This paper focuses on the approximate controllability of Hilfer fractional neutral Volterra integro-differential inclusions via almost sectorial operators. Almost sectorial operators, fractional differential, Leray-Schauder fixed point theorem and multivalued maps are used to prove the result. We start by emphasizing the existence of a mild solution and demonstrate the approximate controllability of the fractional system. In addition, an example is presented to demonstrate the principle.

By refining the standard Riccati substitution technique, integral averaging technique and comparison principle, we obtain new oscillation and asymptotic behavior for a class of third-order neutral differential equations with discrete and distributed delay. These criteria dealing with some cases have not been covered by the existing results in the literature. We present many sufficient conditions and related examples in order to illustrate the main results.

The motivation for this paper is to create new Philos-type oscillation criteria that are established for third-order mixed neutral differential equations with distributed deviating arguments. The key idea of our approach is to use the triple of the Riccati transformation techniques and the integral averaging technique. The established criteria improve, simplify and complement results that have been published recently in the literature. An example is also given to demonstrate the applicability of the obtained conditions.

This paper deals with three-dimensional differential system of nonlinear fractional order problem \begin{document}$ \begin{align*} D^{\alpha}_{0^{+}}\upsilon(\varrho) = f(\varrho,\omega(\varrho),\omega^{\prime}(\varrho),\omega^{\prime\prime}(\varrho),...,\omega^{(n-1)}(\varrho)), \; \varrho \in (0,1),\\ D^{\beta}_{0^{+}}\nu(\varrho) = g(\varrho, \upsilon(\varrho),\upsilon^{\prime}(\varrho),\upsilon^{\prime\prime}(\varrho),...,\upsilon^{(n-1)}(\varrho)), \; \varrho \in (0,1),\\ D^{\gamma}_{0^{+}}\omega(\varrho) = h(\varrho,\nu(\varrho),\nu^{\prime}(\varrho),\nu^{\prime\prime}(\varrho),...,\nu^{(n-1)}(\varrho)), \; \varrho \in (0,1), \end{align*} $\end{document} with the boundary conditions, \begin{document}$ \begin{align*} \upsilon(0) = \upsilon^{\prime}(0) = ... = \upsilon^{(n-2)}(0) = 0,\; \upsilon^{(n-1)}(0) = \upsilon^{(n-1)}(1),\\ \nu(0) = \nu^{\prime}(0) = ... = \nu^{(n-2)}(0) = 0,\; \nu^{(n-1)}(0) = \nu^{(n-1)}(1),\\ \omega(0) = \omega^{\prime}(0) = ... = \omega^{(n-2)}(0) = 0,\; \omega^{(n-1)}(0) = \omega^{(n-1)}(1), \end{align*} $\end{document} where $ D^{\alpha}_{0^{+}}, D^{\beta}_{0^{+}}, D^{\gamma}_{0^{+}} $ are the standard Caputo fractional derivative, $ n-1 < \alpha, \beta, \gamma \leq n, \; n \geq 2 $ and we derive sufficient conditions for the existence of solutions to the fraction order three-dimensional differential system with boundary value problems via Mawhin's coincidence degree theory, and some new existence results are obtained. Finally, an illustrative example is presented.

ON THE OSCILLATION OF A THIRD ORDER NONLINEAR DIFFERENTIAL EQUATIONS WITH NEUTRAL TYPE