A measure of weak noncompactness in \(L^1(\mathbb{R}^N)\) and applications (Q2113571)

In this article a new measure of weak noncompactness in the Banach space \(L^1(\mathbb{R}^N)\) is proposed. It is a generalization of the Banas-Knap measure of weak noncompactness, which in turn is a generalization of the De Blasi measure of weak noncompactness. Then, the multidimensional nonlinear functional integral equation \[ u(t,x)=f(t,x,u(t,x))+ g\left( t,x, \int_0^t \int_{\Omega}\, k(t,x,s,y,u(s,y)) dyds \right),\ t\ge 0,\, x\in \Omega. \] is considered. The existence of solutions to this equation is provided using a recent fixed point theorem of Krasnosel'skii type combined with the new previously introduced measure of weak noncompactness. The above equation easily includes the following Volterra-Fredholm integral equation as a special case \[ u(t,x)=f(t,x,u(t,x))+ \int_0^t \int_{\Omega}\, k(t,x,s,y,u(s,y)) dyds ,\ t\ge 0,\, x\in \Omega. \] Further, also the mixed Volterra-Fredholm integral equation \[ u(t,x)=\phi(x)+\frac1\eta \left( \int_0^t \big( v(s,x)-u(s,x) \big) ds + \int_0^t \int_{\Omega}\, h(s,x,y,u(s,y)) dyds \right) \] can be seen as a particular case of the multidimensional nonlinear functional integral equation under investigation. This last equation is obtained by integration from the integro-differential equation of parabolic type deriving from the neural modeling fields \[ \eta\frac{\partial}{\partial t}u(t,x)+u(t,x)=v(t,x) + \int_{\Omega} h(t,x,y,u(t,y)) dy,\ t\in [0,T],\, x\in \Omega, \] with \(u(0,x)=\phi(x),\ x\in \Omega\).