On Fixed Point Theorem of C Class Functions - B Weak Cyclic Mappings

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1 Joural of Mathematcs ad Statstcs Orgal Research Paper O Fxed Pot Theorem of C Class Fuctos - B Weak Cyclc Mappgs Sahar Mohamed l bou Bakr ad 2 rsla Hoat sar Departmet of Mathematcs, Faculty of Scece, Shams Uversty, Caro, Egypt 2 Departmet of Mathematcs, Kara Brach, Islamc zad Uversty, Kara, Ira rtcle hstory Receved: Revsed: ccepted: Correspodg uthor: Sahar Mohamed l bou Bakr Departmet of Mathematcs, Faculty of Scece, Shams Uversty, Caro, Egypt Emal: saharm al@yahoo.com saharm al@sc.asu.edu.eg bstract: New B-U-cyclc weak cotracto C-class fucto cocept has bee troduced, B-U-cyclc weak F-φ-φ-cotracto types of mappg are defed ad the exstece of fxed pot for such types has bee proved. These results maly geeralze fxed pot theorems some prevous research papers. Keywords: alyss, Fxed Pot Theorem, Cyclc Weak Cotracto Types of Mappgs, Fxed Pots, MSC: 46, 4705, 47H09, 47H0 Itroducto Frst; the class Φ s all o-decreasg mappgs : [0, [0, characterzed by φ(t 0 f ad oly f t 0. Blgl et al. (204; Karapar ad Sadaraga, 202; Du ad Karapar, 203 dscussed the cocept of Φ- weakly cyclc cotracto mappgs ad proved fxed pot theorems for mappgs o Baach spaces. Whle Hara et al. (203 cosdered cyclc weak Φ- cotracto o compact metrc spaces, the cosdered mappg φ eed ot be cotuous, aother fxed pot treatg the cocept s gve (Karapar ad Sadaraga, 202. The fxed pot theorem gve (Karapar ad Sadaraga, 202 focused o a wder class of metrc spaces. Jlel et al. (204; Karapar et al., 202b geeralzed the results to cyclc (φ, -weak cotractos some other metrc spaces. May results have bee proved dfferet stuatos ad settgs for the purpose of geeralzato of the Baach cotracto prcple for cotracto mappgs ad for o-expasve mappgs, (Hardy ad Rogers, 973; Gregus, 980; Kaewcharoe ad Krk, 2006; Kaa, 97; Krk, 965; Park, 980; Rhoades, 977; 200; Sahar Mohamed l bou Bakr, 203; Wog, 975; Rhoades, 2009; Ćrć, Recet results related to cyclc weak (φ-- cotracto mappgs appeared (Sahar Mohamed l bou Bakr, 207 for mappgs wth weak cyclc represetato complete metrc spaces ad weakly complete ormed spaces, the cosdered mappg φ eed ot be addtve, the author gave some examples. O the other sde, Morales ad Roas (2009 defed BZ type mappgs or the B-Zamfrescu mappgs, the mappgs of ay of the followg types: BB-type or B-Baach cotracto BK type or B-Kaa cotracto BC type or B-Chatterea cotracto It s proved the exstece of oly oe fxed pot for such types of mappg for the cotuous, oe to oe ad sub-sequetally coverget mappg B. For geeralzed cyclc weak φ- cotracto types mappgs, (Blgl ad Karapar, 203; Hussa et al., 204; Karapar ad Rakocevc, 203; Karapar et al., 202a; 203. Mathematcal Prelmares Frst, the sequel, (X, d s the space X wth a metrc d ad the class U s the class of all fte collectos of oempty closed subsets of X, { X : { { } : X } U: such that The mappgs B ad S are self mappgs o X. We have the followg deftos. Defto (Blgl ad Karapar, 203 B s kow as weak Φ-cotracto f ad oly f there s a cotuous fucto Φ such that: 207 Sahar Mohamed l bou Bakr ad rsla Hoat sar. Ths ope access artcle s dstrbuted uder a Creatve Commos ttrbuto (CC-BY 3.0 lcese.

2 Sahar Mohamed l bou Bakr ad rsla Hoat sar / Joural of Mathematcs ad Statstcs 207, ( :. DOI: /mssp ( φ( d B x, B y d x, y d x, y foreveryx, y X. Defto 2 (Blgl et al., 204 The elemet { of X f ad oly f: U s B cyclc represetato,,..., ( B B B adb Defto 3 (Jlel et al., 204 B s U-cyclc Φ-weak cotracto o X f ad oly f there are U ad a cotuous fucto Φ satsfyg the two codtos: s a B cyclc represetato of X d(b(x, B(y d(x, y-(d(x,y for every x k, y k+, k,2,..., ad +. Defto 4 B s U-cyclc φ--weak cotracto o X f ad oly f there are U ad φ, Φ, wth φ cotuous such that the followg are true: s B cyclc represetato of X φ(d(b(x,b(y φ(d(x, y-(d(x, y x k ; y k+, k,2,..., ad + Ths paper geeralzes the U-cyclc φ--weak cotracto types to ew B-U-cyclc weak F-φ- cotracto type, ths ew C-class of weak cotracto mappg s defed step by step ext. Defto 5 Let { be a elemet U. The s BScyclc represetato of X f ad oly f: Defto 6 Let { ( ( ( ( ( B S, B S,..., B S adb S U. The the self mappg S o X s B-cyclc weak φ- cotracto mappg o X f ad oly f there are φ, Φ wth φ cotuous such that the followg are true: s a TS-cyclc represetato of X φ(d(b(s(x,b(s(y φ(d(b(x,b(y-(d(b(x,b(y x, y +,, 2,..., ad + Remark The weak type cotracto mappg defed (6 geeralzes the defto of cyclc weak -cotracto of Erdal Karapar, Ksh Sadaraga, cyclc weak (φ- cotracto of Sahar Mohamed l bou Bakr ad TB cotracto mappgs of Jose R. Morales, Edxo Roas (Karapar ad Sadaraga, 202; Sahar Mohamed l bou Bakr, 207; Morales ad Roas, 2009 respectvely, because these are a partcular cases correspodg to takg B ad φ dettes. Defto 7 B s sequetally coverget f t satsfes the codto: If { B( x } coverget. Fally, we have the followg: s coverget, the { x} Defto 8 (sar, 204; sar et al., 206 The real valued mappg F: [0, [0, R s C- class f t s cotuous ad satsfyg the axoms: F(u, v u for all u, v [0;] If F(u; v u, the ether u 0 or v 0 C s the set of C-class fuctos. Meto that some C-class fucto F verfes F(0, 0 0. Examples If h: [0, [0, s a cotuous ad h(v 0 f ad oly f v 0, the F(u, v u-h(v s a C-class fucto, partcular: F(u, v u-v v F(u,v u k + v 2+ v F(u, v u v + v + u v F(u,v u 2+ u + v are C-class fuctos. If h: [0, [0, ] s a cotuous fucto, the F(u,v u h(v s a C-class fucto, partcular we have the followg: F(u, v mu for some m [0,] u F(u, v for some r [0, r ( +v s

3 Sahar Mohamed l bou Bakr ad rsla Hoat sar / Joural of Mathematcs ad Statstcs 207, ( :. DOI: /mssp F(u,v u log v+a a, a > u F(u,v for some r [0, r ( +u I addto to the followg: F(u,v log(v + a u /(+ v, for some a > F(u,v l(+ a u /2, for e>a>. Ideed F(u, v u mples that u 0 r F(u,v ( / + u+ l v -l, l >, for r [0, F(u,v l( + u Now; the geeralzed C-class of TS cyclc weak (φ, -cotracto mappgs are defed as: Defto 9 S s U-B-cyclc F-φ--weak cotracto mappg o X f ad oly f there are { } cotuous ad F C satsfyg: U, φ, Φ wth φ If there s a atural umber 0 such that S(x + S(x 0, the S(x s fxed of S. Ths sures that such fmum s zero. Suppose that B(S(x + B(S(x for all 0,, 2,... The, the cotracto codto yelds: ( d( B( x, B( x+ ( d( BS( x, B( Sx F ( d( B( x, B( x φ d( B( x, B( x d( B( x, B( x ( Sce φ s o-decreasg fucto, we see that: ( (, ( + ( (, ( d B x B x d B x B x N (3. Ths proves that the sequece {d(b(x, B(x + } N s a o-decreasg, hece the lmt: ( ( ( lm d B x, B x + s a BS-cyclc represetato of X ( d( B( S( x, B( S( y ( ( (,, φ( (, F d B x B y d B x B y (2. exsts ad t s equal to the fmum of the sequece, say r: { } ( ( ( + ( ( ( + ( (, ( + N r lm d B x, B x f d B x, B x : N, r d B x B x Remark for every x k, y k+,k, 2,..., ad + The cotracto type mappg defed defto (9 s a geeralzato of the cotracto type defed defto (6, because t s a partcular case whe takg F(u, v u-v. Ma Results The results of ths work are depedg o Propostos ( ad (2 below. Proposto Let S be U-B-cyclc F-φ--weak cotracto o X. The: { d( B( S( x B( x x X} f, : 0 Choose x 0 X ad focus o the terated sequece: ( ( ( ( B x 0,,2,... + B S x B S x for 0 O the other sde, we have the same coclusos for the two sequeces {φ(d(b(x, B(x + } N ad {(d(b(x, B(x + } N, cosder the two postve real umbers 0 ad φ 0 gve as follows: ad: { } ( r 0 lm d( B( x, B( x+ f φ d( B( x, B( x+ : N φ φ φ { } ( r 0 lm d( B( x, B( x+ f d( B( x, B( x+ : N The lmt of the equaltes (3. as gves: ad, therefore: (,lm ( ( (, ( F φ d B x B x ( ( ( ( 0, or lm φ d B x, B x 0

4 Sahar Mohamed l bou Bakr ad rsla Hoat sar / Joural of Mathematcs ad Statstcs 207, ( :. DOI: /mssp Now; suppose that r>0, we have (r>0, thus: ( ( ( 0 < φ r φ d B x, B x N Lettg the last equaltes, we get the followg cotradcto: ( r d( B( x B( x 0< φ lm φ, 0 that s; the assumpto r > 0 s ot true, hece r 0, thus: ( ( + B( x ( ( + ( lm d B x, Hece; { x} { d B x B x N} f, : 0 X such that: ( ( ( B( x lm d B S x, 0 Ths s suffcetly proved that f{d(b(s(x,b(x: x X} 0. Proposto 2 Let S be U-B-cyclc F-φ--weak cotracto mappg o X. The the terated sequece {B(x B(S (x 0 } N s Cauchy. We determe for a gve ϵ>0 a atural umber 0 N satsfyg; f m, > 0 wth x ad x m + for some {, 2,... } (-m ( the d(b(x,b(x m <ϵ gves a cotradcto. Suppose that there s ϵ>0 satsfyg; ay 0 N yelds m, > 0 wth B(x ad B(x m + for some {, 2,..., } (-m ( satsfyg: ( (, ( m ε <d B x B x sce φ ad are o-decreasg, we see that: ( ε d( B( x B( xm 0 <, ad ( ε φ d( B( x B( xm 0 < φ, ad sce φ s cotuous, we see that: (, m d( B( x B( xm (3.2 0< ε lm, (3.3 usg the cotractvty codto of S we see that: ( d( B( x+, B( xm+ + m+ d B( S ( x, B( S ( x 0 0 m d B( S ( x0 B( S ( x 0 F m φ d B( S ( x0, B( S ( x0 d( B( x, B( xm ( ( (,, (3.4 Now; usg equaltes (3.2, (3.3 ad lettg, m (3.4 wth -m ( proves the followg cotradcto: ( ε F( ( ε, ( ε ( ε (3.5 φ Ths proves that F(φ(ϵ, (ϵ φ(ϵ, hece ether φ(ϵ 0 or (ϵ 0, cosequetly ϵ 0, therefore ths sequece s havg Cauchy subsequece, ths fact wth Proposto ( completes the proof. Fally; we have: Theorem If (X, d s complete, B s oe to oe cotuous self sequetally coverget mappg o X ad S s U-Bcyclc F-φ--weak cotracto mappg o X. The S ows fxed pots. Moreover; we have the followg: If { } s represetato of X ad z s fxed of S, the B( z rbtrarly two cosecutve sets of { } cota two dfferet fxed pots caot Usg Proposto (2, the sequece {B(S (x 0 } N s Cauchy, the completeess of X shows that there s a pot y X such that: ( ( 0 lmb S x y (3.6 Such a lmt pot s lyg the set, because each,, 2,..., cotas ftely may members of the fte sequece {B(S (x 0 } N, hece y s a lmt pot for for each, 2,...,, gvg that s closed for each, 2,..., shows that y for each, 2,..., (as closed set cotas all ts lmt pots, thus: y

5 Sahar Mohamed l bou Bakr ad rsla Hoat sar / Joural of Mathematcs ad Statstcs 207, ( :. DOI: /mssp Sce B s sequetally coverget mappg, the sequece { S ( x0 } { S k ( x0 } has a coverget subsequece, hece there s z X such that: ( 0 lms k x k z Usg the cotuty of B gves that: ( k 0 B( z lmb S x k (3.7 Usg (3.6 ad (3.7 proves that B(z y. We wll see that such a z s fxed of S. I fact; there s {, 2,..., } such that z, for the umber + we get N such that: { x, x, x,... 2 } Therefore z ad { x, x, x,... 2 } are lyg two + + cosecutve sets, o the other sde we have:, d( B( S z, B( x + (, B( z + ( ( ( ( d B S z B z + d B x ( d B S z, B S x + d B S x, B z. ( Hece we use the cotracto property of S as: ( (, ( ( d B S z B S x ( ( (,, φ( (, ( d( B( z, B( x d B( z, B( S( x F d B z B x d B z B x ( (3.8 (3.9 Usg the cotuty of φ ad takg the lmt of the equaltes (3.9 as prove the followg: ( d( B( S( z B( S( x ( d( B( z B( S( x ( d( B( z,lmb( S( x lm, lm, ( d( B( z, y d( B( z B( z, 0 0 (3.0 Usg the cotuty of φ oce more wth the equaltes (3.0 shows that: hece: ( ( d B S z B S x ( ( ( ( ( ( ( ( 0 lm, lm d B S z, B S x ( ( ( ( (3. lm d B S z, B S x 0 Usg (3. (3.8 after takg the lmt as gves: ( ( ( ( B( z ( d( B( z B( z ( ( ( d B S z, B z lm d B S z, B S x + lm d B S x, 0 + d y, B z, 0 Hece d(b(s(z, B(z 0, therefore, B(S(z B(z, sce B s oe to oe, we get S(z z ad hece z s fxed of S. To show that two cosecutve sets of { } caot cota two dfferet fxed pots, by cotrary assume that w ad z are two dfferet fxed of S, S(w w ad S(z z those are lyg two cosecutve sets, we have the followg: ( ( ( ( d( B( w, B( z d B( S( w, B( S( z F d( B( w, B( z, φ d( B w, B z hece φ(d(b(w,b(z 0, or (d(b(w,b(z 0, that s; d(b(w,b(z 0, cosequetly B(w B(z, sce B s oe to oe, w z. Ths completes the proof. We also have: Theorem 2 Let (X,. be weakly complete ormed space, C be a closed covex subset of X, { } C be a oempty closed subsets of C, C C. If B ad S are self mappgs o C, { } C s BS represetato of C ad S s weak F-φ-cotracto o C, the S has fxed pots. Usg Proposto (2 the sequece of terates {B(S (x 0 } N s Cauchy, usg the weak completeess assumpto of X there exsts x X such that:

6 Sahar Mohamed l bou Bakr ad rsla Hoat sar / Joural of Mathematcs ad Statstcs 207, ( :. DOI: /mssp ( ( 0 w lmb S x x Sce C s closed covex subset of X, the sequece {B(S (x 0 } N coverges strogly to x ad x C. For the other parts of the proof use Theorem (. Cocluso Ths paper suggests ew C-class of TS cyclc weak (φ, -cotracto mappgs ad proved the exstece of uque fxed pot for such types of mappgs. Competg Iterest The authors have o competg terest. uthor s Cotrbutos The frst author cotrbuted the rato of two-thrd the wrtg of ths paper. Refereces sar,.h., 204. Note o (φ-ψ-cotractve type mappgs ad related fxed pot. Proceedgs of the 2d Regoal Coferece o Mathematcs ad pplcatos, (CM 4, Payame Noor Uversty, pp: sar,.h., P. Kumam ad B. Samet, 206. fxed pot problem wth costrat equaltes va a mplct cotracto. J. Fxed Pot Theory pplc., 9: DOI: 0.007/s Blgl, N. ad E. Karapar, 203. Cyclc cotractos va auxlary fuctos o G-metrc spaces. Fxed Pot Theory pplc., 203: DOI: 0.86/ Blgl, N., I.M. Erha, E. Karapar ad D. Turkoglu, 204. Cyclc cotractos ad related fxed pot theorems o g-metrc spaces. ppled Math. Iform. Sc., 8: DOI: /ams/ Ćrć, L.B., Cotractve type o-self mappgs o metrc spaces of hyperbolc type. J. Math. al. pplc., 37: DOI: 0.06/.maa Du, W.S. ad E. Karapar, 203. ote o carst-type cyclc maps: Related results ad applcatos. Fxed Pot Theory pplc., 203: DOI: 0.86/ Gregus, M., 980. fxed pot theorem baach spaces. Boll. Uoe Math. Italaa, 7: Hardy, G.E. ad T.D. Rogers, 973. geeralzato of a fxed pot theorem of Rech. Caad. Math. Bul., 6: DOI: 0.453/CMB Hara, J., B. Lopez ad K. Sadaraga, 203. Fxed pot theorems for cyclc weak cotractos compact metrc spaces. J. Nolear Sc. pplc., 6: Hussa, N., E. Karapar, S. Sedgh, N. Shobe ad S. Frouza, 204. Cyclc -cotractos uform spaces ad related fxed pot results. bs. ppled al., 204: DOI: 0.55/204/ Jlel, M., E. Karapar ad B. Samet, 204. O cyclc (ψ, -cotractos Kaleva-Sekkala's type fuzzy metrc spaces. J. Itell. Fuzzy Syst., 27: DOI: /IFS-470 Kaewcharoe,. ad W.. Krk, Noexpasve mappgs defed o ubouded domas. Fxed Pot Theory pplc., 2006: DOI: 0.55/FPT/2006/82080 Kaa, R., 97. Some results o fxed pots III. Fudameta Math., 70: Karapar, E. ad K. Sadaraga, 202. Corrgedum to Fxed pot theory for cyclc weak φ-cotracto. ppled Math. Lett. 25: DOI: 0.06/.aml Karapar, E. ad V. Rakocevc, 203. O cyclc geeralzed weakly C-cotractos o partal metrc spaces. bs. ppled al., 203: DOI: 0.55/203/8349 Karapar, E., G. Petruel ad K. Ta, 202a. Best proxmty pot theorems for KT-types cyclc orbtal cotracto mappgs. Fxed Pot Theory, 3: Karapar, E., M. Jlel ad B. Samet, 202b. Fxed pot results for almost geeralzed cyclc (ψ, -weak cotractve type mappgs wth applcatos. bs. ppled al., 202: DOI: 0.55/202/9783 Karapar, E., S. Romaguera ad K. Tas, 203. Fxed pots for cyclc orbtal geeralzed cotractos o complete metrc spaces. Cet. Eur. J. Math., : DOI: /s Krk, W.., 965. fxed pot theorem for mappgs whch do ot crease dstaces. m. Math. Mothly, 72: DOI: / Morales, J.R. ad E. Roas, Some results o T- zamfrescu operators. Revsta Notas de Matematca, 5: Park, S., 980. O geeral cotracto-type codtos. J. Korea Math. Soc., 7: Rhoades, B.E., 977. comparso of varous deftos of cotractve mappgs. Tras. m. Math. Soc., 226: DOI: 0.090/S Rhoades, B.E., 200. Some theorems o weakly cotractve maps. Nolear al., 47: DOI: 0.06/S X(

7 Sahar Mohamed l bou Bakr ad rsla Hoat sar / Joural of Mathematcs ad Statstcs 207, ( :. DOI: /mssp Rhoades, B.E., Necessary ad suffcet codto for commo fxed pot theorems. J. dv. Math. Stud. Sahar Mohamed l bou Bakr., 207. Fxed pot theorem of weak cyclc cotracto types of operators. It. J. Math. Stat. Sahar Mohamed l bou Bakr., 203. Fxed pot theorems of some type cotracto mappgs. It. J. Nolear al. Covex al., 4: Wog, C.S., 975. O Kaa maps. Proc. m. Math. Socety, 47: