Existence and uniqueness of approximate solutions to Cauchy problem of complex fuzzy differential equations

1 Introduction

Since the introduction of fuzzy sets [30], the theory and applications of fuzzy systems has been developed a lot in various aspects, especially in the fields of fuzzy control systems, artificial intelligence, neural networking, symbolic reasoning, medical sciences, social sciences, engineering, management sciences, robotics, decision making and automata theory etc. Fuzzy theory also strengthens the capabilities of human reasoning, and thinking to overcome uncertainties in the knowledge based systems.

For the fuzzy logic systems usually, fuzzy differential equations (FDEs) have been applied to analyze the performance of the system. The notion of fuzzy derivative was introduced by Chang and Zadeh [8]. Later on using the Zadeh’s extension principle, Dubios and Prade [11], modified these notions. The Cauchy problems for fuzzy differential equations were rigorously used by Kaleva [12], Seikala [26], Abbasbandy [2], Allahviranloo [1] and many more [10 , 18]. Related topicswith fuzzy differential equations are; fuzzy Laplace transforms [25], fuzzy integro-differential equations [2], fuzzy partial differential equations [3], generalized and granular differentiability [16], generalized fuzzy differential equations and fuzzy numbers [21,22, 21,22], and many others can be found in the following articles and references therein [9 , 23]. Puri and Ralescu [20] developed fuzzy-valued maps and extended the concept of Hukuhara differentiability (H-Derivative) for set valued maps to fuzzy valued maps. The real motivation of this work has been given in terms of uniqueness and existence criteria for the Cauchy problem in the perspective complex fuzzy differential equations which is given by; x ′ ( τ ) = f ( τ , x ) , x ( τ 0 ) = x 0 , whenever f fulfills the Lipschitz condition.

Recently Wu and Song [27] studied the initial value problem in the perspective of fuzzy differential equations for real fuzzy set-mappings whose values satisfy the conditions of normality, convexity, upper semi-continuity in ℝ n and also utilized Hukuhara derivative of the ς-level sets to generalize the concept of H-differentiability. On the other hand, the notion of levelwise continuous mappings (generalization of continuity of fuzzy set-valued mappings) has been proposed.

In 1987, Buckley [6] introduced the concept of fuzzy complex numbers. After this Nguyen, et al., [17] extended the range of truth values from the interval [0, 1] to ℂ to define the idea for complex numbers. In [24] Ramot et al. defined the notions of complex fuzzy sets, which increased the expressive power of the universe of discourse S, by considering μ s ( x ) = r s ( x ) . e iw s ( x ) , where the real-valued functions r _s (x) and the phase term w _s (x) were exclusively responsible for the fuzziness and crisp information respectively. This range also increase the applications of fuzzy sets in various fields. It was an initiation of Complex Fuzzy Sets and Logics (CFSL). Later on Tamir et al. [28], pointed out the limitations of the mixed fuzzy and crisp definition of [24] and provided an axiomatic approach for complex fuzzy logic and demonstrated its applications to complex monitory systems. For more details about a systematic review of Complex Fuzzy Sets, we refer the readers to the review [29].

Recently in [13], Karpenko et al., highlighted the difference between complex fuzzy sets and fuzzy complex numbers and made the comparisons among the results of [6], fuzzy complex numbers and [28], complex fuzzy sets. Karpenko [13], proved the existence for Cauchy problem for complex fuzzy differential equations using Holder’s continuity and Lipschitz continuity.

In this article we establish a relation between the solutions of complex fuzzy differential equations, with complex membership grades, and approximate solutions by applying the properties of levelwise continuity for (ς, ϑ)-level sets. The notion of H-differentiability is generalized for (ς, ϑ)-level sets in the space E ˆ 2 n . Some results about the existence and uniqueness of the solutions of the Cauchy problem for complex fuzzy differential equations are also included. The relationship between approximate solution and exact solution is presented. Example is given to elaborate the main result.

2 Preliminaries

The symbol P k ( ℝ n ) denotes the family of all non-empty compact convex subsets of ℝ n . Let S and B be two nonempty bounded subsets of ℝ n . The distance between S and B is defined by the Hausdorff metric,

d ( S , B ) = max [ sup s ∈ S inf b ∈ B ‖ s − b ‖ , sup b ∈ B inf s ∈ A , ‖ s − b ‖ ] ,(1)

Euclidean norm in ℝ n is denoted by || . || and it is easy to see that ( P k ( ℝ n ) , d ) is a complete metric space. Let E ⁿ be the space defined as; E n = { ϖ : ℝ n → [ 0 , 1 ] | ϖ satisfies ( i ) - ( iv ) } , where, (i) ϖ is normal, means there is an x 0 ∈ ℝ n in such a way that ϖ (x ₀) =1, (ii) ϖ is fuzzy convex, that is

ϖ ( λ x + ( 1 - λ ) y ) ≥ min { ϖ ( x ) , ϖ ( y ) } , for any x , y ∈ ℝ n and 0 ≤ λ ≤ 1 , (iii) ϖ is upper semicontinuous, (iv) [ ϖ ] 0 = cl { x ∈ ℝ n | ϖ ( x ) > 0 ¯ } is compact, where cl denotes the closure of a set.

For 0 < ς ≤ 1 denote [ ϖ ] ς = { x ∈ ℝ n | ϖ ( x ) ≥ ς ¯ } . Then from (i) - (iv) it follows that the ς- level set

[ ϖ ] ς ∈ P K ( ℝ n ) forall 0 ≤ ς ≤ 1 . According to Zadeh’s extension principle, addition and scalar multiplication in fuzzy number space E ⁿ are as follow: [ ϖ + v ] ς = [ ϖ ] ς + [ v ] ς , [ k ϖ ] ς = k [ ϖ ] ς , for all ϖ , v ∈ E n , k ∈ ℝ and 0 ≤ ς ≤ 1 .

A complex fuzzy set S as defined in [24], on a universe of discourse U, is characterized by a membership function μ _s (x), that assigns any element x ∈ U to a complex-valued grade of membership. By definition, the values μ _s (x) may receive all lie within the unit circle in the complex plane, and thus is of the form r _s (x) . e ^{jw _s (x)}, where j = - 1 , r _s (x) and w _s (x) are both real-valued, and r _s (x) ∈ [0, 1] .

The complex fuzzy set S may be represented as the set of ordered pairs S = { ( x , μ s ( x ) ) : x ∈ U } . The complex membership function μ in [28], is defined as μ ( A , a ) = μ R ( A ) + i μ I ( a ) where A is to be interpreted as a set in a fuzzy set of sets and a as an element of A . We are more interested in the following form of complex fuzzy sets defined in [13] (an extension to ℝ n , as follows) for x ∈ ℝ n , let ζ ( x ) = u ( x ) + iv ( x ) , where u , v : ℝ n → [ 0 , 1 ] .

For ease of notation, denote ζ by (u, v) . Thus, ζ assigns to each x ∈ ℝ n a value in the unit square in ℂ , representing a complex grade of membership. Note that u, v considered individually fuzzy sets in ℝ n .

In [13], Karpenko et al., initiated the notion of (ς, ϑ)-level sets for the ordered pair (u, v), defined as [ ( u , v ) ] ( ς , ϑ ) = [ u ] ς ∩ [ v ] ϑ = { x ∈ ℝ n | u ( x ) ≥ ς , v ( x ) ≥ ϑ , 0 < ς , ϑ ≤ 1 } , and ( u , v ) ] ( ς , 0 ) = { x ∈ ℝ n | u ( x ) ≥ ς , v ( x ) > 0 , 0 < ς ≤ 1 ¯ } , ( u , v ) ] ( 0 , ϑ ) = { x ∈ ℝ n | u ( x ) > 0 , v ( x ) ≥ ϑ , 0 < ϑ ≤ 1 ¯ } , ( u , v ) ] ( 0 , 0 ) = { x ∈ ℝ n | u ( x ) > 0 , v ( x ) > 0 } . ¯ The followings concepts are according to [13], E ˆ 2 n = { ( u , v ) ∈ E n × E n | ∃ x 0 ∈ ℝ n , u ( x 0 ) = v ( x 0 ) = 1 ) } . The set E ˆ 2 n is closed under addition and scalar multiplication defined as: ζ + ξ = ( u ζ + u ξ , v ζ + v ξ ) , c ζ = ( cu ζ , cv ζ ) . for ζ , ξ ∈ E ˆ 2 n , where ζ = (u _ζ , v _ζ ) and ξ = (u _ξ , v _ξ ) and c is a scalar. The product metric D ˆ on E ˆ 2 n is given by D ˆ ( ζ 1 , ζ 2 ) = D ˆ ( ( u 1 , v 1 ) , ( u 2 , v 2 ) ) = max { D ( u 1 , u 2 ) , D ( v 1 , v 2 ) } ,(2) where D : E ⁿ × E ⁿ → [0, ∞), is given by; D ( u 1 , u 2 ) = sup { d ( [ u 1 ] ς , [ u 2 ] ς ) : ς ∈ [ 0 , 1 ] } , d is defined above in ( 1 ) , for ζ 1 , ζ 2 ∈ E ˆ 2 n , where ζ ₁ = (u ₁, v ₁) and ζ ₂ = (u ₂, v ₂). Then ( E ˆ 2 n , D ˆ ) is a complete metric space, [13] and D ˆ preserves linearity. The zero element in E ˆ 2 n is defined as 0 ⌢ 2 ( x ) = ( 0 ⌢ ( x ) , 0 ⌢ ( x ) ) ∈ E ˆ 2 n such as 0 ⌢ 2 ( 0 ) = ( 1 , 1 ) . By the Arens-Eells theorem [4] there exist an embedding E ˆ 2 n → B where B is a Banach space. The concept of differentiability has been defined in terms of Hukuhara difference in [12]. Let I = [ a , b ] ⊂ ℝ be a compact interval. A mapping F : I → E ˆ 2 n is differentiable at τ ₀ ∈ I if there exists some F ′ ( τ 0 ) ∈ E ˆ 2 n such that: lim h → 0 + F ( τ 0 + h ) − F ( τ 0 ) h = F ′ ( τ 0 ) and lim h → 0 + F ( τ 0 ) − F ( τ 0 − h ) h = F ′ ( τ 0 ) . G : I → E ˆ 2 n is defined by G ( τ ) = ∫ a τ F ( ς , ϑ ) ( t ) dt , for all τ ∈ I . = { ∫ a τ sl f ( t ) dt : f is a measurable selection of F ( ς , ϑ ) } , for (ς, ϑ) ∈ [0, 1] ² . It is noted that d d τ G ( τ ) = G ′ ( τ ) = F ( τ ) , f o r a l l τ ∈ I , and we have the Mean Value theorem D ˆ ( F ( b ) , F ( a ) ) ≤ ( b - a ) sup { D ˆ ( F ′ ( τ ) , 0 ⌢ 2 ) : τ ∈ I } . Consider the mapping H : I × E ˆ 2 n → E ˆ 2 n which is continuous. Now consider a Cauchy problem as X ′ ( sl τ ) = H ( sl τ , X ( sl τ ) ) , sl τ ∈ I , X ( a ) = x a . The solution of initial value problem is X if and only if X is continuous and X satisfy complex fuzzy integral equation X ( τ ) = x a + ∫ a τ H ( t , X ( t ) ) dt .(3) For arbitrary H, solution to (3) might not exist. So let us consider only H such that there exists a constant M for which the bound D ˆ ( H ( sl τ , X ) , 0 ˆ 2 ) < M holds for all τ ∈ I and all X ∈ Λ ⊆ E ˆ 2 n , such a bounded H will permit the existence of the solution. Let C ( I , E ˆ 2 n ) denote the set of all continuous maps from I to E ˆ 2 n and let d _C denote the metric on C ( I , E ˆ 2 n ) defined as d C ( X 1 , X 2 ) = sup τ ∈ I { D ˆ ( X 1 ( τ ) , X 2 ( τ ) ) } . Then ( C ( I , E ˆ 2 n ) , d C ) is a complete metric space [13].

The objective of this section is to initiate the study for the existence and uniqueness of solutions of the Cauchy problem for complex fuzzy differential equations and explore the relation between exact and approximate solution under some conditions. We recall the following definitions and basic results from [15], which will be crucial for our main results.

Definition 1. Consider E an open set in ℝ 2 and p ∈ C [ E , ℝ ] . Assume scalar differential equation within initial condition

Suppose v ∈ C [ τ 0 , τ 0 + a ) , ℝ ] , and exists for τ ∈ [τ ₀, τ ₀ + a), and (τ, v (τ)) ∈ E . If v (τ) satisfies the differential inequality v ′ + ( τ ) < p ( τ , v ( τ ) ) , f o r τ ∈ [ τ 0 , τ 0 + a ) with respect to initial problem (4) then, it is said to be an under-function On the other hand, if v + ′ ( τ ) > p ( τ , v ( τ ) ) , τ ∈ [ τ 0 , τ 0 + a ) , then v (τ) is said to be an over-function.

Proposition 1. Consider E an open set in ℝ 2 and p ∈ C [ E , ℝ ] . Consider that v , w ∈ C [ τ 0 , τ 0 + a ) , ℝ ] , (τ, v (τ)), (w, v (τ)) ∈ E, and τ ∈ [τ ₀, τ ₀ + a) . Suppose further that

v ( τ 0 ) < w ( τ 0 ) ,

for τ ∈ [τ ₀, τ ₀ + a), the inequalities D _ v ( τ ) ≤ p ( τ , v ( τ ) ) , and D _ w ( τ ) ≤ p ( τ , w ( τ ) ) ,

hold, then v ( τ ) < w ( τ ) , τ ∈ [ τ 0 , τ 0 + a ) .

Lemma 2. Let v , w ∈ C [ [ τ 0 , τ 0 + a ) , ℝ ] , and for some fixed Dini-derivative D, the inequality D ( v ( τ ) ) ≤ w ( τ ) for τ ∈ [ τ 0 , τ 0 + a ) - S holds, then, D _ v ( τ ) ≤ w ( τ ) for τ ∈ [ τ 0 , τ 0 + a ) .

Definition 2. The solution of the scalar differential equation (4) on τ ∈ [τ ₀, τ ₀ + a) is r (τ). Then r (τ) is said to be a maximum solution of (4) if, for each solution u (τ) of (4) existing on τ ∈ [τ ₀, τ ₀ + a), and the inequality u ( τ ) ≤ r ( τ ) , for τ ∈ [ τ 0 , τ 0 + a ) holds.

Theorem 3. Consider E an open set in ℝ 2 . Let p ∈ C [ E , ℝ ] , and (τ ₀, u ₀) ∈ E . Then equation (4) has a maximum and least solution that can be extended to boundary of E .

Lemma 4. Let the hypothesis of above theorem holds, and let [τ ₀, τ ₀ + a) be the greatest interval of existence of the maximum solution r (τ) of (4) .

Suppose [τ ₀, τ ₁] is a compact sub-interval of [τ ₀, τ ₀ + a) then, there is an ɛ ₀ > 0 such that, for 0 < ɛ < ɛ ₀, the maximal solution r (τ, ɛ) of equation u ′ = p ( τ , u ) + ɛ , u ( τ 0 ) = u 0 + ɛ(5) so that p ( τ , u ) = p ( τ , u ) + ɛ exists over [τ ₀, τ ₁], and lim ɛ → 0 r ( τ , ɛ ) = r ( τ ) exists uniformly on [τ ₀, τ ₁] .

Theorem 5. Consider E an open set in ℝ 2 and p ∈ C [ E , ℝ ] . Assume that [τ ₀, τ ₀ + a) is the maximum interval in which the maximal solution r (τ) of (4) exists.

Let m ∈ C [ ( τ 0 , τ 0 + a ) , ℝ ] , and (τ, m (τ)) ∈ E for τ ∈ [ τ 0 , τ 0 + a ) , m ( τ 0 ) ≤ u 0 , and for a fixed Dini-derivative Dm ( τ ) ≤ p ( τ , m ( τ ) ) , τ ∈ [ τ 0 , τ 0 + a ) - S .(6) Then, m ( τ ) ≤ r ( τ ) , τ ∈ [ τ 0 , τ 0 + a )

Proof. From Lemma 1, it ensue that (6) can be changed by D _ m ( τ ) ≤ p ( τ , m ( τ ) ) , τ ∈ ( τ 0 , τ 0 + a ) .(7) Let τ ₀ < τ < τ ₀ + a . From Lemma 2, the maximum solution r (τ) of (1.3.2) exists on [τ ₀, τ] for all ɛ > 0 sufficiently small, and lim ɛ → 0 r ( τ , ɛ ) = r ( τ )(8) uniformly on [τ ₀, τ] . Using (5) and (7) and apply Theorem 2. We get that m ( τ ) < r ( τ , ɛ ) , τ ∈ [ τ 0 , r ] .(9) The equation (9), together with (8), proves the conditions of the result.□

Definition 3. A mapping f : T × E ˆ 2 n → E ˆ 2 n is said to be levelwise continuous at a point ( τ 0 , w 0 ) ∈ T × E ˆ 2 n provided for any fixed ς, ϑ ∈ [0, 1] and arbitrary ɛ > 0, there exist an δ (ɛ, (ς, ϑ)) >0 in such that d ( [ f ( τ , w ) ] ( ς , ϑ ) , [ f ( τ 0 , w 0 ) ] ( ς , ϑ ) ) < ɛ whenever |τ - τ ₀| < δ ((ɛ, (ς, ϑ)) and d ( [ w ] ( ς , ϑ ) , [ w 0 ] ( ς , ϑ ) ) < δ ( ɛ , ( ς , ϑ ) ) for all τ ∈ T, and w ∈ E ˆ 2 n .

Definition 4. A mapping F : T → E ˆ 2 n at the point τ ₀ ∈ T, is called differentiable. If the set-valued mapping, F _(ς,ϑ) (τ) = [F (τ)] ^(ς,ϑ) at the point τ ₀ is H-differentiable for each (ς, ϑ) ∈ [0, 1] ², and the class { D ˆ F ( ς , ϑ ) ( τ ) : ( ς , ϑ ) ∈ [ 0 , 1 ] 2 } , defines a complex fuzzy number Denote R ₀ = [τ ₀, τ _o + p] × B (w ₀, η), for some p > 0, η > 0 and w 0 ∈ E ˆ 2 n , B ( w 0 , η ) = { w ∈ E ˆ 2 n : D ˆ ( X , X 0 ) ≤ η } .

Theorem 6. Let g ∈ C [ R 0 , E ˆ 2 n ] , r ∈ ( 0 , p ) , w n ∈ C 1 [ [ τ 0 , τ 0 + r ] , B ( w o , η ) ] such that D ˆ ( Z n ( τ ) , 0 ⌢ ) ≤ ɛ n , ∀ τ ∈ [ τ 0 , τ 0 + r ] , n = 1 , 2 , 3 , . . . w n ′ ( τ ) = g ( τ , w n ( τ ) ) + Z n ( τ ) , w n ( τ 0 ) = w 0 , D ˆ ( Z n ( τ ) , 0 ⌢ ) ≤ ɛ n , ∀ τ ∈ [ τ 0 , τ 0 + r ] , n = 1 , 2 , 3 , . . .(10) where ɛ _n > 0, ɛ _n → 0, Z n ∈ C [ [ τ 0 , τ 0 + r ] , E ˆ 2 n ] . For any fixed (ς, ϑ) ∈ [0, 1] ² and each τ ∈ T there exists a δ (τ, (ς, ϑ)) >0, such that the H differences [w _n (τ + h)] ^(ς,ϑ) - [w _n (τ)] ^(ς,ϑ) and [w _n (τ)] ^(ς,ϑ) - [w _n (τ - h)] ^(ς,ϑ) exist for all 0 ≤ h ≤ δ (τ, (ς, ϑ)) and n = 1, 2, 3, …. If D ˆ ( w n ( τ ) , w ( τ ) ) → 0 converges uniformly(11) for all τ ∈ [τ ₀, τ ₀ + r], (as n → ∞), then w ∈ C ¹ [[τ ₀, τ ₀ + r], B (w ₀, η)] and

w ′ ( τ ) = g ( τ , w ( τ ) ) , w ( τ 0 ) = w 0 , τ ∈ [ τ 0 , τ 0 + r ] .(12)

Proof. From (11) we know that w n ( τ ) ∈ C [ [ τ 0 , τ 0 + r ] , B ( w 0 ( τ ) , η ) ] . For fixed τ ₁ ∈ [τ ₀, τ ₀ + r], any τ ∈ [τ ₀, τ ₀ + r], τ > τ ₁, and (ς, ϑ) ∈ [0, 1] ² . Denote F ( ς , ϑ ) ( τ , n ) = j [ w n ( τ ) ] ( ς , ϑ ) - j [ w n ( τ 1 ) ] ( ς , ϑ ) τ - τ 1 - j [ g ( τ 1 , w n ( τ 1 ) ) ] ( ς , ϑ ) - j [ Z n ( τ 1 ) ] ( ς , ϑ ) where j : ( P k ( ℝ n ) , d ( ς , ϑ ) ) ↪ X is embeddable map (see Theorem 2.2 in [7]) . Now consider lim τ → τ 1 + 0 F ( ς , ϑ ) ( τ , n ) = ( j [ w n ( τ 1 ) ] ( ς , ϑ ) ) ′ - j [ g ( τ 1 , w n ( τ 1 ) ) ] ( ς , ϑ ) - j [ Z n ( τ 1 ) ] ( ς , ϑ ) = j D ˆ [ w n ( τ 1 ) ] ( ς , ϑ ) - j [ g ( τ 1 , w n ( τ 1 ) ) ] ( ς , ϑ ) - j [ Z n ( τ 1 ) ] ( ς , ϑ ) = θ ∈ X , where D ˆ [ w n ( τ 1 ) ] ( ς , ϑ ) at the point τ ₁ of [w _n (τ)] ^(ς,ϑ) is the H-derivative and we have

lim n → ∞ F ( ς , ϑ ) ( τ , n ) = j [ w ( τ ) ] ( ς , ϑ ) - j [ w ( τ 1 ) ] ( ς , ϑ ) τ - τ 1 - j [ g ( τ 1 , w ( τ 1 ) ) ] ( ς , ϑ ) .(13) From g ∈ C 1 [ R 0 , E ˆ 2 n ] is such that for any ɛ > 0 there exists δ ₁ > 0 satisfying d ( ς , ϑ ) [ g ( τ , z ) ] ( ς , ϑ ) , [ g ( τ 1 , ( w ( τ 1 ) ) ] ( ς , ϑ ) ) < 1 4 ɛ ,(14) where ɛ is a positive real number, whenever τ ₁ < τ < τ ₁ + δ ₁ along with d ( ς , ϑ ) ( [ z ] ( ς , ϑ ) , [ w ( τ 1 ) ] ( ς , ϑ ) ) < δ 1 , z ∈ B ( w 0 , η ) .

The assumption of theorem, guarantees that there exist natural number N > 0, for which ɛ n < 1 4 ɛ , and d ( ς , ϑ ) ( [ w n ( τ ) ] ( ς , ϑ ) , [ w ( τ ) ] ( ς , ϑ ) ) < 1 2 δ 1 _(15) for any n > N and τ ∈ [τ ₀, τ ₀ + r] . Take δ > 0 such that δ < δ ₁ so that d ( ς , ϑ ) ( [ w ( τ ) ] ( ς , ϑ ) , [ w ( τ 1 ) ] ( ς , ϑ ) ) < 1 2 δ 1 ,(16) whenever τ ₁ < τ < τ ₁ + δ. From definition of F ^(ς,ϑ) (τ, n) along with (10), we get j [ w n ( τ ) ] ( ς , ϑ ) - j [ w n ( τ 1 ) ] ( ς , ϑ ) - ( τ - τ 1 ) j D ˆ [ w n ( τ ) ] ( ς , ϑ ) = ( τ - τ 1 ) F ( ς , ϑ ) ( τ , n ) We take φ ∈ X ^∗ so that ||φ||=1 and φ ( j [ w n ( τ ) ] ( ς , ϑ ) - j [ w n ( τ 1 ) ] ( ς , ϑ ) - ( τ - τ 1 ) j D ˆ [ w n ( τ 1 ) ] ( ς , ϑ ) ) = ‖ j [ w n ( τ ) ] ( ς , ϑ ) - j [ w n ( τ 1 ) ] ( ς , ϑ ) - ( τ - τ 1 ) j D ˆ w n ( τ 1 ) ] ( ς , ϑ ) ‖ . Let Ψ ( τ ) = φ ( j [ w n ( τ ) ] ( ς , ϑ ) ) - ( τ - τ 1 ) φ ( j D ˆ [ w n ( τ 1 ) ] ( ς , ϑ ) ) , consequently, Ψ ′ ( τ ) = φ ( j D ^ [ w n ( τ ) ] ( ς , ϑ ) ) − φ ( j D ^ [ w n ( τ 1 ) ] ( ς , ϑ ) ) hence ‖ j [ w n ( τ ) ] ( ς , ϑ ) - j [ w n ( τ 1 ) ] ( ς , ϑ ) - ( τ - τ 1 ) j D ˆ [ w n ( τ 1 ) ] ( ς , ϑ ) ‖ = Ψ ( τ ) - Ψ ( τ 1 ) = Ψ ′ ( τ ¯ ) ( τ - τ 1 ) = φ ( j D ˆ [ w n ( τ ¯ ) ] ( ς , ϑ ) - j D ˆ w n ( τ 1 ) ] ( ς , ϑ ) ) ( τ - τ 1 ) ≤ ‖ φ ‖ . ‖ j D ˆ [ w n ( τ ¯ ) ] ( ς , ϑ ) - j D ˆ [ w n ( τ 1 ) ] ( ς , ϑ ) ‖ . ( τ - τ 1 ) = ‖ j D ˆ [ w n ( τ ¯ ) ] ( ς , ϑ ) - j D ˆ [ w n ( τ 1 ) ] ( ς , ϑ ) ‖ . ( τ - τ 1 ) , where τ 1 ≤ τ ¯ ≤ τ , In view of (13), we have ‖ F ( ς , ϑ ) ( τ , n ) ‖ ≤ ‖ j D ˆ [ w n ( τ ¯ ) ] ( ς , ϑ ) - j D ˆ [ w n ( τ 1 ) ] ( ς , ϑ ) ‖ ,(17) for τ 1 ≤ τ ¯ ≤ τ . From (16) and (15) we know that d ( ς , ϑ ) ( [ w ( τ ¯ ) ] ( ς , ϑ ) , [ w ( τ 1 ) ] ( ς , ϑ ) ) < 1 2 δ 1 and d ( ς , ϑ ) ( [ w n ( τ ¯ ) ] ( ς , ϑ ) , [ w ( τ 1 ) ] ( ς , ϑ ) ) ≤ d ( ς , ϑ ) ( [ w n ( τ ¯ ) ] ( ς , ϑ ) , [ w ( τ ¯ ) ] ( ς , ϑ ) ) + d ( ς , ϑ ) ( [ w ( τ ¯ ) ] ( ς , ϑ ) , [ w ( τ 1 ) ] ( ς , ϑ ) ) ≤ 1 2 δ 1 + 1 2 δ 1 = δ 1 . Hence by (17) and (14) we have ‖ F ( ς , ϑ ) ( τ , n ) ‖ ≤ ‖ j D ˆ [ w n ( τ ¯ ) ] ( ς , ϑ ) - j D ˆ [ X n ( τ 1 ) ] ( ς , ϑ ) ‖ = ‖ j [ g ( τ ¯ , w n ( τ ¯ ) ) ] ( ς , ϑ ) + j [ Z n ( τ ¯ ) ] ( ς , ϑ ) - j [ g ( τ 1 , w n ( τ 1 ) ) ] ( ς , ϑ ) - j [ Z n ( τ 1 ) ] ( ς , ϑ ) ‖ ≤ ‖ j [ g ( τ ¯ , w n ( τ ¯ ) ) ] ( ς , ϑ ) - j [ g ( τ 1 , w ( τ 1 ) ) ] ( ς , ϑ ) ‖ + ‖ j [ g ( τ 1 , w ( τ 1 ) ) ] ( ς , ϑ ) - j [ g ( τ 1 , w n ( τ 1 ) ) ] ( ς , ϑ ) ‖ + 2 ɛ n = d ( ς , ϑ ) ( [ g ( τ ¯ , w n ( τ ¯ ) ) ] ( ς , ϑ ) , [ g ( τ 1 , w ( τ 1 ) ) ] ( ς , ϑ ) ) + d ( ς , ϑ ) ( [ g ( τ 1 , w ( τ 1 ) ) ] ( ς , ϑ ) , [ g ( τ 1 , w n ( τ 1 ) ) ] ( ς , ϑ ) ) + 2 ɛ n < 1 4 ɛ + 1 4 ɛ + 2 ɛ n < ɛ

whenever n > N, and  τ ₁ < τ < τ ₁ + δ . Assuming n → ∞,  and using equation (6),  we have ‖ j [ w ( τ ) ] ( ς , ϑ ) - j [ w ( τ ) ] ( ς , ϑ ) τ - τ 1 - j [ g ( τ 1 , w ( τ 1 ) ) ] ( ς , ϑ ) ‖ ≤ ɛ ,(18) for  τ ₁ < τ < τ ₁ + δ. Now conditions of Theorem 3, ensure the existence of δ (τ ₁, (ς, ϑ)) ∈ (0, δ) in such a way that H difference [ w n ( τ 1 + h ) ] ( ς , ϑ ) - [ w n ( τ 1 ) ] ( ς , ϑ ) exists, ∀ 0 < h < δ (τ ₁, (ς, ϑ)), n = 1, 2, … Supporting functional δ ( . , K ∗ ) : ℝ n → ℝ ( K ∈ P k ( ℝ n ) ) extracted as δ ( e , K ) = sup { e . k ∗ : k ∗ ∈ K } , usual scalar product of e along with k ^∗ defined as e . k ^∗. Then by Lemma 3.4 in [5] provides a sub-linear function p _m which is positively homogeneous continuous on ℝ n , such that p m ( e ) = δ ( e , [ w m ( τ 1 + h ) ] ( ς , ϑ ) ) - δ ( e , [ w m ( τ 1 ) ] ( ς , ϑ ) ) for some 0 < h < δ (τ ₁, (ς, ϑ)), m = 1, 2, … Now Theorem 2 gives for any e ≠ 0 δ ( e , [ w ( τ 1 ) ] ( ς , ϑ ) ) ≤ δ ( e , [ w m ( τ 1 ) ] ( ς , ϑ ) ) + ‖ e ‖ d ( ς , ϑ ) ( [ w m ( τ 1 ) ] ( ς , ϑ ) , [ w ( τ 1 ) ] ( ς , ϑ ) ) , δ ( e , [ w m ( τ 1 ) ] ( ς , ϑ ) ) ≤ δ ( e , [ w ( τ 1 ) ] ( ς , ϑ ) ) + ‖ e ‖ d ( ς , ϑ ) ( [ w m ( τ 1 ) ] ( ς , ϑ ) , [ w ( τ 1 ) ] ( ς , ϑ ) ) . Using condition (11) it is known that for fixed L > 0, δ ( e , [ w m ( τ 1 ) ] ( ς , ϑ ) ) → δ ( e , [ w ( τ 1 ) ] ( ς , ϑ ) ) uniform convergence for all ‖ e ‖ ≤ L , as m → ∞ . Similarly, δ ( e , [ w m ( τ 1 + h ) ] ( ς , ϑ ) ) → δ ( e , [ w ( τ 1 + h ) ] ( ς , ϑ ) ) uniform convergence for all ‖ e ‖ ≤ L as m → ∞ . Then p ( e ) = δ ( e , [ w ( τ 1 + h ) ] ( ς , ϑ ) ) - δ ( e , [ w ( τ 1 ) ] ( ς , ϑ ) ) = m → ∞ lim p m ( e ) uniform convergence for all ‖ e ‖ ≤ L . So p (e) on ℝ n is continuous positively homogeneous sub-linear function. From Lemma 3.4 in Bradley we conclude that H difference [w (τ ₁ + h)] ^(ς,ϑ) - [w (τ ₁)] ^(ς,ϑ) exists ∀ 0 < h < δ (τ ₁, (ς, ϑ)) . So, from (18) we have d ( ς , ϑ ) ( [ w ( τ ) ] ( ς , ϑ ) - [ w ( τ 1 ) ] ( ς , ϑ ) τ - τ 1 , [ g ( τ 1 , w ( τ 1 ) ) ] ( ς , ϑ ) ) ≤ ɛ , for τ ₁ < τ < τ ₁ + δ . So, τ → τ 1 + 0 lim [ w ( τ ) ] ( ς , ϑ ) - [ X ( τ 1 ) ] ( ς , ϑ ) τ - τ 1 = [ g ( τ 1 , w ( τ 1 ) ) ] ( ς , ϑ ) . Similarly, we have τ → τ 1 - 0 lim [ w ( τ ) ] ( ς , ϑ ) - [ w ( τ 1 ) ] ( ς , ϑ ) τ - τ 1 = [ g ( τ 1 , w ( τ 1 ) ) ] ( ς , ϑ ) . Hence w ′ ( τ 1 ) exists and w ′ ( τ 1 ) = g ( τ 1 , w ( τ 1 ) ) . From τ ₁ ∈ [τ ₀ . τ _o + r] is arbitrary, it is noted that equation (12) holds true along with w ∈ C 1 [ [ τ 0 , τ 0 + r ] , B ( w 0 , η ) ] . Thus, we conclude the proof.□

Corollary 7. If the condition (10) is replace by w n + 1 ′ ( τ ) = g ( τ , w n ( τ ) ) + Z n ( τ ) , w n ( τ 0 ) = w 0 , D ˆ ( Z n ( τ ) , ⌢ 0 ) ≤ ɛ n , ∀ τ ∈ [ τ 0 , τ 0 + r ] , n = 1 , 2 , 3 , . . . and retaining all other conditions, are same, then the conclusions also hold true.

In the following result, uniqueness and existence criteria of the Cauchy problem in the perspective of complex fuzzy differential equations is discussed as follows.

Theorem 8. Let (a) g ∈ C [ R o , E ˆ 2 n ] and D ˆ ( g ( τ , w ) , 0 ˆ ) ≤ M , ∀ ( τ , w ) ∈ R 0 , (b) h ∈ C [ [ τ 0 , τ o + p ] × [ 0 , η ] , ℝ ] , with h (τ, 0) =0 and 0 ≤ h ( τ , v ) ≤ M 1 , ∀ τ ∈ [ τ 0 , τ o + p ] , for 0 ≤ v ≤ η , such that h (τ, v) is nondecreasing w.r.t v, then the initial value problem v ′ ( τ ) = h ( τ , v ( τ ) ) , with v ( τ 0 ) = 0(19) has only the solution v (τ) =0 on [τ ₀, τ ₀ + p], (c) if D ˆ ( g ( τ , w ) , g ( τ , y ) ) ≤ h ( τ , D ˆ ( w , y ) ) , for all  (τ, w), (τ, y) ∈ R ₀, and D ˆ ( w , y ) ≤ η . Then the initial value problem (12) has unique solution w ∈ C ¹ [[τ ₀, τ ₀ + r], B (w ₀, η)] on [τ ₀, τ ₀ + r], where r = min { p , η / M , η / M 1 } , and the successive iterations w 0 ( τ ) = w 0 , w n + 1 ( τ ) = w 0 + ∫ τ 0 τ g ( s , w n ( s ) ) ds , for n = 0 , 1 , 2 , 3 , . .(20) uniformly converges to w (τ) on [τ ₀, τ ₀ + r] .

Proof. From (20) and by inductive process, we get D ˆ ( w n + 1 ( τ ) , w 0 ) ≤ η , forall τ ∈ [ τ 0 , τ 0 + r ] , and n = 0 , 1 , 2 , . . Hence w n + 1 ∈ C 1 [ [ τ 0 , τ 0 + r ] , B ( w 0 , η ) ] also w ′ n + 1 ( τ ) = g ( τ , w n ( τ ) ) with w n ( τ 0 ) = w 0 , f o r n = 0 , 1 , 2 , 3 , ..(21) Let M 2 = max { M , M 1 } Then r = min { p , η / M 2 } and the successive iterations extracted as { v 0 ( τ ) = M 2 ( τ - τ 0 ) for τ 0 ≤ τ ≤ τ 0 + r , v n + 1 ( τ ) = ∫ τ 0 τ h ( s , v n ( s ) ) ds , for τ 0 ≤ τ ≤ τ 0 + r , and n = 0, 1, 2, 3, . ., which instantly provides v 1 ( τ ) = ∫ τ 0 τ h ( s , v 0 ( s ) ) ds ≤ M 1 ( τ - τ 0 ) ≤ v 0 ( τ ) ≤ η , for all τ ∈ [ τ 0 , τ 0 + r ] . So, by induction and the nondecreasing property of h (τ, v) in its second component v, we have 0 ≤ v n + 1 ( τ ) ≤ v n ( τ ) ≤ η , for all τ ∈ [ τ 0 , τ 0 + r ] , and n = 0 , 1 , 2 , 3 , . . . .(22) From Ascoli-Arzela Theorem and inequality (22), we get | v n + 1 ( τ ) | = | h ( τ , v n ( τ ) ) | ≤ M 1 . As {v _n (τ)} uniformly converges to some continuous function v (τ) on [τ ₀, τ ₀ + r], therefore v ( τ ) = ∫ τ 0 τ h ( s , v ( s ) ) ds . Thus v ∈ C 1 [ [ τ 0 , τ 0 + r ] , [ 0 , η ] ] and v is the solution of initial value problem (19) . From condition (b) we obtain v (τ) =0 . Further, we have D ˆ ( w 1 ( τ ) , w 0 ( τ ) ) = D ˆ ( ∫ τ 0 τ g ( s , w 0 ) ds , 0 ˆ ) ) ≤ ( ∫ τ 0 τ D ˆ ( g ( s , w 0 ) , 0 ˆ ) ds ≤ M ( τ - τ 0 ) ≤ v 0 ( τ ) . Now suppose D ˆ ( w k ∗ ( τ ) , w k - 1 ( τ ) ) ≤ v k - 1 ( τ ) , then by condition (c), we have D ˆ ( w k + 1 ( τ ) , w k ( τ ) ) = D ˆ ( ∫ τ 0 τ g ( s , w k ( s ) ) ds , ∫ τ 0 τ g ( s , x k - 1 ( s ) ) ds ) ≤ ∫ τ 0 τ D ˆ ( g ( s , w k ( s ) ) , g ( s , w k - 1 ( s ) ) ) ds ≤ ∫ τ 0 τ h ( s , D ˆ ( w k ( s ) , w k - 1 ( s ) ) ) ds ≤ ∫ τ 0 τ h ( s , v k - 1 ( s ) ) ds = v k . Thus by Mathematical induction, it is clear that D ˆ ( w n + 1 ( τ ) , w n ( τ ) ) ≤ v n ( τ ) , forall τ ∈ [ τ 0 , τ 0 + r ] and n = 0 , 1 , 2 , 3 , . . . Therefore D ˆ ( w n + 1 ( τ ) , w n ( τ ) ) = D ˆ ( g ( τ , w k ( τ ) ) , g ( τ , w k - 1 ( τ ) ) ) ≤ h ( τ , D ˆ ( wn ( τ ) , w n - 1 ( τ ) ) ) ≤ h ( τ , v n - 1 ( τ ) ) .(23) Consider for m ≥ n, from (23) and (22), we get D ˆ ( w n ′ ( τ ) , w m ′ ( τ ) ) ≤ D ˆ ( g ( τ , w n - 1 ( τ ) ) , g ( τ , w n ( τ ) ) ) + D ˆ ( g ( τ , w n ( τ ) ) , g ( τ , w m ( τ ) ) ) + D ˆ ( g ( τ , w m ( τ ) ) , g ( τ , w m - 1 ( τ ) ) ) ≤ h ( τ , v n - 1 ( τ ) ) + h ( τ , D ˆ ( w n ( τ ) , w m ( τ ) ) ) + h ( τ , v m - 1 ( τ ) ) ≤ h ( τ , D ˆ ( w n ( τ ) , w m ( τ ) ) ) + 2 h ( τ , v n - 1 ( τ ) ) . Since h (τ, v _n-1 (τ)) uniformly converges to 0, then for any ɛ > 0, there exists natural number N so that