Vietnam J Maih (2013) 41:323-331 DOI 10.1007/s 10013-013-0024-4
Some Common Fixed Point Results in Partially Ordered Metric Spaces for Generalized Rational Type
Contraction Mappings
Sumit Chandok • Tuisi Dass Narang • Mohamed-Aziz Taoudi
Received 10 August 2012 / Accepted. 7 May 2013 / Published online: 12 June 2013
© Vietnam Academy of Science and Technology (VAST) and Springer Science-i-Business Media Singapore 2013
Abstract Some common fixed point results for generalized contractive mappings satisfying rational type expressions in the framework of partially ordered metric spaces are obtained.
The results proved generalize and extend some known results m the literature.
Keywords Common fixed point Rational type contraction mappings - Compatible mappings • Weakly compatible mappings • Ordered metric spaces
Mathematics Subject Classification (2010) 46X99 41A50 • 47H10 - 54H25
1 Introduction and Preliminaries
The Banach contraction mapping principle is one of the cornerstone results of nonlinear functional analysis. It is a very essential and crucial tool for qualitative nonlinear sciences such as biology, chemistry, physics, various branches of mathematics etc. In particular.
It helps in solving existence and uniqueness problems in different fields of mathematics.
S Chandok (El)
t)cpanment of Mathematics, Khalsa College of Engmeenng & Technology. Punjab Technical University, Ranjii Avenue, Amriisar 143001, India
c-mail.chandhok.sumii@>gmail.com S Chandok
e-mail, chansok s@gmjil com TD Narang
Deparlmeni of Mathematics. Guru Nanak Dev University. Amnrsar 143005. India M.-A Tjoudi
Centre Umversitjirc Polydisciplinairc Kelaa des Srjghna. Unnersiie Cadi Ayyad. B P 263, Kclaa des Sraghnu. MoriKco
M -A Tjoudi
Laborntoire de M.iihcmjiiques el dc Dynamique de Populations. Marrakeiti Morocco
fi Spnnger
324 S.Chandoketjl Due to its importance and applications potential, the Banach contraction mapping principle
has been investigated heavily by many researchers. Consequently, a number of generahza- tions of this celebrated principle have appeared in the literature (see [1-18]).
Ran and Reunngs [18] extended the Banach conbraction principle in partially ordered sets with some applications to linear and nonlinear matrix equations. While Nieto and Rodriguez-Lopez [ 17] extended the result of Ran and Reurings and applied their main theo- rems to obtain a unique solution for a first order ordinary differential equation with periodic boundary conditions. After these initial papers, a number of papers have appeared in this direction (see e.g. [1, 2, 4, 15] and the references cited therein). On the other hand, Gnana Bhaskar and Lakshmtkantham [2] introduced the notion of coupled fixed point via mixed monotone mappings. In this crucial paper, the authors [2] obtained some coupled fixed point results and applied theu^ theorems to solve a first order differential equation with periodic boundary conditions.
The purpose of this paper is to establish some common fixed point results for generalized contraction mappings satisfying a generahzed rational type expression in partially ordered metric spaces.
First, we recall some necessary definitions.
Let M be a nonempty subset of a metric space (X, d) A point JC e M is called a common fixed (coincidence) paint of / and T\fx — fx = Tx (fx = Tx). The set of fixed poinis (respectively, coincidence points) of / and T is denoted by F(f, T) (respectively, C(/, 7")).
The mappings T. f : M ^- M aie called commuting if Tfx = fTx for ai\x sM; compati- ble \i\imd(Tfx„, fTx„) — 0 whenever {x„] is a sequence such that limrj:„ = lim/j:„-l for some r in M; weakly compatible if they commute at their coincidence points, i.e., if fTx = Tfx whenever fx — Tx.
Suppose (X. <) IS a partially ordered set and T. f : X ^- X. T \s said to be f-monotone non-decreasing if for all x.y e X.
fx < fy implies Tx < Ty If / is the identity mapping, then T is monotone non-decreasing.
2 Main Results We start with our main theorem.
Theorem 1 Let [X.<,) be a partially ordered set and suppose thai there exists a metric d on X such that (X. d) is a complete metric space. Suppose ihal T and f are continuous self mappings on X, T(X) C f(X), T is a f-monotone non-decreasing mapping and
+ Y[d{fx. Tx) -V- d(fy, Ty)] + S[d(fx. Ty) -|-d(fy, Tx)]
for all x.y^X with fx > fy, fx / fy and for .'some a, p,y,S€ [0, 1) with a-\-P+2Y*
2S< I.
If there exists XQEX such that fxo < TXQ, T and f are compatible, then T and f hate a coincidence point.
fi Sprmger
Generolized Rational Type Coniraclion Mappings
Proof Since 7 ( X ) c flX). we can ciioose A:i e X so that /jr, = TXQ. Since Tx, e flX).
there exists ;C2 e X such that fx2 — Tx\.By induction, we can construct a sequence ljr„l in X such that /j:n+i — TXn for every /t > 0.
Since fxo < Txti, Txii = fX[, fxo < fx\, r i s a /-monotone non-decreasing mapping, Txti < Tx\. Similarly fx\ < fX2. Txi < Tx2, fxi < /jcs- Continuing this process, we obtain
Txa STx, <TX2<..<Tx„< Tx„+, < • • •
Wesupposethat(i{Tj:„,7"j:„+i) > OforalU.If not, then Ti^+i -Tx„ for somen, Tx.+t - fx.+i, i.e., T and / have a coincidence point j:„+i, and so we have the result.
Consider
/.llfr .. TY ..\rltfr TrW
fdlfx,t,.fx,) dlfx.+,,fx„)
+ rl''lfx.^,.Tx,^,) + illfx..Tx,)]+S[dlfx,^,.Tx.) + dlfx,.Tx.^:)]
^Jdlfx.,,.fx.,2)dlfx,.fx.,,)\
\ dlfx.^ufx.) J
+ y[dlfx.^,.fxnt2) + dlfx,. /A:.+,)] + S[dlfx,t,.fx.^,)
+ d(fx„.fXM)]
= adlfx.t:. fx,^2) + t>dlfx,.^,. fx.) +y[dlfx„^,. fx.^i)
+ dlfx,. /J(,+,)] + S[dlfx,. /Jr.+j)]
< adlfx,f.,, fx.M) + lidlfx,^ufx,) + r[dlfx,+ . fx.t2)
+ dlfx.. fx,t:)]+ildlfx„. fx.^,) + dlfx.t,.fx.t2)]
= loi + r+S)dlfx,+,. fx,M) -I- (^ -I- / + S)dlfx,. /Jt.+i)
= lcx + y+ S)dlTx„. T.\„f,) + lfi + y+ S)dlTx,-i.TxJ
which implies that
dlTx.+t.Tx.)
\-la + y + i)
Using mathematical induction we have+ y-H') dlTx„^uTx„)t
yl - la+ y+S) JPin 1 = J*r*> < 1. We claim that [Tx, | is a Cauchy sequence For m > /i, we have
dlTx„. Tx.) < dlTx... Tx..-t)+dlTx.-i. T.1.,^2) + --- + illTx.tl.Tx.)
<(k"-'-\-k—- + - +k")dlTx,.Tx„)
which i m p l i e s lhat f/(.i,„. .»„) -»• 0 . a s w i . H - - -K:
fi Springer
326 S.Chandokaj.
Thus {Tx„} is a Cauchy sequence in the complete mebnc space X. Therefore theieex- ists u e X such that limrj:„ = «. By the continuity of T, we have lim„..,oo T(Txa) = Tu Since fx„+i — Tx„ -»- u and the pair (7", / ) is compatible, we have 'hm„^a:,d(f(Tx„).
T(fx„)) = 0 . By the triangular inequality, we have
d(Tu, fu) < d{Tu, T(fx„)) +d{T(fx„), f(Tx„)) + d{f(Tx„). fu) Letting n ^ - oc, and using the fact T and / are continuous, we get d(Tu. fu) = 0, i.e.,
Tu = fu and M is a coincidence point of T and / . Q Example I Let X = (1.2.3) and (/(-r.y) = ]j: - i | . Relation is < : = |(1,1), (2,2),(3,3).
(3,1)}, Define 7-(I) = l, r{2) = 2. and T(7,)^\ a n d / ( I ) ^ 1 , / ( 2 ) = 2 a n d / ( 3 ) = 3' Then, for ;c — 3, y = 1, we have
d(Tx.T\)^0 and
(d(fx.Tx)d(fv.Ty)\ ,
<^y-^—^^J^J^~^)+^d(fx,fy)^Y{d(fx,Tx)^d(fy,Ty)\
^h\d(fx, Ty) + d(fy, Tx)] =2[0-\-y+ S].
Hence the inequality holds. On the other hand, it is obvious that 7" is a /-non- mapping with respect to < and there exists xo = 3 such that fxg < Txo and 1 is a fixed point of T and /
Example 2 Let X = | 0 . 1 | AT\dd(x.y) = \x - y\. Relation is ordinary <. Define Tx=Q and fx=\ -X for all xeX. Then
d(Tx,Ty) = 0 and
(d(fx.Tx)d{J\.Ts)\
"V ^nK7jtV~~)'^^''^^'^- ^y'> + >'[^(/-^' Tx)+d(fy. Ty)]
+ 6{d{fx. Ty) + d(fy. Tx)] =p + y + s.
Hence the inequality holds but there is no common fixed point of / and T.
If f = I (the identity mapping) in Theorem 1, then we have the following resuh.
Corollary 1 Lel(X.<)bea partially ordered set and suppose that there exists a metric don X such that {X. d) is a complete metric space. Suppose Ihal T is a continuous self-mappmg on X. T is a monotone non-decreasing mapping and
,,_ _ ^ /d(x.Tx)d(y.T\)\
d(Tx.Ty)<.(^- ^^^J^ - )^m^.y)
+ Y[d(x, Tx) + d(y. Ty)] + S[d(x, Ty) -\-d(y, Tx)] (D for all X. y^X,x>y.x^yandforsomea.^,y.Se[0.\)witha-\-fi-\-2Y+2S<\.
If there exists xo e X with xo < TXQ, then T has a fixed point.
fi Springer
Generalized Rational Type Coniracuon Mappmgs
Examples Let X = {(0,1), (I.O), (1,1)} c M^ with the Euclidean distance t/2- (A'.d,) is obviously a complete metric space. Moreover, we consider in X the partial order given by R={(x..x):xe X] U (((0, I), (1,1))}. LeiT-.X^Xbe given by
7-(0, 1) = (0,1), r ( 1 . 0 ) = ( l , 0 ) and r ( l , l ) = (0,I).
Obviously, T is continuous and non-decreasing mapping since (0,1) < ( I , I) and T(0,1) ^ (0.1) < 7(1.1) ^ (0, 1).
Let j : > y andx ^ y, then necessarily J: = (1,1) and y = (0, 1) and it satisfies the contrac- tive condition (1), i e , d(Tx, Ty) = 0 and
{d(x.Tx)d(y,Ty)\ ,
"V d<!xy) )+^d(x,y) + Y{d{x,Tx) + d(y,Ty)]
-\- 5[d(x, Ty) + d(y. Tx)] ^ (^ + y + ^)
Also, (0, 1) < ^(O, I), so all the conditions of Corollary 1 hold and (0, 1) and (1,0) are fixed points of T.
If J/ T= 0 = 5, we have the following resuh
Corollary 2 (See [15]) Lei (X, <) be a partially ordered set and suppo.se that there exists a metric d on X such that (X, d) is a complete metric space Suppose that T is a continuous self-mapping on X, T is a monotone non-decreasing mapping and
UT T . . (d(x.Tx)d(y,Ty)\ , „ ,^
d(Tx.T\) < a ; \-^fid(x. v) V d(x. y) ) forallx.ysX,x>y,x^yandforsomea.fis[0. 1) witha-i-^ < I.
If there exists x^ e X with xa < Txo. then T has a fixed point.
Example 4 (See [15]) Let X = ((0, I), (1.0), (1. 1)| C R^ with the Euclidean distance t/j.
Notice that elements in X are only comparable to themselves (X. i/i) is obviously a com- plete metric space. Moreover, we consider in X the partial order given by R= {(x. x):x&
X].LetT '.X ^ X be given by
7-(0, l) = (I.O), r ( l , 0 ) - ( 0 , l ) and 7(1.1) = (I. I).
T is tnvially continuous and non-decreasing, and the contractive condition of Corollary 2 is .satisfied since elements in X are only comparable to themselves. Moreover. (1. 1) <
T(\, 1) = (1. I) and. by Corollary 2, T has a fixed point (obviously, this fixed point is ( l . D ) .
lf^ = 0 in Corollary 2, we have the following resuh.
Corollary 3 (See [15]) Let(X. <) be a partially ordered set and suppose that there exists a metric d on X such that (X. d) is a complete metric space. Suppose thai T is a continuous telf-mapping on X. T is a monotone non-decreasmg mapping and
fd(x,T.^)d(y.Tv)\
dlTx.Ty)^ ^^^^,^ - ) foraltx.veX,x > y, x ^ vandforsomea€[Q. 1).
fi Springer
If there exists Xo e X wilh Xfj < TXQ, then T has a fixed point.
If tt = 0 in Corollary 2, we have the following result.
Corollary 4 (See [17]) Let{X, < ) be a partially ordered set and suppose that there exisis a metric d on X such that (X. d) is a complete metric space. Suppose that T is a continuous self-mapping on X.T is a monotone non-decreasing mapping and
d(Tx,Ty)<pd(x,y) for all x,y & X, X > y. x ^y and for some ^ e [0, 1).
If there exisis xo& X with XQ < TXQ, then T has a fixed point.
In what follows, we prove that Theorem 1 is still valid for 7", not necessanly condnuous, assuming the following hypothesis in X:
If {x„] IS anon-decreasing sequence in X such that A:„ - * x, then x„ <x for all«.
Theorem 2 Let (X. <) be a partially ordered set and suppose that there exists a metric d on X such Ihal (X. d) is a complete metric space. Suppose that T and f are self mappings on X, T(X) c f(X), T is a f-monotone non-decreasing mapping and
'''rx.Ty)s.{''f'-;jiy-''')^,dlfx.M
-i- Y[d(fx, Tx) -h d(fy. Ty )] + S[difx, Ty) + d{fy . T.x)]
for all x,y^X for which fx and fy are comparable, fx jt fy and for some a, ^. y. ^ e [0, I) w r / ( a - | - ^ - | - 2 } / - | - 2 t 5 < I .
Also assume lhat f(X) is closed and for any non-decreasing sequence {x„] in X which converges to x we have x„<x for all n If there exists XQ^X such that fxg < Txo, then T and f have a coincidence point.
Proof Following the proof of Theorem 1 we have {Tx„) is a Cauchy sequence and so is Ifx,,]. Since f(X) is closed and X is complete, l\m.,^^Tx„ =]tm„-.oofx„ = fu for some u B X Notice that the sequences \Tx„] and {fx„} are non-decreasing. Then from our assumptions we have Tx„ < fu and fx„ < fu for all n. Keeping in mind that 7 is / - monotone non-decreasing we get Tx„ < Tu for all n. Letting rt to co we obtain fu < Tu.
Suppose fu < Tu (otherwise we are done). Constmct a sequence (M„| as «O = " aJ"*
fu„+, — Tu„ for all n. A similar argument as in the proof of Theorem 1 yields {fu„\ isa non-decreasing sequence and l i m „ ^ ^ fu„ = lim„^-o Tu„ = fv for some v&X. From our assumptions it follows that sup^ fu„ < fv and sup,, Tu„ < fv.
fxn<fu<fui<---sfu„<---< fv.
We distinguish two cases:
Case I. Suppose there is AIQ > 1 with fx„^ = fu„g. Then fx„g ^ fu- fu„^ = fu\ = Tu.
We are done.
fi Springer
Generalized RaUonal Type Coniracuon Mappings
Case 2. Suppose fu„ -^ fx„ for all /i > 1. Then from the contraction assumption we obtain d{fx„+ufu„+,)=d(Tx„,Tu„)
^ /d(fx^,Tx.,)d(fu„,Tu„)\
^i STw;;:^ )+Wx..fn.)
+ Y[difx„. Tx„)-\-d(fu„, Tu„)]
+ S[d(fx„.Tu,)-\-d(fu„,Tx„)].
Letting rt to oo we get d(fu, fv) < (j5 + 2S)d(fu, fv), which implies diat fu = fv since ^ -}- 25 < 1. Hence fu= fv = fu\ = Tu, the proof is complete,
D
If we take / = / (the identity mapping) in Theorem 2 we get the following result.
Theorem 3 Lei (X. <) be a partially ordered sel and suppose thai there exists a metric d on X such lhat (X. d) is a complete metric space. Suppose that T is a self-mapping on X, T is a monotone non-decreasing mapping and
diTx.ry)<_.{^--^^y,dix.y)
-\- y [d(x. Tx) -I- d(y. Ty)] + b[d(x. Ty ) + d(y. T.x)]
forallx,y eX,x> y,x^yandforsomea,p,Y.S e [0, 1) vv/?/j «-j-^-H 2 / -|-2^ < 1, A.s.sume lhat {x„} is a non-decreasmg sequence in X such that x„ -> x, then \„ < x for all n. If there exists Xo€ X with XQ < TXQ, then T has a fixed point.
3 Applications
The aim of this section is to apply our new results to mappings involving contractions of inlegral type. For this purpose, denote by A the set of functions p: [0, oo) —*• [0. c*z) satis- fying Ihe following hypotheses:
(hi) liis a Lebesgue-inlegrable mapping on each compact subset of [0, oo);
(h2) For any e > 0, we have / J ^(f) > 0.
Corollary 5 Lei (X,<) he a partially ordered set and suppo.se that there exists a metric d on X such that (X.d) is a complete metric space. Suppose that T and f are continuous self-mappings on X, T(X) C f(X). T is a monotone f-non-decreasing mapping and
/ f{l)dl<a / ^{t)dt-\-p / ^ll(t)dt J I) Jo Jo
f,ni,.lx)+d(fy.r)) riHI> ly]+d{fy,T,)
-\-y / ^(Ddt-i-S ir(i)dt for all X.Y e X for which fx and fy are comparable, fx # fy. >li € A and for some a.fi.y.Se[0, \) wilh a-^-p-i-2Y + is < L
If there e.\:i\i% XgS X such that fxo < Txa and T and f are compatible, then T and f have a coincidence point.
fi Spnnger
.T\'}+J(y.T:i)
^it)dt
Another consequence of Theorem 2 is the following.
Corollary 6 Let (X. <) be a partially ordered set and suppose lhat there exists a metric d on X such thai (X. d) is a complete metric space. Suppose lhat T is a self-mapping on X.
T is monotone non-decreasing and
/ if(t)dtsa ^(t)dt-^^\ ir{l)dl Jo Jo Jo
+ y j yl/(l)dt-i-Sl
for all x.y eX for which x and y are comparable, xi^y,i/ € A and for some a, ,8, y, 6 e
[0.1)Hi(/jff-H^-i-2y-l-2,5<l.
Assume that for any non-decreasing sequence [x„] in X such that x„ -^ x, we have A-, < X for all n. If there exists XQ&X with XQ<TXO, then T has a fixed point.
If y = 0 — 5. we have the following resuh.
Corollary 7 Let (X.<) be a partially ordered set and suppose that there exists a metric d on X such that (X, d) is a complete metric space. Suppose that T is a self-mapping on X, T is monotone non-decreasing and
/ f(l)dt<a / if(t)dt-\-p / f(t)dt
Jo Jo Jo for all x.y e X for which x and y are comparable, x ^ y.tj/ e A and for some a, ^ £ (0,1) witha-\-'fi<\
Assume that for any non-decreasing sequence {x„} in X such that x„ —> x, we have x„ <xforalln. If there exisis XQ e X wilh Xo < TXQ, then T has a fixed point.
Acknowledgements The authors are thankful to ihe leamed referees for very valuable suggesuons leading to an iniprovemeni of the paper.
The research work of the second author is supported by University Grants Commission, India (F,No.
39-48/2010(SR))
References
1 Agarwal R.P., El-Gebeily. M . A . O'Regan, D,: Generalized contractions in partially ordered metne spates Appl Anal 87, 109-116(2008)
_ Bhaskar T G . Lakshmikantham. V.: Fixed point dieorems in partially ordered menic spaces and appli- cations Nonlinear Anal 65,1379-1393(2006)
3 BranLian A • A fixed point theorem for mappings satisfying a general contractive condition of iniegrsl type Int J Math. Math. Sci 2 9 , 5 3 1 - 5 3 6 ( 2 0 0 2 )
4 Chandok S - Some common fixed point theorems for generalized / - w e a k l y contractive mappings J Appl, Math, Inform, 29, 257-265 (2011)
3, Chandok. S : Some common fixed point theorems for generalized nonlinear contraciive mappings, Com- put Math Appl 62.3692-3699(2011) doi:i0.1016/j,camwa2011.09,009
<3 Chandok, S Common fixed points, invariant approximation and generalized weak contractions Ini J Madi Madi. SCI 2012, 102980(2012) 11 pp
7. Chandok, S.: Common fixed points for generalized nonlinear contractive mappings in meinc spaces.
Mat. Vcsn 6 5 , 2 9 - 3 4 ( 2 0 1 3 )
fi Springer
Generalized Rational Type Contraction Mappings
8. Chandok, S. Some common fixed point results for generalized weak contractive mappings m partially ordered meuic spaces J. Nonlinear Anal. Opum. (2013)
9. Chandok, S ; Common fixed point theorems for generalized contractions mappings. Thai J Math (ac- cepted)
10. Chandok. S.: Coupled common fixed point theorems for a contractive condition of rational type in or- dered metnc spaces. J Appl Math Inform (accepted)
11. Chandok, S.. Karapinar, E,, Some common fixed point results for generalized rational type weak con- traction mappings in partially ordered metnc spaces. Thai J Math 11, 251-260 (2013) 12. Chandok. S , Khan, M S . Rao, K PR.. Some coupled common fixed pomi theorems for a pair of map-
pings satisfying a eoniractive condition of rational type. J Nonlinear Anal. Appl 2013. Jnaa-00174 (2013). 6 pp.
13 Chandok, S , Kim, 3,K : Fixed point theorem in ordered memc spaces for generalized contractions map- pings satisfying rational type expressions, J Nonlinear Funct. Anal, Appl 17, 301-306 (2012) U. Chandok, S , Mustafa, Z.,Postolache.M,: Coupled common fixed point theorems for iraxedg-monolone
mappings m partially ordered G-metnc spaces Umv. Polileh, Sci, Bull., Ser A: Appl. Math Phys (accepted)
15 Harjani, J., L6pez, B , Sadarangani, K A fixed point theorem for mappings satisfying a coniraciive condition of rational type on a partially ordered metnc space Abstr. Appl Anal 2010. 190701 (2010) 8 pp.
16. Jaggi, D.S ; Some unique fixed point theorems Indian J. Pure Appl Math 8 223-230(1977) 17 Nieto, J.J , Rodn'guez-LiSpez, R . Contractive mapping theorems in partially ordered sets and applications
to ordmary differential equations. Order 22, 223-239 (2005)
18, Ran, A C M . , Reurings, M.C.B.. A fixed point theorem m partially ordered sets and some applications to matnx equations Proc. Am. Math. Soc. 132, 1435-1443(2004)
fi Springe
