# Existence and uniqueness of positive solutions for nonlinear fractional mixed problems

## TL;DR

This paper investigates the existence and uniqueness of positive solutions for nonlinear fractional differential equations with mixed boundary conditions, using Green's functions, fixed point theory, and the method of lower and upper solutions.

## Contribution

It provides new conditions for existence and uniqueness of solutions to nonlinear fractional mixed problems involving Riemann-Liouville derivatives.

## Key findings

- Existence of at least one solution under certain asymptotic conditions.
- Development of the method of lower and upper solutions for these problems.
- Conditions under which the Banach contraction principle guarantees uniqueness.

## Abstract

This paper is devoted to study the existence and uniqueness of solutions of a one parameter family of nonlinear fractional differential equation with mixed boundary value conditions. Riemann-Liouville fractional derivative is considered. An exhaustive study of the sign of the related Green's function is done. Under suitable assumptions on the asymptotic behavior of the nonlinear part of the equation at zero and at infinity, and by application of the fixed theory of compact operators defined in suitable cones, it is proved the existence of at least one solution of the considered problem. Moreover it is developed the method of lower and upper solutions and it is deduced the existence of solutions by a combination of both techniques. In some particular situations, the Banach contraction principle is used to ensure the uniqueness of solutions.