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Fractional calculus and fractional differential equations are a field of increasing interest due to their applicability to the analysis of phenomena, and they play an important role in a variety of fields such as rheology, viscoelasticity, electrochemistry, diffusion processes, etc.
For a positive number σ, we denote by $$\begin \mathbb _ \bigl(H_[ \mathbb \mathcal , \mathbb \mathcal ] \bigr)& \le \mathbb _ \biggl( \frac \biggr) \\ & = \mathbb _ \bigl(H_[X, Y] \bigr) - \biggl[ \mathbb _ \bigl(H_[X, Y] \bigr) - \mathbb _ \biggl( \frac \biggr) \biggr].
\end$$ $$\begin \textstyle\begin \mathcal ^(t):= X^(t) \ominus_ \varphi(0) \preceq \varphi(t - ) \ominus_ \varphi(0) = \mathbb \mathcal ^(t), & t \in[ - \sigma, a], \\ \mathcal ^(t) \preceq\frac \int _^= \mathbb \mathcal ^(t), & t \in[a, a p].
\end$$ $$\begin &H \bigl[X(t), Z(t) \bigr] \\ &\quad \le H \bigl[\varphi(0), \psi(0) \bigr] \\ &\qquad H \biggl[\frac \int_^ , \\ &\qquad\frac \int_^ \biggr] \\ &\qquad H \biggl[ \frac \int_^ , \frac \int_^ \biggr] \\ &\qquad H \biggl[ \frac \int_^ , \frac \int_^ \biggr] \\ &\quad \le C(t) \frac \int_^ .
\end$$ □ The following corollary shows a new technique to find the exact solutions of interval-valued delay fractional differential equation by using the solutions of interval-valued delay integer order differential equation. Then a solution of (3.6), $$\begin \textstyle\begin X_'(v) = \lambda_ X_ (k(t,v)-1 ) - \lambda_ ( 2 \sqrt \Gamma(3/2) v - v^ \Gamma^(3/2) ), &v \in [0,1/\Gamma(3/2) ] \\ X_(v) = [k(t,v) -1,k(t,v) ],& v \in[-1,0], \end\displaystyle \end$$ $$\begin \underline_(v)&= (\lambda_ - \lambda_) \biggl( \sqrt \Gamma (3/2)v^ - \frac \biggr) - 2 \lambda_ v-1, \\ \overline_(v)&= (\lambda_ - \lambda_) \biggl( \sqrt \Gamma (3/2)v^ - \frac \biggr) - \lambda_ v.
\end\displaystyle \end$$ is the unique solution to (3.1).
□ The conclusion of Theorem 3.1 is still valid if the existence of a w-monotone lower solution for problem (3.1) is replaced by the existence of a w-monotone upper solution for problem (3.1).
Some of them were detailed further in [8–10, 20, 38–40] and the references therein.
In the following, we denote the space of all nonempty compact intervals of the real line We recall some definitions of Riemann-Liouville and Caputo derivatives for interval-valued functions and some necessary results are given to use in the next section.
Following this direction, the concepts of fuzzy fractional differentiability have been developed and extended in some papers to investigate some results on the existence and uniqueness of solutions to fuzzy differential equations, and have been considered in a wide number of applications of this theory (see, for instance, [15–28]).
It is well known that the Banach fixed point theorem is a useful tool in mathematics and plays an important role in finding solutions to nonlinear, differential and integral equations, among others.