The paper is devoted to the study of the measure-driven differential inclusions dx(t)∈G(t,x(t))dμ(t), x(0)=x0 for arbitrary finite Borel measure μ. This type of results allows one to treat in a similar manner differential and difference inclusions, as well as impulsive problems and therefore to study the evolution of hybrid systems with very complex (including Zeno) behavior. Our method is based on viewing the Borel measures as Lebesgue-Stieltjes measures. We thus obtain, under very general assumptions, the existence of regulated or bounded variation solutions of the considered problem and we indicate some advantages of our approach.

In this paper we prove an existence theorem for the Cauchy problem $$x'\left( t \right) = f\left( {t,x\left( t \right)} \right),x\left( 0 \right) = x_0 ,t \in I_\alpha = \left[ {0,\alpha } \right]$$ using the Henstock-Kurzweil-Pettis integral and its properties. The requirements on the function f are not too restrictive: scalar measurability and weak sequential continuity with respect to the second variable. Moreover, we suppose that the function f satisfies some conditions expressed in terms of measures of weak noncompactness.

Here, we propose a general framework covering a wide variety of fractional operators. We consider integral and differential operators and their role in tempered fractional calculus and study their analytic properties. We investigate tempered fractional integral operators acting on subspaces of $L_1[a,b]$, such as Orlicz or Hölder spaces. We prove that in this case, they map Orlicz spaces into (generalized) Hölder spaces. In particular, they map Hölder spaces into the same class of spaces. The obtained results are a generalization of classical results for the Riemann–Liouville fractional operator and constitute the basis for the use of generalized operators in the study of differential and integral equations. However, we will show the non-equivalence differential and integral problems in the spaces under consideration.

Let E be a Banach space with the topological dual \(E^*\). The aim of this paper is two-fold. On the one hand, we prove some basic properties of Hadamard-type fractional integral operators. These results are related to earlier results about integral operators acting on different function spaces, but for the vector-valued case they are of independent interest. Note that we discuss it in a rather general setting. We study Hadamard–Pettis integral operators in both single and multivalued case. On the other hand, we apply these results to obtain the existence of solutions of the fractional-type problem $$\begin{aligned} \frac{d^\alpha x(t)}{d t^\alpha }= \lambda f(t,x(t)), \quad \alpha \in (0,1),~~t \in [1,e], \quad x(1)+ b x(e)=h \end{aligned}$$ with certain constants \(\lambda , b\), where \(h \in E\) and \(f: [1,e]\times E \rightarrow E\) is Pettis integrable function such that, for every \(\varphi \in E^*\), \(\varphi f\) lies in an appropriate Orlicz spaces. Here \(\frac{d^\alpha }{d t^\alpha }\) stands the Caputo–Hadamard fractional differential operator.