Peter Hess

On an abstract competition model and applications

is a map such that the coordinates of F(x,, x2) represent the population or population densities which have evolved from the state (xi, x2) at a fixed time later. Setting (yi, y2) = F(x,, x2), it is natural to suppose that an increase in x, coupled with a decrease in x2 results in an increase in y, and a decrease in y2 and, symmetrically, a decrease in x1 coupled with an increase in x2 results in a decrease in yr and an increase in yz. We define an order in E, x E2 as follows: If x,, xi E E,, x2, xi E E,, x1 I xi, and xi I x2, then we write (x1 7

3.—A Criterion for the Existence of Solutions of Non-linear Elliptic Boundary Value Problems

1 5 (4 9 x;>.

On the principal eigenvalue of a periodic-parabolic operator

By a new method it is proved that a non-linear elliptic boundary value problem of rather general type admits a weak solution lying between a given weak lower solution ϕ and a given weak upper solution ψ≧ϕ

Nonlinear parabolic boundary value problems with upper and lower solutions

On considere le probleme aux valeurs propres parabolique lineaire Lu=λmu dans Ω×R, u=0 sur ∂Ω×R, u(., 0)=u(., T) sur Ω et des generalisations non lineaires. L=∂/∂t+A(x, t, D), Ω⊂R N (N≥1) ∂Ω de classe C 2+ μ

Existence theorems for nonlinear noncoercive operator equations and nonlinear elliptic boundary value problems

The existence of at least one periodic solution of a very general second order nonlinear parabolic boundary value problem is proved under the assumption that a lower solution ϕ and an upper solution ψ with ϕ≦ψ are known.

On stabilization of discrete monotone dynamical systems

The purpose of this paper is to study the solvability of a nonlinear functional equation f E (A + B)u, where A is a maximal monotone (possibly multivalued) mapping from a reflexive Banach space X to its dual X* and B: X-+ X* satisfies some hind of sign condition. The main feature of the present method is that no asymptotic hypothesis (such as coerciveness, semicoerciveness, asymptotic oddness or homogeneity) is imposed on A + B. In Section 1 we state and prove the abstract result (Theorem 1). In Section 2 we apply Theorem 1 to the discussion of solvability of nonlinear second-order boundary value problems of Neumann type. Such problems were previously considered by Schatzman [6] and Hess [5] by different methods. This research was stimulated by some recent results on the range of the sum of maximal monotone operators announced by BrCzis [3].

An anti-maximum principle for linear elliptic equations with an indefinite weight function

A stabilization theorem for discrete strongly monotone and nonexpansive dynamical systems on a Banach lattice is proved. This result is applied to a periodic-parabolic semilinear initial-boundary value problem to show the convergence of solutions towards periodic solutions.

On a Second-Order Nonlinear Elliptic Boundary Value Problem

we denote a strongly uniformly elliptic linear differential expression of second order with real-valued coefficient functions a,, = akf, a/, a0 > 0 belonging to Ce(@ (0 , and A E R a parameter. In particular, m may change sign in a. Let L, be the differential operator induced by P and the Dirichlet boundary conditions, with domain D(L,) = {u E CZ+e(fi): u = 0 on 80). Note that L, is closable in C(o) (it admits a closed extension in Lp(0), 1 <p < co, having domain Weep n IPp(J2)). We set L := closure of L, in C(n). Then L is invertible. For the further study of L we introduce the real Banach spaces E := Co<.@ := (u E C(a): t) = 0 a*1 x := cgq := {u E Cl@): v = 0 on 8f2}, with C-n:rn (1 . (IE and ?iorm 11 . [IX, respectively. By the LP-theory for linear elliptic boundary value problems, L-’ maps C(a) compactly into Xc E. Let finally M: E --, E c C(n) denote the multiplication operator by the function m. We define u to be a solution of (1) provided

ON BIFURCATION FROM INFINITY FOR POSITIVE SOLUTIONS OF SECOND ORDER ELLIPTIC EIGENVALUE PROBLEMS

Publisher Summary This chapter reviews a second-order nonlinear elliptic boundary value problem. When Φ and ψ are the weak lower and upper solutions of problem (D), respectively, with Φ, ψ ∈ L ∞ (Ω), and such that Φ ≤ ψ a.e. in Ω. If there exist constants c 1 ≥ 0, ɛ > 0 and a function k 1 ∈ L 1 (Ω) such that for a.a. . Then problem (D) admits a weak solution u with Φ ≤ u ≤ φ a.e. in Ω. This theorem generalizes all the classical results by Cohen, Keller, Shampine, Laetsch, Simpson, Sattinger, Amann, and others, which are proved by constructing a monotone iteration scheme. It also extends the main result of Deuel and Hess in the case that Φ, ψ ∈ L ∞ (Ω).

An isoperimetric inequality for the principal eigenvalue of a periodic-parabolic problem

Publisher Summary This chapter focuses on bifurcation from infinity for positive solutions of second-order elliptic eigenvalue problems, and also presents a condition under which a function is a strongly uniformly elliptic linear differential expression of second order having real-valued coefficient functions. It presents an investigation of the existence of positive solutions of the nonlinear eigenvalue problem in the bounded domain. The chapter presents a theorem that can be derived by a standard transformation. It also focuses on the global behavior of Σ∞.