The Bolzano mean-value theorem and partial differential equations

Abstract

We study the existence of solutions to abstract equations of the form 0 = Au + F (u), u ∈ K ⊂ E, where A is an abstract differential operator acting in a Banach space E, K is a closed convex set of constraints being invariant with respect to resolvents of A and perturbations are subject to different tangency condition. Such problems are closely related to the so-called Poicaré-Miranda theorem, being the multi-dimensional counterpart of the celebrated Bolzano intermediate value theorem. In fact our main results can and should be regarded as infinite-dimensional variants of Bolzano and Miranda-Poincaré theorems. Along with single-valued problems we deal with set-valued ones, yielding the existence of the so-called constrained equilibria of set-valued maps. The abstract results are applied to show existence of (strong) steady state solutions to some weakly coupled systems of drift reaction-diffusion equations or differential inclusions of this type. In particular we get the existence of strong solutions to the Dirichlet, Neumann and periodic boundary problems for elliptic partial differential inclusions under the presence of state constraints of different type. Certain aspects of the Bernstein theory for bvp for second order ODE are studied, too. No assumptions concerning structural coupling (monotonicity, cooperativity) are undertaken.

The Poincaré–Miranda theorem and viability condition

“…Our aim is to link assumptions of this theorem to the so-called viability (tangential) condition and to show how some recent generalisations of the 1 Poincaré-Miranda theorem can be deduced from it. Though [12] contains a similar observation (cf. Remarks 2.4 and 2.6 from [12]), it lacks details and refers to [10], while the proof we recall here is much simpler than the one given in [10].…”

“…Though [12] contains a similar observation (cf. Remarks 2.4 and 2.6 from [12]), it lacks details and refers to [10], while the proof we recall here is much simpler than the one given in [10].…”

Degree for weakly upper semicontinuous perturbations of quasi-m-accretive operators

“…The invariance of this type is a subject of intensive research (e.g. [3]). (iv) Let us also address the following question.…”

“…when F is not a self-map of K. It is to observe that the very existence of results are related to some classical issues in analysis (see a survey [1,2], and, e.g. [3]). In [4], motivated by the initial study in [5] (see the references therein), the existence of equilibria of a weakly tangent set-valued F : U E on locally compact L-retracts K has been addressed.…”

Periodic Solutions to Second-Order Nonlinear Differential Equations in Banach Spaces

“…: V,-vol existence for systems of k equations x = f (t, x, x ), where f : [0, 1] × R k × R k → R k is a vector function, subject to various boundary conditions. Recently, the existence of periodic solutions of second-order differential equations on a general Banach space has been treated in several papers; for instance, see [14,17,19]. In [14], operator-valued Fourier multiplier theorems are used to study this problem.…”