Solving An Initial Value Problem
In  Harjani and Sadarangani presented some fixed point generalized theorems involving altering distance functions in the ordered metric spaces, and this result was used to investigate the existence problem of solution to first and second order ordinary differential equations.  used some more generalized fixed point results of weakly contractive mappings in a partial order metric space of fuzzy-valued functions to investigate the existence and uniqueness of fuzzy solutions of the initial-valued problem for integer order fuzzy differential equation in the setting of generalized Hukuhara derivatives.By employing the weakly contractive mapping in the partially ordered space of fuzzy functions, Long et al.\end\displaystyle \end$$ is the unique solution to (3.1).□ The conclusion of Theorem 3.1 is still valid if the existence of a w-monotone lower solution for problem (3.1) is replaced by the existence of a w-monotone upper solution for problem (3.1).Some of them were detailed further in [8–10, 20, 38–40] and the references therein.
It is well known that the Banach fixed point theorem is a useful tool in mathematics and plays an important role in finding solutions to nonlinear, differential and integral equations, among others.
One can find applications of fractional differential equations in signal processing and in the complex dynamic in biological tissues (see [1–3]). Interval analysis and interval differential equation were proposed as an attempt to handle interval uncertainty that appears in many mathematical or computer models of some deterministic real-world phenomena in which uncertainties or vagueness pervade.
To observe some basic information and results of various type of fractional differential equations, one can see the papers and monographs of Samko et al. In the recent time this theory has been developed in theoretical directions, and a wide number of applications of this theory have been considered (see, for instance, [7–12]).
\end$$ $$\begin &H \bigl[X(t), Z(t) \bigr] \ &\quad \le H \bigl[\varphi(0), \psi(0) \bigr] \ &\qquad H \biggl[\frac \int_^ , \ &\qquad\frac \int_^ \biggr] \ &\qquad H \biggl[ \frac \int_^ , \frac \int_^ \biggr] \ &\qquad H \biggl[ \frac \int_^ , \frac \int_^ \biggr] \ &\quad \le C(t) \frac \int_^ .
\end$$ □ 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.