On Existence of Solutions of Parametrized Generalized Equations
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On Existence of Solutions of Parametrized Generalized Equations Asen L. Dontchev1,2 Received: 12 April 2020 / Accepted: 31 August 2020 / © Springer Nature B.V. 2020
Abstract In this paper we study existence of solutions to the generalized equation 0 ∈ f (p, x)+F (x), where f is a function, F is a set-valued mapping, and p is a parameter. Conditions are given, in terms of metric regularity of F , local convex-valuedness of F −1 , and partial calmness of f with respect to x uniformly in p, for the property that, for any p near the reference value, the generalized equation has a solution at a certain distance from the reference solution. Some corollaries and applications of this result are also presented. Keywords Generalized equations · Existence of solutions · Metric regularity · Local selection Mathematics Subject Classification (2010) 49J53 · 49J40 · 49K40 · 90C31 In this paper we consider generalized equations of the form 0 ∈ f (p, x) + F (x), Rd
(1)
where f : × → is a function of a parameter p and solution variable x, and F : Rm → → Rn is a set-valued mapping. The solution mapping of (1) is Rm
Rn
p → S(p) = {x ∈ Rm | 0 ∈ f (p, x) + F (x)}. Throughout we use the notation and terminology from [9]. Specifically, IB a (x) is the closed ball of radius a centered at x. The domain and the graph of the mapping F : Rm → → Rn are denoted by dom F and gph F , respectively, and the range of F is rge F . The distance from a point x ∈ Rm to a set C ⊂ Rm is denoted by d(x, C). The inverse to F is the Dedicated to Terry Rockafellar; mathematician, mentor and friend Supported by the National Science Foundation Award Number CMMI 1562209, the Austrian Science Foundation (FWF) Grant P31400-N32, and the Australian Research Council (ARC) Project DP160100854. Asen L. Dontchev
[email protected] https://sites.google.com/site/adontchev/ 1
Department of Aerospace Engineering, The University of Michigan, Ann Arbor, MI, USA
2
Department of Mathematics, University of Florida, Gainesville, FL, USA
A.L. Dontchev
mapping Rn y → F −1 (y) = {x ∈ Rm | y ∈ F (x)}. For a function f : Rd × Rm → Rn , the partial uniform calmness modulus with respect to x at (p, ¯ x) ¯ ∈ int dom f is defined as f (p, x) − f (p, x) ¯ clm x (f ; (p, ¯ x)) ¯ := lim sup , x − x ¯ p→p, ¯ x→x¯ while the partial uniform Lipschitz modulus with respect to x at (p, ¯ x) ¯ ∈ int dom f is ¯ x)) ¯ := lip x (f ; (p,
lim sup
p→p, ¯ x ,x→x, ¯ x =x
f (p, x ) − f (p, x) . x − x
¯ x)) ¯ < ∞ signals that the function f is calm with respect to x uniformly Then clm x (f ; (p, in p at (p, ¯ x); ¯ that is, there exist a constant c and neighborhoods Q of p¯ and U of x¯ such that f (p, x) − f (p, x) ¯ ≤ cx − x ¯ for every p ∈ Q and x ∈ U, ¯ x)) ¯ < ∞. In particular, when f does not depend on p, and similarly for lip x (f ; (p, then we say that f is calm at x¯ and write clm (f ; x). ¯ If clm x (f − A; (p, ¯ x)) ¯ = 0 for a linear mapping A, then A is the uniform in p around p¯ partial derivative of f with respect to x at (p, ¯ x) ¯ denoted ∇x
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