scholarly journals Tilt Stability, Uniform Quadratic Growth, and Strong Metric Regularity of the Subdifferential

2013 ◽  
Vol 23 (1) ◽  
pp. 256-267 ◽  
Author(s):  
D. Drusvyatskiy ◽  
A. S. Lewis
Keyword(s):  
Metric Regularity ◽  
Quadratic Growth ◽  
Strong Metric Regularity ◽  
Tilt Stability

A sequential implicit function theorem for the chords iteration

2013 ◽  
Vol 63 (5) ◽  
Author(s):  
Diana Nedelcheva
Keyword(s):  
Implicit Function Theorem ◽  
Local Convergence ◽  
Metric Regularity ◽  
Linear Convergence ◽  
Generalized Equation ◽  
Implicit Function ◽  
Strong Metric Regularity ◽  
Additional Assumptions

AbstractIn this paper we study the local convergence of the method $$0 \in f\left( {p,x_k } \right) + A\left( {x_{k + 1} - x_k } \right) + F\left( {x_{k + 1} } \right),$$ in order to find the solution of the generalized equation $$find x \in X such that 0 \in f\left( {p,x} \right) + F\left( x \right).$$ We first show that under the strong metric regularity of the linearization of the associated mapping and some additional assumptions regarding dependence on the parameter and the relation between the operator A and the Jacobian $$\nabla _x f\left( {\bar p,\bar x} \right)$$, we prove linear convergence of the method which is uniform in the parameter p. Then we go a step further and obtain a sequential implicit function theorem describing the dependence of the set of sequences of iterates of the parameter.

scholarly journals Higher differentiability results for solutions to a class of non-autonomous obstacle problems with sub-quadratic growth conditions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Andrea Gentile
Keyword(s):  
Besov Space ◽  
Lipschitz Space ◽  
Growth Conditions ◽  
Obstacle Problems ◽  
Quadratic Growth ◽  
Regularity Assumption ◽  
Lipschitz Regularity ◽  
Fixed Boundary ◽  
Higher Differentiability ◽  
Admissible Functions

Abstract We establish some higher differentiability results of integer and fractional order for solutions to non-autonomous obstacle problems of the form min ⁡ { ∫ Ω f ⁢ ( x , D ⁢ v ⁢ ( x ) ) : v ∈ K ψ ⁢ ( Ω ) } , \min\biggl{\{}\int_{\Omega}f(x,Dv(x)):v\in\mathcal{K}_{\psi}(\Omega)\biggr{\}}, where the function 𝑓 satisfies 𝑝-growth conditions with respect to the gradient variable, for 1 < p < 2 1<p<2 , and K ψ ⁢ ( Ω ) \mathcal{K}_{\psi}(\Omega) is the class of admissible functions v ∈ u 0 + W 0 1 , p ⁢ ( Ω ) v\in u_{0}+W^{1,p}_{0}(\Omega) such that v ≥ ψ v\geq\psi a.e. in Ω, where u 0 ∈ W 1 , p ⁢ ( Ω ) u_{0}\in W^{1,p}(\Omega) is a fixed boundary datum. Here we show that a Sobolev or Besov–Lipschitz regularity assumption on the gradient of the obstacle 𝜓 transfers to the gradient of the solution, provided the partial map x ↦ D ξ ⁢ f ⁢ ( x , ξ ) x\mapsto D_{\xi}f(x,\xi) belongs to a suitable Sobolev or Besov space. The novelty here is that we deal with sub-quadratic growth conditions with respect to the gradient variable, i.e. f ⁢ ( x , ξ ) ≈ a ⁢ ( x ) ⁢ | ξ | p f(x,\xi)\approx a(x)\lvert\xi\rvert^{p} with 1 < p < 2 1<p<2 , and where the map 𝑎 belongs to a Sobolev or Besov–Lipschitz space.

A general metric regularity in asplund banach spaces

1998 ◽  
Vol 19 (3-4) ◽  
pp. 215-226 ◽  
Author(s):  
T. Amahroq ◽  
A. Jourani ◽  
L. Thibault
Keyword(s):  
Banach Spaces ◽  
Metric Regularity

scholarly journals Quantification of leaf greenness and leaf spectral profile in plant diagnosis using an optical scanner

2012 ◽  
Vol 36 (3) ◽  
pp. 309-317 ◽  
Author(s):  
Ryoichi Doi
Keyword(s):  
Spectral Data ◽  
Crop Production ◽  
Response Pattern ◽  
Quadratic Growth ◽  
Spectral Profile ◽  
Growth Data ◽  
Data Set ◽  
Optical Scanner ◽  
Color Scale ◽  
Management Measures

Observation of leaf spectral profile (color) enables suitable management measures to be taken for crop production. An optical scanner was used: 1) to obtain an equation to determine the greenness of plant leaves and 2) to examine the power to discriminate among plants grown under different nutritional conditions. Sweet basil seedlings grown on vermiculite were supplemented with one-fifth-strength Hoagland solutions containing 0, 0.2, 1, 5, 20, and 50 mM NH4+. The 5 mM treatment resulted in the greatest leaf and shoot weights, indicating a quadratic growth response pattern to the NH4+ gradient. An equation involving b*, black and green to describe the greenness of leaves was provided by the spectral profiling of a color scale for rice leaves as the standard. The color scale values for the basil leaves subjected to 0.2 and 1 mM NH4+ treatments were 1.00 and 1.12, respectively. The other treatments resulted in significantly greater values of 2.25 to 2.42, again indicating a quadratic response pattern. Based on the spectral data set consisting of variables of red-green-blue and other color models and color scale values, in discriminant analysis, 81% of the plants were correctly classified into the six NH4+ treatment groups. Combining the spectral data set with the growth data set consisting of leaf and shoot weights, 92% of the plant samples were correctly classified whereas, using the growth data set, only 53% of plants were correctly classified. Therefore, the optical scanning of leaves and the use of spectral profiles helped plant diagnosis when biomass measurements were not effective.

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