In this paper is studied a fuzzy Sturm-Liouville problem with the eigenvalue parameter in the boundary condition. Important notes are given for the problem. Integral equations are found of the problem.
Firstly, Zadeh introduced the concept of fuzzy numbers and fuzzy arithmetic . The major application of fuzzy arithmetic is fuzzy differential equations. Fuzzy differential equations are suitable models to model dynamic systems in which there exist uncertainties or vaguness. Fuzzy differential equations can be examined by several approach, such as Hukuhara differentiability, generalized differentiability, the concept of differential inclusion etc , , , , [6, 7, 8], , , [13, 14],  , , .
In this paper is on a fuzzy Sturm-Liouville problem with the eigenvalue parameter in the boundary condition. Important notes are given for the problem. Integral equations are found of the problem.
Definition 1.  A fuzzy number is a function u :
Let RF denote the space of fuzzy numbers.
Definition 2.  Let uɛℝF. The a-level set of u, denoted,
If a = 0; the support of u is defined
The following remark shows when
Definition 3.  For u,vεℝF and λ ∈R, the sum u+v and the product λ u are defined by
The metric structure is given by the Hausdorff distance
Definition 4.  If A is a symmetric triangular numbers with supports
Definition 5.  u,v 2 ℝF,
Definition 6. [15, 21] Let u,v 2 ℝF.If there exists w 2 ℝF such that u = v+w; then w is called the Hukuhara difference of fuzzy numbers u and v;and it is denoted by w = u⊖v:
Definition 7. [2,15] Let f : [a;b]→RF and t0 ∈ [a,b] :We say that f is Hukuhara differential at t0, if there exists an element f' 0 (t0) 2 ℝF such that for all h > 0 sufficiently small,
Definition 8.  If
is called a fuzzy Sturm-Liouville equation.
3 Findings and Main Results
Consider the fuzzy boundary value problem
where q(x) is continuous function and positive on [-1,1] , λ > 0 and β; γ > 0.
Let u1 (x;λ) and u2 (x;λ) be linearly independent solutions of the classical differential equation τu+λu=0. Then, the general solution of the fuzzy differential equation (3.1) is
be the solution of the equation (3.1) satisfying the conditions
be the solution of the equation (3.1) satisfying the conditions
From here, yields
Also, since u1 (x;λ) and u2 (x;λ) are linearly independent solutions of the classical differential equation
φ(x;λ) be the solution of the classical differential equation τu+lu = 0 satisfying the conditions
From this, a(λ) , b(λ) are obtained as
Again,χ (x,λ) be the solution of the classical differential equation τu+lu = 0 satisfying the conditions
is the solution of the equation (3.1) satisfying the conditions (3.4) and
is the solution of the equation (3.1) satisfying the conditions (3.5), where
are obtained. Computing the value
Considering the 3.7), the value is
and the value
Theorem 1. The Wronskian functions
Proof. Derivating of equations W
are obtained. The proof is complete.
Theorem 2. The eigenvalues of the fuzzy boundary value problem (3.1)-(3.3) if and only if are consist of the zeros of functions
Proof. Let be λ = λ0 is the eigenvalue. We show that
Using the boundary condition (3.2) and using the solution function
are obtained. Again, using (3.4), (3.10), we have
From this, since
Let λ0 be zero of
That is, the functions
Lemma 1. Let λ = s2. The lower and the upper solutions
is the solution of the equation (3.1), the equation
is provided. Using the Hukuhara differentiability and fuzzy arithmetic,
is obtained. From here, yields
Substituing the identity
On integrating by parts twice and using (3.4)
is obtained. Substituing this back into the previous equality yields
From here, we have
Lemma 2. Let λ = s2. The lower and the upper solutions
Proof. Substituing the identity
This paper was supported by Scientific Research Projects Coordination Unit of Giresun University (The project number is FEN-BAP-A-230218-52). Thanks to Scientific Research Projects Coordination Unit of Giresun University.
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