In this paper, we obtain the Gronwall type inequality for generalized fractional operators unifying Riemann-Liouville and Hadamard fractional operators. We apply this inequality to the dependence of the solution of differential equations, involving generalized fractional derivatives, on both the order and the initial conditions. More properties for the generalized fractional operators are formulated and the solutions of initial value problems in certain new weighted spaces of functions are established as well.

<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 &lt; \alpha &lt; 1 $, $ \lambda, \beta &gt; 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>

In this paper, starting from a fixed δ-sequence, we use the generalized Taylor's formula based on conformable derivatives and the neutrix limit to find the powers of the Dirac delta function δ r and (δ ) r for any r ∈ R. MSC: 26A24; 44A15; 46F10; 46F30; 46T30; 58C05

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