A Result on Best Proximity Point

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1 ISSN(Online : ISSN (Print : Engineering and echnology (An ISO 3297: 2007 Certified Organization A Result on Best Proximity Point Sujata Goyal Assistant Professor, Deartment of Mathematics, A.S. College, Khanna, India ABSRAC : S.Karagam and Sushama Aggarwal [3] defined -cyclic contraction to be a ma : satisfying ( (A i, i where A =A such that for some k(0,, d(x,ykd(x,y+(-kdist(,a i, x, y and obtained best roximity oint in one of the sets. heir assumtion forces the distances dist(,, i, to be same.in this aer we have relaxed the second condition in the definition of -cyclic contraction in a way which does not force the distances to be same and obtained the best roximity oint in one of the sets. KEYWORDS : -cyclic contraction, best roximity oint, uniform convex banach sace I. INRODUCION he classical and well-known Banach s contraction rincile states that every contraction on a comlete metric sace has a unique fixed oint that is realizable as the limit of Picard iterates. Numerous interesting extensions and variants of the aforesaid result exist in the literature. However, the maings involved in all these results are self-maings. Fixed oint theory is an indisensable tool for solving the equation x = x for a maing defined on a subset of a metric sace, a normed linear sace or a toological vector sace. As a non-self maing : A B does not necessarily have a fixed oint, one often tries to determine an element x which is in some sense closest to x. Best aroximation theorems and best roximity oint theorems are ertinent in this ersective. A number of authors have studied the conditions which ensure the existence of best roximity oint. See for examle [-7]. Kirk, Srinivasan and Veeramani [] roved II. RELAED WORK heorem.: Let { } i=,2... be nonemty closed subsets of a comlete metric sace and : ( (A be a ma such that i, i where A =A 2For some k(0,, d(x,yk(x,y, x, ya i then there exists a unique fixed oint of Eldered and Veeramani [2], weakened the contraction condition for two set in the setting of uniformly convex banach sace and obtained the following result on best roximity oint: heorem.2:let A and B be nonemty,closed and convex subsets of a uniformly convex banach sace. :A B A B be such that (A B and (B A Coyright to IJIRSE DOI:0.5680/IJIRSE

2 ISSN(Online : ISSN (Print : Engineering and echnology (An ISO 3297: 2007 Certified Organization 2 for some k(0,, xy k x y + (-k dist(a,b, xa and yb then there exists a unique best roximity oint x.e. a oint xa such that d(x,x = dist(a,b S.Karagam and Sushama Aggarwal [3] defined -cyclic contraction as a ma : satisfying ( (A i, i where A =A 2For some k (0,, d(x,ykd(x,y+(-kdist(,, x, y and roved the following theorem : heorem.3:if A,.. A are nonemty,closed,convex subsets of a uniformly convex banach sace and: be a -cyclic contraction ma then there exists an x such that d(x,x=dist(,. Moreover j x is a best roximity oint in A i j. Here the assumtion forces the distances, dist(,, i, to be same. In this aer we have relaxed the condition 2 in the definition of -cyclic contraction in a way which does not the distances to be same and got the best roximity oint in one of the sets. III. PRELIMINARIES We will make use of the following lemmas roved in []: Lemma 2.: Let A be a nonemty closed and convex subset, and B be a nonemty closed subset of a uniformly convex Banach sace. Let {x n } and {z n } be sequences in A and {y n } be a sequence in B satisfying : z y n n dist(a,b 2for every >0 there exists a natural number N such that for all m>nn, x dist(a,b + m y n hen for every >0, there exists N 0, such that for all m>n N 0 x m z n Lemma 2.2: let A be a nonemty closed and convex subset and B be a nonemty closed subset of a uniformly convex Banach sace, let {x n }and {z n }be sequences in A and {y n } be a sequence in B satisfying : x y dist(a,b n n z y 2 n n then dist(a,b x z 0 n n Coyright to IJIRSE DOI:0.5680/IJIRSE

3 ISSN(Online : ISSN (Print : Engineering and echnology (An ISO 3297: 2007 Certified Organization IV. MAIN RESULS heorem 3.:Let A,.. A be nonemty,closed,convex subsets of a uniformly convex banach sace and: ( (A be a ma satisfying i, i where A =A 2For some k (0,, d(x,ykd(x,y+(-kmax{ dist(, : i }, x, y then there exists xa j such that d(x,x=dist(a j,a j where dist(a j,a j = max{ dist(, : i } o rove the theorem we first rove the following lemmas: Lemma 3.2: Let A, A 2,.. A be nonemty, closed subsets of a metric sace and : satisfying the conditions of the theorem 3.,then for x, y A j, d( n x, roof: dist(a j,a j d( n n n x, y k d( thus d( n x, n k[kd( =k 2 d(... n2 n x, n2 n x, n x, n y+(-k dist(a j,a j y+(-k dist(a j,a y+ (-k 2 dist(a j,a j y dist(a j,a j j ]+ (-k dist(a j,a k n d(x,y+(-k n dist(a j,a j dist(a j,a y dist(a j,a j j j Similarly we can rove that d( n n n x, y dist(a j,a j and d( x, n y dist(a j,a j Remark3.3: In view of lemma 2.2 and lemma 3.2 we have that if X is uniformly convex Banach sace and each is convex then for x A j, n n n n n x x dist(a j,a j and x x dist(a j,a j and therefore x x n n n 0 and x x 0 Lemma 3.3 : Let A,.. A be nonemty,closed subsets of a metric sace and: be a ma be a ma satisfying the conditions of the theorem 3. then If for some xa j the sequence{ n x} in A j contains a convergent subsequence { n j x}, converging to A j, then is a best roximity oint of in A j. Coyright to IJIRSE DOI:0.5680/IJIRSE

4 ISSN(Online : ISSN (Print : Engineering and echnology (An ISO 3297: 2007 Certified Organization Proof: we have d( n x, Now dist(a j,a j n d(,= n x dist(a j,a j lim d( n j n x, lim k d( = lim k d( n n lim k[kd( = lim k 2 d( n... lim k n n j n j n j n j 2 n j 2 x, +(-k dist(a j,a j x, n j x+ (-k dist(a j,a x, x, n j j x+(-k dist(a j,a j ] + (-k dist(a j,a j n j d(x,y+(- k dist( A j,a j x+(-k 2 dist(a j,a j n j dist( A j,a Proof of theorem 3.:If dist( A j,a j =0,then by theorem., has a unique fixed oint and the theorem will be n n trivially true. So we assume that dist( A j,a j >0.Let x A j,then by lemma 3.2 x x dist(a j,a j. In view of lemma 3.3,to rove the theorem it is enough to rove that the sequence{ n x} in A j, is Cauchy. Now in view of lemma 2. It will be sufficient to show that for given >0 there exists N such that, for m>nn n x x dist(a j,a j +...( m On contrary suose ( is not true then there exists 0 >0 such that for every natural number k, there exists m k > n k k such that nk x x dist(a j,a j + 0 m k Here we assume that m k is the smallest integer greater than n k to satisfy the above inequality. m k nk mk mk mk nk Now dist(a j,a j + 0 x x x x + x x Now mk this imlies Now mk x x 0 by remark 3.3, nk x x dist(a j,a j + m k nk mk mk mk nk nk nk x x x x + x x + x x m k By remark 3.3 lim k x mk m k nk mk x x 0 and nk x nk x 0 x lim k x mk nk lim k k x j nk x x +(-k dist(a j,a m k j Coyright to IJIRSE DOI:0.5680/IJIRSE

5 ISSN(Online : ISSN (Print : Engineering and echnology (An ISO 3297: 2007 Certified Organization dist(a j,a j + 0 k (dist(a j,a j + 0 +(-k dist(a j,a j k 0,which is a contradiction Hence the theorem. REFERENCES []. Kirk, W. A., Srinivasan, P. S. and Veeramani, P., Fixed oints for maings satisfying cyclic contractive conditions, Fixed Point heory, Vol.4,.79-89, [2]. Eldered, A. A. and Veeramani, P., Existence and convergence of best roximity oints, Journal of Mathematical Analysis and Alications, vol. 323, , [3]. Karagam, S. and Agrawal, S., Existence of best roximity oints of P-Cyclic Contractions, Fixed Point heory, vol.3, , 202. [4]. Basha, S. S., Veeramani, P. and Pai, D. V., Best roximity air theorems, Indian Journal of Pure and Alied Mathematics, vol.32, , 200. [5]. Raj, V. S. and Veeramani, P., Best roximity air theorems for relatively nonexansive maings, Alied General oology, vol. 0,. 2-28, [6]. Srinivasan, P. S. and Veeramani, P., On best roximity air theorems and fixed-oint theorems, Abstract and Alied Analysis, vol. 2003, , [7]. Kosuru, G. S. R. and Veeramani, P., Cyclic contractions and best roximity air theorems, arxiv: ,200. BIOGRAPHY Sujata Goyal received her M.Sc. (Mathematics degree in 202 from Panjab University, Chandigarh (India. Presently, she is working as an Assistant Professor in the Deartment of Mathematics, A. S. College, Khanna (India. She has number of ublications in refereed Journals. Her main research area is Analysis and Algebra. Coyright to IJIRSE DOI:0.5680/IJIRSE