Appl. Math. Mech. -Engl. Ed. 31(4), 415–424 (2010) DOI 10.1007/s10483-010-0402-z c Shanghai University and Springer-Verlag Berlin Heidelberg 2010

Applied Mathematics and Mechanics (English Edition)

Intuitionistic Menger inner product spaces and applications to integral equations∗ Shi-sheng ZHANG ()1 , M. GOUDARZI 2 , R. SAADATI 2 , S. M. VAEZPOUR2 (1. Department of Mathematics, Yibin University, Yibin 644007, Sichuan Province, P. R. China; 2. Department of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Avenue, Tehran 15914, Iran) (Contributed by Shi-sheng ZHANG)

Abstract In this paper, first introduce and define an intuitionistic Menger inner product space, and then, obtain a new fixed point theorem in a complete intuitionistic Menger inner product space. As an application, the results are used to study the existence and uniqueness of the solution to a linear Volterra integral equation. Key words intuitionistic Menger inner product space, Volterra integral equation, h-type t-norm, fixed point theorem Chinese Library Classification O177.91 2000 Mathematics Subject Classification

1

46S40

Introduction

The notion of probabilistic inner product spaces has been introduced in [1] and has been modified in [2] and [3]. Until now, there has been no significant development in the theory of intuitionistic Menger inner product spaces. Presenting a reasonable definition is one of the most important problems in intuitionistic Menger analysis, which can widely be used in intuitionistic Menger optimization and the other fields[4–6] . We need the definition of an intuitionistic Menger inner product space based on the definition of an intuitionistic Menger normed space. Also it leads naturally to pertain to the most important classes of intuitionistic Menger inner product spaces, namely, the intuitionistic Menger Hilbert spaces, which will be defined in the future. Besides, fixed point theorems are the useful instruments in many applied areas such as mathematical economics, non-cooperative game theory, dynamic optimization and stochastic games, functional analysis, and variation calculus[7–11] . Therefore, extending the fixed point theorems to intuitionistic Menger inner product spaces can be applicable in the other fields. In the second part of this paper, we introduce a new fixed point theorem under the given conditions, and apply it to the study of the existence and uniqueness of solution for a linear Volterra integral equation in a complete intuitionistic Menger inner product space. ∗ Received Sept. 25, 2009 / Revised Mar. 10, 2010 Project supported by the Natural Science Foundation of Yibin University (No. 2009Z01) Corresponding author R. SAADATI, E-mail: [email protected]

416

2

Shi-sheng ZHANG, M. GOUDARZI, R. SAADATI, and S. M. VAEZPOUR

Preliminaries Lemma 2.1 (see [12]) Consider the set L∗ and the operation L∗ defined by L∗ = {(x1 , x2 ) : (x1 , x2 ) ∈ [0, 1]2 , x1 + x2  1}.

If it is defined by (x1 , x2 ) L∗ (y1 , y2 ) ⇐⇒ x1  y1 and x2  y2 for every (x1 , x2 ), (y1 , y2 ) ∈ L∗ , then (L∗ , L∗ ) is a complete lattice. For every zα = (xα , yα ) ∈ L∗ , we define ∨(zα ) = (sup xα , inf yα ). Since zα ∈ L∗ , xα + yα  1. Thus, sup xα + inf yα  sup(xα + yα )  1, i.e., ∨(zα ) ∈ L∗ . We denote its units by 0L∗ = (0, 1),

1L∗ = (1, 0).

Classically, a triangular norm (t-norm) T on [0, 1] is defined as an increasing, commutative and associative mapping T : [0, 1]2 → [0, 1] satisfying

T (1, x) = x, ∀x ∈ [0, 1].

A triangular conorm (t-conorm) S is defined as an increasing, commutative and associative mapping S : [0, 1]2 → [0, 1] satisfying S(0, x) = x, ∀x ∈ [0, 1]. Using the lattice (L∗ , L∗ ), these definitions can be straightforwardly extended. Definition 2.1 (see [12]) A triangular norm (t-norm) on L∗ is a mapping T : (L∗ )2 → L∗ satisfying the following conditions: T (x, 1L∗ ) = x,

∀x ∈ L∗

(boundary condition);

∀(x, y) ∈ (L∗ )2

T (x, y) = T (y, x),

T (x, T (y, z)) = T (T (x, y), z), 

(commutativity);

∀(x, y, z) ∈ (L∗ )3







x L∗ x , y L∗ y =⇒ T (x, y) L∗ T (x , y ),

(associativity);

∀(x, x , y, y  ) ∈ (L∗ )4

(monotonicity).

Definition 2.2 (see [12]) A continuous t-norm T on L∗ is called continuous t-representable if and only if there exist a continuous t-norm ∗ and a continuous t-conorm on [0, 1] such that, for all x = (x1 , x2 ), y = (y1 , y2 ) ∈ L∗ , T (x, y) = (x1 ∗ y1 , x2 y2 ). Now, define a sequence T n recursively by T1 =T, T n (x(1) , · · · , x(n+1) ) = T (T n−1 (x(1) , · · · , x(n) ), x(n+1) ),

n  2, x(i) ∈ L∗ .

Definition 2.3 (see [12]) A negator on L∗ is a decreasing mapping N : L∗ → L∗

satisfying

N (0L∗ ) = 1L∗ , N (1L∗ ) = 0L∗ .

Intuitionistic Menger inner product spaces and applications to integral equations

If N (N (x)) = x,

417

∀x ∈ L∗ ,

then N is called an involutive negator. A negator on [0, 1] is a decreasing mapping N : [0, 1] → [0, 1] satisfying

N (0) = 1, N (1) = 0.

Ns denotes the standard negator on [0, 1] defined as Ns (x) = 1 − x. Also, Ns denotes the standard negator on L∗ defined as Ns (x) = (1 − x, x),

∀x ∈ [0, 1].

Throughout this paper, we let R = (−∞, +∞) and R+ = [0, +∞). Definition 2.4 (see [13]) A distance distribution function is a non-decreasing and leftcontinuous mapping F : R → R+ with inf F (t) = 0 and sup F (t) = 1. We will denote D as the t∈R

t∈R

family of all distance distribution functions and H as the special element of D defined by  1, t > 0, H(t) = 0, t  0. Definition 2.5 (see [13]) A non-distance distribution function is a non-increasing and left-continuous mapping L : R → R+ with sup L(t) = 0 and inf L(t) = 1. We will denote by E t∈R

t∈R

the family of all non-distance distribution functions and by G the special element of E defined by  1, t  0, G(t) = 0, t > 0.

3

Main results

Definition 3.1 An intuitionistic distance distribution function is a non-decreasing and left-continuous mapping F: R → L∗ such that sup F (t) = 1L∗ and inf F (t) = 0L∗ . For example, t∈R

t∈R

H(t) =

 1L∗ ,

t > 0,

0L∗ ,

t0

is an intuitionistic distance distribution. Definition 3.2 (see [14]) An intuitionistic Menger set Aζ,η in a universe U is an object Aζ,η = {(ζA (u), ηA (u))|u ∈ U }. Here, for all u ∈ U, ζA (u) ∈ D and ηA (u) ∈ D are, respectively, called the membership map and the non-membership map of u in Aζ,η . Furthermore, they satisfy ζA (u)(t) + ηA (u)(t)  1, ∀t ∈ R. Definition 3.3 Let φ, ϕ be two mappings from X × X to D such that φ(x, y)(t) + ϕ(x, y)(t)  1 for all x, y ∈ X and t > 0. An intuitionistic Menger inner product (IMIP) space is a triplet (X, Fφ,ϕ , T ), where X is a real vector space, T is a continuous t-representable, and Fφ,ϕ is an intuitionistic Menger set on X × X satisfying the following conditions for every x, y, z ∈ X and s, t ∈ R : (IMI-1) Fφ,ϕ (x, y)(0) = 0L∗ and Fφ,ϕ (x, x)(t) >L∗ 0L∗ for each t > 0;

418

Shi-sheng ZHANG, M. GOUDARZI, R. SAADATI, and S. M. VAEZPOUR

(IMI-2) Fφ,ϕ (x, y)(t) = Fφ,ϕ (y, x)(t); (IMI-3) Fφ,ϕ (x, x)(t) = H(t) for all t ∈ R if and only if x = 0; (IMI-4) For any real number α, ⎧ t ⎪ F , α > 0, (x, y) ⎪ φ,ϕ ⎪ ⎨ α α = 0, Fφ,ϕ (αx, y)(t) = H(t), ⎪    ⎪ ⎪ t ⎩ Ns Fφ,ϕ (x, y) − , α < 0; α (IMI-5) sup {T (Fφ,ϕ (x, z)(s), Fφ,ϕ (y, z)(r))} = Fφ,ϕ (x + y, z)(t); s+r=t

(IMI-6) T (Fφ,ϕ (x, x)(t), Fφ,ϕ (y, y)(s)) L∗ Fφ,ϕ (x + y, x + y)(t + s). In this case, Fφ,ϕ is called an intuitionistic Menger inner product. Here, Fφ,ϕ (x, y)(t) = (φ(x, y)(t), ϕ(x, y)(t)). Example 3.1 Let (X, ·, · ) be an ordinary inner product space, and let T (a, b) = (min(a, b), max(a, b)) for all a, b ∈ [0, 1]. For each t ∈ R, define FH,G (x, y)(t) = (H(t − x, y ), G(t − x, y )). Then, we can easily show that (X, FH,G , T ) is an IMIP-space that is called the intuitionistic Menger inner product space induced by the inner product space (X, ·, · ). If 0 < t < s, then k = s − t > 0. We have Fφ,ϕ (x, y)(t) = T (Fφ,ϕ (x, y)(t), 1L∗ ) = T (Fφ,ϕ (x, y)(t), Fφ,ϕ (0, y)(k))  Fφ,ϕ (x, y)(s). Hence, Fφ,ϕ (x, y)(t) is non-decreasing with respect to t. Definition 3.4 Let (X, Fφ,ϕ , T ) be an IMIP-space. (i) A sequence {xn } ⊆ X is said to converge to x ∈ X if for each ε > 0 and for each α ∈ (0, 1), there exists N ∈ N such that Fφ,ϕ (xn − x, xn − x)(ε) >L∗ Ns (α),

∀n  N.

(ii) A sequence {xn } ⊆ X is called a Cauchy sequence if for each ε > 0 and for each α ∈ (0, 1), there exists N ∈ N such that Fφ,ϕ (xn − xm , xn − xm )(ε) >L∗ Ns (α),

∀m, n  N.

(iii) (X, Fφ,ϕ , T ) is called a complete IMIP-space if every Cauchy sequence is convergent. Definition 3.5 A continuous t-norm T on L∗ is h-type if the family of functions {T m (t)} is equicontinuous at t = (1L∗ , 1L∗ ), where T 1 (t) = T (t, t),

t ∈ L∗ ,

T m (t) = (t, T m−1 (t)),

t ∈ L∗ ,

m = 2, 3, · · · .

Theorem 3.1 Let (X, Fφ,ϕ , T ) be a complete IMIP-space and T be a t-representable norm of h-type. Let T : (X, Fφ,ϕ , T ) → (X, Fφ,ϕ , T ) be a linear mapping satisfying the condition  t  Fφ,ϕ (T x, y)(t) L∗ Fφ,ϕ (x, y) k(α, β)

Intuitionistic Menger inner product spaces and applications to integral equations

419

for all x, y ∈ X, t  0, and α, β ∈ (0, ∞). Here, k(α, β) : (0, ∞) × (0, ∞) → (0, 1) is a function. Then, T has exactly one fixed point x∗ ∈ X. Furthermore, for any x0 ∈ X and iterative sequence {T n x0 }, we have Fφ,ϕ

T n x0 −−−→ x∗ . Proof For any x0 ∈ X and m ∈ N, let xm = T m x0 . First, we prove that {xm }∞ m=1 is a Cauchy sequence. By (IMI-5), we have  Fφ,ϕ (x0 − T m x0 , y) = L∗ L∗ = L∗

L∗

t  k(α, β)

t  k(α, β)   t(1 − k(α, β))   tk(α, β)  , Fφ,ϕ (T x0 − T m x0 , y) T Fφ,ϕ (x0 − T x0 , y) k(α, β) k(α, β)   t(1 − k(α, β))   t  , Fφ,ϕ (x0 − T m−1 x0 , y) T Fφ,ϕ (x0 − T x0 , y) k(α, β) k(α, β)  t(1 − k(α, β))   t   , Fφ,ϕ (x0 − T x0 + T x0 − T m−1 x0 , y) T Fφ,ϕ (x0 − T x0 , y) k(α, β) k(α, β)   t(1 − k(α, β))    t(1 − k(α, β))  , T Fφ,ϕ (x0 − T x0 , y) , T Fφ,ϕ (x0 − T x0 , y) k(α, β) k(α, β)  t(k(α, β))  Fφ,ϕ (T x0 − T m−1 x0 , y) k(α, β)   t(1 − k(α, β))    t(1 − k(α, β))  , T Fφ,ϕ (x0 − T x0 , y) , T Fφ,ϕ (x0 − T x0 , y) k(α, β) k(α, β)  t  Fφ,ϕ (x0 − T m−2 x0 , y) k(α, β)

 Fφ,ϕ (x0 − T x0 + T x0 − T m x0 , y)

L∗ · · ·   t(1 − k(α, β))    t(1 − k(α, β))  , T Fφ,ϕ (x0 − T x0 , y) , L∗ T Fφ,ϕ (x0 − T x0 , y) k(α, β) k(α, β)    t(1 − k(α, β))   t   T · · · , T Fφ,ϕ (x0 − T x0 , y) , Fφ,ϕ (x0 − T x0 , y) ··· . k(α, β) k(α, β) Since k(α, β) ∈ (0, 1), we have t(1 − k(α, β)) t  . k(α, β) k(α, β) According to Definition 3.1, it implies that  Fφ,ϕ (x0 − T x0 , y)

 t(1 − k(α, β))  t  L∗ Fφ,ϕ (x0 − T x0 , y) . k(α, β) k(α, β)

420

Shi-sheng ZHANG, M. GOUDARZI, R. SAADATI, and S. M. VAEZPOUR

Also, by properties of T , we obtain  t  Fφ,ϕ (x0 − T m x0 , y) k(α, β)   t(1 − k(α, β))    t(1 − k(α, β))  , T Fφ,ϕ (x0 − T x0 , y) , L∗ T Fφ,ϕ (x0 − T x0 , y) k(α, β) k(α, β)    t(1 − k(α, β))  T · · · , T Fφ,ϕ (x0 − T x0 , y) , k(α, β)  t(1 − k(α, β))   Fφ,ϕ (x0 − T x0 , y) ··· k(α, β)   t(1 − k(α, β))  = T m−1 Fφ,ϕ (x0 − T x0 , y) . k(α, β) Then, for any positive integers m and n, we have

t  k(α, β)   t m n m+n Fφ,ϕ (x0 − T x0 , T x0 − T x0 ) n+1 k (α, β)   t Fφ,ϕ (x0 − T m x0 , x0 − T m x0 ) 2n+1 k (α, β)   t(1 − k(α, β))  T m−1 Fφ,ϕ (x0 − T x0 , x0 − T m x0 ) k 2n+1 (α, β)  t(1 − k(α, β))2    T m−1 T m−1 Fφ,ϕ (x0 − T x0 , x0 − T x0 ) k 2n+1 (α, β)   t(1 − k(α, β))2  . T 2m−2 Fφ,ϕ (x0 − T x0 , x0 − T x0 ) k 2n+1 (α, β)  Fφ,ϕ (T n x0 − T m+n x0 , T n x0 − T m+n x0 )

L∗ L∗ L∗ L∗ =

Then, by (IMI-4) and (IMI-3), we imply t  n→∞ k(α, β)  t(1 − k(α, β))2   lim T 2m−2 Fφ,ϕ (x0 − T x0 , x0 − T x0 ) . n→∞ k 2n+1 (α, β)  lim Fφ,ϕ (T n x0 − T m+n x0 , T n x0 − T m+n x0 )

L∗

However, T is a t-representable norm of h-type, and Fφ,ϕ is non-decreasing with respect to t. Hence, we have lim (T n x0 − T n+m x0 ) = 0. n→∞

m

Therefore, {T x0 } is a Cauchy sequence in X. Since X is complete, we can assume that xn → x∗ as n → ∞. Now, we prove that x∗ is a fixed point of T , and also, we prove the uniqueness of x∗ . Because  t  Fφ,ϕ (xi − T xi , x∗ − T x∗ )(t) L∗ Fφ,ϕ (xi−1 − T xi−1 , x∗ − T x∗ ) k(α, β) L∗ · · ·   t ,  Fφ,ϕ (x0 − T x0 , x∗ − T x∗ ) i k (α, β) we have

 lim Fφ,ϕ (xi − T xi , x∗ − T x∗ )(t) L∗ lim Fφ,ϕ (x0 − T x0 , x∗ − T x∗ )

i→∞

i→∞

 t = 1L∗ k i (α, β)

Intuitionistic Menger inner product spaces and applications to integral equations

and  Fφ,ϕ (x∗ − T x∗ , x∗ − T xi ) L∗

t  k(α, β)   t(1 − k(α, β))   , Fφ,ϕ (xi − T xi , x∗ − T x∗ )(t) . T Fφ,ϕ (x∗ − xi , x∗ − T x∗ ) k(α, β)

However, lim xi = x∗ , T (·, ·) is equicontinuous at (1L∗ , 1L∗ ), and i→∞

Fφ,ϕ (0, x∗ − T x∗ )(t) = 1L∗ ,

∀t > 0.

Thus, we have  lim Fφ,ϕ (x∗ − T xi , x∗ − T x∗ )

i→∞

t  = 1L∗ , k(α, β)

∀t > 0.

Hence, t  k(α, β)   t(1 − k(α, β))   , Fφ,ϕ (T xi − T x∗ , x∗ − T x∗ )(t) T Fφ,ϕ (x∗ − T xi , x∗ − T x∗ ) k(α, β)   t(1 − k(α, β))   t  , Fφ,ϕ (xi − x∗ , x∗ − T x∗ ) . T Fφ,ϕ (x∗ − T xi , x∗ − T x∗ ) k(α, β) k(α, β)

 Fφ,ϕ (x∗ − T x∗ , x∗ − T x∗ ) L∗ L∗ Then,

 Fφ,ϕ (x∗ − T x∗ , x∗ − T x∗ )

t  = 1L∗ , k(α, β)

∀t > 0.

By (IMI-3), we have T x∗ = x∗ . If there exists another point y∗ ∈ X such that T y∗ = y∗ , then Fφ,ϕ (x∗ − y∗ , x∗ − y∗ )(t) = L∗

Fφ,ϕ (T x∗ − T y∗ , T x∗ − T y∗ )(t)   t . Fφ,ϕ (x∗ − y∗ , x∗ − y∗ ) 2 k (α, β)

By the same way, we obtain  Fφ,ϕ (x∗ − y∗ , x∗ − y∗ )(t) L∗ Fφ,ϕ (x∗ − y∗ , x∗ − y∗ ) L∗ · · ·

 L∗ Fφ,ϕ (x∗ − y∗ , x∗ − y∗ )

Then, Fφ,ϕ (x∗ − y∗ , x∗ − y∗ )(t) = 1L∗ , By (IMI-3), we have x∗ = y∗ . Therefore, x∗ is unique. Finally, we prove that Fφ,ϕ

T n x0 −−−→ x∗ ,

∀x0 ∈ X.

∀t > 0.

 t k 2 (α, β)  t . k 2n (α, β)

421

422

Shi-sheng ZHANG, M. GOUDARZI, R. SAADATI, and S. M. VAEZPOUR

Since Fφ,ϕ (x∗ − T n x0 , x∗ − T n x0 )(t) = L∗ = L∗ L∗ 

Fφ,ϕ (T x∗ − T n x0 , T x∗ − T n x0 )(t)  Fφ,ϕ (x∗ − T n−1 x0 , x∗ − T n−1 x0 )

 t k 2 (α, β)   t Fφ,ϕ (T x∗ − T n−1 x0 , T x∗ − T n−1 x0 ) 2 k (α, β)   t Fφ,ϕ (x∗ − T n−2 x0 , x∗ − T n−2 x0 ) 4 k (α, β) ···   t , Fφ,ϕ (x∗ − x0 , x∗ − x0 ) 2n k (α, β)

we have lim Fφ,ϕ (x∗ − T n x0 , x∗ − T n x0 )(t)   t = 1L∗ . lim Fφ,ϕ (x∗ − x0 , x∗ − x0 ) 2n n→∞ k (α, β)

n→∞

L∗ Hence,

lim T n x0 = x∗ .

n→∞

This completes the proof.

4

Applications

As an application, we use the previous theorem to study the existence and uniqueness of solution to a Volterra integral equation in a complete IMIP-space. We recall that L2 [a, b] is a Hilbert space with the inner product x, y =

a

b

∀x, y ∈ L2 [a, b].

x(t)y(t)dt,

Now, we define the space (L2 [a, b], FH , T ), where FH (x, y)(t) = H(t − x, y ), and T is a continuous t-representable norm on L∗ . We know that (L2 [a, b], FH , T ) is an IMIP-space. Also, we have the following theorem: Theorem 4.1 (L2 [a, b], FH , T ) is a complete IMIP-space. Proof Let {xn } be a Cauchy sequence in (L2 [a, b], FH , T ). Then, for any ε > 0 and λ ∈ (0, 1), there exists N ∈ N such that FH (xm − xn , xm − xn )(ε) >L∗ Ns (λ),

∀m, n  N.

Since FH (xm − xn , xm − xn )(ε) = H(ε − xm − xn , xm − xn ) b

 =H ε− [(xm − xn )(t)]2 dt >L∗ Ns (λ), a

Intuitionistic Menger inner product spaces and applications to integral equations

we have



b

a

423

[(xm − xn )(t)]2 dt → 0,

and then, xm − xn → 0. Therefore, {xn } is a Cauchy sequence in L2 [a, b]. By the completeness of L2 [a, b], we have xn → x∗ ∈ L2 [a, b]. We can easily show that xn → x∗ in (L2 [a, b], FH , T ). This shows that (L2 [a, b], FH , T ) is a complete IMIP-space. Theorem 4.2 Suppose the following conditions hold in (L2 [a, b], FH , T ) : (i) t (x(s) − y(s))ds  x(t) − y(t), ∀x, y ∈ L2 [a, b]. a

(ii) T is a linear self-mapping on (L2 [a, b], Fφ,ϕ , T ) defined as t h(t, s)x(s)ds, (T x)(t) = f (t) + λ a

2

where f ∈ L [a, b] is a given function, h(t, s) is a continuous function defined on a  t  b and a  s  t, and λ is a constant. (iii) By putting max h(t, s) = M, we have λM ∈ (0, 1). atb, ast

Then, T has a unique fixed point in L2 [a, b]. Furthermore, for any x0 ∈ L2 [a, b], the iterative sequence {T n x0 } converges to the fixed point in (L2 [a, b], FH , T ). Proof b T x − T y, v = (T x − T y)(t)v(t)dt a



b

= a

t  t  λ h(t, s)x(s)ds − h(t, s)y(s)ds v(t)dt



=λ

b

a



a

a

 h(t, s)(x(s) − y(s))ds v(t)dt.

t

a

By the mean-value theorem and the continuity of h(t, s), there exist s1 and t1 such that b  t  T x − T y, v  λh(t1 , s1 ) (x(s) − y(s))ds v(t)dt = λM

b

a



a t

a

a

 (x(s) − y(s))ds v(t)dt.

By Theorem 4.2 (i), we have T x − T y, v  λM

 a

b

 (x(t) − y(t))v(t)dt = λM x − y, v .

Hence, T x − T y, v  λM x − y, v and H(t − T x − T y, v ) L∗ H(t − λM x − y, v ). Also, by Definition 3.1, we get   t − x − y, v , λM  t  . FH x − y, v, λM

H(t − T x − T y, v ) L∗ H FH (T x − T y, v, t) L∗

424

Shi-sheng ZHANG, M. GOUDARZI, R. SAADATI, and S. M. VAEZPOUR

Besides, by Theorem 3.1, T has a unique fixed point in L2 [a, b]. Moreover, for each x0 ∈ L2 [a, b], the iterative sequence {T nx0 } converges to the fixed point in (L2 [a, b], FH , T ). Acknowledgements The authors would like to thank referees for giving useful suggestions for the improvement of this paper.

References [1] Schweizer, B. and Sklar, A. Probabilistic Metric Spaces, North Holland, New York (1983) [2] Zhang, S. S. On the probabilistic inner product spaces. Appl. Math. Mech. -Engl. Ed. 7(11), 1035–1042 (1986) DOI 10.1007/BF01897206 [3] Alsina, C., Schweizer, B., Sempi, C., and Sklar, A. On the definition of a probabilistic inner product space. Rendiconti di Matematica 17, 115–127 (1997) [4] Dumitresu, C. Un produit scalaire probabiliste. Revue Roumaine de Math´ematique Pures et Appliqu´ees 26(3), 399–404 (1981) [5] Saadati, R. and Park, J. H. On the intuitionistic fuzzy topological spaces. Chaos, Solitons and Fractals 27, 331–344 (2006) [6] Tanaka, Y., Mizno, Y., and Kado, T. Chaotic dynamics in Friedmann equation. Chaos, Solitons and Fractals 24, 407–422 (2005) [7] He, J. H. Nonlinear dynamics and the Nobel prize in physics. International Journal of Nonlinear Science and Numerical Simulation 8(1), 1–4 (2007) [8] Krasnoselskii, M. A. Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, New York (1964) [9] Rahmat, M. R. S. and Noorani, M. S. M. Approximate fixed point theorems in probabilistic normed (metric) spaces. Proceedings of the 2nd IMT-GT Regional Conference on Mathematics, Statistics and Applications, Universiti Sains Malaysia, Penang, June 13–15 (2006) [10] Saadati, R. and Vaezpour, S. M. Some results on fuzzy Banach spaces. J. Appl. Math. Comput. 17, 475–484 (2005) [11] Saadati, R., Sedghi, S., and Shobe, N. Modified intuitionistic fuzzy metric spaces and some fixed point theorems. Chaos, Solitons and Fractals 38(1), 36–47 (2008) DOI 10.1016/j.chaos.2006.11.008 [12] Hossieni, S. B., O‘Regan, D., and Saadati, R. Some results on intuitionistic fuzzy spaces. Iranian Journal of Fuzzy Systems 4(1) 53–64 (2007) [13] Kutukeu, S., Tuna, A., and Yakut, A. T. Generalized contraction mapping principle in intuitionistic Menger spaces and application to differential equations. Appl. Math. Mech. -Engl. Ed. 28(6), 799–809 (2007) DOI 10.1007/s10483-007-0610-z [14] Zhang, S. S., Rassias, J. M., and Saadati, R. The stability of a cubic functional equation in intuitionistic random normed spaces. Appl. Math. Mech. -Engl. Ed. 31(1), 21–26 (2010) DOI 10.1007/s10483-010-0103-6