Existence results for fuzzy Volterra integral equation

1 Introduction

The concept of fuzzy sets was originally introduced by Zadeh [20], led to the definition of fuzzy numbers and their implementations in fuzzy control [7] and approximate reasoning problems [21]. Fuzzy integral equations and fuzzy differential equations play an important role in a variety of fields such as physics, geographic, medical and biology, etc. To obsever some basic information and results of various type fuzzy integral equations and fuzzy differential equations one can see the papers such as [1, 19] and references therein.

The study of fuzzy integral equations was initiated by investigations of Kaleva [12] and Seikkala [17] for fuzzy Volterra integral equations. Many authors applied the fixed point theorems such as the Darbo’s theorem and the classical Banach fixed point theorem [18] and Schauders fixed point theorem [2]. It is well known that the above fixed point theorems are such a useful tool in mathematics and play an important role in the solving of solutions to nonlinear, fuzzy differential and fuzzy integral equations, among others. In the recent time this theorem has been extended and generalized by several authors in various ways, for instance, via Banach’s contraction principle authors [18] proved the existence and uniqueness theorems for certain Volterra integral equations involving fuzzy set-valued mappings. In [2], Allahviranloo et al. also investigated the existence and uniqueness of solution of nonlinear fuzzy Volterra integral equations by using Arzela-Ascoli’s theorem and Schauders fixed point theorem. In [19], using the contractive-like mapping principles, authors discussed the existence and uniqueness of solution of fuzzy differential equation under generalized Hukuhara derivative. In this paper, some recent results of fixed point of contractive mappings on the partially ordered space, established in [15], are used to study the existence and uniqueness of solution for fuzzy Volterra integralequation.

This paper is organized as follows: In Section 2, some properties concerning the partially ordered in the partially ordered space of fuzzy function and some fixed point theorems in partial metric spaces are presented. In Section 3, we prove the existence and uniqueness of solution for two kinds of fuzzy Volterra integral equation. Finally, some examples are also given to illustrate our theory.

2 Preliminaries

In this section, we present some basic notations and necessary preliminaries used throughout the paper. In the following, let K c ( ℝ ) denote the collection of all nonempty compact and convex subsets of ℝ . The addition and scalar multiplication in K c ( ℝ ) , we define as usual, i.e. A , B ∈ K c ( ℝ ) and λ ∈ ℝ , then we have A + B = {a + b|a ∈ A, b ∈ B}, λA = {λa|a ∈ A}. The Hausdorff metric d _H in K c ( ℝ ) is defined as follows d H ( A , B ) = max { sup a ∈ A inf b ∈ B ‖ a - b ‖ ℝ , sup b ∈ B inf a ∈ A ‖ a - b ‖ ℝ } , where A , B ∈ K c ( ℝ ) . It is known that ( K c ( ℝ ) , d H ) is a complete metric space.

Denote E 1 = { u : ℝ → [ 0 , 1 ] such that x (z) satisfies (i)-(iv) stated below}

x is normal, i.e, there exists an z 0 ∈ ℝ such that x (z₀) =1;

For α ∈ (0, 1], denote [ x ] α = { z ∈ ℝ | x ( z ) ≥ α } . We will call this set an α - cut of the fuzzy set x. For x ∈ E¹ one has that [ x ] α ∈ K c ( ℝ ) for every α ∈ [0, 1].

The notation [ x ] α = [ x l α , x r α ] , denotes explicity the α - cut of x ∈ E¹, for α ∈ [0, 1] and we define the diameter of the α - cut of x ∈ E¹ as diam ( [ x ] α ) = x r α - x l α .

The supremum on E¹ is defined as follows: d ∞ ( x 1 , x 2 ) = sup α ∈ [ 0 , 1 ] max { | x 1 , l α - x 2 , l α | , | x 1 , r α - x 2 , r α | } , for every x₁, x₂ ∈ E¹. It is well known (E¹, d_∞) is a complete metric space. For x, y, z ∈ E¹ and λ ∈ ℝ + , we have d ∞ ( x + z , y + z ) = d ∞ ( x , y ) , d ∞ ( λ x , λ y ) = λ d ∞ ( x , y ) , d ∞ ( x , y ) ≤ d ∞ ( x , z ) + d ∞ ( y , z ) .

Next, for x, y ∈ E¹ with [ x ] α = [ x l α , x r α ] , [ y ] α = [ y l α , y r α ] , we can define the following partial orderings: ≤ and ⪯, given by x , y ∈ E 1 , x ≤ y ⇔ x l α ≤ y l α and x r α ≤ y r α , for all α ∈ [ 0 , 1 ] , and x , y ∈ E 1 , x ⪯ y ⇔ [ x ] α ⊆ [ y ] α , for all α ∈ [ 0 , 1 ] , that is, x l α ≥ y l α and x r α ≤ y r α , for all α ∈ [ 0 , 1 ] .

Some properties on the partial ordering ≤ and ⪯ are presented in [16]. Let J = [ a , b ] ⊆ ℝ . The partial ordering ≤ and ⪯ on E¹ can be extended to the space of fuzzy functions, as follows: for all α ∈ [0, 1] x ≤ y if and only if x l α ≤ y l α and x r α ≤ y r α , ∀ t ∈ [ a , b ] x ⪯ y if and only if x l α ≥ y l α and x r α ≤ y r α , ∀ t ∈ [ a , b ]

In this paper we call C ([a, b], E¹) the set of continuous fuzzy functions on [a, b]. It is well known that C ([a, b], E¹) is a complete metric space with respect to the metric D ( x , y ) = sup t ∈ J d ∞ ( x 1 ( t ) , x 2 ( t ) ) , x , y ∈ C ( [ a , b ] , E 1 ) .

Some properties on the partial ordering ≤ and ⪯ of fuzzy functions are presented in [Hoa-Quang-Phu-Tung]. Furthermore, we have

f ⪯ g ⇒ ∫ 0 t f ( s ) ds ⪯ ∫ 0 t g ( s ) ds , for t ∈ J , f , g ∈ C ( J , E 1 ) .

In the sequel, we recall some fixed point theorem in partially ordered space (see [14]) that will be used in next section to analyze the existence of solutions for two kinds of fuzzy Volterra equations.

Theorem 2.1. (see [14]) Let (X, ≤) be a partially ordered set and suppose that there exists a metric d in X such that (X, d) is a complete metric space. Let f : X → X be a monotone non-decreasing mapping such that there exists k ∈ [0, 1) with

d ( f ( x ) , f ( y ) ) ≤ kd ( x , y ) , for all x ≥ y .

Suppose that either f is continuous or X is such that

if a non - decreasing sequence ( x n ) n ∈ ℕ → x in X , then x n ≤ x for all n ∈ ℕ .

If there exists x₀ ∈ X such that x₀ ≤ f (x₀), then f has a fixed point.

Theorem 2.2. (see [14]) Let (X, ≤) be a partially ordered set and suppose that there exists a metric d in X such that (X, d) is a complete metric space. Let f : X → X be a monotone non-decreasing mapping such that there exists k ∈ [0, 1) with d ( f ( x ) , f ( y ) ) ≤ kd ( x , y ) , for all x ≥ y .

Suppose that either f is continuous or X is such that if a non - increasing sequence ( x n ) n ∈ ℕ → x in X , then x n ≥ x for all n ∈ ℕ .

If there exists x₀ ∈ X such that x₀ ≥ f (x₀), then f has a fixed point.

Let (X, ≤) be a partially ordered set, i.e. X is a nonempty set and ≤ is a reflexive, transitive and anti-symmetric relation on X. Denote X ≤ : = { ( x , y ) ∈ X × X | x ≤ y or y ≤ x } .

Theorem 2.3. (see [15]) Let (X, d) be a metric space equipped with a partial ordering ≤. Let f : X → X be an operator. Suppose that

for each x, y ∈ X, there exists z ∈ X such that (x, z) ∈ X_≤ and (y, z) ∈ X_≤, that is, every pair of elements of X has a lower bound or an upper bound.

Then f : X → X is a Picard operator.

Theorem 2.4. (see [14]) Replacing the assumptions (iii) and (iv) in Theorem 2.3 by the following assumptions:

there exists x₀ ∈ X such that x₀ ≥ f (x₀) and

Then, the conclusion of Theorem 2.3 is still valid.

3 Main result

In this section, we consider the following two kinds of fuzzy Volterra integral equation: x ( t ) = f ( t ) + ∫ a t g ( t , s , x ( s ) ) ds(3.1) and x ( t ) = f ( t ) ⊖ ( - 1 ) ∫ a t g ( t , s , x ( s ) ) ds .(3.2) where f : J → E¹ and g : J × J × E¹ → E¹.

Theorem 3.1. Let f : J → E¹ and g : J × J × E¹ → E¹ be continuous and satisfy the following conditions:

(f1) the function g is increasing in the third variable, that is, if x ⪯ y then g (t, s, x) ⪯ g (t, s, y);

(f2) there exists a function φ : ℝ + → ℝ + with φ (λt) ≤ λφ (t), for each t ∈ ℝ + and each λ ≥ 1, such that d ∞ ( g ( t , s , x ) , g ( t , s , y ) ) ≤ φ ( d ∞ ( x , y ) ) , if x ⪯ y , for each (t, s) ∈ J × J, x, y ∈ E¹.

(f3) there exists x₀ ∈ C (J, E¹) such that x 0 ( t ) ⪯ f ( t ) + ∫ a t g ( t , s , x 0 ( s ) ) ds , for any ( t , s ) ∈ J × J .

Then the integral Equation (3.1) has a unique solution on C (J, E¹).

Proof. Let ρ > 0. We consider the space C (J, E¹) equipped with the complete metric D ρ ( x , y ) = sup t ∈ J { d ∞ ( x ( t ) , y ( t ) ) e - ρ ( t - a ) } , x , y ∈ C ( J , E d ) .

It is easy to see that this metric is equivalent to the metric D (u, v), because e - ( b - a ) ρ D ( x , y ) ≤ D ρ ( x , y ) ≤ D ( x , y ) for all x, y ∈ C (J, E¹). Moreover, (C (J, E¹), D _ρ ) is a complete metric space.

Let the operator T : C ( J , E 1 ) → C ( J , E 1 ) be defined by [ T x ] ( t ) = f ( t ) + ∫ a t g ( t , s , x ( s ) ) ds .

Note that x ∈ C (J, E¹) is a fixed point of T if and only if x is a solution of the problem (3.1).

By the assumption (f1), we have [ T x ] ( t ) = f ( t ) + ∫ a t g ( t , s , x ( s ) ) ds ⪯ [ T y ] ( t ) = f ( t ) + ∫ a t g ( t , s , y ( s ) ) ds , whenever x ⪯ y and t ∈ J. Hence, the operator T is increasing.

For each x, y ∈ E¹ with x ⪯ y and using the condition (f2), we have d ∞ ( [ T x ] ( t ) , [ T y ] ( t ) ) = d ∞ ( ∫ a t g ( t , s , x ( s ) ) ds , ∫ a t g ( t , s , y ( s ) ) ds ) ≤ ∫ a t d ∞ ( g ( t , s , x ( s ) ) , g ( t , s , y ( s ) ) ) ds = ∫ a t φ ( d ∞ ( x ( s ) , y ( s ) ) e - ρ ( s - a ) e ρ ( s - a ) ) ds ≤ ∫ a t φ ( d ∞ ( x ( s ) , y ( s ) ) e - ρ ( s - a ) ) e ρ ( s - a ) ds ≤ ∫ a t φ ( D ρ ( x , y ) ) e ρ ( s - a ) ds ≤ e ρ ( t - a ) ρ φ ( D ρ ( x , y ) ) .

Hence, for ρ > 0 we obtain D ρ ( T x , T y ) ≤ 1 ρ φ ( D ρ ( x , y ) ) , for each x, y ∈ E¹ with x ⪯ y.

By the assumption (f3) we have that x 0 ⪯ T x 0 . In this way, since the operator T verifies all conditions of Theorem 2.3, T has a fixed point in C (J, E¹) and it follows that the operator T has a unique fixed point. Thus, x is the unique solution of the problem (3.1). □

Theorem 3.2. Replacing the assumption (f3) in Theorem 3.1 by the following assumption:

(f3)’ there exists x₀ ∈ C (J, E¹) such that x 0 ( t ) ⪰ f ( t ) + ∫ a t g ( t , s , x 0 ( s ) ) ds , for any t , s ∈ J , the conclusion of Theorem 3.1 is still valid.

Proof. The proof of Theorem 3.2 is similar to the proof of Theorem 3.1. □

Theorem 3.3. Let f : J → E¹ and g : J × J × E¹ → E¹ be continuous and satisfy the following conditions: (f1)-(f2) of Theorem 3.1 and

(f3) for any α ∈ [0, 1] and each (t, s) ∈ J × J, diam ( [ f ( t ) ] α ) ≥ ∫ a t diam ( [ g ( t , s , x ( s ) ) ] α ) ds , f l α ( t ) + ∫ a t g r α ( t , s , x ( s ) ) is monotonically increasing in α, for each (t, s) ∈ J × J fixed, f r α ( t ) + ∫ a t g l α ( t , s , x ( s ) ) is monotonically decreasing in α, for each (t, s) ∈ J × J fixed.

(f4) there exists x₀ : J → E¹ such that x 0 ( t ) ⪯ f ( t ) ⊖ ( - 1 ) ∫ a t g ( t , s , x 0 ( s ) ) ds , for any ( t , s ) ∈ J × J .

Then the integral Equation (3.2) has a unique solution on C (J, E¹).

Proof. Notice that the assumption (f3) guarantees the existence of f ( t ) ⊖ ( - 1 ) ∫ a t g ( t , s , x ( s ) ) ds , for any t, s ∈ J.

Now, let the operator Q : C ( J , E 1 ) → C ( J , E 1 ) be defined by [ Q x ] ( t ) = f ( t ) ⊖ ( - 1 ) ∫ a t g ( t , s , x ( s ) ) ds .

If x ∈ C (J, E¹) is a fixed point of Q , then x is a solution of the integral Equation (3.2) and conversely.

The operator Q is increasing. Indeed, by the assumption (f1), we have [ Q x ] ( t ) = f ( t ) ⊖ ( - 1 ) ∫ a t g ( t , s , x ( s ) ) ds ⪯ [ Q y ] ( t ) = f ( t ) ⊖ ( - 1 ) ∫ a t g ( t , s , y ( s ) ) ds , whenever x ⪯ y and t ∈ J.

Besides, for each x, y ∈ E¹ with x ⪯ y, we have D ρ ( Q x , Q y ) = sup t ∈ J { d ∞ ( [ T x ] ( t ) , [ T y ] ( t ) ) e - ρ ( t - a ) } = sup t ∈ J { d ∞ ( ∫ a t g ( t , s , x ( s ) ) ds , ∫ a t g ( t , s , y ( s ) ) ds ) e - ρ ( t - a ) } ≤ sup t ∈ J { ∫ a t d ∞ ( g ( t , s , x ( s ) ) , g ( t , s , y ( s ) ) ) ds e - ρ ( t - a ) } = sup t ∈ J { ∫ a t φ ( d ∞ ( x ( s ) , y ( s ) ) e - ρ ( s - a ) e ρ ( s - a ) ) ds e - ρ ( t - a ) } ≤ sup t ∈ J { ∫ a t φ ( D ρ ( x , y ) ) e ρ ( s - a ) ds e - ρ ( t - a ) } ≤ 1 - e - ρ ( b - a ) ρ φ ( D ρ ( x , y ) ) .

Since ρ > 0, we obtain D ρ ( Q x , Q y ) ≤ φ ( D ρ ( x , y ) ) , for each x, y ∈ E¹ with x ⪯ y.

Using the assumption (f4), we have that x 0 ⪯ Q x 0 . In this way, as the operator Q verifies all conditions of Theorem 2.3, Q has a fixed point in C (J, E¹) and it follows that the operator Q has a unique fixed point. Therefore, x is the unique solution of the integral Equation (3.1). □

Theorem 3.4. Replacing the assumption (f4) in Theorem 3.3 by the following assumption:

(f4)’ there exists x₀ ∈ C (J, E¹) such that x 0 ( t ) ≽ f ( t ) ⊖ ( - 1 ) ∫ a t g ( t , s , x 0 ( s ) ) ds , for any t , s ∈ J . the conclusion of Theorem 3.3 is still valid.

Proof. The proof of Theorem 3.4 is similar to the proof of Theorem 3.3. □

In this section, we present some examples which indicate how our theorems can be applied to concrete problems.

Example 4.1. Consider the following fuzzy Volterra integral equation:

x ( t ) = ( e t , 2 e t , 3 e t ) + ∫ 0 t 2 cos ( t ) x ( s ) ds , ∀ t ∈ [ 0 , π / 2 ] .(4.3)

It is easy to check that the fuzzy integral Equation (4.3) satisfies all the conditions of Theorem 3.1 in [18]. Therefore, the problem has a unique solution on ℝ + . In the following, we shall show that the assertions of Theorem 3.1 are satisfied. Consider the function g : [0, π/2) × [0, π/2) × E¹ → E¹ given by g (t, s, x) =2 cos(t) x (s). We see that the function g is continuous and increasing in the third variable, i.e. if (t, s) ∈ [0, π/2) × [0, π/2) and x ⪯ y, then we have g ( t , s , x ) = 2 cos ( t ) x ( s ) ⪯ 2 cos ( t ) y ( s ) = g ( t , s , y ) .

Let the operator T : C ( [ 0 , π / 2 ) , E 1 ) → C ( [ 0 , π / 2 ) , E 1 ) be defined by [ T x ] ( t ) = ( e t , 2 e t , 3 e t ) + ∫ 0 t 2 cos ( t ) x ( s ) ds .(4.4)

For x ⪯ y, we have D ρ ( T x , T y ) = sup t ∈ [ 0 , π / 2 ) { d ∞ ( ∫ 0 t 2 cos ( t ) x ( s ) ds , ∫ 0 t 2 cos ( t ) x ( s ) ds ) e - ρ t } ≤ sup t ∈ [ 0 , π / 2 ) { ∫ 0 t φ ( 2 cos ( t ) e ρ s D ρ ( x , y ) ) ds e - ρ t } ≤ 2 1 - e - π ρ / 2 ρ φ ( D ρ ( x , y ) ) .(4.5)

It follows that from (4.5) there exists φ : ℝ + → ℝ + with φ ( t ) = 2 e π ρ / 2 ρ t , for ρ > 0, such that D ρ ( T x , T y ) ≤ 2 e π ρ / 2 ρ φ ( D ρ ( x , y ) ) if x ⪯ y .

The assumption (f3) and (4.3a), we obtain x 0 ⪯ T x 0 . Hence, by using Theorem (3.1) the existence of a solution for the fuzzy Volterra integral Equation (4.3) provides the existence of a unique solution.

Example 4.2. Consider the following fuzzy Volterra integral equation: x ( t ) = f ( t ) ⊖ ( - 1 ) ∫ 0 t g ( t , s , x ( s ) ) ds , t ∈ [ 0 , 1 ] ,(4.6) where f (t) = (1 - t, 2 - t, 3 - t) ∈ E¹ and a continuous function g : [0, 1] × [0, 1] × B (f (0), 3) → E¹ defined as follows g ( t , s , x ( s ) ) = 1 1 + s 2 x ( s ) , for every (t, s, x) ∈ [0, 1] × [0, 1] × B ((f (0)), 3), where B (z, 3) = {x ∈ E¹ | d_∞ (z, x) ≤3} .

In this problem, we can see that the right side of (4.6) satisfies condition (f3). Indeed, for x ∈ E¹ and α ∈ [0, 1], we have diam ( [ g ( t , s , x ( s ) ) ] α ) = 1 1 + s 2 diam ( [ x ( s ) ] α ) ≤ diam ( [ x ( s ) ] α ) ≤ 6 .

It is been that diam ( [ f ( t ) ] α ) = 2 ( 1 - α ) ≥ ∫ 0 t diam ( [ g ( t , s , x ( s ) ) ] α ) ds = 6 t , for α ∈ [0, 1] and for t, s ∈ [0, t^*], where t^* = min {1, 1/3 (1 - α)}. Therefore, the Hukuhara difference ( 1 - t , 2 - t , 3 - t ) ⊖ ( - 1 ) ∫ 0 t 1 1 + s 2 x ( s ) ds exists in the case t, s ∈ [0, t^*].

In the sequel, we define an operator Q : C ( [ 0 , t * ] , E 1 ) → C ( [ 0 , t * ] , E 1 ) as follows: [ Q x ] ( t ) = ( 1 - t , 2 - t , 3 - t ) ⊖ ( - 1 ) ∫ 0 t 1 1 + s 2 x ( s ) ds .

Similar to Example 4.1, we can prove that the problem (4.6) satisfies the conditions (f1), (f2), (f4). Hence, using the result of Theorem 3.3, the problem (4.6) has a unique solution.

Example 4.3. We consider the following fuzzy Volterra integral equation: x ( t ) = f ( t ) + ∫ 0 t g ( t , s , x ( s ) ) ds , t ∈ [ 0 , 1 ] ,(4.7) where g ( t , s , x ( s ) ) = 1 1 + t tan - 1 x ( s ) , and [ f ( t ) ] α = ( 0.5 e - t - 0.2 , 0.5 e - t , 0.5 e - t + 0.2 ) .

For (t, s) ∈ [0, 1] × [0, 1] and x ⪯ y, we have g ( t , s , x ( s ) ) = 1 1 + t tan - 1 x ( s ) ⪯ g ( t , s , y ( s ) ) = 1 1 + t tan - 1 y ( s ) .

The operator T : C ( [ 0 , 1 ] , E 1 ) → C ( [ 0 , 1 ] , E 1 ) defined by [ T x ] ( t ) = f ( t ) + ∫ 0 t g ( t , s , x ( s ) ) ds .(4.8)

For x ⪯ y and t, s ∈ [0, 1], c ∈ (0, 1), we have D ρ ( T x , T y ) = sup t ∈ [ 0 , 1 ] { d ∞ ( ∫ 0 t 1 1 + t tan - 1 x ( s ) ds , ∫ 0 t 1 1 + t tan - 1 y ( s ) ds ) e - ρ t } ≤ sup t ∈ [ 0 , 1 ] { ∫ 0 t φ ( 1 1 + t d ∞ ( tan - 1 x ( s ) , tan - 1 y ( s ) ) e - ρ s e ρ s ) ds e - ρ t } ≤ sup t ∈ [ 0 , 1 ] { ∫ 0 t φ ( 1 ( 1 + c 2 ) ( 1 + t ) d ∞ ( x ( s ) , y ( s ) ) e - ρ s e ρ s ) ds e - ρ t } ≤ sup t ∈ [ 0 , 1 ] { 1 ( 1 + c 2 ) ( 1 + t ) φ ( D ρ ( x , y ) ) ∫ 0 t e ρ s ds e - ρ t } ≤ 1 1 + c 2 sup t ∈ [ 0 , 1 ] { 1 - e - ρ t ρ } φ ( D ρ ( x , y ) ) ≤ 1 ρ φ ( D ρ ( x , y ) ) .

Therefore, there exists φ : ℝ + → ℝ + with φ ( t ) = 1 ρ t for ρ > 0, such that D ρ ( T x , T y ) ≤ 1 ρ φ ( D ρ ( x , y ) ) if x ⪯ y .

By using the assumption (f3) and (4.8), we get x 0 ⪯ T x 0 . Since all conditions stated in Theorem 3.1 are satisfied, the problem (4.7) has a unique solution.

In this paper, we use some recent results of fixed point contractive mappings on the partially ordered space to investigate the existence and uniqueness of solution for two kinds of fuzzy Volterra integral equation. In the future works, we will apply this method for fuzzy functional integral equations.