Turkish Joural of Aalysis ad Number Theory, 7, Vol. 5, No., 57-6 Available olie at http://pubs.sciepub.com/tjat/5//3 Sciece ad Educatio Publishig DOI:.69/tjat-5--3 Fixed Poit Theorems for Expasive Mappigs i -metric Spaces Rahim Shah *, Akbar Zada Departmet of Mathematics, Uiversity of Peshawar, Peshawar 5, Pakista *Correspodig author: safeer_rahim@yahoo.com, rahimshahstd@upesh.edu.pk Received October 8, 6; Revised Jauary 5, 7; Accepted February, 7 Abstract I this paper we prove some fixed poit theorems for cotractive as well as for expasive mappigs i -metric space by usig itegral type cotractio. Fially, we preset a example. Keywords: -metric space, fixed poit, itegral type cotractive mappig, expasive mappig Cite This Article: Rahim Shah, ad Akbar Zada, Fixed Poit Theorems for Expasive Mappigs i -metric Spaces. Turkish Joural of Aalysis ad Number Theory, vol. 5, o. (7: 57-6. doi:.69/tjat-5--3.. Itroductio I 6, Mustafa ad Sims [], itroduced the cocept of -metric spaces ad preseted some fixed poit theorems i -metric spaces. Mehdi et al. [9] gave ew approach to -metric spaces ad proved some fixed poit theorems i -metric spaces. Further we ote that may researchers have studied -metric spaces see, [-]. Moreover, I, A. Braciari [8] itroduced the cocept of itegral type cotractive mappigs i complete metric spaces ad study the existece of fixed poits for mappigs which is defied o complete metric space satisfies itegral type cotractio. Recetly A. Zada et al. [], preseted the cocept of itegral type cotractio with respect to -metric spaces ad proved some coupled coicidece fixed poit results for two pairs i such spaces, by usig the otio of itegral type cotractive mappigs give by A. Braciari [8]. I sectio 3, we preseted some fixed poit theorems of itegral type cotractive mappig i settig of -metric spaces. Moreover i sectio 4, we proved some fixed poit theorems of itegral type cotractio for expasive mappig. Also we give suitable example that support our mai results.. Prelimiaries Cosistet with Mustafa & Sims [] ad Braciari [8]. The followig defiitios ad results will be eeded i this paper. Defiitio. [] Let Y be a o-empty set ad : Y Y Y R + is a fuctio that satisfies the followig coditios: abc,, = ifa= b= c, ( ( (3 a, b, c > for all a, b Y with a b, aab,, abc,,, forallabc,, Ywithb c, (4 ( abc ( acb ( bca,, =,, =,, =... symmetry i all three variables, (5 ( aab,, ( ass,, + ( sbc,, (rectagle iequality for all abcs,,, Y. The the fuctio is called a geeralized metric ad the pair (Y, is called a -metric space. Example. [] Let Y= { xy, }. Defie o Y Y Y by = ( yyy = xyy xxx,,,,, xxy,, =,,, = ad exted to Y Y Y by usig the symmetry i the Y, is a -metric space. variables. The it is clear that Defiitio.3 [] Let ( Y, be a -metric space ad ( a a sequece of poits of. Y A poit a Y is said to be the limit of the sequece ( a, if lim m, + ( aa,, am = ad we say that the sequece ( a is -coverget to a. Propositio.4 [] Let ( Y, be a -metric space. The the followig are equivalet: a is -coverget to a. ( ( ( a, a, a as +. (3 ( a, aa, as +. ( ( a, am, a as m, +. Defiitio.5 [] Let ( Y, be a -metric space. A sequece ( a is called -Cauchy if for every >, there is k N such that ( a, am, a l <, for all ml,, k, that is ( a, am, al as ml,, +. Propositio.6 [] Let ( Y, be a -metric space. The the followig are equivalet:
58 Turkish Joural of Aalysis ad Number Theory ( The sequece ( a is -Cauchy. ( For every >, there is k N such that a, a, a <, for all m, k. m m Defiitio.7 [] A -metric space ( Y, is called -complete if every -Cauchy sequece i ( Y, is -coverget i ( Y,. Lemma.8 [] By the rectagle iequality (5 together with the symmetry (4, = ( bba,, ( baa,, ( aba,, ( baa,, abb + = I, Braciari i [8] itroduced a geeral cotractive coditio of itegral type as follows. Yd, be a complete metric space, Theorem.9 [8] Let α ad f : Y Y is a mappig such that for all xy, Y, ( ( t φ α φ t d f x, f y d xy, where φ : [, + [, + is oegative ad Lebesgueitegrable mappig which is summable (i.e., with fiite itegral o each compact subset of [, + such that for each >, ( t φ >, the f has a uique fixed poit a Y, such that for each x Y, lim f x = a. Lemma. By rectagle iequality (5 together with the symmetry (4 of defiitio., = abb,, bba,, baa + baa =.,, aba,, I this paper by usig the otio of itegral type give by Braciari i [8], we preseted some fixed poit theorems i -metric space. 3. Mai Results I this sectio, we prove some fixed poit theorems i geeralize metric space by usig itegral type cotractive mappigs. Our first mai result is follow as, Y, be a -metric space. Suppose Theorem 3. Let H : Y Y be a mappig satisfy the followig coditio for all ab, Y: Ha, Hb, Hb a, Ha, b ϕ( t k ϕ( t, (3. where k [, ad :[, [, ϕ + + is a Lebesgue itegrable mappig which is summable, o-egative ad such that for each >, >. The H has a uique fixed poit i Y. a by Proof. Choose a Y ad defie a : = H a. Notice that if a = a + for some N, the obviously H has a fixed poit. Thus, we suppose that that is, a a for all N. + ϕ t k, a,, a, a, a cotiuig this process, we get + + ϕ t k. a, a, a a, a, a Moreover, for all m, N; < m, (, m, m + + +... + a,,,,, 3, 3 am, am, am + + m a (, a, a (... ϕ k + k + k + + k t k a (, a, a. k So, Thus lim a (, am, am =. m, m, lim a, a, a =. m m This meas that ( a is -Cauchy sequece. Due to completeess of ( Y,, there exists l Y such that a is coverget to l. Suppose that Hl l, the ϕ ( t k. a, Hl, Hl a, a, l Takig limit as, ad usig the fact that fuctio is cotiuous, the ϕ ( l l l t k. l, Hl, Hl,, This cotradictio implies that Hl = l. For uiqueess, let l p such that Hp = p ad use Lemma., the = l,, l p Hl, Hl, Hp lhlp llp (,, k = k, which implies that l = p. The proof is completed. Y, be a -metric space. Suppose Corollary 3. Let H : Y Y be a mappig satisfy the followig coditio for all abc,, Y:
Turkish Joural of Aalysis ad Number Theory 59 Ha Hb Hc ( a, Ha, c ( a, Ha, b α + β where α β + < ad : [, [, ϕ is a Lebesgue itegrable mappig which is summable, o-egative ad such that for each >, >. The H has a uique fixed poit i Y. Theorem 3.3 Let ( Y, be a complete -metric space. Suppose H : Y Y be a mappig satisfy the followig coditio for all ab, Y, where x+ y+ z+ w< Ha Hb H c ( b, Hb, H b ϕ ϕ a Ha H a x t + y t ( a, Ha, Hb b, Hb, H a + z + w. 3 Ad ϕ : [, [, is a Lebesgue itegrable mappig which is summable, o-egative ad such that for each >, >. The H has a uique fixed poit i Y. Proof. Choose a Y. We costruct sequece i the followig way: = Ha for all =,,,... Notice that if a = a + for some N, the obviously H has a fixed poit. Thus, we suppose that a for all N. That is, ( a, a +, a + >. From above coditio, with a = a ad b = a, we have ( Ha, Ha, H a ( a, Ha, H a ( a, Ha, H a ϕ + ϕ 3 ( a, Ha, Ha ( a, Ha, H a x t y t + z + w, implies that a (,, a (, a, a (,, x ϕ( t + y a (, a, a (,, + z ϕ( t + w ϕ( t. So, a (,, a (, a, k, x+ z where k = <. The y w + + ϕ t k, a, a, a a, a, a N for all. From defiitio of -metric space, we kow that ( a, a, ( a,,, with a a +, also by Lemma., we kow that a ( +, a +, a a a a +. The by above iequality, a ( +, a +, a k. Moreover, for all m, N; < m, ( am, am, a a (,, a ( +,, ϕ( t + a ( +, 3, 3 a ( m, am, am + ϕ( t +... + + + 3 m a (, a, a ( k + k + k +... + k k a (, a, a. k So, lim a (, am, am =,. m Thus m, lim a, a, a =. m m This meas that ( a is -Cauchy sequece. Due to completeess of ( Y,, there exists p Y a such that is coverget to p. From the above coditio of theorem, with a = a ad b = p, ( a, Ha, H a ( p, Hp, H p ϕ + ϕ 3 ( a, Ha, Hp ( p, Hp, H a Ha Hp H p x t y t + z + w ϕ t. The ( +,, a Hp H p a (,, phph,, p x ϕ( t + y ( a,, Hp ( p, Hp, 3 + z + w.
6 Turkish Joural of Aalysis ad Number Theory Takig limit as of above iequality, ( z+ w phph,, p pphp,, ϕ( t ( y Now, if Hp. = H p, the H has a fixed poit. Hece, we suppose that Hp H p. Therefore, by defiitio of -metric space, we get ( z+ w ( z+ w ( y p ( y phph,, p pphp,, ϕ( t phph which implies that i.e., p phph, =, p = Hp = H p. The proof is completed. 4. Itegral Type Cotractio for Expasive Mappigs I this sectio of our paper, we prove some fixed poit theorems for expasive mappig of itegral type cotractio i -metric spaces. Theorem 4. Let ( Y, be a complete -metric space. Suppose H : Y Y be a oto mappig satisfy the followig coditio for all ab, Y, where α > holds ( Ha, H a, Hb ( a, Ha, b ϕ( t α ϕ t. Ad ϕ : [, [, is a Lebesgue itegrable mappig which is summable, o-egative ad such that for each >, >. The H has a uique fixed poit i Y. Proof. Choose a Y, as H is oto map, the there exists a Y such that a = Ha. If we cotiue this process, we ca get a = Ha + for all N. I case a = a +, for some N, the clearly a is a fixed poit of H. Next, we suppose that a a + for all. From above coditio of this result, with a = a + ad b = a which implies that (,, Ha a a = a Ha a α = α ( +,, ( +, +, ( +,,, ϕ t h,, a, a a, a, a where h = <. The α + ϕ t h. a, a, a a, a, a By Lemma (., we get a (,, a ( +, a, a a (, a, a h. If we follow the lies of the proof of result 3., we derive that ( a is a -Cauchy sequece. Sice ( Y, is complete, there exists p Y such that a p as. Cosequetly, sice H is oto, so there exists l Y such that p = Hl. From above coditio of this theorem, with a = a + ad b = l, = a, a, p H, H a, Hl a Ha p α a a p = α ( +, +, ( +,,. Takig limit as i above iequality, we get ϕ( t ( = lim =. uul,, a,, That is, u = l. The u = Hl = Hu. For uiqueess, let r s such that Hs = s ad Hr = r. By above coditio of result, we get = r,, r s Hr, H r, Hs r Hr s α rrs α rrs > (,, (,,, which arise cotradictio, Hece r = s. The proof is completed. Theorem 4. Let ( Y, be a complete -metric space. Suppose H : Y Y be a oto mappig satisfy the followig coditio for all ab, Y, where α > ( Ha, Hb, H b ( a, Ha, H a α. (4. Ad ϕ : [, [, is a Lebesgue itegrable mappig which is summable, o-egative ad such that for each >, >. The H has a uique fixed poit i Y.
Turkish Joural of Aalysis ad Number Theory 6 Proof. Choose a Y, as H is oto map, the there exists a Y such that a = Ha. If we cotiue this process, we ca get a = Ha + for all N. I case a = a +, for some N, the clearly a is a fixed poit of H. Next, we suppose that a a + for all. From (4., with a = a + ad b = a implies that ad so, where ( +,, Ha Ha H a α ( +, +, + a Ha H a (,, α, ( +,,, ( +,, (,,, h h = <. By the proof of Theorem 3., we ca α Y, is show that a is a -Cauchy sequece. Sice complete, it exists p Y such that a p as. Cosequetly, sice H is oto, so there exists l Y such that p = Hl. From 4., with a = l ad b = a +, = p, a, a Hl, H, H α lhlh l ϕ( t. O takig limit i above iequality, lhlh,, l =. That is, let r s such that Hs = s ad Hr = r. Now, by usig coditio 4., rrs,, rrs,, ϕ( t > which is cotradictio. Hece r 5. Example l = Hl = H l. For uiqueess,, = s. The proof is completed. I this sectio, we preset a example, which idicates that how our results ca be applied to differet problems. Y =, ad Example 5. Let [, if a = b = c ( abc,, = max { a, b, c}, otherwise be a -metric space o Y. Defie H : Y Y by Ha = a. 5 ad The the coditio of Theorem 3. holds. I fact, ad so,,, ( Ha, Hb, Hb a b b 5 5 5 ϕ( t, ϕ( t = ahab,, max, ϕ( t = ϕ( t { ab}, Ha, Hb, Hb a, Ha, b 4. That is, coditio of Theorem 3. holds with k = [,. 4 Refereces [] R. Agarwal, E, Karapiar, Remarks o some coupled fixed poit theorems i -metric spaces, Fixed Poit Theory Appl., 3, (3. [] H. Aydi, M. Postolache, W. Shataawi, Coupled fixed poit results for (ϕ-ψ-weakly cotractive mappigs i ordered -metric spaces, Comput. Math. Appl., vol. 63(,, pp. 98-39. [3] H. Aydi, W. Shataawi, C. Vetro, O geeralized weakly -cotractio mappigs i -metric spaces, Comput. Math. Appl., vol. 6,, pp. 4-49. [4] H. Aydi, B. Damjaovic, B. Samet, W. Shataawi, Coupled fixed poit theorems for oliear cotractios i partially ordered -metric spaces, Math. Comput. Model., vol. 54,, pp. 443-45. [5] H. Aydi, E. Karapiar, W. Shataawi, Tripled fixed poit results i geeralized metric spaces, J. Appl. Math.,, Article ID 3479. [6] H. Aydi, E. Karapiar, Z. Mustafa. O commo fixed poits i -metric spaces usig (E.A property, Comput. Math. Appl., vol. 64(6,, pp. 944-956. [7] H. Aydi, E. Karapiar, W. Shataawi, Tripled commo fixed poit results for geeralized cotractios i ordered geeralized metric spaces, Fixed Poit Theory Appl.,, (. [8] A. Braciari. A fixed poit theorem for mappigs satisfyig a geeral cotractive coditio of itegral type. Iteratioal Joural of Mathematics ad Mathematical Scieces, vol. 9, o. 9,, pp. 53-536. [9] A. Mehdi, K. Erdal, S. Peyma, A ew approach to -metric ad related fixed poit theorems, J. Ieq. Apps., 3, 3:454. [] Z. Mustafa, B. Sims, A ew approach to geeralized metric spaces, J. Noliear Covex Aal., vol. 7 (, 6, pp. 89-97. [] Z. Mustafa, A ew structure for geeralized metric spaces with applicatios to fixed poit theory, PhD thesis, The Uiversity of Newcastle, Australia, 5. [] Z. Mustafa, M. Khadaqji, W. Shataawi, Fixed poit results o complete -metric spaces, Studia Sci. Math. Hug., vol. 48,, pp. 34-39. [3] Z. Mustafa, H. Aydi, E. Karapiar, Mixed g-mootoe property ad quadruple fixed poit theorems i partially ordered metric space, Fixed Poit Theory Appl.,, 7. [4] Z. Mustafa, Commo fixed poits of weakly compatible mappigs i -metric spaces, Appl. Math. Sci., vol. 6(9,, pp. 4589-46. [5] Z. Mustafa, Some ew commo fixed poit theorems uder strict cotractive coditios i -metric spaces, J. Appl. Math.,, Article ID 48937. [6] Z. Mustafa, Mixed g-mootoe property ad quadruple fixed poit theorems i partially ordered -metric spaces usig (ϕ-ψ cotractios, Fixed Poit Theory Appl.,, 99. [7] Z. Mustafa, W. Shataawi, M. Bataieh, Existece of fixed poit results i -metric spaces, It. J. Math. Math. Sci., 9, Article ID 838 (9.
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