A Family of Measures of Noncompactness in the Space $${L^1_{\text{loc}}(\mathbb{R}_+)}$$ L loc 1 ( R + ) and its Application to Some Nonlinear Volterra Integral Equation

Abstract

The aim of this paper is to study a new family of measures of noncompactness in the space $${L^1_{\text{loc}}(\mathbb{R}_+)}$$ consisting of all real functions locally integrable on $${\mathbb{R}_+}$$ , equipped with a suitable topology. As an example of applications of the technique associated with that family of measures of noncompactness, we study the existence of solutions of a nonlinear Volterra integral equation in the space $${L^1_{\text{loc}}(\mathbb{R}_+)}$$ . The obtained result generalizes several ones obtained earlier with help of other methods.