Reduced Assumption in the Banach Contraction Principle Φ≠ C <

Reduced Assumption in the Banach Contraction Principle

Sahar Mohamed Ali Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt

Abstract: In this study we introduce a novel class of normed spaces called weakly Cauchy normed spaces not necessarily complete in general and proved the existence of a fixed point of contraction mappings in these spaces, this is weaker assumption than the completeness assumption imposed on the given normed space on the other side the contraction condition valid only on a closed convex subset of the given space.

Key words: Reduced assumption, contraction mapping, Banach contraction principle, normed spaces

INTRODUCTION

n Needless to say, fixed point theorem plays a major role Moreover, the sequence {T (y)}n∈ N is strongly in many applications, including variational and linear convergent to x for every y∈X. inequalities, optimization and applications in the field of approximation theory and minimum norm problems. Mathematician in the field of fixed point theory try to Thus mathematical analysis of the fixed point theory improve the results of this theorem in which changing drawn much attention in research. the assumptions imposed on contraction mapping or on We used the concept of weakly Cauchy normed spaces the given topological space, as the case of to prove the existence of the best approximation nonexpansive mappings while compact or Hilbert elements of a convex closed subsets in normed spaces spaces are necessarily or as in the case of Caristi's fixed not necessarily be reflexive Banach space or Hilbert point theorem using a proper convex lower semi space[1]. continuous mappings, for other many other trials one In this study we used the same concept of normed space can see Kirk[3]. to prove the existence of a fixed point of contraction mapping on such spaces in which reduced assumption Notations and Basic Definition: Let X be a linear is imposed. Thus we give the following definition. space. Then a function f from X into (-∞, ∞] is said to be proper lower semicontinuous and convex function if Definition[1] :A normed space X is said to be weakly it satisfies the following conditions: Cauchy space if and only if every Cauchy sequence in 1. There is x ∈ X such that f(x)< ∞ (proper). X is weakly convergent to an element x in X . 2. For any real number , the set {x∈ X: f(x) ≤} is A mapping T on a normed space X into X is said to be closed convex subset of X (lower semicontinuous). contraction if and only if there is a non-negative real 3. For any x, y ∈ X and t∈ [0, , 1], f(t x +(1-t)y)≤ t number r<1 with the property that f(x)+(1-t)f(y)(convex function).

||T(x)-T(y)|| ≤ r ||x-y|| for every x, y ∈ X. Remark:Notice that if X is a normed space and And is said to be nonexpansive if and only if {xn}n∈ N is weakly Cauchy sequence in X (in the sense ≤ ||T(x)-T(y)|| ||x-y|| for every x, y ∈ X. that {f(xn )}n∈ N is Cauchy for every linear bounded functional f in X*) which has a subsequence {xni }i∈ N As T is contraction, it preserves strong convergence, convergent weakly to an element x in X, then the but it does not preserve weak convergence in general. sequence {xn}n∈ N is weakly convergent to the same weak limit. The following is the Banach contraction principle which is the basis theorem of fixed point theory long We have the following lemmas: time ago. Lemma1:Let X be a weakly normed Cauchy space and [2] Theorem :Let X be a complete metric space and let T {Cn}n∈ N be a descending sequence of closed bounded be a contraction of X into itself. Then T has a unique convex subsets of X such that - limn ∞Diam(Cn)=0. ∞ fixed point in the sense that T(x) =x for some x ∈ X. Then the intersection < C ≠ Φ is not empty. n=1 n

Corresponding Author: Sahar Mohamed Ali, Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt 343 J. Math. & Stat., 2 (1): 343-345, 2006

Lemma2:Every reflexive normed space is weakly (x) ∈ N(y) for every n≥ n0, it follows that, Cauchy space. l l n || T (y)- y|| - ε <|| T (y)- T (x)|| for every n≥ n0 . Proof: It is required to show that every Cauchy sequence in a reflexive normed space X is weakly We have two cases, the first is when n0 ≤ l, for such a convergent. So, let {xn}n∈ N be a Cauchy sequence in X. Since case, one has ||f(xn - xm )|| ≤ ||f|| || xn - xm || for every f ∈ X*, l l l {f(xn )}n∈ N is Cauchy sequence of numbers for every || T (y)- y|| - ε <|| T (y)- T (x)||, linear bounded functional f on X, hence it is which in turns showed that, l l l l convergent, let limn ∞ f(xn )= f and define the linear || T (y)- y|| - ε <|| T (y)- T (x)||≤r ||y-x||, bounded functional F on X* by F(f)= f, since X is The other case, if l

m l ≤ l || T (x)- T (x)|| (r /1-r)||T(x)-x||, Corollary1:Let X be a reflexive normed space, C be a

n closed convex subset of X and T contraction mapping and in turns shows that, the sequence {T (x)}n∈ N is from C into itself. Then T has a unique fixed point y in Cauchy sequence in X, since X is weakly Cauchy, the n C. Moreover Cauchy sequence {T (x)}n N is weakly convergent to n ∈ W-liml ∞ T (x) =y for every x ∈ C. some element y ∈ X, since C is convex closed, it contains all its weak limits as well as all its strong Proof: Using Theorem (1) and lemma (2), we get the limits, hence y in C, we claim that y is the unique fixed proof. point of T, to prove that, let l be any arbitrarily natural number and define the proper lower-semicontinuous Corollary2:Let X be a weakly Cauchy normed space convex real valued function µl on C by the following l and T be a contraction mapping on X into itself, if G is formula:µl(x)= || T (y)- x|| the class of all closed convex subsets of X which ≤ for any real number , the set Gl:={ x∈ C: µl(x) < } is closed convex subset of C and in particular, for any invariant under T. Then the intersection C∈G C ≠ Φ ε >0 the set is not empty.

l l G{||Tl(y)-y||- ε }:={x∈ C: ||T (y)-x|| ≤||T (y)-y|| - ε } Proof: Using Theorem (1), the intersection is a set containing at least the unique fixed point of T. is closed convex, hence it is closed in the weak Remarks: Using the generalization of Caristi's fixed topology, therefore its complement point theorem and the characterization of complete c l l G {||Tl(y)-y||- ε }:={x∈ C: ||T (y)-x|| >||T (y)-y|| - ε } metric space introduced by Wataru [4], Theorem (1) is weakly open set which containing y, then there is a c insures that the class of weakly contractive mappings neighborhood N(y) of y such that N(y) ⊂ G {||Tl(y)-y||- on Banach spaces is containing properly the class of all ε }, on the other side, using the fact that y is the weak n n limit point of {T (x)}n∈ N , there is n0∈N such that T contraction mappings. 344 J. Math. & Stat., 2 (1): 343-345, 2006

REFERENCES 1. El-Shobaky, E. M., Sahar Mohammed Ali and 3. Kirk, W. A. and B. Sims, 2001. Handbook of Wataru Takahashi, 2001. On The Projection Metric Fixed Point Theory. Kluwer Academic Constant Problems And The Existence Of The Publishers. Dordrecht. Boston. London. Metric Projections in normed spaces. Abstr. 4. Wataru, T.2000. Nonlinear Functional Appl. Anal. 6 :no.7: 401-410. Analysis, Fixed Point Theory and its 2. Mohamed, A. K and W. A. Kirk, 2001. An Applications. Yokohama Publishers, Introduction to Metric Spaces and Fixed Point Yokohama. Theory”, A Wiley-Interscience Publication. John Wiley and Sons, Inc. New York.

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