Gateaux and Frechet Derivative in Intuitionistic Fuzzy Normed Linear spaces

Fuzzy set theory is a useful tool to describe the situation in which data are imprecise or vague or uncertain. Intuitionistic fuzzy set theory handle the situation by attributing a degree of membership and a degree of non-membership to which a certain object belongs to a set. It has a wide range of application in the field of population dynamics [6], chaos control [16], computer programming [17], medicine [5] etc. The concept of intuitionistic fuzzy set, as a generalisation of fuzzy sets [27] was introduced by Atanassov in [1]. The concept of fuzzy norm was introduced by Katsaras [21] in 1984. In 1992, Felbin [13] introduced the idea of fuzzy norm on a linear space. Cheng-Moderson [7] introduced another idea of fuzzy norm on a linear space whose associated metric is same as the associated metric of Kramosil-Michalek [22]. Latter on Bag and Samanta [3] modified the definition of fuzzy norm of Cheng-Moderson [7] and established the concept of continuity Definition 2.2 [25]. A binary operation ⋄ : [ 0 , 1 ] × [ 0 , 1 ] -→ [ 0 , 1 ] is continuous t-conorm if ⋄ satisfies the following conditions : ( i ) ⋄ is commutative and associative , ( ii ) ⋄ is continuous ,

Definition 2.3 [24] Let * be a continuous t-norm , ⋄ be a continuous t-conorm and V be a linear space over the field F ( = R or C ). An intuitionistic fuzzy norm on V is an object of the form A = { ( ( x , t ) , µ ( x , t ) , ν ( x , t ) ) : ( x , t ) ∈ V × R + } , where µ , ν are f uzzy sets on V × R + , µ denotes the degree of membership and ν denotes the degree of non -membership ( x , t ) ∈ V × R + satisfying the following conditions :

Definition 2.4 [24] If A is an intuitionistic fuzzy norm on a linear space V then (V , A) is called an intuitionistic fuzzy normed linear space.

For the intuitionistic fuzzy normed linear space ( V , A ) , we further assume that µ, ν, * , ⋄ satisfy the following axioms :

, for all t > 0 ⇒ x = θ . (xv) For x = θ, µ(x , .) is a continuous function of R and strictly increasing on the subset { t : 0

) is a continuous function of R and strictly decreasing on the subset { t : 0 < ν(x , t) < 1 } of R. Definition 2.5 [24] A sequence {x n } n in an intuitionistic fuzzy normed linear space (V , A) is said to converge to x ∈ V if for given r > 0, t > 0, 0 < r < 1, there exist an integer n 0 ∈ N such that µ ( x nx , t ) > 1r and ν ( x nx , t ) < r for all n ≥ n 0 . Definition 2.6 [24] Let, ( U , A ) and ( V , B ) be two intuitionistic fuzzy normed linear space over the same field F . A mapping f from ( U , A ) to ( V , B ) is said to be intuitionistic fuzzy continuous at x 0 ∈ U, if for any given ǫ > 0 , α ∈ (0, 1) , ∃ δ = δ(α, ǫ) > 0 , β = β(α, ǫ) ∈ (0, 1) such that for all x ∈ U,

In this section, we shall consider ( R, µ R , ν R , * , ⋄ ) as an intuitionistic fuzzy normed linear space over the field R (the set of all real numbers).

We denote intuitionistic fuzzy derivative of f at x 0 by f ′ (x 0 ).

Alternative definition: Let ( R, µ 1 , ν 1 , * , ⋄ ) and ( R, µ 2 , ν 2 , * , ⋄ ) be two intuitionistic fuzzy normed linear space over the same field R. A mapping

Note 3.2 It is easy to see that these two definitions are equivalent.

Note 3.3 If the intuitionistic fuzzy derivative of f, be f ′ (x 0 ), the intuitionistic fuzzy derivative of f ′ (x 0 ) at x 0 is called second order intuitionistic fuzzy derivative of f at x 0 and is denoted by f ′′ (x 0 ). Similarly, the n-th order intuitionistic fuzzy derivative of f at x 0 exists if f n-1 (x 0 ) is intuitionistic fuzzy differentiable at x 0 and this derivative is denoted by f n (x 0 ) .

Proof. Since f and g are intuitionistic fuzzy differentiable at x 0 , therefore we have for any given ǫ > 0 , α ∈ (0, 1) , ∃ δ = δ(α, ǫ) > 0 , β = β(α, ǫ) ∈ (0, 1) such that for all x( = x 0 ) ∈ R,

and

Definition 3.5 Let ( U , A ) and ( V , B ) be two intuitionistic fuzzy normed linear space over the same field k (= R or C). An operator T from ( U , A ) to ( V , B ) is said to be intuitionistic fuzzy Gateaux differentiable at x 0 ∈ U, if there exists an intuitionistic fuzzy continuous linear operator G : ( U , A ) -→ ( V , B ) (generally depends upon x 0 ) and for any given ǫ > 0 , α ∈ (0, 1) , ∃ δ = δ(α, ǫ) > 0 , β = β(α, ǫ) ∈ (0, 1) such that for every x ∈ U and s

In this case, the operator G is called intuitionistic fuzzy Gateaux derivative of T at x 0 and it is denoted by D f (x 0 ) .

Alternative definition: Let ( U , A ) and ( V , B ) be two intuitionistic fuzzy normed linear space over the same field k (= R or C). An operator T from ( U , A ) to ( V , B ) is said to be intuitionistic fuzzy Gateaux differentiable at x 0 ∈ U, if there exists an intuitionistic fuzzy continuous linear operator G : ( U , A ) -→ ( V , B ) (generally depends upon x 0 ) such that for every x ∈ U, t > 0 and s

In this case, the operator G is called intuitionistic fuzzy Gateaux derivative of T at x 0 and it is denoted by D f (x 0 ) . Note 3.6 It is easy to see that these two definitions are equivalent.

Theor

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