Dragomir and Gosa type inequalities on b-metric spaces

Transcription

1 Karapınar and Noorwali Journal of Inequalitie and Application (019) 019:9 RESEARCH Open Acce Dragomir and Goa type inequalitie on b-metric pace Erdal Karapınar1* and Maha Noorwali * Correpondence: erdalkarapinar@yahoo.com Department of Medical Reearch, China Medical Univerity Hopital, China Medical Univerity, Taichung, Taiwan Full lit of author information i available at the end of the article Abtract 1 In thi paper, we invetigate Dragomir and Goa type inequalitie in the etting of b-metric pace. A an application, we conider ome inequalitie in b-normed pace. We prove that the inequalitie admit geometrical interpretation. Keyword: Dragomir and Goa type inequalitie; b-metric pace; Inequality 1 Introduction and preliminarie It i a natural trend in fixed point theory to refine a tandard metric pace tructure with a weaker one. One of the intereting extenion of the notion of a metric pace i the concept of a b-metric pace which wa introduced by Czerwik [8. Definition 1.1 ([8) Let X be a nonempty et and 1 a given real number. A mapping d : X X [0, ) i aid to be a b-metric if for all x, y, z X the following condition are atified: (bm1 ) d(x, y) = 0 if and only if x = y; (bm ) d(x, y) = d(y, x) (ymmetry); (bm3 ) d(x, z) [d(x, y) + d(y, z) (b-triangle inequality). In thi cae, the pair (X, d) i called a b-metric pace (with contant ). Clearly, any metric pace i a b-metric pace (with contant = 1). Example 1. ([10) Let X = [0, 1 and let d : X X [0, ) be defined by d(x, y) = (x y). Then, clearly, (X, d) i a b-metric pace with =. The following i another contructive example of b-metric. Example 1.3 ([1) Let X = {xi : 1 i M} for ome M N and. Define d : X X a 0 if i = j, d(xi, xj ) = if (i, j) = (1, ) or (i, j) = (, 1), 1 otherwie. The Author() 019. Thi article i ditributed under the term of the Creative Common Attribution 4.0 International Licene ( which permit unretricted ue, ditribution, and reproduction in any medium, provided you give appropriate credit to the original author() and the ource, provide a link to the Creative Common licene, and indicate if change were made.

2 Karapınar and Noorwali Journal of Inequalitie and Application (019) 019:9 Page of 7 Conequently, we derive that d(x i, x j ) [ d(xi, x k )+d(x k, x j ), for all i, j, k {1, M}. Thu,(X, d) formab-metric for > where the ordinary triangle inequality doe not hold. Formore examplefor b-metric, we may refer, e.g., to [1 7, 9, 1 and the correponding reference therein. Example 1.4 (ee, e.g., [6) The pace L p [0, 1 (where 0 < p < 1) of all real function x(t), t [0, 1 uch that 1 0 x(t) p dt <, together with the functional ( 1 ) 1/p d(x, y):= x(t) y(t) p dt, foreachx, y L p [0, 1, 0 i a b-metricpace.noticethat = 1/p. Main reult We tart thi ection by recalling an intereting inequality that wa propoed by Dragomir and Goa in [11. In what follow we invetigate their inequality in the etting of a more general tructure, namely that of b-metric pace. Theorem.1 Let (X, d) be a b-metric pace with contant 1, and x i X, p i 0(i {1,,...,n}) with n p i = 1. Then we have [ n p i p j d(x i, x j ) inf p i d(x i, x). (1) The inequality i harp in the ene that the contant c =1in front of the infimum cannot be replaced by a maller contant. Proof Uing the b-triangle inequality, for any x X, i, j {1,,...,n} we have d(x i, x j ) [ d(x i, x)+d(x, x j ). () If we multiply ()byp i, p j and um over i and j from 1 to n,weget [ n n [ p i p j d(x i, x j ) p i p j d(xi, x)+d(x, x j ). Note that by ymmetry we have n p i p j d(x i, x j )= p i p j d(x i, x j ). (3)

3 Karapınar and Noorwali Journal of Inequalitie and Application (019) 019:9 Page 3 of 7 Now, uing the condition n p i = 1, we can eaily deduce that n [ p i p j d(xi, x)+d(x, x j ) = n p i d(x i, x). So, from (3)wehave p i p j d(x i, x j )= 1 = Therefore, n p i p j d(x i, x j ) [ n [ p i p j d(xi, x)+d(x, x j ) n p i d(x i, x). p i p j d(x i, x j ) n p i d(x i, x) for any x X. Uing the fact that the infimum i the greatet lower bound, we deduce (1). Now, uppoe that there exit c >0uchthat [ n p i p j d(x i, x j ) c inf p i d(x i, x) ; and chooe n =,p 1 = p and p =1 p where p (0, 1). Then, p(1 p)d(x 1, x ) c [ pd(x 1, x)+(1 p)d(x, x ). (4) If we let x = x 1 in (4), we get p(1 p)d(x 1, x ) c(1 p)d(x 1, x ). A d(x 1, x )>0and1 p >0,op c for any p (0, 1). Uing the fact that the upremum i the leat upper bound, we deduce that c 1. The following corollary i a generalization of Corollary 1 in [11tothecaeofab-metric pace. Corollary. Let (X, d) be a b-metric pace with contant 1, and x i X, i {1,,...,n}, then [ n d(x i, x j ) n inf d(x i, x).

4 Karapınar and Noorwali Journal of Inequalitie and Application (019) 019:9 Page 4 of 7 The proof follow directly by taking p i = 1, i {1,,...,n} in the previou theorem. n The above corollary can be interpreted geometrically a follow: The um of all edge and diagonal of a polygon with n vertice in a b-metric pace i le than or equal to n -time the um of the ditance from any arbitrary point in the pace to it vertice. The next corollary i a generalization of Corollary in [11 in the framework of b-metric pace. Corollary.3 Let (X, d) be a b-metric pace with contant and x i X, i {1,,...,n}. If there exit z Xandr>0uchthat the cloed ball B(z, r)={y X : d(z, y) r} contain all the point x i, then for any p i 0 with n p i = 1 we have p i p j d(x i, x j ) r. Proof Uing (1)wehave [ n p i p j d(x i, x j ) inf p i d(x i, x) n p i d(x i, z) r. 3 Application In thi ection we define a new notion of a b-normed pace and tudy ome of it propertie. Definition 3.1 Let X beavectorpaceoverafieldk and let 1 be a contant. A function b : X [0, )iaidtobeab-norm if the following condition hold for every x, y X, c K: (Nb1) x b 0; (Nb) x b =0 x =0; (Nb3) cx b = c log +1 x b (b-homogeneity); (Nb4) x + y b [ x b + y b (b-norm triangle inequality). In thi cae (X, b ) i called a b-normed pace with contant. Here we give an example of a b-normed pace. Example 3. Let X = R and define b : X [0, ) by x b = x p where p (1, ), then, uing the relation (x + y) p p 1 (x + y), we can eaily deduce that (X, b )iabnormed pace with contant = p 1. Remark 3.3 Let (X, b )beab-normedpace with contant 1, x i X, i {1,...,n}. Then it i eay to prove the following generalized b-triangle inequality: n n x i i x i.

5 Karapınar and Noorwali Journal of Inequalitie and Application (019) 019:9 Page 5 of 7 Remark 3.4 Any b-norm with 1defineab-metric a follow: d(x, y)= x y b. Thequetionnowithefollowing:Ianyb-metric induced from a b-norm? The following remark can anwer thi quetion. Remark 3.5 Let X beavectorpaceoverafieldk. Anyb-metric d : X X [0, ) with contant 1inducedfromab-norm mut atify the following propertie for each x, y, z X, c K: (i) d(x + z, y + z) = d(x, y) (tranlation invariance); (ii) d(cx, cy)= c log +1 d(x, y) (b-homogeneity). Propoition 3.6 A b-homogeneou tranlation invariant b-metric d : X X [0, ) with contant 1 can define a b-norm b : X [0, ) a follow: x b = d(x,0) x X. Proof Clearly, (Nb1) and (Nb) are atified. A d i homogeneou, cx = d(cx,0)= c log +1 d(x,0)= c log +1 x b. A d i tranlation invariant, x + y b = d(x + y,0) [ d(x + y, x)+d(x,0) = [ d(y,0)+d(x,0) = [ x b + y b, which prove (Nb3) and (Nb4), repectively. Now, we rewrite inequality (1) intheeneofb-normed pace and obtain ome corollarie. If (X, b )iab-normedpace with contant 1, x i X,andp i 0, i {1,...,n} with n p i = 1,thenby(1)wehave [ n p i p j x i x j inf p i x i x j. (5) The following propoition i a generalization of Propoition in [11 tothecaeofa b-normed pace. Propoition 3.7 Let (X, b ) be a b-normed pace with contant 1, x i Xandp i 0, i {1,...,n} with n p i = 1. Let x p = n p ix i, then 1 n p i x i x p n p i p j x i x j n n p i x i x p. (6)

6 Karapınar and Noorwali Journal of Inequalitie and Application (019) 019:9 Page 6 of 7 Proof A the infimum i a lower bound, the econd part of inequality (6) i trivial. For the firt part, we ue a generalized b-norminequalityafollow: 1 n p i x i x p = 1 = 1 1 n n p i x i n p j x j j=1 n n p i (x i p j x j ) j=1 n p i j x i p j x j n p i p j x i x j = n p i p j x i x j, which complete the proof. We have the following corollary, which ha a nice geometric interpretation. Corollary 3.8 Let (X, b ) be a b-normed pace with contant 1 and x i X, i {1,...,n}. If x = x 1+ +x n i the gravity center of the vector {x n 1,...,x n }, then we have n n x i x n x i x j n n n x i x. Geometrically, the lat corollary mean that the um of the edge and diagonal of a polygon with n vertice in a b-normed pace i le than or equal to n-time the um of the ditance from the gravity center to it vertice and greater than or equal to n n -time thi quantity. 4 Concluion Similarly, we can generalize more inequalitie on metric and normed pace. Funding We declare that funding i not applicable for our paper. Competing interet The author declare that they have no competing interet. Author contribution All author contributed equally and ignificantly in writing thi article. All author read and approved the final manucript. Author detail 1 Department of Medical Reearch, China Medical Univerity Hopital, China Medical Univerity, Taichung, Taiwan. Nonlinear Analyi Reearch Group (NAAM), King Abdulaziz Univerity, Jeddah, Saudi Arabia. Publiher Note Springer Nature remain neutral with regard to juridictional claim in publihed map and intitutional affiliation. Received: 0 November 018 Accepted: 17 January 019

7 Karapınar and Noorwali Journal of Inequalitie and Application (019) 019:9 Page 7 of 7 Reference 1. Alqahtani, B., Fulga, A., Karapınar, E., Özturk, A.: Fiher-type fixed point reult in b-metric pace. Mathematic 7(1), 10 (019). Aydi, H., Bankovic, R., Mitrovic, I., Nazam, M.: Nemytzki Edeltein Meir Keeler type reult in metric pace. Dicrete Dyn. Nat. Soc. 018, Article ID , 7 page (018) Aydi, H., Bota, M., Karapınar, E., Mitrović, S.: A fixed point theorem for et-valued quai-contraction in b-metric pace. Fixed Point Theory Appl. 01,ArticleID88 (01) 4. Aydi, H., Bota, M., Karapınar, E., Moradi, S.: A common fixed point for weak φ-contraction in b-metric pace. Fixed Point Theory 13(), (01) 5. Bota, M., Chifu, C., Karapinar, E.: Fixed point theorem for generalized (α ψ)-ciric-type contractive multivalued operator in b-metric pace. J. Nonlinear Sci. Appl. 9(3), (016) 6. Bota, M., Karapınar, E., Mleşniţe, O.: Ulam Hyer tability for fixed point problem via α φ-contractive mapping in b-metric pace. Abtr. Appl. Anal., 013,Article ID85593 (013) 7. Bota, M.F., Karapinar, E.: A note on Some reult on multi-valued weakly Jungck mapping in b-metric pace. Cent. Eur. J. Math. 11(9), (013) 8. Czerwik, S.: Contraction mapping in b-metric pace. Acta Math. Inform. Univ. Otrav. 1, 5 11 (1993) 9. Czerwik, S.: Nonlinear et-valued contraction mapping in b-metric pace. Atti Semin. Mat. Fi. Univ. Modena 46, (1998) 10. Demmaa, M., Saadatib, R., Vetroa, P.: Fixed point reult on b-metric pace via Picard equence and b-imulation function. Iran. J. Math. Sci. Inform. 11(1), (016) 11. Dragomir, S.S., Go, A.C.: An inequality in metric pace (004) 1. Kutbi, M.A., Karapinar, E., Ahmed, J., Azam, A.: Some fixed point reult for multi-valued mapping in b-metric pace. J. Inequal. Appl. 014, 16 (014)