On the Existence of Equilibria and Fixed Points of Maps under Constraints

“…We call (1) an avoiding cones condition. (For a proof of Theorem 1, see, e.g., [6,8].) In this paper we would like to investigate what happens when Ω is not convex.…”

Section: Theoremmentioning

confidence: 99%

“…There are many other possible definitions of normal cone in the nonconvex case (see [9, page 232] for a clarifying survey), and several theorems on the existence of equilibria are available (see, e.g., the well-written review paper [8]). The main novelty of our paper is allowing the normal cones to vanish at certain points, still recovering the existence result.…”

Section: Theoremmentioning

confidence: 99%

On the topological degree of planar maps avoiding normal cones

Fonda¹,

Klun²

2019

TMNA

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The classical Poincaré-Bohl theorem provides the existence of a zero for a function avoiding external rays. When the domain is convex, the same holds true when avoiding normal cones. We consider here the possibility of dealing with nonconvex sets having inward corners or cusps, in which cases the normal cone vanishes. This allows us to deal with situations where the topological degree may be strictly greater than 1.

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“…Note that in condition (2.10) the Bouligand cone has been replaced by the Clarke cone; there are examples showing that (2.8) is not sufficient (see [35]); however if ϕ = f is a single-valued map, then (2.8) implies (2.10). It is also evident that the weak inwardness in the sense of Clarke cone implies the existence of fixed points.…”

Section: 2mentioning

confidence: 99%

The Bolzano mean-value theorem and partial differential equations

Kryszewski

Siemianowski

2018

Journal of Mathematical Analysis and Applications

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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.

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On the Existence of Equilibria and Fixed Points of Maps under Constraints

Cited by 17 publications

References 62 publications

Inward contractions on metric spaces

Inward contractions on metric spaces

On the topological degree of planar maps avoiding normal cones

The Bolzano mean-value theorem and partial differential equations

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