# Common Fixed Points, Invariant Approximation and Generalized Weak Contractions

Authors: Sumit Chandok
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

Sufficient conditions for the existence of a common fixed point of generalizedf-weakly contractive noncommuting mappings are derived. As applications, some results on the set of best approximation for this class of mappings are obtained. The proved results generalize and extend various known results in the literature.


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## Body

## 1. Introduction and Preliminaries

It is well known that Banach’s fixed point theorem for contraction mappings is one of the pivotal result of analysis. Let(X,d) be a metric space. A mapping T:X→X is said to be contraction if there exists 0≤k<1 such that for all x,y∈X,
(1.1)d(Tx,Ty)≤kd(x,y).
If the metric space (X,d) is complete, then the mapping satisfying (1.1) has a unique fixed point.A natural question is that whether we can find contractive conditions which will imply existence of fixed point in a complete metric space but will not imply continuity. Kannan [1, 2] proved the following result, giving an affirmative answer to the above question.Theorem 1.1.
IfT:X→X, where (X,d) is a complete metric space, satisfies
(1.2)d(Tx,Ty)≤k[d(x,Tx)+d(y,Ty)],
where 0≤k<1/2 and x,y∈X, then T has a unique fixed point.The mappings satisfying (1.2) are called Kannan type mappings. A similar type of contractive condition has been studied by Chatterjea [3] and he proved the following result.Theorem 1.2.
IfT:X→X, where (X,d) is a complete metric space, satisfies
(1.3)d(Tx,Ty)≤k[d(x,Ty)+d(y,Tx)],
where 0≤k<1/2 and x,y∈X, then T has a unique fixed point.In Theorems1.1 and 1.2 there is no requirement of continuity of T.A mapT:X→X is called a weakly contractive (see [4–6]) if for each x,y∈X,
(1.4)d(Tx,Ty)≤d(x,y)-ψ(d(x,y)),
where ψ:[0,∞)→[0,∞) is continuous and nondecreasing, ψ(x)=0 if and only if x=0 and limψ(x)=∞.If we takeψ(x)=(1-k)x, 0≤k<1, then a weakly contractive mapping is called contraction.A mapT:X→X is called f-weakly contractive (see [7]) if for each x,y∈X,
(1.5)d(Tx,Ty)≤d(fx,fy)-ψ(d(fx,fy)),
where f:X→X is a self-mapping, ψ:[0,∞)→[0,∞) is continuous and nondecreasing, ψ(x)=0 if and only if x=0 and limψ(x)=∞.If we takeψ(x)=(1-k)x, 0≤k<1, then a f-weakly contractive mapping is called f-contraction. Further, if f= identity mapping and ψ(x)=(1-k)x, 0≤k<1, then a f-weakly contractive mapping is called contraction.A mapT:X→X is called a generalized weakly contractive (see [5]) if for each x,y∈X,
(1.6)d(Tx,Ty)≤12[d(x,Ty)+d(y,Tx)]-ψ(d(x,Ty),d(y,Tx)),
where ψ:[0,∞)2→[0,∞) is continuous such that ψ(x,y)=0 if and only if x=y=0.If we takeψ(x,y)=(1-(k/2))(x+y), 0≤k<1/2, then inequality (1.6) reduces to (1.3). Choudhury [5] shows that generalized weakly contractive mappings are generalizations of contractive mappings given by Chatterjea (1.3), and it constitutes a strictly larger class of mappings than given by Chatterjea.A mapT:X→X is called a generalizedf-weakly contractive [8] if for each x,y∈X,
(1.7)d(Tx,Ty)≤12[d(fx,Ty)+d(fy,Tx)]-ψ(d(fx,Ty),d(fy,Tx)),
where f:X→X is a self-mapping, ψ:[0,∞)2→[0,∞) is continuous such that ψ(x,y)=0 if and only if x=y=0.Iff=identity mapping, then generalized f-weakly contractive mapping is generalized weakly contractive.For a nonempty subsetM of a metric space (X,d) and x∈X, an element y∈M is said to be a best approximant to x or a bestM-approximant to x if d(x,y)=d(x,M)≡inf{d(x,k):k∈M}. The set of all such y∈M is denoted by PM(x).A subsetM of a normed linear space X is said to be aconvex set if λx+(1-λ)y∈M for all x,y∈M and λ∈[0,1]. A set M is said to be p-starshaped, where p∈M, provided λx+(1-λ)p∈M for all x∈M and λ∈[0,1], that is, if the segment [p,x]={λx+(1-λ)p:0≤λ≤1} joining p to x is contained in M for all x∈M. M is said to be starshaped if it is p-starshaped for some p∈M.Clearly, each convex setM is starshaped but converse is not true.Suppose thatM is a subset of normed linear space X. A mapping T from M to X is said to be demiclosed if for every sequence {xn}⊆M such that xn converges weakly to x∈M and {T(xn)} converges strongly to y∈X imply y=Tx. T is said to be demiclosed at 0, if for every sequence {xn} in M converging weakly to x and {Txn} converging strongly to 0, then Tx=0.For a convex subsetM of a normed linear space X, a mapping T:M→M is said to be affine if for all x,y∈M, T(λx+(1-λ)y)=λTx+(1-λ)Ty for all λ∈[0,1].The ordered pair(T,I) of two self-maps of a metric space (X,d) is called a Banach operator pair [9], if the set F(I) of fixed points of I is T-invariant, that is, T(F(I))⊆F(I). A point x∈X is a coincidence point (common fixed point) of I and T if Ix=Tx(x=Ix=Tx). The set of fixed points (resp., coincidence points) of I and T is denoted by F(I,T) (resp., C(I,T)). The pair (I,T) is called commuting if TIx=ITx for all x∈X. Obviously, commuting pair, (T,I) is a Banach operator pair but not conversely (see [9]). If (T,I) is a Banach operator pair then (I,T) need not be a Banach operator pair (see [9]). If the self-maps T and I of X satisfy d(ITx,Tx)≤kd(Ix,x), for all x∈X and for some k≥0, ITx=TIx whenever x∈F(I), that is, Tx∈F(I), then (T,I) is a Banach operator pair. This class of noncommuting mappings is different from the known classes of noncommuting mappings namely, R-weakly commuting, R-subweakly commuting, compatible, weakly compatible, Cq-commuting, and so forth, existing in the literature.Fixed point theory has gained impetus, due to its wide range of applicability, to resolve diverse problems emanating from the theory of nonlinear differential equations, theory of nonlinear integral equations, game theory, mathematical economics, control theory, and so forth. For example, in theoretical economics, such as general equilibrium theory, a situation arises where one needs to know whether the solution to a system of equations necessarily exists, or, more specifically, under what conditions will a solution necessarily exist. The mathematical analysis of this question usually relies on fixed point theorems. Hence, finding necessary and sufficient conditions for the existence of fixed points is an interesting aspect. Alber and Guerre-Delabriere [4] introduced the concept of weakly contractive mappings and proved the existence of fixed points for single-valued weakly contractive mappings in Hilbert spaces. Thereafter, Rhoades [6] proved a fixed point theorem which is one of the generalizations of Banach's Fixed Point Theorem in 1922, because the weakly contractions contain many contractions as a special case, and he also showed that some results of [4] are true for any Banach spaces. In fact, weakly contractive mappings are closely related to the mappings introduced by Boyd and Wong [10] and Reich [11]. Many other nonlinear contractive type mappings like Chatterjea, Ciric, Kannan, Reich type, and so forth and their generalizations have been investigated by many authors. Fixed point and common fixed point theorems for different types of nonlinear contractive mappings have been investigated extensively by various researchers (see [1–21] and references cited therein). In this paper, sufficient conditions for the existence of a common fixed point of generalized f-weakly contractive noncommuting mappings are obtained. As applications, we also establish some results on the set of best approximation for this class of mappings. The proved results generalize and extend the corresponding results of [5, 8, 9, 12–14, 18–20] and of few others.


## 2. Main Results

The following result is a consequence of the main theorem of Choudhury [5].Lemma 2.1.
LetM be a subset of a metric space (X,d) and T is a self-mapping of M such that clT(M)⊆M. If clT(M) is complete and T satisfies
(2.1)d(Tx,Ty)≤12[d(x,Ty)+d(y,Tx)]-ψ(d(x,Ty),d(y,Tx)),
where ψ:[0,∞)2→[0,∞) is a continuous mapping such that ψ(x,y)=0 if and only if x=y=0, for all x,y∈X, then T has a unique fixed point in M.Corollary 2.2 (see [5]).
LetT be a self-mapping of X, where (X,d) is a complete metric space. If T satisfies
(2.2)d(Tx,Ty)≤12[d(x,Ty)+d(y,Tx)]-ψ(d(x,Ty),d(y,Tx)),
where ψ:[0,∞)2→[0,∞) is a continuous mapping such that ψ(x,y)=0 if and only if x=y=0, for all x,y∈X, then T has a unique fixed point.Ifψ(x,y)=((1/2)-k)(x+y), 0≤k<(1/2), we have Theorem 1.2.Theorem 2.3.
LetM be a subset of a metric space (X,d), and f and T are self-mappings of M such that clT(F(f))⊆F(f). If clT(M) is complete, F(f) is nonempty, and f and T satisfy
(2.3)d(Tx,Ty)≤12[d(fx,Ty)+d(fy,Tx)]-ψ(d(fx,Ty),d(fy,Tx)),
where ψ:[0,∞)2→[0,∞) is a continuous mapping such that ψ(x,y)=0 if and only if x=y=0, for all x,y∈X, then M∩F(T)∩F(f) is a singleton.Proof.
cl
T
(
F
(
f
)
) being a subset of clT(M) is complete and and clT(F(f))⊆F(f). So for all x,y∈F(f), we have
(2.4)d(Tx,Ty)≤12[d(fx,Ty)+d(fy,Tx)]-ψ(d(fx,Ty),d(fy,Tx))=12[d(x,Ty)+d(y,Tx)]-ψ(d(x,Ty),d(y,Tx)).
Thus, by Lemma 2.1, T has a unique fixed point z in F(f) and consequently, M∩F(T)∩F(f) is a singleton.Corollary 2.4.
LetM be a subset of a metric space (X,d), and f and T are self-mappings of M. If clT(M) is complete, (T,f) is a Banach operator pair, F(f) is nonempty and closed, and f and T satisfy
(2.5)d(Tx,Ty)≤12[d(fx,Ty)+d(fy,Tx)]-ψ(d(fx,Ty),d(fy,Tx)),
where ψ:[0,∞)2→[0,∞) is a continuous mapping such that ψ(x,y)=0 if and only if x=y=0, for all x,y∈X, then M∩F(T)∩F(f) is a singleton.Example 2.5.
LetX={p,q,r} and d is a metric defined on X. Let T and f be self-mappings of X such that Tp=fq, Tq=fq, Tr=fp, d(fp,fq)=1, d(fq,fr)=2, d(fr,fp)=1.5 and ψ(a,b)=(1/2)min{a,b}. Then T is a generalized f-weakly contraction, and q is the coincidence point of T and f.
Iff=identity mapping, this example given in [5].Ifψ(x,y)=((1/2)-k)(x+y), 0≤k<1/2, we have the following result.Corollary 2.6.
LetM be a subset of a metric space (X,d), and f and T be self-mappings of M such that clT(F(f))⊆F(f). If clT(M) is complete, F(f) is nonempty, and f and T satisfy
(2.6)d(Tx,Ty)≤k[d(fx,Ty)+d(fy,Tx)],
where 0≤k<1/2, for all x,y∈M, then M∩F(T)∩F(f) is a singleton.Corollary 2.7.
LetM be a subset of a metric space (X,d) and f and T be self-mappings of M. If clT(M) is complete, (T,f) is a Banach operator pair, F(f) is nonempty and closed, and f and T satisfy
(2.7)d(Tx,Ty)≤k[d(fx,Ty)+d(fy,Tx)],
where 0≤k<1/2, for all x,y∈M, then M∩F(T)∩F(f) is a singleton.Theorem 2.8.
LetM be a nonempty subset of a normed (resp., Banach) space X, and Tf be self-mappings of M. Suppose that F(f) is q-starshaped, clT(F(f))⊆F(f) (resp., wclT(F(f))⊆F(f)), clT(M) is compact (resp., wclT(M) is weakly compact, and f-T is demiclosed at 0), and T satisfies
(2.8)∥Tx-Ty∥≤12[dist(fx,[Ty,q])+dist(fy,[Tx,q])]-ψ(dist(fx,[Ty,q]),dist(fy,[Tx,q])),
where ψ:[0,∞)2→[0,∞) is a continuous mapping such that ψ(x,y)=0 if and only if x=y=0, for all x,y∈M. Then M∩F(T)∩F(f)≠∅.Proof.
For eachn, define Tn:M→M by Tnx=(1-kn)q+knTx, x∈M where (kn) is a sequence in (0,1) such that kn→1. Since F(f) is q-starshaped and clT(F(f))⊆F(f) (resp., wclT(F(f))⊆F(f)), we have
(2.9)Tn(x)=(1-kn)q+knTx=(1-kn)fq+knTx∈F(f),
for all x∈F(f) and so clTn(F(f))⊆F(f) (resp., wclTn(F(f))⊆F(f)) for each n. Consider
(2.10)∥Tnx-Tny∥=kn∥Tx-Ty∥≤kn[12[dist(fx,[Ty,q])+dist(fy,[Tx,q])]-ψ(dist(fx,[Ty,q]),dist(fy,[Tx,q]))]≤kn[12[dist(fx,[Ty,q])+dist(fy,[Tx,q])]]≤kn2[∥fx-Tny∥+∥fy-Tnx∥],
for all x,y∈F(f). As clT(M) is compact, clTn(M) is compact for each n and hence complete. Now by Corollary 2.6, there exists xn∈M such that xn is a common fixed point of f and Tn for each n. The compactness of clT(M) implies that there exists a subsequence {Txni} of {Txn} such that Txni→z∈clT(M). Since {Txn} is a sequence in T(F(f)), z∈clT(F(f))⊆F(f). Now, as kni→1, we have(2.11)xni=Tnixni=(1-kni)q+kniTxni→z
and ∥fxni-Txni∥=∥xni-Txni∥→0. Further, we have
(2.12)∥Txni-Tz∥≤12[dist(fxni,[Tz,q])+dist(fz,[Txni,q])]-ψ(dist(fxni,[Tz,q]),dist(fz,[Txni,q]))=12[dist(xni,[Tz,q])+dist(fz,[Txni,q])]-ψ(dist(xni,[Tz,q]),dist(fz,[Txni,q]))≤12[∥xni-Tz∥+∥fz-Txni∥],
on taking limit, we get z=Tz and so M∩F(T)∩F(f)≠∅.
Next, the weak compactness ofwclT(M) implies that wclTn(M) is weakly compact and hence complete. Hence, by Corollary 2.6, for each n∈ℕ, there exists xn∈F(f) such that xn=fxn=Tnxn. The weak compactness of wclT(M) implies that there is subsequence {Txni} of {Txn} such that Txni⇀y∈wclT(M). Since {Txn} is a sequence in T(F(f)), y∈wclT(F(f))⊆F(f). Also, we have fxni-Txni=xni-Txni→0.
Iff-T is demiclosed at 0, then fy=Ty=y and so M∩F(T)∩F(f)≠∅.LetM be a nonempty subset of a metric space (X,d). Suppose that C=PM(u)∩CMf(u), where CMf(u)={x∈M:fx∈PM(u)}.Corollary 2.9.
LetX be a normed (resp., Banach) space, and T, f are self-mappings of X. If u∈X, D⊆C, G=D∩F(f) is q-starshaped, clT(G)⊆G (resp., wclT(G)⊆G), clT(D) is compact (resp., wclT(D) is weakly compact and f-T is demiclosed at 0), and T satisfies the inequality (2.8) for all x,y∈D, then PM(u)∩F(f,T) is nonempty.Corollary 2.10.
LetX be a normed (resp., Banach) space and T,f are self-mappings of X. If u∈X, D⊆PM(u), G=D∩F(f) is q-starshaped, clT(G)⊆G (resp., wclT(G)⊆G), clT(D) is compact (resp., wclT(D) is weakly compact and f-T is demiclosed at 0), and T satisfies the inequality (2.8) for all x,y∈D, then PM(u)∩F(f,T) is nonempty.Remark 2.11.
Theorem2.8 extends and generalizes the corresponding results of [8, 9, 13, 14, 18, 19].
LetG0 denote the class of closed convex subsets of a normed space X containing 0. For M∈G0 and p∈X, let Mp={x∈M:∥x∥≤2∥p∥}. Then PM(p)⊂Mp∈G0 (see [12, 20]).Theorem 2.12.
LetX be a normed (resp., Banach) space, and T, g are self-mappings of X. If p∈X and M∈G0 such that T(Mp)⊆M, clT(Mp) is compact (resp., wclT(Mp) is weakly compact), and ∥Tx-p∥≤∥x-p∥ for all x∈Mp, then PM(p) is nonempty, closed, and convex with T(PM(p))⊆PM(p). If, in addition, D is a subset of PM(p), G=D∩F(g) is q-starshaped, clT(G)⊆G (resp., wclT(G)⊆G and g-T is demiclosed at 0), and T satisfies inequality (2.8) for all x,y∈D, then PM(p)∩F(g,T) is nonempty.Proof.
Ifp∈M, then the results are obvious. So assume that p∉M. If x∈M∖Mp, then ∥x-x0∥>2∥p-x0∥ and so ∥p-x∥≥∥x∥-∥p∥>∥p∥≥dist(p,M). Thus α=dist(p,M)≤∥p∥. Since cl(T(Mp)) is compact, and by the continuity of norm, there exists z∈cl(T(Mp)) such that β=dist(p,cl(T(Mp))=∥z-p∥.
On the other hand, ifwcl(T(Mp)) is weakly compact, then using Lemma 5.5 of Singh et al. [22, page 192], we can show that there exists z∈wcl(T(Mp)) such that β=dist(p,wcl(T(Mp)))=∥z-p∥.
Hence, in both cases, we have(2.13)α=dist(p,M)≤dist(p,cl(T(Mp)))=β=dist(p,T(Mp))≤∥Tx-p∥≤∥x-p∥,
for all x∈Mp. Therefore, α=β=dist(p,M), that is, dist(p,M)=dist(p,cl(T(Mp))=d(p,z), that is, z∈PM(p) and so PM(p) is nonempty. The closedness and convexity of PM(p) follow from that of M. Now to prove T(PM(p)⊆PM(p), let y∈T(PM(p)). Then y=Tx for x∈PM(p). Consider
(2.14)d(p,y)=d(p,Tx)≤d(p,x)=dist(p,M),
and so y∈PM(p) as PM(p)⊂Mp⇒T(PM(p))⊂M, that is, y∈M.
The compactness ofclT(Mp) (resp., weakly compactness of wclT(Mp)) implies that clT(D) is compact (resp., wclT(D) is weakly compact). Hence, the result follows from Corollary 2.10.Corollary 2.13.
LetX be a normed (resp., Banach) space, and T, g are self-mappings of X. If p∈X and M∈G0 such that T(Mp)⊆M, clT(Mp) is compact (resp., wclT(Mp) is weakly compact), and ∥Tx-p∥≤∥x-p∥ for all x∈Mp, then PM(p) is nonempty, closed, and convex with T(PM(p))⊆PM(p). If, in addition, D is a subset of PM(p), G=D∩F(g) is q-starshaped and closed (resp., weakly closed and g-T is demiclosed at 0), (T,g) is a Banach operator pair on D, and T satisfies inequality (2.8) for all x,y∈D, then PM(p)∩F(g,T) is nonempty.Remark 2.14.
Theorem2.12 extends and generalizes the corresponding results of Al-Thagafi [12], Al-Thagafi and Shahzad [13], Habiniak [18], Narang and Chandok [20], and Shahzad [21].The following result will be used in the sequel.Lemma 2.15.
LetC be a nonempty subset of a metric space (X,d), f, g self-maps of C, clT(F(f)∩F(g))⊆F(f)∩F(g). Suppose that cl(T(C)) is complete, and T, f, g satisfy for all x,y∈C and 0≤k<1,
(2.15)d(Tx,Ty)≤kmax{d(fx,gy),d(Tx,fx),d(Ty,gy),d(Tx,gy),d(Ty,fx)}.
If F(f)∩F(g) is nonempty and clT(F(f)∩F(g))⊆F(f)∩F(g), then there is a common fixed point of T,f and g.Proof.
cl
T
(
F
(
f
)
∩
F
(
g
)
), being a closed subset of the complete set clT(C), is complete. Further for all x,y∈F(f)∩F(g), we have
(2.16)d(Tx,Ty)≤kmax{d(fx,gy),d(Tx,fx),d(Ty,gy),d(Tx,gy),d(Ty,fx)}=kmax{d(x,y),d(Tx,x),d(Ty,y),d(Tx,y),d(Ty,x)}.
Hence, T is a generalized contraction on F(f)∩F(g) and clT(F(f)∩F(g))⊆F(f)∩F(g). So by Lemma 3.1 of [13], T has a unique fixed point y in F(f)∩F(g) and consequently F(T)∩F(f)∩F(g) is a singleton.Remark 2.16.
Iff=g, then Theorem 3.2 of Al-Thagafi and Shahzad [13] is a particular case of Lemma 2.15.
The following result extends and improves the corresponding results of [9, 12–14, 18, 20].Theorem 2.17.
LetT,g,h be self-mappings of a Banach space X. If p∈F(T)∩F(g)∩F(h) and M∈G∘ such that T(Mp)⊆g(M)⊆M⊆h(M), clg(Mp) is compact, and ∥Tx-p∥≤∥gx-hp∥, ∥gx-u∥≤∥x-u∥, ∥hx-u∥=∥x-u∥ for all x∈Mp, then(i)
PM(p) is nonempty, closed, and convex,(ii)
T(PM(p))⊆g(PM(p))⊆PM(p)=h(PM(p)),(iii)
PM(p)∩F(T)∩F(g)∩F(h)≠∅, provided T is continuous, F(g) is q-starshaped, clg(F(h))⊆F(h), and the pair (g,h) satisfies the inequality (2.8) for all x,y∈PM(p), F(g) is q-starshaped with q∈PM(p)∩F(g)∩F(h)=G, clT(F(g)∩F(h))⊆F(g)∩F(h) and T,g,h satisfy for all q∈F(g)∩F(h) and x,y∈PM(p)(2.17)d(Tx,Ty)≤max{d(hx,gy),dist(hx,[q,Tx]),dist(gy,[q,Ty]),dist(hx,[q,Ty]),<max=dist(gy,[q,Tx])},then there is a common fixed point of PM(p), T, g, and h.Proof.
Proceeding as in Theorem2.12, we can prove (i) and (ii).
By (ii), the compactness ofclg(Mp) implies that clg(PM(p)) and clT(PM(p)) are compact. Hence, Theorem 2.8 implies that F(g)∩F(h)∩PM(p)≠∅.
For eachn∈ℕ, define Tn:X→X by Tnx=(1-kn)q+knTx, for each x∈X, where {kn} is a sequence in (0,1) such that kn→1. Then each Tn is a self-mapping of C. Since clT(F(g)∩F(h))⊆F(g)∩F(h), F(g)∩F(h) is q-starshaped with q∈G, so clTn(F(g)∩F(h))⊆F(g)∩F(h) for each n. Consider
(2.18)∥Tnx-Tny∥=kn∥Tx-Ty∥≤knmax{∥hx-gy∥,dist(hx,[q,Tx]),dist(gy,[q,Ty]),dist(hx,[q,Ty]),kdist(gy,[q,Tx])}≤knmax{∥hx-gy∥,∥hx-Tnx∥,∥gy-Tny∥,∥hx-Tny∥,∥gy-Tnx∥},
for all x,y∈PM(p). As cl(T(PM(p))) is compact, cl(Tn(PM(p))) is compact for each n and hence complete. Now by Lemma 2.15, there exists xn∈M such that xn is a common fixed point of g,h and Tn for each n. The compactness of cl(T(PM(p))) implies that there exists a subsequence {Txni} of {Txn} such that Txni→z∈clT(PM(p)). Since {Txn} is a sequence in T(F(g)∩F(h)), and clT(F(g)∩F(h))⊆F(g)∩F(h), then z∈F(g)∩F(h). Now, as kni→1, we have
(2.19)xni=Tnixni=(1-kni)q+kniTxni→z,T is continuous, we have Tz=z and hence PM(p)∩F(T)∩F(g)∩F(h)≠∅.Remark 2.18.
(i) LetM=[0,1]. Let d be defined by d(x,y)=|x-y|. We set Tx=x/4 and fx=x/2 for all x∈M. Define ψ:[0,∞)×[0,∞)→[0,∞) by
(2.20)ψ(t,s)=t+s8.
Then for x,y∈M, we have
(2.21)d(Tx,Ty)=d(x4,y4)=|x4-y4|=14|x-y|,(2.22)12[d(fx,Ty)+d(fy,Tx)]-ψ(d(fx,Ty),d(fy,Tx))=12[|x2-y4|+|y2-x4|]-ψ(|x2-y4|,|y2-x4|)=12[|x2-y4|+|y2-x4|]-18[|x2-y4|+|y2-x4|]=38[|x2-y4|+|y2-x4|].
From (2.21) and (2.22), without loss of generality assume that x≥y. Hence, we have two cases:
Case  1. If 2y≥x, from (2.22), we have
(2.23)12[d(fx,Ty)+d(fy,Tx)]-ψ(d(fx,Ty),d(fy,Tx))=38[|x2-y4|+|y2-x4|]=38[(x2-y4)+(y2-x4)]=38(x4+y4)=332(x+y)≥14(x-y).
Case  2. If 2y<x, from (2.22), we have
(2.24)12[d(fx,Ty)+d(fy,Tx)]-ψ(d(fx,Ty),d(fy,Tx))=38[|x2-y4|+|y2-x4|]=38[(x2-y4)+(-y2+x4)]=38(3x4+3y4)=932(x-y)≥14(x-y).
Thus inequality (2.3) is satisfied and by Theorem 2.3; 0 is a common fixed point of T and f.
(ii) It may be noted that the assumption of linearity or affinity forI is necessary in almost all known results about common fixed points of maps T, I such that T is I-nonexpansive under the conditions of commuting, weakly commuting, R-subweakly commuting, or compatibility (see [9, 12, 14, 20, 21] and the literature cited therein), but our results in this paper are independent of the linearity or affinity.
(iii) ConsiderM=ℝ2 with usual metric d((x1,y1),(x2,y2))=|x1-x2|+|y1-y2|, (x1,y1), (x2,y2)∈ℝ2. Define T and I on M as T(x,y)=((x-2)/2,(x2+y-4)/2) and I(x,y)=((x-2)/2,x2+y-4). Obviously, T is I-nonexpansive, I-asymptotically nonexpansive, but I is not linear or affine. Moreover, F(T)=(-2,0), F(I)={(-2,y):y∈ℝ} and C(I,T)={(x,y):y=4-x2,x∈ℝ}. Thus, clT(F(I))⊂F(I), which is not a compatible pair (see [9]), F(I) is convex, starshaped for any z∈F(I), and (-2,0) is a common fixed point of I and T.



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*Source: 102980-2012-09-29.xml*