DEMONSTRATIO MATHEMATICA
Vol. XLVII
No 4
2014
Valeriu Popa
A GENERAL COINCIDENCE
AND COMMON FIXED POINT THEOREM
FOR TWO HYBRID PAIRS OF MAPPINGS
Abstract. In this paper, a general coincidence and common fixed points theorem for
two hybrid pairs of mappings satisfying property pE.Aq is proved, generalizing the main
results from [3] and [9].
1. Introduction
Sessa [18] introduced the concept of weakly commuting mappings. Jungck
[5] defined the notion of compatible mappings in order to generalize the concept of weak commutativity and showed that weak commuting mappings
are compatible but the converse is not true. In recent years, a number of
fixed point theorems and coincidence theorems have been obtained by various authors using this notion. Jungck further weakened the notion of weak
compatibility in [6] and Jungck and Rhoades further extended weak compatibility for set-valued mappings. Pant [11], [12], [13] initiated the study of
noncompatible mappings and Singh and Mishra [19] introduced the notion
of pI, T q-commuting.
More recently, Aamri and Moutawakil [1] defined property pE.Aq for self
mappings of a metric space.
Liu et al. [10] extended the notion of pE.Aq property for two pairs of mappings. The class of mappings satisfying pE.Aq property contains the class of
noncompatible mappings. Kamran [8] extends the property pE.Aq for hybrid pairs of single and multi-valued mappings and generalize the notion of
pI.T q-commutativity for hybrid pairs. Recently, Sintunavarat and Kumam
[21] established new coincidence and common fixed point theorems for hybrid
strict contraction maps by dropping the assumption “f is T weakly commut2010 Mathematics Subject Classification: 47H10, 54H25.
Key words and phrases: metric space, hybrid mappings, implicit relation, coincidence
point, fixed point.
DOI: 10.2478/dema-2014-0077
c Copyright by Faculty of Mathematics and Information Science, Warsaw University of Technology
972
V. Popa
ing for a hybrid pair pf, T q of single and multivalued maps" in Theorem 3.10
[8]. Quite recently, some fixed point theorems for hybrid pairs are obtained
in [9] and [3].
The study of fixed points for mappings satisfying implicit relations is
initiated in [14], [15] for single valued mappings. In [16], [17] and in other
papers, the study of fixed points for hybrid pairs of mappings satisfying
implicit relations is initiated.
Actually, the method is used in the study of fixed points and coincidence
points in metric spaces, symmetric spaces, quasi-metric spaces, ultra metric
spaces, reflexive spaces, in two or three metric spaces for single valued mappings, hybrid pairs of mappings and set valued mappings. Quite recently,
the method is used in the study of fixed points for mappings satisfying a
contractive condition of integral type, in fuzzy metric spaces, intuitionistic
metric spaces and G-metric spaces. The method unified some results from
literature.
The purpose of this paper is to prove a general coincidence and fixed
point theorem for two hybrid pair of mappings with common pE.Aq-property
satisfying an implicit relation generalizing main results by [9] and [3].
2. Preliminaries
Let pX, dq be a metric space. We denote by CLpXq (respectively, CBpXq),
the set of all nonempty closed (respectively, nonempty closed bounded) subsets of pX, dq and by H, the Hausdorff–Pompeiu metric, i.e.
!
)
HpA, Bq “ max sup dpx, Bq, sup dpx, Aq ,
xPA
xPB
where A, B P PpXq and
dpx, Aq “ inf tdpx, yqu .
yPA
Definition 2.1. Let f : pX, dq Ñ pX, dq and F : pX, dq Ñ CLpXq.
(1) A point x P X is said to be a coincidence point of f and F if f x P F x.
The set of all coincidence points of f and F is denoted by Cpf, F q.
(2) A point x P X is a common fixed point of f and F if x “ f x P F x.
Definition 2.2. Let f : pX, dq Ñ pX, dq and T : pX, dq Ñ CLpXq. Then
f is said to be T -weakly commuting at x P X [19] if f f x P T f x.
Definition 2.3. Let f : pX, dq Ñ pX, dq and T : pX, dq Ñ CBpXq. f and
T satisfy property pE.Aq [8] if there exists a sequence txn u in X such that
limnÑ8 f xn “ t P A “ limnÑ8 T xn .
Liu et al. [10] extend Definition 2.3 for two hybrid pair of mappings.
A general coincidence and common fixed point theorem
973
Definition 2.4. Let f, g : pX, dq Ñ pX, dq and F, G : pX, dq Ñ CBpXq.
The pairs pf, F q and pg, Gq are said to satisfy a common property pE.Aq if
there exist two sequences txn u and tyn u in X, some t P X and A, B P CBpXq
such that limnÑ8 F xn “ A, lim Gyn “ B and limnÑ8 f xn “ limnÑ8 gxn “
t P A X B.
Motivated by Berinde and Berinde [2], Kamran [9] proved the following
theorem.
Theorem 2.1. Let f : pX, dq Ñ pX, dq and T : pX, dq Ñ CLpXq be such
that
(i) f and T satisfy property pE.Aq,
(ii) for all x ‰ y,
"
*
dpf x, T xq`dpf y, T yq dpf y, T xq`dpf x, T yq
HpT x, T yq ă max dpf x, f yq,
,
2
2
`Ldpf x, T yq,
where L ≥ 0.
If f pXq is a closed subset of X, then f and T have a coincidence point
a P X. Further, if f is T -weakly commuting at a and f a “ f f a, then f and
T have a common fixed point.
Quite recently, Hashim and Abbas [3] extended Theorem 2.1 for two pairs
of hybrid mappings dropping the condition "f is T -weakly commuting".
Theorem 2.2. [3] Let f, g : pX, dq Ñ pX, dq and F, G : pX, dq Ñ CLpXq
such that
(i) pf, F q and pg, Gq satisfy the common property pE.Aq,
(ii) for all x ‰ y,
HpF x, Gyq ă maxtdpf x, gyq, αrdpf x, F xq ` dpgy, Gyqs,
αrdpf x, Gyq ` dpf y, Gxqsu
`L mintdpf x, F xq, dpgy, Gyq, dpf x, Gyq, dpgy, F xqu,
where 0 ă α ă 1 and L ≥ 0.
If f pXq and gpXq are closed subsets of X then Cpf, F q ‰ ∅ and Cpg, Gq
‰ ∅. Further,
(a) if f v “ f f v for v P Cpf, F q then f and F have a common fixed point,
(b) if gv “ ggv for v P Cpg, Gq then g and G have a common fixed point,
(c) if (a) and (b) are both true, then f, g, F and G have a common fixed
point.
This theorem generalizes Theorem 2.3 [10].
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V. Popa
Theorem 2.3. [3] Let f, g : pX, dq Ñ pX, dq and F, G : pX, dq Ñ CLpXq
such that
(i) pf, F q and pg, Gq satisfy the common property pE.Aq,
(ii) for all x ‰ y,
"
HpF x, Gyq ă max dpf x, gyq, αdpf x, F xq, αdpgy, Gyq,
*
dpf x, Gyq ` dpgy, F xq
α
2
`L mintdpf x, F xq, dpgy, Gyq, dpf x, Gyq, dpgy, F xqu,
where 0 ă α ă 1 and L ≥ 0.
If f pXq and gpXq are closed subsets of X, then the conclusion of Theorem 2.2 follows.
This theorem extends and generalizes Theorem 1 [20].
3. Implicit relations
Definition 3.1. Let FC be the set of all continuous functions F pt1 , . . . , t6 q :
R6` Ñ R satisfying the following conditions:
pF1 q : F is nondecreasing in variable t1 ,
pF2 q : F pt, 0, 0, t, t, 0q ą 0, @t ą 0,
pF3 q : F pt, 0, t, 0, 0, tq ą 0, @t ą 0.
Example 3.1.
F pt1 , . . . , t6 q “ t1 ´ maxtt2 , αpt3 ` t4 q, αpt5 ` t6 qu ´ L mintt2 , t4 , t5 , t6 u,
where α P p0, 1q and L ≥ 0.
pF1 q: Obviously,
pF2 q : F pt, 0, 0, t, t, 0q “ tp1 ´ αq ą 0, @t ą 0,
pF3 q : F pt, 0, t, 0, 0, tq “ tp1 ´ αq ą 0, @t ą 0.
Example 3.2.
*
"
t5 ` t6
´ L mintt2 , t4 , t5 , t6 u,
F pt1 , . . . , t6 q “ t1 ´ max t2 , αt3 , αt4 , α
2
where α P r0, 1q and L ≥ 0.
pF1 q: Obviously,
pF2 q : F pt, 0, 0, t, t, 0q “ tp1 ´ αq ą 0, @t ą 0,
pF3 q : F pt, 0, t, 0, 0, tq “ tp1 ´ αq ą 0, @t ą 0.
Example 3.3.
F pt1 , . . . , t6 q “ t1 ´ k maxtt2 , t3 , t4 , t5 , t6 u ´ L mintt3 , t4 , t5 , t6 u,
where k P p0, 1q and L ≥ 0.
A general coincidence and common fixed point theorem
975
pF1 q: Obviously,
pF2 q : F pt, 0, 0, t, t, 0q “ tp1 ´ kq ą 0, @t ą 0,
pF3 q : F pt, 0, t, 0, 0, tq “ tp1 ´ kq ą 0, @t ą 0.
Example 3.4. F pt1 , . . . , t6 q “ t1 ´ at2 ´ bt3 ´ ct4 ´ dt5 ´ et6 , where
a, b, c, d, e ≥ 0, c ` d ă 1 and b ` e ă 1.
pF1 q: Obviously,
pF2 q : F pt, 0, 0, t, t, 0q “ tr1 ´ pc ` dqs ą 0, @t ą 0,
pF3 q : F pt, 0, t, 0, 0, tq “ tr1 ´ pb ` eqs ą 0, @t ą 0.
t2 `t2
3
4
Example 3.5. Fpt1 , . . . , t6 q “ t21 ´ t22 ´ a 1´mintt
, where a P p0, 1q.
5 ,t6 u
pF1 q: Obviously,
pF2 q : F pt, 0, 0, t, t, 0q “ t2 p1 ´ aq ą 0, @t ą 0,
pF3 q : F pt, 0, t, 0, 0, tq “ t2 p1 ´ aq ą 0, @t ą 0.
Example 3.6. F pt1 , . . . , t6 q “ t1 ´ α maxtt2 , t3 , t4 u ´ p1 ´ αqpat5 ` bt6 q,
where α P p0, 1q, a, b ≥ 0 and maxta, bu ă 1.
pF1 q: Obviously,
pF2 q : F pt, 0, 0, t, t, 0q “ tp1 ´ αqp1 ´ aq ą 0, @t ą 0,
pF3 q : F pt, 0, t, 0, 0, tq “ tp1 ´ αqp1 ´ bq ą 0, @t ą 0.
Example 3.7. F pt1 , . . . , t6 q “ t1 ´ maxtct2 , ct3 , ct4 , at5 ` bt6 u, where
a, b, c ≥ 0 and maxta, b, cu ă 1.
pF1 q: Obviously,
pF2 q : F pt, 0, 0, t, t, 0q “ tr1 ´ maxta, cus ą 0, @t ą 0,
pF3 q : F pt, 0, t, 0, 0, tq “ tr1 ´ maxtb, cus ą 0, @t ą 0.
Example 3.8. F pt1 , . . . , t6 q “ t21 ´
t23 t24 `t25 t26
1`t2 .
pF1 q: Obviously,
pF2 q : F pt, 0, 0, t, t, 0q “ t2 ą 0, @t ą 0,
pF3 q : F pt, 0, t, 0, 0, tq “ t2 ą 0, @t ą 0.
4. Main results
Theorem 4.1. Let pX, dq be a metric space, f, g : X Ñ X and F, G : X Ñ
CLpXq such that
p4.1q pf, F q and pg, Gq satisfy the common property pE.Aq,
p4.2q
φpHpF x, Gyq, dpf x, gyq, dpf x, F xq,
dpgy, Gyq, dpf x, Gyq, dpgy, F xqq ă 0,
for all x, y P X, where φ P FC .
If f pXq and gpXq are closed subsets of X then Cpf, F q ‰ ∅ and Cpg, Gq
‰ ∅.
976
V. Popa
Further,
(a) if f v “ f f v for v P Cpf, F q then f and F have a common fixed point,
(b) if gw “ ggw for w P Cpg, Gq then g and G have a common fixed point,
(c) if f v “ f f v for v P Cpf, F q and gw “ ggw for w P Cpg, Gq then f , g,
F and G have a common fixed point.
Proof. Since pf, F q and pg, Gq satisfy the common property pE.Aq A, B P
CLpXq such that
lim F xn “ A, lim Gyn “ B and lim f xn “ lim gyn “ u P A X B.
nÑ8
nÑ8
nÑ8
nÑ8
Since f pXq and gpXq are closed, we have u “ f v and u “ gw for some
v, w P X. First we prove that gw P Gw.
By p4.2q, we have
φpHpF xn , Gwq, dpf xn , gwq, dpf xn , F xn q,
dpgw, Gwq, dpf xn , Gwq, dpgw, F xn qq ă 0.
Letting n tend to infinity, we have successively
φpHpA, Gwq, dpf v, gwq, dpf v, Aq, dpgw, Gwq, dpf v, Gwq, dpgw, Aqq ≤ 0,
φpHpA, Gwq, 0, 0, dpgw, Gwq, dpgw, Gwq, 0q ≤ 0.
Since gw P A then dpgw, Gwq ≤ HpA, Gwq. Then by pF1 q, it follows that
φpdpgw, Gwq, 0, 0, dpgw, Gwq, dpgw, Gwq, 0q ≤ 0,
a contradiction with pF2 q if dpgw, Gwq ą 0. Hence, dpgw, Gwq “ 0 which
implies that gw P Gw. Hence Cpg, Gq ‰ ∅.
Now, we prove that f v P F v. By p4.2q, we have
φpHpF v, Gyn q, dpf v, gyn q, dpf v, F vq,
dpgyn , Gyn q, dpf v, Gyn q, dpgyn , F vqq ă 0.
Letting n tend to infinity, we have successively
φpHpF v, Bq, dpf v, gwq, dpf v, F vq, dpgw, Bq, dpf v, Bq, dpgw, F vqq ≤ 0.
Since f v P B then dpf v, F vq ≤ HpF v, Bq. Then by pF1 q, it follows that
φpdpf v, F vq, 0, dpf v, F vq, 0, 0, dpf v, F vqq ≤ 0,
a contradiction with pF3 q if dpf v, F vq ą 0. Hence, dpf v, F vq “ 0 which
implies f v P Gv. Therefore Cpf, F q ‰ ∅ and Cpg, Gq ‰ ∅. Further, let
f v “ f f v and z “ f v P F v. Then f z “ f f v “ f v “ z and z is a fixed point
of f .
Again, by p4.2q, we have
φpHpF z, Gwq, dpf z, gwq, dpf z, F zq, dpgw, Gwq, dpf z, Gwq, dpgw, F zqq ă 0.
A general coincidence and common fixed point theorem
977
Since dpf z, F zq “ dpf v, F zq “ dpgw, F zq ≤ HpGw, F zq then by pF1 q, we
obtain
φpdpf z, F zq, 0, 0, dpf z, F zq, 0, 0, dpf z, F zqq ă 0,
a contradiction with pF3 q if dpf z, F zq ą 0. Hence dpf z, F zq “ 0, which
implies z “ f z P F z. Therefore, z is a common fixed point of f and F .
Further, let gw “ ggw and z “ f v “ gw P Gw. Then gz “ ggw “ gw “ z
and z is a fixed point of g.
Again, by p4.2q, we have
φpHpF v, Gzq, dpf v, gzq, dpf v, F vq, dpgz, Gzq, dpf v, Gzq, dpgz, F vqq ă 0.
Since dpgz, Gzq “ dpgw, Gzq “ dpf v, Gzq ≤ HpF v, Gzq then by pF1 q, we
obtain
φpdpgz, Gzq, 0, 0, dpgz, Gzq, dpgz, Gzq, 0q ă 0,
a contradiction with pF2 q if dpgz, Gzq ą 0. Hence dpgz, Gzq “ 0, which
implies z “ gz P Gz. Therefore, z is a common fixed point of g and G.
If f f v “ f v for v P Cpf, F q and ggw “ gw for w P Cpg, Gq, then f, g, F
and G have a common fixed point.
Remark 4.1. 1. By Theorem 4.1 and Example 3.1, we obtain Theorem 2.2.
2. By Theorem 4.1 and Example 3.2, we obtain Theorem 2.3.
3. By Theorem 4.1 and Examples 3.3–3.8, we obtain new particular results.
If f “ g and F “ G, we obtain
Theorem 4.2. Let pX, dq be a metric space, f : X Ñ X and F : X Ñ
CLpXq such that
p4.3q pf, F q satisfy property pE.Aq,
p4.4q φpHpF x, F yq, dpf x, f yq, dpf x, F xq,
dpf y, F yq, dpf x, F yq, dpf y, F xqq ă 0
for all x, y P X, where φ satisfies pF1 q and pF2 q.
If f pXq is a closed subset of X then Cpf, F q ‰ ∅. Further, if f v “ f f v
for v P Cpf, F q, then f and F have a common fixed point.
Proof. The proof follows by the first part of the proof of Theorem 4.1.
Remark 4.2. By Theorem 4.2 and Example 3.1, we obtain a generalization
of Theorem 2.1.
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V. Popa
“VASILE ALECSANDRI" UNIVERSITY OF BACĂU,
157 CALEA MĂRĂŞEŞTI
BACĂU, 600115, ROMANIA
E-mail: [email protected]
Received August 7, 2013; revised version March 17, 2014.
Communicated by W. Domitrz.