Generalized Dynamic Process for Generalized Multivalued F-contraction of Hardy Rogers Type in b-metric Spaces

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1 Turkish Joural of Aalysis a Number Theory, 08, Vol. 6, No., Available olie at Sciece a Eucatio Publishig DOI:0.69/tjat-6-- Geeralize Dyamic Process for Geeralize Multivalue F-cotractio of Hary Rogers Type i b-metric Spaces Abullah Shoaib,*, Awais Asif, Muhamma Arsha, Eskaar Ameer Departmet of Mathematics a Statistics, Riphah Iteratioal Uiversity, Islamaba , Pakista Departmet of Mathematics, Iteratioal Islamic Uiversity, H-0, Islamaba 44000, Pakista *Correspoig author: abullahshoaib5@yahoo.com Receive September 7, 06; Revise April 07, 08; Accepte April 8, 08 Abstract The aim of this paper is to establish commo fixe poit results for multivalue mappigs satisfyig geeralize F-cotractive coitios of Hary Rogers type with respect to geeralize yamic process i b-metric space. Our results improve a geeralize several well kow results i the existig literature. Keywors: fixe poit, geeralize F-cotractio, b-metric space, geeralize yamic process, Hausorff metric Cite This Article: Abullah Shoaib, Awais Asif, Muhamma Arsha, a Eskaar Ameer, Geeralize Dyamic Process for Geeralize Multivalue F-cotractio of Hary Rogers Type i b-metric Spaces. Turkish Joural of Aalysis a Number Theory, vol. 6, o. (08): oi: 0.69/tjat Itrouctio a Prelimiaries Let X be a o empty set, f : X X be a mappig. A poit x X is calle a fixe poit of f if x = fx. Fixe poits results of mappigs, which satisfies some specific cotractive coitios o some space have bee very useful i research activity (see [-30]). Recetly, Warowski [30] itrouce a ew cocept of cotractio ame F-cotractio a prove a fixe poit theorem which geeralizes Baach cotractio priciple. Klim et al. [] further establishe fixe poit result for F-cotractive mappig i yamic process. Cosetio et al. [6] further geeralize this cocept as F-Cotractive Mappigs of Hary-Rogers-Type. Arsha et al. [4] prove fixe poit result i α GF cotractio of Hary-Rogers-type. Followig this irectio of research, i this paper, we will preset some fixe poit results of Hary-Rogers-type for multivalue mappigs i b-metric space with geeralize yamic process. This paper cotai commo fixe poit results for two mappigs. Throughout our paper R +, N a CB( X ) represet set of real umbers, set of atural umbers a family of oempty close boue subsets X respectively. Defiitio [0] Let X be a o-empty set a let s be a give real umber. A fuctio : X X R + is calle a b-metric provie that, for all xyz,, X ) xy (, ) = 0 iff x = y ) xy (, ) = yx (, ) 3) ( x, z) s[ ( x, y) + ( y, z)]. The pair ( X, ) is calle a b-metric space. Defiitio [5] Let ( X, ) be a b-metric space. The a sequece { x } i X is calle a Cauchy sequece if a oly if for all ε > 0 there exist ( ε ) N for each m, > ε ( ), we have ( x, xm) < ε. Defiitio 3 [30] Let F : R+ R be a mappig satisfyig: (F) F is strictly icreasig. (F) for each sequece { a } R + of positive umbers lim a 0 lim F a = = if a oly if ( ) (F3) there exists k ( 0,) lim a 0 + k afa ( ) = 0. We eote with the family of all fuctios F that satisfy the coitios (F)-(F3). Let ( X, ) be a metric space. A self-mappig T o X is calle a F-cotractio if there exist τ + F( ( Tx, Ty) F( ( x, y)), for all x, y X with ( Tx, Ty) > 0. τ R+ Theorem 4 [30] Let ( X, ) be a complete metric space a let T : X X be a F-cotractio. The T has a uique fixe poit x X a for every x0 X a sequece { T x0} N is coverget to x. Defiitio 5 [6] Let ( X, ) be a metric space. A selfmappig T o X is calle a geeralize F-cotractio of Hary-Rogers-type if there exist F a R τ + τ + F( ( Tx, Ty)) F( α( x, y) + β( x, Tx) + γ( y, Ty) + δ( x, Ty) + L( y, Tx)),

2 Turkish Joural of Aalysis a Number Theory 44 for all xy, X with ( Tx, Ty ) > 0, αβδ,, [0,), α + β + γ + δ =, γ a L 0. Theorem 6 [6] Let ( X, ) be a complete metric space a let T be a self-mappig o X. Assume that there exist F a τ R+ T is a F-cotractio of Hary-Rogers-type, that is, τ + F( ( Tx, Ty)) F( α( x, y) + β( x, Tx) + γ( y, Ty) + δ( x, Ty) + L( y, Tx)), for all xy, X, Tx Ty, αβδ,, [0,), α + β + γ + δ =, γ a L 0. The T has a fixe poit. Moreover, if α + δ + L, the the fixe poit of T is uique. Defiitio 7 [0] Let ( X, ) be a metric space. For A, B CB( X ), the Hausorff metric H o CB( X ) iuce by metric is give as: H( AB, ) = max sup ( xb, ),sup ( ya, ) x A y B ( x, B) = if{ ( x, y): y B} a H : CB( X ) CB( X ) R + is the Hausorff metric iuce by. Defiitio 8 [] Let x0 X a let F : R+ R satisfies (F)-(F3). The mappig T : X C( X) is calle a set-value F-cotractio with respect to a yamic process { x} DT (, x0 ) if there exists a fuctio τ : R+ R+ for all N ( ( x, x + ) > 0 τ ( ( x, x)) + F( ( x, x+ )) F( ( x, x))). I the above iscussio C( X ) eotes the collectio of all o empty close subsets of X a DT (, x 0) is calle a yamic process of T startig at x 0. The yamic process ( x) N {0} will be writte simply as ( x ) DT (, x0 ) = {( x) N {0} X : x Tx, for all N}. Defiitio 9 [3] Let f : X X, T : X CB( X ) a x 0 be a arbitrary but fixe elemet i X. The set D( f, T, x0 ) = {( x) N {0} : x+ = fx Tx for all N} is calle a geeralize yamic process of f a T startig at x 0. The geeralize yamic process D( f, T, x 0) will simply be writte as ( fx ). The sequece ( x ) for which ( fx ) is a geeralize yamic process is calle f iterative sequece of T startig at x 0. Lemma 0 [3] Suppose ( M, ) be a b-metric space a { x } be a sequece i M ( x, x ) λ( x, x ), = 0,,, λ <. The the sequece { x } is a Cauchy sequece i M provie that s. λ <.. Mai Result Defiitio Let F : R + R be a mappig satisfyig: (F) F is strictly icreasig. (F) for each sequece { a } R + of positive umbers a = if a oly if lim F ( a ) =. lim 0 We eote with the family of all fuctios F that satisfy the coitios (F) a (F). Let ( X, ) be a b-metric space. A self-mappig T o X is calle a F -cotractio if there exist τ R+ such that τ + F ( ( Tx, Ty) F ( ( x, y)), for all x, y X with ( Tx, Ty) > 0. () Defiitio Let ( X, ) be a b-metric space, f : X X is cotiuous, T : X CB( X ) a x 0 be a arbitrary poit i X. A mappig f is calle a geeralize multivalue F -cotractio of Hary-Rogers-type with respect to a geeralize yamic process D( f, T, x 0) if there exist F a τ : R+ R+ is o ecreasig τ M x x + F fx fx+ F M x x ( (, )) ( (, )) ( (, )) M( x, x) = α( fx, fx) + β( fx, Tx ) + γ( fx, Tx) + δ ( fx, Tx) + L( fx, Tx ) for all xy, DfTx (,, 0) with ( fx, fy ) > 0, α + β + γ + sδ <, γ, L 0, ( α + β + sδ ) = λ a ( γ sδ) s. λ <. Now we state a prove our mai result. Theorem 3 Let ( X, ) be a complete b-metric space a T : X CB( X ). If f is a geeralize multivalue F -cotractio of Hary-Rogers-type with respect to a geeralize yamic process D( f, T, x 0). The f a T have a commo fixe poit. Proof: Let x0 X be a arbitrary poit, by efiitio of geeralize multivalue F -cotractio of Hary-Rogerstype with respect to a geeralize yamic process D( f, T, x 0), we have F( + ) = F( ( x+, x+ )) = F( ( fx, fx+ )) F[( α( fx, fx) + β( fx, Tx ) + γ( fx, Tx) + δ( fx, Tx) + L( fx, Tx ))] τ[( α( fx, fx) + β( fx, Tx ) + γ( fx, Tx) + δ ( fx, Tx) + L( fx, Tx ))]

3 45 Turkish Joural of Aalysis a Number Theory F[( α( fx, fx) + β( fx, fx) + γ( fx, fx+ ) + δ( fx, fx+ ) + L( fx, fx))] τ[( α( fx, fx) + β( fx, Tx ) + γ( fx, Tx) + δ ( fx, Tx) + L( fx, Tx ))] = F[( α( x, x+ ) + β( x, x+ ) + γ( x+, x+ ) + δ( x, x+ ) + L( x+, x+ )] τ[( α( fx, fx) + β( fx, Tx ) + γ( fx, Tx) + δ ( fx, Tx) + L( fx, Tx ))] F[( α( x, x+ ) + β( x, x+ ) + γ( x+, x+ ) + δ s[ ( x, x+ ) + ( x+, x+ )] + L( x+, x+ )] τ[( α( fx, fx) + β ( fx, Tx ) + γ ( fx, Tx) + δ ( fx, Tx) + L( fx, Tx ))] = F[( α + β + sδ) + ( γ + sδ) + ] τ[( α( fx, fx) + β( fx, Tx ) + γ( fx, Tx) + δ ( fx, Tx) + L( fx, Tx ))] F ( ) < F[( α + β + sδ) + ( γ + sδ) ]. + + As F is strictly icreasig, therefore < ( α + β + sδ) + ( γ + sδ), + + ( γ sδ) < ( α + β + sδ). + As α + β + γ + sδ < a γ, we euce that γ sδ > 0 a so + < [( α + β + sδ) / ( γ sδ)] =. This implies that + < λ. Cotiuig this process, we ca easily say that + < λ 0. Now, to show that { x } is a Cauchy sequece i X. Let m>, 0 with m> ( x, xm) s[ ( x, x ) + ( x, x )] + + m 3 (, + ) ( +, + ) ( +, + 3) ( 0, ) ( 0, ) ( 0, )... 3 sλ 0 0 s x x + s x x + s x x + sλ x x + s λ x x + s λ x x + = sλ ( x, x )[ + sλ+ ( sλ) + ( sλ)...] = ( x, x ). sλ Usig Lemma 0, a takig limit, we get lim ( x, xm) = 0. which proves that { x } is a Cauchy, so there exist some x X lim x =. Now we prove that x x = fx Tx. For this, we have lim[ α( fx, fx) + β( fx, Tx ) + γ( fx, Tx) + δ( fx, Tx) + L( fx, Tx ))] lim[ α( fx, fx) + β( fx, fx) + γ( fx, fx+ ) + δ[ ( fx, fx) + ( fx, fx+ )] = α( fx, fx ) + β( fx, fx ) + γ( fx, fx ) + δ[ ( fx, fx ) + ( fx, fx ) = 0. By (F) lim F[ α( fx, fx) + β( fx, Tx ) + γ( fx, Tx) + δ ( fx, Tx) + L( fx, Tx ))] =. Therefore lim F( M ( fx, fx + )) =. From above we ca write lim F( ( x+, fx+ ) = lim F( ( fx, fx+ )) =. Agai by (F) lim ( x+, fx+ ) = 0. Therefore ( x, fx ) = 0. So x = fx. Moreover lim ( x, Tx ) lim ( fx, fx ) = ( fx, fx ) = Hece, ( x, Tx ) = 0 x Tx, so x = fx Tx, that is x is the commo fixe poit of f a T. Puttig α = δ = L = 0, we obtai a geeralize form of Kaa's result i yamic process. Corollary 4 Let ( X, ) be a complete b-metric space a T : X CB( X ). Assume that there exist τ R+ a F F is cotiuous satisfyig: τ( β( fx, Tx ) + γ( fx, Tx )) + F ( ( fx, fx )) + F ( β( fx, Tx ) + γ( fx, Tx )) for all xy, DfTx (,, 0) with ( fx, fy ) > 0, β β + γ <, γ, L 0, γ = λ a s. λ <. The f a T have a commo fixe poit. Choosig δ = L = 0, we obtai a geeralize versio of Reich's result. Corollary 5 Let ( X, ) be a complete b-metric space a T : X CB( X ). Assume that there exist R a F F is cotiuous satisfyig: τ + τ[( α( fx, fx) + β( fx, Tx ) + γ ( fx, Tx)] + F( ( fx, fx+ )) F[( α( fx, fx) + β( fx, Tx ) + γ( fx, Tx)] for all x, y DT (, x0 ) with ( fx, fy ) > 0, α + β + γ <, γ, ( α + β ) = λ a s. λ <. The f ( γ ) a T have a commo fixe poit.

4 Turkish Joural of Aalysis a Number Theory 46 Theorem 6 Let ( X, ) be a complete b-metric space, f : X X is cotiuous a T : X CB( X ). Assume that there exist τ R+ a F F is cotiuous from the right. Now if D( f, T, x 0) is a geeralize yamic process ( (, )) ( (, )) ( (, )) τ M x y + F H Tx Ty F M x y () M ( x, y) = α( fx, fy) + β( fx, Tx) + γ( fy, Ty) + δ( fx, Ty) + L( fy, Tx) for all xy, DfTx (,, 0) with ( fx, fy ) > 0, α + β + γ + sδ <, γ, L 0, ( α + β + sδ ) = λ a ( γ sδ) s. λ <.. The f a T have a commo fixe poit. Proof: Let x0 X be a arbitrary poit of X. By efiitio of geeralize yamic process fx Tx0. Sice F is cotiuous from the right, there exists a real umber h > F ( hh ( Tx, Tx )) < F ( H ( Tx, Tx )) + τ ( M ( x, x )) Now, from ( fx, Tx) < hh ( Tx0, Tx), we euce that there exists fx Tx ( fx, fx) hh ( Tx0, Tx). Cosequetly, we get which implies F ( ( fx, fx )) F ( hh ( Tx, Tx )) 0 < F ( H ( Tx, Tx )) + τ ( M ( x, x )), 0 0 τ ( M( x0, x)) + F( ( x, x3)) = τ ( M ( x0, x)) + F( ( fx, fx)) τ( M ( x0, x)) + F( H ( Tx0, Tx)) + τ( M ( x0, x)) = F( α( x, x) + β( x, Tx0) + γ( x, Tx) + δ( x, Tx) + L( x, Tx0)) + τ( M ( x0, x)) F( α( x, x) + β( x, fx) + γ( x, fx) + δ( x, fx) + L( x, fx)) + τ ( M ( x0, x)) = F( α( x, x) + β( x, x) + γ( x, x3) + δ( x, x3) + L( x, x)) + τ ( M ( x0, x)) F[ α( x, x) + β( x, x) + γ( x, x3) + s[ δ( ( x, x) + ( x, x3))] + τ ( M( x0, x)) = F[( α + β + sδ) + ( γ + sδ) ] + τ( M( x0, x )) τ( M( x, x )) + F (( γ sδ) ) F[( α + β + sδ) ]. 0 As F is strictly icreasig we euce ( γ sδ) ( α + β + sδ). As α + β + γ + sδ < a γ hece we euce that γ sδ > 0 a so ( α + β + sδ) = λ, ( γ sδ) cosequetly, λ. Cotiuig this way we get a hece + < λ, + < λ 0. Proceeig this as i Theorem 3, we obtai that { x } is a Cauchy sequece. Sice X is a complete metric space, there exists some x X lim x = x. Now we prove that x = fx Tx. For this, sice lim[ α( fx, fx) + β( fx, Tx ) + γ( fx, Tx) + δ ( fx, Tx) + L( fx, Tx ))] lim[ α( fx, fx) + β( fx, fx) + γ( fx, fx+ ) + δ[ ( fx, fx) + ( fx, fx+ )] = α( fx, fx ) + β( fx, fx ) + γ( fx, fx ) + δ[ ( fx, fx ) + ( fx, fx )] = 0. By (F) lim F[ α( fx, fx) + β( fx, Tx ) + γ( fx, Tx) + δ( fx, Tx) + L( fx, Tx ))] =. x Therefore, lim F( H ( Tx, Tx)) =. Hece lim F( ( x+, fx+ )) lim F( hh ( Tx, Tx)) < lim [ F( H ( Tx, Tx)) + τ ( M ( x, x))] =. By usig (F) lim ( x, fx ) = 0 which implies = fx. Also + + lim F( ( fx, Tx)) lim F( hh ( Tx, Tx)) < lim [ F( H ( Tx, Tx)) + τ ( M ( x0, x))] =. Therefore lim ( fx, Tx) = 0, hece fx = Tx, that is x = fx Tx. Replacig b-metric space by metric space i the above result we get the followig corollary: Corollary 7 Assume that ( X, ) is a complete metric space, f : X X is cotiuous a T : X CB( X ). Assume that there exist τ R+ a F F is cotiuous from the right. Now if D( f, T, x 0) is a geeralize yamic process i such a way that τ ( M ( x, y)) + F ( H ( Tx, Ty)) F ( M ( x, y)) M ( x, y) = α( fx, fy) + β( fx, Tx) + γ( fy, Ty) + δ( fx, Ty) + L( fy, Tx)

5 47 Turkish Joural of Aalysis a Number Theory for each x, y ( fx ) with ( fx, fy ) > 0, α + β + γ + δ =, γ a L 0. The f a T possess a commo fixe poit. Example 8 Assume that X = [0, ) a is efie by 0 if x = y xy (, ) = 80 h if x y h >. Defie f : X X, T : X CB( X ) a F : R+ R by fx =, x [0, x] Tx = a F( x) = l( x). Defie a sequece { x } by for all N, with x 0 = a r = The a so o. Here x = = f( x) T( x0) = [0,] x3 = = f( x) T( x) = [0,] x4 = = f( x3) T( x) = [0, ] 4 D( f, T,) = {,,,...} 4 x 0 = xr Fix α = β = γ =, δ =, L = a s = 3, 0 80 clearly s. λ <. We ca check that is a b-metric. Now checkig for all xy, DfT (,,) we ca fi some τ : R+ R+ that satisfy the iequality () i such a way that H ( Tx, Ty ), Mxy (, ) > 0 for each xy, DfT (,, ). Moreover 0 = f(0) T(0) is fou to be the commo fixe poit of f a T. Puttig β = γ = δ = L = 0 a α = i the above result we get the followig corollary: Corollary 9 Assume that ( X, ) is a complete b-metric space, f : X X is cotiuous, T : X CB( X ), there exist a fuctio τ : R+ R+ a let F : R + R satisfy (F)-(F). Now if ( fx ) is a geeralize yamic process i such a way that for all x, y ( fx ) ( ( Tx, Ty) > 0 τ ( ( fx, fy)) + F( H ( Tx, Ty)) F( ( fx, fy))), the there is a commo fixe poit of f a T i.e. x = fx Tx. Puttig f as a ietity fuctio i Corollary 9 we get: Corollary 0 Assume that ( X, ) is a complete metric space, f : X X is cotiuous T : X CB( X ), there exist a fuctio τ : R+ R+ a suppose F : R + R satisfy (F)-(F). Now if ( x ) is a yamic process i such a way that: ( Tx, Ty) > 0 τ ( ( x, y)) + F ( H ( Tx, Ty)) F ( ( x, y)), the there exists a fixe poit of T. Iterests The authors eclare that they have o competig iterests. Refereces [] M. Abbas, B. Ali a S. Romaguera, Fixe a perioic poits of geeralize cotractios i metric spaces, Fixe Poit Theory a Applicatios 03, 03: 43. [] I. Altu, G. Durmaz, Some fixe poit theorems o orere coe metric spaces, Reicoti el Circolo Matematico i Palermo 58 (009) [3] M. Arsha, M. Abbas, A. Hussai a N. Hussai, Geeralize Dyamic Process for Geeralize (f,l)-almost F-Cotractio with Applicatios, J. Noliear Sci. Appl. 9 (06), [4] M. Arsha, E. Ameer a A.Hussai, Hary-Rogers-Type Fixe Poit Theorems for α-gf-cotractios, Archivum Mathematicum (BRNO) Tomus 5 (05), 9-4. [5] M. Arsha, A. Shoaib, I. Beg, Fixe poit of a pair of cotractive omiate mappigs o a close ball i a orere complete islocate metric space, Fixe Poit Theory a Appl. (03), 03:5, 5 pages. [6] M. Arsha, A. Shoaib, a P. Vetro, Commo Fixe Poits of a Pair of Hary Rogers Type Mappigs o a Close Ball i Orere Dislocate Metric Spaces, Joural of Fuctio Spaces, 03 (03), Article ID [7] M. Arsha, A. Shoaib, M. Abbas a A. Azam, Fixe Poits of a pair of Kaa Type Mappigs o a Close Ball i Orere Partial Metric Spaces, Miskolc Mathematical Notes, 4(3), 03, [8] M. Arsha, A. Azam, M. Abbas a A. Shoaib, Fixe poit results of omiate mappigs o a close ball i orere partial metric spaces without cotiuity U.P.B. Sci. Bull., Series A, 76(), 04. [9] A. Azam, M. Arsha, I. Beg, Commo fixe poits of two maps i coe metric spaces, Reicoti el Circolo Matematico i Palermo 57 (008) [0] I.A. Bakhti, The cotractio mappig priciple i quasi-metric spaces, Fuct. Aal. Uiaowsk Gos. Pe. Ist. 30 (989), [] I. Beg, M. Arsha, A. Shoaib, Fixe Poit o a Close Ball i orere islocate Metric Space, Fixe Poit Theory, 6(), 05. [] V. Berie, F. Vetro, Commo fixe poits of mappigs satisfyig implicit cotractive coitios, Fixe Poit Theory a Applicatios 0:05 (0). [3] V. Berie, F. Vetro, Fixe poit for cyclic weak (Ψ, C)- cotractios i 0-complete partial metric spaces, Filomat 7 (03) [4] A. Bhatt a H. Chara, Commo fixe poits for JH operators a occasioally weakly g-biase pairs uer relaxe coitio o probabilistic metric space, Joural of Fuctio Spaces a Applicatios, vol. 03, Article ID 84635, 6 pages, 03. [5] M. Boriceau, Fixe Poit theory for multivalue geeralize cotractio o a set with two b-metrics, stuia Uiv Babes, Bolya: Math. LIV (3) (009), -4. [6] M. Cosetio, P. Vetro, Fixe Poit Results for F-Cotractive Mappigs of Hary-Rogers-Type, Filomat 8:4 (04), [7] N. Hussai, J. Ahma a A. Azam, Geeralize fixe poit theorems for multi-value α - ψ -cotractive mappigs, J. Iequal. Appl., 04, 04:348. [8] N. Hussai, S. Al-Mezel a P. Salimi, Fixe poits for ψ - graphic cotractios with applicatio to itegral equatios, Abstract a Applie Aalysis, Volume 03, Article ID [9] N. Hussai, M. Arsha, A. Shoaib a Fahimui, Commo Fixe Poit results for α - ψ -cotractios o a metric space eowe with graph, J. Iequalities a Appl., 04, 04:36. [0] M. Jleli, H. Kumar, B. Samet a C. Vetro, O multivalue weakly Picar operators i partial Hausorff metric spaces, Fixe Poit Theory a Applicatios 05, 05: 5. [] Z. Kaelburg, L. Pauović, S. Raeović, A ote o fixe poit theorems for weakly T-Kaa a weakly T-Chatterjea cotractios i b-metric spaces, Gulf Joural of Mathematics 3 (05)

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