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International Journal of Mathematics and Mathematical Sciences Volume 2012, Article ID 102980,11 pages doi:10.1155/2012/102980

Research Article

Common Fixed Points, Invariant Approximation and Generalized Weak Contractions

Sumit Chandok

Department of Mathematics, Khalsa College of Engineering & Technology (Punjab Technical University), Ranjit Avenue, Amritsar-143001, India

Correspondence should be addressed to Sumit Chandok, chansok.s@gmail.com Received 23 March 2012; Revised 23 August 2012; Accepted 31 August 2012 Academic Editor: Harvinder S. Sidhu

Copyright © 2012 Sumit Chandok. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Sufficient conditions for the existence of a common fixed point of generalized /-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.

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 e X,

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. If T : X ^ X, where (X, d) is a complete metric space, satisfies

d(Tx,Ty) < kd(x,y).

d(Tx,Ty) < k[d(x,Tx) + d(y,Ty)],

where 0 < k < 1/2 and x,y e 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. If T : X ^ X, where (X, d) is a complete metric space, satisfies

d(Tx,Ty) < k[d(x,Ty) + d(y,Tx)], (1.3)

where 0 < k < 1/2 and x,y e X, then T has a unique fixed point.

In Theorems 1.1 and 1.2 there is no requirement of continuity of T. A map T : X ^ X is called a weakly contractive (see [4-6]) if for each x,y e X,

d(Tx,Ty) < d(x,y) - f(d(x,y)), (1.4)

where f : [0, to) ^ [0, to) is continuous and nondecreasing, f (x) = 0 if and only if x = 0 and lim f (x) = to.

If we take f (x) = (1 - k)x, 0 < k < 1, then a weakly contractive mapping is called contraction.

A map T : X ^ X is called f -weakly contractive (see [7]) if for each x,y e X,

d(Tx, Ty) < dfx, fy) - f(d(fx, fy)), (1.5)

where f : X ^ X is a self-mapping, f : [0, to) ^ [0, to) is continuous and nondecreasing, f (x) = 0 if and only if x = 0 and lim f (x) = to.

If we take f (x) = (1 - k)x, 0 < k < 1, then a f-weakly contractive mapping is called f-contraction. Further, if f = identity mapping and f (x) = (1 - k)x,0 < k< 1,thena f-weakly contractive mapping is called contraction.

A map T : X ^ X is called a generalized weakly contractive (see [5]) if for each x,y e X, 1

d(Tx,Ty) < 2 [d(x,Ty) + d(y,Tx)] - f\d(x,Ty),d(y,Tx)), (1.6)

where f : [0, to)2 ^ [0, to) is continuous such that f(x, y) = 0 if and only if x = y = 0.

Ifwetake f(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 map T : X ^ X is called a generalized f-weakly contractive [8] if for each x,y e X, 1

d(Tx,Ty) < ^ [d(fx,Ty) + d(fy,Tx)] - f\d(fx,Ty),d(fy,Tx)), (1.7)

where f : X ^ X is a self-mapping, f : [0, to)2 ^ [0, to) is continuous such that f (x,y) = 0 if and only if x = y = 0.

If f = identity mapping, then generalized f-weakly contractive mapping is generalized weakly contractive.

For a nonempty subset M of a metric space (X, d) and x e X, an element y e M is said to be a best approximant to x or a best M-approximant to x if d(x,y) = d(x,M) = inf{d(x,k) : k e M}. The set of all such y e M is denoted by PM(x).

A subset M of a normed linear space X is said to be a convex set if Xx + (1 - X)y e M for all x,y e M and X e [0,1]. A set M is said to be p-starshaped, where p e M, provided Xx + (1 - X)p e M for all x e M and X e [0,1], that is, if the segment [p, x] = {Xx + (1 - X)p : 0 < X < 1} joining p to x is contained in M for all x e M. M is said to be starshaped if it is p-starshaped for some p e M.

Clearly, each convex set M is starshaped but converse is not true.

Suppose that M 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} c MM such that xn converges weakly to x e and {T(xn)} converges strongly to y e 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 subset M of a normed linear space X, a mapping T : M ^ M is said to be affine if for all x,y e M, T(Xx + (1 - X)y) = XTx + (1 - X)Ty for all X e [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)) c F(I). A point x e 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 e 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 e X and for some k > 0, ITx = TIx whenever x e F (I), that is, Tx e 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, K-weakly commuting, K-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 / -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. Let M be a subset of a metric space (X,d) and T is a self-mapping of M such that cl T(M) c M. If cl T(M) is complete and T satisfies

d(Tx,Ty) < 2 [d(x,Ty) + d(y,Tx)] - f(d(x,Ty),d(y,Tx)), (2.1)

where f : [0, to)2 ^ [0, to) is a continuous mapping such that f(x, y) = 0 if and only if x = y = 0, for all x,y e X, then T has a unique fixed point in M.

Corollary 2.2 (see [5]). Let T be a self-mapping of X, where (X,d) is a complete metric space. If T satisfies

d(Tx,Ty) < ^ [d(x,Ty) + d(y,Tx)] - f(d(x,Ty),d(y,Tx)), (2.2)

where f : [0, to)2 ^ [0, to) is a continuous mapping such that f(x, y) = 0 if and only if x = y = 0, for all x,y e X, then T has a unique fixed point.

If f (x,y) = ((1/2) - k)(x + y), 0 < k < (1/2), we have Theorem 1.2.

Theorem 2.3. Let M be a subset of a metric space (X, d), and f and T are self-mappings of M such

that cl T (F (f)) c F(f). If cl T (M) is complete, F(f) is nonempty, and f and T satisfy

d(Tx, Ty) < 2 [d(fx, Ty) + dfy, Tx)] - f(d(fx, Ty), dfy, Tx)), (2.3)

where f : [0, to)2 ^ [0, to) is a continuous mapping such that f(x, y) = 0 if and only if x = y = 0, for all x,y e X, then M n F (T) n F(f) is a singleton.

Proof. clT(F(f)) being a subset of clT(M) is complete and and clT(F(f)) c F(f). So for all x,y e F (f), we have

d(Tx,Ty) < - [d(fx,Ty) + d(fy,Tx)] - f(dfx,Ty),dfy,Tx))

= 2 [d(x,Ty) + d(y,Tx)] - f(d(x,Ty),d(y,Tx)).

Thus, by Lemma 2.1, T has a unique fixed point z in F(f) and consequently, M n F (T) n F(f) is a singleton. □

Corollary 2.4. Let M be a subset of a metric space (X,d), and / and T are self-mappings of M. If cl T(M) is complete, (T,/) is a Banach operator pair, F(/) is nonempty and closed, and / and T satisfy

d(Tx, Ty) < ^ [d(/x, Ty) + d/y, Tx)] - f(d(/x, Ty), d{/y, Tx)), (2.5)

where f : [0, to)2 ^ [0, to) is a continuous mapping such that f(x, y) = 0 if and only if x = y = 0, for all x,y e X, then M n F (T) n F(/) is a singleton.

Example 2.5. Let X = {p,q,r} and d is a metric defined on X. Let T and / be self-mappings of X such that Tp = /q, Tq = /q, Tr = /p, d(/p,/q) = 1, d(/q,/r) = 2, d(/r,/p) = 1.5 and f (a,b) = (1/2) min{a,b}. Then T is a generalized /-weakly contraction, and q is the coincidence point of T and /.

If / =identity mapping, this example given in [5].

If f (x,y) = ((1/2) - k)(x + y), 0 < k < 1/2, we have the following result.

Corollary 2.6. Let M be a subset of a metric space (X, d), and / and T be self-mappings of M such that cl T (F (/)) c F(/). If cl T (M) is complete, F(/) is nonempty, and / and T satisfy

d(Tx,Ty) < k[d(/x, Ty) + d/y,Tx)], (2.6)

where 0 < k < 1/2, for all x,y e M, then M n F (T) n F(/) is a singleton.

Corollary 2.7. Let M be a subset of a metric space (X,d) and / and T be self-mappings of M. If cl T(M) is complete, (T,/) is a Banach operator pair, F(/) is nonempty and closed, and / and T satisfy

d(Tx,Ty) < k[d/x,Ty) + d{/y,Tx)], (2.7)

where 0 < k < 1/2, for all x,y e M, then M n F (T) n F(/) is a singleton.

Theorem 2.8. Let M be a nonempty subset of a normed (resp., Banach) space X, and T/ be self-mappings of M. Suppose that F(/) is q-starshaped, cl T(F(/)) c F(/) (resp., wcl T(F(/)) c F(/)), cl T(M) is compact (resp., w cl T(M) is weakly compact, and / - T is demiclosed at 0), and T satisfies

\\Tx - Ty\\ < 2 [dist(/x, [Tfrq]) + dist(/y, [Tx,q])]

2 (2.8) - f(dist(^ [Ty,q]),dist(/y, [Tx,q])),

where f : [0, to)2 ^ [0, to) is a continuous mapping such that f(x, y) = 0 if and only if x = y = 0, for all x,y e M. Then M n F (T) n F(/) = 0.

Proof. For each n, define Tn : M ^ M by Tnx = (1 - kn)q + knTx, x e M where (kn) is a sequence in (0,1) such that kn ^ 1. Since F(f) is q-starshaped and cl T(F(f)) c F(f) (resp., wcl T(F(f)) c F(f)), we have

Tn(x) = (1 - kn)q + knTx = (1 - kn)fq + knTx e Ff), (2.9)

for all x e F(f) and so cl Tn(F(f)) c F(f) (resp., wcl Tn(F(f)) c F(f)) for each n. Consider |Tnx - Tny|| = kn\\Tx - TyW

- f (dist(fx, [Ty,q]),dist(fy, [Tx,q]))] (2.10)

2[dist(fx, [Ty,q]) + dis^fy, [Tx,q])]

^[disKfx, [Ty^]) + dist(fy, iTx,q])]

< y[llfx- Tny|| + ||fy-Tnx\\],

for all x,y e F(f ).AsclT(M) is compact, cl Tn(M) is compact for each n and hence complete. Now by Corollary 2.6, there exists xn e 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 e clT(M). Since {Txn} is a sequence in T(F(f)), z e clT(F(f)) c F(f). Now, as kni ^ 1, we have

xni = Tnixni = (1 - kni)q + knTxni —> z (2.11)

and \\fxni - Txni || = ||xni - Txni || ^ 0. Further, we have

||Txn - Tz|| < 2[dist(fxm, [Tz,q]) + dis^fz, [Txm,q])]

- f(dist(fxm, [Tz,q]),dis^fz, [Txm,q])) 1

= ^[dist(xm, [Tz,q]) + distfz, [Txni,q])] (2.12)

- f( dist (xni, [Tz,q]), dis^fz, [Txni,q])) 1

< 2 [Wxn, - Tz|| + ||fz - Txm |,

on taking limit, we get z = Tz and so M n F(T) n F(f) = 0.

Next, the weak compactness of wclT(M) implies that wclTn(M) is weakly compact and hence complete. Hence, by Corollary 2.6, for each n e N, there exists xn e F(f) such that xn = fxn = Tnxn. The weak compactness of w cl T(M) implies that there is subsequence {Txni} of {Txn} such that Txni ^ y e wclT(M). Since {Txn} is a sequence in T(F(f)), y e wcl T(F(f)) c F(f). Also, we have fxni - Txni = xni - Txni ^ 0.

If f - T is demiclosed at 0, then fy = Ty = y and so M n F(T) n F(f) = 0. □

Let M beanonempty subset of a metric space (X, d). Suppose that C = PM(u) n C^M (u), where C^M(u) = {x e M : /x e PM(u)}.

Corollary 2.9. Let X be a normed (resp., Banach) space, and T, / are self-mappings of X. If u e X, D c C, G = D n F(/) is q-starshaped, cl T(G) c G (resp., w cl T(G) c G), cl T(D) is compact (resp., w cl T(D) is weakly compact and / - T is demiclosed at 0), and T satisfies the inequality (2.8) for all x,y e D, then PM (u) n F (/, T) is nonempty.

Corollary 2.10. Let X be a normed (resp., Banach) space and T,/ are self-mappings of X. If u e X, D c PM(u), G = D n F(/) is q-starshaped, cl T(G) c G (resp., w cl T(G) c G), cl T(D) is compact (resp., w cl T(D) is weakly compact and / - T is demiclosed at 0), and T satisfies the inequality (2.8) for all x,y e D, then PM(u) n F(/, T) is nonempty.

Remark 2.11. Theorem 2.8 extends and generalizes the corresponding results of [8, 9, 13, 14, 18,19].

Let Go denote the class of closed convex subsets of a normed space X containing 0. For M e G0 and p e X, let Mp = {x e M : ||x|| < 2||p||}. Then Pm(p) c Mp e G0 (see [12, 20]).

Theorem 2.12. Let X be a normed (resp., Banach) space, and T, g are self-mappings of X. If p e X and M e G0 such that T(Mp) c M, cl T(Mp) is compact (resp., w cl T(Mp) is weakly compact), and HTx - pH < ||x - pH for all x e Mp, then PM(p) is nonempty, closed, and convex with T(PM(p)) c PM(p). If, in addition, D is a subset of PM(p), G = D n F(g) is q-starshaped, cl T(G) c G (resp., w cl T(G) c G and g - T is demiclosed at 0), and T satisfies inequality (2.8) for all x,y e D, then PM(p) n F(g, T) is nonempty.

Proof. If p e M, then the results are obvious. So assume that p / M. If x e M \ Mp, then ||x - x0|| > 2||p - x0|| and so Hp - x|| > ||x||-||p|| > Hp|| > dist(p,M). Thus a = dist(p,M) < ||p||. Since cl(T(Mp)) is compact, and by the continuity of norm, there exists z e cl(T(Mp)) such that ¡5 = dist(p,cl(T(Mp)) = ||z - p||.

On the other hand, if wcl(T(Mp)) is weakly compact, then using Lemma 5.5 of Singh et al. [22, page 192], we can show that there exists z e wcl(T(Mp)) such that ¡5 = dist(p, wcl(T(Mp))) = ||z - p||. Hence, in both cases, we have

a = dist(p,M) < dist(p,cl(T(Mp)))

= dist (p,T (Mp)) (2.13)

< \ Tx - p\

< \ x - p\ ,

for all x e Mp. Therefore, a = 5 = dist(p, M), that is, dist(p, M) = dist(p, cl(T(Mp)) = d(p, z), that is, z e 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) c PM(p), let y e T(PM(p)). Then y = Tx for x e PM(p). Consider

d(p,y) = d(p, Tx) < d(p,x) = dist(p, M), (2.14)

and so y e PM(p) as PM(p) c Mp ^ T(PM(p)) c M, that is, y e M.

The compactness of cl T(Mp) (resp., weakly compactness of w cl T(Mp)) implies that clT(D) is compact (resp., wclT(D) is weakly compact). Hence, the result follows from Corollary 2.10. □

Corollary 2.13. Let X be a normed (resp., Banach) space, and T, g are self-mappings of X. If p e X and M e Go such that T(Mp) c M, cl T(Mp) is compact (resp., w cl T(Mp) is weakly compact), and ||Tx - p|| < ||x - p|| for all x e Mp, then PM(p) is nonempty, closed, and convex with T(PM(p)) c PM(p). If, in addition, D is a subset of PM(p), G = DnF(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 e D, then PM (p) n F(g, T) is nonempty.

Remark 2.14. Theorem 2.12 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. Let C be a nonempty subset of a metric space (X,d), f, g self-maps of C, cl T (F (f) n F (g)) c F (f) n F(g). Suppose that cl(T (C)) is complete, and T, f, g satisfy for all x,y e C and 0 < k< 1,

d(Tx,Ty) < k max{ dfx, gy) ,d(Tx, fx),d(Ty, gy) ,d(Tx, gy) ,d(Ty, fx)}. (2.15)

If F(f) n F (g) is nonempty and cl T (F(f) n F (g)) c F(f) n F(g), then there is a common fixed point ofT, f and g.

Proof. cl T(F(f) nF(g)), being a closed subset of the complete set cl T(C), is complete. Further for all x,y e F(f) n F(g), we have

d(Tx,Ty) < k max{ d(fx,gy),d(Tx,fx),d(Ty,gy),d(Tx,gy),d(Ty,fx)}

(2.16)

= k max{ d(x,y),d(Tx,x),d(Ty,y),d(Tx,y),d(Ty, x)}.

Hence, T is a generalized contraction on F(f) n F(g) and clT(F(f) n F(g)) c F(f) n F(g). So by Lemma 3.1 of [13], T has a unique fixed point y in F(f) n F(g) and consequently F(T) n F(f) n F(g) is a singleton. □

Remark 2.16. If f = 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,

Theorem 2.17. Let T,g,h be self-mappings of a Banach space X. If p e F(T) n F(g) n F(h) and M e G0 such that T(Mp) c g(M) c M c h(M), clg(Mp) is compact, and ||Tx - p|| < ||gx - hp||, ||gx - uW < ||x - u||, ||hx - u|| = ||x - u|| for all x e Mp, then

(i) PM (p) is nonempty, closed, and convex,

(ii) T(Pm(p)) c g(Pm(p)) c Pm(p) = h(PM(p)),

(iii) PM(p) n F(T) n F(g) n F(h)/ 0, provided T is continuous, F(g) is q-starshaped, clg(F(h)) c F(h), and the pair (g,h) satisfies the inequality (2.8) for all x,y e PM(p), F(g) is q-starshaped with q e Pm(p) nF(g) nF(h) = G, cl T(F(g) nF(h)) c F(g) nF(h) and T, g, h satisfy for all q e F(g) n F(h) and x,y e PM(p)

d(Tx,Ty) < max{d(hx,gy), dist(hx, [q,Tx]), dist (gy, [q,Ty]), dist (hx, [q,Ty]),

dist(gy, [q,Tx])},

(2.17)

then there is a common fixed point of PM(p), T, g, and h.

Proof. Proceeding as in Theorem 2.12, we can prove (i) and (ii).

By (ii), the compactness of clg(Mp) implies that clg(PM(p)) and clT(PM(p)) are compact. Hence, Theorem 2.8 implies that F(g) n F(h) n PM(p) = 0.

For each n e N, define Tn : X ^ X by Tnx = (1 - kn)q + knTx, for each x e X, where {kn} is a sequence in (0,1) such that kn ^ 1. Then each Tn is a self-mapping of C. Since cl T(F(g) nF(h)) c F(g) nF(h), F(g) nF(h) is q-starshaped with q e G,so cl Tn(F(g) nF(h)) c F(g) n F(h) for each n. Consider

\\Tnx - Tny\\ = kn\\Tx - Ty\\

< kn max{\\hx - gy\\, dist (hx, [q,Tx]), dist (gy, [q,Ty]), dist (hx, [q,Ty]),

dist (gy, [q,Tx])}

< kn max{\\hx - gy\\ h - Tnxl \\gy - Tny\\, \\hx - Tny\\, \\gy - Tnx\\},

(2.18)

for all x,y e 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 e 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 Txm ^ z e cl T(PM(p)). Since {Txn} is a sequence in T(F(g) n F(h)), and cl T(F(g) n F(h)) c F(g) n F(h), then z e F(g) n F(h). Now, as kni ^ 1, we have

xni = Tnixni = (1 - kni)q + kniTxni —> z, (2.19)

T is continuous, we have Tz = z and hence PM (p) n F(T) n F(g) n F(h) / 0. □

Remark 2.18. (i) Let M = [0,1]. Let d be defined by d(x,y) = \x - y|. We set Tx = x/4 and /x = x/2 for all x e M. Define f : [0, to) x [0, to) ^ [0, to) by

n \ t + s f(t,s) = ■

(2.20)

Then for x,y e M, we have

d(Tx,Ty) ■ d(X y = |X - yl= 1 Ix - yl'

(2.21)

2 [d(fx,Ty) + d(fy'Tx)] - y(d(fx'Ty)'d(fy'Tx))

1 N x y1

8 I-1 2

4 1 y 1

I2 - xII - ix -!M! - xI)

1 2 1 y

1 f1 x

(2.22)

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 1

2[dfx,Ty) + d(fy,Tx)] - f(d(fx,Ty),d(fy,Tx))

■ 3112 - y 1+If - xII ■ 81(2 - 4)+(2 - 4)1 ^

= 3(x + y) = A ( ^ 1

^4 + 4) = 32(x + y - 4-

Case 2. If 2y < x, from (2.22), we have

2 [d(fx,Ty) + d(fy,Tx)] - f(d(fx'Ty)'d(fy'Tx))

■ 3fix - 4I+12 - XII - 8f(x - 4)+(-2+4)1 ("«)

3(3x 3y\ 9 , N 1 / x

3\3T +■ 32 - y -1 - yy

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 for I 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, K-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) Consider M = R2 with usual metric d((xi,yi), (x2,yf)) = |xi - x2| + |yi - y2|, (xi,yi), (xf,yf) e 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 e R} and C(I,T) = {(x,y) : y = 4 - x2,x e R}. Thus, clT(F(I)) c F(I), which is not a compatible pair (see [9]), F(I) is convex, starshaped for any z e F(I), and (-2,0) is a common fixed point of I and T.

Acknowledgment

The author is thankful to the learned referees for the valuable suggestions. References

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