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Isolated zeros of lipschitzian metrically regular -Functions [PDF]

Optimization, 2001
Given a metrically regular locally Lipschitzian function sending $_{ℝn}$ into $_{ℝm}$,the structure of the preimages will be studied. In particular, for the case of m =n, it will be shown that all preimages are locally finite sets provided that the Lipschitzian function in question is directionally differentiable. Some consequences of this fact will be
Fusek, Peter
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Lipschitzian composition operators in some function spaces

Nonlinear Analysis: Theory, Methods & Applications, 1997
There are several function spaces \(X\) with the property that, whenever the Nemytskij operator \(F\phi(x)= f(x,\phi(x))\) is Lipschitz continuous in the norm of \(X\), the generating function \(f\) must be affine, i.e. \(f(x, y)= g(x)y+ h(x)\) with \(g,h\in X\). For example, in case \(X= \text{Lip}\) by the author [Funkc. Ekvacioj Ser. Int.
Janusz Matkowski
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On the qualitative approximation of Lipschitzian functions

Nonlinear Analysis: Theory, Methods & Applications, 2004
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Alonso, María, Marín, Luis Rodríguez
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A note on locally Lipschitzian functions

Mathematical Programming, 1995
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Pritchard, G., Gürkan, G., Ozge, A.Y.
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Subdifferential Regularity of Directionally Lipschitzian Functions

Canadian Mathematical Bulletin, 2000
AbstractFormulas for the Clarke subdifferential are always expressed in the form of inclusion. The equality form in these formulas generally requires the functions to be directionally regular. This paper studies the directional regularity of the general class of extended-real-valued functions that are directionally Lipschitzian.
Bounkhel, M., Thibault, L.
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Subgradient of distance functions with applications to Lipschitzian stability

Mathematical Programming, 2005
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Boris S. Mordukhovich, Nguyen Mau Nam
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Paraconvexity of the graphs of lipschitzian functions

Journal of Mathematical Sciences, 1996
Following \textit{E. Michael} [Math. Scand. 7, 372-376 (1960; Zbl 0093.12001)] a closed subset \(P\) of a Banach space \(B\) is called \(\alpha\)-paraconvex if for \(x\in B\), \(r> \text{dist} (x,P)\) and \(y\in\text{conv} (P\cap K(x,r))\) we have \(\text{dist} (y,P)\leq \alpha \cdot r\), where \(K(x,r): =\{z\in B:|z-x |\leq r\}\).
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A trust region algorithm for minimization of locally Lipschitzian functions

Mathematical Programming, 1994
The authors prove the global convergence of the classical trust region algorithm in the non-smooth case where the objective function is only locally Lipschitzian. The result is interesting.
Liqun Qi 0001, Jie Sun 0001
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Dini Derivatives of the Marginal Function of a Non-Lipschitzian Program

SIAM Journal on Optimization, 1996
Summary: Upper and lower bounds are establised for the Dini directional derivatives of the marginal function of a parametric mathematical program. In this program, the equality constraint functions are assumed to be strictly differentiable, but the objective and inequality constraint functions can belong to a large class of non-Lipschitzian functions ...
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Lipschitzian semigroups and abstract functional differential equations

Nonlinear Analysis: Theory, Methods & Applications, 2010
The authors consider the abstract functional differential equation \[ (FDE)\quad u'(t)=Au(t)+\Phi u_t, \quad t>0,\quad u(0)=x,\quad u_0=f, \] where \(A\) is a closed and densely defined linear operator, \(\Phi:L^p([-1,0];X)\to X\) is a globally Lipschitz operator, \(f\in L^p([-1,0];X)\) and \(u_t(\sigma):=u(t+\sigma)\). By assuming that the space \(X\)
Song, Xueli, Peng, Jigen
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