Vietnam J. Math. (2015) 43-621-627 DOI I0.1007/SI0013-014-0105-Z
Common Fixed Point Theorems for Pairs of Mappings in ^/-Complete Topological Spaces
Hakan Karayilan • Mustafa Telci
Received. 12 September 2013 / Accepled, 24 May 2014 / Published online 5 November 2014
© Vieinam Academy of Science and Technology (VAST) and Spnnger Science+Business Media Singapore 2014
Abstract Some common fixed theorems which generahze the result of (Hicks Int J. Math.
Math Sci. 15: 435-439, 1992) and (Karayilan, Telci, Sci. Stud. Res.. Ser. Math. Inform. 20, 39-48, 2010) are proved for pairs of mappings defined on d-complete topological spaces.
Two examples related to our results are also given.
Keywords Common fixed point • (/-Complete topological spaces Mathematics Subject Classification (2010) 54H10 • 54H25
1 Introduction
Hicks m (5], Hicks and Rhoades in [6] and [7], Hicks and Sahga in [8], Harder and Saliga [4), and Saliga in [17J proved several metnc space fixed point theorems in a large class of non-metnc space so-called rf-complete topological space. Also, many authors in [1-3, 13-16] have proved several fixed point theorems in this space
Let (A", r ) be a topological space andd : X x X -* [0, oo), such that d(x, y) — 0 if and only if .V = y. Then, X is said to be t/-complete if Xiii^i ^i^n.Xn+y) < oo implies that the sequence {x„) is convergent in (X, r ) Complete metric spaces and complete quasi-metric spaces are examples of (/-complete topological spaces. The (/-complete semi-metric spaces form an important class of examples of (/-complete topological spaces (see [5]). The basic idea of a (/-complete topological space goes back to Kasahara [11, 12), Iseki [9], and their /.-spaces.
Now, let S, 7" ; X -»• X be two mappings. The set
OsTi.r. oo) = [x, Tx, STx (ST)''x. T(ST)"x....]
H. Karayilan - M. Telci {><)
Department of Mathematics. Faculty of Science. Trakya University. 22030 Edune. Turkey e-mail mielci@irakyaedu,ir
H, Karayilan
e-mail: [email protected],tr
^ S p n i
H. Karayilan, M Telci is called ST"-orbit of x. A real-valued function G : X ^- [0, CXJ) is said to be ST-orbitaliy lower semicontinuous at z relative to x if and only if {x„] is a sequence in OST(X. oo) and limn_oo J^n — z implies Giz) < limn^-oo inf G(j:n). A function T : X —>• X is w- continuous a t x if jTn -» A: implies Tx,, -^ Tx asn -> DC
2 Main Results
In this section, we state and prove our main results. Now, we give the first main result as follows:
Theorem 1 Let (X, z) be a d-complete topological space and S. T be self-mappings ofX.
Suppose there exists an xg e X, such that
d(STy.Ty) < /((/(Ty. y)), (1) diTSy.Sy) < yidiSy.y)) (2) for all y e OST(XO, OO), where y : [0, oo) -> [0, DC) IS a nondecreasing function with
12'^^oY"it) < ooforallt > 0. Then,
(i) lim„_,co(S7')"jco = hm„_.co TiST)''xo = x' exists.
(ii) n ' = Sx' = x' ifand only if Fix) = d(Sx, x) or Gix) = diTx.x) is ST-orbitally lower semicontinuous at x' relative to XQ .
Note that function / satisfies yi^) = 0.
Proof (i) Consider the sequence {x„] defined by X2„ = Sx2„-i = iST)"xQ and X2„+] = TXIJ, — TiST)"xo forn = 1 , 2 , . . . , where JTJ = 7'jto.Then. using inequality (I), wehave
diX2n+2,X2„+l) = diSx2n+l,Tx2n) = diSTx2„,Tx2„)
< YidiTx2n,X2„)) = yidixi„+i,X2„)). (3) Similarly, using inequality (2), we have
dix2„+[,X2„) - diTx2„, Sx2„-]) = diTSx2„-], Sxin-i)
< yidiSx2„-uX2„-i)) ^ y(dix2„,X2„-i)). (4) Then, from inequalities (3) and (4), we obtain
dix„+i,x„) ^y(dix„,x„-i)) for w — 1, 2 and in general.
dix„+i.Xa)<Y"(d(xuxo)) (5) forn ^ 1 . 2 , . . .
Using inequality (5). we have
Sn = X l ' ' < ^ ' + ' ' ^ ' * = '^*-^''•^0*"'•'''•^2.J:l) + •••^-d(:^:„^.l,;c„) 1=0
< d(xi. xo) + yidixi. xo)) + • • • + y"(dixi, XQ))
(=0 >=l)
^ Springer
Common Fixed Pomt Theorems for Pairs of Mappings
Hence, {S„] is bounded above and also non-decreasing and so is convergent. Thus, we have ^ , ^ Q ( / ( J : , _ I _ I , J : J < oo. Since (,Y, r ) is (/-complete, there exists JT' e X, such that h m „ _ ^ J:„ = JC'. Therefore,
hm(ST)''xo= lim 7'(57')"J:O = JC'.
(ii) Assume that Tx' = Sx' = x' and {ji:„} is a sequence in Osrixo.oo) with linin-too ^n = x'. Then, we have
F(x')^d(Sx',x')=0< hm infdiSx„,x„)= lim infF(jv„) and
Gix') = diTx',x')=0< hm infdiTx„,Xn)= lim infG(jc„) and so F and G aie ST-orbitally lower semi continuous at x' relative to Jto-
Suppose that Gix) = diTx.x) is Sr-orbitally lower semicontinuous al x' relative to XQ. Now. {j>:2nl is a sequence in 057(^:0.00), such that lim„_ooJ":2n = x'. Then
x' e OsTixo. 00).
Since ^jl(,(/(jc,+i,.j:,) < 00, we get lim„^^(/(j:2n-i-i,->:2/,) = Oand so 0<d(Tx'.x') = G(x') < lim infG(jr2„) = lim infd(Txi„.X2n)
lim inf(/(;c2n-i-i,Jr2n) = 0.
Thus, diTx'. x') ^ 0 and so Tx' = x'.
Using inequality (1), we have
diSx', x') - diSTx'. Tx') < yidiTx', x')) ^ y(0) = 0.
Therefore, Sx' = x' = Tx'.
Similarly, considering the 57'-orbitally lower semicontinuity of F, it can be seen that 5
and T have a common fixed point and this completes the proof. D Note that Theorem 1 generalizes Corollary 2,3 given in [10].
Define y : [0, 00) -*• [0, 00) by yit) = kt where k < 1 in Theorem 1, Then, we obtain the following corollary for pair of mappings which is a generalization of CoroUary 1 given in [5],
Corollary 1 Let (X, r) be a d-coinplete topological space and S. T be self-mappings ofX.
Suppose there exists an XQ e X, such that
diSTy.Tv) < kdiTy.y).
diTSy. Sy) < kd(Sy. y) for all y € Osrixo. 00), where 0 < ft < 1, Then,
(i) lim„..*oo(S7")''jro = linin-^oo T{ST)"xo =x' exists.
(11) Tx' = Sx' =x' ifand only if Fix) = diSx,x)orGix) = diTx. x) is ST-orbitally lower semicontinuous at x' relative to XQ .
We now prove the following common fixed point theorem by using the to-continuily.
^ Springer
H. Karayilan, M Telci Theorem 2 Let (X. z) be a d-complete Hausdorff lopological space and S. T be self- mappings ofX satisfying the following inequalities:
diSTx,Tx) < (pidiTx.x)), (6) diTSx,Sx) < ^id(Sx.x)) (7) for all JC e X, where (p : [0, oo) -*• [0, oo) is a nondecreaUng function with ipiO) ^ 0,
Suppose that S or T is w-continuous. Then, S arui T have a common fixed point ifand only if there exists an X e X withYfT:^\'(^'idiSx.x)) < oo or''£f^^^lp''(diTx.x)) < oo.
Proof IS Sz ^ Tz = z, then diSz. z) = diTz. z) = 0. Since ^"(0) = 0, the following inequalities
Y^<p"idiSz,z)) <oo and J i ] ^ " ( ( / ( 7 ; , ;)) < oo n = l n = l are satisfied.
Conversely, suppose that there exists an JC in X with Yl'nL] <p''id(Sx,x)) < oo. Let X = XQ. Define the sequence [x„ ] inductively by
J:2„-|-| = Sxi„ and j:2n+2 = TX2„+1 f o r n = 0 , l , 2 , . . . .
Using inequality (7), we have
diX2„+2, X2n+\) = diTx2n-\-l,SX2„) = diTSx2„,Sx2n)
< <pid(Sxin.X2n)) = <pid(X2n+\,X2n)). (8) Similarly, using inequality (6), we have
diX2„ + ],X2„) = diSx2n,TX2n-l) ^ d(STx2n-],TX2n-l)
< <pidiTx2n-l,X2„-i))=^<PidiX2a,X2„-i)). (9) Then, from inequalities (8) and (9), we obtain
dix„+i,X„) < <pidiXa,X„-l)) and in general,
dix„+i,x„) < <p''id(x],xo)) =<p''idiSxa,xo)) for /I = 0, 1, 2, . , since <p is nondecreasing. It follows from assumption that
Y^d{x„+i,x„) <'^ip''idiSxo.xo)) < 0
^dix„+,,Xn) < 00.
Since (X, r ) is (/-complete, limn-,coJCn — z exists. If S is u;-continuous, then since (X, r ) is Hausdorff, we have
z = lim j:2n+i = lim Sxin = Sz Using inequality (7), we have
diTz, z) = d(TSz. Sz) < <pid(Sz, z)) = <p(0) = 0 which implies Tz ~ z and so ^ is a common fixed point of S and T
^ Springer
Common Fixed Pomi Theorems for Pairs of Mappings
Similarly, if T is uj-continuous, then since (X, i ) is Hausdorff, we have z = hm j:2n-r2 = lim TX2J,+\ = Tz.
Thus, using inequality (6). we have
diSz. z) = diSTz. Tz) < ipidiTz. z)) = ^{0) = 0 and s o ; = 5 ; — Tz.
If there exists an J: in X with X ! ^ i <p"id(Tx. x)) < oc. similarly considering the sequence |jc„ ] inductively by
X2n+\ = Tx2„ and X2„+i = Sxin^\
for n = 0, 1 , 2 , . . . . it can be proved that S and T have a common fixed point and this
completes the proof. D Note that putting 5 = T in Theorem 2, then we obtain Theorem 2 given in [5], Therefore,
Theorem 2 of [5] is a special case of Theorem 2.
We now illustrate our results by the following examples.
Example I Let X = R, p be a metric on X and let r be a metnc topology on X induced by /3. Define (/ : X x X -» [O,oo)hy dix.y) = |f^sn(-r-3')_i|_ where sgn is a function defined by sgn(x) = &^ if J: jl^ 0 and sgn(O) = 0. Since d does not satisiy the symmetry property, d IS neither a metric nor a semi-metric on X, Also, (X, r ) is a (/-complete topological space with d.
Now, define S, 7 : X ^ X by
o/ > n -r, -. [ 0 if J: < 1, 5(^) = 0, TO=ji , f ^ ^ j
Takex = 1/2. Then, we get OSTC^IXOO) ^ (1/2,0, 0 , 0 , . . . |, Thus, Sand T satisfy inequalities (1) and (2) for all y e 05^(1/2, oo), where / is any function as in Theorem I.
Also, F and G are Sr-orbitally lower semicontinuous at JC = 0 relative to XQ = 1/2. Thus, all the conditions of Theorem 1 are satisfied and ;c — 0 is a common fixed point of S and T.
To show that S and T satisfy the condition of Theorem 2, we consider the funciion ip : [0, oo) -^ [0, 00) defined hy ipix) = £ • ' - ! .
IfjT < 1, then wehave
diSTx. Tx) = diO, 0) = 0 < <pidiO, x)) = (pidiTx.x)).
diTSx, Sx) = diO. 0) = 0 < ^((/(O, x)) ^ <pidiSx. x)) If jr > I, then we have
diSTx,Tx)=diO.\) = | e ^ s " < - u _ i | ^ | ^ - i _ , |
< e''^'-''>-i=<pid(l.x)) = <p(d(Tx,x)), and
diTSx. S.x) = diO. 0) = 0 < <pid(0. X)) = ipidiSx.x)).
Also. S IS w-continuous and for .i: = 0
^(()"((/(S.v,Jc)) < oc and ^ip"(diTx.x)) <.-x.
,1-1 n=l Thus, all the coiKliiions of Theorem 2 are satisfied
^ Springer
H. Karayilan. M. Telci
dix, y) .
Example 2 hstX = [i/n : n = 1. 2 . . . . | , p be a metric on X and let T be a metric topology on X induced by p Define d : X x X -> [0, oo) by
1 ifj: = ^ . y = ; 7 q : Y o r j : = ^ , y — ^ ,
0 ifji: = y = i ,
2" if JC = ^ , y — ^ where n ^ m,m ^ n-i-1 and n ^ m-\- I.
Then d is neither a metric nor a quasi-metric on X. Also d is not a semi-metric, as well.
Because d does not satisfy the symmetry property and triangle inequality.
Though (X, T ) is a (/-complete topological space with d, (X. z) is not complete in according to r .
Now. define S, T : X ^ X by
f 1 ifjc= 1, I 1 if:c = l.
^ < ^ > - i ^ i f j c = l . « ^ l , ^f-^> = { j ^ . f . = l . „ / l .
Takejc = 1/2. Then, weget 6»57-(l/2.oo) = { 1 / 2 , 1 . 1 , I , . . . } . Thus, S and r satisfy inequalities (1) and (2) for all y e OsT(i/2, oo), where y is any funciion as in Theorem 1. Also, F and G are 5r-orbitaIly lower semicontinuous al J: = 1 relative to XQ = 1/2 Therefore, all the conditions of Theorem I are satisfied, and J: = 1 is a common fixed point ofSandT",We now show that S and T satisfy the condition of Theorem 2. Consider the function ip : (0, oo) ->• [0, oo) defined by ^o(jc) — e ' - I as in Example 1 Then, we have
d(STx, Tx) = dil. 1) = 0 < ipidil.x)) ^ ipidiTx.x)) forjf = l,;i: ^ 1/2, and
diSTx, Tx) = d(\ln, l/(« - 1)) = 1
< e - 1 = <p(\) = <pidi\lin - 1), l / « ) ) = ipidiTx, x)) for JT = l/fi,n — 3,4 Also, forjc = I, we have
(/(TSJC, SJC) ^ (/(I, 1) = 0 ^ P((/(SJC, ;c)) and for JC = l / n , « = 2, 3 we have
diTSx, Sx) = (/(l/n, l/(rt -1- D ) = I
< e - 1 = ^(1) = ipiddlin + 1), \ln)) = .pidiSx. JC)).
Further, S and T are m-continuous and
Y^ip"id{Sx.x)) < o o and Y^'p"idiTx,x)) < oo for J: = 1. Thus, all the conditions of Theorem 2 are satisfied.
Acknowledgements The auihors would like to thank the referees for their valuable suggestions
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