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 study existence and uniqueness of solutions for the Hammerstein equation $$u\left(x\right)=v\left(x\right)+\lambda {\int }_{I_a^b}K(x,y)f(y,u\left(y\right))dy,x\in I_a^b:=[a_1,b_1]\times [a_2,b_2],$$ in the space $B{V}_{\varphi }^{ℝ}\left(I_a^b\right)$ of function of bounded total $\varphi -$variation in the sense of Riesz, where $\lambda \in ℝ$, $K:I_a^b\times I_a^b\to ℝ$ and $f:I_a^b\times ℝ\to ℝ$ are suitable functions.

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.

We prove that a set of weak solutions of the nonlinear Volterra integral equation has the Kneser property. The main condition in our result is formulated in terms of axiomatic measures of weak noncompactness.

In this paper we consider a model of a one-dimensional body where strain depends on the history of stress. We show local existence for large data and global existence for small data of classical solutions and convergence of the displacement, strain and stress to zero for time going to infinity.