Balazs, Margareta-Eliza

Miskolc Mathematical Notes

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Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 18 (2017), No. 1, pp. 71–81 DOI: 10.18514/MMN.2017.1875A MAIA TYPE FIXED POINT THEOREM FOR PRESˇIC´-KANNANOPERATORSMARGARETA-ELIZA BALAZSReceived 16 November, 2015Abstract. Starting from Presˇic´’s fixed point theorem, established in [Presˇic´, Slavisˇa B. Sur uneclasse d’ine´quations aux diffe´rences finies et sur la convergence de certaines suites. (French)Publ. Inst. Math. (Beograd) (N.S.) 5 (19) 1965 75–78] on the one hand, and Maia’s fixed pointtheorem obtained in [Maia, Maria Grazia. Un’osservazione sulle contrazioni metriche. (Italian)Rend. Sem. Mat. Univ. Padova 40 1968 139–143] on the other hand, two different extensions ofBanach fixed point theorem are extended, generalised and unified.2010 Mathematics Subject Classification: 47H10; 54H25Keywords: fixed point, metric space, Presic-Kannan operator, extension of Banach fixed pointtheorem1. INTRODUCTION AND PRELIMINARIESIn 1965, S. Presˇic´ [4] extended the famous Banach contraction principle [1] tothe case of product spaces and obtained convergence results for some particularsequences. Presˇic´ type operators have applications in solving nonlinear differenceequations, cyclic systems and in the study of convergence of sequences. Presˇic´’sresult may be stated as follows:Theorem 1 ([5]). Let .X;d/ be a complete metric space, k a positve integer andf WXk!X a Presˇic´ operator, it is, a mapping for which there exists ˛1;˛2; : : : ;˛k 2RC;kPiD1˛i D ˛ < 1 such that:d f .x0; : : : ;xk 1/;f .x1; : : : ;xk/ kXiD1˛id.xi 1;xi /; (1.1)for all x0; : : : ;xk 2X:Then:1/ f has a unique fixed point xI2/ the sequence fyngn0;c 2017 Miskolc University Press72 MARGARETA-ELIZA BALAZSynC1 D f .yn;yn; : : : ;yn/;n 0;converges to xI3/ the sequence fxngn0 with x0; : : : ;xk 1 2X andxn D f .xn k;xn kC1; : : : ;xn 1/;n k;also converges to x; with a rate estimated byd.xnC1;x/ ˛d.xn;x/CM n;n 0;where M > 0 is constant.The following result was given by M. G. Maia in 1968 and is also a generaliza-tion of Banach contraction mapping principle for sets endowed with two comparablemetrics.Theorem 2 ([8], [3]). Let X be a nonempty set, d and  two metrics on X andf WX !X an operator. We suppose that:(i) d.x;y/ .x;y/ for all x;y 2X I(ii) .X;d/ is a complete metric space;(iii) f W .X;d/! .X;d/ is continuous;(iv) f W .X;/! .X;/ is an ˛-contraction.Then:(a) Ff D fxgI(b) f n.x/d ! x as n!1; for all x 2X I(c) f n.x/ ! x as n!1; for all x 2X I(d) .x;x/ 11 ˛.x;f .x//; for each x 2X:Remark 1. Particular cases:1. From Maia’s fixed point theorem when d  , we get Banach’s fixed point the-orem.2. From Presˇic´’s fixed point theorem when k D 1, we get Banach’s fixed point the-orem.3. A Presˇic´-Maia type fixed point theorem has been obtained by Shukla, S. andRadenovic´, S. in [9].Presˇic´’s fixed point theorem has been extended by Pa˘curar to the case when theBanach type contraction condition .1:1/ is replaced by a Presˇic´-Kannan type con-traction.Theorem 3 ([5],[2]). Let .X;d/ be a complete metric space, k a positive integer,a 2 R a constant such that 0  a  k  .kC 1/ < 1 and f W Xk ! X an operatorsatisfying the following condition:A MAIA TYPE FIXED POINT THEOREM FOR PRESˇIC´-KANNAN OPERATORS 73d f .x0;x1; : : : ;xk 1/;f .x1;x2; : : : ;xk/ a d.x0;f .x0; : : : ;x0//Ca d.x1;f .x1; : : : ;x1//C CCa d.xk;f .xk; : : : ;xk//; (1.2)for any x0;x1; : : : ;xk 2X .Then:1/ f has a unique fixed point xI2/ the sequence fyngn0;ynC1 D f .yn;yn; : : : ;yn/;n 0;converges to xI3/ the sequence fxngn0 with x0; : : : ;xk 1 2X andxn D f .xn k;xn kC1; : : : ;xn 1/;n k;also converges to x; with a rate estimated byd.xnC1;x/ C n;n 0;where C is a positive constant.Starting from these results, the aim of this paper is to extend Theorem 3 to the caseof a set endowed with two comparable metrics.We shall need in our proof the following lemma given by S. Presˇic´ in [4].Lemma 1. Let k 2N;k ¤ 0 and ˛1;˛2; : : : ;˛k 2 RC such thatkPiD1˛i D ˛ < 1: Iffngn1 is a sequence of positive numbers satisfyingnCk  ˛1nC˛2nC1C C˛knCk 1;n 1; (1.3)then there exist L> 0 and  2 .0;1/ such thatn  L n; for al l n 1: (1.4)Remark 2. The following consideration shall simplify our approach in this proof[7].For any operator f W Xk ! X , k a positive integer, we can define its associateoperator F WX !X byF.x/D f .x; : : : ;x/; x 2X:Obviously, x 2 X is a fixed point of f W Xk ! X if and only if it is a fixed point ofits associate operator F . This enables the study of f by means of the operator F .74 MARGARETA-ELIZA BALAZS2. THE MAIN RESULTTheorem 4 (Maia fixed point theorem for Presˇic´-Kannan operators). Let X be anonempty set, d and  two metrics on X and k a positive integer, a 2 R a constantsuch that 0 a k  .kC1/ < 1 and f WXk!X an operator satisfying the followingcondition: f .x0;x1; : : : ;xk 1/;f .x1;x2; : : : ;xk/ a .x0;f .x0; : : : ;x0//Ca .x1;f .x1; : : : ;x1//CC Ca .xk;f .xk; : : : ;xk//; (2.1)for any x0;x1; : : : ;xk 2X .We suppose that:.i/ d.x;y/ .x;y/ for all x;y 2X I.i i/ .X;d/ is a complete metric space;.i i i/ f W .Xk;d /! .X;d/ is continuous;Then:.a/ f has a unique fixed point x, Ff D x;f .x;x; : : : ;x/D xI.b/ the sequence fxngn0 with x0; : : : ;xk 1 2X andxnC1 D f .xn;xn 1; : : : ;xn kC1/;n k 1;converges to x with respect to d ;.c/ the sequence fyngn0;ynC1 D f .yn;yn; : : : ;yn/;n 0;converges to x with respect to I.d/ the following estimate holds.xnC1;x/ C n;n 0; (2.2)where C is a positive constant.Proof. .c/: From fyngn0, ynC1 D f .yn;yn; : : : ;yn/; n  0 and considering theassociate operator F WX !X , F.x/D f .x;x; : : : ;x/, for any x;y 2X we have: F.x/;F.y//D  f .x;x; : : : ;x/;f .y;y; : : : ;y/  f .x;x; : : : ;x/;f .x;x; : : : ;y/CC f .x;x; : : : ;y/;f .x; : : : ;x;y;y/CC CC f .x;y; : : : ;y/;f .y;y; : : : ;y/:By condition (1.2) and inequality (2.5) it follows F.x/;F.y// a  k  .kC1/2  x;F.x/C y;F.y/A MAIA TYPE FIXED POINT THEOREM FOR PRESˇIC´-KANNAN OPERATORS 75for any x;y 2X . As a was assumed to satisfy 0 a k  .kC1/ < 1; it follows that0 a  k  .kC1/2<12;so F is a Kannan operator.According to Kannan’s theorem [2], there exists a unique x 2X such thatF.x/ D x, namely that x D f .x;x; : : : ;x/ and this can be obtained as limitof the sequence of successive approximations of F , which is exactly the sequencefyngn0, defined by ynC1 D f .yn;yn; : : : ;yn/; n 0, solimn!1F.yn/D x; F .yn/ converges to x D f .x;x; : : : ;x/ in .X;/andynC1 D f .yn;yn; : : : ;yn/ converges to x D f .x;x; : : : ;x/:.a/; .b/ W Let fxngn0, xnC1 D f .xn;xn 1; : : : ;xn kC1/;n k 1.For n 0 we have:.xnC1;x/D  f .xn;xn 1; : : : ;xn kC1/;f .x;x; : : : ;x/  f .xn;xn 1; : : : ;xn kC1/;f .xn 1; : : : ;xn kC1;x/CC f .xn 1; : : : ;xn kC1;x/;f .xn 2; : : : ;xn kC1;x;x/CC CC f .xn kC1;x; : : : ;x/;f .x;x; : : : ;x/:Let F WX!X , F.x/D f .x;x; : : : ;x/;x 2X and using the condition (1.2) in theinequality above, we get:.xnC1;x/ a  xn;F .xn/C CC xn kC1;F .xn kC1/C x;F .x/CCa   xn 1;F .xn 1/C C xn kC1;F .xn kC1/CC x;F .x/C x;F .x/CC CCa   xn kC1;F .xn kC1/C x;F .x/C C x;F .x/:Since  x;F .x/D 0, this implies:.xnC1;x/ a 1  xn;F .xn/C2  xn 1;F .xn 1/C Ck  xn kC1;F .xn kC1/ (2.3)For each j 2N the following holds: xj ;F .xj / .xj ;x/C x;F .xj / (2.4)and  x;F .xj /D  F.x/;F .xj / F.x/;F .xj /76 MARGARETA-ELIZA BALAZSa   x;f .x;x; : : : ;x/C C x;f .x;x; : : : ;x/„ ƒ‚ …k timesCC xj ;f .xj ;xj ; : : : ;xj /CCa   x;f .x;x; : : : ;x/C C x;f .x;x; : : : ;x/„ ƒ‚ …k 1 timesCC xj ;f .xj ;xj ; : : : ;xj /C xj ;f .xj ;xj ; : : : ;xj /„ ƒ‚ …2 timesCC CCa   x;f .x;x; : : : ;x/CC xj ;f .xj ;xj ; : : : ;xj /C C xj ;f .xj ;xj ; : : : ;xj /„ ƒ‚ …k timesso F.x/;F .xj / a  x;f .x;x; : : : ;x/  kC .k 1/C C1CCa  xj ;f .xj ;xj ; : : : ;xj /  1C2C Ckand finally F.x/;F .xj / a  k  .kC1/2  x;f .x;x; : : : ;x/C xj ;f .xj ;xj ; : : : ;xj /:Thus F.x/;F .xj / a  k  .kC1/2  x;F .x/C xj ;F .xj / (2.5)and x;F .xj /D  F.x/;F .xj / a  k  .kC1/2  x;F .x/C xj ;F .xj /DD a  k  .kC1/2  xj ;F .xj /:Thus (2.5) becomes: xj ;F .xj / .xj ;x/Ca  k  .kC1/2  xj ;F .xj /:By denoting AD a  k.kC1/2we have xj ;F .xj / 11 A .xj ;x/ (2.6)Using (2.6) in inequality (2.3) we obtain.xnC1;x/A MAIA TYPE FIXED POINT THEOREM FOR PRESˇIC´-KANNAN OPERATORS 77 a1 A .xn;x/C 2 a1 A .xn 1;x/C C k a1 A .xn kC1;x/ (2.7)Now by denotingn D .xn;x/;n 0˛i D i a1 A; i D 1;k;the inequality (2) becomesnC1  ˛1 nC˛2 n 1C C˛k n kC1; n k: (2.8)The coefficients ˛1;˛2; : : : ;˛k are all positive, as 0 < a < 1k.kC1/ :BesideskXiD1˛i DkXiD1i a1 A Da1 A kXiD1i D a1 A k  .kC1/2D A1 A :So from0 < a k  .kC1/ < 10 <a k  .kC1/2<120 < A <12< 1we havekXiD1˛i D A1 A < 1 :Now the conditions required in Lemma .1:1/ are fulfilled. Consequently thereexist L 0 and  2 .0;1/ such that n  L n; n 1, namely such that.xn;x/ L n; n 1: (2.9)Using .2:9/ in the inequality (2) we have:.xnC1;x/ a1 A L nC 2 a1 A L n 1C C k a1 A L n kC1  n  a L1 A  .1C2  1C Ck 1 k/DD n  a L1 A kXiD1i 1 i ; L 0;  > 0:This inequality shall lead to estimate .d/, where:C D a L1 A kXiD1i 1 i > 078 MARGARETA-ELIZA BALAZS.xnC1;x/ C n; n 0 : (2.10)It follows immediately that .xnC1;x/! 0 as n!1, so the sequence fxngn0converges to x in .X;/, and x is the unique fixed point of the operator f .For n 0, from .i/ and (2.10) we obtaind.xnC1;x/ .xnC1;x/ < ":Since  2 .0;1/, it follows that fxngn0 is a Cauchy sequence in .X;d/, which inthe complete metric space .X;d/ is also convergent.From the continuity of f in .X;d/,limn!1d f .xn;xn 1; : : : ;xn kC1/;f .x;x; : : : ;x/D 0we getlimn!1f .xn;xn 1; : : : ;xn kC1/D f .x;x; : : : ;x/ in .X;d/and fxngn0 converges to x in .X;d/ and Ff D x, x D f .x;x; : : : ;x/:Remark 3. We have the following important particular cases of Theorem 4:1. If k D 1, by Theorem 4 we get a Maia fixed point theorem corresponding to Kan-nan contraction condition. Kannan fixed point theorem is obtained in the particularcase when d D . Note that the original Maia’s fixed point theorem is established forthe case of Banach’s contraction condition.2. If d D , by Theorem 4 we get Pa˘curar’s fixed point theorem [6].Example 1. Let X D Œ0;1 endowed with two metrics d and . For k D 2 definef WX2!X byf .x;y/D8<:xCy6;.x;y/ 2 Œ0;1;x  y;0; otherwise :(2.11)Then f is a Presˇic´-Kannan contraction.Proof. In the first part of the proof we will show that f is a Presˇic´-Kannan oper-ator. In this particular case condition (2.1) becomes:.f .x0;x1/;f .x1;x2// a .x0;f .x0;x0//CCa .x1;f .x1;x1//Ca .x2;f .x2;x2//;with x0 > x1 > x2.If k D 2 then 0 a < 16.For x0 D 1;x1 D x2 D 0.f .x0;x1/;f .x1;x2//D .f .1;0/;f .0;0//D .16;0/D 136A MAIA TYPE FIXED POINT THEOREM FOR PRESˇIC´-KANNAN OPERATORS 79 a .1;f .1;1//Ca .0;f .0;0//Ca .0;f .0;0//DD a  Œ.1; 13/C0C0D a  29so136 a  29it follows thata  18:The conclusion is that f given by .2:11/ is a Presˇic´-Kannan operator, that is, itsatisfies (2.1) for any x0;x1;x2 2 Œ0;1;x0 > x1 > x2, with constant a 218; 16:Let fxngn0, xnC2 D fxnC1;xng;n 0. For any x;y 2X we have:.xnC2;x/ C n;n 0; (2.12)whereC D a L1 3 a  .1C2 1/ > 0:It follows immediately that .xnC2;x/ ! 0 when n ! 1, so the sequencefxngn0 converges to x in .X;/, and x D f .x;x/ is the unique fixed pointof the operator f .d.xnC2;x/ .xnC2;x/ < .Since  2 .0;1/, it follows that fxngn0 is a Cauchy sequence in .X;d/, which inthe complete metric space .X;d/ is also convergent.From the continuity of f in .X;d/,limn!1d.f .xnC1;xn/;f .x;x//D 0we getlimn!1f .xnC1;xn/D f .x;x/ in .X;d/and fxngn0 converges to x in .X;d/ and Ff D x, x D f .x;x/:From fyngn0, ynC2 D f .yn;yn/; n  0 and considering the associate operatorF WX !X , F.x/D f .x;x/, for any x;y 2X we have: F.x/;F.y// 3 a   x;F.x/C y;F.y/for any x;y 2X . As a was assumed to satisfy 0 a k  .kC1/ < 1; it follows that0 3 a < 12;so F is a Kannan operator and due to Kannan’s theorem, there exist a unique fixedpoint x 2X such that F.x/D x, so fyngn0 converges to x D f .x;x/.80 MARGARETA-ELIZA BALAZSThe following theorem is an extension of Theorem 4 to the case of metric spacesendowed with a partial order.Theorem 5. LetX be a nonempty set equipped with partial order , d and  twometrics onX and k a positive integer, a 2R a constant such that 0 a k .kC1/ < 1and f WXk!X an operator satisfying the following condition: f .x0;x1; : : : ;xk 1/;f .x1;x2; : : : ;xk/ a .x0;f .x0; : : : ;x0//Ca .x1;f .x1; : : : ;x1//CC Ca .xk;f .xk; : : : ;xk//; (2.13)for any x0;x1; : : : ;xk 2X with x0  x1      xk :We suppose that:.i/ d.x;y/ .x;y/ for all x;y 2X with x  yI.i i/ .X;d/ is a complete metric space;.i i i/ f W .Xk;d /! .X;d/ is continuous and nondecreasing with respect to ” ”;Then:.a/ f has a unique fixed point x, Ff D x;f .x;x; : : : ;x/D xI.b/ the sequence fxngn0 with x0; : : : ;xk 1 2X andxnC1 D f .xn;xn 1; : : : ;xn kC1/;n k 1;converges to x;.c/ the sequence fyngn0;ynC1 D f .yn;yn; : : : ;yn/D F.yn/;for all n 0, y0 2X such that y0  x, converges to x as well;.d/ the following estimate holds.xnC1;x/ C n;n 0; (2.14)where C is a positive constant and  2 .0;1/:Proof of this theorem follows from a similar process as used in the proof of the-orem 4.REFERENCES[1] S. Banach, “Sur les o´perations dans les ensembles abstraits et leur application aux e´quationsinte´grales,” Fund. Math, vol. 3, pp. 133–181, 1922.[2] R. Kannan, “Some results on fixed points,” Bull. Calcutta Math. Soc, vol. 10, pp. 71–76, 1968.[3] M. G. Maia, “Un’osservazione sulle contrazioni metriche,” Rend. Sem. Mat. Univ. Padova, vol. 40,pp. 139–143, 1968.[4] S. B. Presˇic´, “Sur une classe d’ine´quations aux diffe´rences finies et sur la convergence de certainessuites,” Publ. Inst. Math. (Beograd) (N.S.), vol. 5, no. 19, pp. 75–78, 1965.[5] M. Pa˘curar, Iterative methods for fixed point approximation. Risoprint, 2009.A MAIA TYPE FIXED POINT THEOREM FOR PRESˇIC´-KANNAN OPERATORS 81[6] M. Pa˘curar, “A multi-step iterative method for approximating fixed points of presˇic´-kannan operat-ors,” Acta Math. Univ. Comenian. (N.S.), vol. 79, no. 1, pp. 77–88, 2010.[7] I. A. Rus, “An abstract point of view in the nonlinear difference equations,” Conf. on An., FunctionalEquations, App. and Convexity, Cluj-Napoca, pp. 272–276, 1999.[8] I. A. Rus, A. Petrus¸el, and G. Petrus¸el, Fixed Point Theory. Cluj University Press, 2008.[9] S. Shukla and S. Radenovic´, “Presˇic´-maia type theorems in ordered metric spaces,” G. J. Math.,vol. 2, no. 2, pp. 73–82, 2014.Author’s addressMargareta-Eliza BalazsTechnical University of Cluj-Napoca, North University Center at Baia Mare

A Maia type fixed point theorem for Presic-Kannan operators

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