Solving An Initial Value Problem

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]).

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Finally, a new technique to find the exact solutions of problem (1.1) is provided and two examples are given to illustrate this technique.

In this section, we present basic notations and necessary preliminaries used throughout the paper.

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].

In the recent time this theorem has been extended and generalized by several authors in various ways, for instance, the results on the existence of a fixed point in partially ordered sets for first order ordinary differential equations, Fredholm and Volterra type integral equations, among others, have been studied.

In particular, some applications of fixed points in partially ordered sets to resolution of matrix equation were presented by Ran and Reurings in [29], and the applicability of the existence of a unique fixed point for mappings defined in partially ordered sets to the study of the existence of a unique solution for periodic boundary condition problems for integer order ordinary differential equations was shown by Nieto and Rodríguez-López in [30, 31].

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\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).


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