Equilibria of set-valued maps on nonconvex domains

Abstract

We present new theorems on the existence of equilibria (or zeros) of convex as well as nonconvex set-valued maps defined on compact neighborhood retracts of normed spaces. The maps are subject to tangency conditions expressed in terms of new concepts of normal and tangent cones to such sets. Among other things, we show that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a compact neighborhood retract with nontrivial Euler characteristic in a Banach space <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E"> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding="application/x-tex">E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi colon upper K long right-arrow 2 Superscript upper E"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Φ </mml:mi> <mml:mo>:</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false"> ⟶ </mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>E</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding="application/x-tex">\Phi :K\longrightarrow 2^E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is an upper hemicontinuous set-valued map with nonempty closed convex values satisfying the tangency condition <disp-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper Phi left-parenthesis x right-parenthesis intersection upper T Subscript upper K Superscript r Baseline left-parenthesis x right-parenthesis not-equals normal empty-set for all x element-of upper K comma"> <mml:semantics> <mml:mrow> <mml:mi mathvariant="normal"> Φ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ∩ </mml:mo> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mi>K</mml:mi> <mml:mi>r</mml:mi> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ≠ </mml:mo> <mml:mi mathvariant="normal"> ∅ </mml:mi> <mml:mtext> for all </mml:mtext> <mml:mi>x</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>K</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\begin{equation*} \Phi (x)\cap T_K^r(x)\neq \emptyset \text { for all }x\in K, \end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> then there exists <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x 0 element-of upper K"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo> ∈ </mml:mo> <mml:mi>K</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">x_0\in K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="0 element-of normal upper Phi left-parenthesis x 0 right-parenthesis period"> <mml:semantics> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo> ∈ </mml:mo> <mml:mi mathvariant="normal"> Φ </mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>x</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">0\in \Phi (x_0).</mml:annotation> </mml:semantics> </mml:math> </inline-formula> Here, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper T Subscript upper K Superscript r Baseline left-parenthesis x right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mi>K</mml:mi> <mml:mi>r</mml:mi> </mml:msubsup> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">T_K^r(x)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes a new concept of retraction tangent cone to <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> at <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x"> <mml:semantics> <mml:mi>x</mml:mi> <mml:annotation encoding="application/x-tex">x</mml:annotation> </mml:semantics> </mml:math> </inline-formula> suited for compact neighborhood retracts. When <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is locally convex at <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x comma upper T Subscript upper K Superscript r Baseline left-parenthesis x right-parenthesis"> <mml:semantics>