Hyers-Ulam Stability and Existence Criteria for the Solution of Second-Order Fuzzy Differential Equations

Abstract

In this paper, existence, uniqueness, and Hyers-Ulam stability for the solution of second-order fuzzy differential equations (FDEs) are studied. To deal a physical model, it is required to insure whether unique solution of the model exists. The natural transform has the speciality to converge to both Laplace and Sumudu transforms only by changing the variables. Therefore, this method plays the rule of checker on the Laplace and Sumudu transforms. We use natural transform to obtain the solution of the proposed FDEs. As applications of the established results, some nontrivial examples are provided to show the authenticity of the presented work.

1. Introduction

Zadeh [1] introduced the concept of fuzziness in the set theory. The complexity of uncertainty as ambiguity in a real-life scenario is dealt properly with fuzzy set theory. The mathematical tool of fuzzy set theory deals with uncertainty in the real-life problem in a better way. The fuzzy set theory has suggestions for nonclassical and higher order fuzzy sets for different specialized purposes. In this direction, Chang and Zadeh used fuzzy sets and initiate the concept of fuzzy mapping and control [2]. The work of many researchers on fuzzy mappings and control puts the foundation of elementary fuzzy calculus. For detail, see [3–7]. For the last two decades, the valuable interest of fuzzy integral and differential equations has extended classical calculus to modern fuzzy calculus. Fuzzy differential and integral equations are the well-equipped mathematical tools to deal properly with physical models in the fuzzy environments. The solutions of every fuzzy differential and integral equation do not exist. Therefore, some strategy is required to insure whether the solution of FDEs exists or not. The existence theory is one of the best research areas in the field of fuzzy differential equations (FDEs). Before dealing a physical model, it is important to know whether its solution exists. Now, if unique solution of a physical model exists, then a physical model is dealt properly. The existence of a unique solution of fuzzy differential equation properties of differentiable fuzzy mappings was studied by Kaleva [5]. Liu and Liu [8] introduced self-duality credibility measure for the measurement of fuzzy event. Moreover, they studied the existence of unique solutions of FDEs. The existence and uniqueness result for FDEs with linear growth and Lipschitz conditions was discussed by Fei et al. [9]. The uniqueness result for the FDEs with non-Lipschitz coefficients was investigated by Chen and Qin [10] for more detail (see [6, 11, 12]).

Stability analysis of differential equations (DEs) is another most important and remarkable area in the qualitative theory. The stability of DEs has been studied by various researchers with different concepts like Lyapunov stability, asymptotic stability, and Ulam stability. The first effort of Ulam stability was initiated by Ulam [13], and just after one year, Hyers [14] studied the stability of the linear functional equation known as Hyers-Ulam stability. Oblaza [15] proved the Hyers-Ulam stability of linear DEs; for further detail of Hyers-Ulam, see [16–19]. Shen [20] investigated the Ulam stability of first-order linear FDEs.

The Laplace and Sumudu transforms are commonly used for the solutions of differential equations. The natural transform introduced by Khan and Khan [21] has speciality to converge to both Laplace and Sumudu transforms only by changing the variables. Therefore, the natural transform method plays the rule of checker on the Laplace and Sumudu transforms. The applications of the natural transform method turn out to be well for solutions of differential equations, see [21–23].

The aim of this work is to study the existence, uniqueness, and Hyers-Ulam stability of second-order FDEs. For this purpose, the corresponding second-order FDEs are reduced to equivalent systems of fuzzy integral equations. Using the concept of Hukuhara generalized differentiability, existence, uniqueness, and Hyers-Ulam stability of the equivalent system of integral equations are discussed. We use the natural transform method to solve second-order FDEs. The last Hyers-Ulam stability of the numerical problem is discussed. Two nontrivial examples are given to show the authenticity of the presented work.

2. Preliminaries

Here, some basic results are provided from the existing literature.

Definition 1 (see [24]). Let satisfy the conditions, where (set of real numbers) (i) is upper semicontinuous(ii)(iii) such that (iv) is compactThen, is a fuzzy number.
Throughout this paper, represent the set that contains all fuzzy numbers.

Definition 2 (see [25]). The fuzzy number can be written as in order pair form, with , and holds the following conditions: (i)Nondecreasing bounded function, is left-continuous in (ii)Nonincreasing bounded function, is left-continuous in (iii) is a crisp nunmber when .

Theorem 3. Let the mapping is contractive with , where is a generalized complete metric space. If for , with and , then (i) converge to (ii)The point is in (iii)If , then

Definition 4. The mapping is a generalized metric on if and only if, and .
(F₁) if and only if
(F₂)
(F₃) . Then, is a metric space

Definition 5 (see [26]). The mapping , defined by The pair is a generalized complete metric space. Moreover, and hold
(D₁)
(D₂)
(D₃)

Definition 6 (see [27]). The fuzzy function, , at , is generalized Hukuhara differentiable if there exist such that (i)For , sufficiently small, the Hukuhara difference, , , and limits exist in the complete metric space (ii)For , sufficiently small, the Hukuhara difference, , , and limits exist in the complete metric space The first one is referred to (i)-differentiable and second one to (ii)-differentiable.

Definition 7 (see [27]). The fuzzy function, , at , is generalized Hukuhara second-order differentiable if there exist, such that (i)For , sufficiently small, the Hukuhara difference, , , and limits exist in the complete metric space (ii)For , sufficiently small, the Hukuhara difference, , , and limits exist in the complete metric space The first one is referred to (i)-differentiable and second one to (ii)-differentiable.

Definition 8 (see [28]). The crisp set is the -level set of fuzzy number . Moreover, the -level set is bounded and closed with upper and lower bond, and , respectively, denoted by .

Theorem 9 (see [29]). Let a continuous fuzzy function , such that, , and : (i) is (i)-differentiable, then (ii) is (ii)-differentiable, then

Corollary 10 (see [26]). The continuous mapping is integrable, where is an interval.

Theorem 11 (see [27]). The generalized differentiable mapping has integrable derivative on . (i)When is (i)-differentiable, , we have(ii)When is (ii)-differentiable, , we have

Definition 12 (see [21]). The natural transform of is given by the following formula: where and are transform parameter which is real and positive.

Lemma 13 (see [22]). The duality relation of natural and Laplace transform is given by where is the Laplace transform. The natural transform converts to Laplace transform by taking parameter and Sumudu transform by taking parameter .

Definition 14 (see [23]). The natural transform of th order derivative of is The natural transform of and first- and second-order derivative of is given by

3. Existence, Uniqueness, and Stability Analysis

Before dealing a physical model, it is important to know whether its solution exists. Now, if unique solution of a physical model exists, then the model is dealt properly. But if the solution is not unique then who is to say that the solution we found is what will actually happen in real life. Therefore, we need additional conditions to get unique solution. In this section, we carry out results for the existence of unique solutions of second-order fuzzy differential equations by using the contraction principle.

Let us consider FDEs where and , and .

Definition 15. The fuzzy-valued function, , is the solution of differential Equation (13) if satisfies Equation (13).
Let , which is the solution of differential Equation (13), we reduce Equation (13) to an equivalent system of fuzzy-valued integral equations. Let , which is continuous fuzzy-valued function, such that ; therefore, we deduce Using Corollary 10, Theorem 11, and initial condition, then the following four systems of equations are obtained: (1)If u(t) and are (i)-differentiable.(2)If u(t) is (i)-differentiable and is (ii)-differentiable.(3)If u(t) is (ii)-differentiable and is (i)-differentiable.(4)If u(t) and are (ii)-differentiable.

Theorem 16. Let the continuous fuzzy-valued function and such that , with where be continuous mapping; then, is a generalized complete metric space.

Proof. The first two conditions and are easy to show; therefore, we only show . Assume that for every , there exist , such that Hence, is a generalized metric space.
Now, we need to show that is complete. Let us consider the Cauchy sequence, on , then , for all , using definition, (19), one can get Since is a complete metric space, then there is , such that Cauchy sequence, and converges to and respectively.
When , then (21) produced Hence, converge to in Therefore, is a complete generalized metric space.

Theorem 17. The fuzzy problem (13) has a unique and Hyers-Ulam stable solution , defined by If the following conditions hold for some nonnegative and (i),where be fuzzy-valued continuous function and (ii),where are (i)-differentiable (iii)

Proof. Let and such that , with where be continuous mapping, then is a generalized complete metric space, in view of Theorem 16. Let us define a self operator and , by First, we show Let and Hence, is well defined. Now, we can show for all the operator is strictly contractive on To multiply , on both sides of the above inequality, we have Now, from the definition of , for all , one can get This indicates that is strictly contractive on Therefore, there exist unique solution of problem (13).
The and Hukuhara differences exist for where and are -differentiable, then using condition , we have To multiply on both sides of the above inequality, we have Now, from the definition of and substitute for all we can obtain From, Theorem 3 condition , there exist unique solution of problem (13), such that Using definition of one can get Hence, for all we can get Hence, the fuzzy solutions of the problem (13) are Hyers-Ulam stable.

Theorem 18. Let be the fuzzy-valued continuous function such that If the functions, are (i)-differentiable and is (ii)-differentiable, the following condition holds: Then, there exist unique solution of the problem (13), defined by The fuzzy solution is Hyers-Ulam stable, if the following conditions are satisfied: where is nonnegative constant and

Proof. The proof can be easily obtained on the similar procedure of Theorem 17.

Theorem 19. Let be the fuzzy-valued continuous function such that If the function is (ii)-differentiable and is (i)-differentiable, the following condition holds: Then, there exist unique solutions of the problem (13), defined by The fuzzy solution is Hyers-Ulam stable, if the following conditions are satisfied: where is nonnegative constant and

Proof. The proof can be easily obtained on the similar procedure of Theorem 17.

Theorem 20. Let be the fuzzy-valued continuous function such that If the functions are (ii)-differentiable, the following conditions hold: Then, there exist unique solution of the problem (13), defined by The fuzzy solution is Hyers-Ulam stable, if the following conditions are satisfied: where is nonnegative constant and

Proof. Using a procedure similar to Theorem 17, one can easily prove.

4. Natural Transforms and Solutions of the Proposed Fuzzy Differential Equations

Theorem 21. Let fuzzy-valued function be continuous such that , , and exist and are continuous. Moreover, they are Riemann integrable on ; then, (A)If and are (i)-differentiable, then(B)If is (i)-differentiable and is (ii)-differentiable, then(C)If is (ii)-differentiable and is (i)-differentiable, then(D)If and are (ii)-differentiable, then

Proof. (A)Let and are (i)-differentiable, thenUsing (52), this identity yields (B)If, and are (i)-differentiable and (ii)-differentiable, respectively, thenUsing (54), this identity yields (C)If and are (i)-differentiable and (ii)-differentiable, respectively, thenUsing (56), this identity yields (D)If and are (ii)-differentiable, thenUsing (58), this identity yields

Next, we have to solve problem (13), with natural transform. The fuzzy solution of the concerned problem can be discussed in four cases. Taking natural transform of (13), we can get

Case I. Let and are (i)-differentiable; then, using Theorem 21, in (60), one can get where Assume that solving Equation (61), satisfying the conditions and one can obtain the solutions as follows: Taking inverse fuzzy natural transform of Equation (63), one can find and as follows:

Case II. Let is (i)-differentiable and is (ii)-differentiable, then using Theorem 21, in (60), one can get Using initial condition in (65), and one can get Assume that by solving Equation (66), we can get the solutions as follows: Taking inverse fuzzy natural transform of Equation (67), one can find and

Case III. Let is (ii)-differentiable and is (i)-differentiable, then using Theorem 21, in (60), one can get Using initial condition in (69), and one can get Assume that by solving Equation (70), one can obtain the solution as follows: Taking inverse fuzzy natural transform of Equation (71), one can find

Case IV. Let and are (ii)-differentiable, then using Theorem 21, in (60), one can get Using initial condition in (73), and