Biomedical subjects
Publications and source records attributed to F E Browder.
A classical degree function is constructed for pseudomonotone mappings from a reflexive Banach space to its dual, using Galerkin approximations. This generalizes the Leray-Schauder degree when the Banach space is a Hilbert space and yields a flexible analytical tool for the study of nonlinear elliptic problems of higher order in divergence form.
The classical degree function constructed earlier for pseudomonotone mappings has been used to develop a broader degree theory of classical type for the sum of a maximal monotone map from a reflexive Banach space to its dual together with a bounded pseudomonotone map. The proof uses the generalized Yosida approximation of the maximal monotone mapping.
The classical degree function developed earlier for pseudomonotone operators from a reflexive Banach space to its dual is extended for the special case of Sobolev spaces to the sum of a pseudomonotone mapping (representing a nonlinear elliptic operator) and a strongly nonlinear lower order term.
An existence and uniqueness result is established for a general class of variational inequalities for parabolic partial differential equations of the form partial differentialu/ partial differentialt + A(u) + g(u) = f with g nondecreasing but satisfying no growth condition. The proof is based upon a type of compactness result for solutions of variational inequalities that should find a variety of other applications.
An existence and uniqueness result is presented for the solution of a parabolic initial-boundary value problem under Dirichlet null boundary conditions for a general parabolic equation of order 2m with a strongly nonlinear zeroth-order perturbation. This is the parabolic generalization of a class of elliptic results considered earlier by the writers and others and is based upon a new compactness theorem.
If K is a compact convex subset of a locally convex topological vector space X, we consider a continuous mapping f of K into X. A fixed-point theorem is proved for such a map f under the assumption that for a given continuous realvalued function p on K x X with p(x,y) convex in y and for each point x in K not fixed by f, there exists a point y in the inward set I(K)(x) generated by K at x with p(x,y - f(x)) less than p(x,x - f(x)). For X a Banach space, in particular, this yields a sharp extension and a drastic simplification of the fixed point theory of weakly inward (and weakly outward) mappings. The result comes close in the domain of mappings of compact convex sets to the thrust of fixed point conditions of the Leray-Schauder type for compact maps of sets with interior in X.
A general boundary value problem of variational type is considered for a general quasi-linear elliptic partial differential operator of order 2m in generalized divergence form. Such problems are considered on an arbitrary domain in a Euclidean space without hypotheses of boundedness on the domain or smoothness on its boundary. Contrary to the prevailing doctrine in the literature, it is shown that the corresponding operator between Banach spaces is pseudo-monotone, and that a wide variety of existence results can be derived from this fact.
This paper studies the solution of the nonlinear Hammerstein equation u(x) + k(x,y)f[y,u(y)]mu(dy) = h(x) in the singular case, i.e., where the linear operator K with kernel k(x,y) is not defined for all the range of the nonlinear mapping F given by Fu(y) = f[y,u(y)] over the whole class X of functions u which are potential solutions of the equation. An existence theorem is derived under relatively minimal assumptions upon k and f, namely that (Ku,u) >/= 0, that K maps L(1) into L(1) (loc) and is compact from L(1) [unk] L(infinity) into L(1) (loc), that f(y,s) has the same sign as s for s >/= R, and that for each constant r > 0, f(y,s) </= g(r)(y) for s </= r where g is bounded and summable. The proof is obtained by combining a priori bounds, a truncation procedure, and a convergence argument using the Dunford-Pettis theorem.
For a continuous self mapping f of a locally convex topological vector space which is locally compact (i.e., f maps a neighborhood of each point into a relatively compact set), it is shown that a sufficient condition for the existence of a fixed point is the existence of a compact attractor K(0) such that each orbit under f has a point of K(0) in its closure. The proof is based upon the circle of ideas of the Lefschetz fixed point theorem.
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