New Results on Multiple Solutions for http://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

  • A Khastan1, 2Email author,

    Affiliated with

    • F Bahrami1, 2 and

      Affiliated with

      • K Ivaz1, 2

        Affiliated with

        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 http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq6_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq7_HTML.gif is characterized by its membership function http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq8_HTML.gif Thus, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq10_HTML.gif in the fuzzy set http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq11_HTML.gif for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq12_HTML.gif

        Let us denote by http://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., http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq14_HTML.gif ) satisfying the following properties:

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

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

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

        (iv) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq22_HTML.gif is compact where http://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 http://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, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq25_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq26_HTML.gif denote http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq27_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq28_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq29_HTML.gif belongs to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq30_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq31_HTML.gif -level set http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq33_HTML.gif . The notation
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ1_HTML.gif
        (2.1)

        denotes explicitly the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq34_HTML.gif -level set of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq35_HTML.gif . One refers to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq36_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq38_HTML.gif , respectively. The following remark shows when http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq39_HTML.gif is a valid http://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 http://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) http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq43_HTML.gif and right-continuous for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq44_HTML.gif ,

        (ii) http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq46_HTML.gif and right-continuous for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq47_HTML.gif ,

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

        For http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq49_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq50_HTML.gif , the sum http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq51_HTML.gif and the product http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq52_HTML.gif are defined by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq53_HTML.gif where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq55_HTML.gif and http://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 http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ2_HTML.gif
        (2.2)
        by
        http://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) http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq58_HTML.gif

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

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

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

        In this paper the sign " http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq69_HTML.gif " stands always for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq70_HTML.gif -difference and let us remark that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq71_HTML.gif in general. Usually we denote http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq72_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq73_HTML.gif , while http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq74_HTML.gif stands for the http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq77_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq78_HTML.gif

        Definition 3.1.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq79_HTML.gif be a fuzzy function. One says, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq80_HTML.gif is differentiable at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq81_HTML.gif if there exists an element http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq82_HTML.gif such that the limits
        http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq85_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq86_HTML.gif is a fuzzy number and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq87_HTML.gif is a function with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq88_HTML.gif , then http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq90_HTML.gif and fix http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq91_HTML.gif One says http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq92_HTML.gif is (1)-differentiable at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq93_HTML.gif , if there exists an element http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq94_HTML.gif such that for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq95_HTML.gif sufficiently near to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq96_HTML.gif , there exist http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq97_HTML.gif and the limits (in the metric http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq98_HTML.gif )
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ5_HTML.gif
        (3.2)
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq99_HTML.gif is (2)-differentiable if for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq100_HTML.gif sufficiently near to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq101_HTML.gif , there exist http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq102_HTML.gif and the limits (in the metric http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq103_HTML.gif )
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ6_HTML.gif
        (3.3)

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

        Example 3.3.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq109_HTML.gif and define http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq110_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq111_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq112_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq113_HTML.gif is differentiable at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq114_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq115_HTML.gif is generalized differentiable on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq116_HTML.gif and we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq117_HTML.gif . For instance, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq118_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq119_HTML.gif is (1)-differentiable. If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq120_HTML.gif then http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq122_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq123_HTML.gif we have http://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, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq126_HTML.gif be fuzzy function, where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq127_HTML.gif for each http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq128_HTML.gif .

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

          If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq133_HTML.gif is (2)-differentiable, then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq134_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq135_HTML.gif are differentiable functions and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq138_HTML.gif ot http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq139_HTML.gif : http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq140_HTML.gif and http://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: http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq142_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq143_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq144_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq145_HTML.gif respectively.

        Definition 3.7.

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq146_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq147_HTML.gif . One says say http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq148_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq149_HTML.gif -differentiable at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq150_HTML.gif , if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq151_HTML.gif exists on a neighborhood of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq153_HTML.gif -differentiable at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq154_HTML.gif . The second derivatives of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq155_HTML.gif are denoted by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq156_HTML.gif for http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq158_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq159_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq160_HTML.gif -differentiable simultaneously at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq161_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq162_HTML.gif is (1)- and (2)-differentiable around http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq163_HTML.gif . By remark in [9], http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq165_HTML.gif .

        Theorem 3.9.

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

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

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

           
        2. (iii)

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

           
        3. (iv)

          If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq181_HTML.gif is (2)-differentiable, then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq182_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq183_HTML.gif are differentiable functions and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq185_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq186_HTML.gif , we have

        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ7_HTML.gif
        (3.4)
        and multiplying by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq187_HTML.gif we have
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ8_HTML.gif
        (3.5)
        Similarly, we obtain
        http://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
        http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq189_HTML.gif to denote the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq190_HTML.gif th-derivatives of http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq191_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq192_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq193_HTML.gif for http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq196_HTML.gif are http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq197_HTML.gif th-order generalized differentiable at http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq199_HTML.gif is generalized differentiable of order http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq200_HTML.gif at http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq201_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq204_HTML.gif is (1,1)-differentiable and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq205_HTML.gif is (2,1)-differentiable or http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq206_HTML.gif is (1,2)-differentiable and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq207_HTML.gif is (2,2)-differentiable or http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq208_HTML.gif is (2,1)-differentiable and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq209_HTML.gif is (1,1)-differentiable or http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq210_HTML.gif is (2,2)-differentiable and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq211_HTML.gif is (1,2)-differentiable on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq212_HTML.gif . If the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq213_HTML.gif -difference http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq214_HTML.gif exists for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq215_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq216_HTML.gif is second-order generalized differentiable and
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ11_HTML.gif
        (3.8)

        for all http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq218_HTML.gif is (1)-differentiable and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq219_HTML.gif is (2)-differentiable on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq220_HTML.gif , by [10, Theorem 4], http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq221_HTML.gif is (1)-differentiable and we have http://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 http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ12_HTML.gif
        (3.9)

        The http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq225_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq226_HTML.gif be two differentiable functions ( http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq228_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq229_HTML.gif is (1)-differentiable, then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq230_HTML.gif is (1)-differentiable and
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ13_HTML.gif
        (3.10)
        1. (ii)
          If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq231_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq232_HTML.gif is (2)-differentiable, then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq233_HTML.gif is (2)-differentiable and
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq234_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq235_HTML.gif be second-order differentiable functions ( http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq237_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq238_HTML.gif is (1,1)-differentiable then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq239_HTML.gif is (1,1)-differentiable and
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ15_HTML.gif
        (3.12)
        1. (ii)
          If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq240_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq241_HTML.gif is (2,2)-differentiable then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq242_HTML.gif is (2,2)-differentiable and
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq243_HTML.gif and http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq245_HTML.gif is (1)-differentiable and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq247_HTML.gif and define http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq248_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq249_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq250_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq251_HTML.gif is differentiable on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq252_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq253_HTML.gif is differentiable on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq254_HTML.gif , with http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq255_HTML.gif . By Theorem 3.12, if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq256_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq257_HTML.gif is (1)-differentiable on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq258_HTML.gif . Also if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq259_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq260_HTML.gif is (2)-differentiable on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq261_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq262_HTML.gif , by [9, Theorem 10], we have http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq264_HTML.gif be twice differentiable on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq265_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq266_HTML.gif and define http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq267_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq268_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq269_HTML.gif .

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

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

           
        2. (iii)

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

           
        3. (iv)

          If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq283_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq284_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq285_HTML.gif is (2,2)-differentiable with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq287_HTML.gif , by Remark 3.14, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq288_HTML.gif is (1)-differentiable and we have http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq289_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq290_HTML.gif . Also, since http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq291_HTML.gif , then http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq293_HTML.gif is a fuzzy number and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq294_HTML.gif where
        http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq295_HTML.gif : http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq296_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq297_HTML.gif then by (iv), http://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, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq299_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq300_HTML.gif ,

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

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

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

        (v)for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq316_HTML.gif : we have http://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 http://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 http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ20_HTML.gif
        (4.1)

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq320_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq321_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq323_HTML.gif . The interval http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq324_HTML.gif can be http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq325_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq326_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq327_HTML.gif . In this paper, we suppose http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq329_HTML.gif be a fuzzy function and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq330_HTML.gif One says http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq331_HTML.gif is an http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq332_HTML.gif -solution for problem (4.1) on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq333_HTML.gif , if http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq334_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq335_HTML.gif exist on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq336_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq337_HTML.gif .

        Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq338_HTML.gif be an http://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 http://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:

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

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

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

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

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

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

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

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

        Theorem 4.2.

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

        Proof.

        Suppose http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq352_HTML.gif is the http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq354_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq357_HTML.gif -system corresponding to http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq359_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq360_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq361_HTML.gif solve the http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq362_HTML.gif -system on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq363_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq364_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq365_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq366_HTML.gif has valid level sets on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq367_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq368_HTML.gif exists, then http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq369_HTML.gif is an http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq371_HTML.gif is ( http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq373_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq374_HTML.gif according to http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq375_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq376_HTML.gif . Due to the fact that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq377_HTML.gif solve http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq379_HTML.gif is an http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ25_HTML.gif
        (4.6)

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

        If http://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

        http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ27_HTML.gif
        (4.8)
        We see that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq385_HTML.gif are valid level sets for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq386_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ28_HTML.gif
        (4.9)

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

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

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

        (2,1)-system yields

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

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

        Finally, (2-2)-system gives

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

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq401_HTML.gif has valid level sets for all http://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 http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_Equ32_HTML.gif
        (4.13)

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

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

        where http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq408_HTML.gif has valid level sets for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq409_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq410_HTML.gif . From Theorem 3.15, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq411_HTML.gif is (1,2)-differentiable on http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq412_HTML.gif , then by Remark 3.8, http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq413_HTML.gif is not http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq414_HTML.gif -differentiable on http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq416_HTML.gif .

        For (1,2)-solutions we deduce

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

        we see that http://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 http://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

        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq419_HTML.gif has valid level sets for http://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 http://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

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

        that http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq422_HTML.gif has valid level sets for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq423_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq424_HTML.gif is (2,2)-differentiable on http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq428_HTML.gif solutions for an http://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

        References

        1. Kandel A, Byatt WJ: Fuzzy differential equations. Proceedings of the International Conference on Cybernetics and Society, 1978, Tokyo, Japan 1213–1216.
        2. Kandel A, Byatt WJ: Fuzzy processes. Fuzzy Sets and Systems 1980, 4(2):117–152. 10.1016/0165-0114(80)90032-9MATHMathSciNetView Article
        3. Buckley JJ, Feuring T: Fuzzy differential equations. Fuzzy Sets and Systems 2000, 110(1):43–54. 10.1016/S0165-0114(98)00141-9MATHMathSciNetView Article
        4. Puri ML, Ralescu DA: Differentials of fuzzy functions. Journal of Mathematical Analysis and Applications 1983, 91(2):552–558. 10.1016/0022-247X(83)90169-5MATHMathSciNetView Article
        5. Kaleva O: Fuzzy differential equations. Fuzzy Sets and Systems 1987, 24(3):301–317. 10.1016/0165-0114(87)90029-7MATHMathSciNetView Article
        6. Diamond P, Kloeden P: Metric Spaces of Fuzzy Sets. World Scientific, Singapore; 1994:x+178.MATHView Article
        7. Hüllermeier E: An approach to modelling and simulation of uncertain dynamical systems. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 1997, 5(2):117–137. 10.1142/S0218488597000117MATHMathSciNetView Article
        8. Bede B, Gal SG: Almost periodic fuzzy-number-valued functions. Fuzzy Sets and Systems 2004, 147(3):385–403. 10.1016/j.fss.2003.08.004MATHMathSciNetView Article
        9. Bede B, Gal SG: Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations. Fuzzy Sets and Systems 2005, 151(3):581–599. 10.1016/j.fss.2004.08.001MATHMathSciNetView Article
        10. Bede B, Rudas IJ, Bencsik AL: First order linear fuzzy differential equations under generalized differentiability. Information Sciences 2007, 177(7):1648–1662. 10.1016/j.ins.2006.08.021MATHMathSciNetView Article
        11. Chalco-Cano Y, Román-Flores H: On new solutions of fuzzy differential equations. Chaos, Solitons & Fractals 2008, 38(1):112–119. 10.1016/j.chaos.2006.10.043MATHMathSciNetView Article
        12. Georgiou DN, Nieto JJ, Rodríguez-López R: Initial value problems for higher-order fuzzy differential equations. Nonlinear Analysis: Theory, Methods & Applications 2005, 63(4):587–600. 10.1016/j.na.2005.05.020MATHMathSciNetView Article
        13. Buckley JJ, Feuring T: Fuzzy initial value problem for http://static-content.springer.com/image/art%3A10.1155%2F2009%2F395714/MediaObjects/13661_2009_Article_845_IEq430_HTML.gif th-order linear differential equations. Fuzzy Sets and Systems 2001, 121(2):247–255. 10.1016/S0165-0114(00)00028-2MATHMathSciNetView Article
        14. Chang SSL, Zadeh LA: On fuzzy mapping and control. IEEE Transactions on Systems Man Cybernetics 1972, 2: 30–34.MATHMathSciNetView Article
        15. Dubois D, Prade H: Towards fuzzy differential calculus—part III: differentiation. Fuzzy Sets and Systems 1982, 8(3):225–233. 10.1016/S0165-0114(82)80001-8MATHMathSciNetView Article
        16. Goetschel R Jr., Voxman W: Elementary fuzzy calculus. Fuzzy Sets and Systems 1986, 18(1):31–43. 10.1016/0165-0114(86)90026-6MATHMathSciNetView Article
        17. Lakshmikantham V, Nieto JJ: Differential equations in metric spaces: an introduction and an application to fuzzy differential equations. Dynamics of Continuous, Discrete & Impulsive Systems. Series A 2003, 10(6):991–1000.MATHMathSciNet

        Copyright

        © A. Khastan et al. 2009

        This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.