Ulam-Hyers Stability and Ulam-Hyers-Rassias Stability for Fuzzy Integrodifferential Equation
In this paper, we establish the Ulam-Hyers stability and Ulam-Hyers-Rassias stability for fuzzy integrodifferential equations by using the fixed point method and the successive approximation method.
Fuzzy differential and integrodifferential equation (FD-FIDE) are a natural way to model dynamical systems subject to uncertainties. In the past few years, the study of fuzzy integrodifferential equations is an area of mathematics that has recently received a lot of attention (see, e.g., [1–11]). Alikhani et al.  studied the existence and uniqueness of global solutions for fuzzy initial value problems via integrodifferential operators of Volterra type. Alikhani et al.  introduced a new concept of upper and lower solutions for FIDE. Using this concept, authors proved the existence and uniqueness of global solutions of an initial value problem for first-order nonlinear FIDE under generalized differentiability. Additionally, authors  studied the two kinds of solutions to FIDE with delay by the usage of two different concepts of fuzzy derivative.
In recent years, accompanied by the development of the Ulam-Hyers stability of differential equation (see, e.g., [12–18]). Ulam-Hyers stability of fuzzy differential equation has attracted the attention of reseachers; the reader is refferred to [19–22]. Ren  studied the Ulam-Hyers stability of the Hermite fuzzy differential equation associated with the inhomogeneous Hermite fuzzy differential equation under some suitable conditions. However, the fixed point method has been successfully used to study the Ulam stability of fuzzy differential equation by [19–21]. In [19, 20], Shen considered the Ulam stability of the first order linear (partial) fuzzy differential equation under generalized differentiability. Using the fixed point technique, authors  studied the Ulam stability of fuzzy differential equations and we see that Ulam stability of this problem requires various prerequisites under different types of differentiability.
To the best of our knowledge, up to now, the number of papers dealing with Ulam stability for FD-FIDE is rather limited as opposed to the amount of publications concerning FD-FIDE. From this reason, we choose the study of Ulam stability of FIDE by using the fixed point technique and the method of successive approximation.
The rest of this paper is organized as follows. In Section 2, some notations and preparation results are given. In Section 3, Ulam-Hyers stability and Ulam-Hyers-Rassias stability criteria for FIDE defined on a bounded and closed interval are obtained in terms of the method of successive approximation and the fixed point theorem in .
In this section, we introduce some basic definitions, theorems, and lemmas, which are required throughout this paper.
Definition 1 (see ). A function is called a generalized metric on if and only if satisfies (1) if and only if ;(2) for all ;(3) for all .
Theorem 2 (see ). Let be a generalized metric on and is a generalized complete metric space. Assume that is a strictly contractive operator with the Lipschitz constant . If there exists a nonnegative integer such that for some , then the following are true: (i)the sequence converges to a fixed point of ;(ii) is the unique fixed point of in (iii)if , then we have
Lemma 3. Let be a continuous function. We define the set equipped with the metric Then, is a complete generalized metric space.
Proof. The proof of this lemma can be found in Shen and Wang .
Denote by the class of fuzzy sets with the following properties: (i) is normal, i.e., there exists such that ; (ii) is fuzzy convex; that is, for any , and ; (iii) is upper semi-continuous; (iv) is compact, where denotes the closure of a set.
Usually, the set is called the space of fuzzy numbers and it is easy to see that . For , we denote and . Then it follows from the conditions (i)-(iv) that the -level set is a non-empty compact interval for all and each . For any and , the addition and scalar multiplication can be defined, levelwise, by and for all .
The supremum metric between and is defined by It is easy to see that is a complete metric space. It is well known that the supremum metric has the following properties: (D1) for any ; (D2) for any , ; (D3) for any .
Definition 4 (see ). Let . If there exists such that , then is called the H-difference of and , and it is denoted by .
Throughout this paper, the symbol “” always stands for the H-difference. In general, .
Definition 5 (see ). Let and . We say is generalized differential at , if there exists an element , such that (1)for all sufficiently small, there exists , and then limits (in metric ) (2)for all sufficiently small, there exists , and then limits (in metric )
Note that Bede and Gal  considered four cases in the definition of derivative. In this paper, we consider only the two first cases of Definition 5 in . In the other cases, the derivative reduces to a crisp element; that is, .
Theorem 6 (see ). Let and denote for each , . (i)If is (1)-differentiable at all , then and are differentiable functions and we have (ii)If is (2)-differentiable at all , then and are differentiable functions and we have
In Theorem 6, we see that if is (1)-differentiable, then it is not (2)-differentiable and vice versa.
Theorem 7 (see ). Let be differentiable on and assume that derivative is integrable over . For each , we have (i)If is (1)-differentiable, then (ii)If is (2)-differentiable, then
Let (with ) be a compact interval of . We denote by On the space , we consider the supremum metric as follows: In next section, we consider the following fuzzy integrodifferential equation:where the symbol is generalized Hukuhara derivative, the mapping is continuous on , and is a continuous function on .
Lemma 9 (see ). Let be a continuous function on . Problem (14) is equivalent to one of the following the fuzzy integro integral equations: (S1)if is (1)-differentiable on , then (S2)if is (2)-differentiable on , then
3. Main Results
Definition 10. We say that problem (14) is Ulam-Hyers stable if there exists a real number such that for and for each to the problem there exists a solution to problem (14) with for all . We call a Ulam-Hyers stability constant of (14).
Firstly, we prove that problem (14) is Hyers-Ulam stable via the method of successive approximation. We consider the following inequality:
Definition 12. We say that(a)A function is a (S1)-solution of the inequality (21) if and only if there exists a function such that(i) for any ;(ii) for any .(b)A function is a (S2)-solution of the inequality (21) if and only if there exists a function such that(i) for any ;(ii) for any .
Theorem 13. Assume that and are a continuous function that satisfies the following conditions: (i) there exists a constant such that for any , ; (ii) for each , if a continuously (2)-differentiable function satisfiesthen there exists a (S2)-solution of (14) with such thatwhere is a maximum balancing constant.
Proof. For each , let a continuously (2)-differentiable function satisfy inequality (21) for any , . By the part (b) of Definition 12, we have andfor any and .
If a function is continuous and (2)-differentiable on , then by Lemma 9 it satisfies equivalently the following fuzzy integrointegral equation: Let us define and sequence functions , of successive approximations as followsBy virtue of the properties of and assumption (i), we have Therefore, Observe that for and for , one has In particular, and so and for we havewhere are balancing constants.
We choose (called is a maximum balancing constant) and then estimation (35) can be rewritten as follows: Further, if we assume thatthen we obtain By the principle of mathematical induction that (37) holds for every and now using estimation (37) for any , we get The series is convergent for every . Hence, for every we infer that the series is uniformly convergent on with respect to metric and For , we haveand the following estimationCombining estimation (41) and inequality (42), we obtain that converges to 0 uniformly as . Therefore, is a (S2)-solution of (14) with initial condition .
Finally, we shall prove that problem (14) has a unique (S2)-solution. Assume that is another (S2)-solution of (14) with initial condition . Then, we have for any If we let for any , then Applying Lemma 2.3 in Hoa et al. , we obtain for any . This completes the proof.
Example 14. Consider the following fuzzy integro differential equation:where , and .
Let It is easy to see that satisfy Lipschitz condition with Lipschitz constant . Indeed, for any and Hence, by Theorem 15, the problem (46)-(47) has unique (S1)-solution or (S2)-solution on .
Moreover, if a continuously (2)-differentiable function satisfies the following inequation: then as shown in Theorem 15, there exists a (S2)-solution of (46) such that where is the maximum balancing constant.
Secondly, we shall prove the Ulam-Hyers-Rassias stability of the FIDE (14) defined on a bounded and closed interval.
Theorem 15. Assume that and are continuous function satisfying the following conditions: (i) there exists a constant such that for each ; (ii) there exists a constant such that . Let be a continuous function and increasing on withIf a continuously (1)-differentiable function satisfies the following inequalityfor any , then there exists a unique (S1)-solution of (14) such thatandfor any .
Proof. Let us define a set of all continuous fuzzy functions by equipped with the metric It is easy to see that is also a complete generalized metric space (see Lemma 3).
The operator is defined as follows:Based on Lemma 3.2 and 3.3 in , we infer that is (1)-differentiable and so .
The operator is strict contractive on . Indeed, for any and letting be an arbitrary constant with , that is, by the definition of the metric , we haveFrom the definition of the metric and assumptions (i)-(ii) of Theorem 15 and inequality (59), we infer that for any .
For each and by the definition of metric , we get for any . Hence, by (59), we can conclude that for any .
It follows form the definitions of and the operator that, for arbitrary , there exists a constant such that for any , since , , and are bounded on , and . Thus, by definition of it is implied that Therefore, according to Theorem 2, there exists a continuous function on such that as in the space and ; that is, satisfies (58) for each .
Observe that Indeed, for any , there exists a constant such that since and are bounded on and . It follows from the preceding inequality that for all . Hence, we obtained that .
From Theorem 2, we infer that is a unique fixed point of in . It is obvious that is a unique fuzzy function in which satisfies equation .
On the other hand, we have for any . This means thatFinally, by Theorem 2 and inequation (69), we deduce that for any , which means that inequality (55) is true for all .
Corollary 16. Assume that and satisfy the assumption (i) of Theorem 15 and . For a given , if a continuously (1)-differentiable function satisfies the following inequality for any , then there exists a unique (S1)-solution of (14) such that and for any .
Theorem 17. Suppose the functions and satisfy all conditions as in Theorem 15. If a continuously (2)-differentiable function satisfies inequality (53) in Theorem 15 for any , then there exists a unique (S2)-solution of (14) such thatand the estimation (55) as in Theorem 15 on .
Proof. Similar to the proof of Theorem 15. Consider the operator defined byfor all . Based on Lemmas 3.2 and 3.3 in , it is easy to see that is (2)-differentiable and so .
We check the operator is strict contractive on . Let and let be an arbitrary constant with ; we haveFrom the assumptions (i)-(ii) in Theorem 15 and (76), we get for any . Hence, by (76), we conclude that