Research Article | Open Access
Constructive Existence of (1,1)-Solutions to Two-Point Value Problems for Fuzzy Linear Multiterm Fractional Differential Equations
In this paper, we consider the following two-point boundary value problems of fuzzy linear fractional differential equations: , , and , where , , , , and . Our existence result is based on Banach fixed point theorem and the approximate solution of our problem is obtained by applying the Haar wavelet operational matrix.
A lot of researchers have studied fuzzy differential equations especially fuzzy boundary value problems (FBVPs) because they are effective tools for modeling processes. FBVPs arise in many applications such as modeling of fuzzy optimal control problem  and HIV infection . Many theoretical researches have been carried out on fractional differential equations over the last years [3–7].
Currently, the approximative methods for solving fuzzy fractional differential equation include the operational matrix method based on orthogonal functions [8–10], linearization formula , Homotopy Analysis Method [12–14], and Variation of constant formula .
Lakshmikantham et al.  considered Riemann–Liouville differentiability concept based on the Hukuhara differentiability to solve fuzzy fractional differential equations. Prakash  considered initial value problems for differential equations of fractional order with uncertainty. Mazandarania  investigated the solution to fuzzy fractional initial value problem (FFIVP) under Caputo-type fuzzy fractional derivatives by a modified fractional Euler method. As we can see, fuzzy initial value problems were studied by many researchers, but few fuzzy boundary value problems were considered in special cases. Nieto  considered second order fuzzy differential by the sense of (1, 1), (1, 2), (2, 1), (2, 2)–derivatives. Also Nieto  investigated the existence and uniqueness of solutions for a first-order linear fuzzy differential equation with impulsive boundary value condition. Ngo et al.  proved the existence and uniqueness results of the solution to initial value problem of Caputo–Katugampola (CK) fractional differential equations in fuzzy setting and  present that a fractional fuzzy differential equation and a fractional fuzzy integral equation are not equivalent in general. Wang  considered the existence and uniqueness of solution for a class of FFDEs:where is the fuzzy fractional Caputo derivative, is generalized Hukuhara derivative of , is the space of fuzzy number, is a continuous fuzzy valued function, is a real number, and .
Gasilov  presented a new approach to a nonhomogeneous fuzzy boundary value problem.
But researchers who studied fuzzy differential equations by using -cut did not consider if the solutions of -cut equations constitute intervals. So they had to recheck whether the solutions of -cut equations constitute intervals or not after solving a problem. For instance, in  they only consider the existence of solutions of -cut equation. And the existence of fuzzy solutions was considered in specific example. In that specific example, they noted that the fuzzy solutions do not exist even if the solutions of -cut equation exist.
These facts lead to the following: the existence of solutions of fuzzy problem is not equivalent to the existence of solutions of corresponding -cut equation. So, it is necessary to obtain a new -cut problem that guarantees the existence of fuzzy solutions.
Motivated by the results mentioned above, we consider the following fuzzy boundary value problem:where , , , , , , is fuzzy fractional Caputo derivative, and is the space of fuzzy number.
We obtain existence result by using Banach fixed point theorem and obtain its approximate solution by applying the Haar wavelet operational matrix. Also we present a new -cut problem which involves inequalities to obtain the conditions of existence of fuzzy solutions and prove that these inequalities guarantee that the solutions of -cut equations constitute fuzzy solutions. Our paper is organized as follows: In Section 2, we recall some definitions and basic results and prove some lemmas that will be useful to our main results. Section 3 investigated the constructive existence of solutions to our problem. In Section 4, a method to find out the solutions is given. Section 5 presented two examples to illustrate our results. In Section 6, we summarize our main results.
2. Preliminaries and Basic Results
Definition 1 (see ). We denote the set of all fuzzy numbers on by . A fuzzy number is a mapping with the following properties:(i) is normal, i.e., ; (ii) is a convex fuzzy subset, i.e.,(iii) is upper semicontinuous on (iv)The set is compact in (where ) Then is called the space of fuzzy numbers.
Definition 2 (see ). Let , , . The distance structure is defined as follows: Let . If there exists such that , then is called the H-difference (Hukuhara difference) of and it is denoted as .
Definition 3 (see ). Let and fix .
(i) We say that is (1)-differentiable at , if there exists an element such that, for all sufficiently near to , , and the limits exist.
(ii) We say that is (2)-differentiable at , if there exists an element such that, for all sufficiently near to , , and the limits exist.
If is differentiable at , we denote its first derivatives by for .
Lemma 4 (see ). Let be fuzzy valued function, where for each . (i)If is (1)-differentiable, then and are differentiable functions and .(ii)If is (2)-differentiable, then and are differentiable functions and .
Definition 5 (see ). Let and . We say that is differentiable at , if exists, and is differentiable at . The second derivative of is denoted by for .
Lemma 6 (see ). Let be fuzzy valued function and denote its level sets by for each .(i)If is (1)-differentiable, then are differentiable and(ii)If is (2)-differentiable, then are differentiable and
Definition 7. The Riemann-Liouville fractional integral operator of order of a real function is defined aswhere is the Euler gamma function.
Definition 8. Let , the Caputo fractional derivative of order , , , be defined as
Definition 9. The fuzzy fractional Caputo differentiability of fuzzy valued function is defined as follows: where , , . Then we say that is -differentiable.
Lemma 11 (see ). Let be a family of real intervals such that the following three conditions are satisfied:(i) is a nonempty compact interval for all (ii)if then (iii)given any nondecreasing sequence with it is Then there exists a unique fuzzy number such that for all and .
Let us consider the following fractional integral equation:where , .
Lemma 12. There exists a unique solution of (14) in and a positive real number for the solution to satisfy the following inequality:where is any positive number that satisfies .
Proof. Let .
Then equation (14) is as follows:Now, we use the following norm equivalent to the norm .
That is For all , we have the following inequalities:Therefore, we get Thus, we obtainSince the number satisfieswe obtain the following inequality:Since is complete, we haveThat is, (14) has unique solution .
Then the following equation holds:and we obtain the following inequality: so we haveTherefore, we getSo if there is a positive number satisfying then we get and so we obtain Thus we have
Consider the integral equation where , , , .
We define an operator by .
Lemma 13. If , then the integral equation (32) has one nonnegative solution, where is the identity operator.
Lemma 14. , and then the integral equation has one nonpositive solution.
3. Constructive Existence of (1,1)-Solution of Two-Point Value Problem for the Fuzzy Linear Multiterm Fractional Differential Equation
Let us consider the following fuzzy boundary value problem:where , , , , , .
Definition 15. A fuzzy valued function is called a (1,1)-solution of the problem (38), (39) if it satisfies (38), (39) and .
By using r-cuts, we can obtain andBy the operation of intervals, we can have In particular, the inequalities imply that and constitute interval and it is the same for and
Let us denote the -cut representation of by , . Then the cut forms of boundary conditions (39) are Now the expression (42)-(44) is called cut problem of (38)-(39).
Proof. Suppose is a solution of the problem (42)-(44).
Then, we can getFrom the boundary condition (44), we have Thus, we obtain and so we have Using Green’s function, (50) can be expressed as follows:From (43), (51), and (52), we can obtain (45).
On the other hand, since , we have At last, from the inequality , we can have Conversely, let us suppose that the pair satisfies (45). From (46), we can get Therefore, we have , and so we can get and We can easily obtain the other inequalities and prove the boundary condition (44).
Now, let us consider the integral equation
Assumption 1. .
Denote the following:
We are going to consider a the scheme of successive approximation