Random Fuzzy Differential Equations with Impulses

Abstract

We consider the random fuzzy differential equations (RFDEs) with impulses. Using Picard method of successive approximations, we shall prove the existence and uniqueness of solutions to RFDEs with impulses under suitable conditions. Some of the properties of solution of RFDEs with impulses are studied. Finally, an example is presented to illustrate the results.

1. Introduction

Impulsive differential equations (IDEs) are a new branch of differential equations. IDEs can find numerous applications in different branches of optimal control, electronics, economics, physics, chemistry, and biological sciences. We refer to [1–4] and the references therein. As we know, the real systems are often faced with two kinds of uncertainties (fuzziness and randomness). Therefore, this topic has extensively been studied by mathematicians in recent years. Investigations of dynamic systems with fuzziness have been developed in connection with fuzzy differential equations (FDEs). Evidence of FDEs for such areas as control theory, differential inclusions, and fuzzy differential equations can be found in the papers of [5–8], the books and monographs [9], and references therein. In [10], Lakshmikantham and McRae combined the theories of impulsive differential equations and fuzzy differential equations. There are a few papers on the latter topic; see [10–12].

Moreover, the class of random fuzzy differential equations (RFDEs) could be applicable in the investigation of numerous engineering and economics problems where the phenomena are simultaneously subjected to two kinds of uncertainties, that is, fuzziness and randomness, simultaneously (see, e.g., Malinowski [13–16], Feng [17, 18], and Fei [19, 20]). Feng [17] introduced the concepts by the mean-square derivative and mean-square integral of second-order fuzzy stochastic processes. Using the results, the author [18] investigated the properties of solutions of the fuzzy stochastic differential systems, including the existence and uniqueness of solution, the dependence of the solution of the initial condition, and the continuity and the boundedness of solution of systems when there are perturbations of the coefficients and the initial conditions. In [19, 20], Fei proved the existence and uniqueness of solution of fuzzy random differential equation (FRDE). The author also discussed the dependence of solution to FRDE on initial values. Finally, the nonconfluence property of the solution for FRDE is studied.

In [13], Malinowski considered the following random fuzzy differential equations:where and the symbol denotes the fuzzy Hukuhara derivative. The author proved the existence and uniqueness of the solution for RFDEs under Lipschitz condition. Malinowski [14, 15] studied two kinds of solutions to the RFDEs with two kinds of fuzzy derivatives. For both cases the author established the existence and uniqueness of local solutions to RFDEs. In addition, the author also presented some examples being simple illustrations of the theory of RFDEs.

Inspired and motivated by Fei [19], Feng [18], Malinowski [14], and other authors as in [3, 10, 21], in this paper, we consider the RFDEs with impulses under Hukuhara derivative. The paper is organized as follows: in Section 2, we summarize some preliminary facts and properties of the fuzzy set space, fuzzy differentiation, and integration. We also recall the notions of fuzzy random variable and fuzzy stochastic process. In Section 3, we discuss the RFDEs with impulses. Under suitable conditions, we prove the existence and uniqueness of solutions to RFDEs with impulses. In Section 4, we give some examples to illustrate these results.

2. Preliminaries

In this section, we give some definitions and properties and introduce the necessary notation which will be used throughout the paper. We denote , where(i) is normal; that is, there exists an such that ;(ii) is fuzzy convex; that is, for , , for any ;(iii) is upper semicontinuous;(iv) is compact set.Then is called the space of fuzzy numbers.

For , we denote and . For and from conditions (i)–(iv), we infer that the -level cut of , denoted by , is a bounded closed interval for any and , and , where and are the lower and upper branches of .

For , the Hausdorff distance between and is defined byand is a complete metric space.

If we define by the expressionthen it is well-known that is metric in and is also a complete metric space.

Some properties are well-known for the metric Hausdorff defined on as follows:for every and .

Definition 1 (see [22]). Let . If there exists such that , then is called the Hukuhara difference of and it is denoted by .

Definition 2 (see [22]). Let and . We say that is differentiable at if there exists an element such that the limitsexist and are equal to .

Definition 3 (see [22]). Let . The integral of on , denoted by , is defined levelwise by the equationfor all .

Definition 4 (see [22]). A fuzzy mapping is integrable if is integrable bounded and strongly measurable.

The following are some properties of integrability of fuzzy mapping (see [22]):(a)If is continuous then it is integrable.(b)If is integrable and then .(c)Let be integrable and . Then(i),(ii),(iii) is integrable and .

Let be a complete probability space. A function is called a fuzzy random variable, if the set-valued mapping is a measurable multifunction for all ; that is,for every closed set .

Definition 5 (see [13]). A mapping is said to be a fuzzy stochastic process if is a fuzzy-set-valued function with any fixed and is a fuzzy random variable for any fixed .

Definition 6 (see [13]). A fuzzy stochastic process is called continuous if there exists with and such that, for every , the trajectory is a continuous function on with respect to the metric .

For convenience, from now on, we shall write to replace for short, where , are random elements, and similarly for inequalities. Also we shall write to replace for short, where , are some stochastic processes, and similarly for inequalities.

3. Existence and Uniqueness for RFDEs with Impulses

In this section, we consider the following random fuzzy differential equation with impulses:where , is continuous with , and , , are points of impulses such that and is fuzzy random variable.

Lemma 7. Let be a fuzzy stochastic process. Then is the solution of problem (8) if and only if is a continuous fuzzy stochastic process and satisfy the following random impulsive fuzzy integral equation:

Proof. We divide the proof into two steps.
Step  1. If satisfies problem (8), then it will be expressed as (9). Indeed, for every we haveBy Lemma  3.1 in [13], we obtain If and by Lemma  3.1 in [13], we haveIf we assume that then we have It follows by mathematical induction that (13) holds for any .
Step  2. Conversely, if a fuzzy stochastic process satisfies the random fuzzy integral equation (9), then it is equivalent to problem (8). Indeed, if we easily see that and the Hukuhara difference exists, with . By Lemma  3.2 in [13] we haveLet small enough such that for every ; we have Similarly, let small enough such that for every ; we obtainMultiplying both sides of (16) and (17) by and passing to the limit with , we obtain This allows us to claim that is differentiable on and consequently By mathematical induction, if , , we get Also, we can easily show thatThe proof is complete.

Lemma 8. Let be a probability space. Let , , , and stochastic processes be such that(i) is nonnegative and continuous with and are the points of discontinuity of the first of with ,(ii) is locally Lebesgue integrable with .Ifthen we have

Now, we show the main results of this paper.

Theorem 9. Let the mapping be continuous with and . Assume the following conditions hold:(A1)There exists a nonnegative constant such that , for every and with .(A2)There exists a nonnegative constant such that , for , for every and with .(A3)There exists a nonnegative constant such that , for every and with .Then the random fuzzy differential equation with impulses (9) has a unique solution, provided that

Proof. Define a sequence of the functions , as follows: for every let us putFor every and , we have it follows that . Furthermore, by assumptions (A1)-(A2) and (25), we can find thatwhich implies thatNow, we need to prove that for all with the following inequality holds: for any ,Indeed, inequality (29) holds for . Further, if inequality (29) is true for any , then using (25) and assumptions (A1)-(A2), we haveThus, inequality (29) is true for every with .
Next, we see that does not depend on and for the right-side continuity of , one obtainsFrom the assumption (A3) and as with , we imply that as with .
For every , we deduce thatUsing inequality (29) and assumption (A3), we getSimilar for the left-side continuity, we have   . Hence the functions , , are continuous with .
For and the function defined by (25) is fuzzy random variable. Indeed, is measurable multifunction for every ; it remains to show the same for the mapping which is a measurable multifunction with every , , and . Let be fixed. By virtue of the definition of fuzzy integral and theorem of Nguyen [23] we obtainAs the integrand is a multifunction continuous in and measurable in , with any , the mapping is a measurable multifunction for . Therefore, for every , the sequence is a sequence of fuzzy random variable. Consequently, is a sequence of fuzzy stochastic process.
In the sequel, for any , we shall prove that the sequence is a Cauchy sequence uniformly on the variable with and then is uniformly convergent with .
For any and by inequality (29), we obtainNotice now that, for every , we haveFor large enough, it follows from the above inequalities with thatSince is a complete metric space and (38) holds, then , which means that there exists such that and for every the sequence is uniformly convergent.
In the following, we shall show that is solution of the random impulsive fuzzy integral equation (8). Let . Observe thatSince the sequence converges uniformly to on the variable with as , Thus for any there is large enough such that, for all , we derive Therefore,On the other hand, we have Thus, in view of the convergence of the two previous equations and (41), one obtains thatIt means the fuzzy stochastic process is solution of problem (8).
To prove the uniqueness, let us assume that are the two continuous fuzzy stochastic processes which are solutions of problem (8). Note that By Lemma 8, we getThe uniqueness is proved. The proof is complete.

4. Some of the Properties of Solution of RFDEs with Impulses

Theorem 10. Suppose that the mappings and satisfy all the conditions of Theorem 9. Then we havewhere , are constants nonnegative for any .

Proof. Let be solution of problem (8). For every and , we haveFor every , , and , we have If we let , , and , then we haveBy virtue of Lemma 8, one obtains The proof is complete.

Theorem 11. Suppose that the mappings and satisfy all the conditions of Theorem 9. Then we havewhere , are constants nonnegative for any .

Proof. Let and be solutions of problem (8). For every and , we haveFor every , , and , we have If we let , , and , then we haveBy virtue of Lemma 8, one obtains The proof is complete.

5. Illustrative Examples

In this section, we shall consider two examples. First, we give an example to illustrate the existence and uniqueness results obtained in Section 3. Second, we will find explicit representation of solutions RFDEs with impulses.

Example 1. Let , -Borel -algebra of subsets of , and -Lebesgue measure on . Let us consider the problem as follows:where is a fuzzy stochastic process.
SetFor every , , , we have where .
Using a similar calculation as above, for every , , , we obtainBy a direct calculation, one obtains thatand for any ,

We can see that conditions (A1)–(A4) are satisfied. Hence, by Theorem 9, problem (56) has a solution defined on .

Example 2. Let , -Borel -algebra of subsets of , and -Lebesgue measure on . Consider the RFDEs with impulses as follows:where is a random variable and is a fuzzy stochastic process. In this example, we suppose that and with and for every and initial conditions , where are the crisp stochastic process.
Problem (62) can translate this into the following system of random differential equation with impulses:Solving system (64) on , we obtain It is easy to see that the diameter of solution of (62) is an increasing function with for every . Hence we infer that the solution to (62) is as follows: orNote that the existence of a unique solution is guaranteed. Therefore, this procedure can be continued to be the solution on each , for every , .

6. Conclusion

Under suitable conditions, we investigated the existence and uniqueness of solutions to random fuzzy differential equation with impulses by using the method of successive approximations. Moreover, we studied some of the properties of solution of RFDEs with impulses. Finally, some examples are given to illustrate the main theorems. In the future, we shall study the class of random fuzzy differential equations in the quotient space of fuzzy numbers, introduced by Qiu et al. in [24].

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.