GENERALIZED CONTRACTION MAPPING PRINCIPLE IN LOCALLY CONVEX TOPOLOGICAL VECTOR SPACES YANXIA TANG1, JINYU GUAN1, PENGCHENG MA1, YONGCHUN XU1;¤, YONGFU SU1;2 1. Department of Mathematics, Hebei North University, Zhangjiakou 075000, China 2. Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China Abstract: The purpose of this paper is to present the concept of contraction mapping in a locally convex topological vector spaces and to prove the generalized contraction mapping principle in such spaces. The neighborhood-type error estimate formula was also established. The results of this paper improve and extend Banach contraction mapping principle in new idea. Keywords: Contraction mapping principle; Locally convex; Topological vector spaces; Fixed point; Error estimate formula. 1. Introduction Banach contraction mapping principle is one of the important tool (or method) in nonlinear analysis and other mathematical ¯eld. Weak contractions are generalizations of Banach contraction mappings which have been studied by several authors. Let (X; d) be a metric space and Á : [0; +1) ! [0; +1) be a function. We say that T : X ! X is a Á-contraction if d(T x; T y) · Á(d(x; y)); 8 x; y 2 X: In 1968, Browder [1] proved that if Á is non-decreasing and right continuous and (X; d) is ¤ n ¤ complete, then T has a unique ¯xed point x and lim T x0 = x for any given x0 2 X. n!1 Subsequently, this result was extended in 1969 by Boyd and Wong [2] by weakening the hypothesis on Á, in the sense that it is su±cient to assume that Á is right upper semi- continuous. For a comprehensive study of relations between several such contraction type conditions, see [3-6]. In 1973, Geraghty [5] introduced the Geraghty-contraction and obtained the ¯xed point theorem. Let (X; d) be a metric space. A mapping T : X ! X is said to be a Geraghty- contraction if there exists ¯ 2 ¡ such that for any x; y 2 X d(T x; T y) · ¯(d(x; y))d(x; y); where the class ¡ denotes those functions ¯ : [0; +1) ! [0; +1) satisfying the following condition: ¯(tn) ! 1 ) tn ! 0. On the other hand, in 2015, Su and Yao [7] proved the following generalized contraction mapping principle. Theorem 1.1. Let (X; d) be a complete metric space. Let T : X ! X be a mapping such that Ã(d(T x; T y)) · Á(d(x; y)); 8 x; y 2 X (2:1); where Ã; Á : [0; +1) ! [0; +1) are two functions satisfying the conditions: (1):Ã(a) · Á(b) ) a · b; 1 2YANXIA TANG1, JINYU GUAN1, PENGCHENG MA1, YONGCHUN XU1;¤, YONGFU SU1;2 ( Ã(a ) · Á(b ) (2): n n ) " = 0: an ! "; bn ! " n Then, T has a unique ¯xed point and, for any given x0 2 X, the iterative sequence T x0 converges to this ¯xed point. De¯nition 1.2. Let (X; d) be a metric space, T : XN ! X be a N-variables mapping, an element p 2 X is called a multivariate ¯xed point (or a ¯xed point of order N, see [8]) of T if p = T (p; p; ¢ ¢ ¢; p): Recently [8], Yongfu Su, A. Petru»seland Jen-Chih Yao proved a multivariate ¯xed point theorem for the N-variables contraction mappings which further generalizes Banach Contraction Principle. In particular, the study of the ¯xed points for weak contractions and generalized con- tractions was extended to partially ordered metric spaces in [9-19]. Among them, some results involve altering distance functions. Such functions were introduced by Khan et al. in [20], where some ¯xed point theorems are presented. The purpose of this paper is to present the concept of contraction mapping in a lo- cally convex topological vector spaces and to prove the generalized contraction mapping principle in such spaces. The results of this paper improve Banach contraction mapping principle in Banach. 2. Contraction mapping principle in locally convex spaces Let us recall some concepts and results on the topological vector spaces. De¯nition 2.1. A Hausdor® topology ¿ on a real vector space X over R is said to be a vector space topology for X if addition and scalar-multiplication are continuous, i.e., the mappings (x; y) 7! x + y from X £ X into X and (®; x) 7! ®x from R £ X into X are continuous, where X £ X and R £ X are equipped with the respective product topolo- gies. X itself, or more precisely (X; ¿) is then called a topological vector space. Remark 2.2. Continuity of addition means: For every neighborhood W of x0 + y0 there exist neighborhood U of x0 and V of y0 such that U + V ½ W . Continuity of scalar- multiplication means: For every neighborhood W of ®0x0 there exist a ± > 0 and a neighborhood U of x0 such that ®U ½ W; 8 j® ¡ ®0j < ±: De¯nition 2.3. A topological vector space (X; ¿) is said to be locally convex, if there exists a basis of neighborhood of zero ­ such that every U 2 ­ is convex set. Conclusion 2.4. Let (X; ¿) be a locally convex topological vector space. For any convex neighborhood of zero U 2 ­, there exists a balanced convex neighborhood of zero V such that V ½ U. Proof. For any convex neighborhood of zero U 2 ­, there exists a balanced neighborhood of zero W such that W ½ U. Let \ A = ®U; j®j=1 CONTRACTION MAPPING PRINCIPLE IN LOCALLY CONVEX SPACES 3 then A and A0 are convex. Since W is balanced, we have W = ®W ½ ®U; 8 j®j = 1; which implies W ½ A; W ½ A0: Hence A0 is a neighborhood of zero. Next, we show A0 is balanced. In fact that, for any j¸j · 1, we have \ \ \ ¸A = ¸®U = j¸j®U ½ ®U = A; j®j=1 j®j=1 j®j=1 which implies A is balanced, so is A0. Let V = A0, we have V is a balanced convex neighborhood of zero such that V ½ U. This completes the proof.¤ From Conclusion 2.4, we can get the following result. Conclusion 2.5. Let (X; ¿) be a locally convex topological vector space. Then (X; ¿) there exists a basis of balanced convex neighborhood of zero ­. Furthermore, each U 2 ­ is absorbing, balanced and convex. De¯nition 2.6. Let (X; ¿) be a locally convex topological vector space with a basis of balanced convex neighborhood of zero ­. (1) A mapping T : X ! X is said to be contractive, if there exists a constant h 2 (0; 1) such that for any U 2 ­ and any x; y 2 X x ¡ y 2 tU implies T x ¡ T y 2 htU; for any t > 0. (2) A mapping T : X ! X is said to be (Ã; Á)- contractive, if there exist two functions à : [0; +1) ! [0; +1);Á : [0; +1) ! [0; +1) such that for any U 2 ­ and any x; y 2 X x ¡ y 2 Á(t)U implies T x ¡ T y 2 Ã(t)U; for any t > 0. De¯nition 2.7. Let (X; ¿) be a topological vector space with a basis of balanced convex neighborhood of zero ­, a net net fx¸g¸2I ½ X is said to be Cauchy, if for any U 2 ­, there exists a ¸0 2 I such that x¸1 ¡ y¸2 2 U; 8 ¸1; ¸2 ¸ ¸0: The topological vector space (X; ¿) is said to be complete, if every Cauchy net is conver- gent. The following results are well-known in the theory of topological vector space. Conclusion 2.8. Let (X; ¿) be a locally convex topological vector space with a basis of balanced convex neighborhood of zero ­. For any U 2 ­, the Minkowski functional of U is de¯ned by MU (x) = infft > 0 : x 2 tUg; 8 x 2 X: Then the following hold: (1) MU (x) ¸ 0, for any x 2 X, and x = 0 implies MU (x) = 0; (2) MU (¸x) = j¸jMU (x) for any x 2 X; ¸ 2 R; (3) MU (x + y) · MU (x) + MU (y) for any x; y 2 X. (4) net fx¸g¸2I ½ X convergent to x0 2 X if and only if lim¸2I MU (x¸ ¡ x0) = 0. (5) net fx¸g¸2I ½ X is a Cauchy net if and only if for any U 2 ­ lim MU (x¸1 ¡ x¸2 ) = 0: ¸1;¸22I 4YANXIA TANG1, JINYU GUAN1, PENGCHENG MA1, YONGCHUN XU1;¤, YONGFU SU1;2 Remark 2.9. In fact that, for any U 2 ­, the Minkowski functional MU (¢) is a semi-norm on the X. Theorem 2.10. (Generalized contraction mapping principle) Let (X; ¿) be a complete locally convex topological vector space with a basis of balanced convex neighborhood of zero ­. Let T : X ! X be a (Ã; Á)-contractive mapping satisfying the following conditions: (1) Ã(t);Á(t) are continuous and strictly increasing; (2) Ã(0) = Á(0) and Ã(t) < Á(t) for all t > 0. n Then T has a unique ¯xed point and for any given x0 2 X, the iterative sequence T x0 converges to this ¯xed point. Proof. Since T is a (Ã; Á)-contractive mapping, we have, for any U 2 ­, that ¡1 ¡1 à (MU (T x ¡ T y)) = à (infft > 0 : T x ¡ T y 2 tUg) = á1(inffÃ(t) > 0 : T x ¡ T y 2 Ã(t)Ug) = á1(Ã(infft > 0 : T x ¡ T y 2 Ã(t)Ug)) = infft > 0 : T x ¡ T y 2 Ã(t)Ug · infft > 0 : x ¡ y 2 Á(t)Ug (2:1) = Á¡1(Á(infft > 0 : x ¡ y 2 Á(t)Ug)) = Á¡1(Á(Á¡1 inffÁ(t) > 0 : x ¡ y 2 Á(t)Ug)) = Á¡1(inffÁ(t) > 0 : x ¡ y 2 Á(t)Ug) = Á¡1(infft > 0 : x ¡ y 2 tUg) ¡1 = Á (MU (x ¡ y)); 8 x; y 2 X: For any given x0 2 X, we de¯ne an iterative sequence as follows x1 = T x0; x2 = T x1; ::::; xn+1 = T xn; :::: (2:2) Then, for each integer n ¸ 1, from(2.1) and (2.2) we get ¡1 ¡1 ¡1 à (MU (xn+1 ¡ xn)) = à (MU (T xn ¡ T xn¡1)) · Á (MU (xn ¡ xn¡1)): (2:3) Using the condition (2) we have, á1(t) ¸ Á¡1(t), therefore MU (xn+1 ¡ xn) · MU (xn ¡ xn¡1) for all n ¸ 1.