The nonlinear differential equation \(dz/dt+f(z,t)=0\) with Hilbert space valued continuous operator function need not have any solutions; the book is devoted to the monotonic and accretive subcases of the operator f which ensure the existence of effective approximative methods. It contains a large amount of very explicit results both of theoretical and of practical importance and may be extremely useful for specialists in nonlinear oscillations. The exposition is precise and brief. The book is concluded with comments and well-selected references.
The first third deals with preparatory information (which is of independent interest) including upper and lower solutions (in the plane), special differential inequalities, consideration of several types of monotonic many-valued operators (explicit criteria), subgradients and several types of accretive operators, subtle convexity properties of Banach spaces, theory of the operator equation \(Fx=0\) with the existence of solutions and estimates of the iterations \(x_{k+1}=x_ k-b_ kFx_ k.\)
The second third of the book is concerned with the main theme, the above mentioned differential equation. Existence and uniqueness, periodic, almost periodic, bounded and stable solutions, dependence on parameter, averaging methods are considered. The indefinitely monotonic operators f and the existence of relatively stable manifolds of solutions are also included. The nonlinear boundary value problem \(z(0)=Az(T)\) with precise estimates and certain higher order differential equations conclude this part. The last third contains the Galerkin method (also in the indefinite case) with exact estimates of the approximations. Finally, several pages are devoted to the accretive differential equations including the criteria for the existence of periodic, almost periodic and uniformly bounded solutions.