On coincidence point results of a pair of rational type contractions in quasi-symmetric spaces

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Soliman, Cogent Mathematic (2017, 4: 1301181 PURE MATHEMATICS RESEARCH ARTICLE On coincidence point reult of a pair of rational type contraction in quai-ymmetric pace Ahmed H Soliman 1 * Received: 25

1 Soliman, Cogent Mathematic (2017, 4: PURE MATHEMATICS RESEARCH ARTICLE On coincidence point reult of a pair of rational type contraction in quai-ymmetric pace Ahmed H Soliman 1 * Received: 25 September 2016 Accepted: 21 February 2017 Firt Publihed: 03 March 2017 *Correponding author: Ahmed H Soliman, Faculty of Science, Department of Mathematic, Al-Azhar Univerity, Aiut 71524, Egypt aholimanm@gmailcom Reviewing editor: Angelamaria Cardone, Univerita degli Studi di Salerno, Italy Additional information i available at the end of the article Abtract: In thi work, we etablih ome coincidence and common point theorem for a pair of rational type contraction in quai-ymmetric pace Subject: Science; Mathematic & Statitic; Advanced Mathematic; Analyi - Mathematic; Functional Analyi Keyword: ymmetric pace; quai-ymmetric pace; generalized metric pace; rational contraction mapping; fixed point AMS ubject claification: 47H09; 47H10; 47H20; 46T99 1 Introduction and preliminarie It i ignificant that the prominent contraction principle i a remarkable outcome in fixed point area which ha been utilized broadly in numerou field In cutting edge year many author have talked about variou idea of generalized metric pace, quai metric, dilocated metric, dilocated quai metric, ymmetric pace and quai-ymmetric pace (ee, Hitzler, 2001; Kumari, 2012; Sarma & Kumari, 2012; Schroeder, 1999; Shen & Lu, 2010 in different way It waot immediately oberved that uch pace may fail to atify propertie of metric pace uch a a unique limit of convergence equence, every convergent equence i a Cauchy equence and other thing Hence, in ome of lat paper, the author implicitly ued ome of propertie of metric pace, o that their reult were inaccurate One of generalized metric pace which will be under conideration in thi paper i quai-ymmetric pace It i worth to mention that the notion of quaiymmetric pace and their topological apect are introduced by Sumati Kumari, Ramana, and Zoto (2014 ABOUT THE AUTHOR Ahmed H Soliman received the BSc degree in Mathematic from Faculty of Science, Al-Azhar Univerity at Auit, Egypt, in 1996, the MSc degree in pure Mathematic from Auit univerity at Auit, Egypt, in 2002 and PhD degree in pure Mathematic from Al-Azhar Univerity at Auit, Egypt, in 2006 Currently, he i an aociate profeor, King Khalid Univerity Faculty of Science, Department of Mathematic Abha, Saudi Arabia He ha upervied everal PhD and mater tudent in different area of pure mathematic and ha publihed a lot of article in reputed international journal of mathematical cience Epecially he ha publihed many in metric pace and it generalization Hi reearch interet are functional analyi, fixed point theory, applied mathematic and harmonic analyi He i referee of mathematical journal PUBLIC INTEREST STATEMENT The theory of fixed point i one of the extreme important tool to the tudy of nonlinear phenomena Fixed point theory ha been applied in uch divere field a Differential Equation, Topology, Functional Analyi, Economic, Biology, Phyic, Engineering, Game Theory, Chemitry, Dynamic and Optimal Control Many of the mot important nonlinear problem of applied Mathematic minimize to olving a given equation which in turn may be reduced to finding the fixed point in certain pace Furthermore, metric pace and contractive conditionaturally arie for everal of thee problem we invetigate the fixed point of a new generalization of metric pace called quai-ymmetric pace in order to extend the theory of fixed point due to Banach and introduce a new fixed point reult in the pace of quai-ymmetric pace 2017 The Author( Thi open acce article i ditributed under a Creative Common Attribution (CC-BY 40 licene Page 1 of 9

2 Soliman, Cogent Mathematic (2017, 4: In thi paper, we introduce coincidence point theorem for two contraction elf-mapping of rational type in quai-ymmetric pace Our reult generalize the reult due to Almeida, Roldan- Lopez-de-Hierro and Sadarangani (2015 Next, we preent ome preliminarie and notation related to ymmetric pace and rational type contraction Definition 11 (Hick & Rhoade, 1999 Suppoe that X be a nonempty et and S:X X [0, be a ditance function uch that: (i S(x, y =0 x = y (ii S(x, y =S(y, x, for all x, y X We mean by a pair (X, S with a ymmetric pace Downloaded by [ ] at 17:20 09 January 2018 Definition 12 (Sumati Kumari et al, 2014 Suppoe that X be a nonempty et and :X X [0, be a ditance function uch that: (x, y =0 x = y for all x, y X We mean by a pair (X, with a quai-ymmetric pace Example 11 (Sumati Kumari et al, 2014 For any nonempty et X and any :X X [0, uch that q (x, x =0, x X, (x, y 0, x, y X, x y Then i quai-ymmetric on X Example 12 (Sumati Kumari et al, 2014 (x, y = x + y 2, for all x, y R (the et of all real number Then (R, i quai-ymmetric but not ymmetric Definition 13 (Sumati Kumari et al, 2014 Let (X, be a quai-ymmetric pace (1 A equence in X converge to a point x X if lim q (, x =lim q (x, =0 (2 A equence in X i -Cauchy equence if lim q (, +r =lim q (+r, =0, r N (the et of all natural number (3 (X, i -complete if for every -Cauchy equence, there exit x in X with lim q (x, x =lim q (x, x =0 (4 A:X X i -continuou if lim q (, x =lim q (x, =0 implie lim q (A, Ax = lim q (Ax, Ax =0 We need the following propertie in a quai-ymmetric pace (X, (W (Wilon, 1931 Given x 3 n, y and x in X, lim q (x, x =lim q (x, x =0 and lim q (x, y =lim q (y, x =0 imply that x = y (W (Wilon, 1931 Given x, y 4 n n and x in X, lim =0 imply that lim lim q (x, y =lim q (y, x n n (y n q (x, x =lim q (x, x =0 and, x =lim (x, y n =0 Page 2 of 9

3 Soliman, Cogent Mathematic (2017, 4: (1C (Imdad, Chauhan, Soliman, & Ahmed, 2014 A function q i 1-continuou if lim q (x, x =lim q (x, x =0 lim q (x, y =lim q (y, x =q (x, y =q (y, x Definition 14 (Roa & Vetro, 2014 Let A, B:X X and β:x X [0, The mapping A i B β-admiible if, for all x, y X uch that β(bx, By > 1, we have β(ax, Ay > 1 If B i the identity mapping, then A i called β-admiible Definition 15 (Roa & Vetro, 2014 Let (X, be a ymmetric pace and β:x X [0, X i β regular if, for each equence in X uch that β(, +1 > 1 for all n N and lim x n = x, then there exit a ubequence k of uch that β(k, x > 1 k N Definition 16 (Roa & Vetro, 2014 Let f 1 :X X two mapping defined on a nonempty et X We ay that f 1 weakly compatible if they commute at their coincidence point (ie f 1 f 2 x = f 2 f 1 x whenever f 1 x = f 2 x A point u X i called point of coincidence of f 1 if there exit a point a X uch that u = f 1 a = f 2 a Downloaded by [ ] at 17:20 09 January Main reult In thi ection we introduce ome coincidence point reult for two rational contraction elf-mapping on ymmetric pace Property (GW 4 Suppoe that (X, q be a ymmetric pace If x, y, z n n n three equence in X, uch that lim q (x, y =lim q (y, x n n =0 and lim q (y, z =lim q (z, y =0, n n then lim q (x, z =lim q (z, x =0 n n Lemma 21 (GW 4 (W 4 Proof Put y n = x in (GW Then the proof i obtained 4 Theorem 21 Suppoe that (X, be a ymmetric pace atify (GW and 4 (1C Let f and f 1 2 be elfmapping on X uch that f 1 X f 2 X Suppoe that X, i a -complete quai-ymmetric pace and the following condition hold: x y φ(m(x, y + C min x x, y y, x y, y x x, y X, C 0, where M(x, y i defined by M(x, y =max x y, q (f x, f x(q (f y, f y , q (f y, f y(q (f x, f x x y x y and φ: [0, [0, be a continuou, nondecreaing function and lim φ n (t =0 t > 0 Then f 1 have a unique point of coincidence in X Moreover if f 1 are weakly compatible, then f 1 have a unique common fixed point (1 Proof Let X be an arbitrary point Define the equence and z n in X defined by z n = f 2 +1 = f 1 Page 3 of 9

4 Soliman, Cogent Mathematic (2017, 4: If z n = z n+1, then z n+1 i a point of coincidence of f 1 Conequently, we can uppoe that z n z n+1 for all n N Now, by (1, we have +1 φ(m(, +1 + C min, +1 +1, +1, +1 = φ(m(, +1 (2 Downloaded by [ ] at 17:20 09 January 2018 where M(, +1 =max +1, q (f x, f x (q (f x, f x +1 2 n 1 n 2 n+1 1 n+1, ( = max 1, q (z, z (1 + q (z, z 1 n n+1, q 1 +1, we conider the following cae If M(, +1 = 1 from (2 we have +1 φ( 1 < 1 If M(, +1 = q (z, z (1 + q (z, z 1 n n+1 from (2 we obtain 1 ( q 1 ( φ 1 Hence +1 < 1, that i (4 hold Similarly, one can get that +1 < 1, < q (z, z (1 + q (z, z 1 n n+1 1 (3 (4 If M(, +1 = +1 from (2 we get +1 < +1, and imilarly one can get that +1 < +1, which i impoible In any cae, we proved that (4 hold Since +1 i decreaing and bounded from below Hence, it converge to a nonnegative number, c 0 If c > 0, then letting n + in (2, we deduce ( c(1 + c c φ max c, 1 + c, c = φ(c < c, which implie that c = 0, that i Page 4 of 9

5 Soliman, Cogent Mathematic (2017, 4: lim q (z, z =0 n+1 Similarly, one can have that lim q (z, z =0 +1 n Similarly of (5 one can deduce that lim q (z, z =0 +1 n+2 From (4, (5 and by uing (GW we have 4 lim q (z, z =0 n+2 For any integer number r and by uing (GW we obtain that 4 lim q (z, z =lim q (z, z =0, n+r +r n Downloaded by [ ] at 17:20 09 January 2018 which implie that z n i -Cauchy equence Since X, i -complete, there exit z f 2 X uch that lim z n = z Let u X be uch that f 2 u = z, applying (1 and uing the (1C we get u=lim u =lim u where lim [φ(m(u, + C min, u u, u, u ] = lim [φ(m(u, + C min 1, u u, 1 u, (z ] = lim [φ(m(u, ] < u, M(u, =max, q (f u, f u(q (f x, f x n 1 n, q (f x, f x (q (f u, f u+1 2 n 1 n 2 1 = max (z 1, q (f u, f u(q (z, z n, q (z, z (q (f u, f u+1 1 n 2 1 u 1 u 1 = u u a From (5 and it imilarity we obtain that u u= u=0, that i, z = f 2 u = f 1 u and o z i a coincidence point for f 1 (5 Now, we prove that z i the unique point of coincidence of f 1 Let x and y be arbitrary point of coincidence of f 1 uch that x = f 1 u = f 2 u and y = f 1 v = f 2 v Uing the condition (1, it follow that ( (x, y = u v φ max v, q (f u, f u(q (f v, f v , q (f v, f v(q (f u, f u v v + C min v v, u u, v u, u v imilarly one can obtain that (y, x =0, = φ( v < v= (x, y, which implie that x = y and f 1 have a unique point of coincidence Next, we prove that z = f 1 z = f 2 z If z i the coincidence point of f 1 a f 1 weakly compatible, we obtain that f 1 z = f 1 f 1 u = f 1 f 2 u = f 2 f 1 u = f 2 z and o z = f 1 z = f 2 z Conequently, z i unique common fixed point of f 1 Page 5 of 9

6 Soliman, Cogent Mathematic (2017, 4: Corollary 21 Replacing the condition (1 in Theorem 21 with the following condition: x y a 1 x y+a 2 x x( y y+1 x y + C min x x, y y, x y, y x, + a 3 y y( x x+1 x y (6 where a 1, a 2, a 3, C 0, and a 1 + a 2 + a 3 < 1 Then f 1 have a unique point of coincidence in X Moreover if f 1 are weakly compatible, then f 1 have a unique common fixed point Corollary 22 Putting f 2 = I (the identity mapping in Theorem 21 Then one can get a unique fixed point of f 1 Remark 21 (Almeida et al, 2015, Theorem 7 I pecial cae of Theorem 21 Downloaded by [ ] at 17:20 09 January 2018 Next, we introduce ome coincidence point theorem for two (α, ψ, φ-contraction elf-mapping of rational type in -complete quai-ymmetric pace Theorem 22 Let (X, be a quai-ymmetric pace atify (W and 4 (1C Let f 2 :X X be two elf-mapping atify the following condition: φ(β x y x y φ(m(x, y ψ(m(x, y x, y X, where M(x, y a in Theorem 21 X f 2 X, and X, i a -complete Conider alo that the next condition hold: (i X uch that β 1, (ii f 1 i f 2 -β-admiible, (iii X i β-regular and β(x m, 1, m, n N, m n, (iv either β x y 1 or β y x 1 whenever f 2 x = f 1 x y = f 1 y, (iiv φ:[0, [0, be a continuou, nondecreaing and φ(t =0 t = 0, and ψ:[0, [0, be a lower emi-continuou function and ψ(t = 0 t = 0 Then f 1 have a unique point of coincidence in X Moreover if f 1 are weakly compatible, then f 1 have a unique common fixed point (7 Proof Suppoe that X, β 1 Define z n and be two equence in X uch that z n = f 2 +1 = f 1, n = 0, 1, 2, 3, If z n = z n+1, then f 2 +1 = f 1 +1 which implie that +1 i a coincidence point of f 1 Conequently, we can uppoe that z n z n+1 for all n N From (i, we get that β =β x 1 1 Alo, by (ii we have that β x 1 =β x 1 x 2 1, β x 1 x 2 =β x 2 x 3 1 Continuou with thi proce we obtain that β +1 1 Now, by uing (7, we get φ( +1 φ(β φ(m(, +1 ψ(m(, +1 (8 where M(, +1 =max +1, q (f x, f x (q (f x, f x +1 2 n 1 n 2 n+1 1 n+1, ( = max 1, q (z, z (1 + q (z, z 1 n n+1, q 1 +1, Page 6 of 9

7 Soliman, Cogent Mathematic (2017, 4: we conider the following cae If M(, +1 = 1 from (9 we have φ( +1 φ( 1 ψ( 1 <φ( 1 (9 Downloaded by [ ] at 17:20 09 January 2018 Since φ iondecreaing we have +1 < 1 Similarly, one can have +1 < 1 If M(, +1 = q (z, z (1 + q (z, z 1 n n+1 from (9 we obtain 1 φ ( +1 ( q 1 ( +1 φ 1 ( q 1 ( +1 <φ 1 The nondecreaing property of φ implie that Similarly, we have Hence, (10 and (11 are obtained If M(, +1 = +1 By (9 we obtain thi i a contradiction ψ ( q 1 ( < 1 ( < < 1 φ( +1 φ( +1 ψ( +1 <φ( +1, +1 < 1 (10 (11 (12 In any cae, we proved that (10 and (11 hold Since +1 i decreaing and bounded from below Hence, it converge to a nonnegative number, c 0 If c > 0, then letting n + in (9, we deduce ( c(1 + c φ(c φ max c, 1 + c, c = φ(c, which lead to contradiction Hence, c = 0, that i lim q (z, z =0 n+1 Alo, one can get that (13 lim q (z, z =0 +1 n By (W 4 we get that z n i a -Cauchy equence Since X, d i -complete, there exit z f 2 X uch that lim z n = z Let w X be uch that f 2 u = z, applying (8 we have φ( u = lim φ( u lim [φ(m(u, φ(m(u, ], (14 (15 Page 7 of 9

8 Soliman, Cogent Mathematic (2017, 4: where M(u, =max, q (f u, f u(q (f x, f x n 1 n, q (f x, f x (q (f u, f u+1 2 n 1 n 2 1 = max (z 1, q (f u, f u(q (z, z +1 q n (z, z (q (f u, f u+1 nk 1 n 2 1, u 1 u 1 = u u a We get from (13 and (15 that φ( u u <φ( u u, which implie a contradiction, then u u= u=0, that i, z = f 2 u = f 1 u and o z i a coincidence point for f 1 Downloaded by [ ] at 17:20 09 January 2018 Now, we prove that z i the unique point of coincidence of f 1 Let x and y be arbitrary coincidence point of f 1 uch that x = f 1 u = f 2 u and y = f 1 v = f 2 v Uing the condition (7, it follow that φ( (x, y = φ( u v ( φ max v, q (f u, Au(q (f v, f v v ( ψ max, q (f v, f v(q (f u, f u v v, q (f u, f u(q (f v, f v , q (f v, f v(q (f u, f u v v = φ( v ψ( v <φ( v = φ( (x, y, hence, we have (x, y = (x, y =0 Thu, x = y and f 1 have a unique coincidence point A in the concluion in lat paragraph of the proof of Theorem 21 and the weakly compatible property of f 1, we obtain that f 1 i unique common fixed point Corollary 23 Putting f 2 = I in Theorem 22 Then one can get a unique fixed point of f 1 Remark 22 (Almeida et al, 2015, Theorem 16 I pecial cae of Theorem 22 In Branciari (2000 introduced a new concept of generalized metric pace a follow: Definition 21 (Branciari, 2000 Suppoe that X be a nonempty et and d: X X [0, be a ditance function uch that for all w, a, b, c X and w a b c, (i d(w, a =0 w = x, (ii d(w, a =d(a, w, (iii d(w, a d(a, b +d(b, c +d(c, w (quadrilateral inequality Then we ay that (X, d generalized metric pace (GMS, for hort Propoition 21 Let (X, d be a GMS Then (GW and 4 (1C are atified Definition 22 Aume that X be a nonempty et and S:X X [0, be a ditance function atify the condition (i and (ii in Definition 21 Then (X, S i called ymmetric generalized metric pace (SGMS, for hort Propoition 22 Let (X, d be a SGMS Then (GW and 4 (1C are atified Remark 23 Theorem 21 and 22 are correct in SGMS and quai-sgms Page 8 of 9

Soliman, Cogent Mathematic (2017, 4: 1301181 Downloaded by [463202222] at 17:20 09 January 2018 Acknowledgement The author thank the anonymou reviewer for their careful reading of thi paper and their

9 Soliman, Cogent Mathematic (2017, 4: Downloaded by [ ] at 17:20 09 January 2018 Acknowledgement The author thank the anonymou reviewer for their careful reading of thi paper and their many inightful comment and uggetion Funding The author received no direct funding for thi reearch Author detail Ahmed H Soliman 1 aholimanm@gmailcom 1 Faculty of Science, Department of Mathematic, Al-Azhar Univerity, Aiut 71524, Egypt Citation information Cite thi article a: On coincidence point reult of a pair of rational type contraction in quai-ymmetric pace, Ahmed H Soliman, Cogent Mathematic (2017, 4: Reference Almeida, A, Roldan-Lopez-de-Hierro, A F, & Sadarangani, K (2015 On a fixed point theorem and it application in dynamic programing Applicable Analyi and Dicrete Mathematic, 9, Branciari, A (2000 A fixed point theorem of Banach- Caccioppoli type on a cla of generalized metric pace Publicatione Mathematicae Debrecen, 57, Hick, T L, & Rhoade, B E (1999 Fixed point theory in ymmetric pace with application to probabilitic pace Nonlinear Analyi, 36, The Author( Thi open acce article i ditributed under a Creative Common Attribution (CC-BY 40 licene You are free to: Share copy and reditribute the material in any medium or format Adapt remix, tranform, and build upon the material for any purpoe, even commercially The licenor cannot revoke thee freedom a long a you follow the licene term Hitzler, P (2001, January Generalized metric and topology in logic programming emantic (PhD Thei, School of Mathematic, Applied Mathematic and Statitic, National Univerity Ireland, Univerity College Cork Imdad, M, Chauhan, S, Soliman, A H, & Ahmed, M A (2014 Hyprid fixed theorem in ymmetic pace via common limit rang property Demontratio Mathematica, XLVII, Kumari, P S (2012 On dilocated quai metric Journal of Advanced Studie in Topology, 3, Roa, V L, & Vetro, P (2014 Common fixed point for α ψ ф contraction in generalized metric pace Nonlinear Analyi: Modelling and Control, 19, Sarma, I R, & Kumari, P S (2012 On dilocated metric pace International Journal of Mathematical Archive, 3(1, 1 6 Schroeder, V (1999 Quai metric and metric pace, conformal geometry and dynamic, Volume 10, Page (December 26, 2006 application to probabilitic pace Nonlinear Analyi, 36, Shen, Y, & Lu, M (2010 Fixed point theorem in ymmetric pace and application to probabilitic pace Nonlinear Analyi, 72, Sumati Kumari, P, Ramana, C V, & Zoto, K (2014 On quaiymmetric pace Indian Journal of Science and Technology, 7, Wilon, W A (1931 On emi-metric pace American Journal of Mathematic, 53, Under the following term: Attribution You mut give appropriate credit, provide a link to the licene, and indicate if change were made You may do o in any reaonable manner, but not in any way that ugget the licenor endore you or your ue No additional retriction You may not apply legal term or technological meaure that legally retrict other from doing anything the licene permit Cogent Mathematic (ISSN: i publihed by Cogent OA, part of Taylor & Franci Group Publihing with Cogent OA enure: Immediate, univeral acce to your article on publication High viibility and dicoverability via the Cogent OA webite a well a Taylor & Franci Online Download and citation tatitic for your article Rapid online publication Input from, and dialog with, expert editor and editorial board Retention of full copyright of your article Guaranteed legacy preervation of your article Dicount and waiver for author in developing region Submit your manucript to a Cogent OA journal at wwwcogentoacom Page 9 of 9

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