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# Existence results for fractional differential equations in presence of upper and lower solutions

• * Corresponding author: Universidade de Santiago de Compostela, Instituto de Matemáticas, Santiago de Compostela, 15782, Spain

The second author is supported by grant numbers MTM2016-75140-P (AEI/FEDER, UE) and ED431C 2019/02 (GRC Xunta de Galicia)

• In this paper, we study some existence results for fractional differential equations subject to some kind of initial conditions. First, we focus on the linear problem and we give an explicit form of solutions by reduction to an integral problem. We analyze some properties of the solutions to the linear problem in terms of its coefficients. Then we provide examples of application of the mentioned properties. Secondly, with the help of this theory, we study the nonlinear problem subject to initial value conditions. By using the upper and lower solutions method and the monotone iterative algorithm, we show the existence and localization of solutions to the nonlinear fractional differential equation.

Mathematics Subject Classification: Primary: 34A08, 26A33; Secondary: 35G55.

 Citation: • • Figure 1.  Graph of the solution to (17)

Figure 2.  Graph of the solution for a negative forcing term

Table 1.  Sufficient conditions for the nonnegativity of solutions

 $\begin{array}{c} f \geq 0 \mbox{ on } [0,b] \\ c\geq 0 \\ \lambda \geq 0 \\ R\geq 0 \end{array}$ $\begin{array}{c} f \geq 0 \mbox{ on } [0,b] \\ c\geq 0 \\ \lambda < 0 \\ R\leq 0\end{array}$

Table 2.  Sufficient conditions for the nonpositivity of solutions

 $\begin{array}{c} f \leq 0 \mbox{ on } [0,b] \\ c\leq 0 \\ \lambda \geq 0 \\ R\geq 0 \end{array}$ $\begin{array}{c} f \leq 0 \mbox{ on } [0,b] \\ c\leq 0 \\ \lambda\geq 0 \\ R\leq 0\end{array}$
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