We present an existence theorem for integral equations of Urysohn-Volterra type involving fuzzy set valued mappings. A fixed point theorem due to Schauder is the main tool in our analysis.

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.