ОРИГИНАЛНИ НАУЧНИ ЧЛАНЦИ ОРИГИНАЛЬНЫЕ НАУЧНЫЕ СТАТЬИ ORIGINAL SCIENTIFIC PAPERS 0 8 ANSARI’S METHOD IN 1-2 6 GENERALIZATIONS OF SOME 2 p. p RESULTS IN THE FIXED POINT y, e v THEORY: SURVEY ur S Tatjana M. Došenovića, Stojan N. Radenovićb ory: a University of Novi Sad, Faculty of Technology, he Novi Sad, Republic of Serbia, nt t e-mail: [email protected], oi p ORCID ID: http://orcid.org/0000-0002-3236-4410 d b University of Belgrade, Faculty of Mechanical Engineering, xe Belgrade, Republic of Serbia, e fi e-mail: [email protected], h ORCID ID: http://orcid.org/0000-0001-8254-6688 n t s i http://dx.doi.org/10.5937/vojtehg66-16045 ult s e FIELD: Mathematics e r ARTICLE TYPE: Original Scientific Paper m o ARTICLE LANGUAGE: English s of Abstract: s n o The aim of this paper is to show that the C-class function introduced by ati z A. H. Ansari is a powerful weapon for the generalization of many ali important results in the theory of fixed points. er n Keywords: C class function, Metric space, Cauchy sequence, Common ge fixed point, Fixed point. n d i o h Introduction et m s With the advent of S. Banach paper (Banach, 1922), the ari’ development of the theory of fixed point is moved upwards. A huge s n nguemnebrearli zeo fB asnciaecnhti sctosn, trfaocrt iomn oprrein ctihpalen (A9b0b ayse a&r sJ,u nhgacvke, 2m00a8n,a pgpe.d4 16to- T., A ć, 420), (Altun et al, 2010, pp.2238-2242), (Boyd & Wong, 1969, pp.458- vi o 464), (Đorić, 2009, 1896-1990), (Geraghty, 1973, pp.604-608), n e š (Amini-Harandi & Emami, 2010, pp.2238-2242), (Hussain et al, 2013), o D ACKNOWLEDGMENT: The first author is grateful for the financial support from the Ministry of Education and Science and Technological Development of the Republic of Serbia (Matematički modeli nelinearnosti, neodređenosti i odlučivanja, 174009) and from the Provincial Secretariat for Higher Education and Scientific Research, Province of Vojvodina, Republic of Serbia, project no. 142-451-2838/2017-01. 261 2 (Harjani & Sadarangani, 2009, 3403-3410), (Jachymski, 2011, pp.768- e u 774), (Jungck, 1976, pp.261-263), (Karapinar & Salimi, 2012), (Khan et s s 6, I al, 1984, pp.1-9), (Liu et al, 2015), (Rhoades, 1977, pp.257-290), 6 (Rhoades, 2001, pp.2683-2693), (Radenović & Kadelburg, 2010, ol. pp.1776-1783), (Radenović et al, 2012, pp.625-645), (Salimi et al, 2013), V 8, (Samet et al, 2012, pp.2154-2165). In 2014 A. H. Ansari (Ansari, 2014a, 1 0 pp.373-376), (Ansari, 2014b, pp.377-380) introduced the concept of C- 2 R, class functions which cover a large class of contractive conditions, see E RI also (Ansari, 2014a, pp.373-376), (Ansari, 2014b, pp.377-380), (Ansari U et al, 2017, pp.2657-2673), (Ansari & Chandok, 2016, pp.65-71). O C L A Definition 1 (Ansari et al, 2017, pp.2657-2673) A C-class function C NI is a continuous function F :0,0, R such that for any H EC x,y[0,), the following conditions hold: T Y (C1) F(x,y) x; R A (C2) F(x,y)= x implies that either x=0 or y = 0. T MILI An extra condition on F that F0,00 could be imposed in some K / cases if required. By C we will denote the class of all C- functions. NI S A L Example 1 (Ansari et al, 2017, pp.2657-2673) The following G KI functions belong to the class C: Č NI H 1. F(x,y)= x y. E OT 2. F(x,y)= mx, for some m(0,1). N J x O 3. F(x,y)= for some r(0,). V (1 y)r log(yax) 4. F(x,y)= , for some a>1. 1 y r 5. F(x,y)=(xl)(1/(1y) ) l, l >1, for r(0,). y 6. F(x,y)= x . k y 7. F(x,y)= x(x), where :[0,)[0,) is a continuous function such that (t)=0 if and only if t =0. 8. F(x,y)= n ln(1xn). We start this section with the following definitions and notions: 262 Definition 2 (Ansari, 2014b, pp.377-380) A mapping F:[0,)2 R has a property C , if there exists an C 0 such that 80 F F 2 (C 1) Fx,y>C implies x> y; 61- F F 2 (C 2) Fy,yC , for all y[0,). pp. F F y, For more examples of Cclass functions that have the property C e F v see (Ansari, 2014b, pp.377-380) Here we announce the following three Sur examples y: a) Fx,y= x y,C =r,r[0,); eor 2Fyy nt th b) Fx,y= x ,C =0; oi 1 y F d p e x c) Fx,y= x ,k 1,C = r ,r 2. e fi 1ky F 1k n th Let  denote the class of all functions :[0,)[0,),  s i ult denote the class of all functions :[0,)[0,) and F elements of C es satisfying the following conditions: me r so (i)  is non-decreasing and continuous; of s n (ii)  is non-decreasing and continuous; o ati (iii) (t)F(s),(s)>0 for all t > 0 and s t or s = 0. aliz er n e The condition (iii) generalizes (2.3) from (Karapinar & Salimi, 2012, g n p.9). d i ho Definition 3 (Ansari et al, 2017, pp.2657-2673) A subclass of type met I is a function H :R[0,) R if it is continuous and s s1 implies H1,t Hs,t for all t[0,). sari’ n A T., Example 2 (Ansari et al, 2017, pp.2657-2673) We have the ć, following functions of the subclass of the type I : ovi n e š o  Hs,t=tls, l >1, D  Hs,t=slt, l >1,  Hs,t= stn, nN, 263 2  Hs,t= st, e ssu  Hs,t=t, 6, I s1 ol. 6  Hs,t= 2 t, V 8, 2s1 1  Hs,t= t, 0 2 3 R, E  n  RI sni  U O  Hs,t= i=0 t, L C  n1  A   C   NI H t C  n  E sni  T Y  Hs,t= i=0 l , l >1. R A  n1  LIT   MI   K / NI Definition 4 (Ansari et al, 2017, pp.2657-2673) We say that the pair S A F,H is an upclass of the type I if F :[0,)[0,) R is L G KI continuous, H is a function of the subclass of the type I and satisfies: Č (1) 0 x1 implies Fx,y F1,y NI H     E (2) H 1,y  F x,y implies y  xy , T 1 2 1 2 O for all x,y,y ,y [0,). N 1 2 J O V Example 3 (Ansari et al, 2017, pp.2657-2673) Below are listed the functions of the upclass of the type I, for all sR, t,y[0,),   x 0,1 :  Hs,t=tls, l >1,Fx,y= xyl,  Hs,t=tls, l >1,Fx,y= 1lxy,  Hs,t= stn,Fx,y= xnyn,      H s,t = st,F x,y = xy. 264 Definition 5 (Ansari et al, 2017, pp.2657-2673) We say that the pair   F,H is a special upclass of the type I if F :[0,)[0,) R is 0 continuous, H is a function of the subclass of the type I and satisfies: 8 2 1- 6     2 (1) 0 x1 implies F x,y  F 1,y p. p (2) H1,t F1,y implies t  y, y, e v for all y,t[0,). ur S y: Example 4 (Ansari et al, 2017, pp.2657-2673) The following eor h functions are a special upclass of the type I, for all sR,t,x,y[0,): nt t oi p  Hs,t tk lsn, l 1, Fx,y xmyk l, xed e fi  Hs,t sn ltk, l 1, Fx,y 1lxmyk, n th  Hs,t sntk, Fx,y xpyk, ults i s e  Hs,t st, Fx,y xy. e r m o s Remark 1 (Ansari et al, 2017, pp.2657-2673) Every pair F,H of of s n the upclass of the type I also belongs to the class of a special upclass o ati of the type I , but converse is not true. z Assertions similar to the following lemma were used (and proved) in erali the course of proofs of several fixed point results in various papers en g (Radenović et al, 2012, pp.625-645). n d i o   h Lemma 1 (Radenović et al, 2012, pp.625-645) Let X,d be a et m metric space and let x  be a sequence in X such that s n ari’   s limd x ,x =0. n n n1 A n T., If x  is not a Cauchy sequence, then there exist >0 and two ć, n vi     o sequences m and n of positive integers such that n >m >k and n k k k k e š the following sequences tend to  when k : Do         d x ,x ,d x ,x ,d x x ,d x ,x . mk nk mk nk1 mk1, nk mk1 nk1 265 2 Definition 6 (Abbas & Jungck, 2008, pp.416-420) Let f and g be e su self maps of a set X. If = fx = gx for some xX, then x is called a s 6, I coincidence point of f and g, and  is called a point of coincidence of 6 ol. f and g. The pair f,g of self maps is weakly compatible if they V 8, commute at their coincidence points. 1 0 2 R, Proposition 1 (Abbas & Jungck, 2008, pp.416-420) Let f and g E RI be weakly compatible self maps of a set X. If f and g have a unique U O point of coincidence = fx = gx, then  is the unique common fixed C L point of f and g. A C NI H Main results C E Y T Previously described functions attracted the attention of authors and R now there are various generalizations of the results from the fixed point A T theory, not only in a metric space, but also in the partial metric spaces, LI MI metric like-spaces, G-metric spaces,...(Ansari, 2014a, pp.373-376), K / (Ansari, 2014b, pp.377-380), (Ansari et al, 2017, pp.2657-2673), (Isik et NI al, 2015, pp.703-708), (Ansari & Chandok, 2016, pp.65-71). In this paper, S A we will present some of these results. Also, we shall prove some new L G results, which generalize already known ones, by using the Cclass KI Č functions introduced recently by A.H. Ansari (Ansari, 2014a, pp.373-376), NI (Ansari, 2014b, pp.377-380). In this review paper, we will use only C- H E class functions. T O Our first (probably new) result is the following: N OJ V   Theorem 1 Let X,d be a complete metric space. Suppose that the mappings f,g:X  X satisfy the following condition           d fx, fy  Fd gx,gy ,d gx,gy (1) for all x,yX where , and FC. If the range of g     contains the range of f and f X or g X is a closed subset of X, then f and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point. 266         Remark 2 Putting in (1): t =t,t = 1k t,k 0,1 and   F s,t = st for all s,t[0,) we get a well-known Jungck’s result from 0 8 (Jungck, 1976, pp.261-263). Hence, Theorem 1 is a genuine 2 1- generalization of the old and important Jungck’s result in several 6 2 directions (see all assumptions in (Jungck, 1976, pp.261-263). p. p Further, putting in (1): t=t,Fs,t=ss for all s,t[0,) y, e v where :[0,)[0,1) such that t 0 whether t 1,g = I ur n n X S (identity mapping on X ), we get a well-known Geraghty type result y: or (Geraghty, 1973, pp.604-608). Hence, Theorem 1 is a new generalization e h of this old and important result in the fixed point theory in the framework nt t of complete metric spaces. oi p Proof. Let us prove that the point of coincidence of f and g is ed x unique in the case that it exists. Suppose that  and  are two distinct e fi h points of coincidence of f and g. From this follows that there exist two n t points u and u such that fu= gu== fu = gu. Now, (1) ults i s implies e d,=dfu , fu  me r   so  Fdgu,gu,dgu,gu ns of o = Fd,,d,d,, ati z ali that is er n Fd,,d,=d,. (2) ge n From (2) according to the property of the function F follows that d i o       h either d , =0 or d , =0. In both cases, we get a et m contradiction. s Further, let x be an arbitrary point in X. Let us choose a point ari’ 0 s n x1X such that y0 = fx0 = gx1. This can be done, since the range of g T., A contains the range of f . Continuing this process, having chosen x in ć, n vi o X, we obtain x in X such that y = fx = gx . Now consider the n n1 n n n1 e š following two possible cases: o D 10 y = y for some kN. Hence, gx = fx is a (unique) k k1 k1 k1 point of coincidence and then the proof of Theorem 1 is finished. 267 e 2 20 Thus, suppose that yn  yn1 for all nN0. In this case, we u s have s 6, I dy ,y =dfx , fx  6 Vol.  Fdng1x n,g2x ,dgnx1 ,gnx2  8, n1 n2 n1 n2 01 = Fdy ,y ,dy ,y  R, 2 n n1 n n1    E d y ,y . RI n n1 U     O Since  we get that d y ,y d y ,y , i.e., C n1 n2 n n1 AL dy ,y , r  0. We prove now that r =0. Indeed, if r >0, then C n n1 NI passing to the limit in the previous relation when n, we obtain that H C          E r  Fr ,r r , T Y            R that is Fr ,r =r . This implies that either r =0 or r =0. A T   MILI In both cases we get a contradiction. Hence, limnd yn,yn1 =0.   K / We next prove that yn is a Cauchy sequence in a complete metric NI space X,d. If that is not case, then by using Lemma 1 we get that S LA there exist >0 and two sequences m  and n  of positive integers G k k KI and sequences Č         NI d y y ,d y y ,d y y ,d y y , EH mk nk mk nk1 mk1 nk mk1 nk1 T O all tend to  when k . By applying condition (1) to the elements N J O x= x and y = x and since y = fx = gx for each n0, we get V mk nk1 n n n1 that           d y y  Fd y y ,d y y . (3) mk nk1 mk1 nk mk1 nk Letting k  in (3), we obtain          F, , which is a contradiction because >0. This shows that       y = fx = gx is a Cauchy sequence in a complete metric space n n n1   X,d .     Since g X is closed in a complete metric space X,d , then it is a complete metric space. Therefore, there exists u,vX such that v = gu 268 and lim gx = gu =v. We shall show that also fu =v= gu. Indeed, n n putting x= x ,y =u in (1) we get n 0 8 dfx , fu Fdgx ,gu,dgx ,gu. (4) 1-2 n n n 6 2 Letting n in (4) and applying the properties of all three p. p functions F, and , we get y, e v dgu, fu Fdgu,gu,dgu,gudgu,gu=0=0, ur S y: i.e., fu = gu is a (unique) point of coincidence of the functions f and g. or e By the Proposition 1 f and g have the unique common fixed point. nt th In the case when fX is a closed subset in X,d, the proof is oi p d similar. e x       s2t e fi Putting t =t =t,F s,t = ,g = I the identity mapping of h 1st X n t X in Theorem 1, we get the following result: ults i es Corollary 1 Let X,d be a complete metric space. Suppose e r m o mappings f :X  X satisfies s of d3x,y ns dfx, fy1d2x,y (5) atio aliz for all x,yX . Then f has a unique fixed point in X . er n Putting t=t=t,Fs,t=s, where :[0,)[0,) is ge n upper semicontinuous from the right, satisfying t<t for t >0 as well d i o h as 0=0,g = I the identity mapping of X in Theorem 1 we get the et X m following well-known Boyd and Wong result (Boyd & Wong, 1969, ari’s pp.458-464). s n A Corollary 2 Let X,d be a complete metric space. Suppose that a T., ć, mapping f :X  X satisfies the following condition vi o n dfx, fydx,y (6) Doše for all x,yX. Then f has a unique fixed point, say uX and f nxu as n for each xX. 269 2 Putting t=t=t,Fs,t= ss, where :[0,)[0,) is a e u ss continuous function such that (t)=0 if and only if t = 0, g = I the 6, I identity mapping of X in Theorem 1 we get the following well-kXnown 6 ol. B.E. Rhoades result (Rhoades, 2001, pp.2683-2693). V 8, 01 Corollary 3 Let X,d be a complete metric space. Suppose that 2 R, the mappings f :X  X satisfies the following condition E URI dfx, fydx,ydx,y (7) O C for all x,yX. Then f has a unique fixed point, say uX and L A C f nxu as n for each xX. NI H In the sequel of this section we shall consider two results which C E provide the existence of a coincidence point and a common fixed point T Y for three mappings satisfying the generalized F,,contractive R A T condition. These results are addressed in the following theorems. MILI K / Theorem 2 Let X,d be a metric space, and let f,g,S:X  X NI be three mappings such that for all x,yX S A GL dfx,gy Fmx,y,m x,y, (8) KI 1 Č for some ,  and F C, where NI H E  1  OT mx,y= maxdSx,Sy,dSx, fx,dSy,gy, dSx,gydSy, fx N  2  J O V and          m x,y =max d Sx,Sy ,d fx,Sx ,d gy,Sy . 1   If fX gX  SX and S(X) is a complete subspace of X,d ,   then f,g and S have a unique point of coincidence. Moreover, if f,S   and g,S are weakly compatible, then f,g and S have a unique common fixed point. The proof of the following theorem is similar to that of Theorem 1.   Theorem 3 Let X,d be a complete metric space, and let f,g,S:X  X be three mappings such that for all x,yX 270