This paper is mainly concerned with the existence of mild solutions and approximate controllability for a class of fractional semilinear systems with instantaneous and non-instantaneous impulses. By applying the Kuratowski measure of noncompactness and $\rho$-set contractive fixed-point theorem, the results for the considered system were obtained. In the end, an example was studied to support the main results.

This paper studies the existence of solutions for fractional dynamical systems with two damping terms in Banach space. First, we generalize the well-known Gronwall inequality. Next, according to fixed-point theorems and inequalities, the existence results for the considered system are obtained. At last, an example is used to support the main results.

In this paper, we mainly study the following boundary value problems of fractional discontinuous differential equations with impulses:

$$\{\begin{array}{l}{}_t^C{\mathcal{D}}_{0^{+}}^{\mathfrak{R}}\mathrm{\Lambda }(t)=\mathcal{E}(t)ϝ(t,\mathrm{\Lambda }(t)),a.e.t\in Q,\\ \mathrm{△}\mathrm{\Lambda }{|}_{t=t_{\kappa }}={\mathrm{\Phi }}_{\kappa }(\mathrm{\Lambda }(t_{\kappa })),\kappa =1,2,\cdots ,m,\\ \mathrm{△}{\mathrm{\Lambda }}^{\prime }{|}_{t=t_{\kappa }}=0,\kappa =1,2,\cdots ,m,\\ \vartheta \mathrm{\Lambda }(0)-\chi \mathrm{\Lambda }(1)={\int }_0^1{\varrho }_1(\upsilon )\mathrm{\Lambda }(\upsilon )d\upsilon ,\\ \zeta {\mathrm{\Lambda }}^{\prime }(0)-\delta {\mathrm{\Lambda }}^{\prime }(1)={\int }_0^1{\varrho }_2(\upsilon )\mathrm{\Lambda }(\upsilon )d\upsilon ,\end{array}$$

where $\vartheta >\chi >0,\zeta >\delta >0$, $\text{Phi\_{{kappa}}in C(text{}$\mathbb{R} $\text{}^{+}, text{}$\mathbb{R} $\text{}^{+})}$, $\mathcal{E},{\varrho }_1,{\varrho }_2\ge 0$ a.e. on $Q=[0,1]$, $\mathcal{E},{\varrho }_1,{\varrho }_2\in L^1(0,1)$ and $\text{digamma:[0, 1]times text{}$\mathbb{R} $\text{}^{+}rightarrow text{}$\mathbb{R} $\text{}^{+}}$, $\text{text{}$\mathbb{R} $\text{}^{+} = [0, +infty)}$. By using Krasnosel skii's fixed point theorem for discontinuous operators on cones, some sufficient conditions for the existence of single or multiple positive solutions for the above discontinuous differential system are established. An example is given to confirm the main results in the end.