Open Access

New Results on Multiple Solutions for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq1_HTML.gif th-Order Fuzzy Differential Equations under Generalized Differentiability

Boundary Value Problems20092009:395714

DOI: 10.1155/2009/395714

Received: 30 April 2009

Accepted: 1 July 2009

Published: 20 July 2009

Abstract

We firstly present a generalized concept of higher-order differentiability for fuzzy functions. Then we interpret https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq2_HTML.gif th-order fuzzy differential equations using this concept. We introduce new definitions of solution to fuzzy differential equations. Some examples are provided for which both the new solutions and the former ones to the fuzzy initial value problems are presented and compared. We present an example of a linear second-order fuzzy differential equation with initial conditions having four different solutions.

1. Introduction

The term "fuzzy differential equation" was coined in 1987 by Kandel and Byatt [1] and an extended version of this short note was published two years later [2]. There are many suggestions to define a fuzzy derivative and in consequence, to study fuzzy differential equation [3]. One of the earliest was to generalize the Hukuhara derivative of a set-valued function. This generalization was made by Puri and Ralescu [4] and studied by Kaleva [5]. It soon appeared that the solution of fuzzy differential equation interpreted by Hukuhara derivative has a drawback: it became fuzzier as time goes by [6]. Hence, the fuzzy solution behaves quite differently from the crisp solution. To alleviate the situation, Hüllermeier [7] interpreted fuzzy differential equation as a family of differential inclusions. The main shortcoming of using differential inclusions is that we do not have a derivative of a fuzzy-number-valued function.

The strongly generalized differentiability was introduced in [8] and studied in [911]. This concept allows us to solve the above-mentioned shortcoming. Indeed, the strongly generalized derivative is defined for a larger class of fuzzy-number-valued functions than the Hukuhara derivative. Hence, we use this differentiability concept in the present paper. Under this setting, we obtain some new results on existence of several solutions for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq3_HTML.gif th-order fuzzy differential equations. Higher-order fuzzy differential equation with Hukuhara differentiability is considered in [12] and the existence and uniqueness of solution for nonlinearities satisfying a Lipschitz condition is proved. Buckley and Feuring [13] presented two different approaches to the solvability of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq4_HTML.gif th-order linear fuzzy differential equations.

Here, using the concept of generalized derivative and its extension to higher-order derivatives, we show that we have several possibilities or types to define higher-order derivatives of fuzzy-number-valued functions. Then, we propose a new method to solve higher-order fuzzy differential equations based on the selection of derivative type covering all former solutions. With these ideas, the selection of derivative type in each step of derivation plays a crucial role.

2. Preliminaries

In this section, we give some definitions and introduce the necessary notation which will be used throughout this paper. See, for example, [6].

Definition 2.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq5_HTML.gif be a nonempty set. A fuzzy set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq6_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq7_HTML.gif is characterized by its membership function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq8_HTML.gif Thus, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq9_HTML.gif is interpreted as the degree of membership of an element https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq10_HTML.gif in the fuzzy set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq11_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq12_HTML.gif

Let us denote by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq13_HTML.gif the class of fuzzy subsets of the real axis (i.e., https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq14_HTML.gif ) satisfying the following properties:

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq15_HTML.gif is normal, that is, there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq16_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq17_HTML.gif

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq18_HTML.gif is convex fuzzy set (i.e., https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq19_HTML.gif ),

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq20_HTML.gif is upper semicontinuous on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq21_HTML.gif ,

(iv) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq22_HTML.gif is compact where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq23_HTML.gif denotes the closure of a subset.

Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq24_HTML.gif is called the space of fuzzy numbers. Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq25_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq26_HTML.gif denote https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq27_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq28_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq29_HTML.gif belongs to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq30_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq31_HTML.gif -level set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq32_HTML.gif is a nonempty compact interval for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq33_HTML.gif . The notation
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ1_HTML.gif
(2.1)

denotes explicitly the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq34_HTML.gif -level set of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq35_HTML.gif . One refers to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq36_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq37_HTML.gif as the lower and upper branches of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq38_HTML.gif , respectively. The following remark shows when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq39_HTML.gif is a valid https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq40_HTML.gif -level set.

Remark 2.2 (see [6]).

The sufficient conditions for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq41_HTML.gif to define the parametric form of a fuzzy number are as follows:

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq42_HTML.gif is a bounded monotonic increasing (nondecreasing) left-continuous function on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq43_HTML.gif and right-continuous for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq44_HTML.gif ,

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq45_HTML.gif is a bounded monotonic decreasing (nonincreasing) left-continuous function on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq46_HTML.gif and right-continuous for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq47_HTML.gif ,

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq48_HTML.gif

For https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq49_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq50_HTML.gif , the sum https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq51_HTML.gif and the product https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq52_HTML.gif are defined by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq53_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq54_HTML.gif means the usual addition of two intervals (subsets) of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq55_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq56_HTML.gif means the usual product between a scalar and a subset of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq57_HTML.gif

The metric structure is given by the Hausdorff distance:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ2_HTML.gif
(2.2)
by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ3_HTML.gif
(2.3)

The following properties are wellknown:

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq58_HTML.gif

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq59_HTML.gif

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq60_HTML.gif

and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq61_HTML.gif is a complete metric space.

Definition 2.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq62_HTML.gif . If there exists https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq63_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq64_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq65_HTML.gif is called the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq66_HTML.gif -difference of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq67_HTML.gif and it is denoted https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq68_HTML.gif .

In this paper the sign " https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq69_HTML.gif " stands always for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq70_HTML.gif -difference and let us remark that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq71_HTML.gif in general. Usually we denote https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq72_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq73_HTML.gif , while https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq74_HTML.gif stands for the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq75_HTML.gif -difference.

3. Generalized Fuzzy Derivatives

The concept of the fuzzy derivative was first introduced by Chang and Zadeh [14]; it was followed up by Dubois and Prade [15] who used the extension principle in their approach. Other methods have been discussed by Puri and Ralescu [4], Goetschel and Voxman [16], Kandel and Byatt [1, 2]. Lakshmikantham and Nieto introduced the concept of fuzzy differential equation in a metric space [17]. Puri and Ralescu in [4] introduced H-derivative (differentiability in the sense of Hukuhara) for fuzzy mappings and it is based on the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq76_HTML.gif -difference of sets, as follows. Henceforth, we suppose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq77_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq78_HTML.gif

Definition 3.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq79_HTML.gif be a fuzzy function. One says, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq80_HTML.gif is differentiable at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq81_HTML.gif if there exists an element https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq82_HTML.gif such that the limits
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ4_HTML.gif
(3.1)

exist and are equal to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq83_HTML.gif Here the limits are taken in the metric space https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq84_HTML.gif

The above definition is a straightforward generalization of the Hukuhara differentiability of a set-valued function. From [6, Proposition 4.2.8], it follows that Hukuhara differentiable function has increasing length of support. Note that this definition of derivative is very restrictive; for instance, in [9], the authors showed that if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq85_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq86_HTML.gif is a fuzzy number and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq87_HTML.gif is a function with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq88_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq89_HTML.gif is not differentiable. To avoid this difficulty, the authors [9] introduced a more general definition of derivative for fuzzy-number-valued function. In this paper, we consider the following definition [11].

Definition 3.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq90_HTML.gif and fix https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq91_HTML.gif One says https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq92_HTML.gif is (1)-differentiable at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq93_HTML.gif , if there exists an element https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq94_HTML.gif such that for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq95_HTML.gif sufficiently near to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq96_HTML.gif , there exist https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq97_HTML.gif and the limits (in the metric https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq98_HTML.gif )
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ5_HTML.gif
(3.2)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq99_HTML.gif is (2)-differentiable if for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq100_HTML.gif sufficiently near to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq101_HTML.gif , there exist https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq102_HTML.gif and the limits (in the metric https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq103_HTML.gif )
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ6_HTML.gif
(3.3)

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq104_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq105_HTML.gif -differentiable at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq106_HTML.gif , we denote its first derivatives by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq107_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq108_HTML.gif

Example 3.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq109_HTML.gif and define https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq110_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq111_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq112_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq113_HTML.gif is differentiable at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq114_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq115_HTML.gif is generalized differentiable on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq116_HTML.gif and we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq117_HTML.gif . For instance, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq118_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq119_HTML.gif is (1)-differentiable. If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq120_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq121_HTML.gif is (2)-differentiable.

Remark 3.4.

In the previous definition, (1)-differentiability corresponds to the H-derivative introduced in [4], so this differentiability concept is a generalization of the H-derivative and obviously more general. For instance, in the previous example, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq122_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq123_HTML.gif we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq124_HTML.gif .

Remark 3.5.

In [9], the authors consider four cases for derivatives. Here we only consider the two first cases of [9, Definition 5]. In the other cases, the derivative is trivial because it is reduced to crisp element (more precisely, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq125_HTML.gif . For details, see [9, Theorem 7]).

Theorem 3.6.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq126_HTML.gif be fuzzy function, where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq127_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq128_HTML.gif .

(i)If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq129_HTML.gif is (1)-differentiable, then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq130_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq131_HTML.gif are differentiable functions and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq132_HTML.gif .
  1. (ii)

    If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq133_HTML.gif is (2)-differentiable, then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq134_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq135_HTML.gif are differentiable functions and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq136_HTML.gif .

     

Proof.

See [11].

Now we introduce definitions for higher-order derivatives based on the selection of derivative type in each step of differentiation. For the sake of convenience, we concentrate on the second-order case.

For a given fuzzy function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq137_HTML.gif , we have two possibilities (Definition 3.2) to obtain the derivative of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq138_HTML.gif ot https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq139_HTML.gif : https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq140_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq141_HTML.gif . Then for each of these two derivatives, we have again two possibilities: https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq142_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq143_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq144_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq145_HTML.gif respectively.

Definition 3.7.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq146_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq147_HTML.gif . One says say https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq148_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq149_HTML.gif -differentiable at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq150_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq151_HTML.gif exists on a neighborhood of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq152_HTML.gif as a fuzzy function and it is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq153_HTML.gif -differentiable at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq154_HTML.gif . The second derivatives of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq155_HTML.gif are denoted by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq156_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq157_HTML.gif .

Remark 3.8.

This definition is consistent. For example, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq158_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq159_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq160_HTML.gif -differentiable simultaneously at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq161_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq162_HTML.gif is (1)- and (2)-differentiable around https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq163_HTML.gif . By remark in [9], https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq164_HTML.gif is a crisp function in a neighborhood of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq165_HTML.gif .

Theorem 3.9.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq166_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq167_HTML.gif be fuzzy functions, where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq168_HTML.gif .

(i)If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq169_HTML.gif is (1)-differentiable, then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq170_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq171_HTML.gif are differentiable functions and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq172_HTML.gif .
  1. (ii)

    If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq173_HTML.gif is (2)-differentiable, then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq174_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq175_HTML.gif are differentiable functions and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq176_HTML.gif .

     
  2. (iii)

    If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq177_HTML.gif is (1)-differentiable, then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq178_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq179_HTML.gif are differentiable functions and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq180_HTML.gif .

     
  3. (iv)

    If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq181_HTML.gif is (2)-differentiable, then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq182_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq183_HTML.gif are differentiable functions and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq184_HTML.gif .

     

Proof.

We present the details only for the case (i), since the other cases are analogous.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq185_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq186_HTML.gif , we have

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ7_HTML.gif
(3.4)
and multiplying by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq187_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ8_HTML.gif
(3.5)
Similarly, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ9_HTML.gif
(3.6)
Passing to the limit, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ10_HTML.gif
(3.7)

This completes the proof of the theorem.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq188_HTML.gif be a positive integer number, pursuing the above-cited idea, we write https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq189_HTML.gif to denote the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq190_HTML.gif th-derivatives of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq191_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq192_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq193_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq194_HTML.gif . Now we intend to compute the higher derivatives (in generalized differentiability sense) of the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq195_HTML.gif -difference of two fuzzy functions and the product of a crisp and a fuzzy function.

Lemma 3.10.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq196_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq197_HTML.gif th-order generalized differentiable at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq198_HTML.gif in the same case of differentiability, then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq199_HTML.gif is generalized differentiable of order https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq200_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq201_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq202_HTML.gif . (The sum of two functions is defined pointwise.)

Proof.

By Definition 3.2 the statement of the lemma follows easily.

Theorem 3.11.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq203_HTML.gif be second-order generalized differentiable such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq204_HTML.gif is (1,1)-differentiable and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq205_HTML.gif is (2,1)-differentiable or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq206_HTML.gif is (1,2)-differentiable and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq207_HTML.gif is (2,2)-differentiable or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq208_HTML.gif is (2,1)-differentiable and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq209_HTML.gif is (1,1)-differentiable or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq210_HTML.gif is (2,2)-differentiable and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq211_HTML.gif is (1,2)-differentiable on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq212_HTML.gif . If the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq213_HTML.gif -difference https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq214_HTML.gif exists for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq215_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq216_HTML.gif is second-order generalized differentiable and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ11_HTML.gif
(3.8)

for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq217_HTML.gif .

Proof.

We prove the first case and other cases are similar. Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq218_HTML.gif is (1)-differentiable and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq219_HTML.gif is (2)-differentiable on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq220_HTML.gif , by [10, Theorem 4], https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq221_HTML.gif is (1)-differentiable and we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq222_HTML.gif . By differentiation as (1)-differentiability in Definition 3.2 and using Lemma 3.10, we get https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq223_HTML.gif is (1,1)-differentiable and we deduce
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ12_HTML.gif
(3.9)

The https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq224_HTML.gif -difference of two functions is understood pointwise.

Theorem 3.12.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq225_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq226_HTML.gif be two differentiable functions ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq227_HTML.gif is generalized differentiable as in Definition 3.2).

(i)If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq228_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq229_HTML.gif is (1)-differentiable, then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq230_HTML.gif is (1)-differentiable and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ13_HTML.gif
(3.10)
  1. (ii)
    If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq231_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq232_HTML.gif is (2)-differentiable, then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq233_HTML.gif is (2)-differentiable and
    https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ14_HTML.gif
    (3.11)
     

Proof.

See [10].

Theorem 3.13.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq234_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq235_HTML.gif be second-order differentiable functions ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq236_HTML.gif is generalized differentiable as in Definition 3.7).

(i)If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq237_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq238_HTML.gif is (1,1)-differentiable then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq239_HTML.gif is (1,1)-differentiable and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ15_HTML.gif
(3.12)
  1. (ii)
    If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq240_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq241_HTML.gif is (2,2)-differentiable then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq242_HTML.gif is (2,2)-differentiable and
    https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ16_HTML.gif
    (3.13)
     

Proof.

We prove (i), and the proof of another case is similar. If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq243_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq244_HTML.gif is (1)-differentiable, then by Theorem 3.12 we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ17_HTML.gif
(3.14)

Now by differentiation as first case in Definition 3.2, since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq245_HTML.gif is (1)-differentiable and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq246_HTML.gif then we conclude the result.

Remark 3.14.

By [9, Remark 16], let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq247_HTML.gif and define https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq248_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq249_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq250_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq251_HTML.gif is differentiable on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq252_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq253_HTML.gif is differentiable on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq254_HTML.gif , with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq255_HTML.gif . By Theorem 3.12, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq256_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq257_HTML.gif is (1)-differentiable on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq258_HTML.gif . Also if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq259_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq260_HTML.gif is (2)-differentiable on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq261_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq262_HTML.gif , by [9, Theorem 10], we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq263_HTML.gif . We can extend this result to second-order differentiability as follows.

Theorem 3.15.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq264_HTML.gif be twice differentiable on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq265_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq266_HTML.gif and define https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq267_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq268_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq269_HTML.gif .

(i)If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq270_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq271_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq272_HTML.gif is (1,1)-differentiable and its second derivative, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq273_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq274_HTML.gif ,
  1. (ii)

    If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq275_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq276_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq277_HTML.gif is (1,2)-differentiable with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq278_HTML.gif ,

     
  2. (iii)

    If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq279_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq280_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq281_HTML.gif is (2,1)-differentiable with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq282_HTML.gif ,

     
  3. (iv)

    If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq283_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq284_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq285_HTML.gif is (2,2)-differentiable with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq286_HTML.gif .

     

Proof.

Cases (i) and (iv) follow from Theorem 3.13. To prove (ii), since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq287_HTML.gif , by Remark 3.14, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq288_HTML.gif is (1)-differentiable and we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq289_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq290_HTML.gif . Also, since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq291_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq292_HTML.gif is (2)-differentiable and we conclude the result. Case (iii) is similar to previous one.

Example 3.16.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq293_HTML.gif is a fuzzy number and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq294_HTML.gif where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ18_HTML.gif
(3.15)
is crisp second-order polynomial, then for
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ19_HTML.gif
(3.16)

we have the following

(i)for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq295_HTML.gif : https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq296_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq297_HTML.gif then by (iv), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq298_HTML.gif is (2-2)-differentiable and its second derivative, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq299_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq300_HTML.gif ,

(ii)for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq301_HTML.gif : https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq302_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq303_HTML.gif then by (ii), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq304_HTML.gif is (1-2)-differentiable with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq305_HTML.gif ,

(iii)for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq306_HTML.gif : https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq307_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq308_HTML.gif then by (iii), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq309_HTML.gif is (2-1)-differentiable and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq310_HTML.gif ,

(iv)for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq311_HTML.gif : https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq312_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq313_HTML.gif then by (i), https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq314_HTML.gif is (1-1)-differentiable and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq315_HTML.gif ,

(v)for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq316_HTML.gif : we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq317_HTML.gif , then by [9, Theorem 10] we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq318_HTML.gif , again by applying this theorem, we get https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq319_HTML.gif

4. Second-Order Fuzzy Differential Equations

In this section, we study the fuzzy initial value problem for a second-order linear fuzzy differential equation:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ20_HTML.gif
(4.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq320_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq321_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq322_HTML.gif is a continuous fuzzy function on some interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq323_HTML.gif . The interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq324_HTML.gif can be https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq325_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq326_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq327_HTML.gif . In this paper, we suppose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq328_HTML.gif Our strategy of solving (4.1) is based on the selection of derivative type in the fuzzy differential equation. We first give the following definition for the solutions of (4.1).

Definition 4.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq329_HTML.gif be a fuzzy function and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq330_HTML.gif One says https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq331_HTML.gif is an https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq332_HTML.gif -solution for problem (4.1) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq333_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq334_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq335_HTML.gif exist on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq336_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq337_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq338_HTML.gif be an https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq339_HTML.gif -solution for (4.1). To find it, utilizing Theorems 3.6 and 3.9 and considering the initial values, we can translate problem (4.1) to a system of second-order linear ordinary differential equations hereafter, called corresponding https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq340_HTML.gif -system for problem (4.1).

Therefore, four ODEs systems are possible for problem (4.1), as follows:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq341_HTML.gif -system

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ21_HTML.gif
(4.2)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq342_HTML.gif -system

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ22_HTML.gif
(4.3)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq343_HTML.gif -system

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ23_HTML.gif
(4.4)

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq344_HTML.gif -system

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ24_HTML.gif
(4.5)

Theorem 4.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq345_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq346_HTML.gif be an https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq347_HTML.gif -solution for problem (4.1) on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq348_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq349_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq350_HTML.gif solve the associated https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq351_HTML.gif -systems.

Proof.

Suppose https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq352_HTML.gif is the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq353_HTML.gif -solution of problem (4.1). According to the Definition 4.1, then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq354_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq355_HTML.gif exist and satisfy problem (4.1). By Theorems 3.6 and 3.9 and substituting https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq356_HTML.gif and their derivatives in problem (4.1), we get the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq357_HTML.gif -system corresponding to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq358_HTML.gif -solution. This completes the proof.

Theorem 4.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq359_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq360_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq361_HTML.gif solve the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq362_HTML.gif -system on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq363_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq364_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq365_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq366_HTML.gif has valid level sets on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq367_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq368_HTML.gif exists, then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq369_HTML.gif is an https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq370_HTML.gif -solution for the fuzzy initial value problem (4.1).

Proof.

Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq371_HTML.gif is ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq372_HTML.gif )-differentiable fuzzy function, by Theorems 3.6 and 3.9 we can compute https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq373_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq374_HTML.gif according to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq375_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq376_HTML.gif . Due to the fact that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq377_HTML.gif solve https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq378_HTML.gif -system, from Definition 4.1, it comes that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq379_HTML.gif is an https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq380_HTML.gif -solution for (4.1).

The previous theorems illustrate the method to solve problem (4.1). We first choose the type of solution and translate problem (4.1) to a system of ordinary differential equations. Then, we solve the obtained ordinary differential equations system. Finally we find such a domain in which the solution and its derivatives have valid level sets and using Stacking Theorem [5] we can construct the solution of the fuzzy initial value problem (4.1).

Remark 4.4.

We see that the solution of fuzzy differential equation (4.1) depends upon the selection of derivatives. It is clear that in this new procedure, the unicity of the solution is lost, an expected situation in the fuzzy context. Nonetheless, we can consider the existence of four solutions as shown in the following examples.

Example 4.5.

Let us consider the following second-order fuzzy initial value problem
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ25_HTML.gif
(4.6)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq381_HTML.gif are the triangular fuzzy number having https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq382_HTML.gif -level sets https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq383_HTML.gif

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq384_HTML.gif is (1,1)-solution for the problem, then

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ26_HTML.gif
(4.7)
and they satisfy (1,1)-system associated with (4.1). On the other hand, by ordinary differential theory, the corresponding (1,1)-system has only the following solution:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ27_HTML.gif
(4.8)
We see that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq385_HTML.gif are valid level sets for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq386_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ28_HTML.gif
(4.9)

By Theorem 3.15, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq387_HTML.gif is (1,1)-differentiable for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq388_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq389_HTML.gif defines a (1,1)-solution for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq390_HTML.gif .

For (1,2)-solution, we get the following solutions for (1,2)-system:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ29_HTML.gif
(4.10)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq391_HTML.gif has valid level sets for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq392_HTML.gif How ever-also https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq393_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq394_HTML.gif is (1,2)-differentiable. Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq395_HTML.gif gives us a (1,2)-solution on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq396_HTML.gif .

(2,1)-system yields

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ30_HTML.gif
(4.11)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq397_HTML.gif has valid level sets for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq398_HTML.gif We can see https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq399_HTML.gif is a (2,1)-solution on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq400_HTML.gif

Finally, (2-2)-system gives

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ31_HTML.gif
(4.12)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq401_HTML.gif has valid level sets for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq402_HTML.gif and defines a (2,2)-solution on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq403_HTML.gif .

Then we have an example of a second-order fuzzy initial value problem with four different solutions.

Example 4.6.

Consider the fuzzy initial value problem:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ32_HTML.gif
(4.13)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq404_HTML.gif is the fuzzy number having https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq405_HTML.gif -level sets https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq406_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq407_HTML.gif

To find (1,1)-solution, we have

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ33_HTML.gif
(4.14)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq408_HTML.gif has valid level sets for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq409_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq410_HTML.gif . From Theorem 3.15, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq411_HTML.gif is (1,2)-differentiable on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq412_HTML.gif , then by Remark 3.8, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq413_HTML.gif is not https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq414_HTML.gif -differentiable on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq415_HTML.gif . Hence, no (1,1)-solution exists for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq416_HTML.gif .

For (1,2)-solutions we deduce

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ34_HTML.gif
(4.15)

we see that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq417_HTML.gif has valid level sets and is (1,1)-differentiable for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq418_HTML.gif . Since the (1,2)-system has only the above solution, then (1,2)-solution does not exist.

For (2,1)-solutions we get

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ35_HTML.gif
(4.16)

we see that the fuzzy function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq419_HTML.gif has valid level sets for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq420_HTML.gif and define a (2,1)-solution for the problem on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq421_HTML.gif

Finally, to find (2,2)-solution, we find

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ36_HTML.gif
(4.17)

that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq422_HTML.gif has valid level sets for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq423_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq424_HTML.gif is (2,2)-differentiable on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq425_HTML.gif .

We then have a linear fuzzy differential equation with initial condition and two solutions.

5. Higher-Order Fuzzy Differential Equations

Selecting different types of derivatives, we get several solutions to fuzzy initial value problem for second-order fuzzy differential equations. Theorem 4.2 has a crucial role in our strategy. To extend the results to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq426_HTML.gif th-order fuzzy differential equation, we can follow the proof of Theorem 4.2 to get the same results for derivatives of higher order. Therefore, we can extend the presented argument for second-order fuzzy differential equation to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq427_HTML.gif th-order. Under generalized derivatives, we would expect at most https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq428_HTML.gif solutions for an https://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq429_HTML.gif th-order fuzzy differential equation by choosing the different types of derivatives.

Declarations

Acknowledgments

We thank Professor J. J. Nieto for his valuable remarks which improved the paper. This research is supported by a grant from University of Tabriz.

Authors’ Affiliations

(1)
Department of Applied Mathematics, University of Tabriz
(2)
Research Center for Industrial Mathematics, University of Tabriz

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Copyright

© A. Khastan et al. 2009

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