Some remarks on differential equations of quadratic type (Q1176691)

Let \(C\) be a closed, bounded convex subset of a Banach space \(X\) and \(f:C\to C\) a Lipschitz continuous map. The subject of the paper is the study of the differential equation (1) \(x'(t)+x(t)=f(x(t))\), \(x(0)=x_ 0\in C\). This can be considered the abstract formulation of partial differential equations as the Boltzmann model \[ {\partial Y\over \partial t}(t,x)+Y(t,x)=\int^ \infty_ 0\int^ \infty_ 0P(y,z\mid x)Y(t,z)dy dz. \tag{2} \] Equation (1) is studied by means of the theory of nonexpansive nonlinear semigroups and several sharp results are obtained for the existence and asymptotic behaviour of the solutions. A more detailed treatment is given to the case (as (2)) in which \(f(x)=Q(x,x)\) with \(Q:x\times x\to X\) a symmetric, continuous and bilinear map. A precise explicit formula is derived for the Lipschitz constant of \(f\) when \(C\) is the unit simplex of \(\mathbb{R}^ n\) with the \(\ell_ 1\) norm.