Available online at www.isr-ublications.com/jnsa J. Nonlinear Sci. Al., 10 2017), 429 435 Research Article Journal Homeage: www.tjnsa.com - www.isr-ublications.com/jnsa A shar generalization on cone b-metric sace over Banach algebra Huaing Huang a,, Stojan Radenović b, Guantie Deng a a School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Comlex Systems, Ministry of Education, Beijing, 100875, China. b Faculty of Mechanical Engineering, University of Belgrade, Kraljice Marije 16, 11120, Beograd, Serbia. Communicated by Y. J. Cho Abstract The aim of this aer is to generalize a famous result for Banach-tye contractive maing from ρk) [0, s 1 ) to ρk) [0, 1) in cone b-metric sace over Banach algebra with coefficient s 1, where ρk) is the sectral radius of the generalized Lischitz constant k. Moreover, some similar generalizations for the contractive constant k from k [0, s 1 ) to k [0, 1) in cone b-metric sace and in b-metric sace are also obtained. In addition, two examles are given to illustrate that our generalizations are in fact real generalizations. c 2017 All rights reserved. Keywords: Cone b-metric sace over Banach algebra, fixed oint, c-sequence, iterative sequence. 2010 MSC: 47H10, 54H25. 1. Introduction and reliminaries Since Bakhtin [1] or Czerwik [4] introduced b-metric sace, also called metric tye sace by some authors, as a large generalization of metric sace, eole have aid close attention to fixed oint results in such saces for many years. Recently, Jovanović et al. [10] roved Banach-tye version of a fixed oint result see [2]) for contractive maing including the contractive constant k [0, 1 s ) in b-metric sace with coefficient s 1. Subsequently, Huang and Xu [8] exanded the work of [10] into cone b-metric sace with coefficient s 1, where the contractive constant k also satisfies k [0, 1 s ). Later on, Huang and Radenović [7] gave a further generalization from cone b-metric sace with coefficient s 1 to cone b-metric sace over Banach algebra with the same coefficient. They considered the Banach-tye version of a fixed oint result with the generalized Lischitz constant k satisfying ρk) [0, 1 s ), where ρk) is the sectral radius of k. So far, there have been some oen questions whether the result in b-metric sace or in cone b-metric sace is true for k [0, 1), and whether the result in cone b-metric sace over Banach algebra is true for ρk) [0, 1). In this aer, by using a new method of roof, we rove that the answers to the above questions are ositive. Corresonding author Email addresses: mathhh@163.com Huaing Huang), radens@beotel.net Stojan Radenović), denggt@bnu.edu.cn Guantie Deng) doi:10.22436/jnsa.010.02.09 Received 2016-10-29
H. Huang, S. Radenović, G. Deng, J. Nonlinear Sci. Al., 10 2017), 429 435 430 Definition 1.1 [1, 4]). Let X be a nonemty) set and s 1 be a given real number. A function d : X X [0, ) is called a b-metric on X, if for all x, y, z X the following conditions hold: b1) d x, y) = 0 iff x = y; b2) d x, y) = d y, x); b3) d x, z) s [d x, y) + d y, z)]. In this case, the air X, d) is called a b-metric sace. For some concets such as b-convergence, b-cauchy sequence and b-comleteness in the setting of b-metric saces, the reader refers to [10, 13, 15]. In the following examle we correct some errors from several aers see [3, 12 16]) for b-metric sace with the false coefficient s = 2 1. As a matter of fact, its correct coefficient should be s = 2 1 1. Examle 1.2. The set l R) with 0 < < 1, where { } l R) := {x n } R x n <, together with the maing d : l R) l R) [0, ) defined by dx, y) = x n y n, for each x = {x n }, y = {y n } l R) is a b-metric sace with coefficient s = 2 1 1. In fact, we only need to rove that condition b3) in Definition 1.1 is satisfied. To this end, let x = {x n }, y = {y n }, z = {z n } l R), we shall show that x n z n 2 1 1 x n y n + y n z n. 1.1) Denote a n = x n y n, b n = y n z n, then x n z n = a n + b n, so 1.1) becomes a n + b n 2 1 1 a n + b n. 1.2) In order to rove 1.2), we notice that the following inequalities: then a + b) a + b, a, b 0, 0 < 1), a + b) 2 1 a + b ), a, b 0, 1), a n + b n = a n + [ a n + b n ) ] 1 b n 2 1 1 [ a n + b n ) a n + ] 1 b n.
H. Huang, S. Radenović, G. Deng, J. Nonlinear Sci. Al., 10 2017), 429 435 431 Let A be a real Banach algebra, be its norm and θ be its zero element. A nonemty closed subset P of A is called a cone, if P 2 = P P P, P P) = {θ} and λp + µp P for all λ, µ 0. We denote intp as the interior of P. If intp, then P is said to be a solid cone. Define a artial ordering with resect to P by u v, iff v u P. Define u v, iff v u intp. In the sequel, unless otherwise secified, we always suose that A is a real Banach algebra with a unit e, P is a solid cone in A, and and are artial orderings with resect to P. Definition 1.3 [7]). Let X be a nonemty) set, s 1 be a constant and A be a Banach algebra. Suose that a maing d : X X A satisfies for all x, y, z X, d1) θ dx, y) and dx, y) = θ, iff x = y; d2) dx, y) = dy, x); d3) dx, z) s[dx, y) + dy, z)]. Then d is called a cone b-metric on X, and X, d) is called a cone b-metric sace over Banach algebra. For some examles on cone b-metric sace over Banach algebra, the reader refers to [6, 7]. Definition 1.4 [6]). A sequence {u n } in a solid cone P is said to be a c-sequence, if for each c θ, there exists a natural number N such that u n c for all n > N. Definition 1.5. Let X, d) be a cone b-metric sace over Banach algebra and {x n } a sequence in X. We say that i) {x n } b-converges to x X, if {dx n, x)} is a c-sequence; ii) {x n } is a b-cauchy sequence, if {dx n, x m )} is a c-sequence for n, m; iii) X, d) is b-comlete, if every b-cauchy sequence in X is b-convergent. Lemma 1.6 [7]). Let {u n } and {v n } be two c-sequences in a solid cone P. If α, β P are two arbitrarily given vectors, then {αu n + βv n } is a c-sequence. Lemma 1.7 [9]). If u v and v w, then u w. Lemma 1.8 [6]). Let A be a Banach algebra with a unit e, then the sectral radius ρk) of k A holds ρk) = lim n kn 1 n = inf k n 1 n. If ρk) < 1, then e k is invertible in A, moreover, e k) 1 = i=0 ki. Lemma 1.9 [6]). Let A be a Banach algebra with a unit e. Let k A and ρk) < 1. Then {k n } is a c-sequence. 2. Main results Theorem 2.1. Let X, d) be a b-comlete cone b-metric sace over Banach algebra with coefficient s 1. Suose that T : X X is a maing such that for all x, y X it holds: dtx, Ty) kdx, y), 2.1) where k P is a generalized Lischitz constant with ρk) < 1. Then T has a unique fixed oint in X. And for any x X, the iterative sequence {T n x} n N) b-converges to the fixed oint. Proof. Let x 0 X and x n+1 = Tx n for all n N. We divide the roof into three cases.
H. Huang, S. Radenović, G. Deng, J. Nonlinear Sci. Al., 10 2017), 429 435 432 Case 1: Let ρk) [0, 1 s ) s > 1). By 2.1), we have dx n, x n+1 ) = dtx n 1, Tx n ) kdx n 1, x n ) = kdtx n 2, Tx n 1 ) k 2 dx n 2, x n 1 ). k n dx 0, x 1 ). In view of ρk) < 1 s, then ρsk) = sρk) < 1, so by Lemma 1.8, we get that e sk is invertible and e sk) 1 = i=0 sk)i. Thus for any n > m, it follows that dx m, x n ) s[dx m, x m+1 ) + dx m+1, x n )] sdx m, x m+1 ) + s 2 [dx m+1, x m+2 ) + dx m+2, x n )] sdx m, x m+1 ) + s 2 dx m+1, x m+2 ) + s 3 [dx m+2, x m+3 ) + dx m+3, x n )] sdx m, x m+1 ) + s 2 dx m+1, x m+2 ) + s 3 dx m+2, x m+3 ) + + s n m 1 dx n 2, x n 1 ) + s n m 1 dx n 1, x n ) sk m dx 0, x 1 ) + s 2 k m+1 dx 0, x 1 ) + s 3 k m+2 dx 0, x 1 ) + + s n m 1 k n 2 dx 0, x 1 ) + s n m 1 k n 1 dx 0, x 1 ) sk m e + sk + s 2 k 2 + + s n m 2 k n m 2 + s n m 1 k n m 1 )dx 0, x 1 ) [ ] sk m sk) i dx 0, x 1 ) i=0 = sk m e sk) 1 dx 0, x 1 ). Note that ρk) < 1 s < 1 and Lemma 1.9, it is easy to see that {km } is a c-sequence. Therefore, by using Lemma 1.6 and Lemma 1.7, we claim that {x n } is a b-cauchy sequence. Since X, d) is b-comlete, then there exists x X such that {x n } b-converges to x. Next, let us show that x is a fixed oint of T. Indeed, by 2.1), we have dx n+1, Tx ) kdx n, x ). 2.2) Since {dx n, x )} is a c-sequence, then by Lemma 1.6, it is not hard to verify that {kdx n, x )} is a c- sequence. Hence, by Lemma 1.7, 2.2) imlies that {dx n+1, Tx )} is also a c-sequence, which means that {x n } b-converges to Tx. By the uniqueness of limit of a b-convergent sequence, we get Tx = x. That is to say, x is a fixed oint of T. Further, x is the unique fixed oint of T. Actually, assume that there is another fixed oint y, then by 2.1), it is obvious that dx, y ) = dtx, Ty ) kdx, y ) k 2 dx, y ) k n dx, y ). Now that {k n } is a c-sequence, then by Lemma 1.6 and Lemma 1.7, we obtain dx, y ) = θ. This leads to x = y. Case 2: Let ρk) [ 1 s, 1) s > 1). In this case, we have [ρk)]n 0 as n, then there is n 0 N such that [ρk)] n 0 < 1 s. Notice that ρk n 0 ) = lim n kn 0 ) n 1 n
H. Huang, S. Radenović, G. Deng, J. Nonlinear Sci. Al., 10 2017), 429 435 433 = lim n kn } {{ k n } n 0 terms 1 n lim k n k n n }{{} n 0 terms n ) ) = lim n kn n 1 lim n kn n 1 }{{} n 0 terms = [ρk)] n 0 < 1 s, and by 2.1) for all x, y X, it follows that dt n 0 x, T n 0 y) = d TT n 0 1 x), TT n 0 1 y) ) kdt n 0 1 x, T n 0 1 y) = kd TT n 0 2 x), TT n 0 2 y) ) k 2 dt n 0 2 x, T n 0 2 y). k n 0 dx, y). Then by Case 1, we claim that the maing T n 0 has a unique fixed oint x X. Now we rove that x is also a fixed oint of T. As a matter of fact, on account of T n 0x = x, we have T n 0 Tx ) = T n 0+1 x = TT n 0 x ) = Tx, then Tx is also a fixed oint of T n 0. Thus, by the uniqueness of fixed oint of T n 0, it ensures us that Tx = x. In other words, x is also a fixed oint of T. Finally, we show that the fixed oint of T is unique. Virtually, we suose for absurd that there exists another fixed oint x of T, that is, Tx = x, Tx = x, then T n 0 x = T n 0 1 Tx ) = T n 0 1 x = = Tx = x, T n 0 x = T n 0 1 Tx ) = T n 0 1 x = = Tx = x, which imly that x and x are two fixed oints of T n 0. Because the fixed oint of T n 0 is unique, we claim that x = x. Case 3: s = 1. Since ρk) < 1, reeat the rocess of Case 1, then the claim holds. Corollary 2.2. Let X, d) be a b-comlete cone b-metric sace with coefficient s 1. Suose that T : X X is a maing such that for all x, y X, it holds: dtx, Ty) kdx, y), where k [0, 1) is a real constant. Then T has a unique fixed oint in X. And for any x X, the iterative sequence {T n x} n N) b-converges to the fixed oint. Proof. Choose k R in Theorem 2.1, then the roof is comleted. Corollary 2.3. Let X, d) be a b-comlete b-metric sace with coefficient s 1. Suose that T : X X is a maing such that for all x, y X, it holds: dtx, Ty) kdx, y), where k [0, 1) is a real constant. Then T has a unique fixed oint in X. And for any x X, the iterative sequence {T n x} n N) b-converges to the fixed oint.
H. Huang, S. Radenović, G. Deng, J. Nonlinear Sci. Al., 10 2017), 429 435 434 Remark 2.4. Theorem 2.1 greatly generalizes [7, Theorem 2.1] from ρk) [0, 1 s ) to ρk) [0, 1). Corollary 2.2 greatly generalizes [8, Theorem 2.1] from k [0, 1 s ) to k [0, 1). Corollary 2.3 greatly generalizes [10, Theorem 3.3] from k [0, 1 s ) to k [0, 1). Remark 2.5. Regarding the imrovement of contractive coefficients, there have been some articles dealing with them. For instance, comared with [11], [5] generalizes the range of the coefficient λ from λ 0, 1 2 ) to λ 0, 1) for quasi-contraction, which is an interesting generalization. Whereas, our results generalize some famous results on Banach-tye contractions for the coefficient k from ρk) [0, 1 s ) to ρk) [0, 1), as well as from k [0, 1 s ) to k [0, 1). Consequently, our generalizations are indeed shar generalizations. The following examles illustrate our conclusions. Examle 2.6. Let X = [0, 1], A = C 1 R X) and define a norm on A by u = u + u. Define multilication in A as just ointwise multilication. Then A is a real Banach algebra with a unit e = 1 et) = 1 for all t X). The set P = {u A : ut) 0, t X} is a non-normal solid cone see [9]). Define a maing d : X X A by dx, y)t) = x y 2 e t. We have that X, d) is a b-comlete cone b- metric sace over Banach algebra A with coefficient s = 2. Define a self-maing T on X by Tx = 2 2 x. Put k = 1 2 + 1 4 t. Then dtx, Ty) kdx, y) for all x, y X. Simle calculations show that 1 s = 1 2 < ρk) = 3 4 < 1. Clearly, ρk) / [0, 1 s ), but ρk) [ 1 s, 1). Hence, [7, i) of Theorem 2.1] is not satisfied. That is to say, [7, Theorem 2.1] cannot be used in this examle. However, our Theorem 2.1 is satisfied. Accordingly, T has a unique fixed oint x = 0. Examle 2.7. Let X = [0, 3 5 ], E = R2 and 5 be a constant. Put P = {x, y) E : x, y 0}. We define d : X X E as dx, y) = x y, for all x, y X. Then X, d) is a b-comlete b-metric sace with coefficient s = 2 1. Define a self-maing T on X by Tx = 1 2 cos x 2 x 1 2 ), for all x X. Hence, for all x, y X, we seculate that dtx, Ty) = Tx Ty = 1 2 cos x 2 cos y ) x 1 2 2 y 1 ) 2 1 cos x 2 2 cos y ) + x y 2 1 ) x + y 2 x y + x y 8 0.575 x y. In view of 5, then k = 0.575 / [0, 1 s ), but k = 0.575 [ 1 s, 1). Thus, [10, Theorem 3.3] does not hold in this case. In other words, [10, Theorem 3.3] is not alicable in this examle. However, our Corollary 2.3 can be utilized in this case. To sum u, x 0 X satisfied with 0.472251591454 < x 0 < 0.472251591479 is the unique fixed oint of T. Acknowledgment The research was artially suorted by the National Natural Science Foundation of China 11271045). References [1] I. A. Bakhtin, The contraction maing rincile in almost metric sace, Russian) Functional analysis, Ulyanovsk. Gos. Ped. Inst., Ulyanovsk, 1989), 26 37. 1, 1.1 [2] S. Banach, Sur les oérations dans les ensembles abstraits et leur alication aux équations intégrales, Fund. Math., 3 1922), 133 181. 1 [3] M. Boriceanu, M. Bota, A. Petruşel, Multivalued fractals in b-metric saces, Cent. Eur. J. Math., 8 2010), 367 377. 1 [4] S. Czerwik, Contraction maings in b-metric saces, Acta Math. Inform. Univ. Ostraviensis, 1 1993), 5 11. 1, 1.1
H. Huang, S. Radenović, G. Deng, J. Nonlinear Sci. Al., 10 2017), 429 435 435 [5] L. Gajić, V. Rakočević, Quasi-contractions on a nonnormal cone metric sace, Funct. Anal. Al., 46 2012), 62 65. 2.5 [6] H.-P. Huang, S. Radenović, Common fixed oint theorems of generalized Lischitz maings in cone b-metric saces over Banach algebras and alications, J. Nonlinear Sci. Al., 8 2015), 787 799. 1, 1.4, 1.8, 1.9 [7] H.-P. Huang, S. Radenović, Some fixed oint results of generalized Lischitz maings on cone b-metric saces over Banach algebras, J. Comut. Anal. Al., 20 2016), 566 583. 1, 1.3, 1, 1.6, 2.4, 2.6 [8] H.-P. Huang, S.-Y. Xu, Correction: Fixed oint theorems of contractive maings in cone b-metric saces and alications, Fixed Point Theory Al., 2014 2014), 5 ages. 1, 2.4 [9] S. Janković, Z. Kadelburg, S. Radenović, On cone metric saces: a survey, Nonlinear Anal., 74 2011), 2591 2601. 1.7, 2.6 [10] M. Jovanović, Z. Kadelburg, S. Radenović, Common fixed oint results in metric-tye saces, Fixed Point Theory Al., 2010 2015), 15 ages. 1, 1, 2.4, 2.7 [11] Z. Kadelburg, S. Radenović, V. Rakočević, Remarks on Quasi-contraction on a cone metric sace, Al. Math. Lett., 22 2009), 1674 1679. 2.5 [12] P. K. Mishra, S. Sachdeva, S. K. Banerjee, Some fixed oint theorems in b-metric sace, Turkish J. Anal. Number Theory, 2 2014), 19 22. 1 [13] W. Sintunavarat, Fixed oint results in b-metric saces aroach to the existence of a solution for nonlinear integral equations, ev. R. Acad. Cienc. Exactas Fs. Nat. Ser. A Math. RACSAM, 110 2016), 585 600. 1 [14] W. Sintunavarat, Nonlinear integral equations with new admissibility tyes in b-metric saces, J. Fixed Point Theory Al., 18 2016), 397 416. [15] O. Yamaod, W. Sintunavarat, Y. J. Cho, Common fixed oint theorems for generalized cyclic contraction airs in b-metric saces with alications, Fixed Point Theory Al., 2015 2015), 18 ages. 1 [16] O. Yamaod, W. Sintunavarat, Y. J. Cho, Existence of a common solution for a system of nonlinear integral equations via fixed oint methods in b-metric saces, Oen Math., 14 2016), 128 145. 1