Volume 2009, Article ID 804734,8pages doi:10.1155/2009/804734
Research Article
Some Common Fixed Point Theorems for Weakly Compatible Mappings in Metric Spaces
M. A. Ahmed
Department of Mathematics, Faculty of Science, Assiut University, Assiut 71516, Egypt
Correspondence should be addressed to M. A. Ahmed,[email protected] Received 23 October 2008; Accepted 18 January 2009
Recommended by William A. Kirk
We establish a common fixed point theorem for weakly compatible mappings generalizing a result of Khan and Kubiaczyk1988. Also, an example is given to support our generalization. We also prove common fixed point theorems for weakly compatible mappings in metric and compact metric spaces.
Copyrightq2009 M. A. Ahmed. 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.
1. Introduction
In the last years, fixed point theorems have been applied to show the existence and uniqueness of the solutions of differential equations, integral equations and many other branches mathematics see, e.g., 1–3. Some common fixed point theorems for weakly commuting, compatible, δ-compatible and weakly compatible mappings under different contractive conditions in metric spaces have appeared in 4–15. Throughout this paper, X, dis a metric space.
Following9,16, we define, 2X
A⊂X:Ais nonempty ,
BX
A∈2X:Ais bounded
. 1.1
For allA, B∈BX, we define
δA, B sup
da, b:a∈A, b∈B , DA, B inf
da, b:a∈A, b∈B , HA, B inf
r >0 :Ar ⊃B, Br ⊃A ,
1.2
whereAr {x∈ X : dx, a < r, for somea∈ A}andBr {y ∈X :dy, b< r, for some b∈B}.
IfA{a}for somea∈A, we denoteδa, B,Da, BandHa, BforδA, B,DA, B andHA, B, respectively. Also, ifB {b}, then one can deduce thatδA, B DA, B HA, B da, b.
It follows immediately from the definition ofδA, Bthat, for everyA, B, C∈BX, δA, B δB, A≥0, δA, B≤δA, C δC, B, δA, B 0,
iffAB{a}, δA, A diamA. 1.3
We need the following definitions and lemmas.
Definition 1.1see16. A sequenceAnof nonempty subsets ofXis said to be convergent to A⊆Xif:
ieach pointainAis the limit of a convergent sequencean, wherean is inAn for n∈ {0} ∪NN:the set of all positive integers,
iifor arbitrary >0, there exists an integermsuch thatAn⊆Aforn > m, whereA
denotes the set of all pointsxinXfor which there exists a pointainA, depending onx,such thatdx, a< .
Ais then said to be the limit of the sequenceAn.
Definition 1.2see9. A set-valued functionF :X → 2Xis said to be continuous if for any sequencexninXwith limn→ ∞xnx, it yields limn→ ∞HFxn, Fx 0.
Lemma 1.3see16. IfAnandBnare sequences inBXconverging toAand BinBX, respectively, then the sequenceδAn, Bnconverges toδA, B.
Lemma 1.4 see 16. Let An be a sequence in BX and let y be a point in X such that δAn, y → 0. Then the sequenceAnconverges to the set{y}inBX.
Lemma 1.5see9. For anyA, B, C, D ∈BX, it yields thatδA, B ≤HA, C δC, D HD, B.
Lemma 1.6see17. LetΨ:0,∞ → 0,∞be a right continuous function such thatΨt< t for everyt > 0. Then, limn→ ∞Ψnt 0 for everyt > 0, whereΨn denotes the n-times repeated composition ofΨwith itself.
Definition 1.7see15. The mappingsI :X → XandF:X → BXare weakly commuting onXifIFx∈BXandδFIx, IFx≤max{δIx, Fx,diamIFx}for allx∈X.
Definition 1.8see13. The mappingsI : X → X andF : X → BXare said to beδ- compatible if limn→ ∞δFIxn, IFxn 0 wheneverxnis a sequence in X such thatIFxn ∈ BX,Fxn → {t}andIxn → tfor somet∈X.
Definition 1.9see13. The mappingsI :X → X andF :X → BXare weakly compatible if they commute at coincidence points, that is, for each pointu∈Xsuch thatFu{Iu}, then FIuIFunote that the equationFu{Iu}implies thatFuis a singleton.
If F is a single-valued mapping, thenDefinition 1.7 resp., Definitions1.8 and 1.9 reduces to the concept of weak commutativityresp., compatibility and weak compatibility for single-valued mappings due to Sessa18 resp., Jungck11,12.
It can be seen that
weakly commuting⇒δ-compatible andδ-compatible⇒weakly compatible, 1.4
but the converse of these implications may not be truesee,13,15.
Throughtout this paper, we assume thatΦis the set of all functions φ : 0,∞5 → 0,∞satisfying the following conditions:
iφ is upper semi-continuous continuous at a point 0 from the right, and non- decreasing in each coodinate variable,
iiFor eacht >0,Ψt max{φt, t, t, t, t, φt, t, t,2t,0, φt, t, t,0,2t}< t.
Theorem 1.10see19. LetF, Gbe mappings of a complete metric spaceX, dintoBXand I be a mapping ofXinto itself such thatI, F andGare continuous,FX ⊆ JX,GX ⊆ IX, IFFI,IGGIand for allx, y∈X,
δFx, Gy≤φdIx, Iy, δIx, Fx, δIy, Gy, DIx, Gy, DIy, Fx, 1.5 whereφsatisfies (i) andφt, t, t, at, bt< tfor eacht >0, anda≥0,b≥0 withab≤2. ThenI, F andGhave a unique common fixed pointusuch thatuIu∈Fu∩Gu.
In the present paper, we are concerned with the following:
1replacing the commutativity of the mappings in Theorem 1.10 by the weak compatibility of a pair of mappings to obtain a common fixed point theorem metric spaces without the continuity assumption of the mappings,
2giving an example to support our generalization ofTheorem 1.10,
3establishing another common fixed point theorem for two families of set-valued mappings and two single-valued mappings,
4proving a common fixed point theorem for weakly compatible mappings under a strict contractive condition on compact metric spaces.
2. Main Results
In this section, we establish a common fixed point theorem in metric spaces generalizing Theorems 1.10. Also, an example is introduced to support our generalization. We prove a common fixed point theorem for two families of set-valued mappings and two single-valued mappings. Finally, we establish a common fixed point theorem under a strict contractive condition on compact metric spaces.
First we state and prove the following.
Theorem 2.1. LetI, J be two sefmaps of a metric spaceX, dand let F, G : X → BXbe two set-valued mappings with
∪FX⊆JX, ∪GX⊆IX. 2.1
Suppose that one ofIXandJXis complete and the pairs{F, I}and{G, J}are weakly compatible.
If there exists a functionφ∈Φsuch that for allx, y∈X,
δFx, Gy≤φdIx, Jy, δIx, Fx, δJy, Gy, DIx, Gy, DJy, Fx, 2.2
then there is a pointp∈Xsuch that{p}{Ip}{Jp}FpGp.
Proof. Let x0 be an arbitrary point in X. By 2.1, we choose a point x1 in X such that Jx1 ∈ Fx0 Z0and for this pointx1 there exists a pointx2 inX such thatIx2 ∈ Gx1 Z1. Continuing this manner we can define a sequencexnas follows:
Jx2n1∈Fx2nZ2n, Ix2n2∈Gx2n1Z2n1, 2.3 forn∈ {0} ∪N. For simplicity, we putVn δZn, Zn1forn∈ {0} ∪N. By2.2and2.3, we have that
V2nδ
Z2n, Z2n1 δ
Fx2n, Gx2n1
≤φ d
Ix2n, Jx2n1 , δ
Ix2n, Fx2n
, δ
Jx2n1, Gx2n1 , D
Ix2n, Gx2n1 , D
Jx2n1, Fx2n
≤φ δ
Z2n−1, Z2n
, δ
Z2n−1, Z2n
, δ
Z2n, Z2n1 , δ
Z2n−1, Z2n
δ
Z2n, Z2n1 ,0 φ
V2n−1, V2n−1, V2n, V2n−1V2n,0.
2.4
IfV2n > V2n−1, then
V2n≤φV2n, V2n, V2n,2V2n,0≤ΨV2n< V2n. 2.5
This contradiction demands that V2n≤φ
V2n−1, V2n−1, V2n−1,2V2n−1,0
≤Ψ V2n−1
. 2.6
Similarly, one can deduce that V2n1≤φ
V2n, V2n, V2n,0,2V2n
≤Ψ V2n
. 2.7
So, for eachn∈ {0} ∪N, we obtain that Vn1≤Ψ
Vn
≤Ψ2 Vn−1
≤ · · · ≤Ψn V1
, 2.8
whereV1δZ1, Z2 δFx2, Gx1≤φV0, V0, V0,0,2V0. By2.8andLemma 1.6, we obtain that limn→ ∞Vnlimn→ ∞δZn, Zn1 0. Since
δ Zn, Zm
≤δ
Zn, Zn1 δ
Zn1, Zn2
· · ·δ
Zm−1, Zm
, 2.9
then limn,m→ ∞δZn, Zm 0. Therefore,Znis a Cauchy sequence.
Let zn be an arbitrary point in Zn for n ∈ {0} ∪N. Then limn,m→ ∞dzn, zm ≤ limn,m→ ∞δZn, Zm 0 andznis a Cauchy sequence. We assume without loss of generality thatJXis complete. Letxnbe the sequence defined by2.3. ButJx2n1 ∈Fx2n Z2nfor alln∈ {0} ∪N. Hence, we find that
d
Jx2m−1, Jx2n1
≤δ
Z2m−2, Z2n
≤V2m−2δ
Z2m−1, Z2n
−→0, 2.10
asm, n → ∞. So,Jx2n1is a Cauchy sequence. Hence,Jx2n1 → p Jv ∈JXfor some v∈X. ButIx2n∈Gx2n−1Z2n−1by2.3, so thatdIx2n, Jx2n1≤δZ2n−1, Z2n V2n−1 → 0.
Consequently,Ix2n → p. Moreover, we have, forn∈ {0}∪N, thatδFx2n, p≤δFx2n, Ix2n dIx2n, p ≤ V2n−1 dIx2n, p. Therefore,δFx2n, p → 0. So, we have byLemma 1.4that Fx2n → {p}. In like manner it follows thatδGx2n1, p → 0 andGx2n1 → {p}.
Since, forn∈ {0} ∪N,
δ
Fx2n, Gv
≤φ d
Ix2n, Jv , δ
Ix2n, Fx2n
, δ
Jv, Gv , D
Ix2n, Gv , D
Jv, Fx2n
≤φ d
Ix2n, Jv , δ
Ix2n, Fx2n
, δ
Jv, Gv , δ
Ix2n, Gv , δ
Jv, Fx2n
, 2.11
andδIx2n, Gv → δp, Gvasn → ∞, we get fromLemma 1.3that δp, Gv≤φ
0,0, δp, Gv, δp, Gv,0
≤Ψ
δp, Gv
< δp, Gv. 2.12 This is absurd. So,{p} Gv {Jv}. But∪GX ⊆ IX, so∃u ∈X such that{Iu} Gv {Jv}. IfFu /Gv,δFu, Gv/0, then we have
δFu, p δFu, Gv
≤φ
dIu, Jv, δIu, Fu, δJv, Gv, DIu, Gv, DJv, Fu
≤φ
dIu, Jv, δIu, Fu, δJv, Gv, δIu, Gv, δJv, Fu φ
0, δFu, p,0,0, δFu, p
≤Ψ
δFu, p
< δFu, p.
2.13
We must conclude that{p}FuGv{Iu}{Jv}.
SinceFu{Iu}and the pair{F, I}is weakly compatible, soFpFIuIFu{Ip}.
Using the inequality2.2, we have δFp, p≤δFp, Gv
≤φ
dIp, Jv, δIp, Fp, δJv, Gv, DIp, Gv, D
Jv, Fp
≤φ
δFp, p,0,0, δFp, p, δFp, p
≤Ψ
δFp, p
< δFp, p.
2.14
This contradiction demands that {p} Fp {Ip}. Similarly, if the pair {G, J} is weakly compatible, one can deduce that{p} Gp {Jp}. Therefore, we get that{p} Fp Gp {Ip}{Jp}.
The proof, assuming the completeness ofIX, is similar to the above.
To see thatpis unique, suppose that{q}FqGq{Iq}{Jq}. Ifp /q, then dp, q δFp, Gq≤φ
dp, q,0,0, dp, q, dp, q
≤Ψ
dp, q
< dp, q, 2.15 which is inadmissible. So,pq.
Now, we give an example to show the greater generality of Theorem 2.1 over Theorem 1.10.
Example 2.2. Let X 0,1 endowed with the Euclidean metric d. Assume that φt1, t2, t3, t4, t5 t1/3 for every t1, t2, t3, t4, t5 ∈ 0,∞. Define F, G : X → BX and I, J:X → Xas follows:
Fx 1 2
if x∈X, Gx 1 2
ifx∈ 0,1
2 , Gx
3 8,1
2 ifx∈ 1 2,1 , Ix 1
2 ifx∈
0,1
2 , Ix x1
4 ifx∈ 1
2,1 , Jx1−x ifx∈
0,1 2 , Jx0 ifx∈
1 2,1 .
2.16 We have that∪FX {1/2} {J1/2} ⊆ JXand ∪GX 3/8,1/2 IX.
Moreover,δFx, Gy 0 ify∈0,1/2. Ify∈1/2,1, thenδFx, Gy≤1/8 anddIx, Jy≥ 3/8. So, we obtain that
δFx, Gy≤ 1
3dIx, Jy 1 3φ
dIx, Jy, δIx, Fx, δJy, Gy, DIx, Gy, DJy, Fx , 2.17 for all x, y ∈ X. It is clear that X is a complete metric space. Since JX 1/2,1 ∪ {0} is a closed subset of X, so JX is complete. We note that {F, I} is a δ-compatible
pair and therefore a weakly compatible pair. Also, G1/2 {J1/2} and GJ1/2 JG1/2 {1/2}, that is, G and J are weakly compatible. On the other hand, if xn 1/2− 2−n, so that δGJxn, JGxn → 1/8/0 even though Gxn,{Jxn} → {1/2}, that is, {G, J} is not a δ-compatible pair. We know that 1/2 is the unique common fixed point of I, J, F and G. Hence the hypotheses of Theorem 2.1 are satisfied. Theorem 1.10 is not applicable becauseGJx /JGxfor allx ∈ X, and the maps I, J and G are not continuous at x1/2.
InTheorem 2.1, if the mappingsFandGare replaced byFαandGα,α∈ΛwhereΛis an index set, we obtain the following.
Theorem 2.3. LetX, dbe a metric space, and letI, Jbe selfmaps ofX, and forα∈Λ,Fα, Gα:X → BXbe set-valued mappings with∪∪α∈ΛFαX⊆JXand∪∪α∈ΛGαX⊆IX. Suppose that one ofIXandJXis complete and forα∈Λthe pairs{Fα, I}and{Gα, J}are weakly compatible.
If there exists a functionφ∈Φsuch that, for allx, y∈X,
δ
Fαx, Gαy
≤φ
dIx, Jy, δ
Ix, Fαx , δ
Jy, Gαy , D
Ix, Gαy , D
Jy, Fαx
, 2.18
then there is a pointp∈Xsuch that{p}{Ip}{Jp}FαpGαpfor eachα∈Λ.
Proof. UsingTheorem 2.1, we obtain for anyα∈Λ, there is a unique pointzα ∈Xsuch that IzαJzαzαandFαzαGαzα{zα}. For allα, β∈Λ,
d zα, zβ
≤δ
Fαzα, Gβzβ
≤φ d
Izα, Jzβ
, δ
Izα, Fαzα
, δ
Jzβ, Gβzβ
, D
Izα, Gβzβ
, D
Jzβ, Fαzα
≤φ d
zα, zβ
,0,0, d zα, zβ
, d zβ, zα
≤Ψ d
zα, zβ
< d zα, zβ
.
2.19
This yields thatzαzβ.
Inspired by the work of Chang9, we state the following theorem on compact metric spaces.
Theorem 2.4. LetX, dbe a compact metric space,I, Jselfmaps ofX, F, G:X → BXset-valued functions with∪FX⊆JXand∪GX⊆IX.Suppose that the pairs{F, I},{G, J}are weakly compatible and the functionsF,Iare continuous. If there exists a functionφ∈Φ, and for allx, y∈X, the following inequality:
δFx, Gy< φ
dIx, Jy, δIx, Fx, δJy, Gy, DIx, Gy, DJy, Fx
, 2.20
holds whenever the right-hand side of 2.20is positive, then there is a unique pointuinXsuch that FuGu{u}{Iu}{Ju}.
Acknowledgment
The author wishes to thank the refrees for their comments which improved the original manuscript.
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