# On a Fractional Oscillator Equation with Natural Boundary Conditions

## TL;DR

This paper establishes the existence of solutions for a nonlinear fractional oscillator equation with natural boundary conditions using a transformation approach and monotonicity properties of fractional derivatives.

## Contribution

It introduces a novel method combining transformation and upper-lower solutions to prove existence for fractional oscillator equations with mixed derivatives.

## Key findings

- Existence of solutions proven for the fractional oscillator equation.
- Transformation reduces the problem to a lower order fractional boundary value problem.
- Monotonicity of the right Caputo derivative is established.

## Abstract

We prove existence of solutions for a nonlinear fractional oscillator equation with both left Riemann-Liouville and right Caputo fractional derivatives subject to natural boundary conditions. The proof is based on a transformation of the problem into an equivalent lower order fractional boundary value problem followed by the use of an upper and lower solutions method. To succeed with such approach, we first prove a result on the monotonicity of the right Caputo derivative.