The problem with fuzzy eigenvalue parameter in one of the boundary conditions

In this work, we study the problem with fuzzy eigenvalue parameter in one of the boundary conditions. We ﬁnd fuzzy eigenvalues of the problem using the Wronskian functions W α ( λ ) and W α ( λ ). Also, we ﬁnd eigenfunctions associated with eigenvalues. We draw graphics of eigenfunctions


Introduction
Fuzzy logic is studied in many areas [1,2]. To solve many problems, Sturm-Liouville Theory is used in mathematical physics [3,4]. Sturm-Liouville fuzzy problem was defined by Gültekin Ç itil and Altınışık [5]. They studied Sturm-Liouville fuzzy problems with reel and fuzzy coefficients in the boundary conditions under the Hukuhara differentiability [6,7]. Also, fuzzy eigenvalue problems were investigated under the approach of generalized differentiability in many papers [8,9]. In the other hand, the fuzzy problem with eigenvalue parameter in the boundary condition was studied [10,11]. But, eigenvalue parameter was not fuzzy in these papers. The problem with fuzzy eigenvalue parameter was defined and investigated by Gültekin Ç itil [12]. This paper is on the problem with fuzzy eigenvalue parameter in one of the boundary conditions. That is, we concern the fuzzy eigenvalue problem where We show the space of fuzzy sets with R F . Definition 2. [14] Let u ∈ R F . The α-level set of u is defined as The α-level set of u is denoted as Definition 4. [14] For u, v ∈ R F and λ ∈ R, the sum u + v and the product λu are defined by where means the usual addition of two intervals (subsets) of R and λ [u] α means the usual product between a scalar and a subset of R.
The product uv is defined by then w is called the Hukuhara difference of fuzzy numbers u and v,and it is denoted by w = u ⊖ v.
We say that f is Hukuhara differentiable at t 0 , if there exists an element f ′ (t 0 ) ∈ R F such that for all h > 0 sufficiently small,

The fuzzy eigenvalues and fuzzy eigenfunctions of the problem
In this section, we investigate the fuzzy eigenvalues and the fuzzy eigenfunctions of the problem (1)-(3).
In Figure 1

Conclusion
In this work, we study the problem with fuzzy eigenvalue parameter in one of the boundary conditions. We find infinitely many eigenvalues for each α ∈ [0, 1]. Also, we find solutions associated with eigenvalues. We draw graphics of solutions. But solutions are not valid α−level sets every time. That is, solutions are valid fuzzy functions different interval for each α ∈ [0, 1]. Thus, found solutions are solutions only in interval which they are valid fuzzy function. That is, found solutions are eigenfunctions only in interval which they are valid fuzzy function.