Admissible Function

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Xiyin Zheng - One of the best experts on this subject based on the ideXlab platform.

  • well posedness and generalized metric subregularity with respect to an Admissible Function
    Science China-mathematics, 2019
    Co-Authors: Binbin Zhang, Xiyin Zheng
    Abstract:

    In the framework of complete metric spaces, this paper provides several sufficient conditions for the well-posedness with respect to an Admissible Function, which improves some known results on error bounds. As applications, we consider the generalized metric subregularity of a closed multiFunction between two complete metric spaces with respect to an Admissible Function φ. Even in the special case when φ(t) = t, our results improve (or supplement) some results on error bounds in the literature.

  • Stable Well-posedness and Tilt stability with respect to Admissible Functions
    ESAIM: Control Optimisation and Calculus of Variations, 2017
    Co-Authors: Xiyin Zheng
    Abstract:

    Note that the well-posedness of a proper lower semicontinuous Function f can be equivalently described using an Admissible Function. In the case when the objective Function f undergoes the tilt perturbations in the sense of Poliquin and Rockafellar, adopting Admissible Functions ϕ and ψ , this paper introduces and studies the stable well-posedness of f with respect to ϕ (in brief, ϕ -SLWP) and tilt-stable local minimum of f with respect to ψ (in brief, ψ -TSLM). In the special case when ϕ ( t ) = t 2 and ψ ( t ) = t , the corresponding ϕ -SLWP and ψ -TSLM reduce to the stable second order local minimizer and tilt stable local minimum respectively, which have been extensively studied in recent years. We discover an interesting relationship between two Admissible Functions ϕ and ψ : ψ ( t ) = ( ϕ ′) -1 ( t ), which implies that a proper lower semicontinuous Function f on a Banach space has ϕ -SLWP if and only if f has ψ -TSLM. Using the techniques of variational analysis and conjugate analysis, we also prove that the strong metric ϕ ′-regularity of ∂f is a sufficient condition for f to have ϕ -SLWP and that the strong metric ϕ ′-regularity of ∂ [co( f + δ B X [ x,r ] ) ] for some r > 0 is a necessary condition for f to have ϕ -SLWP. In the special case when ϕ ( t ) = t 2 , our results cover some existing main results on the tilt stability.

  • error bound and well posedness with respect to an Admissible Function
    Applicable Analysis, 2016
    Co-Authors: Xiyin Zheng
    Abstract:

    Using techniques of variational analysis and in terms of subdifferential, we study the generalized error bound defined by an Admissible Function, an interesting extension of the Holder error bound and the usual error bound. With the help of a chain rule of subdifferential established in the paper, without the solvability assumption, we provide a sufficient condition for a nonconvex inequality to have a global generalized error bound. We also provide some sufficient and/or necessary conditions for an inequality to have Holder error bounds. As applications, we consider well-posedness with respect to an Admissible Function.

  • Stable Well-posedness and Tilt stability with respect to Admissible Functions
    arXiv: Optimization and Control, 2016
    Co-Authors: Xiyin Zheng
    Abstract:

    Note that the well-posedness of a proper lower semicontinuous Function $f$ can be equivalently described using an Admissible Function. In the case when the objective Function $f$ undergos the tilt perturbations in the sense of Poliquin and Rockafellar, adopting Admissible Functions $\varphi$ and $\psi$, this paper introduces and studies the stable well-posedness of $f$ with respect to $\varphi$ (in breif, $\varphi$-SLWP) and tilt-stable local minimum of $f$ with respect to $\psi$ (in brief, $\psi$-TSLM). In the special case when $\varphi(t)=t^2$ and $\psi(t)=t$, the corresponding $\varphi$-SLWP and $\psi$-TSLM reduce to the stable second local minimizer and tilt stable local minimum respectively, which have been extensively studied in recent years. We discover an interesting relationship between two Admissible Functions $\varphi$ and $\psi$: $\psi(t)=(\varphi')^{-1}(t)$, which implies that a proper lower semicontinous Function $f$ on a Banach space has $\varphi$-SLWP if and only if $f$ has $\psi$-TSLM. Using the techniques of variational analysis and conjugate analysis, we also prove that the strong metric $\varphi'$-regularity of $\partial f$ is a sufficient condition for $f$ to have $\varphi$-SLWP and that the strong metric $\varphi'$-regularity of $\partial\overline{\rm co}(f+\delta_{B[\bar x,r]})$ for some $r>0$ is a necessary condition for $f$ to have $\varphi$-SLWP. In the special case when $\varphi(t)=t^2$, our results cover some existing main results on the tilt stability.

  • generalized metric subregularity and regularity with respect to an Admissible Function
    Siam Journal on Optimization, 2016
    Co-Authors: Xiyin Zheng
    Abstract:

    In this paper, adopting an Admissible Function $\varphi$, we consider a kind of generalized metric subregularity/regularity of a multiFunction $F$ with respect to $\varphi$, which is a natural generalization of the Holder metric regularity. In the special case when $F$ is the subdifferential mapping of a proper lower semicontinuous Function $f$, it is known that such a generalized metric subregularity is very closely related to the well-posedness of $f$. Using the technique of variational analysis and in terms of the coderivative, we established some sufficient conditions for a multiFunction to be metrically subregular/regular with respect to an Admissible Function $\varphi$. In particular, we extend some existing results on the metric regularity and Holder metric regularity.

Yuan Zhou - One of the best experts on this subject based on the ideXlab platform.

  • Localized BMO and BLO spaces on RD-spaces and applications to Schrödinger operators
    Communications on Pure and Applied Analysis, 2010
    Co-Authors: Dachun Yang, Yuan Zhou
    Abstract:

    An RD-space χ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling condition holds in χ. Let ρ be an Admissible Function on RD-space χ. The authors first introduce the localized spaces BMO ρ (χ) and BLO ρ (χ) and establish their basic properties, including the John-Nirenberg inequality for BMO ρ (χ), several equivalent characterizations for BLOρ(χ), and some relations between these spaces. Then the authors obtain the boundedness on these localized spaces of several operators including the natural maximal operator, the Hardy-Littlewood maximal operator, the radial maximal Functions and their localized versions associated to ρ, and the Littlewood-Paley g-Function associated to ρ, where the Littlewood-Paley g-Function and some of the radial maximal Functions are defined via kernels which are modeled on the semigroup generated by the Schrödinger operator. These results apply in a wide range of settings, for instance, to the Schrödinger operator or the degenerate Schrödinger operator on ℝ d , or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups.

  • Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrödinger operators
    Nagoya Mathematical Journal, 2010
    Co-Authors: Dachun Yang, Yuan Zhou
    Abstract:

    Let be a space of homogeneous type in the sense of Coifman and Weiss, and let be a collection of balls in . The authors introduce the localized atomic Hardy space the localized Morrey-Campanato space and the localized Morrey-Campanato-BLO (bounded lower oscillation) space with α ∊ ℝ and p ∊ (0, ∞) , and they establish their basic properties, including and several equivalent characterizations for In particular, the authors prove that when α > 0 and p ∊ [1, ∞), then and when p ∈(0,1], then the dual space of is Let ρ be an Admissible Function modeled on the known auxiliary Function determined by the Schrödinger operator. Denote the spaces and , respectively, by and when is determined by ρ . The authors then obtain the boundedness from of the radial and the Poisson semigroup maximal Functions and the Littlewood-Paley g -Function, which are defined via kernels modeled on the semigroup generated by the Schrödinger operator. These results apply in a wide range of settings, for instance, the Schrödinger operator or the degenerate Schrödinger operator on ℝ d , or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups.

  • Endpoint properties of localized riesz transforms and fractional integrals associated to schrödinger operators
    Potential Analysis, 2009
    Co-Authors: Dachun Yang, Yuan Zhou
    Abstract:

    Let $\{{\backslash}mathcal L\}{\backslash}equiv-{\}Delta+V$ be the Schr{ö}dinger operator in ${\{{\backslash}mathbb R}^n}$ , where V is a nonnegative Function satisfying the reverse H{ö}lder inequality. Let $ρ$ be an Admissible Function modeled on the known auxiliary Function determined by V. In this paper, the authors characterize the localized Hardy spaces $H^1\_{\backslash}rho({\{{\backslash}mathbb R}^n})$ in terms of localized Riesz transforms and establish the boundedness on the BMO-type space $\{{\backslash}mathop{\}mathrm{BMO\_{\backslash}rho(\{{\backslash}mathbb R}^n)}}$ of these operators as well as the boundedness from $\{{\backslash}mathop{\}mathrm{BMO\_{\backslash}rho(\{{\backslash}mathbb R}^n)}}$ to $\{{\backslash}mathop{\}mathrm{BLO\_{\backslash}rho(\{{\backslash}mathbb R}^n)}}$ of their corresponding maximal operators, and as a consequence, the authors obtain the Fefferman--Stein decomposition of $\{{\backslash}mathop{\}mathrm{BMO\_{\backslash}rho(\{{\backslash}mathbb R}^n)}}$ via localized Riesz transforms. When $ρ$ is the known auxiliary Function determined by V, $\{{\backslash}mathop{\}mathrm{BMO\_{\backslash}rho(\{{\backslash}mathbb R}^n)}}$ is just the known space \${\backslash}mathop{\}mathrm{BMO}_\{{\backslash}mathcal L}({\{{\backslash}mathbb R}^n})$ , and $\{{\backslash}mathop{\}mathrm{BLO\_{\backslash}rho(\{{\backslash}mathbb R}^n)}}$ in this case is correspondingly denoted by \${\backslash}mathop{\}mathrm{BLO}_\{{\backslash}mathcal L}({\{{\backslash}mathbb R}^n})$ . As applications, when n\thinspace≥\thinspace3, the authors further obtain the boundedness on \${\backslash}mathop{\}mathrm{BMO}_\{{\backslash}mathcal L}({\{{\backslash}mathbb R}^n})$ of Riesz transforms \${\backslash}nabla\{{\backslash}mathcal L}^{-1/2}$ and their adjoint operators, as well as the boundedness from \${\backslash}mathop{\}mathrm{BMO}_\{{\backslash}mathcal L}({\{{\backslash}mathbb R}^n})$ to \${\backslash}mathop{\}mathrm{BLO}_\{{\backslash}mathcal L}({\{{\backslash}mathbb R}^n})$ of their maximal operators. Also, some endpoint estimates of fractional integrals associated to $\{{\backslash}mathcal L}$ are presented.

Dachun Yang - One of the best experts on this subject based on the ideXlab platform.

  • Boundedness of Littlewood-Paley Operators Associated with Gauss Measures
    Journal of Inequalities and Applications, 2010
    Co-Authors: Dachun Yang
    Abstract:

    Modeled on the Gauss measure, the authors introduce the locally doubling measure metric space Open image in new window , which means that the set Open image in new window is endowed with a metric Open image in new window and a locally doubling regular Borel measure Open image in new window satisfying doubling and reverse doubling conditions on Admissible balls defined via the metric Open image in new window and certain Admissible Function Open image in new window . The authors then construct an approximation of the identity on Open image in new window , which further induces a Calderon reproducing formula in Open image in new window for Open image in new window . Using this Calderon reproducing formula and a locally variant of the vector-valued singular integral theory, the authors characterize the space Open image in new window for Open image in new window in terms of the Littlewood-Paley Open image in new window -Function which is defined via the constructed approximation of the identity. Moreover, the authors also establish the Fefferman-Stein vector-valued maximal inequality for the local Hardy-Littlewood maximal Function on Open image in new window . All results in this paper can apply to various settings including the Gauss measure metric spaces with certain Admissible Functions related to the Ornstein-Uhlenbeck operator, and Euclidean spaces and nilpotent Lie groups of polynomial growth with certain Admissible Functions related to Schrodinger operators.

  • Localized BMO and BLO spaces on RD-spaces and applications to Schrödinger operators
    Communications on Pure and Applied Analysis, 2010
    Co-Authors: Dachun Yang, Yuan Zhou
    Abstract:

    An RD-space χ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling condition holds in χ. Let ρ be an Admissible Function on RD-space χ. The authors first introduce the localized spaces BMO ρ (χ) and BLO ρ (χ) and establish their basic properties, including the John-Nirenberg inequality for BMO ρ (χ), several equivalent characterizations for BLOρ(χ), and some relations between these spaces. Then the authors obtain the boundedness on these localized spaces of several operators including the natural maximal operator, the Hardy-Littlewood maximal operator, the radial maximal Functions and their localized versions associated to ρ, and the Littlewood-Paley g-Function associated to ρ, where the Littlewood-Paley g-Function and some of the radial maximal Functions are defined via kernels which are modeled on the semigroup generated by the Schrödinger operator. These results apply in a wide range of settings, for instance, to the Schrödinger operator or the degenerate Schrödinger operator on ℝ d , or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups.

  • Localized Morrey-Campanato spaces on metric measure spaces and applications to Schrödinger operators
    Nagoya Mathematical Journal, 2010
    Co-Authors: Dachun Yang, Yuan Zhou
    Abstract:

    Let be a space of homogeneous type in the sense of Coifman and Weiss, and let be a collection of balls in . The authors introduce the localized atomic Hardy space the localized Morrey-Campanato space and the localized Morrey-Campanato-BLO (bounded lower oscillation) space with α ∊ ℝ and p ∊ (0, ∞) , and they establish their basic properties, including and several equivalent characterizations for In particular, the authors prove that when α > 0 and p ∊ [1, ∞), then and when p ∈(0,1], then the dual space of is Let ρ be an Admissible Function modeled on the known auxiliary Function determined by the Schrödinger operator. Denote the spaces and , respectively, by and when is determined by ρ . The authors then obtain the boundedness from of the radial and the Poisson semigroup maximal Functions and the Littlewood-Paley g -Function, which are defined via kernels modeled on the semigroup generated by the Schrödinger operator. These results apply in a wide range of settings, for instance, the Schrödinger operator or the degenerate Schrödinger operator on ℝ d , or the sub-Laplace Schrödinger operator on Heisenberg groups or connected and simply connected nilpotent Lie groups.

  • Endpoint properties of localized riesz transforms and fractional integrals associated to schrödinger operators
    Potential Analysis, 2009
    Co-Authors: Dachun Yang, Yuan Zhou
    Abstract:

    Let $\{{\backslash}mathcal L\}{\backslash}equiv-{\}Delta+V$ be the Schr{ö}dinger operator in ${\{{\backslash}mathbb R}^n}$ , where V is a nonnegative Function satisfying the reverse H{ö}lder inequality. Let $ρ$ be an Admissible Function modeled on the known auxiliary Function determined by V. In this paper, the authors characterize the localized Hardy spaces $H^1\_{\backslash}rho({\{{\backslash}mathbb R}^n})$ in terms of localized Riesz transforms and establish the boundedness on the BMO-type space $\{{\backslash}mathop{\}mathrm{BMO\_{\backslash}rho(\{{\backslash}mathbb R}^n)}}$ of these operators as well as the boundedness from $\{{\backslash}mathop{\}mathrm{BMO\_{\backslash}rho(\{{\backslash}mathbb R}^n)}}$ to $\{{\backslash}mathop{\}mathrm{BLO\_{\backslash}rho(\{{\backslash}mathbb R}^n)}}$ of their corresponding maximal operators, and as a consequence, the authors obtain the Fefferman--Stein decomposition of $\{{\backslash}mathop{\}mathrm{BMO\_{\backslash}rho(\{{\backslash}mathbb R}^n)}}$ via localized Riesz transforms. When $ρ$ is the known auxiliary Function determined by V, $\{{\backslash}mathop{\}mathrm{BMO\_{\backslash}rho(\{{\backslash}mathbb R}^n)}}$ is just the known space \${\backslash}mathop{\}mathrm{BMO}_\{{\backslash}mathcal L}({\{{\backslash}mathbb R}^n})$ , and $\{{\backslash}mathop{\}mathrm{BLO\_{\backslash}rho(\{{\backslash}mathbb R}^n)}}$ in this case is correspondingly denoted by \${\backslash}mathop{\}mathrm{BLO}_\{{\backslash}mathcal L}({\{{\backslash}mathbb R}^n})$ . As applications, when n\thinspace≥\thinspace3, the authors further obtain the boundedness on \${\backslash}mathop{\}mathrm{BMO}_\{{\backslash}mathcal L}({\{{\backslash}mathbb R}^n})$ of Riesz transforms \${\backslash}nabla\{{\backslash}mathcal L}^{-1/2}$ and their adjoint operators, as well as the boundedness from \${\backslash}mathop{\}mathrm{BMO}_\{{\backslash}mathcal L}({\{{\backslash}mathbb R}^n})$ to \${\backslash}mathop{\}mathrm{BLO}_\{{\backslash}mathcal L}({\{{\backslash}mathbb R}^n})$ of their maximal operators. Also, some endpoint estimates of fractional integrals associated to $\{{\backslash}mathcal L}$ are presented.

Zhou Ding - One of the best experts on this subject based on the ideXlab platform.

  • the application of a type of new Admissible Function to the vibration of rectangular plates
    Computers & Structures, 1994
    Co-Authors: Zhou Ding
    Abstract:

    Abstract A new fast converging series consisting of static beam Functions under point load is used as Admissible Functions in the Rayleigh-Ritz method to study the problem of the flexural vibration of thin, isotropic rectangular plates. The Admissible sets of displacement Functions are obtained by varying the location of the point load applied to the beam. Which type of Admissible Function to be selected depends on the boundary conditions of the plate. Some numerical results are given for the rectangular plates with various aspect ratios and boundary conditions. It is demonstrated that the method may be used to tackle such plate problems and has high accuracy and good convergence compared with the available results, the calculations of mass and rigid matrices being very simple.

Jaime Navarro - One of the best experts on this subject based on the ideXlab platform.