0 Fixed Point Theory and Applications

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Some common fixed-point and invariant approximation results with generalized almost contractions

Savita Rathee and Anil Kumar*

Correspondence: anilkshk84@gmail.com Department of Mathematics, Maharshi Dayanand University, Rohtak, Haryana 124001, India

Abstract

In this paper, the concept of a generalized almost (f,g)-contraction is introduced and we establish some common fixed-point results for the noncommuting generalized almost (f,g)-contraction in the setup of metric spaces and normed linear spaces, where the set of fixed points of f and g need not be starshaped. As applications, invariant approximation results are proved. Supporting examples are also given.

Keywords: best approximation; Banach operator pair; generalized almost contraction; property (N); jointly continuous contractive family

1 Introduction

The classical Banach contraction principle is a very popular tool for solving problems in nonlinear analysis. It has various applications to operator theory, variational analysis, and approximation theory, so it has been extended in many ways (see, e.g., [1-30]).

In 2004, Berinde [1] defined the notion of a weak contraction mapping, which is more general than a contraction mapping. However, in [2] Berinde renamed it as an almost contraction, which is more appropriate.

Definition 1.1 Let (X, d) be a complete metric space. A map T: X ^ X is called an almost contraction if there exist a constant S e (0,1) and some L > 0 such that

d(Tx, Ty) < Sd(x,y) + Ld(y, Tx) for all x,y e X. (1.1)

Berinde [1] proved some fixed-point theorems for almost contractions in a complete metric space which generalized the results of Kannan [3], Chatterjea [4], and Zamfirescu [5].

In 2008, Babu etal. [6] defined the class ofmappings satisfying 'condition (B)' as follows.

Definition 1.2 Let (X, d) be a metric space. A map T: X ^ X is said to satisfy 'condition (B)' if there exist a constant S e (0,1) and some L > 0 such that

ft Springer

d(Tx, Ty) < Sd(x,y) + Lmin{d(x, Tx),d(y, Ty),d(x, Ty),d(y, Tx)} (1.2)

for all x, y e X.

©2014 Rathee and Kumar; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

They prove that any map T satisfying 'condition (B)' has a unique fixed point in complete metric spaces. They also discuss quasi-contraction, almost contraction, and the class of mappings that satisfy 'condition (B)' in detail.

Afterwards Berinde [7] generalized the above definition and proved the following fixed-point result.

Theorem 1.3 Let (X, d) be a complete metric space and let T: X ^ X be a mapping for which there exist S e (0,1) and some L > 0 such that for all x, y e X

d(Tx, Ty) < SMi(x,y) + Lmin{d(x, Tx),d(y, Ty),d(x, Ty),d(y, Tx)}, (1.3)

M1(x, y) = ma^ d(x, y), d(x, Tx), d(y, Ty),1 [d(x, Ty) + d(y, Tx)

Then T has a unique fixed point.

The contractive condition (1.3) is termed as generalized almost contraction.

Recently, Abbas and Ilic in [15] introduced the following definition.

Definition 1.4 Let T and f be two self-maps of a metric space (X, d). A map T is called a generalized almost f -contraction if there exist S e (0,1) and some L > 0 such that

d(Tx, Ty) < SM1(x,y) + Lminjdfx, Tx),d(fy, Ty),dfx, Ty),d(fy, Tx)), (1.4)

M1(x, y) = max j dfx,fy), dfx, Tx), dfy, Ty), - [dfx, Ty) + dfy, Tx)

Iff = identity map, then condition (1.3) can be obtained as particular case of condition (1.4). However, in [15] Abbas and Ilic obtained various common fixed-point and invariant approximation results for such mappings under the assumption of weak compatibility of maps.

Recently, Chen and Li [10] introduced the class of Banach operator pairs, as a new class of noncommuting mappings and obtained some common fixed-point and invariant approximation results for this class of maps. This class of noncommuting maps is different from the class of noncommuting maps (viz. ^-subcommuting, ^-sub-weakly commuting, Cq-commuting, compatible, weakly compatible etc.) studied in [11-13,15,17-19, 27-29]. So, it has been further studied by various authors (see, e.g., [16, 21, 22, 24]).

In this article, we introduce the class of generalized almost f,g)-contraction and consequently establish some common fixed-point results for the noncommuting generalized almost f,g)-contraction in the framework of metric spaces and normed linear spaces, where the set of fixed points of f and g need not be starshaped. As an application, invariant approximation results are proved. The proved results generalize and extend the corresponding results of Chen and Li [10], Al-Thagafi and Shahzad [16], Akbar etal. [22], Chandok and Narang [24], Al-Thagafi [25] and Jungck and Sessa [26], Shahzad [28] to the class of generalized almost f,g)-contractions.

2 Preliminaries

First, we introduce some well-known notations and definitions that will be needed in the sequel.

Let (X, d) be a metric space, M be a subset of X and f, T be self-maps of M. A point x e M is a coincidence point (common fixed point) off and T iffx = Tx (fx = Tx = x). The set of coincidence points off and T is denoted by Cf, T) and the set of fixed points off is denoted by Ff). The pair f, T} is called

(1) commuting if Tfx = fTx for all x e M,

(2) compatible [8] if limn—TO d(Tfxn,fTxn) = 0 whenever {xn} is a sequence in M such that limn—mfxn = limn—TO Txn = t for some t e M,

(3) weakly compatible [9] if Tfx = fTx for all x e Cf, T),

(4) a Banach operator pair [10] ifthe set F(f) is T-invariant, namely T(F(f)) c Ff). Obviously, a commuting pair (T,f) is a Banach operator pair but not conversely. If (T,f) is a Banach operator pair, then (f, T) need not be Banach operator pair (see [10]).

Let M be a subset of a normed space (X, || • ||). The set BM(p) = {x e M : ||x -p\\ = dist(p,M)} is called the set of best approximants to p e X out of M, where dist(p,M) = inf{||y -p|| :p e M}. We denote by N and cl(M) (wcl(M)) the set of positive integers and the closure (weak closure) of a set M in X, respectively.

The set M is said to be (a) q-starshaped if there exists q e M such that the line segment [q,x] = {(1 - k)q + kx: 0 < k < 1} joining q to x is contained in M for all x e M; (b) convex if kx + (1 - k)y e M for all x, y e M. The mapf defined on a set M is called

(1) affine [11] if M is convex and f ((1 - k)y + kx) = (1 - k)fy + kfx, for all x, y e M,

(2) q-affine [11] if M is q-starshaped and f ((1 - k)q + kx) = (1 - k)q + kfx, for all x e M. Suppose that M is q-starshaped with q e F(f) and is both T- andf-invariant. Then T

and f are called

(1) Cq-commuting [11] iffTx = Tfx for all x e Cq(f, T), where Cq(f, T) = U{C(f, Tk): 0 < k < 1} where Tk(x) = (1 - k)q + kTx,

(2) R-subcommuting on M [12] if, for all x e M, there exists a real number R >0 such that || Tfx -fTx|| < R WkTx + (1 - k)q -/x||, 0 < k < 1,

(3) R-sub-weakly commuting on M [13] if, for all x e M, there exists a real number R >0 such that || Tfx -fTx! < R dist(fx,[q, Tx]).

A Banach space X is said to satisfy Opial's condition if, whenever {xn} is a sequence in X such that {xn} converges weakly to x e X, the inequality

liminf ||xn -x|| < liminf ||xn -y||

n—n—

holds for all y = x. A Hilbert space and the space lp (1 < p < to) satisfy Opial's condition. The map T: M — X is said to be demiclosed at zero if, whenever {xn} is a sequence in M such that {xn} converges weakly to x e M and {Txn} converges to 0, then Tx = 0.

The following important extension of the concept of starshapedness was defined by Naimpally et al. [14] and has been studied by many authors.

Definition 2.1 A subset M of a linear space X is said to have property (N) with respect to T if

(1) T: M — M,

(2) (1 - kn)q + knTx e M, for some q e M and a fixed sequence of real numbers kn (0 < kn < 1) converging to 1 and for each x e M.

It is to be noted that each T-invariant q-starshaped set has property (N) but converse does not hold in general. This is shown by the following example.

Example 2.2 Let X = R be the set of real numbers and M = {1/n, where n is a natural number} be endowed with the usual norm. Define Tx = 1 for each x e M. Then clearly M is not q-starshaped but has property (N) with respect to T, for q = 1, kn = 1 - 1/n.

3 Main results

First we introduce the notion of a generalized almost (f,g)-contraction.

Definition 3.1 Let (X, d) be a metric space andf, g be self-maps of X. A mapping T: X ^ X is said to be a generalized almost (f, g)-contraction if there exist S e (0,1) and some L > 0 such that

d(Tx, Ty) < SMi(x,y) + LNi(x,y) for all x,y e X, (3.1)

M1(x, y) = maxj dfx,gy), dfx, Tx), d(gy, Ty), 1 [dfx, Ty) + d(gy, Tx)

N1 (x,y) = min{dfx, Tx),d(gy, Ty),dfx, Ty),d(gy, Tx)}.

If g = f, then Definition 1.4 is a particular case of Definition 3.1. If g = f = I (identity operator), then equation (1.3) can be obtained as a special case of equation (3.1).

Here we observe that if T satisfies 'condition (B)' then T is a generalized almost contraction but its converse need not be true. This is shown by the following example.

Example 3.2 Let X = [0, to) be endowed with the Euclidean metric d(x,y) = |x -y|. We define a mapping T: X ^ X by

T (x) =

if 0 < x < 1,

if 0 < x < to.

Then T is a generalized almost contraction with S = | and L = 0. But T does not satisfy 'condition (B)' at x = |, y = 1 for any S e (0,1) and L > 0.

In (3.1) ifL = 0, then T is called a generalized (f,g)-contraction. Obviously, a generalized (f,g)-contraction implies a generalized almost (f,g)-contraction, but the converse is not true in general.

Example 3.3 Let X = {0,1,2} with the usual metric and f,g : X ^ X be given by f (x) = g(x) = 1 for all x e X. Also define a mapping T: X ^ X as

T (x)={°, x e{0,2},

Then T is a generalized almost (f,g)-contraction with any S e (0,1) and L > 2. But T is not a generalized (f,g)-contraction at x = 0, y = 1 or x = 1, y = 2 for any S e (0,1).

The following lemma is a particular case of the main theorem of Abbas and Ilic [15].

Lemma 3.4 Let M be a nonempty subset of a metric space (X, d), and T be a self-map of M. Assume that cl(T(M)) c M, cl(T(M)) is complete, and T is a generalized almost contraction. Then M n F(T) is singleton.

Now, we start with the following common fixed-point result, which will be used in sequel.

Theorem 3.5 Let M be a nonempty subset of a metric space (X, d), and T, f andg be self-maps ofM. Assume thatF(f) n F(g) is nonempty, cl(T(F(f) n F(g))) c F(f) n F(g), cl(T(M)) is complete, and T is a generalized almost (f,g)-contraction. Then M n F(T) n F(f) n F(g) is singleton.

Proof The completeness of cl(T(M)) implies that of cl(T(Ff) n F(g))). Further, by a generalized almost (f,g)-contraction of T, for all x,y e Ff) n F(g), we have

Hence T is a generalized almost contraction mapping on F (f) n F (g) and cl(T (F (f) n F (g))) Ç F f) n F (g). By Lemma 3.4, T has a unique fixed point z in F f) n F (g) and con-

Corollary 3.6 Let M be a nonempty subset of a metric space (X, d), and T, f and g be self-maps of M such that (T,f) and (T,g) are Banach operator pairs on M. Assume that cl(T(M)) is complete, T is a generalized almost (f,g)-contraction and F(f) n F(g) is nonempty and closed. Then M n F(T) n F(f) n F(g) is singleton.

In Theorem 3.5 if we take L = 0, then we easily obtain the following result, which improves and extends Lemma 3.1 of Chen and Li [10] and Theorem 2.2 of Al-Thagafi and Shahzad [16].

Corollary 3.7 Let M be a nonempty subset of a metric space (X, d), and T, f, and g be self-maps on M. Assume that F(f ) n F(g) is nonempty, cl(T(F(f ) n F(g))) c F(f) n F(g), cl(T (M)) is complete, and T is a generalized (f, g)-contraction. ThenM n F(T) n F (f) n F(g) is singleton.

Remark 3.8 By comparing Theorem 2.1 of Shahzad [17] with Corollary 3.7 (when g = f), their assumptions that M is closed, T(M) c f (M), T is continuous and (T,f) is ^-weakly commuting pair on M are replaced with 'F(f) is nonempty, cl(T(F(f))) c F(f).

d(Tx, Ty) < SMi(x,y)+LN1(x,y)

+ Lmin{d(x, Tx),d(y, Ty),d(x, Ty),d(y, Tx)}.

sequently M n F (T) n F f) n F (g) is singleton.

Theorem 3.9 Let M be a nonempty subset of a normed (respectively, Banach) space X and T, f, and g be self-maps of M. IfF(f) n F(g) has the property (N) with respect to T, cl(T (F(f) n F (g))) c F (f) n F (g) (respectively, wcl(T (F (f) n F(g))) c F (f) n F (g)), and there exists a constant L > 0 such that

||Tx - Ty|| < m(x, y)+Ln(x, y) forallx, y e M, (3.2)

m(x, y) = max j |fx - gy||, dist(fx,[q, Tx]), dist(gy,[q, Ty]), 2 [dist(gy,[q, Tx]) + dist(fx,[q, Ty])] |

n(x,y) = min{dist(f;,[q, Tx]),dist(gy,[q, Ty]),dist(gy,[q, Tx]),distfx,[q, Ty])}

thenM n F(T) n F(f) n F(g) = $,provided cl(T(M)) is compact (respectively, wcl(T(M)) is weakly compact) and T is continuous (respectively, I - T is demiclosed at 0, where I stands for identity map).

Proof As T(F(f) n F(g)) c F(f) n F(g) and F(f) n F(g) has the property (N) with respect to T, for each n e N, we can define Tn : F(f) n F(g) — F(f) n F(g) by Tnx = (1 - kn)q + kn Tx for all x e F f) n F (g) and a fixed sequence of real numbers kn (0 < kn < 1) converging to 1. Since Ff) n F(g) has the property (N) with respect to T, and cl(T(F(f) n F(g))) c F(f) n F(g) (respectively, wcl(T(F(f) n F(g))) c F(f) n F(g)), we have cl(Tn(F(f) n F(g))) c F(f) n F(g) (respectively, wcl(Tn(F(f) n F(g))) c F(f) n F(g)) for each n e N. Also, by the inequality (3.2),

||Tnx - Tny| = knWTx - Ty||

< kn [m (x, y)+ Ln (x, y) ] = knm(x, y) +Lnn(x, y),

m(x,y) = ma^J ||/x -gy||, dist(fx,[q, Tx]), dist(gy,[q, Ty]), 2[dist(gy,[q, Tx]) + dist(fx,[q, Ty])]|

< ma^J |fx - gy ||, ||fx - Tnx||, |gy - Tny 2 [f - Tny| + |gy - Tnx|]J

n(x,y) = min{distfx,[q, Tx^, dis^gy,[q, Ty^,dist(gy,[q, Tx]), distfx,[q, Ty])}

< mi^ ||/x - Tnx||gy - Tny|, ||/x - Tny|, |gy - Tnx|}

for all x,y e Ff) n F(g), Ln := knL, and 0 < kn < 1. Thus, for each n e N, Tn is a generalized (f, g)-almost contraction.

If cl(T(M)) is compact, then, for each n e N, cl(Tn(M)) is compact and hence complete. By Theorem 3.5, for each n > 1, there is a unique xn in M such that xn = f (xn) = g(xn) = Tn(xn). The compactness of cl(T(M)) implies that there exists a subsequence {Txm} of {Txn} such that Txm — z e cl(T(M)). Since {Txm} is a sequence in T(Ff) n F(g)) and cl(T(F(f) n F(g))) c F(f ) n F(g), we have z e Ff) n F(g). Moreover,

xm = Tm (xm ) = (1 - km)q + kmTxm — z.

As T is continuous on M, we have Tz = z. Thus M n F(T) n F(f) n F(g) = $.

Next, the weak compactness of wcl(T(M)) implies that wcl(Tn(M)) is weakly compact and hence complete due to completeness of X. From Theorem 3.5, for each n > 1, there is a unique xn in M such that xn = f (xn) = g(xn) = Tn(xn). The weak compactness of wcl(T(M)) implies that there is a subsequence {Txm} of {Txn} such that Txm converges weakly to z e wcl(T(M)). Since {Txm} is a sequence in T(Ff) n F(g)) and wcl(T(F(f) n F(g))) c Ff) n F(g), therefore z e F(f) n F(g). Also we have (I - T)xm — 0 as m — to. Further, demiclosedness of I - T at 0 implies z = Tz, thus M n F (T) n Ff) n F (g) = $. □

Corollary 3.10 Let M be a nonempty subset of a normed (respectively, Banach) space X and T, f, and g be self-maps of M. IfFf) n F(g) has the property (N) with respect to T and is closed (respectively, weakly closed), (T,f) and (T, g) are Banach operator pairs and satisfy (3.2) for all x,y e M. Then M n F(T) n F(f) n F(g) = $, provided cl(T(M)) is compact (respectively, wcl(T(M)) is weakly compact) and T is continuous (respectively, I - T is demiclosed at 0, where I stands for the identity map).

Remark 3.11 (1) By comparing Theorem 2.2 of Shahzad [17] with the first case of Theorem 3.9 (when g = f, L = 0), their assumptions 'q e F(f), M is closed and q-starshaped, f is linear and continuous on M, T(M) c f (M) and (T,f) is R-sub-weakly commuting pair on M' are replaced with 'M is a nonempty subset, Ff) has the property (N) with respect to T, cl(T(Ff))) c Ff).

(2) By comparing Theorem 2.2(i) of Hussain and Jungck [18] with the first case of Theorem 3.9 (when L = 0), their assumptions 'M is complete and q-starshaped, f and g are continuous and affine on M, T(M) c f (M) n g(M), q e F(f) n F(g), and (T,f) and (T,g) are R-sub-weakly commuting pair on M' are replaced with 'F(f) n F(g) has the property (N) with respect to T, cl(T(Ff) n F(g))) c F(f) n F(g)'.

(3) By comparing Theorem 2.2(ii) of Hussain and Jungck [18] with the second case of Theorem 3.9 (when L = 0), their assumptions M is weakly compact and q-starshaped, f andg are affine and continuous onM, T(M) c f (M) ng(M), q e F(f ) nF(g), and (T,f) and (T,g) are R-sub-weakly commuting pair on M, andf - T is demiclosed at 0' are replaced with 'wcl(T(M)) is weakly compact, Ff) n F(g) has the property (N) with respect to T, wcl(T (F (f) n F(g))) c F (f) n F (g) and I - T is demiclosed at 0.

Remark 3.12 If the contractive condition (3.2) in Theorem 3.9 is replaced with the stronger contractive condition

||Tx - Ty|| < m(x,y) +Ln(x,y)

for all x, y e M and some L > 0, where

m(x,y) = maij \fx -gy||, 2 [dist(fx,[q, Tx]) + dist(gy,[q, Ty])], 2[dist(gy,[q, Tx]) + dist(fx,[q, Ty])]|

n(x, y) = min{dist(f;,[q, Tx]), dist(gy,[q, Ty]), dist(gy,[q, Tx]), distfx,[q, Ty])}, then continuity of T can be relaxed in the first case of Theorem 3.9.

Proof The proof will be similar to the first case of Theorem 3.9. To prove Tz = z, instead of continuity of T, using (3.3) we have

\\Txm - Tz\\ < m(xm,z) + Ln(xm,z), (3.4)

m(xm,z) = maxj\fxm -gz\\, 2[distfrn,[q, Txm]) + dist(gz,[q, Tz])], 2 [dist(gz,[q, Txm]) + dist(fxm,[q, Tz])] J

< ma^J \\xm - z\\, 2 [\xm - Txm\\ + \\z - Tz\\],

< ma^j \\xm -z\\, — [\\xm - Txm\\ + \\z- Txm\\ + \\Txm - Tz\\],

2 [\\z - Txm\\ + \\xm - Tz\\

2 [ \ \z Txm \\ + \\xm Txm \\ + \\ Txm Tz\\

n(xm, z) = minjdistfxmjq, Txm]), dis^gz,[q, Tz]), dis^gz,[q, Txm]), distfxm,[q, Tz])} < mi^ \\xm - Txm\\, \\z - Tz\\, \\z - Txm\\, \ixra - Tz\\).

Now taking m ^to in (3.4) we can write 1

lim \\Txm - Tz\\ <- lim \\Txm - Tz\\.

m—2 m—

This is possible only if Txm — Tz as m — to, which implies Tz = z. □

Let C = Bm(p) n CfMg(p), where CfMg(p) = {x e M: fx e Bm(p),gx e Bm(p)}.

Corollary 3.13 LetX be a normed (respectively, Banach) space and let T, f, andg be self-maps ofX. If p e X and D c C, D0 := D n Ff) n F(g) has the property (N) with respect to T, cl(T(D0)) c D0 (respectively, wcl(T(D0)) c D0), cl(T(D)) is compact (respectively, wcl(T(D)) is weakly compact), T is continuous on D (respectively, I - T is demiclosed at 0, where I stands for identity map) and (3.2) holds for allx, y e D, then BM (p) n F (T) n F f) n F (g) = <p.

Corollary 3.14 LetX be a normed (respectively, Banach) space and let T, f, andg be self-maps ofX. Ifp e X and D c BM(p), D0 := D n F f) n F (g) has the property (N) with respect to T, cl(T(D0)) c D0 (respectively, wcl(T(D0)) c D0), cl(T(D)) is compact (respectively, wcl(T(D)) is weakly compact), T is continuous on D (respectively, I - T is demiclosed at 0, where I stands for the identity map) and (3.2) holds for all x, y e D, then BM (p) n F (T) n F (f) n F (g) = 4>.

Remark 3.15 Corollaries 3.13 and 3.14 improve and develop Theorems 2.8-2.11 of Hussain and Jungck [18] and Theorems 3.1-3.4 of Song [19] to the non-starshaped domain.

Denote by the class of closed convex subsets of X containing 0. For M e L0, we define Mp = {x e M: \\x\\ < 2\\p\\}. Clearly Bm(p) c Mp e A.

The following invariant approximation result constitutes an extension of Theorem 2.6 of Al-Thagafi and Shahzad [16] and Corollary 2.10 of [29] to a non-starshaped domain.

Theorem 3.16 LetX be a normed (respectively, Banach) space and T,f,g: X — X. Ifp e X and M e such that T(Mp) c M, cl(T(Mp)) is compact (respectively, wcl(T(Mp)) is weakly compact), and \\Tx - p\\ < \\x - p\\for allx e Mp, thenBM(p) is nonempty, closed, and convex with T(BM(p)) c BM(p). If, in addition, Disa subset ofBM(p), D0 := D n Ff) n F(g) has the property (N) with respect to T, cl(T(D0)) c D0 (respectively, wcl(T(D0)) c D0), T is continuous on D (respectively, I - T is demiclosed at 0, where I stands for the identity map) and (3.2) holds for allx, y e D, then Bm (p) n F (T) n F(f) n F(g) =

Proof We may assume thatp e M.Ify e M\Mp, then \\y\\ > 2\\p\\ and, so

\\y-p\\>\\y\\ - \\p\> \\p\ > dist(p,M).

Thus dist(p, Mp) = dist(p, M). Assume that cl(T(Mp)) is compact, then by the continuity of the normthereexists z e cl(T(Mp)) such that \\z -p\\ = dist(p,clT(Mp)).

If we assume that wcl(T (Mp)) is weakly compact, then by using Lemma 5.5 of [20, p.192] we can show the existence of z e wcl(T(Mp)) such that \\z -p\\ = dist(p, wcl T(Mp)). Thus in both cases, we have

dist(p,Mp) < dist(p,cl T(Mp)) < dist(p, T(Mp)) < \\Tx -p\\ < \\x-p\\

for all x e Mp. It follows that \\z-p\\ = dist(p, M). Thus BM(p) is nonempty, closed, and convex with T(BM(p)) c BM(p). The compactness of cl(T(Mp)) (respectively, weak compactness of wcl(T(Mp))) implies that cl(T(D)) is compact (respectively, wcl(T(D)) is weakly compact). Then by Corollary 3.14, Bm(p) n F(T) n F(f) n F(g) = 4>. □

Now, we present some non-trivial examples in support of Theorem 3.9.

Example 3.17 Let X = R be the set of real numbers with the usual norm and M = [0,1). We define mappings f, g, T: M — M by

,, , [0 if 0 < x <|, if if 0 < x <|,

f (x) = 4 3 g(x)= 2 .f 2 ^ 3

if -x if2<x<1, 13 if2<x<1,

and T(x) = 2 ,for 0 < x <1.

Here we observe that F(f) n F(g) = {0, §}, cl(T(F(f) n F(g))) = {2} c F(f) n F(g) and cl(T(M)) = {3} is compact. Clearly Ff) n F(g) is not starshaped but has property (N) with respect to T, for q = f and kn = 1 - 1/n. Further, the mappings T,f, andg satisfy the contractive condition (3.2) and also T is continuous. Hence all the conditions of the first case of Theorem 3.9 are satisfied and consequently T, f, and g have a common fixed point,

x = 3.

Remark 3.18 In Example 3.17, it is interesting to note that Theorem 2.19 of Hussain and Cho [21], and Corollary 3.10 of Akbar et al. [22] cannot apply, since Ff) n F(g) is not q-starshaped.

Example 3.19 Let X = R be the set of real numbers with the usual norm and M = [0,1]. Define f,g, T: M — M by

x, x is rational in M, f (x)= < g(x) = x for all x e M

0, otherwise,

T (x) =

1 if 0 < x <1, 0 if x = 1.

Clearly F f) n F(g) = {x,x is rational in M} has property (N) with respect to T, for q = 0, kn = 1 - 1/n. Further, cl(T(Ff) n F(g))) = {0,1} c F(f) n F(g), cl(T(M)) = {0,1} is compact and T, f, and g satisfy the contractive condition (3.2). Hence all the conditions of the first case of Theorem 3.9 are satisfied except the continuity of T. Note that F (T) n F f) n F (g) = $.

Remark 3.20 It is to be noted that the maps T, f, and g given in Example 3.19 do not satisfy the contractive condition (3.3) at the point x = 2, y = 1.

4 Results with joint contractive family

Dotson [23] proved some results concerning the existence of fixed points of nonexpansive mappings on a certain class of non-convex sets. For proving these results, he extends the concept of starshapedness by introducing the following class of non-convex set.

Let M be a subset of a normed space X and T = {hx : x e M} be a family of functions from [0,1] to M such that hx(1) = x for each x e M. The family T is said to be contractive if there exists a function y : (0,1) — (0,1) such that for all x, y e M and all t e (0,1), we have

||hx(t)-hy(t)|| < y(t)||x -y||.

Such a family T is said to be jointly continuous (jointly weakly continuous) if t — t0 in [0,1] and x — xo (x — xo weakly) in M; then hx(t) — hx0 (t) (hx(t) — hx0 (t) weakly) in M.

We observe that if M is q-starshaped subset of a normed linear space X and hx(t) = (1 - t)q + tx, for each x e M, q e M and t e [0,1], then T is a contractive jointly continuous and jointly weakly continuous family with y(t) = t. Thus the class of subsets of X with the property of contractiveness and joint continuity contains the class of starshaped sets which in turns contains the class of convex sets. We shall denote YJqx = {hB(k): 0 < k < 1} where q = hjx(0).

The following results properly contain Theorems 3.2 and 3.3 of [10], Theorems 1 and 2 of [24] and improves Theorem 2.2 of [25], Theorem 6 of [26].

Theorem 4.1 LetM be a nonempty subset of a normed (respectively, Banach) space X and T, f andg be self-maps ofM. Suppose F f) n F(g) is nonempty and has a contractive, jointly continuous (respectively, jointly weakly continuous) family of functions T = {hx: x e F f) n F(g)}, cl(T(Ff) nF(g))) c F(f) nF(g) (respectively, wcl(T(Ff) nF(g))) c F(f) nF(g)), and there exists a constant L > 0 such that

\\Tx - Ty|| < m(x,y) +Ln(x,y) (.1)

for all x, y e M, where

m(x, y) = maxj |fx - gy||, distfx, Yf), dist(gy, Yf), 2 [dist(gy, Yf) + distfx, Yf)] J

n(x, y) = minjdistfx, YT), dist(gy, Yf), dist(gy, Yf), distfx, YT)}.

Then M n F(T) n F(f) n F(g) = $, provided cl(T(M)) is compact (respectively, wcl(T(M)) is weakly compact) and T is continuous (respectively, T is weakly continuous).

Proof For each natural number n, let kn = n+j. Define Tn : F(f) n F(g) — F f) n F(g) by Tn(x) = hTx(kn) for all x e F f) n F(g). Since F f) n F(g) has a contractive family and cl(T(F(f) n F(g))) c F(f) n F(g) (respectively, wcl(T(F(f) nF(g))) c F(f) n F(g)), so for each n e N, cl(Tn(F(f) n F(g))) c F(f) n F(g) (respectively, wcl(Tn(F(f) n F(g))) c F(f) n F(g)). We have

IITnx - Tny| = \hjx(kn)-hxy(kn) |

< ^(kn) || Tx - Ty || since T is a contractive family

< ^(kn^m(x,y) +Ln(x,y)} using (4.1) = ^ (kn )m(x, y)+Lnn(x, y)

for each x,y e Ff) n F(g), where Ln = L^(kn), ^(kn) e (0,1),

m(x,y) = ma^j\fx -gy||, dist(fx, Y^*),dist(gy, Yf), 2 [dist(gy, YT^) + dist(fx, Y?Ty)] J < max! |fx -gy||, \\/x - Tnx||, |gy - TnyУ,f [|gy - Tnx| + ||fx - Tny||]|

n(x, y) = minjdistfx, If), dist(gy, Yf), dist(gy, Yf), distf, Yf)} < min{ \\fx - Tnx\\gy- Tny\\, \\gy- Tnx\\fx - Tny\\).

Thus, for each n e N, Tn is a generalized almost f,g)-contraction.

If cl(T(M)) is compact, then, for each n e N, cl(Tn(M)) is compact and hence complete. By Theorem 3.5, for each n > 1, there exists a unique xn e F f) n F (g) such that xn = f (xn) = g(xn) = Tn(xn). Again the compactness of cl(T(M)) implies that there exists a subsequence {Txm} of {Txn} such that Txm — z e cl(T(M)). Since {Txm} is a sequence in T(F(f ) n F(g)) and cl(T (F(f) n F (g))) c F(f) n F (g), we have z e F(f) n F(g). Further, the joint continuity of family Y implies that

xm = Tmxm = hjxm (km) — hz (1) = z as m — to.

By the continuity of T, we obtain z = T(z). Thus, M n F(T) n F(f) n F(g) = $.

The weak compactness of wcl(T(M)) implies that wcl(Tn(M)) is weakly compact and hence complete due to completeness of X. From Theorem 3.5 for each n > 1, there exists a unique xn e F(f) n F(g) such that xn = f (xn) = g(xn) = Tn(xn). The weak compactness of wcl(T(M)) implies that there is a subsequence {Txm} of {Txn} such that Txm converges weakly to z e wcl(T(M)) as m — to. Since {Txm} is a sequence in T(F(f) n F(g)) and wcl(T(Ff) n F(g))) c F(f) n F(g), we have z e F(f) n F(g). By the joint weak continuity of the family we obtain

xm = Tm xm = hjxm (km) — hz(1) = z (weakly) as m — to.

Since the weak topology is Hausdorff, by weak continuity of T, we have z = T(z). Thus,

Remark 4.2 By comparing Theorem 2.2(i) of Chandok and Narang [27] with the first case of Theorem 4.1 (when L = 0), their assumptions 'M is complete and has a contractive jointly continuous family Y with g(hx(k)) = hgx(k) and f (hx(k)) = hfx(k) for k e (0,1), cl(T (M)) c f (M) n g(M), the pairs (T,f) and (T, g) are Cq-commuting andf, g are continuous on M are replaced with 'M is nonempty subset, Ff) n F(g) is nonempty and has a contractive jointly continuous family Y, and cl(T(F(f ) n F(g))) c F(f ) n F(g)'.

Corollary 4.3 LetM be a nonempty subset of a normed (respectively, Banach) space X and T, f, and g be self-maps of M. Suppose Ff) n F(g) is q-starshaped, cl(T(Ff) n F(g))) c F(f) n F(g) (respectively, wcl(T(F(f) n F(g))) c F(f) n F(g)), and there exists a constant L > 0 such that

\\Tx - Ty\ < m(x,y) +Ln(x,y) (.2)

for all x, y e M, where

m(x,y) = maxj \\fx -gy\,distfx,[q, Tx}), dis^gy,[q, Ty}),

M n F(T) n F(f) n F(g)= $.

n(x,y) = min{distf%,[q, Tx]),dist(gy,[q, Ty]),dist(gy,[q, Tx]),dist{fx,[q, Ty])\.

Then M n F(T) n Ff) n F(g) = $, provided cl(T(M)) is compact (respectively, wcl(T(M)) is weakly compact) and T is continuous (respectively, T is weakly continuous).

Remark 4.4 (1) By comparing Theorem 2.3(i) of Abbas and Ilic [15] with the first case of Corollary 4.3 (wheng = f), their assumptions 'M is q-starshaped, cl(T(M)) c f (M), f and T are weakly compatible on M' are replaced with' Ff) is q-starshaped, cl(T(F(f))) c F(f).

(2) By comparing Theorem 2.3(ii) of Abbas and Ilic [15] with the second case of Corollary 4.3 (wheng = f), their assumptions 'M is q-starshaped, cl(T(M)) c f (M), f and T are weakly compatible on M, f is weakly continuous andf - T is demiclosed at 0' are replaced with 'F(f) is q-starshaped, cl(T(F(f))) c F(f) and T is weakly continuous.

(3) By comparing Theorem 2.4 of Song [19] with the first case of Corollary 4.3 (when L = 0), their assumptions 'M is q-starshaped, cl(T(M)) cf (M) ng(M), the pairs (T,f) and (T,g) are Cq-commuting, f and g are q-affine and continuous on M' are replaced with 'F(f) n F(g) is q-starshaped, cl(T(F(f) n F(g))) c F(f) n F(g).

Corollary 4.5 Let M be a nonempty subset of a normed (respectively, Banach) space X and T, f, andg be self-maps ofM. IfM has a contractive jointly continuous (respectively, jointly weakly continuous) family Y = {hx: x e M} such thatg(hx(k)) = hgx(k) andf (hx(k)) = hfx(k) for all x e M, k e [0,1]. Suppose F(f) n F(g) is nonempty, closed (respectively, weakly closed), cl(T(M)) is compact (respectively, wcl(T(M)) is weakly compact), T is continuous (respectively, weakly continuous), (T ,f) and (T, g) are Banach operator pair on M and satisfy (4.1). Then M n F(T) n F(f) n F(g) = $.

Proof For each natural number n, define Tn : M — M by Tn(x) = hTx(kn), for all x e M. Clearly, for each n > 1, Tn is a self-map on M. Since (T,f) is Banach operator pair on M, for each x e Ff), we have Tx e F(f). Consider

f (Tnx) =f(hjx (kn)) = hjrx(kn) = hjx (kn) = Tnx.

This implies that Tnx e F(f) for each x e F(f). Thus for each n e N, (Tn,f) is a Banach operator pair on M. Similarly, for each n e N, (Tn,g) is a Banach operator on M. Now the result follows from Theorem 4.1. □

Corollary 4.6 Let X be a normed (respectively, Banach) space and let T, f, and g be self-maps ofX. If p e X and D c C, D0 := D n F(f) n F(g) is nonempty, has a contractive jointly continuous (respectively, jointly weakly continuous) family of functions Y = {hx: x e D0}, cl(T(D0)) c D0 (respectively, wcl(T(D0)) c D0), cl(T(D)) is compact (respectively, wcl(T(D)) is weakly compact), T is continuous on D (respectively, T is weakly continuous) and (4.1) holds for all x, y e D, then Bm (p) n F (T) n F(f) n F (g) = $.

Corollary 4.7 LetX be a normed (respectively, Banach) space and T, f, andg be self-maps ofX. Ifp e X and D c BM (p), D0 := D n Ff) n F (g) is nonempty, has a contractive jointly continuous (respectively, jointly weakly continuous) family of Y = {hx: x e D0}, cl(T(D0)) c

D0 (respectively, wcl(T(D0)) c D0), cl(T(D)) is compact (respectively, wcl(T(D)) is weakly compact), T is continuous on D (respectively, T is weakly continuous) and (4.1) holds for allx,y e D, then Bm(p) n F(T) n Ff) n F(g) =

Remark 4.8 (1) Theorems 4.1 and 4.2 of Chen and Li [10], Theorems 3 and 4 of Chandok and Narang [24] are particular cases of Corollaries 4.6 and 4.7.

(2) By Proposition 2.2 of Chen and Li [10], it can be concluded that Corollary 4.5 extends and generalizes Corollary 2.1 of Shahzad [28].

Now we present two examples in support of Theorem 4.1 and Theorem 3.5, respectively.

Example 4.9 Let X = R be the set of real numbers with the usual norm and M = [0,1]. Assume T(x) = 2, for every x in M and define f,g: M — M by

. x, x is rational, f (x)= < g(x) = x for all x e M.

I 1 - x, x is irrational,

Then F(f) n F(g) = {x,x is rational in M}, cl(T(F(f) n F(g))) = {2} c Ff) n F(g) and cl(T(M)) = {2} is compact. Suppose that T = {hx: x e F(f) n F(g)} is a family of functions from [0,1] into F(f ) n F(g), defined by

hx(t)= , „

1, x e F(f) n F(g), t e M\F(f) n F(g), t2x, x, t e F(f) n F(g).

We observe that the family T is contractive jointly continuous for y (t) = t2, t e (0,1). Thus all the conditions of Theorem 4.1 are satisfied. Consequently T,f, andg have a common fixed point. Here it is seen that x = 2 is the common fixed point of T,f, andg.

Remark 4.10 (1) Theorem 2.2(i) of Chandok and Narang [27] cannot apply to Example 4.9, since f is not continuous.

(2) It is interesting to note that the results of Akbar et al. [22] cannot apply to Example 4.9, since Ff) n F(g) is not q-starshaped.

Example 4.11 Let X = M = {a, f, y, 5} and let d: X x X — R be given as

d(a, f) = d(f, a) = 0.5, d(a, y) = d(y, a) = 2.5, d(a, 5) = d(5, a) = 1.6, d(f, y ) = d(y, f ) = 2.5, d(f, 5)=d(5, f) = 1.5, d(Y, 5)=d(5, y ) = 2 and d(a, a) = d(f, f ) = d(y, y )= d(5,5) = (0,0).

Then (X, d) is a metric space. Let T,f,g: M — M is defined, respectively, as follows:

T (x) = If, * = ',

5, x = y

fa = f, ff = f, fy = a, f5 = f, ga = 5, gf = f, gy = y, g 5 = a.

Clearly Ff) n F(g) = {f} and cl(T(F(f) n F(g))) = {f} c F(f) n F(g). Further T is a generalized almost (f,g)-contraction for 5 = 20 and L = 0. Hence, all the conditions of Theorem 3.5 are satisfied. Consequently T,f, and g have a unique common fixed point. Here it is seen that x = f is the unique common fixed point of T, f, and g.

Remark 4.12 (1) In Example 4.11, f (M) = {a, f}, g(M) = {a, f, y, 5} and T(M) = {f, 5}, therefore cl T(M) is not contained inf (M) ng(M). Hence Theorem 2.1 of Song [19] cannot apply to Example 4.11.

(2) In Example 4.11, if we take g(x) =f (x) = {^ xx=y, then T and f does not satisfy the contractive condition of Lemma 3.1 of [10] and Theorem 2.2 of [16] atx = y, y = a. Hence Lemma 3.1 of [10] and Theorem 2.2 of [16] cannot apply to Example 4.11.

Remark 4.13 (1) Example 3.3 satisfies all the conditions of Theorem 3.5 except the condition cl(T(F(f) n F(g))) c F(f) n F(g). Note that F(T) n F(f) n F(g) =

Competing interests

The authors declare that they have no competing interests. Authors' contributions

Both authors contributed equally and significantly in writing this paper. Both authors read and approved the final manuscript.

Acknowledgements

The author would like to thank the referees for their valuable suggestions, which helped to improve the presentation of the paper.

Received: 7 July 2013 Accepted: 5 January 2014 Published: 24 Jan 2014 References

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10.1186/1687-1812-2014-23

Cite this article as: Rathee and Kumar: Some common fixed-point and invariant approximation results with generalized almost contractions. Fixed Point Theory and Applications 2014, 2014:23

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