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# Issue:Intuitionistic fuzzy fractional boundary value problem

 shortcut http://ifigenia.org/wiki/issue:nifs/23/1/31-41
Title of paper: Intuitionistic fuzzy fractional boundary value problem
Author(s):
 Said Melliani Laboratoire de Mathématiques Appliquées & Calcul Scientifique, Sultan Moulay Slimane University, BP 523, 23000 Beni Mellal, Morocco said.melliani@gmail.com M. Elomari Laboratoire de Mathématiques Appliquées & Calcul Scientifique, Sultan Moulay Slimane University, BP 523, 23000 Beni Mellal, Morocco A. Elmfadel Laboratoire de Mathématiques Appliquées & Calcul Scientifique, Sultan Moulay Slimane University, BP 523, 23000 Beni Mellal, Morocco
Presented at: 4th International Intuitionistic Fuzzy Sets and Contemporary Mathematics Conference, 3–7 May 2017, Mersin, Turkey
Published in: "Notes on IFS", Volume 23, 2017, Number 1, pages 31—41
Abstract: In this paper we investigate the existence and uniquness of intuitionistic fuzzy solution for three-point boundary value problem for fractional differential equation:

$\displaystyle{ \left\{ \begin{array}{lrr} D^{\alpha}X(t)=F(t, X_t, D^{\beta}X(t)) & t\in J:=[0, 1]\\ X(t)=\phi(t)&t\in[-r, 0]\\ X(1)=X(\xi)& \end{array}, \right. }$ where $\displaystyle{ D^{\alpha}, D^{\beta} }$ are the standard Riemann--Liouville fractional derivatives (α−β>0) and (1<α < 2), $\displaystyle{ (\xi\in[0, 1[) }$,$\displaystyle{ F:J \times C_0\times\mathbf{IF}^1 \longrightarrow \mathbf{IF}^1 }$ is an intuitionistic fuzzy function, $\displaystyle{ \phi\in C_0 }$, $\displaystyle{ \phi(0)=0_{IF} }$ and $\displaystyle{ C_0=C\left([-r, 0], IF^1\right) }$. We denote by $\displaystyle{ X_t }$ the element of $\displaystyle{ C_0 }$ defined by $\displaystyle{ X_t(\theta)=X(t+\theta) }$, $\displaystyle{ \theta\in[-r, 0]. }$

Keywords: Intuitionistic fuzzy sets, Distance between intuitionistic fuzzy sets, Intuitionistic fractional derivative.
AMS Classification: 03E72.
References:
1. Bede, B., & Gal, S. G. (2005). Generalizations of the differentiability of fuzzy-numbervalued functions with applications to fuzzy differential equations. Fuzzy Sets and Systems, 151, 581–599.
2. Kilbas, A. A. A., Srivastava, H. M., & Trujillo, J. J. (2006). Theory and Applications of Fractional Sifferential Equations, North-Holland Mathematical studies 204, Ed van Mill, Amsterdam.
3. Melliani, S., Elomari, M. Chadli, L. S., & Ettoussi, R. (2015). Intuitionistic fuzzy metric space, Notes on Intuitionistic Fuzzy Sets, 21(1), 43–53.
4. Salahshour, S., Allahviranloo, T., Abbasbandy, S., & Baleanu, D. (2012). Existence and uniqueness results for fractional differential equations with uncertainty. Adv. Diff. Equ., 112, doi:10.1186/1687-1847-2012-112.
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