# Common Fixed Points of Generalized Meir-Keeler Type Condition and Nonexpansive Mappings

Authors: R. K. Bisht
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

---

## Abstract

The aim of the present paper is to obtain common fixed point theorems by employing the recently introduced notion of weak reciprocal continuity. The new notion is a proper generalization of reciprocal continuity and is applicable to compatible mappings as well as noncompatible mappings. We demonstrate that weak reciprocal continuity ensures the existence of common fixed points under contractive conditions, which otherwise do not ensure the existence of fixed points. Our results generalize and extend Banach contraction principle and Meir-Keeler-type fixed point theorem.


---

## Body

## 1. Introduction

In his earlier works, Pant [1, 2] introduced the notion of reciprocal continuity and obtained the first results that established a situation in which a collection of mappings has a fixed point, which is a point of discontinuity for all the mappings. These papers are the genesis of a large number of papers (e.g., [3–16]) that employ or deal with reciprocal continuity to study fixed points of discontinuous mappings in various settings. Imdad and Ali [4] used this concept in the setting of non-self-mappings. Singh et al. [9, 10] have obtained applications of reciprocal continuity for hybrid pair of mappings. Balasubramaniam et al. [14] (see also [15]) extended the study of reciprocal continuity to fuzzy metric spaces. Kumar and Pant [16] studied this concept in the setting of probabilistic metric space. Muralisankar and Kalpana [11] established a common fixed point theorem in an intuitionistic fuzzy metric space using contractive condition of integral type.In 1986, Jungck [17] generalized the notion of weakly commuting maps by introducing the concept of compatible maps.Definition 1.1.
Two self-mapsf and g of a metric space (X,d) are called compatible [17] if limnd(fgxn,gfxn)=0, whenever {xn} is a sequence in X such that limnfxn=limngxn=t for some t in X.
The definition of compatibility implies that the mappingsf and g will be noncompatible if there exists a sequence {xn} in X such that limnfxn=limngxn=t for some t in X but limnd(fgxn,gfxn) is either nonzero or nonexistent.Definition 1.2.
Two self-mapsf and g are called pointwise R-weakly commuting [1] (see also [18, 19]) on X if given x in X there exists R>0 such that d(fgx,gfx)≤Rd(fx,gx).Definition 1.3.
Two self-mapsf and g are called pointwise R-weakly commuting of type (Af) [20] (see also [21]) on X if given x in X there exists R>0 such that d(fgx,ggx)≤Rd(fx,gx).Definition 1.4.
Two self-mapsf and g are called pointwise R-weakly commuting of type (Ag) [20] on X if given x in X there exists R>0 such that d(ffx,gfx)≤Rd(fx,gx).Definition 1.5.
A pair(f,g) of self-mappings defined on a nonempty set X is said to be weakly compatible [22] if the pair commutes on the set of coincidence points, that is, fx=gx(x∈X) implies fgx=gfx.It is well known now that pointwiseR-weak commutativity and analogous notions of pointwise R-weak commutativity of type (Af) or pointwise R-weak commutativity of type (Ag) are equivalent to commutativity at coincidence points and in the setting of metric spaces these notions are equivalent to weak compatibility. On the other hand, pointwise R-weak commutativity and analogous notions of pointwise R-weak commutativity of type (Af) or (Ag) are more useful in establishing common fixed point theorems since they not only imply commutativity at coincidence points but may also help in the determination of coincidence points [19, 21].In a recent work, Al-Thagafi and Shahzad [23] generalized the notion of nontrivial weakly compatible maps by introducing the notion of occasionally weakly compatible mappings.Definition 1.6.
A pair(f,g) of self-mappings defined on a nonempty set X is said to be occasionally weakly compatible [23] (in short owc) if there exists a point x in X, which is a coincidence point of f and g at which f and g commute.Definition 1.7.
Two self-mappingsf and g of a metric space (X,d) are called conditionally commuting [24] if they commute on a nonempty subset of the set of coincidence points whenever the set of their coincidences is nonempty.
From the definition itself, it is clear that if two maps are weakly compatible or owc then they are necessarily conditionally commuting; however, the conditionally commuting mappings are not necessarily weakly compatible or owc [24].Definition 1.8.
Letf and g(f≠g) be two self-maps of a metric space (X,d), then f is called g-absorbing [25] if there exists some positive real number R such that d(gx,gfx)≤Rd(fx,gx) for all x in X. Similarly, g will be called f-absorbing if there exists some positive real number R such that d(fx,fgx)≤Rd(fx,gx) for all x in X.
It is well known that the absorbing maps are neither a subclass of compatible maps nor a subclass of noncompatible maps [25].Definition 1.9.
Two self-mappingsf and g of a metric space (X,d) are called reciprocally continuous [1, 2] if and only if fgxn→ft and gfxn→gt whenever {xn} is a sequence such that limnfxn=limngxn=t for some t in X.Iff and g are both continuous, then they are obviously reciprocally continuous but the converse is not true [1, 2]. The notion of reciprocal continuity is mainly applicable to compatible mapping satisfying contractive conditions [7]. To widen the scope of the study of fixed points from the class of compatible mappings satisfying contractive conditions to a wider class including compatible as well as noncompatible mappings satisfying contractive, nonexpansive, or Lipschitz-type condition Pant et al. [7] generalized the notion of reciprocal continuity by introducing the new concept of weak reciprocal continuity as follows.Definition 1.10.
Two self-mappingsf and g of a metric space (X,d) are called weakly reciprocally continuous [7] iff fgxn→ft or gfxn→gt, whenever {xn} is a sequence in X such that limnfxn=limngxn=t for some t in X.We now give examples of compatible and weakly reciprocally continuous mappings with or without common fixed points.Example 1.11.
LetX=[0,1] and d be the usual metric on X. Define f,g:X→X by
(1.1)fx=x,∀x,gx=x2ifx>0,g(0)=1.
Then it can be verified that f and g are compatible as well as weakly reciprocally continuous mappings but do not have a common fixed point.Example 1.12.
LetX=[0,1] and d be the usual metric on X. Define f,g:X→X by
(1.2)fx=(1-x),∀x,gx=fractional part  of(1-x).
It may be noted that f and g are compatible as well as weakly reciprocally continuous mappings and have infinitely many common fixed points. Examples of noncompatible weakly reciprocally continuous mappings are given on the following pages.
Iff and g are reciprocally continuous, then they are obviously weakly reciprocally continuous but, as shown in Example 2.2 below, the converse is not true. As an application of weak reciprocal continuity we prove common fixed point theorems under contractive conditions that extend the scope of the study of common fixed point theorems from the class of compatible continuous mappings to a wider class of mappings, which also includes noncompatible and discontinuous mappings. Our results also demonstrate the usefulness of the notion of the absorbing maps in fixed point considerations.


## 2. Main Results

Theorem 2.1.
Letfand g be weakly reciprocally continuous pointwise R-weakly commuting of type (Af) self-mappings of a complete metric space (X,d) such that(i)
fX⊆gX,(ii)
d(fx,fy)≤kd(gx,gy),k∈[0,1).Ifg is f-absorbing or f is g-absorbing, then f and g have a unique common fixed point.Proof.
Letx0 be any point in X. Define sequences {xn} and {yn} in X by
(2.1)yn=fxn=gxn+1.
We claim that {yn} is a Cauchy sequence. Using (ii), we obtain
(2.2)d(yn,yn+1)=d(fxn,fxn+1)≤kd(gxn,gxn+1)=kd(yn-1,yn)≤⋯≤knd(y0,y1).
Moreover, for every integer p>0, we get
(2.3)d(yn,yn+p)≤d(yn,yn+1)+d(yn+1,yn+2)+⋯+d(yn+p-1,yn+p)≤d(yn,yn+1)+kd(yn,yn+1)+⋯+kp-1d(yn,yn+1)=(1+k+k2+⋯+kp-1)d(yn,yn+1)≤(11-k)d(yn,yn+1)≤(kn1-k)d(y0,y1).
This means that d(yn,yn+p)→0 as n→∞. Therefore, {yn} is a Cauchy sequence. Since X is complete, there exists a point t in X such that yn→t. Moreover, yn=fxn=gxn+1→t.
Suppose thatg is f-absorbing. Now, weak reciprocal continuity of f and g implies that fgxn→ft or gfxn→gt. Let gfxn→gt. By virtue of (2.1), this also yields ggxn+1=gfxn→gt. Since g is f-absorbing, d(fxn,fgxn)≤Rd(fxn,gxn). On letting n→∞, we obtain fgxn→t. Using (ii), we get d(ft,fgxn)≤kd(gt,ggxn). On making n→∞ we get fgxn→ft. Hence t=ft. Since fX⊆gX, there exists u in X such that t=ft=gu. Now using (ii), we obtain d(fxn,fu)≤kd(gxn,gu). On letting n→∞, we get fu=gu. Since f and g are pointwise R-weak commutative of type (Af), we have d(fgu,ggu)≤R1d(fu,gu)=0 for some R1>0, that is, fgu=ggu. Thus fgu=gfu=ggu=ffu. Finally using (ii), we obtain d(fu,ffu)≤kd(gu,gfu)=kd(fu,ffu), that is, (1-k)d(fu,ffu)=0. Hence fu=ffu=gfu and fu is a common fixed point of f and g.
Next suppose thatfgxn→ft. Since g is f-absorbing, d(fxn,fgxn)≤Rd(fxn,gxn). On letting n→∞, we get t=ft. Since fX⊆gX, there exists u in X such that t=ft=gu. Now using (ii), we obtain d(fxn,fu)≤kd(gxn,gu). On letting n→∞, we get fu=t. Thus fu=gu. Since f and g are pointwise R-weak commutative of type (Af), we have d(fgu,ggu)≤R1d(fu,gu)=0 for some R1>0, that is, fgu=ggu. Thus fgu=gfu=ggu=ffu. Finally using (ii), we obtain d(fu,ffu)≤kd(gu,gfu)=kd(fu,ffu), that is, (1-k)d(fu,ffu)=0. Hence fu=ffu=gfu and fu is a common fixed point of f and g.
Finally suppose thatf is g-absorbing. Now, weak reciprocal continuity of f and g implies that fgxn→ft or gfxn→gt. Let us first assume that gfxn→gt. Since f is g-absorbing, d(gxn,gfxn)≤Rd(fxn,gxn). On making n→∞, we get t=gt. Using (ii) we get d(fxn,ft)≤kd(gxn,gt). On letting n→∞, we get fxn→ft. Hence t=ft=gt and t is a common fixed point of f and g.
Next suppose thatfgxn→ft. Then fX⊆gX implies that ft=gu for some u∈X and fgxn→gu. By virtue of (2.1), this also yields ffxn-1→gu. Since f is g-absorbing, d(gxn,gfxn)≤Rd(fxn,gxn). On letting n→∞, we get gfxn→t. Now, using (ii), we get d(fxn,ffxn)≤kd(gxn,gfxn). On making n→∞, we obtain t=gu. Again, by virtue of (ii), d(fxn,fu)≤kd(gxn,gu). Making n→∞ we get fu=t. Hence t=fu=gu. Since f and g are pointwise R-weak commutative of type (Af), we have d(fgu,ggu)≤R1d(fu,gu)=0 for some R1>0, that is, fgu=ggu. Thus fgu=gfu=ggu=ffu. Finally using (ii), we obtain d(fu,ffu)≤kd(gu,gfu)=kd(fu,ffu), that is, (1-k)d(fu,ffu)=0. Hence fu=ffu=gfu and fu is a common fixed point of f and g.Uniqueness of the common fixed point theorem follows easily in each of the two cases.We now give an example to illustrate the above theorem.Example 2.2.
LetX=[2,20] and d be the usual metric on X. Define f,g:X→X as follows:
(2.4)fx=2ifx=2orx>5,fx=6if2<x≤5,g2=2,gx=12if2<x≤5,gx=(x+1)3ifx>5.
Thenf and g satisfy all the conditions of Theorem 2.1 and have a unique common fixed point at x=2. It can be verified in this example that f and g satisfy the contraction condition (ii) for k=4/5. The mappings f and g are pointwise R-weakly commuting of type (Af) maps as they commute at their only coincidence point x=2. Furthermore, f is g-absorbing with R=29/18. It can also be noted that f and g are weakly reciprocally continuous. To see this, let {xn} be a sequence in X such that fxn→t,gxn→t for some t. Then t=2 and either xn=2 for each n from some place onwards or xn=5+εn, where εn→0 as n→∞. If xn=2 for each n from some place onwards, fgxn→2=f2 and gfxn→2=g2. If xn=5+εn, then fxn→2,gxn=(2+∈n/3)→2,fgxn=f(2+∈n/3)→6≠f2, and gfxn→g2=2. Thus limn→∞gfxn=g2 but limn→∞fgxn≠f2. Hence f and g are weakly reciprocally continuous. It is also obvious that f and g are not reciprocally continuous mappings.Remark 2.3.
Puttingg equal to identity map, we get the famous Banach fixed point theorem as a particular case of the above theorem.We now establish a common fixed point theorem for a pair of mappings satisfying an(∈,δ) type contractive condition. It is now well known (e.g., Example 2.4 below) that an (∈,δ) contractive condition does not ensure the existence of a fixed point.Example 2.4 (see [26]).
LetX=[0,2] and d be the Euclidean metric on X. Define f:X→X by
(2.5)fx=(1+x)2ifx<1,fx=0ifx≥1.
Thenf satisfies the contractive condition
(2.6)ε≤max⁡{d(x,y),d(x,fx),d(y,fy)}<ε+δ⟹d(fx,fy)<ε
with δ(ε)=1 for ε≥1 and δ(ε)=1-ε for ε<1 but f does not have a fixed point.In view of the above example, the next theorem demonstrates the usefulness of weak reciprocal continuity and shows that the new notion ensures the existence of a common fixed point under an(ε,δ) contractive condition.Theorem 2.5.
Letf and g be weakly reciprocally continuous pointwise R-weakly commuting of type (Af) self-mappings of a complete metric space (X,d) such that(i)
fX⊆gX;(ii)
d(fx,fy)<d(gx,gy) whenever gx≠gy;(iii)
givenε>0 there exists δ>0 such that
(2.7)ε<d(gx,gy)<ε+δ⟹d(fx,fy)≤ε.If g is f-absorbing or f is g-absorbing, then f and g have a unique common fixed point.Proof.
Letx0 be any point in X. Define sequences {xn} and {yn} in X by
(2.8)yn=fxn=gxn+1.
We claim that {yn} is a Cauchy sequence. Using (ii), we obtain
(2.9)d(yn,yn+1)=d(fxn,fxn+1)<d(gxn,gxn+1)=d(yn-1,yn).
Thus {d(yn,yn+1)} is a strictly decreasing sequence of positive real numbers and, therefore, tends to a limit r≥0, that is, limn→∞d(yn,yn+1)=r,r≥0. We assert that r=0. For, if not, suppose that r>0. Then given δ>0, no matter small δ may be, there exists a positive integer N such that for each n≥N, we have
(2.10)r<d(yn,yn+1)=d(fxn,fxn+1)<r+δ,
that is,
(2.11)r<d(gxn+1,gxn+2)<r+δ.
Selecting δ in (2.11) in accordance with (iii), for each n≥N, we get d(fxn+1,fxn+2)≤r, that is, d(yn+1,yn+2)≤r, a contradiction to (2.11). Therefore, limn→∞d(yn,yn+1)=0. We now show that {yn} is a Cauchy sequence. Suppose it is not. Then there exist an ε>0 and a subsequence {yni} of {yn} such that d(yni,yni+1)≥2ε. Select δ in (iii) so that 0<δ≤ε. Since limn→∞d(yn,yn+1)=0, there exists an integer N such that d(yn,yn+1)<δ/6 whenever n≥N.
Letni≥N. Then, there exist integers mi satisfying ni<mi<ni+1 such that d(yni,ymi)≥ε+(δ/3). If not, then
(2.12)d(yni,yni+1)≤d(yni,yni+1-1)+d(yni+1-1,yni+1)<ε+(δ3)+(δ6)<2ε,
a contradiction. Let mi be the smallest integer such that d(yni,ymi)≥ε+(δ/3). Then d(yni,ymi-2)<ε+(δ/3) and
(2.13)ε+(δ3)≤d(yni,ymi)≤d(yni,ymi-2)+d(ymi-2,ymi-1)+d(ymi-1,ymi)<ε+(δ3)+(δ6)+(δ6)<ε+(2δ3),
that is, ε<ε+(δ/3)≤d(gxni+1,gxmi+1)<ε+(2/3)δ. In view of (iii), this yields d(yni+1,ymi+1)≤ε. But then
(2.14)d(yni,ymi)≤d(yni,yni+1)+d(yni+1,ymi+1)+d(ymi+1,ymi)<(δ6)+ε+(δ6)=ε+(δ3),
which contradicts (2.13). Hence {yn} is a Cauchy sequence. Since X is complete, there exists a point t in X such that yn→t. Moreover, yn=fxn=gxn+1→t.
Suppose thatg is f-absorbing. Now, weak reciprocal continuity of f and g implies that fgxn→ft or gfxn→gt. Let gfxn→gt. By virtue of (2.8), this also yields ggxn+1=gfxn→gt. Since g is f-absorbing, d(fxn,fgxn)≤Rd(fxn,gxn). On letting n→∞, we get fgxn→t. Using (ii), we get d(ft,fgxn)<d(gt,ggxn). On making n→∞, we get fgxn→ft. Hence t=ft. Since fX⊆gX, there exists u in X such that t=ft=gu. Now using (ii), we obtain d(fxn,fu)<d(gxn,gu). On letting n→∞, we get fu=t. Thus fu=gu. Since f and g are pointwise R-weak commutative of type (Af), we have d(fgu,ggu)≤R1d(fu,gu)=0 for some R1>0, that is, fgu=ggu. Thus fgu=gfu=ggu=ffu. If fu≠ffu, then using (ii) we get d(fu,ffu)<d(gu,gfu)=d(fu,ffu), a contradiction. Hence fu=ffu=gfu and fu is a common fixed point of f and g.
Next suppose thatfgxn→ft. Since g is f-absorbing, d(fxn,fgxn)≤Rd(fxn,gxn). On letting n→∞, we get t=ft. Since fX⊆gX, there exists u in X such that t=ft=gu. Now using (ii), we obtain d(fxn,fu)<d(gxn,gu). On letting n→∞, we get fu=t. Thus fu=gu. Since f and g are pointwise R-weak commutative of type (Af), we have d(fgu,ggu)≤R1d(fu,gu)=0 for some R1>0, that is, fgu=ggu. Thus fgu=gfu=ggu=ffu. If fu≠ffu then using (ii) we get d(fu,ffu)<d(gu,gfu)=d(fu,ffu), a contradiction. Hence fu=ffu=gfu and fu is a common fixed point of f and g.
Whenf is assumed g-absorbing, the proof follows on similar lines as in the corresponding part of Theorem 2.1.We now give an example to illustrate Theorem2.5.Example 2.6.
LetX=[2,20] and d be the usual metric on X. Define f,g:X→X as follows:
(2.15)fx=2ifx=2orx>5,fx=6if2<x≤5,g2=2,gx=(x+31)3if2<x≤5,gx=(x+1)3ifx>5.
Then f and g satisfy all the conditions of Theorem 2.5 and have a unique common fixed point at x=2. It can be seen in this example that f and g satisfy the condition (ii) and the condition
(2.16)ε<d(gx,gy)<ε+δ⟹d(fx,fy)≤ε
with δ(ε)=1 for ε≥4 and δ(ε)=4-ε for ε<4. Furthermore, f is g-absorbing with R=2. It can also be noted that f and g are weakly reciprocally continuous. To see this, let {xn} be a sequence in X such that fxn→t,gxn→t for some t. Then t=2 and either xn=2 for each n from some place onwards or xn=5+εn where εn→0 as n→∞. If xn=2 for each n from some place onwards, fgxn→2=f2 and gfxn→2=g2. If xn=5+εn, then fxn→2,gxn=(2+∈n/3)→2,fgxn=f(2+∈n/3)→6≠f2, and gfxn→g2=2. Thus limn→∞gfxn=g2 but limn→∞fgxn≠f2. Hence f and g are weakly reciprocally continuous. It is also obvious that f and g are not reciprocally continuous mappings. Further, f and g are pointwise R-weakly commuting of type (Af) maps as they commute at their only coincidence point x=2.Remark 2.7.
Theorem2.5 generalizes the well-known fixed point theorem of Meir and Keeler [27].
It may be observed that the mappingsf and g in Examples 2.2 and 2.6 are noncompatible mappings. However, in the case of noncompatible mappings there is an alternative method of proving the existence of fixed points [6, 7, 11, 19, 24, 26, 28–32]. This alternative method was introduced by Pant [19, 26, 28–30] and is also applicable under strictly contractive [19, 26, 31–33], nonexpansive [7], and Lipschitz-type conditions [6, 24, 30]. The existence of such a method is important since there is no general method for studying the fixed points of nonexpansive or Lipschitz-type mapping pairs in ordinary metric spaces.
In the area of fixed point theory, Lipschitz type mappings constitute a very important class of mappings and include contraction mappings, contractive mappings and, nonexpansive mappings as subclasses. The next theorem provides a good illustration of the applicability of recently introduced notions of conditional commutativity and weak reciprocal continuity to establish a situation in which a pair of mappings may possess common fixed points as well as coincidence points, which may not be common fixed points.Theorem 2.8.
Letf and g be weakly reciprocally continuous noncompatible self-mappings of a metric space (X,d) satisfying(i)
fX⊆gX,(ii)
d(fx,fy)≤kd(gx,gy),k≥0.
Iff and g are conditionally commuting and g is f-absorbing or f is g-absorbing, then f and g have a common fixed point.Proof.
Sincef and g are noncompatible maps, there exists a sequence {xn} in X such that fxn→t and gxn→t for some t in X but either limnd(fgxn,gfxn)≠0 or the limit does not exist. Since fX⊆gX, for each xn there exists yn in X such that fxn=gyn. Thus fxn→t,gxn→t and gyn→t as n→∞. By virtue of this and using (ii), we obtain fyn→t. Therefore, we have
(2.17)fxn=gyn⟶t,gxn⟶t,fyn⟶t.
Suppose that g is f-absorbing. Then d(fxn,fgxn)≤Rd(fxn,gxn) and d(fyn,fgyn)≤Rd(fyn,gyn). On letting n→∞, these inequalities yield
(2.18)fgxn⟶t,fgyn(=ffxn)⟶t.
Weak reciprocal continuity of f and g implies that fgxn→ft or gfxn→gt. Let gfxn→gt. By virtue of (ii), we get d(ffxn,ft)≤kd(gfxn,gt). On letting n→∞, we get ffxn→ft. In view of (2.18), this yields t=ft. Since fX⊆gX, there exists u in X such that t=ft=gu. Now using (ii), we obtain d(fxn,fu)≤kd(gxn,gu). On letting n→∞, we get fu=t. Thus fu=gu. Conditional commutativity of f and g implies that f and g commute at u, or there exists a coincidence point v of f and g at which f and g commute. Suppose f and g commute at the coincidence point v. Then fv=gv and fgv=gfv. Also ffv=fgv=gfv=ggv. Since g is f-absorbing d(fv,fgv)≤Rd(fv,gv). This yields fv=fgv. Hence fv=ffv=gfv and fv is a common fixed point of f and g.
Next suppose thatfgxn→ft. In view of (2.18), we get t=ft. Since fX⊆gX, there exists u in X such that t=ft=gu. Now using (ii), we obtain d(fxn,fu)≤kd(gxn,gu). On letting n→∞, we get fu=t. Thus fu=gu. This, in view of conditional commutativity and f-absorbing property of g, implies that f and g have a common fixed point.
Now suppose thatf is g-absorbing. Then d(gxn,gfxn)≤Rd(fxn,gxn) and d(gyn,gfyn)≤Rd(fyn,gyn). On letting n→∞, these inequalities yield
(2.19)gfxn(=ggyn)⟶t,gfyn⟶t.
Weak reciprocal continuity of f and g implies that fgyn→ft or gfyn→gt. Let us first assume that gfyn→gt. In view of (2.19), this yields t=gt. Using (ii) we get d(fxn,ft)≤kd(gxn,gt). On letting n→∞, we obtain t=ft. Hence t=ft=gt and t is a common fixed point of f and g.
Next suppose thatfgyn→ft. Then fX⊆gX implies that ft=gu for some u∈X. Therefore, fgyn→ft=gu. Using (ii) and in view of (2.19), we get d(fyn,fgyn)≤kd(gyn,ggyn). On letting n→∞, we get t=gu. Again, by virtue of (ii), we obtain d(fyn,fu)≤kd(gyn,gu). Making n→∞, we get t=fu. Hence fu=gu. Conditional commutativity of f and g implies that f and g commute at u or there exists a coincidence point v of f and g at which f and g commute. Suppose f and g commute at the coincidence point v. Then fv=gv and fgv=gfv. Also ffv=fgv=gfv=ggv. Since f is g-absorbing, d(gv,gfv)≤Rd(fv,gv). This yields gv=gfv. Hence fv=ffv=gfv and fv is a common fixed point of f and g. This completes the proof of the theorem.We now give examples to illustrate Theorem2.8.Example 2.9.
LetX=[0,1] and d be the usual metric on X. Define f,g:X→X by fx=(1/2)-|x-(1/2)|,
(2.20)gx=23(1-x).
Thenf and g satisfy all the conditions of the above theorem and have two coincidence points x=1,2/5 and a common fixed point x=2/5. It may be verified in this example that f(X)=[0,1/2],g(X)=[0,2/3] and fX⊆gX. Also that f and g are noncompatible but conditionally commuting maps. Furthermore, f and g are conditionally commuting since they commute at their coincidence point 2/5. To see that f and g are noncompatible, let us consider the sequence {xn} given by xn=1-(1/n). Then fxn→0,gxn→0,fgxn→0, and gfxn→2/3. Hence f and g are noncompatible. It may also be verified that f and g are not pointwise R-weakly commuting of type (Af) as they do not commute at the coincidence point x=1, since f(g(1))=0 and g(f(1))=2/3. It is also easy to verify that f and g satisfy the Lipschitz-type condition d(fx,fy)≤(3/2)d(gx,gy) together with f-absorbing condition d(fx,fgx)≤d(fx,gx) for all x. It can also be noted that f and g are weakly reciprocally continuous since both f and g are continuous.In Example2.9, f and g are not pointwise R-weakly commuting of type (Af) as they do not commute at the coincidence point x=1. We now give an example of pointwise R-weakly commuting of type (Af) maps satisfying Theorem 2.8.Example 2.10.
LetX=[0,1] and d be the usual metric on X. Define f,g:X→X as follows:
(2.21)fx=12-|x-12|,gx=(23)fractional  part  of(1-x).
Then f and g satisfy all the conditions of the above theorem and have three coincidence points x=0,2/5,1 and two common fixed point x=0,2/5. It may be verified in this example that f(X)=[0,1/2],g(X)=[0,2/3) and fX⊆gX. Also, f and g are pointwise R-weakly commuting of type (Af) maps, hence also conditionally commuting, since they commute at each of their coincidence points, namely, x=0,2/5,1. To see that f and g are noncompatible, let us consider the sequence {xn} given by xn=1-1/n. Then fxn→0,gxn→0,fgxn→0, and gfxn→2/3. Hence f and g are noncompatible. It is also easy to verify that f and g satisfy the Lipschitz-type condition d(fx,fy)≤(3/2)d(gx,gy). The mapping g is f-absorbing since d(fx,fgx)≤d(fx,gx) for all x. It can also be noted that f and g are weakly reciprocally continuous. To see this, let {xn} be a sequence in X such that fxn→t,gxn→t for some t. Then t=0 and either xn=0 for each n or xn→1. If xn=0 for each n, then fxn→0,gxn→0,fgxn→0=f(0), and gfxn→0=g(0). If xn→1, then fxn→0,gxn→0,fgxn→0=f(0), and gfxn→2/3≠g(0). Thus limn→∞fgxn=f(0) but limn→∞gfxn≠g(0). Hence f and g are weakly reciprocally continuous.
Puttingk=1 in Theorem 2.8, we get a common fixed point theorem for a non-expansive-type mapping pair.Corollary 2.11.
Letf and g be weakly reciprocally continuous noncompatible self-mappings of a metric space (X,d) satisfying(i)
fX⊆gX,(ii)
d(fx,fy)≤d(gx,gy).
Iff and g are conditionally commuting and g is f-absorbing or f is g-absorbing, then f and g have a common fixed point.



---

*Source: 786814-2012-07-16.xml*