ON BI-SHADOWING OF SUBCLASSES OF ALMOST CONTRACTIVE TYPE MAPPINGS

Transcription

1 Vol. 9, No., pp , Jue 015 Olie ISSN: ; Prit ISSN: Available olie at ON BI-SHADOWING OF SUBCLASSES OF ALMOST CONTRACTIVE TYPE MAPPINGS Awar A. Al-Badareh Departmet of Mathematics ad Statistics, Faculty of Sciece, Mutah Uiversity Mu tah 61710, Karak, Jorda ABSTRACT A result o bi-shadowig for cotiuous sigle-valued cotractios satisfyig certai coditios has bee give by Al- Badareh (015). I this paper, we cotiue the discussio of asymptotic properties of trajectories for mappigs belogig to more subclasses of almost cotractios. I particular, we prove that Zamfirescu Mappigs, Quasi- Cotractios, Geeralised Cotractios, ad Hardy-Rogers Cotractios are bi-shadowig. Keywords: Bi-Shadowig, almost cotractio, zamfirescu mappig, quasi-cotractio, geeralised cotractio, hardyrogers cotractio. 010 AMS Mathematics Subject Classificatio: 37C50, 47H07 INTRODUCTION The study of asymptotic behaviour of trajectories of discrete-time dyamical systems usually eeds techiques ad tools that ca capture specific properties of the system. These tools iclude the properties of shadowig ad iverse shadowig which are importat especially for validatig umerical computatios of the system (Palmer, 000; Pilyugi, 00; Pilyugi, 1999). Geerally, shadowig ad iverse shadowig esures the existece of true trajectories ear pseudo-trajectories, ad vice versa. I this paper we shall cosider a more geeral property of shadowig ad iverse shadowig, called bi-shadowig. It was itroduced by Diamod et al. (1995) ad applied for systems geerated by semi-hyperbolic mappigs (Diamod et al., 01). Later, it was cosidered by Kloede ad Ombach (1997) ad applied for ifiite dimesioal dyamical systems by Al-Nayef et al. (1997). It was also established by Al-Badareh (015) that β-cotractios, Kaa mappigs, Chatterjea mappigs ad Reich mappigs are all bi-shadowig. I the ext sectio, we give some defiitios ad prelimiaries eeded throughout the paper. The results of bi-shadowig for the subclasses cosistig of Zamfirescu Mappigs, Quasi-Cotractive Mappigs, Geeralised Cotractive Mappigs, ad Hardy-Rogers Cotractive *Correspodig author awar@mutah.edu.jo Mappigs will be cosidered i the Mai Sectio Result. DEFINITION AND PRELIMINARIES Throughout the paper, will always deote a metric space with metric d. We shall cosider a dyamical system o geerated by iteratios of a cotiuous mappig f : ad idetify f with the correspodig dyamical system. A sequece {x } =0 is called a (true) trajectory of f if x =f(x ), for = 0,1,,. While, a sequece {y } +1 =0 is called a γ-pseudo-trajectory of f if d(y,f(y )) γ for +1 = 0, 1,, ad for γ > 0. A mappig f : is called a k-cotractio, or a cotractio, if there exists a costat 0 < k < 1 such that d(f(x),f(y)) k d(x,y), for all x,y. (.1) The mappig f is called (λ, L)-cotractio or almost cotractio (sometimes weak cotractio is used), if there exist costats 0 λ < 1 ad L 0 such that d((f(x)), f(y)) λ d(x,y) + L d(y,f(x)) for all x, y. Almost cotractive mappigs are more geeral tha cotractios as every β-cotractive mappig is almost cotractio with λ=k ad L=0.

2 3450 We say that f has the shadowig property o if give ε>0 there exists γ>0 such that for ay give γ- pseudotrajectory {y } there exists a true trajectory {x } satisfyig d( y, x ) ε, for all =1,, We ow give a defiitio of the cocept of bi-shadowig i the cotext of a geeral metric space. Defiitio.1 A cotiuous mappig f: is called bi shadowig with respect to a compariso class of mappigs C() cosistig of cotiuous mappigs o ad with positive parametersα ad β if for ay give γ- The followig two theorems of bi-shadowig for cotractive mappigs ad almost cotractios i a metric space are give by Al-Badareh (015). Theorem.1 If f: is a k-cotractive mappig o, the f is bi-shadowig o with respect to the compariso class C() ad with positive parameters α ad β give by α = ad β = 1 k 1 k Theorem. Let f: be a cotiuous almost cotractio with costats 0<λ<1 ad L 0 such that d((f(x)),f(y)) λ d(x,y)+l d(y,f(x)), for all x,y. Assume that λ+l<1, ad that f satisfyig the followig two coditios: i) For every γ-pseudo-trajectory {z } =0 of f with γ < (1 λ L)/, the followig series is coverget: S: = d( f( z), z). = 0 ii) For every cotiuous mappig g: satisfyig sup d(g( x), f( x)) (1 λ L) /, the followig iequality is satisfied: d(f(x),f(y)) δ d(x,f(x))+δ d(x,y). LS < γ + sup d(g( x), f( x)). (.5) (5) The f is bi-shadowig o with respect to the class C() ad with parameters α ad β give by α = ad β 1 λ L 1 λ L = (.6) As a cosequece of this theorem, it was established by Al-Badareh (015) that a cotiuous Kaa mappig, Chatterjea mappig ad Reich mappig that satisfyig certai coditios are all bi-shadowig with appropriate costats. RESULTS AND DISCUSSION I this sectio, we cosider various subclasses of almost cotractios ad prove that the systems geerated by mappigs i these classes are bi-shadowig. pseudo-trajectory {y } =0 of f with 0 γ β ad ay Zamfirescu Mappigs C() satisfyig A mappig f: is called a Zamfirescu mappig if there exist real umbers β,α ad c with 0 β<1, 0 α<1/ γ + sup d( φ( x), f( x)) β (.) ad 0 c<1/ such that for all x,y, at least oe of the there exists a true trajectory {x } followig holds: =0 of such that i) d(f(x),f(y)) β d(x,y), d( y, x ) αγ ( + sup d( φ( x), f( x))), =0,1,, (.3) ii) d(f(x),f(y)) α [d(x,f(x))+d(y,f(y))], (3) iii) d(f(x),f(y)) c [d(x,f(y))+d(y,f(x))]. Note that the coditios i), ii) ad iii) above are equivalet to the followig coditio. d( x, f( x)) + d(y,f(y)) d( f(x), f ( y) ) h max d( x, y),, d( x, f (y)) + d(y,f(x)), for all x,y, where 0<h<1, see Ciric (1974). It was show i Beride (003) that Zamfirescu (197) mappigs are almost cotractios, thus, usig Theorem (.4) (4)., we have the followig result. Theorem 3.1 Let a cotiuous mappig f: be a Zamfirescu mappig with oegative costats β, α ad c ad assume that the coditios i) ad ii) of Theorem. are satisfied. The f is bi-shadowig o with respect to the class C() provided that α c δ : = max β,, < 1/5. 1 α 1 c Here, the positive parameters α ad β are give by α = ad β = 1 5δ (3.7) 1 5δ Proof: First, we should cosider the cases i i), ii) ad iii) above ad the uify the three costats i oe. By Beride (006), we have Thus d(f(x),f(y)) δ d(x,f(x))+δ d(x,y) (8)

3 Al-Badareh 3451 δ d(x,y)+δ d(y,f(x))+δ d(x,y) = 3δ d(x,y)+δ d(y,f(x)). This shows that f is almost cotractio with λ=3δ ad L=δ. Sice the coditios i) ad ii) of Theorem. are assumed to be satisfied, Theorem. implies that f is bishadowig with respect to the class C() provided that λ+l=5δ<1, that is δ<1/5. The values i (3.7) are easily obtaied by substitutig the values of λ=3δ ad L=δ i the relatios (.6). This eds the proof of the theorem. Quasi-Cotractios A mappig f : is called quasi-cotractio, see Ciric (1974), if there exists 0<h<1 such that d( f(x), f(y)) h M(x,y), for all x, y, where M(x,y)={d(x,y),d(x,f(x)),d(y,f(y)),d(x,f(y)),d(y,f(x))}. (3.8) The followig theorem of Beride (004) implies that quasi cotractios are almost cotractios for the value of the costat h satisfyig 0<h<1/. Theorem 3. Ay quasi-cotractio with 0<h<1/ is a almost cotractio. Usig Theorems 3. ad., we have the followig result. Theorem 3.3 Let a cotiuous mappig f: be quasicotractio with 0<h<1/ ad assume that the coditios i) ad ii) of Theorem. are satisfied. The f is bishadowig o with respect to the class C(). Proof: For a cotiuous quasi-cotractio f:, Theorem 3. implies that f is almost cotractio with appropriate values of λ ad L. It follows from the proof of Propositio 3 of Beride (004) that the values of λ ad L deped o what the maximum value i (3.) is. For example, if M(x,y)=d(x,f(y)), the h d(f(x),f(y)) 1 h d(x,y)+ h 1 h d(y,f(x)). h Thus λ=l= ad sice the coditios i) ad ii) of 1 h Theorem. are assumed to be satisfied, Theorem. implies that f is bi-shadowig with respect to the class C() provided that λ+l= h 1 h <1, that is h<1/3. I this case, the values of α ad β are: 1 h 1 3h α = ad β =. 1 3 h 1 h The other cases are treated similarly. The theorem is proved. Geeralised Cotractios A mappig f: is called geeralised cotractio, or Cirić cotractio, see (Cirić, 1971), if there exist oegative costats α,β,γ ad δ with α+β+γ+δ<1 such that d(f(x),f(y)) α d(x,y)+β d(x,f(x))+(y,f(y)) +δ[d(x,f(y))+d(y,f(x))], for all x,y. We ow show that a geeralised cotractio is almost cotractio with appropriate costats. Lemma 3.1 Let f: be a geeralised cotractio (Cirić) with α+β+γ+δ<1. The f is almost cotractio with costats α + β + δ β + γ + δ λ = ad L = (3.9) 1 γ δ 1 γ Proof: The followig estimates are direct. d(f(x),f(y)) α d(x,y)+β d(x,f(x))+ (9) γ d(y,f(y))+ δ d(x,f(y))+δ d(y,f(x)) α d(x,y)+β [d(x,y)+d(y,f(x))]+ δ [d(x,y)+d(y,f(y))]+δ d(y,f(x)) (α+β+δ) d(x,y)+(β+γ+δ) d(y,f(x))+ (γ+δ) d(f(x),f(y)). It follows that d(f(x),f(y)) α+β+δ β+γ+δ 1 γ d(x,y)+ 1 γ d(y,f(x)). If we take λ= α+β+δ 1 γ ad L= β+γ+δ 1 γ, the clearly, L 0 ad λ < 1 sice by assumptio α+β+γ+δ<1. This shows that f is almost cotractio. The lemma is proved. As a cosequece of Lemma 3.1 ad Theorem. we have the followig result. Theorem 3.4 Let a cotiuous mappig f: be a geeralised (Cirić) cotractio ad assume that the coditios i) ad ii) of Theorem. are satisfied. The f is bi-shadowig o with respect to the class C() provided that α+β+γ+4δ<1 ad with positive parameters: 1 γ α = 1 γ 4 δ α β ad 1 γ 4δ α β β = (3.10) 1 γ Proof: If f is a geeralised cotractio such that the coditios i) ad ii) of Theorem. are satisfied ad sice by Lemma 3.1 a geeralised cotractio is almost cotractio, Theorem. implies that f is bi-shadowig o with respect to the class C() for the values of λ ad L satisfyig the relatio λ+l= α+β+γ+3δ 1 γ <1,

4 345 that is α+β+γ+4δ<1. The values i (3.10) ca be obtaied by substitutig the values of λ ad L give by (3.9) i the relatios (.6). The theorem is proved. Hardy-Rogers Cotractios We ow cosider aother subclass of almost cotractios. A mappig f: is called Hardy-Rogers cotractio, see Hardy ad Rogers (1973), if there exist oegative costats a1, a, a3, a4, a 5 with a 1 + a + a 3 + a 4 + a 5 < 1 such that d( f( x), f( y)) a d( x, y) + a d( x, f( x)) + a d( y, f( y)) ad( x, f(y)) + ad(y, f( x)) for all x, y. a1 + a + a3 + a4 + 3a5 < 1. The values i (3.1) are obtaied by substitutig the values of λ ad L of (3.11) i the relatios (.6).This eds the proof of the theorem. CONCLUSION We utilised a result o bi-shadowig for cotractios that satisfy some coditios give by Al-Badareh (015). We studied the asymptotic properties for dyamical systems via bi-shadowig i view of the result metioed above. I particular, we proved that Zamfirescu Mappigs, Quasi-Cotractios, Geeralised Cotractios, ad Hardy-Rogers Cotractios are all bi-shadowig. REFERENCES Clearly if we take, as a special case, a = a3 = a4 = a5 = 0 the the iduced mappig is the usual a1 -cotractio mappig. While, if we take a1 = a4 = a5 = 0 the we obtai a cotractio coditio i the defiitio of Kaa mappigs, see Kaa (1969), ad so o. Thus, Hardy- Rogers mappigs are more geeral tha most of the cotractio type mappigs Proof: The proof follows the same lies as i the proof of Lemma 3.1 for geeralised cotractios (Ciric). Lemma 3. ad Theorem. imply the followig result. Theorem 3.5 Let a cotiuous mappig f: be Hardy-Rogers cotractio ad assume that the coditios i) ad ii) of Theorem. are satisfied. The f is bishadowig o with respect to the class C() provided that a1 + a + a3 + a4 + 3a5 < 1 ad with positive parameters 1 a3 a5 α = 1 a a a a 3 a, β = / α (3.1) Proof: Let f be a cotiuous Hardy-Rogers cotractio ad assume that the coditios i) ad ii) of Theorem. are satisfied. Lemma 3. ad Theorem. imply that f is bi-shadowig o with respect to the class C() provided that a1 + a + a3 + a4 + a5 λ + L = < 1, 1 a3 a5 that is Al-Badareh, AA Bi-Shadowig of Some Classes of Sigle-Valued Almost Cotractios. Applied Mathematical Scieces. 9(58): Al-Nayef, AA., Kloede, PE. ad Pokrovskii, AP Semi-hyperbolic mappigs, codesig operators ad eutral delay equatios. J. Differetial Equatios. 137: Beride, M Approximate fixed poit theorems, Lemma 3. Let f: be a Hardy-Rogers cotractio Studia Uiv. Babeş-Bolyai. Mathematica. 1:11-5. with a1 + a + a3 + a4 + a5 < 1. The f is almost cotractio Beride, V Approximatig fixed poits of weak with costats cotractios usig the Picard iteratio. Noliear a1 + a + a5 a + a3 + a4 + a5 λ = ad L = (3.11) Aalysis Forum. 9(1): () 1 a a 1 a a Beride, V O the approximatio of fixed poits of weak cotractive mappigs. Carpathia J. Math. 19(1):7-. Cirić, LB A geeralizatio of Baach s cotractio priciple. Proc. Am. Math. Soc. 45: Cirić, LB Geeralized cotractios ad fixed-poit theorems. Publicatios de l Istitut Mathématique. 1(6):19-6. Diamod, P., Kloede, P., Kozyaki, V. ad Pokrovskii, A Computer Robustess of semi-hyperbolic mappigs. Radom ad computatioal Dyamics. 3: Diamod, P., Kloede, P., Kozyaki, V. ad Pokrovskii, A. 01. Semi-Hyperbolicity ad Bi-Shadowig. AIMS Series o Radom & Computatioal Dyamics o. 1, America Istitute of Mathematical Scieces. Hardy, GE. ad Rogers, TD A Geeralizatio of a fixed poit theorem of Reich, Caad. Math. Bull. 16: Kaa, P Some results o fixed poits. II, The America Mathematical Mothly. 76(4):

5 Kloede, P. ad Ombach, J Hyperbolic homeomorphisms ad bi-shadowig. A. Polo. Math. 65(): Palmer, K Shadowig i Dyamical Systems. Theory ad Applicatios. Kluwer Acad. Publ. Pilyugi, S. Yu. 00. Iverse shadowig by cotiuous methods. Discr. Cot. Dy. Syst. 8(1):9-38. Pilyugi, S. Yu Shadowig i Dyamical Systems. Lecture Notes i Mathematics, 1706, Spriger-Verlag, Berli. Reich S Kaa fixed poit theorem. Boll. U. Math. Ital. 4:1-11. Zamfirescu, T Fix poit theorems i metric spaces. Arch. Math. (Basel). 3:9-98. Al-Badareh 3453 Received: March 8, 015; Accepted: April 1, 015