We prove some existence theorems for nonlinear integral equations of the Urysohn type $x\left(t\right)=\phi \left(t\right)+\lambda {\int }_0^{a}f(t,s,x\left(s\right))ds$ and Volterra type $x\left(t\right)=\phi \left(t\right)+{\int }_0^{t}f(t,s,x\left(s\right))ds$, $t\in I_a=[0,a]$, where f and φ are functions with values in Banach spaces. Our fundamental tools are: measures of noncompactness and properties of the Henstock-Kurzweil integral.

In this paper we investigate weakly continuous solutions of some integral equations in Banach spaces. Moreover, we prove a fixed point theorem which is very useful in our considerations.

In this paper, we prove an existence theorem for the pseudo-non-local Cauchy problem $x^{\text{'}}\left(t\right)+Ax\left(t\right)=f(t,x\left(t\right),{\int }_{t₀}^{t}k(t,s,x\left(s\right))ds)$, x₀(t₀) = x₀ - g(x), where A is the infinitesimal generator of a C₀ semigroup of operator ${T\left(t\right)}_{t>0}$ on a Banach space. The functions f,g are weakly-weakly sequentially continuous and the integral is taken in the sense of Pettis.

The set of solutions of a Volterra equation in a Banach space with a Carathéodory kernel is proved to be an ${ℛ}_{\delta }$, in particular compact and connected. The kernel is not assumed to be uniformly continuous with respect to the unknown function and the characterization is given in terms of a B₀-space of continuous functions on a noncompact domain.