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

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.