Searcharxiv⌕ Search

arXiv subjects

Jinlu Li

Publications and source records attributed to Jinlu Li.

At least 37 records · Page 2Linked to original sources
arXiv2025

Frechet and Mordukhovich Derivative (Coderivative) and Covering Constant for Single-Valued Mapping in Euclidean Space with Application (II)

We continue the study in part I for calculating the Frechet derivatives and Mordukhovich derivatives (coderivatives) and covering constants for single-valued mappings in Euclidean spaces (It is part I). In this paper, we particularly consider a norm-reserved mapping f: R^2 to R^2 that is defined by (1.1) in Section 1. We will find the precise solutions of Frechet derivative and Mordukhovich derivative at every point in R^2. By using these solutions, we will find the covering constant for this mapping f is exact 1 at every point in R^2 except the origin. Then we extend this mapping to R^4. Finally, by using the covering constant for f and by applying the Arutyunov Mordukhovich and Zhukovskiy Parameterized Coincidence Point Theorem, we will solve some parameterized equations.

math.FA↗
arXiv2025

Ill-posedness issue on the Oldroyd-B model in the critical Besov spaces

It is proved in \cite[J. Funct. Anal., 2020]{AP} that the Cauchy problem for some Oldroyd-B model is well-posed in $\B^{d/p-1}_{p,1}(\R^d) \times \B^{d/p}_{p,1}(\R^d)$ with $1\leq p<2d$. In this paper, we prove that the Cauchy problem for the same Oldroyd-B model is ill-posed in $\B^{d/p-1}_{p,r}(\R^d) \times \B^{d/p}_{p,r}(\R^d)$ with $1\leq p\leq \infty$ and $1< r\leq\infty$ due to the lack of continuous dependence of the solution.

math.AP↗
arXiv2024

Continuity of Metric Projection Operator from C[0, 1] onto Pn with Applications to Mordukhovich Derivatives

Let C[0, 1] be the Banach space of all continuous real valued functions on [0, 1]. For an arbitrarily given nonnegative integer n, let Pn denote the set of all polynomials with degree less than or equal to n. Pn is a closed subspace of C[0, 1]. In this paper, we first prove (in details) that the metric projection operator from C[0, 1] to Pn is a single-valued mapping and it is (norm to norm) continuous. Then, we use the continuity of Pn to investigate the Gateaux directional derivatives, some properties and fixed-point properties of the Mordukhovich derivatives of the metric projection operator.

math.FA↗
arXiv2024

Covering Constants for Metric Projection Operator with Applications to Stochastic Fixed-Point Problems

In this paper, we use the Mordukhovich derivatives to precisely find the covering constants for the metric projection operator onto nonempty closed and convex subsets in uniformly convex and uniformly smooth Banach spaces. We consider three cases of the subsets: closed balls in uniformly convex and uniformly smooth Banach spaces, closed and convex cylinders in l, and the positive cone in L, for some p. By using Theorem 3.1 in [2] and as applications of covering constants obtained in this paper, we prove the solvability of some stochastic fixed-point problems. We also provide three examples with specific solutions of stochastic fixed-point problems.

math.FA↗
arXiv2024

Fixed-point properties of the Mordukhovich differential operator

In this paper, we investigate some fixed-point properties of the Mordukhovich differential operator of set valued mappings (or, single valued mappings) on Banach spaces. In particular, we study the fixed-point properties of the Mordukhovich differential operator for the metric projection operator onto some closed and convex subsets in Banach spaces, such as, closed balls in Banach spaces, positive cones in real spaces l2 and l1 and sets of polynomials in C[0, 1].

math.FA↗
arXiv2024

Global existence and blow-up for the Euler-Poincaré equations with a class of initial data

In this paper we investigate the Cauchy problem of d-dimensional Euler-Poincaré equations. By choosing a class of new and special initial data, we can transform this d-dimensional Euler-Poincaré equations into the Camassa-Holm type equation in the real line. We first obtain some global existence results and then present a new blow-up result to the system under some different assumptions on this special class of initial data.

math.AP↗
arXiv2024

Strict Frechet Differentiability of the Metric Projection Operator in Hilbert Spaces

In this paper, we prove strict Frechet differentiability of the metric projection operator onto closed balls in Hilbert spaces and onto positive cones in Euclidean spaces. We find the exact expressions for Frechet derivatives. Since Frechet differentiability implies Gateaux directional differentiability, the results obtained in this paper strengthen the results obtained in [8] and [10] about the directional differentiability of the metric projection operator onto closed balls in Hilbert spaces and positive cones in Euclidean spaces and in the real Hilbert space l2.

math.FA↗
arXiv2024

Well-posedness and no-uniform dependence for the Euler-Poincaré equations in Triebel-Lizorkin spaces

In this paper, we study the Cauchy problem of the Euler-Poincaré equations in $\R^d$ with initial data belonging to the Triebel-Lizorkin spaces. We prove the local-in-time unique existence of solutions to the Euler-Poincaré equations in $F^s_{p,r}(\R^d)$. Furthermore, we obtain that the data-to-solution of this equation is continuous but not uniformly continuous in these spaces.

math.AP↗
arXiv2024

Nowhere-uniform continuity of the solution map of the Camassa-Holm equation in Besov spaces

In the paper, we gave a strengthening of our previous work in [32] (J. Differ. Equ. 269 (2020)) and proved that the data-to-solution map for the Camassa-Holm equation is nowhere uniformly continuous in $B^s_{p,r}(\R)$ with $s>\max\{1+1/{p},3/2\}$ and $(p,r)\in [1,\infty]\times[1,\infty)$. The method applies also to the b-family of equations which contain the Camassa-Holm and Degasperis-Procesi equations.

math.AP↗
arXiv2024

Mordukhovich derivatives of the set-valued metric projection operator in general Banach spaces

In this paper, we investigate the properties and the precise solutions of the Mordukhovich derivatives of the set-valued metric projection operator onto some closed balls in some general Banach spaces. In the Banach space c, we find the properties of Mordukhovich derivatives of the set-valued metric projection operator onto the closed subspace c0. We show that the metric projection from C[0, 1] to polynormal with degree less than or equal to n is a single-valued mapping. We investigate its Mordukhovich derivatives and Gateaux directional derivatives.

math.FA↗
arXiv2024

Non-uniform dependence on initial data for the Camassa--Holm equation in Besov spaces: Revisited

In the paper, we revisit the uniform continuity properties of the data-to-solution map of the Camassa--Holm equation on the real-line case. We show that the data-to-solution map of the Camassa--Holm equation is not uniformly continuous on the initial data in Besov spaces $B_{p, r}^s(\mathbb{R})$ with $s>\frac{1}{2}$ and $1\leq p, r< \infty$, which improves the previous works [Himonas et al., Asian J. Math., 11 (2007)], [Li et al., J. Differ. Equ., 269 (2020)] and [Li et al., J. Math. Fluid Mech., 23 (2021)]. Furthermore, we present a strengthening of our previous work in [Li et al., J. Differ. Equ., 269 (2020)] and prove that the data-to-solution map for the Camassa--Holm equation is nowhere uniformly continuous in $B^s_{p,r}(\mathbb{R})$ with $s>\max\{1+1/{p},3/2\}$ and $(p,r)\in [1,\infty]\times[1,\infty)$. The method applies also to the b-family of equations which contain the Camassa--Holm and Degasperis--Procesi equations.

math.AP↗
arXiv2024

Generalized differentiability of the metric projection operator in Hilbert spaces

In this paper, we study the generalized differentiability of the metric projection operator in Hilbert spaces. We find exact expressions for Mordukhovich derivatives for the metric projection operator onto closed balls in Hilbert spaces and positive cones in Euclidean spaces and in real Hilbert space l2. We investigate the connections between Frechet differentiability, Gateaux directional differentiability and the Mordukhovich derivatives of the metric projection operator in Hilbert spaces.

math.FA↗
arXiv2024

On the ill-posedness for the Navier--Stokes equations in the weakest Besov spaces

It is proved in \cite{IO21} that the Cauchy problem for the full compressible Navier--Stokes equations of the ideal gas is ill-posed in $\dot{B}_{p, q}^{2 / p}(\mathbb{R}^2) \times \dot{B}_{p, q}^{2 / p-1}(\mathbb{R}^2) \times \dot{B}_{p, q}^{2 / p-2}(\mathbb{R}^2) $ with $1\leq p\leq \infty$ and $1\leq q<\infty$. In this paper, we aim to solve the end-point case left in \cite{IO21} and prove that the Cauchy problem is ill-posed in $\dot{B}_{p, \infty}^{d / p}(\mathbb{R}^d) \times \dot{B}_{p, \infty}^{d / p-1}(\mathbb{R}^d) \times \dot{B}_{p, \infty}^{d / p-2}(\mathbb{R}^d)$ with $1\leq p\leq\infty$ by constructing a sequence of initial data which shows that the solution map is discontinuous at zero. As a by-product, we demonstrate that the incompressible Navier--Stokes equations is also ill-posed in $\dot{B}_{p,\infty}^{d/p-1}(\mathbb{R}^d)$, which is an interesting open problem in itself.

math.AP↗
arXiv2024

Strict Frechet and generalized differentiability of the metric projection operator onto balls in Hilbert spaces

In this paper, we prove strict Frechet differentiability of the metric projection operator onto closed balls in Hilbert spaces, and we find exact expressions for Frechet derivatives. Since Frechet differentiability implies Gateaux directional differentiability, the results obtained in this paper strengthens the results in [8] and [10] about the directional differentiability of the metric projection operator onto closed balls in Hilbert spaces. We apply the Frechet differentiability of the metric projection onto closed balls in Hilbert spaces to study the generalized differentiability of the metric projection operator.

math.FA↗
arXiv2024

Frechet differentiability of the metric projection operator in Banach spaces

In this paper, we prove Frechet differentiability of the metric projection operator onto closed balls, closed and convex cylinders and positives cones in uniformly convex and uniformly smooth Banach spaces. With respect to these closed and convex subsets, we find the exact expressions for Frechet derivatives and Gateaux directional derivatives of the metric projection operator.

math.FA↗
arXiv2023

OpenPerf: A Benchmarking Framework for the Sustainable Development of the Open-Source Ecosystem

Benchmarking involves designing scientific test methods, tools, and frameworks to quantitatively and comparably assess specific performance indicators of certain test subjects. With the development of artificial intelligence, AI benchmarking datasets such as ImageNet and DataPerf have gradually become consensus standards in both academic and industrial fields. However, constructing a benchmarking framework remains a significant challenge in the open-source domain due to the diverse range of data types, the wide array of research issues, and the intricate nature of collaboration networks. This paper introduces OpenPerf, a benchmarking framework designed for the sustainable development of the open-source ecosystem. This framework defines 9 task benchmarking tasks in the open-source research, encompassing 3 data types: time series, text, and graphics, and addresses 6 research problems including regression, classification, recommendation, ranking, network building, and anomaly detection. Based on the above tasks, we implemented 3 data science task benchmarks, 2 index-based benchmarks, and 1 standard benchmark. Notably, the index-based benchmarks have been adopted by the China Electronics Standardization Institute as evaluation criteria for open-source community governance. Additionally, we have developed a comprehensive toolkit for OpenPerf, which not only offers robust data management, tool integration, and user interface capabilities but also adopts a Benchmarking-as-a-Service (BaaS) model to serve academic institutions, industries, and foundations. Through its application in renowned companies and institutions such as Alibaba, Ant Group, and East China Normal University, we have validated OpenPerf's pivotal role in the healthy evolution of the open-source ecosystem.

cs.SE↗