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

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.


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][4][5][6][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][17][18][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][22][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.

Preliminaries
Here, some basic results are provided from the existing literature.
Definition 1 (see [24]). Let s : R → ½0, 1 satisfy the conditions, where y 0 , y 1 , y 2 , z ∈ R (set of real numbers) (i) s is upper semicontinuous Then, s is a fuzzy number. Throughout this paper,F R represent the set that contains all fuzzy numbers.
Definition 2 (see [25]). The fuzzy number can be written as ðkðqÞ, kðqÞÞ in order pair form, with 0 ≤ q ≤ 1, and holds the following conditions:   [26]). The mapping d H : The pair ðd H ,F R Þ is a generalized complete metric space. Moreover, ∀x 1 , Definition 6 (see [27] (ii) For r > 0, sufficiently small, the Hukuhara difference, χðyÞ ⊖ χðy + rÞ, χðy − rÞ ⊖ χðyÞ, 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 fe ∈ RjμðeÞ ≥ γg is the γ -level set of fuzzy number μ ∈F R . Moreover, the γ-level set is bounded and closed with upper and lower bond, μðqÞ and μðqÞ, respectively, denoted by ðμðqÞ, μðqÞÞ.
Theorem 11 (see [27]). The generalized differentiable mapping V : I →F R has integrable derivative V ′ on I.
Definition 12 (see [21]). The natural transform of gðtÞ is Rðs, pÞ given by the following formula: where s and p are transform parameter which is real and positive.
Lemma 13 (see [22]). The duality relation of natural and Laplace transform is given by where L is the Laplace transform. The natural transform converts to Laplace transform by taking parameter p = 1 and Sumudu transform by taking parameter s = 1.
Definition 14 (see [23]). The natural transform of nth order derivative of gðtÞ is The natural transform of g′ðtÞand g″ðtÞ firstand second-order derivative of gðtÞ is given by

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 Definition 15. The fuzzy-valued function, u ∈ CðY,F R Þ, is the solution of differential Equation (13) if uðtÞ satisfies Equation (13). Let u ∈ CðY,FRÞ, which is the solution of differential Equation (13), we reduce Equation (13) to an equivalent system of fuzzy-valued integral equations. Let v : Y → FˆR, which is continuous fuzzy-valued function, such that u ' ðtÞ = vðtÞ; therefore, we deduce 3 Journal of Function Spaces Using Corollary 10, Theorem 11, and initial condition, then the following four systems of equations are obtained: Theorem 16. Let the continuous fuzzy-valued function u ∈ CðY,F R Þ and χ = CðY, F∧ R Þ 2 such that H : where b ϕ :Ŷ → ð0, ∞Þ be continuous mapping; then, ðχ, HÞ is a generalized complete metric space.
Proof. The first two conditions F 1 and F 2 are easy to show; therefore, we only show F 3 . Assume that for every Hence, ðχ, HÞ is a generalized metric space. Now, we need to show that ðχ, HÞ is complete. Let us consider the Cauchy sequence, ðu n , v n Þ on ðχ, HÞ, then Hððu n , v n Þ, ðu m , v m Þ ≤ ε, for all n, m ≥ NðεÞ, using definition, (19), one can get Since ðF R , d H Þ is a complete metric space, then there is ∃uðtÞ, vðtÞ: Y →F R , such that Cauchy sequence, u n ðtÞ and v n ðtÞ converges to uðtÞ and vðtÞ, respectively.
If the following conditions hold for some nonnegative L, ε and T = t − t 0 : where b ϕ :Ŷ → ð0, ∞Þ be continuous mapping, then ðχ, HÞ is a generalized complete metric space, in view of Theorem 16. Let us define a self operator G : χ → χ, ∀ðu, vÞ ∈ χ and t ∈ Y, by First, we show Gðu, vÞ ∈ χ: Let ðp 0 , q 0 Þ ∈ χ and t ∈ Y, Hence, GðuðtÞ, vðtÞÞ = ðGðuÞ, GðvÞÞ is well defined. Now, we can show for all ðu, vÞ ∈ χ, the operator G is strictly contractive on ðχ, HÞ: To multiply e −ðL+1Þðt−t 0 Þ , on both sides of the above inequality, we have Now, from the definition of H, for all t ∈ Y, one can get This indicates that G is strictly contractive on ðχ, HÞ: Therefore, there exist unique solution of problem (13).
The uðtÞ ⊖ uðt 0 Þ and vðtÞ ⊖ u′ðt 0 Þ, Hukuhara differences exist for t ∈ Y, where u and v are ðiÞ-differentiable, then using condition ðiiÞ, we have To multiply e −ðL+1Þðt−t 0 Þ , on both sides of the above inequality, we have Now, from the definition of H and substitute T = t − t 0 , for all t ∈ Y, we can obtain From, Theorem 3 condition ðiiiÞ, there exist unique solution ðû,vÞ, of problem (13), such that 5

Journal of Function Spaces
Using definition of H, one can get Hence, for all t ∈ Y, we can get Hence, the fuzzy solutions of the problem (13) are Hyers-Ulam stable.
If the functions, u : Y →F R , are (i)-differentiable and v : Y →F R is (ii)-differentiable, the following condition holds: Then, there exist unique solution of the problem (13),û, v : Y →F R , defined bŷ The fuzzy solution is Hyers-Ulam stable, if the following conditions are satisfied: where L is nonnegative constant and T = t − t 0 : Proof. The proof can be easily obtained on the similar procedure of Theorem 17.

Theorem 19.
Let F : Y ×F R ×F R →F R , be the fuzzy-valued continuous function such that If the function u : Y →F R is (ii)-differentiable and v : Y →F R is (i)-differentiable, the following condition holds: Then, there exist unique solutions of the problem (13),û, v : Y →F R , defined bŷ The fuzzy solution is Hyers-Ulam stable, if the following conditions are satisfied: where L is nonnegative constant and T = t − t 0 : Proof. The proof can be easily obtained on the similar procedure of Theorem 17.
Theorem 20. Let F : Y ×F R ×F R →F R be the fuzzy-valued continuous function such that If the functions u, v : Y →F R are (ii)-differentiable, the following conditions hold: Then, there exist unique solution of the problem (13),û, v : Y →F R , defined bŷ The fuzzy solution is Hyers-Ulam stable, if the following conditions are satisfied: where L is nonnegative constant and T = t − t 0 :

Journal of Function Spaces
Proof. Using a procedure similar to Theorem 17, one can easily prove.

Examples
Here, we discuss the numerical problem.
where p, q : Y → R + be continuous functions such that for nonnegative L,jpðwÞj ≤ L,jqðwÞj ≤ L − 1, and c : Y →F R be continuous function for all w ∈ Y, while Y = ½0, 3/L: If the functions χ : Y →F R , the following conditions for some nonnegative ε hold: Assume that, on setting Fðw, χðwÞ, χ′ðwÞÞ = pðwÞχðwÞ + qðwÞχ ′ ðwÞ + wðwÞ, it is easy to obtain condition (i), as follows Then, there exist unique solution of the problem (77) This shows that the fuzzy solution of differential equations, (77), is Hyers-Ulam stable on Y.