Langevin differential equation in frame of ordinary and Hadamard fractional derivatives under three point boundary conditions

Abstract

<abstract><p>In this paper, we study a type of Langevin differential equations within ordinary and Hadamard fractional derivatives and associated with three point local boundary conditions</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \mathcal{D}_{1}^{\alpha} \left( \mathrm{D}^{2} + \lambda^{2}\right) x(t) = f\left( t, x(t), \mathcal{D}_1^{\alpha} \left[ x\right] (t) \right), $\end{document} </tex-math></disp-formula></p> <p>$ \mathrm{D}^{2} x\left(1 \right) = x(1) = 0 $, $ x(e) = \beta x(\xi) $, for $ t\in \left(1, e\right) $ and $ \xi \in (1, e] $, where $ 0 < \alpha < 1 $, $ \lambda, \beta > 0 $, $ \mathcal{D}_1^\alpha $ denotes the Hadamard fractional derivative of order $ \alpha $, $ \mathrm{D} $ is the ordinary derivative and $ f:[1, e]\times C([1, e], \mathbb{R})\times C([1, e], \mathbb{R})\rightarrow C([1, e], \mathbb{R}) $ is a continuous function. Systematical analysis of existence, stability and solution's dependence of the addressed problem is conducted throughout the paper. The existence results are proven via the Banach contraction principle and Schaefer fixed point theorem. We apply Ulam's approach to prove the Ulam-Hyers-Rassias and generalized Ulam-Hyers-Rassias stability of solutions for the problem. Furthermore, we investigate the dependence of the solution on the parameters. Some illustrative examples along with graphical representations are presented to demonstrate consistency with our theoretical findings.</p></abstract>