Herbert Amann

Fixed Point Equations and Nonlinear Eigenvalue Problems in Ordered Banach Spaces

This paper gives a survey over some of the most important methods and results of nonlinear functional analysis in ordered Banach spaces. By means of iterative techniques and by using topological tools, fixed point theorems for completely continuous maps in ordered Banach spaces are deduced, and particular attention is paid to the derivation of multiplicity results. Moreover, solvability and bifurcation problems for fixed point equations depending nonlinearly on a real parameter are investigated.In order to demonstrate the importance of the abstract results, there are given some nontrivial applications to nonlinear elliptic boundary value problems. But, of course, the abstract techniques and results of this paper apply also to a variety of other problems which are not considered here.This paper presents in a unified manner most of the recent work in this field. In addition, by making consequent use of the fixed point index for compact maps, short and simple proofs are obtained for most of the “classical” r...

Nonhomogeneous Linear and Quasilinear Elliptic and Parabolic Boundary Value Problems

It is the purpose of this paper to describe some of the recent developments in the mathematical theory of linear and quasilinear elliptic and parabolic systems with nonhomogeneous boundary conditions. For illustration we use the relatively simple set-up of reaction-diffusion systems which are — on the one h and — typical for the whole class of systems to which the general theory applies and — on the other h and — still simple enough to be easily described without too many technicalities. In addition, quasilinear reaction-diffusion equations are of great importance in applications and of actual mathematical and physical interest, as is witnessed by the examples we include.

Parabolic evolution equations and nonlinear boundary conditions

On etudie le probleme aux valeurs limites et initiales: u˙+#7B-A(t)u=f(t,u), #7B-B(t)u=g(t,u), 0<t≤T, u(0)=u 0 qui peut etre considere comme une contrepartie abstraite a des problemes paraboliques semilineaires

Dual semigroups and second order linear elliptic boundary value problems

It is shown that general second order elliptic boundary value problems on bounded domains generate analytic semigroups onL1. The proof is based on Phillips’ theory of dual semigroups. Several sharp estimates for the corresponding semigroups inLp, 1≦p<∞, are given.

On the number of solutions of nonlinear equations in ordered Banach spaces

Abstract Let E be an ordered Banach space and A a continuous operator mapping some bounded order interval [ v , w ] ⊂ E into itself. This paper is concerned with the number of fixed points of A on [ v , w ]. There are given conditions on A and the ordering which guarantee the existence of no fixed point, precisely one, two, and more than two, distinct fixed points. The nonexistence and uniqueness theorems are completely elementary. The multiplicity results are based on the fixed-point index for α-set contractions. All of these results have applications to nonlinear integral equations and to mildly nonlinear elliptic boundary-value problems.

Periodic Solutions of Semilinear Parabolic Equations

Publisher Summary This chapter describes the methods of nonlinear functional analysis, namely, fixed-point theorems in ordered Banach spaces, to prove existence and multiplicity result for periodic solutions of semilinear parabolic differential equations of the second order. The oldest method for the study of periodic solutions of differential equations is to find fixed points of the Poincare operator. Subsequently in the case of parabolic equations, it turns out that the Poincare operator is compact in suitable function spaces. Even in the case of the general semilinear parabolic equations, this operator is strongly increasing. Having seen that the Poincare operator is strongly increasing, it is clear that the problem can be included in the general framework of nonlinear equations in ordered Banach spaces. Hence, by applying other general fixed-point theorems for equations of this type, it is possible to obtain further existence and multiplicity results.

Invariant sets and existence theorems for semilinear parabolic and elliptic systems

INTRODUCTION In this paper we prove a g2obaZ existence theorem for classical solutions of second order semilinear parabolic systems of the form g + A(x, t, D) u =f(x, t, u, Du) in Q x (0, Tl, B(x, D) u = 0 on aJ x (0, q, u(., 0) = 240 on 0. Here u = (ul,..., @-‘): fi

Dynamic theory of quasilinear parabolic equations—I. Abstracts evolution equations

On etablit les fondements abstraits pour une theorie dynamique generale pour des problemes aux valeurs limites et initiales quasilineaires

Periodic Solutions of Asymptotically Linear Hamiltonian Systems.

We prove existence and multiplicity results for periodic solutions of time dependent and time independent Hamiltonian equations, which are assumed to be asymptotically linear. The periodic solutions are found as critical points of a variational problem in a real Hilbert space. By means of a saddle point reduction this problem is reduced to the problem of finding critical points of a function defined on a finite dimensional subspace. The critical points are then found using generalized Morse theory and minimax arguments.

Maximal regularity for nonautonomous evolution equations

We derive sufficient conditions, perturbation theorems in particular, for nonautonomous evolution equations to possess the property of maximal Lp regularity.