Existance of solutions of nonlinear integral equations and Henstock–Kurzweil integrals

We prove an existence theorems for the nonlinear integral equation $$x(t)=f(t)+{\int }_0^a{k}_1(t,s)x(s)ds+{\int }_0^a{k}_2(t,s)g(s,x(s))ds,t\in I_a=[0,a],a\in {\mathbb{R}}_{+},$$ where $f,g,x$ are functions with values in Banach spaces. Our fundamental tools are: measures of noncompactness and properties of the Henstock-Kurzweil integral.