On a Generalized Class of Nonlinear Equations Defined by Elliptic Symbols

The aim of this paper is to study the nonlinear and nonlocal equations of infinite order \(f(-\Lambda (D)) u(x)=U(x, u(x))\) $f(-\mathrm{\Lambda }(D))u(x)=U(x,u(x))$ on \({\mathbb {R}}^n\) ${\mathbb{R}}^n$ , in which \(f:{\mathbb {R}} \rightarrow {\mathbb {R}}\) $f:\mathbb{R}\to \mathbb{R}$ is a measurable function that satisfies a kind of ellipticity condition, \(\Lambda : {\mathbb {R}}^n \rightarrow {\mathbb {R}}\) $\mathrm{\Lambda }:{\mathbb{R}}^n\to \mathbb{R}$ is a positive continuous function and \(\Lambda (D)\) $\mathrm{\Lambda }(D)$ is a Fourier multiplier. We introduce a class of symbols to which f belongs and an Hilbert space in which we will find the solutions of these equations. In addition, we give conditions on the functions f and \(\Lambda \) $\mathrm{\Lambda }$ in order that the solutions we find are in fact real-analytic.