A new technique to solve fuzzy differential equations

1 Introduction and preliminaries

Due to their immense applications in various fields, the study of differential equations is a well established knowledge area of Mathematics [11, 23]. However, in many situations, these equations seem unable to describe the real life phenomena. For example, in mathematical modeling for biological phenomena and quality control problems, the parameters are presented in the form of fuzzy subsets and sometimes the use of fuzzy numbers in the calculations is required [4, 21]. In these cases the basic initial value problem can be stated as follows: X ′ ( t ) = f ( t , X ( t ) ) , X ( 0 ) = X 0 ,(1) where f is continuous in a crisp domain and X₀ is a fuzzy set.

Since its inception in 1965, Zadeh’s extension principle has been a bridge between crisp and fuzzy concepts [8, 12]. In its simplest form, it works as follows: Given a function f from a crisp universe X to a crisp universe Y and a fuzzy set on the universe X, the extension principle helps in defining a fuzzy set on the universe Y with the help of f. This provides a mathematical approach for extending classical functions to fuzzy mappings. The extension principle is an important tool in the development of fuzzy arithmetic. But in some cases, the extension principle may not be appropriate for pessimistic or conservative decision since it uses supremum in its construction. A new extension principle was hence proposed in to solve fuzzy differential equation.

In [19], differential equations with fuzzy parameters and initial conditions (1) were studied. That work dealt with introducing the notion of fuzzy solutions by applying Zadeh’s extension principle to the deterministic solution and a presented numerical algorithm, based on monotonicity properties of f. More valuable research can be found in [1–3, 16]. In this paper, we show that a solution for (1) can also be obtained through Zadeh’s averaging extension principle, which is although similar to [19] and [18] but is obtained with the help of averaging extension principle defined in instead of Zadeh’s extension principle. We also show the existence of a fuzzy solution which is strongly dependent on the choice of both fuzzy initial condition and parameter. Moreover, we conclude that this fuzzy solution coincides with the solution obtained by using Hü llermeier’s interpretation, via differential inclusions as well as one obtained by differential inclusions [5]. In this way, we present a novel technique for solving Fuzzy Differential Equations (FDE).

Let X be a Banach space with the norm || . ||. The norm induces the d (. , .) metric on X given by: d (x, y) = ||x - y||, for all x, y ∈ X. If x ∈ X and A is a non empty subset of X, then the distance from x to A is defined by d ( x , y ) = inf a ∈ A ‖ x - a ‖ .

A set A ⊂ X is called convex if (1 - λ) a + λb ∈ A for all a, b ∈ A and all λ ∈ [0, 1]. We denote by K (X) the family of all non empty compact subsets of X and by K _c  (X) the family of all non empty compact convex subsets of X.

Let A, B be non empty subsets of X and λ ∈ ℝ . We define the following Minkowski addition and multiplication, respectively, by: A + B = { x + y | x ∈ A , y ∈ B } and λ A = { λ x | x ∈ A } .

If for fixed elements A, B ∈ K _c  (X) there is a C ∈ K _c  (X) such that A = B + C then C is called the Hukuhara difference of A and B and is denoted by A - B.

The function e : P b ( X ) × P b ( X ) → ℝ given by: e ( A , B ) = sup x ∈ A d ( x , B ) = sup x ∈ A inf y ∈ B ‖ x - y ‖ is called the excess function, where P _b  (X) stands for the family of nonempty bounded subsets of X. The real number e (A, B) is called the excess of A over B. If A, B ∈ P _b  (X) we define the Hausdorff - Pompeiu distance between A and B by: d HP ( A , B ) = max { e ( A , B ) , e ( B , A ) } .

The notion of fuzzy set was defined by Zadeh in a seminal paper in 1965 [22]. Let X be a non empty set called universe. A fuzzy set U in X is a function from the universe X to the unit interval [0, 1] that maps an element x ∈ X to U (x) given by U : X → [0, 1] , where U (x) is the degree of membership of x in U.

Let X be a universe of discourse and let α ∈ (0, 1]. An α -level set or an α-cut of a fuzzy set U : X → [0, 1] is a crisp subset of X denoted by [u]  ^α and is defined by: [ U ] α = { x ∈ X | U ( x ) ≥ α } .

The crisp subset of X defined by [ U ] 0 = { x ∈ X | U ( X ) > 0 } ¯ is called the support of fuzzy set U.

Let X be a normed space over the field ℝ of real numbers. A fuzzy set U : X → [0, 1] is called fuzzy convex if the crisp set [U]  ^α is a convex set for all α ∈ (0, 1]. A fuzzy set u : X → [0, 1] is called fuzzy compact if [U]  ^α is a compact set for all α ∈ [0, 1]. By F ( X ) we will denote the space of all fuzzy sets on X while F C ( X ) would stand for compact and convex fuzzy sets on X.

If u ∈ F C ( ℝ ) , then u is called a fuzzy interval and the α-level set [u]  ^α is a nonempty compact interval for all α ∈ [0, 1]. We denote by (a, b, c) the triangular fuzzy number with support [a, c].

The operations of addition and scalar multiplication on F C ( R n ) are defined as ( u + v ) ( x ) = sup y ∈ ℝ n { u ( y ) ∧ v ( x - y ) } and ( λ . u ) ( x ) = { u ( x λ ) , if λ ≠ 0 ; χ { 0 } ( x ) , if λ = 0 , where χ_{0} (x) is the characteristic function of {0}. If u , v ∈ F C ( ℝ n ) and λ ∈ ℝ , then the following properties are true:

[u + v]  ^α  = [u]  ^α  + [v]  ^α  and  [λ . u]  ^α  = λ [u]  ^α , ∀ α ∈ [0, 1] . We can extend the Hausdorff metric H on K c n to F C ( R n ) by means of D ( u , v ) = sup α ∈ [ 0 , 1 ] H ( [ u ] α , [ v ] α ) , ∀ u , v ∈ F ( ℝ n ) .

Zadeh’s extension principle provides a mathematical approach for extending classical functions to fuzzy mappings ([12]). In its simplest form it may be undertood as: Let X be a set, P ( X ) be the power set of X, and F ( x ) be the fuzzy power set of X. That is, P ( X ) = { U | U ⊂ X } = { U | U : X → { 0 , 1 } } and F ( X ) = { U | U : X → [ 0 , 1 ] } .

Let f be a mapping from a set X to another set Y. The mapping f ˜ max from F ( X ) to F ( Y ) can be induced by f as following: f ˜ max : F ( X ) → F ( Y ) , u ↦ f ˜ max ( u ) , where f ˜ max ( u ) ( y ) = { ⋁ f ( x ) = y u ( x ) , if { x | f ( x ) = y , x ∈ X } ≠ ∅ , 0 , otherwise . ∀ y ∈ Y .

This is the most popular form of Zadeh’s extension principle but as can be een easily it is an optimistic construction as it uses ⋁_f(x)=y the supremum. For the sake of pessimistic decision or conservative decision other extension principles have been considered in [12]. Let’s introduce average extension principle.

Let f : X → Y be a classical mapping and for any y ∈ Y, f^-1 (y) is a finite set. The average extension principle is defined by f ˜ p : F ( X ) → F ( Y ) , u ↦ f ˜ p ( u ) , where f ˜ p ( u ) ( y ) = { 1 n Σ f ( x ) = y u ( x ) , if | { x | f ( x ) = y , x ∈ X } | = n , 0 , if { x | f ( x ) = y , x ∈ X } = ∅ . ∀ y ∈ Y .

Clearly, if u ∈ P ( X ) , then u = u₀. Moreover, f ˜ ( u ) ( y ) > 0 ⇔ ∃ x ∈ f - 1 ( y ) , u ( x ) = 1 ⇔ ∃ x ∈ u ∩ f - 1 ( y ) , where f ˜ is either f ˜ p or f ˜ max . This shows that f ˜ p and f ˜ max are the same in classical sets under the meaning above.

The idea is that each function f : X → Y induces another function f ˜ p : F ( X ) → F ( Y ) defined as above for each fuzzy set u in X. The function f ˜ p is obtained from f by average extension principle. If f : ℝ n → ℝ n is a continuous function, then f ˜ p : F ( ℝ n ) → F ( ℝ n ) is a well-defined function and [ f ˜ p ( u ) ] α = f ( [ u ] α ) , ∀ α ∈ [ 0 , 1 ](2) ∀ u ∈ F ( ℝ n ) where f (A) = {f (a) |a ∈ A} for some nonempty compact convex subset A of X.

We establish now the following result.

Theorem 2.1. If f : ℝ n → ℝ n is a continuous function, then f ˜ p : F ( ℝ n ) → F ( ℝ n ) is also a continuous function under the metric D ( f ˜ p ( χ { x } ) , f ˜ p ( χ { y } ) ) = ‖ f ( x ) - f ( y ) ‖ ∀ x , y ∈ ℝ n .

Proof. We note that ∥x - y ∥ = H ({x} , {y}) = D (χ_{x}, χ_{y}) and D ( f ˜ p ( χ { x } ) , f ˜ p ( χ { y } ) ) = ∥ f ( x ) - f ( y ) ∥ ∀ x , y ∈ ℝ n .

(ii) ⇒ (i): We consider D : u _q  → u₀ where u q , u 0 ∈ F ( ℝ n ) . We choose X ∈ K ( ℝ n ) such that ⋃ α ∈ [ 0 , 1 ] , q ≥ 0 [ u q ] α ⊆ X(3) by Lemma 3.2 [10]. For given ϵ > 0 there exists δ > 0 such that H ( A , B ) < δ ⇒ H ( f ( A ) , f ( B ) ) < ϵ(4) for A, B ∈ K (X), see Lemma 3.1 [10]. As D : u _q  → u₀, therefore, there exists p δ = p ( δ ) ∈ ℕ such that D ( u q , u 0 ) = sup α ∈ [ 0 , 1 ] H ( [ u q ] α , [ u 0 ] α ) < δ , ∀ p ≥ p _δ  . Thus, from Equations (3) and (4), we obtain H ([u _q ]  ^α , [u₀]  ^α ) < δ ⇒ H (f ([u _q ]  ^α ) , f ([u₀]  ^α )) < ϵ, ∀ p ≥ p _δ  . Consequently, we have D ( f ˜ p ( u q ) , f ˜ p ( u 0 ) ) = sup α ∈ [ 0 , 1 ] H ( [ f ˜ p ( u q ) ] α , [ f ˜ p ( u 0 ) ] α ) = sup α ∈ [ 0 , 1 ] H ( f ( [ u q ] α ) , f ( [ u 0 ] α ) ) ≤ ϵ , ∀ p ≥ p _δ  . by Equation (2). Hence, D : f ˜ p ( u q ) → f ˜ p ( u 0 ) , i.e., f ˜ p is D-continuous. □

We introduce a new method for solving fuzzy initial value problem X ′ ( t ) = f ( t , X ( t ) ) , X ( 0 ) = X 0 ,(5) where f : [ 0 , T ] × F ( U ) → F ( ℝ n ) is obtained by average extension principle from a continuous function g : [ 0 , T ] × U → ℝ n , where U ⊂ ℝ n . Note that f is continuous because g is continuous [10] and by (2) we have [ f ( t , X ) ] α = g ( t , [ X ] α ) , where g ( t , A ) = { g ( t , a ) | a ∈ A } .

Now, consider the associated deterministic differential equation x ′ ( t ) = g ( t , x ( t ) ) , x ( 0 ) = c ,(6) where x′ (t) is the derivative (crisp) of a function x : [ 0 , T ] → ℝ n .

Suppose that the Problem (6) has the solution x (t, c). Then applying the average extension principle to x (t, c) in relation to parameter c, we obtain the extension X ( t ) = x ˜ p ( t , X 0 ) , for each t fixed, which is a fuzzy solution of Problem (5).

In this way we establish the following result.

Theorem 3.1. Let O be an open set in ℝ n and [X₀]  ^α  ⊂ O. Suppose that g is continuous and that for each c ∈ O there exists a unique solution x (. , c) of the Deterministic Problem (6) and that x (t, .) is continuous on O for each t ∈ [0, T] fixed. Then, there exists a unique fuzzy solution X ( t ) = x ˜ p ( t , X 0 ) of FDE (5).

Proof. Since Problem (6) has a unique solution x (t, c) and it is continuous on O, x ( t , . ) : O → ℝ n is well defined and it is continuous for each t ∈ [0, T] fixed. Then, by (2), x ˜ p ( t , . ) : F ( O ) → F ( ℝ n ) is a continuous function and it is well defined. Therefore there exists a unique solution of the form X ( t ) = x ˜ p ( t , X 0 ) for the Fuzzy Differential Equation (5). □

Let O be an open subset in ℝ n such that there exists a solution x (t, x₀) of (6) with x₀ ∈ O in the interval [t₀, T], and for all t ∈ [t₀, T], x (t, x₀) is continuous on O. We can define the operator L t = O → ℝ n by L ^t  (x₀) = x (t, x₀), which is the unique solution of (6) and is continuous relative to x₀, see [18]. The application of the average extension principle to L ^t provides us the extension L ˜ p t : F ( O ) → F ( ℝ n ) ,(7) which is the solution of problem (5), via average extension principle, with initial condition X 0 ∈ F ( O ) . Note that the existence of L ˜ p t is guaranteed by (2), since the function L ^t is continuous.

According to Hüllermeier’s interpretation, we can write the Fuzzy Initial Value Problem (2) as a family of differential inclusions y α ′ ( t ) = g ( t , y α ( t ) ) , y α ( 0 ) ∈ [ X 0 ] α ,(8) 0 ≤ α ≤ 1 . Following Diamond [9], the attainable sets A α t = { y α ( t ) | y α } is a solution of (8)} are the α-levels of a fuzzy set A α t ( x 0 ) , where A α t ⊂ ℝ n , for each α and for each t ∈ [t₀, T].

We establish now a result relating Hüllermeier’s solution to the one obtained through the average extension principle.

Theorem 3.2. Let O be an open set in ℝ n and X 0 ∈ F ( O ) . Suppose that g is continuous, that for each x₀ ∈ O there exists one unique solution x (t, x₀) of the Problem (6) and that x (t, x₀) is continuous in O for each t ∈ [t₀, T]. Then, there exists L ˜ p t ( X 0 ) such that L ˜ p t ( X 0 ) = A t ( X 0 )(9) for all t₀ ≤ t ≤ T. That is, the attainable sets for problem (5) can be obtained as the image of the fuzzy initial condition by the average extension of the deterministic solution.

Proof. We intend to prove that [ L ˜ p t ( X 0 ) ] α = A t ( [ X 0 ] α ) ∀ α ∈ [0, 1] . Since f is continuous and for each x₀ ∈ O there exists a unique solution for (6) in the interval [t₀, T]. Therefore, L t : O → ℝ n , given by L ^t  (x₀) = x (t, x₀), is well defined for each t ∈ [t₀, T] and is continuous. Consequently, L ˜ p t : F ( O ) → F ( ℝ n ) is well defined and continuous by (2). So, given α ∈ [0, 1], we have [ L ˜ p t ( X 0 ) ] α = L t ( [ X 0 ] α ) = { L t ( x 0 ) | x 0 ∈ [ X 0 ] α } = { x ( t , x 0 ) | x 0 ∈ [ X 0 ] α } .

On the other hand, the α-levels of the attainable set for Problem (8) are given by A t ( [ X 0 ] α ) = ⋃ x 0 ∈ [ X 0 ] α A t ( x 0 ) = { x ( t , x 0 ) | x 0 ∈ [ X 0 ] α } .

Hence the result. □

We define now the generalized derivatives, see [7, 6].

Definition 3.3. x : T → F ( ℝ n ) is said to be differentiable at t₀ ∈ T if

there exists an element x ′ ( t 0 ) ∈ F ( ℝ n ) such that there are the limits lim h → 0 + x ( t 0 + h ) - x ( t 0 ) h = lim h → 0 + x ( t 0 ) - x ( t 0 - h ) h = x ′ ( t 0 ) in D-metric for all h > 0 sufficiently near to 0. or

Theorem 3.4. [7] Let x : T → F ( ℝ ) be a function and denote [x (t)]  ^α  = [f _α  (t) , g _α  (t)] for each α ∈ [0, 1]. Then,

If x is differentiable in the first form (I), then f _α and g _α are differentiable functions and [ x ′ ] α = [ f α ′ ( t ) , g α ′ ( t ) ] .

The last result gives us a procedure to solve Fuzzy Differential Equation (5) where X′ (t) is the generalized derivative in the first form (I) or second form (II), see [7, 14]. Now, we establish the relationships between the fuzzy solution for problem (5) proposed in Theorem 3.1 and the solution of problem (5) when X′ (t) is the generalized derivative.

Theorem 3.5. Let O be an open set in ℝ n and X 0 ∈ F ( O ) . Suppose that g is continuous, that for each c ∈ O there exists one unique solution x (. , c) of problem (6) and that x (t, .) is continuous in O for each t ∈ [0, T]. Then,

If g is nondecreasing, the fuzzy solution of (5) and the solution of (5) via the derivative in the first form (I) are identical.

Proof. If g is nondecreasing, then from Theorem 3 of [14], the solution of (5) and the solution by differential inclusions are identical. Thus, the result is direct consequence of Theorem 3.4. Similarly, if g is nonincreasing, then from Theorem 4 of [7], the solution of (5) and the solution by differential inclusions are identical. Hence the result from Theorem 3.4. □

Example 3.6. Consider the fuzzy initial value problem X ′ ( t ) = 1 1 + t 2 X ( t ) , X ( 0 ) = X 0 .(10)

The deterministic problem associated with (10) is x ′ ( t ) = 1 1 + t 2 x ( t ) , x ( 0 ) = x 0 ,(11) and the solution to this problem is L t ( x 0 ) = x 0 e tan - 1 t .(12)

Note that L ^t  (x₀) is continuous in x 0 ∈ ℝ for each t ≥ 0 fixed. We apply the average extension principle to L ^t  (x₀) in relation to x₀, for each t ≥ 0 fixed. Then we obtain the unique fuzzy solution L ˜ p t ( X 0 ) of problem (10) for any initial condition X₀, with X₀ a fuzzy interval, which is given by L ˜ p t ( X 0 ) = X 0 . e tan - 1 t , t ≥ 0 .(13)

Now, we set [ X ( t ) ] α = [ x α ( t ) , y α ( t ) ] , [ X 0 ] α = [ x α 0 , y α 0 ] and [f (t, X (t))]  ^α  = [g (t, x _α  (t) , y _α  (t)) , h (t, x _α  (t) , y _α  (t))] , where f ( t , X ( t ) ) = 1 1 + t 2 X ( t ) . We transform the problem (10) into the following parameterized differential system { x α ′ ( t ) = g ( t , x α ( t ) , y α ( t ) ) , x α ( 0 ) = x α 0 y α ′ ( t ) = h ( t , x α ( t ) , y α ( t ) ) , y α ( 0 ) = y α 0(14)

In particular, { x α ′ ( t ) = 1 1 + t 2 x α ( t ) , x α ( 0 ) = x α 0 y α ′ ( t ) = 1 1 + t 2 y α ( t ) , y α ( 0 ) = y α 0(15)

The solution is [ X ( t ) ] α = [ x α ( t ) , y α ( t ) ] = e tan - 1 t [ x α 0 , y α 0 ] .(16)

So, the fuzzy solution L ˜ p t ( X 0 ) of problem (10) is coincident with the solution obtained by fuzzy differential inclusions and is also coincident with the solution obtained in [7] by considering X′ (t) the generalized derivative.

Example 3.7. We consider the fuzzy initial value problem X ′ ( t ) = λ X 2 ( t ) , X ( 0 ) = X 0(17) with λ > 0. The deterministic problem associated with (10) is x ′ ( t ) = λ x 2 ( t ) , x ( 0 ) = x 0 .(18)

Note that the the deterministic solution is given by L t ( x 0 ) = x 0 1 - λ tx 0 , which is continuous with respect to x₀ on the open interval J = (0, z) for each t ∈ [ 0 , 1 λ z ) fixed. Therefore, for any fuzzy interval X₀ the attainable set L ˜ p t ( X 0 ) exists for each t ∈ [ 0 , 1 λ z ) , where z = sup k ∈ [ X 0 ] 0 k .

In [14] Kaleva studied the problem (17) for λ = 1. That is, X ′ ( t ) = X 2 ( t ) , X ( 0 ) = X 0 ,(19) where the initial condition X₀ is the triangular fuzzy number X 0 ( y ) = { 3 - y , if 2 ≤ y ≤ 3 , y - 1 , if 1 ≤ y ≤ 2 , 0 , elsewhere .(20)

In this case there exists an attainable set L ˜ p t ( X 0 ) for t ∈ [0, 1/3). Note that the function L ^t  (x₀) is nondecreasing with respect to x₀ for each t ∈ [0, 1/3). Thus, for each α ∈ [0, 1] we have [ L ˜ p t ( X 0 ) ] α = L t ( [ X 0 ] α ) = L t [ 1 + α , 3 - α ] = [ L t ( 1 + α ) , L t ( 3 - α ) ] = [ 1 + α 1 - t - t α , 3 - α 1 - 3 t - t α ] .

In [14] Kaleva studied the problem (17) for λ = 1 considering X′ (t) the Hukuhara derivative and the initial condition X₀ as a triangular fuzzy number. In that case he obtained the same solution which was obtained here via differential inclusion. Therefore, from Theorem 3.2, the three processes studied lead to the same solution for Example 3.7.

In this paper an attempt is made to use an alternative form of of Zadeh’s Extension Principle in solution of Fuzzy initial value problems. The related theorems are proved successfully. The acheivement of the theory presented is demonstrated in examples where the solution by differential inclusions and by Average Extension Principle is compared and are found to be the same, while obviousely the extension principle technique is much easier to apply.