Ekeland's Variational Principle for Set-Valued Functions

Abstract

Abstract. We establish several set-valued function versions of Ekeland's variational principle and hence provide some sufficient conditions ensuring the existence of error bounds for inequality systems defined by finitely many lower semicontinuous functions.

On error bounds for systems

“…The (γ , A)-descent condition is a generalization of γ -descent condition defined in Liu and Ng (2011), Section 4. In fact, for a family F, if the γ -descent condition holds at some point x, formulas (4.6) and (4.7) in Liu and Ng (2011) imply (10) and (11), with (γ , A) replaced by (γ /n, I), where I is the n × n identity matrix.…”

“…In fact, for a family F, if the γ -descent condition holds at some point x, formulas (4.6) and (4.7) in Liu and Ng (2011) imply (10) and (11), with (γ , A) replaced by (γ /n, I), where I is the n × n identity matrix. Therefore, the γ -descent condition at x implies that F satisfies the the (γ /n, I)-descent condition at x.…”

“…The error bound property of a single convex (piecewise) polynomial also gains many researcher's attention (see Yang 2008;Li 2010Li , 2013. Sufficient conditions ensuring error bound property for systems of general functions have been provided in terms of some abstract subdifferentials by Wu and Ye (2001), and in terms of upper Dinidirectional derivatives by Liu and Ng (2011).…”

“…In Liu and Ng (2011), some sufficient conditions were discussed ensuring error bounds for family of functions. Along this line we provide some further relaxed conditions in the present paper so as to make use of the powerful matrices machinery in light of applications.…”

An equilibrium version of vectorial Ekeland variational principle and its applications to equilibrium problems

“…In the last four decades, the famous EVP emerged as one of the most important results of nonlinear analysis and it has significant applications in optimization, optimal control theory, game theory, fixed point theory, nonlinear equations, dynamical systems, etc. Motivated by its wide applications, many authors have been interested in extending EVP to the case with vector-valued maps or set-valued maps, see for example [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] and the references therein. Motivated by its wide applications, many authors have been interested in extending EVP to the case with vector-valued maps or set-valued maps, see for example [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24] and the references therein.…”