On Parametric S-Metric Spaces and Fixed-Point Theorems for R-Weakly Commuting Mappings
by Rajvir Kaur*, Tejwant Singh, Saurabh Manro,
- Published in Journal of Advances and Scholarly Researches in Allied Education, E-ISSN: 2230-7540
Volume 14, Issue No. 2, Jan 2018, Pages 1391 - 1397 (7)
Published by: Ignited Minds Journals
ABSTRACT
In this paper, we prove some common fixed point theorems for variants of R-weakly commuting mappings, R-weakly commuting of type in parametric S-metric spaces.
KEYWORD
parametric S-metric spaces, fixed-point theorems, R-weakly commuting mappings, common fixed point theorems, variants
1. INTRODUCTION
In 1922 Stefan Banach proved a common fixed point theorem, which ensures the existence and uniqueness of a fixed point under appropriate conditions. This result of Banach is known as Banach fixed point theorem or contraction mapping principle.
These contractive conditions have been used in various fixed-point theorems for some generalized metric spaces. Recently, the notion of an S-metric has been studied by some mathematicians. This notion was introduced by Sedghi et al. in 2012 [5] as follows, Definition 1.1[5]Let X be a non-empty set and let S: [0,)XXXbe a function. Then S is called an S-metric on X if, (S1)(,,)0Sabcif an only if abc, (S2)(,,)(,,)(,,)(,,)SabcSaaxSbbxSccx, for each ,,,.abcxX The pair of (,)XSis called an S-metric space. Definition 1.2 [3]Let X be a non-empty set and let :(0,)[0,)PXXbe a function. P is called a parametric metric on X if, (P1) (,,)0Pabtif and only if ,ab (P2) (,,)(,,),PabtPbat (P3) (,,)(,,)(,,),PabtPaxtPxbtfor each ,,abxXand all t> 0. The pair of (,)XPis called a parametric metric space. Definition 1.3 [3]Let(,)XPbe a parametric metric space andletnabe a sequence in X : (1) naconverges to xif and only if there exist 0nℕ such that (,,)nPaxt< ε, for all n0nand all t> 0; that is,lim(,,)0.nnPaxt (2) nais called a Cauchy sequence if, for all t> 0, ,lim(,,)0.nmnmPaat (3) (,)XPis called complete if every Cauchy sequence is convergent. Definition 1.4Let X be a non- empty set and let :(0,)[0,)SPXXXbe a function.SPis called a parametric S-metric on X if, (PS1) (,,,)0SPabctif and only if ,abc (PS2)
(,,,)SPabct
(,,,)(,,,)(,,,),SSSPaaxtPbbxtPccxtfor each Lemma 1.1Let (,)SXPbe a parametric S-metric space.then we have (,,,)(,,,),SSPaabtPbbat for each ,abX and all t> 0. Proof: using (PS2), we obtain, (,,,)2(,,,)(,,,)SSSPaabtPaabtPbbat =(,,,),SPbbat (,,,)2(,,,)(,,,)SSSPbbatPbbbtPaabt =(,,,).SPaabt From the above inequalities, we have (,,,)(,,,).SSPaabtPbbat Definition 1.5Let (,)SXPbe a parametric S-metric space and let nabe a sequence in X: (1) naconverges to xif and only if there exists 0nℕ such that (,,,)SnnPaaxt<ε,for all t>0; that is lim(,,,)0.SnnnPaaxt It is denoted by lim.nnax (2) nais called a Cauchy sequence if, for all t>0, ,lim(,,,)0SnnmnmPaaat (3) (,)SXPis called complete if every Cauchy sequence is convergent. Lemma 1.2Let (,)SXPbe a parametric S-metric space. If naconverges to x, then xis unique. Proof: Letlimnnaxand limnnaywith .xyThen there exists 12,nnℕ such that
(,,,)SnnPaayt<2
, For each ε> 0, all t> 0, and n>12,.nnif we take 012max,nnnthen using (PS2) and lemma 1.1 , we get (,,,)2(,,,)(,,,)SSnSnPxxytPxxatPyyat
<,22
For each0nn. Therefore (,,,)0SPxxytand .xy
2. FIXED POINT RESULT FOR R- WEAKLY COMMUTING MAPPINGS
In this section, we prove some fixed point results for R-weakly commuting mappings in parametric S-metric space. Definition 2.1A pair (,)fgof self-mappings of a parametric S-metric space (,)SXPis said to be R- weakly commuting at a point aX if (,,,)SPfgafgagfatR(,,,)SPfafagatfor some R> 0. Definition 2.2 A pair (,)fgof self-mappings of a parametric S-metric space (,)SXP is said to be point wise R-weakly commuting on X if given ,aXthere exists R >0 such that (,,,)SPfgafgagfatR(,,,)SPfafagat. Theorem 2.1 Let (,)SXPbe a complete parametric S-metric space and let fand gbe R-weakly commuting self-mapping ofXsatisfying the following conditions: (i) ()()fXgX; (ii) for gis continuous;
every ,abXand 0q<1. Then fandghave a unique fixed point in X. Proof: Let 0a be an arbitrary point in X. By (i) one can choose a point 1aX such that 01faga. In general, choose 1na such that 1nnnbfaga. Now we show that nbis aSP- Cauchy sequence in X. For proving this we take 1,nnaabain (iii), we have 1(,,,)SnnnPfafafatq1(,,,)SnnnPgagagat= q11(,,,)SnnnPfafafat Continuing in the same way, we get 1(,,,)nSnnnPfafafatq001(,,,)SPfafafat. This implies, 1001(,,,)(,,,)nSnnnSPbbbtqPbbbt. Therefore, for all ,,nmNn
= (1)
nq
q001(,,,)0SPbbbtas n. Thus nb is a SP- Cauchy sequence in X. since (,)SXPis complete S-metric space, therefore, there exists a point cXsuch that limnnb= limnnga= limnnfa=c.Let us suppose that the mapping fis continuous. Therefore limnnfga = limnnffa= fc. Since fand g are R-weakly commuting, where R > 0. On lettingnwe get limnngfa = limnnfga = fc. Now we prove that cfc. Suppose cfcthen (,,,)SPcfcfct>0 On setting ,nnaabfain (iii), we have (,,,)SnnnPfafaffatq(,,,)SnnnPgagagfat Letting limit as nwe get (,,,)SPccfctq(,,,)SPccfct<(,,,)SPccfct. Is a contradiction. Therefore cfcsince ()()fXgXwe can find 1cXsuch that1cfcgc. Now put 1,nafabccin (iii), we have 1(,,,)SnnPffaffafctq1(,,,)SnnPgfagfagct. Taking limit as nwe get
1(,,,)SPfcfcfctq1(,,,)SPfcfcgct=q
(,,,)0SPfcfcfct, Which implies that 1fcfci.e., 11cfcfcgc. Also by using the definition og R-weakly commutativity,
111(,,,)(,,,)SSPfcfcgctPfgcfgcgfctR
111(,,,)0SPfcfcgct, Implies that fcgcc. Thus cis a common fixed point of fand g. Uniqueness: assume that ()dcbe another common fixed point of fand g. Then (,,,)SPccdt>0 and q(,,,)SPccdt<(,,,)SPccdt. Which is a contradiction. Therefore cd. Hence uniqueness follows. Example 2.1 Let XRand let the function :(0,)[0,)SPXXXbe defined by (,,,)()(SPabctgtabbcacfor each ,,abcRand all t>0, where :(0,)(0,)gis a continuous function. Then SPis a parametric S-metric and the pair (,)SRPis a parametric S-metric space. Define self-mappings f and g on X by fxx and 21gxx, for all xX.Here we note that, (i) ()()fXgX; (ii) f is continuous on X ; (iii) (,,,)SPfafbfctq(,,,)SPgagbgct, holds for all ,,abcX,
1
2q<1.
Further, the mappings f and g are R-weakly commuting. Thus, all conditions of the theorem 2.1 are satisfied and x = 1 is the unique common fixed point of f and g.
3. FIXED POINT RESULT FOR VARIANTS OF R-WEAKLY COMMUTING MAPPINGS BY USING WEAK
Reciprocal continuity In this section, we prove some fixed point results for R-weakly commuting mappings by using weak reciprocal continuity in parametric S-metric space. Definition 3.1Apair(,)fgof self-mappings of a parametric S-metric space (,)SXPis said to be (i) R-weakly commuting of type()gAif there exists some R>0 such that(,,,)SPffaffagfatR(,,,)SPfafagat, for all .aX (ii) R-weakly commuting of type ()fAif there exists some R>0 such
S, for all .aX
Definition 3.2 A pair (,)fgof self-mappings of a parametric S-metric space (,)SXPis said to be R-weakly commuting of type (P) if there exists some R>0 such that (,,,)SPffaffaggatR(,,,)SPfafagat, for all .aX Theorem 3.1letfandgbe weakly reciprocally continuous self-mappings of a complete parametric S-metric space (,)SXPsatisfying the following conditions: (i) ()()fXgX; (ii) (,,,)SPfafafbtq(,,,)SPgagagbt, for every ,abXand 0q<1. If fandgare either compatible or R-weakly commuting of type ()gAor R-weakly commuting type ()fAor R-weakly commuting of type (P), then fandghave a unique common fixed point. Proof. Let 0abe an arbitrary point in X. As ()()fXgX, therefore there exists a sequence of points nasuch that 1.nnfaga Define a sequence nbin X by 1nnnbfaga. (3.1) Sequence nbis SP- Cauchy sequence in X . (the proof follows the same lines as in Theorem 2.1). Now since (,)SXPis complete parametric S-metric space, therefore there exists a point cX such that limnnbc. Hence limlimlimnnnnnnbfagac. Suppose that fandgare compatible mappings. Weak reciprocal continuity of fandgimplies that limnnfgafcor limnngfagc. Let
then the compatibility ofandgives: lim(,,,)0SnnnnPfgafgagfat i.e., (lim,,,)0SnnnPfgafgagct. Hence, limnnfgagc. By (2.1) we have 1limlimnnnnfgaffagc. Therefore by the use of (ii) we get (,,,)SnPfcfcffatq(,,,)SnPgcgcgfat. Letting non both sides we have (,,,)SPfcfcgctq(,,,)0SPgcgcgct This gives fcgc. Again compatibility of fand gimplies commutativity at a coincidence point. Hence gfcfgcffcggc. Now, we claim that fcffc. Suppose fcffc, then by using (ii) we obtain, (,,,)SPfcfcffctq(,,,)SPgcgcgfct (,,,)SPfcfcffctq(,,,)SPfcfcffct, Which is a contradiction Since [0,1)q. Hence,fcffcgfcandfcis a common fixed point of fand.g Next suppose that limnnfgafc. Then()()fXgX implies fcgufor some uXand therefore lim.nnfgagu Compatibility of fandgimplies,limnngfagu. By virtue of (3.1), we have 1limlim.nnnnfgaffagu Using (ii), we get (,,,)SnPfufuffatq(,,,)SnPgugugfat. Letting non both sides, we have This gives fugu. Compatibility of fand gyields fgugguffugfu. Finally we claim that .fuffusuppose that fuffu, then by using (ii) we obtain, (,,,)SPfufuffutq(,,,)SPgugugfut (,,,)SPfufuffutq(,,,)SPfufuffut which again gives a contradiction, since [0,1)q. Hence .fuffugfuTherefore fuis a common fixed point of fand g. Now suppose that fand gare R-weakly commuting type of ()gA. Now, weak reciprocal continuity of fand g implies that limnnfgafcor lim.nngfagcLet us first assume that lim.nngfagcThen R-weakly commutativity of type ()gAof fand gyields (,,,)SnnnPffaffagfatR(,,,)SnnnPfafagatwhere R>0. Letting non both sides, we have (lim,,,)SnnnPffaffagctR(,,,)0.SPccct This gives lim.nnffagcAlso using (ii) we get (,,,)SnPfcfcffatq(,,,)SnPgcgcgfat. Letting non both sides we have (,,,)SPfcfcgctq(,,,)0SPgcgcgct. Hence we get fcgc. Again by using R-weakly commutativity of type ()gA, (,,,)SPffcffcgfctR(,,,)0SPfcfcgct that.fcffcSuppose fcffc Using (ii) we get (,,,)SPfcfcffctq(,,,)SPgcgcgfct (,,,)SPfcfcffctq(,,,)SPfcfcffct A contradiction. Hence fcffcgfcand fcis common fixed point of fand g. Similarly, we an prove if lim.nnfgafc On the other hand if f and gare R-weakly commuting mappings of type ()fA, then by following the similar steps as presented above, it can easily be proved that fcis a common fixed point of fand g. Finally now, suppose thatfand gare R-weakly commuting of type (P). Weak reciprocal continuity of fand gimplies that limnnfgafc or lim.nngfagcLet us assume that lim.nngfagcsince pair (,)fgR-weakly commuting of type (P), we have (,,,)SnnnPffaffaggatR(,,,)SnnnPfafagat where R>0. Letting non both sides we get,
(lim,lim,lim,)SnnnnnnPffaffaggatR
(,,,)0Psccct. This gives (lim,lim,lim,)0SnnnnnnPffaffaggat. Using (i) and (3.1) we have 1nngfaggagc as nthis gives, nffagcn. Also, by using (ii) we have (,,,)SnPfcfcffatq(,,,)SnPgcgcgfat Letting non both sides we get This implies that.fcgcAgain by using R-weakly commutativity of type (P), (,,,)SPffcffcggctR(,,,)0SPfcfcgctwhere R>0 . This yields .fcggcTherefore.ffcfgcgfcggc Lastly, we claim that .fcffc Suppose that fcffc. Using (ii) we have (,,,)SPfcfcffctq(,,,)SPgcgcgfct (,,,)SPfcfcffctq(,,,)SPfcfcffct, Which is a contradiction. Therefore .fcffc Hence fcffcgfcand fcis a common fixed point of fandg. This results holds good even if limnnfgafcis considered instead of lim.nngfagc Uniqueness of the common fixed point in each of the three types of mappings can easily be obtained by using (ii). The following example shows the validity of theorem 3.1. Example 3.1 Let(,)SXPbe S-metric space, where X = [2,20] and for all ,,abcX(SPabbcac). Define f, g : XXby and gait can be easily verified that (i) ()()fXgX; (ii) f and g satisfies condition (ii) of theorem 3.1;
gA; (iv) f and g are weakly reciprocally continuous for sequences 2naor
15nan
for each in X. Thus, f and g satisfy all the conditions of theorem 3.1 and have a unique common fixed point at x = 2.
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Corresponding Author Rajvir Kaur*
Department of Mathematics, Desh Bhagat University, Mandi, Gobindgarh