Int. Journal of Math. Analysis, Vol. 4, 21, no. 5, 233-242 A Fixed Point Theorem for ϕ - Weakly Contractive Mapping in Metric Spaces Nguyen Van Luong and Nguyen Xuan Thuan Deparment of Natural Sciences Hong Duc University, Thanh Hoa, Viet Nam luongk6ahd4@yahoo.com, thuannx@yahoo.com.vn Abstract In this paper, we study the existence of fixed point for mapping T defined on complete metric space such that is R ϕ - weakly contractive mapping. Mathematics Subject Classification: 47H1, 54H25 Keywords: fixed point, integral type, weakly contractive 1 Introduction Banach s contraction mapping principle is one of the pivotal results of analysis. It widely considered as the source of metric fixed point theory. Also its significance lies in its vast applicability in a number of branches of mathematics Theorem 1.1. (Banach s contraction mapping principle Let (X,d be a complete metric space and f : X X be a contraction (there exists k (, 1 such that for each x, y X, d(fx,fy kd(x, y. Then f has a unique fixed point in X, and for each x X the sequence of iterates f n (x converges to this fixed point. There have been many theorems dealing with mappings satisfying various types of contractive inequalities. Such conditions involve linear and nonlinear expressions (rational, irrational, and general type. The intrested reader who wants to know more about this matter is recommended to go deep into the survey articles by Rhoades [13-15] and Meszaros [1], and into the references therein. Another result on fixed points for contractive-type mapping is generally attributed to Edelstein (1962 who actually obtained slightly more general versions.
234 Nguyen Van Luong and Nguyen Xuan Thuan In [4], A. Branciari obtained a fixed point result for a single mapping an analogue of Banach s contraction principle for an integral type inequality, stated as follows, Theorem 1.2. Let (X, d be a complete metric space, α [, 1, f : X X a mapping such that, for each x, y X, d(fx,fy α where ϕ :[, + [, + is a Lebesgue - integrable mapping which is summable, nonnegative and such that ε >, for each ε> Then f has a unique fixed point z X such that for each x X, lim f n x = z Since then there have have been many theorems dealing with mappings satisfying a general contractive condition of integral type. Some of these works are noted in [1], [3-4], [8-9], [11-12], [16], [19]. Another generalisation of the contraction principle was suggested by Alber and Guerre - Delabriere [2] in Hlibert space. Rhoades [17] has shown that the result which Alber - Delabriere have proved in [2] is also valid in complete metric space. We state the result of Rhoades in the following Definition 1.3. (weakly contractive mapping A mapping T : X X, where (X, d is a metric space, is said to be weakly contractive if d(tx,ty d(x, y φ(d(x, y where x, y X, and φ :[, + [, + is continuous and nondecreasing function such that φ(t =if and only if t =. Theorem 1.4. [17] If T : X X is a weakly contractive mapping, where (X,d is a metric space, then T has a unique fixed point. Weakly contractive mapping have been dealt with in a number of paper. Some of these works are noted in [5-6], [17]. Also, a generalisation of Banach contraction mapping principle was suggested P.N. Dutta and B.S. Choudhury. The result of P.N. Dutta and B.S. Choudhury [7] state as follows Theorem 1.5. Let (X, d be a complete metric space and let T : X X be sefl - mapping satisfying the inequality (d(tx,ty (d(x, y φ(d(x, y
A fixed point theorem for R ϕ - weakly contractive mapping 235 where, ϕ :[, [, are both continuous and monotone nondecreasing functions with (t ==φ(t if and only if t =. Then T has a unique fixed point. In this paper, we study the existence of fixed point for mapping T defined on complete metric space such that is R ϕ- weakly contractive mapping 2 The main results The following theorem (theorem 2.3 is the main result of this paper. In the first, we define definition of R ϕ- weakly contractive mapping. Definition 2.1. Let (X,d be a metric space and ϕ :[, + [, + be a Lebesgue - integrable mapping. A mapping T : X X is said to be R ϕ- weakly contractive if for all x, y X, ( ( d(t x,t y ( φ (2.1 where :[, + [, + is a continuous and nondecreasing function and φ :[, + [, + is a lower semi -continuous and nondecreasing function such that (t ==φ(t if and only if t =. Remark 2.2. By taking (t =t, for all t [, +, we can defined ϕ - weakly contractive mapping. And weakly contractive mapping is special case of R ϕ- weakly contractive mapping. Theorem 2.3. Let (X,d be a complete metric space and T : X X is R ϕ - weakly contractive mapping where ϕ :[, + [, + is a Lebesgue - integrable mapping which is summable, nonnegative and such that ε Then T has a unique fixed point >, for each ε > (2.2 Proof. Let x be an arbitrary point in X. We construct a sequence x n in X by x n = Tx n 1, n =1, 2, 3,... Taking x := x n and y := x n 1 in (2.1, we have ( ( ( d(xn+1,x n φ (
236 Nguyen Van Luong and Nguyen Xuan Thuan which implies d(xn+1,x n d(xn,x n 1 (using monotone property of - function. Set y n := d(x n+1,x n, then y n y n 1 for all n>. It follows that the sequence {y n } is monotone decreasing and lower bounded. Therefore, there exists r such that lim d(xn+1,x n = lim y n = r Then (by the lower semi - continuity of φ ( φ(r lim inf φ Suppose that r> Taking upper limit as n on either side of the following inequality ( ( d(xn+1,x n ( φ we get ( (r (r lim inf φ which is a contradiction. Thus r =, i.e Therefore, we have lim d(xn+1,x n (r φ(r = lim y n = lim d(x n+1,x n =, see for detail [4] (2.3 We now prove that {x n } is a Cauchy sequence. Suppose it is not so. Then there exists an ε > and subsequences {x m(k } and {x n(k } of {x n } with n(k >m(k >ksuch that d(x n(k,x m(k ε (2.4 Further, corresponding to m(k, we can choose n(k in such a way that it is the smallest integer n(k >m(k and satisfying (2.4. Then d(x n(k 1,x m(k <ε (2.5
A fixed point theorem for R ϕ - weakly contractive mapping 237 Then we have ε d(x n(k,x m(k d(x n(k,x n(k 1 +d(x n(k 1,x m(k <ε+ d(x n(k,x n(k 1 Now <δ:= ε d(xn(k,x m(k Taking k and using (2.3, we get lim k By the triangular inequality, d(xn(k,x m(k ε+d(xn(k,x n(k 1 = δ (2.6 d(x n(k,x m(k d(x n(k,x n(k 1 +d(x n(k 1,x m(k 1 +d(x m(k 1,x m(k d(x n(k 1,x m(k 1 d(x n(k 1,x n(k +d(x n(k,x m(k +d(x m(k,x m(k 1 and so d(xn(k,x m(k d(xn(k,x n(k 1 +d(x n(k 1,x m(k 1 +d(x m(k 1,x m(k d(xn(k 1,x m(k 1 d(xn(k 1,x n(k +d(x n(k,x m(k +d(x m(k,x m(k 1 Taking k in the above two inequalities and using (2.3, (2.6, we get d(xn(k 1,x m(k 1 lim = δ (2.7 k Taking x := x n(k 1 and y := x m(k 1 in (2.1, we have ( ( d(xn(k 1,x m(k 1 d(xn(k,x m(k ( d(xn(k 1,x m(k 1 φ
238 Nguyen Van Luong and Nguyen Xuan Thuan then taking k, using (2.6, (2.7 and property of, φ, we obtain (δ (δ φ(δ which is a contradiction because δ>. Thus {x n } is a Cauchy sequence. Since X be a complete metric, there exists u in X such that x n u as n (2.8 Taking x := x n 1 and y := u in (2.1, we have ( d(xn,t u ( d(xn 1,u φ ( d(xn 1,u Taking n, using (2.8 and property of, φ, we have ( d(u,t u ( φ ( = which implies d(u,t u =. Thus d(u,tu = (since (2.2, that is, u = Tu. Now we prove u is the unique fixed point of T. Let us suppose that u and v are two fixed points of T. Putting x := u and y := v in (2.1, ( d(t u,t v ( d(u,v φ ( d(u,v or ( d(u,v ( d(u,v φ ( d(u,v or φ ( d(u,v or equivalently d(u,v = which implies d(u, v =, that is, u = v. This proves the uniqueness of the fixed point.
A fixed point theorem for R ϕ - weakly contractive mapping 239 Example 2.4. Let X =[, 1] and d be usual metric d(x, y = x y.let T : X X be given by Tx = 1 x for all x [, 1] 2 Let, φ, ϕ :[, + [, + be given by Then, for each ε>, (t =t, φ(t = 1 2 t2 and ϕ(t =2t, for all t [, + For all x, y X, we have and d(t x,t y ε = = = ε 2 1 2 x y x y = 1 x y 2 4 = x y 2 So ( ( φ = x y 2 1 x y 4 2 x y 2 1 2 x y 2 = 1 x y 2 ( 2 1 d(t x,t y 4 x y 2 = (since x y 1 That is, the condition of theorem 2.1 holds and the is unique fixed point of T. Corollary 2.5. Let (X, d be a complete metric space and T : X X be a sefl - map defined on X satisfying the following condition d(t x,t y ( φ
24 Nguyen Van Luong and Nguyen Xuan Thuan for all x, y X. where ϕ :[, + [, + is a Lebesgue -integrable mapping which is summable, nonnegative and such that ε >, for each ε> and φ : [, + [, + is monotone nondecreasing and lower semi - continuous functions with φ(t > for t> and φ( =. Then T has a unique fixed point. Proof. By using theorem 2.1, where (t =t, for all t [, +. Acknowledgement. The authors are thankful to the referee for his valuable comments and suggestions. References [1] M. Abbas and B. E. Rhoades, Common fixed point theorems for hybrid pairs of occasionally weakly compatible mappings satisfying generalized contractive condition of integral type, Fixed Point Theory and Applications,Volume 27, Article ID 5411, 9 pages. [2] Ya. I. Alber and S. Guerre-Delabriere, Principle of weakly contractive maps in Hilbert spaces, in New Results in Operator Theory and Its Applications, I. Gohberg and Y. Lyubich, Eds., vol. 98 of Operator Theory: Advances and Applications, pp. 7-22, Birkhäuser, Basel, Switzerland, 1997. [3] I. Altun, D. Turkoglu, B.E. Rhoades, Fixed points of weakly compatible maps satisfying a general contractive condition of integral type, Fixed Point Theory and Applications, Volume 27, Article ID 1731, 9 pages. [4] A. Branciari, A fixed point theorem for mapping satisfying a general contractive condition of general type, International Journal of Mathematics and Mathematical Sciences, 29:9 (22 531-536. [5] I. Beg and M. Abbas, Coincidence point and invariant approximation for mappings satisfying generalized weak contractive condition, Fixed Point Theory and Applications, Vol. 26, Article ID 7453, 7 pages, 26. [6] C. E. Chidume, H. Zegeye, and S. J. Aneke, Approximation of fixed points of weakly contractive nonself maps in Banach spaces,journal of Mathematical Analysis and Applications, vol. 27, no. 1, pp. 189-199, 22.
A fixed point theorem for R ϕ - weakly contractive mapping 241 [7] P. N. Dutta and B. S. Choudhury, A generalisation of contraction principle in metric spaces, Fixed Point Theory and Applications, Volume 28, Article ID 46368, 8 pages. [8] U. C. Gairola and A. S. Rawat, A fixed point theorem for integral type inequality, Int. Journal of Math. Analysis, Vol. 2, 28, no. 15, 79-712. [9] S. Kumar, R. Chugh and R. Kumar, Fixed point theorem for compatible mapping satisfying a contractive condition of integral type, Soochow Journal of Mathematics, Volume 33, No. 2, pp. 181-185, April 27. [1] J. Meszaros, A comparison of various definitions of contractive type mappings, Bull. Calcutta Math. Soc, 84(1992, no. 2, 167-194. [11] M. O. Olatinwo,A result for approximating fixed points of generalized weak contraction of the integral-type by using Picard iteration, Revista Colombiana de Matemáticas. Volumen 42(282, páginas 145-151. [12] H.K. Pathak, R. Tiwari, M.S. Khan, A common fixed point theorem satifying integral type implicit relations, Applied Mathematics E - Notes, 7(27, 222-228. [13] B. E. Rhoades, A Comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc. 226(1977, 257-29 [14] B. E. Rhoades, Contractive definitions revisited, Topological Methods in Nonlinear Functional Analysis (Toronto, Ont.,1982, Contemp. Math., Vol. 21, American Mathematical Society, Rhode Island, 1983, pp. 189-23. [15] B. E. Rhoades, Contractive definitions, Nonlinear Analysis, World Science Publishing, Singapore, 1987, pp. 513-526. [16] B.E. Rhoades, Two fixed point theorems for mappings satisfying a general contractive condition of integral type, International Journal of Mathematics and Mathematical Sciences, 23:63, 47-413. [17] B. E. Rhoades, Some theorems on weakly contractive maps, Nonlinear Analysis: Theory, Methods & Applications, Vol. 47, no. 4, pp. 2683-2693, 21. [18] T. Suzuki, Meir Keeler contractions of integral are still Meir - Keeler contractions, International Journal of Mathematics and Mathematical Sciences, Volume 27, Article ID 39281, 6 pages.
242 Nguyen Van Luong and Nguyen Xuan Thuan [19] P. Vijayaraju, B. E. Rhoades and R. Mohanraj, A fixed point theorem for a pair of maps satisfying a general contractive condition of integral type, International Journal of Mathematics and Mathematical Sciences, 25:15 (25 2359-2364. Received: May, 29