In this paper, we consider the following kind of fractional evolution equation driven by measure with nonlocal conditions: $$ \textstyle\begin{cases} {}^{C} D_{0+}^{\alpha }x(t)=Ax(t)\,dt+ (f(t,x(t))+Bu(t) )\,dg(t), \quad t\in (0, b],\\ x(0)+p(x)=x_{0}. \end{cases} $${D0+αCx(t)=Ax(t)dt+(f(t,x(t))+Bu(t))dg(t),t∈(0,b],x(0)+p(x)=x0. The regulated proposition of fractional equation is obtained for the first time. By noncompact measure method and fixed point theorems, we obtain some sufficient conditions to ensure the existence and nonlocal controllability of mild solutions. Finally, an illustrative example is given to show practical usefulness of the analytical results.

This article is devoted to the study of the existence and uniqueness of mild solution to a class of Riemann–Liouville fractional stochastic evolution equations driven by both Wiener process and fractional Brownian motion. Our results are obtained by using fractional calculus, stochastic analysis, and the fixed-point technique. Moreover, an example is provided to illustrate the application of the obtained abstract results.

In this paper, we study a kind of fuzzy differential equation with Hilfer–Katugampola fractional derivative and nonlocal condition. By using successive approximation method, we obtain some sufficient conditions to ensure the existence and uniqueness of solution. An illustrative example is given to show the practical usefulness of the analytical results.

The study is concerned with the Hadamard sequential fractional hybrid differential inclusions with two-point hybrid integral boundary conditions. With the help of the Dhage fixed-point theorem for the product of two operators and the Covitz-Nadler fixed-point theorem in the case of fractional inclusions, we obtain the existence results of solutions for Hadamard sequential fractional hybrid differential inclusions. Finally, two examples are presented to illustrate the main results.

In the paper, we investigate a kind of Hilfer fractional q-difference equations with nonlocal condition. Firstly, the existence and uniqueness results of solutions are obtained by using topological degree theory and Banach fixed point theorem. Subsequently, the existence of extremal solutions in an ordered Banach space is discussed by monotone iterative method. In that following, we consider the Ulam stability results for equations. Finally, two examples are given to illustrate the effectiveness of theory results.