Fixed point theorems of contractive mappings in cone b-metric spaces and applications

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1 Huang an Xu Fixe Point Theory an Applications 2013, 2013:112 R E S E A R C H Open Access Fixe point theorems of contractive mappings in cone b-metric spaces an applications Huaping Huang 1 an Shaoyuan Xu 2* * Corresponence: xushaoyuan@126.com 2 School of Mathematics an Computer, Gannan Normal University, Ganzhou, , China Full list of author information is available at the en of the article Abstract In this paper we present some new examples in cone b-metric spaces an prove some fixe point theorems of contractive mappings without the assumption of normality in cone b-metric spaces. The results not only irectly improve an generalize some fixe point results in metric spaces an b-metric spaces, but also expan an complement some previous results in cone metric spaces. In aition, we use our results to obtain the existence an uniqueness of a solution for an orinary ifferential equation with a perioic bounary conition. Keywors: coneb-metric space; fixe point; perioic bounary problem 1 Introuction Fixe point theory plays a basic role in applications of many branches of mathematics. Fining a fixe point of contractive mappings becomes the center of strong research activity. There are many works about the fixe point of contractive maps see, for example, [1, 2]. In [2], Polish mathematician Banach prove a very important result regaring a contraction mapping, known as the Banach contraction principle, in In [3], Bakhtin introuce b-metric spaces as a generalization of metric spaces. He prove the contraction mapping principle in b-metric spaces that generalize the famous Banach contraction principle in metric spaces. Since then, several papers have ealt with fixe point theory or the variational principle for single-value an multi-value operators in b-metric spaces see [4 6] an the references therein. In recent investigations, the fixe point in nonconvex analysis, especially in an orere norme space, occupies a prominent place in many aspects see [7 10]. The authors efine an orering by using a cone, which naturally inuces a partial orering in Banach spaces. In [7], Huang an Zhang introuce cone metric spaces as a generalization of metric spaces. Moreover, they prove some fixe point theorems for contractive mappings that expane certain results of fixe points in metric spaces. In [10], Hussain an Shah introuce cone b-metric spaces as a generalization of b-metric spaces an cone metric spaces. They establishe some topological properties in such spaces an improve some recent results about KKM mappings in the setting of a cone b-metric space. Throughout this paper, we firstly offer some new examples in cone b-metric spaces, then obtain some fixe point theorems of contractive mappings without the assumption of normality in cone b-metric spaces. Furthermore, we supportour results by an example. The results greatly generalize an improve the work of [3, 4, 7, 8]an[10] Huang an Xu; licensee Springer. This is an Open Access article istribute uner the terms of the Creative Commons Attribution License which permits unrestricte use, istribution, an reprouction in any meium, provie the original work is properly cite.

2 Huangan Xu Fixe Point Theory an Applications 2013, 2013:112 Page 2 of 10 As some applications, we show the existence an uniqueness of a solution for a first-orer orinary ifferential equation with a perioic bounary conition. Consistent with Huang an Zhang [7], the following efinitions an results will be neee in the sequel. Let E be a real Banach space an P be a subset of E.Byθ we enote the zero element of E an by int P the interior of P.ThesubsetP iscalleaconeifanonlyif: i P is close, nonempty, an P {θ}; ii a, b R, a, b 0, x, y P ax + by P; iii P P={θ}. On this basis, we efine a partial orering with respect to P by x y if an only if y x P. Weshallwritex < y to inicate that x y but x y, whilex y will stan for y x int P.Write as the norm on E.TheconeP is calle normal if there is a number K > 0 such that for all x, y E, θ x y implies x K y. The least positive number satisfying the above is calle the normal constant of P.It is well known that K 1. In the following, we always suppose that E is a Banach space, P is a cone in E with int P an is a partial orering with respect to P. Definition 1.1 [7] Let X be a nonempty set. Suppose that the mapping : X X E satisfies: 1 θ < x, y for all x, y X with x y an x, y=θ if an only if x = y; 2 x, y=y, x for all x, y X; 3 x, y x, z+z, y for all x, y, z X. Then is calle a cone metric on X an X, is calle a cone metric space. Definition 1.2 [10] Let X be a nonempty set an s 1 be a given real number. A mapping : X X E is sai to be cone b-metric if an only if, for all x, y, z X, the following conitions are satisfie: i θ < x, y with x y an x, y=θ if an only if x = y; ii x, y =y, x; iii x, y s[x, z+z, y]. The pair X, iscalleacone b-metricspace. Remark 1.3 The class of cone b-metricspacesislargerthantheclassofconemetricspaces since any cone metric space must be a cone b-metric space. Therefore, it is obvious that cone b-metric spacesgeneralize b-metric spaces an cone metric spaces. We can present a number of examples, as follows, which show that introucing a cone b-metric space instea of a cone metric space is meaningful since there exist cone b-metric spaces which are not cone metric spaces. Example 1.4 Let E = R 2, P = {x, y E : x, y 0} E, X = R an : X X E such that x, y= x y p, α x y p, where α 0anp >1aretwoconstants.ThenX, is aconeb-metric space, but not a cone metric space. In fact, we only nee to prove iii in Definition 1.2asfollows: Let x, y, z X.Setu = x z, v = z y,sox y = u + v.fromtheinequality a + b p 2 max{a, b} p 2 p a p + b p for all a, b 0,

3 Huangan Xu Fixe Point Theory an Applications 2013, 2013:112 Page 3 of 10 we have x y p = u + v p u + v p 2 p u p + v p =2 p x z p + z y p, which implies that x, y s[x, z+z, y] with s =2 p >1.But x y p x z p + z y p is impossible for all x > z > y. Inee, taking account of the inequality a + b p > a p + b p for all a, b >0, we arrive at x y p = u + v p =u + v p > u p + v p =x z p +z y p = x z p + z y p, for all x > z > y. Thus, 3 in Definition 1.1 is not satisfie, i.e., X, is not a cone metric space. Example 1.5 Let X = l p with 0 < p <1,wherel p = {{x n } R : n=1 x n p < }. Let : X X R +, x, y= n=1 x n y n p 1 p, where x = {x n }, y = {y n } l p.thenx, isab-metric space see [5]. Put E = l 1, P = {{x n } E : x n 0, for all n 1}. Letting the mapping : X X E be efine by x, y ={ x,y 2 n } n 1,weconcluethatX, isaconeb-metric space with the coefficient s =2 1 p > 1, but it is not a cone metric space. Example 1.6 Let X = {1, 2, 3, 4}, E = R 2, P = {x, y E : x 0, y 0}.Define : X X E by x y 1, x y 1, if x y, x, y= θ, ifx = y. Then X, isaconeb-metric space with the coefficient s = 6. But it is not a cone metric 5 space since the triangle inequality is not satisfie. Inee, 1, 2 > 1, 4 + 4, 2, 3, 4 > 3, 1 + 1, 4. Definition 1.7 [10] Let X, beaconeb-metric space, x X an {x n } be a sequence in X.Then i {x n } converges to x whenever, for every c E with θ c, there is a natural number N such that x n, x c for all n N. We enote this by lim n x n = x or x n x n.

4 Huangan Xu Fixe Point Theory an Applications 2013, 2013:112 Page 4 of 10 ii {x n } is a Cauchy sequence whenever, for every c E with θ c,thereisanatural number N such that x n, x m c for all n, m N. iii X, is a complete cone b-metric space if every Cauchy sequence is convergent. The following lemmas are often use in particular when ealing with cone metric spaces in which the cone nee not be normal. Lemma 1.8 [9] Let P be a cone an {a n } be a sequence in E. If c int Panθ a n θ as n, then there exists N such that for all n > N, we have a n c. Lemma 1.9 [9] Let x, y, z E, if x yany z, then x z. Lemma 1.10 [10] Let P be a cone an θ u cforeachc int P, then u = θ. Lemma 1.11 [11] Let P be a cone. If u Panu ku for some 0 k <1,then u = θ. Lemma 1.12 [9] Let P be a cone an a b + cforeachc int P, then a b. 2 Main results In this section, we will present some fixe point theorems for contractive mappings in the setting of cone b-metric spaces. Furthermore, we will give examples to support our main results. Theorem 2.1 Let X, be a complete cone b-metric space with the coefficient s 1. Suppose the mapping T : X X satisfies the contractive conition Tx, Ty λx, y, for x, y X, where λ [0, 1 is a constant. Then T has a unique fixe point in X. Furthermore, the iterative sequence {T n x} converges to the fixe point. Proof Choose x 0 X. We construct the iterative sequence {x n },wherex n = Tx n 1, n 1, i.e., x n+1 = Tx n = T n+1 x 0.Wehave x n+1, x n =Tx n, Tx n 1 λx n, x n 1 λ n x 1, x 0. For any m 1, p 1, it follows that x m+p, x m s [ x m+p, x m+p 1 +x m+p 1, x m ] = sx m+p, x m+p 1 +sx m+p 1, x m sx m+p, x m+p 1 +s 2[ x m+p 1, x m+p 2 +x m+p 2, x m ] = sx m+p, x m+p 1 +s 2 x m+p 1, x m+p 2 +s 2 x m+p 2, x m sx m+p, x m+p 1 +s 2 x m+p 1, x m+p 2 +s 3 x m+p 2, x m+p s p 1 x m+2, x m+1 +s p 1 x m+1, x m sλ m+p 1 x 1, x 0 +s 2 λ m+p 2 x 1, x 0 +s 3 λ m+p 3 x 1, x 0 +

5 Huangan Xu Fixe Point Theory an Applications 2013, 2013:112 Page 5 of 10 + s p 1 λ m+1 x 1, x 0 +s p 1 λ m x 1, x 0 = sλ m+p 1 + s 2 λ m+p 2 + s 3 λ m+p s p 1 λ m+1 x 1, x 0 + s p 1 λ m x 1, x 0 = sλm+p [sλ 1 p 1 1] x 1, x 0 +s p 1 λ m x 1, x 0 s λ sp λ m+1 s λ x 1, x 0 +s p 1 λ m x 1, x 0. Let θ c be given. Notice that sp λ m+1 s λ x 1, x 0 +s p 1 λ m x 1, x 0 θ as m for any k. Making full use of Lemma 1.8,wefinm 0 N such that s p λ m+1 s λ x 1, x 0 +s p 1 λ m x 1, x 0 c, for each m > m 0.Thus, x m+p, x m sp λ m+1 s λ x 1, x 0 +s p 1 λ m x 1, x 0 c for all m > m 0 an any p. So, by Lemma 1.9, {x n } is a Cauchy sequence in X,. Since X, is a complete cone b-metric space, there exists x * X such that x n x *.Taken 0 N such that x n, x * c sλ+1 for all n > n 0.Hence, Tx *, x * s [ Tx *, Tx n + Txn, x *] s [ λ x *, x n + xn+1, x *] c, for each n > n 0. Then, by Lemma 1.10,weeucethatTx *, x * =θ, i.e., Tx * = x *.Thatis, x * is a fixe point of T. Now we show that the fixe point is unique. If there is another fixe point y *,bythe given conition, x *, y * = Tx *, Ty * λ x *, y *. By Lemma 1.11, x * = y *. The proof is complete. Example 2.2 Let X = [0, 1], E = R 2 an p > 1 be a constant. Take P = {x, y E : x, y 0}. We efine : X X E as x, y= x y p, x y p for all x, y X. Then X, is a complete cone b-metricspace.letusefinet : X X as Tx = 1 2 x 1 4 x2 for all x X. Therefore, Tx, Ty = Tx Ty p, Tx Ty p 1 = 2 x y 1 p x yx + y 4, 1 2 x y 1 p x yx + y 4

6 Huangan Xu Fixe Point Theory an Applications 2013, 2013:112 Page 6 of 10 = x y p p x + y 4, x y p p x + y p x y p, x y p = 1 x, y. 2p Here 0 X istheuniquefixepointoft. Theorem 2.3 Let X, be a complete cone b-metric space with the coefficient s 1. Suppose the mapping T : X X satisfies the contractive conition Tx, Ty λ 1 x, Tx+λ 2 y, Ty+λ 3 x, Ty+λ 4 y, Tx, for x, y X, where the constant λ i [0, 1 an λ 1 + λ 2 + sλ 3 + λ 4 <min{1, 2 }, i =1,2,3,4.Then T has a s unique fixe point in X. Moreover, the iterative sequence {T n x} converges to the fixe point. Proof Fix x 0 X an set x 1 = Tx 0 an x n+1 = Tx n = T n+1 x 0.Firstly,wesee x n+1, x n =Tx n, Tx n 1 It follows that Seconly, λ 1 x n, Tx n +λ 2 x n 1, Tx n 1 +λ 3 x n, Tx n 1 +λ 4 x n 1, Tx n λ 1 x n, x n+1 +λ 2 x n 1, x n +sλ 4 [ xn 1, x n +x n, x n+1 ] =λ 1 + sλ 4 x n, x n+1 +λ 2 + sλ 4 x n, x n 1. 1 λ 1 sλ 4 x n+1, x n λ 2 + sλ 4 x n, x n x n+1, x n =Tx n, Tx n 1 =Tx n 1, Tx n This establishes that λ 1 x n 1, Tx n 1 +λ 2 x n, Tx n +λ 3 x n 1, Tx n +λ 4 x n, Tx n 1 λ 1 x n 1, x n +λ 2 x n, x n+1 +sλ 3 [ xn 1, x n +x n, x n+1 ] =λ 2 + sλ 3 x n, x n+1 +λ 1 + sλ 3 x n, x n 1. 1 λ 2 sλ 3 x n+1, x n λ 1 + sλ 3 x n, x n Aing up 2.1an2.2yiels x n+1, x n λ 1 + λ 2 + sλ 3 + λ 4 2 λ 1 λ 2 sλ 3 + λ 4 x n, x n 1. Put λ = λ 1+λ 2 +sλ 3 +λ 4 2 λ 1 λ 2 sλ 3 +λ 4,itiseasytoseethat0 λ <1.Thus, x n+1, x n λx n, x n 1 λ n x 1, x 0.

7 Huangan Xu Fixe Point Theory an Applications 2013, 2013:112 Page 7 of 10 Following an argument similar to that given in Theorem 2.1, thereexistsx * X such that x n x *.Letc θ be arbitrary. Since x n x *,thereexistsn such that x n, x * 2 sλ 1 sλ 2 s 2 λ 3 s 2 λ 4 c for all n > N. 2s 2 +2s Next we claim that x * is a fixe point of T.Actually,ontheonehan, Tx *, x * s [ Tx *, Tx n + Txn, x *] = s Tx *, Tx n + s xn+1, x * s [ λ 1 x *, Tx * + λ 2 x n, Tx n +λ 3 x *, Tx n + λ4 x n, Tx *] + s x n+1, x * = s [ λ 1 x *, Tx * + λ 2 x n, x n+1 +λ 3 x *, x n+1 + λ4 x n, Tx *] + s x n+1, x * sλ 1 x *, Tx * + s 2 λ 2 x n, x * + s 2 λ 2 x *, x n+1 + sλ3 x *, x n+1 + s 2 λ 4 x n, x * + s 2 λ 4 x *, Tx * + s x n+1, x * = sλ 1 + s 2 λ 4 x *, Tx * + s 2 λ 2 + s 2 λ 4 xn, x * + s 2 λ 2 + sλ 3 + s x *, x n+1, which implies that 1 sλ1 s 2 λ 4 x *, Tx * s 2 λ 2 + s 2 λ 4 xn, x * + s 2 λ 2 + sλ 3 + s x *, x n On the other han, x *, Tx * s [ x *, Tx n + Txn, Tx *] = s x *, x n+1 + s Txn, Tx * s x * [, x n+1 + s λ1 x n, Tx n +λ 2 x *, Tx * + λ 3 x n, Tx * + λ 4 x * ], Tx n = s x * [, x n+1 + s λ1 x n, x n+1 +λ 2 x *, Tx * + λ 3 x n, Tx * + λ 4 x * ], x n+1 s x *, x n+1 + s 2 λ 1 x n, x * + s 2 λ 1 x *, x n+1 + sλ2 x *, Tx * + s 2 λ 3 x n, x * + s 2 λ 3 x *, Tx * + sλ 4 x *, x n+1 = sλ 2 + s 2 λ 3 x *, Tx * + s 2 λ 1 + s 2 λ 3 xn, x * + s 2 λ 1 + sλ 4 + s x *, x n+1, which means that 1 sλ2 s 2 λ 3 x *, Tx * s 2 λ 1 + s 2 λ 3 xn, x * + s 2 λ 1 + sλ 4 + s x *, x n Combining 2.3an2.4yiels 2 sλ1 sλ 2 s 2 λ 3 s 2 λ 4 x *, Tx * s 2 λ 1 + λ 2 + λ 3 + λ 4 x n, x * + s 2 λ 1 + s 2 λ 2 + sλ 3 + sλ 4 +2s x *, x n+1 s 2 x n, x * + s 2 +2s x *, x n+1.

8 Huangan Xu Fixe Point Theory an Applications 2013, 2013:112 Page 8 of 10 Simple calculations ensure that x *, Tx * s2 x n, x * +s 2 +2sx *, x n+1 2 sλ 1 sλ 2 s 2 λ 3 s 2 λ 4 c. It is easy to see from Lemma 1.10 that x *, Tx * =θ, i.e., x * is a fixe point of T. Finally, we show the uniqueness of the fixe point. Inee, if there is another fixe point y *,then x *, y * = Tx *, Ty * λ 1 x *, Tx * + λ 2 y *, Ty * + λ 3 x *, Ty * + λ 4 y *, Tx * sλ 3 [ x *, y * + y *, Ty *] + sλ 4 [ y *, x * + x *, Tx *] = sλ 3 + λ 4 x *, y *. Owing to 0 sλ 3 + λ 4 < 1, we euce from Lemma 1.11 that x * = y *. Therefore, we complete the proof. Remark 2.4 Theorem 2.1 extens the famous Banach contraction principle to that in the setting of cone b-metric spaces. Remark 2.5 Any fixe point theorem in the setting of a metric space, a b-metric space or a cone metric space cannot cope with Example 2.2. So, Example 2.2 shows that the fixe point theory of cone b-metric spaces offers inepenently a strong tool for stuying the positive fixe points of some nonlinear operators an the positive solutions of some operator equations. Remark 2.6 The main results are some valuable aitions to the available references regaring cone b-metric spacessince we have known few fixepoint theoremsof contractive mappings in the setting of cone b-metric spaces. 3 Applications In this section we shall apply Theorem2.1 to the first-orer perioic bounary problem x = Ft, xt, t x0 = ξ, 3.1 where F :[ h, h] [ξ δ, ξ + δ] is a continuous function. Example 3.1 Consier the bounary problem 3.1 withthecontinuousfunctionf an suppose Fx, y satisfies the local Lipschitz conition, i.e.,if x h, y 1, y 2 [ξ δ, ξ + δ], it inuces Fx, y 1 Fx, y 2 L y 1 y 2. Set M = max [ h,h] [ξ δ,ξ+δ] Fx, y such that h 2 < min{δ/m 2,1/L 2 }, then there exists a unique solution of 3.1.

9 Huangan Xu Fixe Point Theory an Applications 2013, 2013:112 Page 9 of 10 Proof Let X = E = C[ h, h] an P = {u E : u 0}. Put : X X E as x, y = f t max h t h xt yt 2 with f :[ h, h] R such that f t =e t.itisclearthatx, is a complete cone b-metric space. Note that 3.1 is equivalent to the integral equation xt=ξ + t 0 F τ, xτ τ. Define a mapping T : C[ h, h] R by Txt=ξ + t 0 Fτ, xτ τ.if xt, yt Bξ, δf { ϕt C [ h, h] : ξ, ϕ δf }, then from Tx, Ty =f t max t h t h 0 t = f t max h t h 0 F τ, xτ τ t 0 F τ, yτ τ [ F τ, xτ F τ, yτ ] τ 2 h 2 f t max F τ, xτ F τ, yτ 2 h τ h h 2 L 2 f t max xτ yτ 2 = h 2 L 2 x, y, h τ h 2 an Tx, ξ=f t max h t h t 0 F τ, xτ 2 τ h 2 f max F τ, xτ 2 h 2 M 2 f δf, h τ h we speculate T : Bξ, δf Bξ, δf is a contractive mapping. Finally, we prove that Bξ, δf, is complete. In fact, suppose {x n } is a Cauchy sequence in Bξ, δf. Then {x n } isalsoacauchysequenceinx.sincex, is complete, there is x X such that x n x n. So, for each c int P,thereexistsN,whenevern > N,weobtain x n, x c. Thus, it follows from ξ, x x n, ξ+x n, x δf + c an Lemma 1.12 that ξ, x δf,whichmeansx Bξ, δf, that is, Bξ, δf, iscomplete. Owing to the above statement, all the conitions of Theorem 2.1 are satisfie. Hence, T has a unique fixe point xt Bξ, δf. That is to say, there exists a unique solution of 3.1. Competing interests The authors eclare that they have no competing interests. Authors contributions All authors contribute equally an significantly in writing this paper. All authors rea an approve the final manuscript.

10 Huangan Xu Fixe Point Theory an Applications 2013, 2013:112 Page 10 of 10 Author etails 1 School of Mathematics an Statistics, Hubei Normal University, Huangshi, , China. 2 School of Mathematics an Computer, Gannan Normal University, Ganzhou, , China. Acknowlegements The authors thank the eitor an the referees for their valuable comments an suggestions which improve greatly the quality of this paper. The research was supporte by the National Natural Science Founation of China an partly supporte by the Grauate Initial Fun of Hubei Normal University 2008D36. Receive: 14 August 2012 Accepte: 19 November 2012 Publishe: 26 April 2013 References 1. Deimling, K: Nonlinear Functional Analysis. Springer, Berlin Banach, S: Sur les operations ans les ensembles abstrait et leur application aux equations, integrals. Funam. Math. 3, Bakhtin, IA: The contraction mapping principle in almost metric spaces. Funct. Anal., Gos. Pe. Inst. Unianowsk 30, Czerwik, S: Nonlinear set-value contraction mappings in b-metric spaces. Atti Semin. Mat. Fis. Univ. Moena 46, Boriceanu, M, Bota, M, Petrusel, A: Mutivalue fractals in b-metric spaces. Cent. Eur. J. Math. 82, Bota, M, Molnar, A, Csaba, V: On Ekelan s variational principle in b-metric spaces. Fixe Point Theory 12, Huang, L-G, Zhang, X: Cone metric spaces an fixe point theorems of contractive mappings. J. Math. Anal. Appl. 3322, Rezapour, S, Hamlbarani, R: Some notes on the paper Cone metric spaces an fixe point theorems of contractive mappings. J. Math. Anal. Appl. 345, Janković, S, Kaelburg, Z, Raenović, S: On cone metric spaces: a survey. Nonlinear Anal. 47, Hussian, N, Shah, MH: KKM mappings in cone b-metric spaces. Comput. Math. Appl. 62, Cho, S-H, Bae, J-S: Common fixe point theorems for mappings satisfying property E.A on cone metric space. Math. Comput. Moel. 53, oi: / Cite this article as: Huang an Xu: Fixe point theorems of contractive mappings incone b-metric spaces an applications. Fixe Point Theory an Applications :112.