On Parametric S-Metric Spaces and Fixed-Point Theorems for R-Weakly Commuting Mappings

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,)XXXbe a function. Then S is called an S-metric on X if, (S1)(,,)0Sabcif 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,)PXXbe a function. P is called a parametric metric on X if, (P1) (,,)0Pabtif and only if ,ab (P2) (,,)(,,),PabtPbat (P3) (,,)(,,)(,,),PabtPaxtPxbtfor each ,,abxXand all t> 0. The pair of (,)XPis called a parametric metric space. Definition 1.3 [3]Let(,)XPbe a parametric metric space andletnabe a sequence in X : (1) naconverges to xif and only if there exist 0nℕ such that (,,)nPaxt< ε, for all n0nand 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,)SPXXXbe a function.SPis called a parametric S-metric on X if, (PS1) (,,,)0SPabctif and only if ,abc (PS2)

(,,,)SPabct

(,,,)(,,,)(,,,),SSSPaaxtPbbxtPccxtfor 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 0nℕ 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: Letlimnnaxand limnnaywith .xyThen there exists 12,nnℕ such that

(,,,)SnnPaayt<2



, For each ε> 0, all t> 0, and n>12,.nnif we take 012max,nnnthen using (PS2) and lemma 1.1 , we get (,,,)2(,,,)(,,,)SSnSnPxxytPxxatPyyat

<,22



For each0nn. Therefore (,,,)0SPxxytand .xy

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 ,aXthere 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 ,abXand 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,nnaabain (iii), we have 1(,,,)SnnnPfafafatq1(,,,)SnnnPgagagat= q11(,,,)SnnnPfafafat Continuing in the same way, we get 1(,,,)nSnnnPfafafatq001(,,,)SPfafafat. This implies, 1001(,,,)(,,,)nSnnnSPbbbtqPbbbt. Therefore, for all ,,nmNn

= (1)

q001(,,,)0SPbbbtas n. Thus nb is a SP- Cauchy sequence in X. since (,)SXPis complete S-metric space, therefore, there exists a point cXsuch that limnnb= limnnga= limnnfa=c.Let us suppose that the mapping fis continuous. Therefore limnnfga = limnnffa= fc. Since fand g are R-weakly commuting, where R > 0. On lettingnwe get limnngfa = limnnfga = fc. Now we prove that cfc. Suppose cfcthen (,,,)SPcfcfct>0 On setting ,nnaabfain (iii), we have (,,,)SnnnPfafaffatq(,,,)SnnnPgagagfat Letting limit as nwe get (,,,)SPccfctq(,,,)SPccfct<(,,,)SPccfct. Is a contradiction. Therefore cfcsince ()()fXgXwe can find 1cXsuch that1cfcgc. Now put 1,nafabccin (iii), we have 1(,,,)SnnPffaffafctq1(,,,)SnnPgfagfagct. Taking limit as nwe get

(,,,)0SPfcfcfct, Which implies that 1fcfci.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 ()dcbe 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 XRand let the function :(0,)[0,)SPXXXbe defined by (,,,)()(SPabctgtabbcacfor each ,,abcRand all t>0, where :(0,)(0,)gis 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

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.

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

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 ,abXand 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 limnnfgafcor 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 non 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,fcffcgfcandfcis a common fixed point of fand.g Next suppose that limnnfgafc. Then()()fXgX implies fcgufor some uXand therefore lim.nnfgagu Compatibility of fandgimplies,limnngfagu. By virtue of (3.1), we have 1limlim.nnnnfgaffagu Using (ii), we get (,,,)SnPfufuffatq(,,,)SnPgugugfat. Letting non both sides, we have This gives fugu. Compatibility of fand gyields fgugguffugfu. Finally we claim that .fuffusuppose that fuffu, then by using (ii) we obtain, (,,,)SPfufuffutq(,,,)SPgugugfut (,,,)SPfufuffutq(,,,)SPfufuffut which again gives a contradiction, since [0,1)q. Hence .fuffugfuTherefore 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 limnnfgafcor lim.nngfagcLet us first assume that lim.nngfagcThen R-weakly commutativity of type ()gAof fand gyields (,,,)SnnnPffaffagfatR(,,,)SnnnPfafagatwhere R>0. Letting non both sides, we have (lim,,,)SnnnPffaffagctR(,,,)0.SPccct This gives lim.nnffagcAlso using (ii) we get (,,,)SnPfcfcffatq(,,,)SnPgcgcgfat. Letting non both sides we have (,,,)SPfcfcgctq(,,,)0SPgcgcgct. Hence we get fcgc. Again by using R-weakly commutativity of type ()gA, (,,,)SPffcffcgfctR(,,,)0SPfcfcgct that.fcffcSuppose fcffc Using (ii) we get (,,,)SPfcfcffctq(,,,)SPgcgcgfct (,,,)SPfcfcffctq(,,,)SPfcfcffct A contradiction. Hence fcffcgfcand 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.nngfagcLet us assume that lim.nngfagcsince pair (,)fgR-weakly commuting of type (P), we have (,,,)SnnnPffaffaggatR(,,,)SnnnPfafagat where R>0. Letting non both sides we get,

(lim,lim,lim,)SnnnnnnPffaffaggatR

(,,,)0Psccct. This gives (lim,lim,lim,)0SnnnnnnPffaffaggat. Using (i) and (3.1) we have 1nngfaggagc as nthis gives, nffagcn. Also, by using (ii) we have (,,,)SnPfcfcffatq(,,,)SnPgcgcgfat Letting non both sides we get This implies that.fcgcAgain by using R-weakly commutativity of type (P), (,,,)SPffcffcggctR(,,,)0SPfcfcgctwhere R>0 . This yields .fcggcTherefore.ffcfgcgfcggc Lastly, we claim that .fcffc Suppose that fcffc. Using (ii) we have (,,,)SPfcfcffctq(,,,)SPgcgcgfct (,,,)SPfcfcffctq(,,,)SPfcfcffct, Which is a contradiction. Therefore .fcffc Hence fcffcgfcand fcis a common fixed point of fandg. This results holds good even if limnnfgafcis 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 : XXby and gait 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 2naor

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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Department of Mathematics, Desh Bhagat University, Mandi, Gobindgarh