Research Journal of Applied Sciences, Engineering and Technology 3(5): 968-970, 2011 ISSN: 2040-7467 © Maxwell Scientific Organization, 2011 Submitted: July 13, 2011 Accepted: August 27, 2011 Published: September 20, 2011 A Common Fixed Point Theorem with Pseudo Property in Menger Space Hojat Allah Ebadizadeh and Hamid Reza Hagh Bayan Department of Mathematics, Faculty of Science, Islamic Azad University North Tehran Branch, Tehran, Iran Abstract: The main purpose of this study is to generalize the definition of the E.A property between maps in this way that don’t need to one point such that the limit of two sequence is equal to that, and give a common fixed point theorem in manger space with the above condition. Key words: Common fixed point, menger space, weakly compatible maps INTRODUCTION Fixed point theorems give the conditions under which maps (single or multivalued) have solutions. The theory itself is a beautiful mixture of analysis, topology, and geometry. Over the last 50 years or so the theory of fixed points has been revealed as a very powerful and important tool in the study of nonlinear phenomena. In particular fixed point techniques have been applied in such diverse fields as biology, chemistry, economics, engineering, game theory, and physics. Common fixed point theory is a stronger version of fixed point theory, that provide a number of operator have a common fixed point. The concept of probabilistic metric space was first introduced and studied by Menger (Menger, 1942) , which is a generalization of the metric space and also the study of this space was expanded rapidly with the pioneering works of Schweizer and Sklar (1960, 1983) and Kubiaczyk and Sharma (2008) and others. The theory of probabilistic space is of fundamental importance in probabilistic functional analysis. Jungck (1986) introduced the notion of compatible mappings and since that time many authors use this definition in his studies. Now in this study we introduce the concept of pseudo E. A, property and then give a common fixed point theorem. a ≥ b, c ≥ d ⇒ T ( a , b)(a , b, c, d , ∈[ 0,1]) C T (a , T ( b, c) ) = T (T ( a , b) , c)(a , b, c ∈[ 0,1]) Definition 2: A probabilistic metric space is a pair (X, F) F, where X is a nonempty set and is a mapping from L X×X to L (the set of all distribution function). The distribution function assumed to satisfy the following conditions: C C C C F(u, v)(x) for every x > 0 iff u = v . F(u v) (0) for every u , v 0 X F(u, v)(x) = F(v, u)(x) for every u, v 0 X If F(u, v)(x) = 1 and F(v, w)(y) = 1 then F(v, w) (x+y) = 1 for every u, v, w 0 X Definition 3: A Menger space is a triple (X, F, T) where (X, F) is a probabilistic metric space and T is a T- norm with the following property : F (u, w)( x + y ) ≥ T[ F (u, v )( x ), F (v , w)( x )] for every u, v , w ∈ X and ≥ 0 . Definition 4: Let (X, F, T) be a menger space with continuous T-norm. a sequence {pn} in X is said to be convergent to p and show with limn→ ∞ pn = p if for every g > 0, 8 > 0, there exists an integer N such that F(pn, p) > 1 - 8 for all n $ N. BASIC DEFINITION Definition 1: A mapping T: [ 0,1] × [ 0,1] → [ 0,1] is called a T norm (or triangular norm) if if the following conditions are satisfied: C C C Definition 5: Two self mappings S, T are said to be weakly compatible if they commute at their coincidence points, i.e, if Tu = Su for some u 0 X, then Tsu = STu. Definition 6: Let S,T be two self mappings of a Menger space (X, F, T). We say that S,T satisfy the property E.A if there exists a sequence {xn} in X such that: T(a,1) = a for every a ∈[ 0,1] T(a,b) = T(b,a) for every a,b , [0,1] Corresponding Author: Hojat Allah Ebadizadeh, Department of Mathematics, Faculty of Science, Islamic Azad University North Tehran Branch, Tehran, Iran 968 Res. J. Appl. Sci. Eng. Technol., 3(9): 968-970, 2011 C limn → ∞ Sxn = limn→ ∞ Txn = z for some z , X. limn→ ∞ Sxn = limn→ ∞ Txn Definition 7: Let S , T be two self mappings of a Menger space (X, F, T). We say that S, T are pseudo E. A if there exists a sequence {xn} in X such that: limn→ ∞ Sxn = limn→ ∞ Txn C C Example 1: Let X = [0, 4) and define S , A: X → X by: Tx = x 4 , Sx = C C 3x (x ∈ X) 4 C Consider the sequence {xn = 1/n}. Clearly lim n→ ∞ Tx n = 0 . So S and T satisfy the E. A property and pseudo E. A property. Example 2: Let X = [0, ∞ ) and define S , T: X → X by: F ( Txn , Ta )( kx ) ≥ min {F(Sxn, Sa)(x), F(Sxn, Txn)(x), F(Sa, Ta)(x), F(Sa, Txn)(x), F(Sxn, Ta)(x)} Letting n → ∞ , yields, F(Sc, Ta)(kx) $ F(Sa, Ta)(x) Which is a contradiction by definition 3. Therefore Sa = Ta. Since T and S are weakly compatible, Sa = Ta and therefore, Tta = Tsa = Sta = SSa(*) F ( Ta , TTa )( kn) ≥ min{F ( Sa , STa )( x ) Consider the sequence {xn = n}. Clearly F ( Sa , Txn)( xn) , F ( Sa , Ta )( x ) , lim n→ ∞ Sx n = lim n → ∞Tx n = ∞ . F ( STa , Ta )( x ) , F ( Sa , TTa )( x )} Therefore, S and T satisfy the pseudo E. A property. Note that if we regard X f [0, 4], then 4 óX. So the class of operator that satisfy the pseudo E. A property is bigger that the class of operator that satisfy the E. A property. By (*), from the above inequalities we have, F(Ta, Tta)(kx) $ F(Ta, TTa) This is a contradiction. Therefore Ta = TTa. Thus STa = TTa = Ta. When TX is assumed to be closed subset of X, the proof is similar, since TX f SX. If Sz = Tz = z, and Ta … z, we have, RESULTS AND DISCUSSION Now in this section we give theorems between mappings that have pseudo E. A property. F ( Ta , Tz )( kx ) ≥ min{F ( Sa , Sz )( x ) Theorem 1: Let (X, F, T) be a menger space with T(x, y_ = min(x, y) for all x, y 0 [0, 1]. Let S and T be weakly compatible mappings X of into itself such that, F ( Sa , Ta )( x ) , F ( Sz , Tz )( x ) , F ( Sz, Ta )( x ) , F ( Sa , Tz )( x )} = F ( Ta , Tz )( x ) S and T satisfy the pseudo E. A property There exists a number k 0 (0, 1) such that, which is a contradiction. So the common fixed point Ta is unique. F ( Tu, Tv )( kx ) ≥ min{F ( Su, Sv )( x ) , F ( Su, Tu)( x ) , F ( Sv , Tv )( x ) , F ( Sv , Tu)( x ) , F ( Su, Tv )( x )} C C Suppose that is closed. Then we SX have limn → ∞ Sxn = SA for some a 0 X. Also limn → ∞ Txn = Sa . We show that Ta = Sa. Suppose that Ta ≠ Sa . By condition ii) we have, Finally we show that Ta is a common fixed point of S and T. Suppose that Ta … TTa, then by condition ii) we have, Tx = x , Sx = x + 1( x ∈ X ) C C Proof: Since S and T satisfy the pseudo E. A property, there exists a sequence {Xn} in X such that: CONCLUSION for all u, v 0 X TX f SX If TX or SX be a closed subset of X, then S and T have a unique common fixed point. In this study we define the concept of Pseudo E.A. property and then a common fixed point theorem with its proof stated. Note that this definition make us more flexible in dealing with real-world problems. 969 Res. J. Appl. Sci. Eng. Technol., 3(9): 968-970, 2011 Kubiaczyk, I. and S. Sharma, 2008. some common fixed point theorems in Menger space under strict contractive conditions. Sou. Asi. Bull. Math. 32: 117-124. Menger, K., 1942. Statistical metric, Proc. Natl. Acad. Sci. USA, 28(12): 535-537. Schweizer, B. and A. Sklar, 1960. Statistical spaces. Pacific. J. Math., 10: 313-334. Schweizer, B. and A. Sklar, 1983. Probabilistic Metric Space. Vol. 5, North- Holland, Series in Probability and Applied Math, NY, USA. ACKNOWLEDGMENT The authors are grateful than referee(s) for valuable comment for improving this work and Islamic Azad University, North Tehran branch, Tehran, Iran for supporting this work and Dr. M. Foozooni for his deep insight of fix point that help us. REFERENCES Jungck, G., 1986. Compatible mappings and common fixed points. Internat. J. Math. Sci., 9(4): 771-779. 970