Research Article
Some fixed point results for nonlinear mappings in convex metric spaces
Chao Wang
School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, P. R. China Communicated by P. Kumam
Abstract
In this paper, we consider an iteration process to approximate a common random fixed point of a finite family of asymptotically quasi-nonexpansive random mappings in convex metric spaces. Our results extend and improve several known results in recent literature.
Keywords: Asymptotically quasi-nonexpansive random mappings, random iteration process, common random fixed point, convex metric spaces.
2010 MSC: 47H09, 47H10.
1. Introduction and Preliminaries
Random fixed point theorems are stochastic generalizations of classical fixed point theorems, which are usually used to obtain the solutions of nonlinear random systems [3]. Some random fixed point theorems for random mappings on separable metric spaces were first proved by Spacek [18] and Hans [7]. Itoh [8] introduced multivalued random contractive mappings on separable metric spaces and considered some random fixed point theorems for the mappings. Choudhury [5] gave a random Ishikawa iteration process to converge to fixed points of the given random mappings. After that, many authors [1, 2, 5, 11, 12, 13, 14, 17, 16] have worked on random iterative algorithms for contractive and asymptotically nonexpansive random mappings in separable normed spaces, Banach spaces and uniformly convex Banach spaces.
In 1970, Takahashi [19] introduced a notion of convex metric space which is a more general space, and each linear normed space is a special example of a convex metric space. Recently [4, 10, 21, 22] have discussed different iteration processes to obtain fixed point of asymptotically quasi-nonexpansive mappings in convex metric spaces.
∗Corresponding author
Email address: [email protected](Chao Wang) Received 2015-2-4
Inspried and motived by the above facts, we will construct an iteration process which converges strongly to a common random fixed point of a finite family of asymptotically quasi-nonexpansive random mappings in convex metric spaces. The results extend and improve the corresponding results in [1, 2, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 17, 16, 20, 21, 22].
Let (Ω,Σ) be a mesurable space with Σ being a σ-algebra of subsets of Ω, and let K be a nonempty subset of a metric space (X, d).
Definition 1.1 ([1]). (i) A mappingξ : Ω→X is measurable ifξ−1(U)∈Σ for each open subset U of X;
(ii) The mappingT : Ω×K →X is a random mapping if and only if for each fixedx∈K, the mapping T(·, x) : Ω → X is measurable, and it is continuous if for each ω ∈ Ω, the mapping T(ω,·) : K → X is continuous;
(iii) A measurable mappingξ : Ω→K is a random fixed point of the random mappingT : Ω×K→X if and only if T(ω, ξ(ω)) =ξ(ω) for each ω∈Ω.
We denote by N the set of natural numbers,F(T) the set of all random fixed points of a random map T and Tn(ω, x) thenth iterationT(ω, T(ω, T(ω,· · ·T(ω, x)· ··))) ofT for each ω ∈Ω. The letterI denotes the random mappingT : Ω×K→K defined byI(ω, x) =x and T0=I for each ω ∈Ω.
Next, we introduce some random mappings in metric spaces.
Definition 1.2. LetK be a nonempty subset of a separable metric space (X, d) and T : Ω×K→ K be a random mapping. The mappingT is said to be
(i) a nonexpansive random mapping if
d(T(ω, x), T(ω, y))≤d(x, y) for each ω∈Ω and x, y∈K;
(ii) an asymptotically nonexpansive random mapping if there exists a sequence of measurable mappings {rn(ω)}: Ω→[0,∞) with lim
n→∞rn(ω) = 0 such that
d(Tn(ω, x), Tn(ω, y))≤(1 +rn(ω))d(x, y) for each ω∈Ω, n∈Nand x, y∈K;
(iii) an asymptotically quasi-nonexpansive random mapping if there exists a sequence of measurable mappings {rn(ω)}: Ω→[0,∞) with lim
n→∞rn(ω) = 0 such that
d(Tn(ω, η(ω)), ξ(ω))≤(1 +rn(ω))d(η(ω), ξ(ω))
for each ω∈Ω and n∈N, whereξ ∈F(T)6=∅and η: Ω→K is any measurable mapping.
(iv) an semicompact random mapping if for any sequence of measurable mappings {ξn(ω)} : Ω → K, with lim
n→∞d(T(ω, ξn(ω)), ξn(ω)) = 0 for each ω ∈ Ω and n ∈ N, there exists a subsequence {ξnj} of {ξn} which converges pointwise toξ, where ξ: Ω→K is a measurable mapping.
Remark 1.3. It is easy to see that if T is an asymptotically nonexpansive random mapping and F(T)6=∅, thenT is an asymptotically quasi-nonexpansive random mapping.
Definition 1.4 ([19]). A convex structure in a metric space (X, d) is a mapping W :X×X×[0,1]→X satisfying, for eachx, y, u∈X and each λ∈[0,1]
d(u, W(x, y;λ))≤λd(u, x) + (1−λ)d(u, y).
A metric space together with a convex structure is called a convex metric space.
A nonempty subsetK of X is said to be convex ifW(x, y;λ)∈K for all (x, y;λ)∈K×K×[0,1]. The mapping W :K×K ×[0,1]→ K is said to be a measurable convex structure if for any two measurable mappings ξ, η: Ω→K and each fixedλ∈[0,1], the mappingW(ξ(·), η(·);λ) : Ω→K is measurable.
In Banach spaces, Khan et al. [9] introduced the following iteration process for common fixed points of asymptotically quasi-nonexpansive mappings {Ti :i∈J ={1,2,· · ·, k}}: any initial pointx1∈K,
xn+1 = (1−αkn)xn+αknTkny(k−1)n,
y(k−1)n= (1−α(k−1)n)xn+α(k−1)nT(k−1)n y(k−2)n, y(k−2)n= (1−α(k−2)n)xn+α(k−2)nT(k−2)n y(k−3)n,
...
y1n= (1−α1n)xn+α1nT1ny0n,
(1.1)
where y0n = xn and {αin} are real sequences in [0,1] for all n ∈ N. And then, Khan and Ahmed [10]
considered the iteration process (1.1) in convex metric spaces as follows:
xn+1 =W(Tkny(k−1)n, xn;αkn),
y(k−1)n=W(Tk−1n y(k−2)n, xn;α(k−1)n), y(k−2)n=W(Tk−2n y(k−3)n, xn;α(k−2)n),
...
y1n=W(T1ny0n, xn;α1n),
(1.2)
wherey0n=xn and {αin} are real sequences in [0,1] for alln∈N.
From (1.1) and (1.2), we investigate the following random iteration process in convex metric space.
Definition 1.5. Let{Ti :i∈J}be a finite familiy of asymptotically quasi-nonexpansive random mappings from Ω×K toK , whereK is a nonempty closed convex subset of a separable convex metric space (X, d).
Letξ1 : Ω→K be a measurable mapping, for eachω∈Ω, the sequence {ξn(ω)}is defined as follows:
ξn+1(ω) =W(Tkn(ω, η(k−1)n(ω)), ξn(ω);αkn),
η(k−1)n(ω) =W(Tk−1n (ω, η(k−2)n(ω)), ξn(ω);α(k−1)n), η(k−2)n(ω) =W(Tk−2n (ω, η(k−3)n(ω)), ξn(ω);α(k−2)n),
...
η1n(ω) =W(T1n(ω, η0n(ω)), ξn(ω);α1n),
(1.3)
whereη0n(ω) =ξn(ω) and {αin} are real sequences in [0,1] for alln∈N. We need the following two results for proving our main results.
Lemma 1.6 ([20]). Let X be a separable metric space and Y be a metric space. If f : Ω×X → Y is measurable in ω ∈Ω and continuous in x∈X, and if x : Ω→ X is measurable, then f(·, x(·)) : Ω →Y is measurable.
Lemma 1.7 ([15]). Let {βn} and {γn} be sequences of nonnegative real numbers satisfying the following conditions:
βn+1≤(1 +γn)βn,
∞
X
n=1
γn<∞
We have (i) lim
n→∞βn exists;
(ii) if lim inf
n→∞ βn= 0, then lim
n→∞βn= 0.
2. Main results
In this section, we give some conditions for the convergence of the random iteration process (1.3) to a common random fixed point of a finite family asymptotically quasi-nonexpansive random mappings {Ti, i∈J}. We first prove the following lemma.
Lemma 2.1. Let K be a nonempty closed convex subset of a separable complete convex metric space(X, d).
Let {Ti : i ∈ J} : Ω×K → K be a finite family of asymptotically quasi-nonexpansive random mappings with rin(ω) : Ω → [0,∞) for each ω ∈ Ω. Suppose that the sequence {ξn(ω)} is defined as (1.3) and P∞
n=1αkn<∞. IfF =Tk
i=1F(Ti)6=∅, then (i) there exists a constant M0>0 such that
d(ξn+1(ω), ξ(ω))≤(1 +αknM0)d(ξn(ω), ξ(ω)) for allξ(ω)∈F and n∈N;
(ii) there exists a constant M1>0 such that
d(ξn+m(ω), ξ(ω))≤M1d(ξn(ω), ξ(ω)) for allξ(ω)∈F and n, m∈N.
Proof. (i) Since {Ti :i∈J}: Ω×K →K be a finite family of asymptotically quasi-nonexpansive random mappings with rin : Ω → [0,∞) for each ω ∈ Ω, there exists a measurable mapping rn(ω) =max{r1n(ω), r2n(ω),· · ·, rkn(ω)}for each ω ∈Ω with lim
n→∞rn(ω) = 0 , such that d(Tin(ω, η(ω)), ξ(ω))≤(1 +rn(ω))d(η(ω), ξ(ω))
wherei∈J and η: Ω→K is any measurable mapping. By (1.3), we have d(η1n(ω), ξ(ω)) =d(W(T1n(ω, η0n(ω)), ξn(ω);α1n), ξ(ω))
≤α1nd(T1n(ω, η0n(ω)), ξ(ω)) + (1−α1n)d(ξn(ω), ξ(ω))
≤α1n(1 +rn(ω))d(ξn(ω), ξ(ω)) + (1−α1n)d(ξn(ω), ξ(ω))
≤(1 +α1n(1 +rn(ω)))d(ξn(ω), ξ(ω)).
Since rn(ω) : Ω→[0,∞) and lim
n→∞rn(ω) = 0, there exists a constantL >0 such that L= sup
n≥1
{1 +rn(ω)}<∞.
Therefore,
d(η1n(ω), ξ(ω))≤(1 +L)d(ξn(ω), ξ(ω)).
Assume that
d(ηin(ω), ξ(ω))≤(1 +L)id(ξn(ω), ξ(ω)) holds for some 1≤i≤k−1. Then
d(η(i+1)n(ω), ξ(ω)) =d(W(Ti+1n (ω, ηin(ω)), ξn(ω);α(i+1)n), ξ(ω))
≤α(i+1)nd(Ti+1n (ω, ηin(ω)), ξ(ω)) + (1−α(i+1)n)d(ξn(ω), ξ(ω))
≤α(i+1)n(1 +rn(ω))d(ηin(ω), ξ(ω)) + (1−α(i+1)n)d(ξn(ω), ξ(ω))
≤(1−α(i+1)n+α(i+1)nL(1 +L)i)d(ξn(ω), ξ(ω))
≤(1 +L(1 +L)i)d(ξn(ω), ξ(ω))
≤(1 +L)i+1d(ξn(ω), ξ(ω))
So, by induction, we obtain
d(ηin(ω), ξ(ω))≤(1 +L)id(ξn(ω), ξ(ω)) for all 1≤i≤k. Now, by (1.3) and the above inequality, we get
d(ξn+1(ω), ξ(ω)) =d(W(Tkn(ω, η(k−1)n(ω)), ξn(ω);αkn), ξ(ω))
≤αknd(Tkn(ω, η(k−1)n(ω)), ξ(ω)) + (1−αkn)d(ξn(ω), ξ(ω))
≤αkn(1 +rn(ω))d(η(k−1)n(ω), ξ(ω)) + (1−αkn)d(ξn(ω), ξ(ω))
≤(1−αkn+αknL(1 +L)k)d(ξn(ω), ξ(ω))
≤(1 +αknM0)d(ξn(ω), ξ(ω)) whereM0 = (1 +L)k>0.
(ii)Notice that 1 +x≤ex for all x≥0. Using this and P∞
n=1αkn<∞, we have d(ξn+m(ω), ξ(ω))≤(1 +αk(n+m−1)M0)d(ξn+m−1(ω), ξ(ω))
≤eαk(n+m−1)M0(1 +αk(n+m−2)M0)d(ξn+m−2(ω), ξ(ω))
≤e[αk(n+m−1)+αk(n+m−2)]M0d(ξn+m−2(ω), ξ(ω))
· · · ·
≤eM0Σ∞j=1αkj
d(ξn(ω), ξ(ω))
≤M1d(ξn(ω), ξ(ω)), whereM1 =eM0Σ∞j=1αkj
>0 .
Theorem 2.2. Let K be a nonempty closed convex subset of a separable complete convex metric space (X, d) with a measurable convex structure W. Let {Ti : i ∈ J} : Ω ×K → K be a finite family of continuous asymptotically quasi-nonexpansive random mappings with rin(ω) : Ω → [0,∞) for each ω ∈Ω.
Suppose that the sequence {ξn(ω)} is defined as (1.3) and P∞
n=1αkn < ∞. If F = Tk
i=1F(Ti) 6= ∅, then {ξn(ω)} converges to a common fixed point of {Ti : i ∈ J} if and only if lim inf
n→∞ d(ξn(ω), F) = 0, where d(ξn(ω), F) = inf{d(ξn(ω), η(ω)) :∀η(ω)∈F} for each ω∈Ω.
Proof. The necessity is obvious. Thus, we only need prove the sufficiency. From Lemma 2.1 (i), we have d(ξn+1(ω), F)≤(1 +αknM0)d(ξn(ω), F).
By Lemma 1.7 andP∞
n=1αkn<∞, we know that
n→∞lim d(ξn(ω), F) exists. Since lim inf
n→∞ d(ξn(ω), F) = 0, we obtain
n→∞lim d(ξn(ω), F) = 0 for each ω∈Ω.
Next, We show that {ξn(ω)} is a Cauchy sequence. Indeed, for any ε > 0, there exists a constant N0
such that for alln≥N0, we have
d(ξn(ω), F)≤ ε 2M1.
In particular, there exist a p1(ω)∈F and a constantN1 > N0 such that d(ξN1(ω), p1(ω))≤ ε
2M1.
It follows from Lemma 2.1 (ii) that forn > N1, we have
d(ξn+m(ω), ξn(ω))≤d(ξn+m(ω), p1(ω)) +d(p1(ω), ξn(ω))
≤M1d(ξN1(ω), p1(ω)) +M1d(ξN1(ω), p1(ω))
≤2M1 ε 2M1 =ε.
This implies that {ξn} is a Cauchy sequence in closed convex subset of a complete convex metric space.
Therefore, {ξn(ω)}converges to a point in K.
Suppose lim
n→∞ξn(ω) =p(ω) for each ω ∈ Ω. SinceTi are continuous, by Lemma 1.6, we know that for any measurable mapping f : Ω→ K, Tin(ω, f(ω)) : Ω → K are measurable mappings. Thus, {ξn(ω)} is a sequence of measurable mappings. Hence,p(ω) : Ω→K is also measurable. Notice that
d(p(ω), F)≤d(ξn(ω), p(ω)) +d(ξn(ω), F), together with lim
n→∞d(ξn(ω), F) = 0 and lim
n→∞d(ξn(ω), p(ω)) = 0, we can conclude that d(p(ω), F) = 0.
Therefore, p(ω)∈F.
Remark 2.3. (i) Theorem 2.2 extends the corresponding results in [1, 2, 5, 6, 8, 11, 12, 13, 14, 17, 16] to the convex metric space, which is a more general space;
(ii) Theorem 2.2 extends the corresponding results in [4, 9, 10, 20, 21, 22] to a finite family of asymp- totically quasi-nonexpansive random mappings, which are stochastic generalizations of asymptotically quasi-nonexpansive mappings;
(iii) In Theorem 2.2, we remove the condition: “ P∞
n=1rin <∞, i∈J”, which is required in many other papers (see, e.g., [1, 2, 4, 9, 10, 16, 20, 22]). And the condition “P∞
n=1αin <∞, i ∈J” is replaced with “P∞
n=1αkn<∞”.
By Remark 1.3, we can get the following result:
Corollary 2.4. LetKbe a nonempty closed convex subset of a separable complete convex metric space(X, d) with a measurable convex structure W. Let {Ti :i∈J} : Ω×K → K be a finite family of asymptotically nonexpansive random mappings withrin(ω) : Ω→[0,∞) for eachω ∈Ω. Suppose that the sequence{ξn(ω)}
is defined as (1.3) and P∞
n=1αkn <∞. If F =Tk
i=1F(Ti)6=∅, then {ξn(ω)} converges to a common fixed point of{Ti :i∈J}if and only iflim inf
n→∞ d(ξn(ω), F) = 0, whered(ξn(ω), F) = inf{d(ξn(ω), η(ω)) :∀η(ω)∈F} for each ω∈Ω.
Theorem 2.5. Let K be a nonempty closed convex subset of a separable complete convex metric space (X, d) with a measurable convex structureW. Let{Ti:i∈J}: Ω×K→K be a finite family of continuous asymptotically quasi-nonexpansive random mappings with rin(ω) : Ω → [0,∞) for each ω ∈ Ω. Suppose that the sequence {ξn(ω)} is defined as (1.3) ,P∞
n=1αkn <∞ and F =Tk
i=1F(Ti) 6=∅. If for some given 1≤l≤k and eachω ∈Ω,
(i) lim
n→∞d(Tl(ω, ξn(ω)), ξn(ω)) = 0,
(ii) there exists a constant M2>0 such that
d(Tl(ω, ξn(ω)), ξn(ω))≥M2d(ξn(ω), F).
Then{ξn(ω)} converges to a common fixed point of{Ti :i∈J}.
Proof. From the conditions (i) and (ii), we have
n→∞lim d(ξn(ω), F) = 0.
Therefore, from the proof of Theorem 2.2, we know that {ξn(ω)} converges to a common fixed point of {Ti:i∈J}
Theorem 2.6. Let K be a nonempty closed convex subset of a separable complete convex metric space (X, d) with a measurable convex structureW. Let{Ti:i∈J}: Ω×K→K be a finite family of continuous asymptotically quasi-nonexpansive random mappings withrin(ω) : Ω→[0,∞) for eachω ∈Ω. Suppose that the sequence{ξn(ω)} is defined as (1.3), P∞
n=1αkn<∞ and F =Tk
i=1F(Ti)6=∅. If (i) for all1≤i≤k and eachω∈Ω, lim
n→∞d(Ti(ω, ξn(ω)), ξn(ω)) = 0; (ii) for some 1≤l0 ≤k, Tl0 is semicompact.
Then{ξn(ω)} converges to a common fixed point of{Ti :i∈J}.
Proof. Since Tl0 is semicompact and limn→∞d(Tl0(ω, ξn(ω)), ξn(ω)) = 0, there exists a subsequence {ξnj(ω)} ⊂ {ξn(ω)} such that limj→∞ξnj(ω) = ξ0(ω) for each ω ∈ Ω. Since Ti are continuous, it fol- lows that{ξn} is a sequence of measurable mappings. Therefore,ξ0(ω) : Ω→K is also measurable. Hence, it follows from
d(Ti(ω, ξ0(ω)), ξ0(ω)) = lim
n→∞d(Ti(ω, ξnj(ω)), ξnj(ω)) = 0 thatξ0(ω)∈F. By Lemma 2.1 (i), we have
d(ξn+1(ω), ξ0(ω))≤(1 +αknM0)d(ξn(ω), ξ0(ω)).
According to Lemma 1.7 andP∞
n=1αkn<∞, there exists a constant δ≥0 such that
n→∞lim d(ξn(ω), ξ0(ω)) =δ.
Since lim
j→∞ξnj(ω) =ξ0(ω), we have δ = 0. Therefore, {ξn(ω)} converges to a common fixed point of {Ti:i∈J}.
Acknowledgements:
This work was partially supported by the NSF of China (No.11126290) and University Science Research Project of Jiangsu Province (No.13KJB110021).
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