded.

Let p∈Γ, from (19), Lemmas 2.4 and 3.6, we have

We also have from (19) and (8) that

which implies that

From (19), (21), (23) and Lemma 2.4, we obtain

Thus, we obtain from Lemma 2.4 that

Therefore, {xn} is bounded. Hence, {yn} and {zn} are also bounded.

Next, we show that

From (19), (24) and Lemma 2.4, we have that