The pandemic SARS-CoV-2 has become an undying virus to spread a sustainable disease named COVID-19 for upcoming few years.Mortality rates are rising rapidly as approved drugs are not yet available.Isolation from the i...The pandemic SARS-CoV-2 has become an undying virus to spread a sustainable disease named COVID-19 for upcoming few years.Mortality rates are rising rapidly as approved drugs are not yet available.Isolation from the infected person or community is the preferred choice to protect our health.Since humans are the only carriers,it might be possible to control the positive rate if the infected population or host carriers are isolated from each other.Isolation alone may not be a proper solution.These are the resolutions of previous research work carried out on COVID-19 throughout the world.The present scenario of the world and public health is knocking hard with a big question of critical uncertainty of COVID-19 because of its imprecise database as per daily positive cases recorded all over the world and in India as well.In this research work,we have pre-sented an optimal control model for COVID-19 using granular differentiability based on fuzzy dynamical systems.In the first step,we created a fuzzy Susceptible-Exposed-Infected-Asymptomatic-Hospitalized-Recovered-Death(SEIAHRD)model for COVID-19,analyzed it using granular differentiability,and reported disease dynamics for time-independent disease control parameters.In the second step,we upgraded the fuzzy dynamical system and granular differentiability model related to time-dependent disease control parameters as an optimal control problem invader.Theoretical studies have been validated with some practical data from the epidemic COVID-19 related to the Indian perspective during first wave and early second wave.展开更多
Let X be a real uniformly convex and uniformly smooth Banach space and C a nonempty closed and convex subset of X.Let Π_(C):X→C denote the generalized metric projection operator introduced by Alber in[1].In this pap...Let X be a real uniformly convex and uniformly smooth Banach space and C a nonempty closed and convex subset of X.Let Π_(C):X→C denote the generalized metric projection operator introduced by Alber in[1].In this paper,we define the Gâteaux directional differentiability of Π_(C).We investigate some properties of the Gâteaux directional differentiability of Π_(C).In particular,if C is a closed ball,or a closed and convex cone(including proper closed subspaces),or a closed and convex cylinder,then,we give the exact representations of the directional derivatives of Π_(C).By comparing the results in[12]and this paper,we see the significant difference between the directional derivatives of the generalized metric projection operator Π_(C) and the Gâteaux directional derivatives of the standard metric projection operator PC.展开更多
Characterizations of differentiability are obtained for continuous convex functions defined on nonempty open convex sets of Banach spaces as a generalization and application of a mumber of mathematicians several years...Characterizations of differentiability are obtained for continuous convex functions defined on nonempty open convex sets of Banach spaces as a generalization and application of a mumber of mathematicians several years effort, and a characteristic theorem is given for Banach spaces which are (weak) Asplund spaces.展开更多
The aim of this paper is to investigate the differentiability(Gateaux differentiabllity and subdifferentiability) of continuous convex functions on locally convex spaces and to study the behaviour of some important re...The aim of this paper is to investigate the differentiability(Gateaux differentiabllity and subdifferentiability) of continuous convex functions on locally convex spaces and to study the behaviour of some important results for this research area in locally convex spaces.展开更多
The aim of this paper is to prove the following theorem concerning the term by term differentiation the-orem of Walsh-Kaczmarz series. Let (ck) be a decreasing real sequence withare integrable functions and f(x) is a....The aim of this paper is to prove the following theorem concerning the term by term differentiation the-orem of Walsh-Kaczmarz series. Let (ck) be a decreasing real sequence withare integrable functions and f(x) is a. e. dyadic (or Butzer and Wagner) differentiate withThe function Kk means the kth Walsh-Kaczmarz function.展开更多
A locally convex space is said to be a Gateaux differentiability space (GDS) provided every continuous convex function defined on a nonempty convex open subset D of the space is densely Gateaux differentiable in .D.Th...A locally convex space is said to be a Gateaux differentiability space (GDS) provided every continuous convex function defined on a nonempty convex open subset D of the space is densely Gateaux differentiable in .D.This paper shows that the product of a GDS and a family of separable Prechet spaces is a GDS,and that the product of a GDS and an arbitrary locally convex space endowed with the weak topology is a GDS.展开更多
This paper considers the differentiability of C 0 semigroups with respect to (w.r.t.) parameters contained in their infinitesimal generators.It is proved that the generalized continuity and strong differentiability ...This paper considers the differentiability of C 0 semigroups with respect to (w.r.t.) parameters contained in their infinitesimal generators.It is proved that the generalized continuity and strong differentiability of their infinitesimal generators w.r.t.parameters imply the differentiability of the C 0 semigroups.The results are applied to the differentiability of the solution of a linear delay differential equation w.r.t.its delays.展开更多
Many papers on a wide range of control problems for Pritchard-Salamon systems have appeared and many of its important mathematical and system theoretical properties have been revealed. This paper deals with the differ...Many papers on a wide range of control problems for Pritchard-Salamon systems have appeared and many of its important mathematical and system theoretical properties have been revealed. This paper deals with the differentiability of the Pritchard-Salamon system with admissible state-feedback. Spectrum analysis showed that under definite condition, the unbounded perturbation semigroup of the Pritchard-Salamon system is eventually differentiable.展开更多
It is shown that an arbitrary function from D Rn to Rm will become C0,a-continuous in almost every x∈ D after restriction to a certain subset with limit pointx. For n 〉 m differentiability can be obtained. Example...It is shown that an arbitrary function from D Rn to Rm will become C0,a-continuous in almost every x∈ D after restriction to a certain subset with limit pointx. For n 〉 m differentiability can be obtained. Examples show the Ho1der exponent a=min{1,n/m}is optimal.展开更多
The sufficient conditions of Holder continuity of two kinds of fractal interpolation functions defined by IFS (Iterated Function System) were obtained. The sufficient and necessary condition for its differentiability ...The sufficient conditions of Holder continuity of two kinds of fractal interpolation functions defined by IFS (Iterated Function System) were obtained. The sufficient and necessary condition for its differentiability was proved. Its derivative was a fractal interpolation function generated by the associated IFS, if it is differentiable.展开更多
Some basic concepts for functions defined on subsets of the unit sphere,such as the s-directional derivative,s-gradient and s-Gateaux and s-Frechet differentiability etc,are introduced and investigated.These concepts ...Some basic concepts for functions defined on subsets of the unit sphere,such as the s-directional derivative,s-gradient and s-Gateaux and s-Frechet differentiability etc,are introduced and investigated.These concepts are different from the usual ones for functions defined on subsets of Euclidean spaces,however,the results obtained here are very similar.Then,as applications,we provide some criterions of s-convexity for functions defined on unit spheres which are improvements or refinements of some known results.展开更多
In this paper,we study the differentiability of solutions on the boundary for equartions of type L_λu=~2u/x^2+|x|^(2λ)~2u/y^2=f(x,y),whereλis an arbitrary positive number.By introducing a proper metric that is rela...In this paper,we study the differentiability of solutions on the boundary for equartions of type L_λu=~2u/x^2+|x|^(2λ)~2u/y^2=f(x,y),whereλis an arbitrary positive number.By introducing a proper metric that is related to the elliptic operator L_λ,we prove the differentiability on the boundary when some well-posed boundary conditions are satisfied.The main difficulty is the construction of new barrier functions in this article.展开更多
The purpose of this paper is to verify the Smulyan lemma for the support function, and also the Gateaux differentiability of the support function is studied on its domain. Moreover, we provide a characterization of Fr...The purpose of this paper is to verify the Smulyan lemma for the support function, and also the Gateaux differentiability of the support function is studied on its domain. Moreover, we provide a characterization of Frechet differentiability of the support function on the extremal points.展开更多
This paper studies the smoothness of solutions of the higher dimensional polynomial-like iterative equation. The methods given by Zhang Weinian([7]) and by Kulczvcki M, Tabor j.([3]) are improved by constructing a new...This paper studies the smoothness of solutions of the higher dimensional polynomial-like iterative equation. The methods given by Zhang Weinian([7]) and by Kulczvcki M, Tabor j.([3]) are improved by constructing a new operator for the structure of the equation in order to apply fixed point theorems. Existence, uniqueness and stability of continuously differentiable solutions are given.展开更多
In this paper,we study the differentiability of the solutions of stochastic differential equations driven by the G-Brownian motion with respect to the initial data and the parameter.
This paper presents a type of variational tinuous functions on certain subsets in duals principles for real valued ω^＊ lower semicon- of locally convex spaces, and resolve a problem concerning differentiability of c...This paper presents a type of variational tinuous functions on certain subsets in duals principles for real valued ω^＊ lower semicon- of locally convex spaces, and resolve a problem concerning differentiability of convex functions on general Banach spaces. They are done through discussing differentiability of convex functions on nonlinear topological spaces and convexification of nonconvex functions on topological linear spaces.展开更多
Let X and Y be real Banach spaces.Suppose that the subset sm[S1(X)] of the smooth points of the unit sphere [S1(X)] is dense in S1(X).If T0 is a surjective 1-Lipschitz mapping between two unit spheres,then,under some ...Let X and Y be real Banach spaces.Suppose that the subset sm[S1(X)] of the smooth points of the unit sphere [S1(X)] is dense in S1(X).If T0 is a surjective 1-Lipschitz mapping between two unit spheres,then,under some condition,T0 can be extended to a linear isometry on the whole space.展开更多
This paper focuses on the higher order fractional differentiability of weak solution pairs to the following nonlinear stationary Stokes system{div A(x-Du)-■π=divF,inΩdivu=0,inΩ.In terms of the difference quotient ...This paper focuses on the higher order fractional differentiability of weak solution pairs to the following nonlinear stationary Stokes system{div A(x-Du)-■π=divF,inΩdivu=0,inΩ.In terms of the difference quotient method,our first result reveals that if F∈B_(p,q.loc)^(β)(Ω,R^(n))for p=2 and 1≤q≤2n/n-2β,then such extra Besov regularity can transfer to the symmetric gradient Du and its pressureπwith no losses under a suitable fractional differentiability assumption on x■A(x,ξ).Furthermore,when the vector field A(x,Du)is simplified to the full gradient■u,we improve the aforementioned Besov regularity for all integrability exponents p and q by establishing a new Campanato-type decay estimates for(■u,π).展开更多
文摘The pandemic SARS-CoV-2 has become an undying virus to spread a sustainable disease named COVID-19 for upcoming few years.Mortality rates are rising rapidly as approved drugs are not yet available.Isolation from the infected person or community is the preferred choice to protect our health.Since humans are the only carriers,it might be possible to control the positive rate if the infected population or host carriers are isolated from each other.Isolation alone may not be a proper solution.These are the resolutions of previous research work carried out on COVID-19 throughout the world.The present scenario of the world and public health is knocking hard with a big question of critical uncertainty of COVID-19 because of its imprecise database as per daily positive cases recorded all over the world and in India as well.In this research work,we have pre-sented an optimal control model for COVID-19 using granular differentiability based on fuzzy dynamical systems.In the first step,we created a fuzzy Susceptible-Exposed-Infected-Asymptomatic-Hospitalized-Recovered-Death(SEIAHRD)model for COVID-19,analyzed it using granular differentiability,and reported disease dynamics for time-independent disease control parameters.In the second step,we upgraded the fuzzy dynamical system and granular differentiability model related to time-dependent disease control parameters as an optimal control problem invader.Theoretical studies have been validated with some practical data from the epidemic COVID-19 related to the Indian perspective during first wave and early second wave.
文摘Let X be a real uniformly convex and uniformly smooth Banach space and C a nonempty closed and convex subset of X.Let Π_(C):X→C denote the generalized metric projection operator introduced by Alber in[1].In this paper,we define the Gâteaux directional differentiability of Π_(C).We investigate some properties of the Gâteaux directional differentiability of Π_(C).In particular,if C is a closed ball,or a closed and convex cone(including proper closed subspaces),or a closed and convex cylinder,then,we give the exact representations of the directional derivatives of Π_(C).By comparing the results in[12]and this paper,we see the significant difference between the directional derivatives of the generalized metric projection operator Π_(C) and the Gâteaux directional derivatives of the standard metric projection operator PC.
文摘Characterizations of differentiability are obtained for continuous convex functions defined on nonempty open convex sets of Banach spaces as a generalization and application of a mumber of mathematicians several years effort, and a characteristic theorem is given for Banach spaces which are (weak) Asplund spaces.
文摘The aim of this paper is to investigate the differentiability(Gateaux differentiabllity and subdifferentiability) of continuous convex functions on locally convex spaces and to study the behaviour of some important results for this research area in locally convex spaces.
基金Research supported by the Hungarian"M uvel odesi es Kozoktatosi Miniszterium",grant no.FKFP0182/200O,the Bolyai Fellowship of the Hungarian Academy of Science and the Hungarian National Foundation for Scientific Research(OTKA),grant no.M 36511/2001.
文摘The aim of this paper is to prove the following theorem concerning the term by term differentiation the-orem of Walsh-Kaczmarz series. Let (ck) be a decreasing real sequence withare integrable functions and f(x) is a. e. dyadic (or Butzer and Wagner) differentiate withThe function Kk means the kth Walsh-Kaczmarz function.
基金Supported by the NSF of China (10071063 and 10471114)
文摘A locally convex space is said to be a Gateaux differentiability space (GDS) provided every continuous convex function defined on a nonempty convex open subset D of the space is densely Gateaux differentiable in .D.This paper shows that the product of a GDS and a family of separable Prechet spaces is a GDS,and that the product of a GDS and an arbitrary locally convex space endowed with the weak topology is a GDS.
文摘This paper considers the differentiability of C 0 semigroups with respect to (w.r.t.) parameters contained in their infinitesimal generators.It is proved that the generalized continuity and strong differentiability of their infinitesimal generators w.r.t.parameters imply the differentiability of the C 0 semigroups.The results are applied to the differentiability of the solution of a linear delay differential equation w.r.t.its delays.
基金Project (No. 10271111) supported partially by the National Natural Science Foundation of China
文摘Many papers on a wide range of control problems for Pritchard-Salamon systems have appeared and many of its important mathematical and system theoretical properties have been revealed. This paper deals with the differentiability of the Pritchard-Salamon system with admissible state-feedback. Spectrum analysis showed that under definite condition, the unbounded perturbation semigroup of the Pritchard-Salamon system is eventually differentiable.
文摘It is shown that an arbitrary function from D Rn to Rm will become C0,a-continuous in almost every x∈ D after restriction to a certain subset with limit pointx. For n 〉 m differentiability can be obtained. Examples show the Ho1der exponent a=min{1,n/m}is optimal.
文摘The sufficient conditions of Holder continuity of two kinds of fractal interpolation functions defined by IFS (Iterated Function System) were obtained. The sufficient and necessary condition for its differentiability was proved. Its derivative was a fractal interpolation function generated by the associated IFS, if it is differentiable.
基金Supported by the National Natural Science Foundation of China(12071334,11671293)
文摘Some basic concepts for functions defined on subsets of the unit sphere,such as the s-directional derivative,s-gradient and s-Gateaux and s-Frechet differentiability etc,are introduced and investigated.These concepts are different from the usual ones for functions defined on subsets of Euclidean spaces,however,the results obtained here are very similar.Then,as applications,we provide some criterions of s-convexity for functions defined on unit spheres which are improvements or refinements of some known results.
基金Research supported by the National Natural Science Foundation of China(11671243)the Shaanxi natural science basic research project of China(2018JM1020)
文摘In this paper,we study the differentiability of solutions on the boundary for equartions of type L_λu=~2u/x^2+|x|^(2λ)~2u/y^2=f(x,y),whereλis an arbitrary positive number.By introducing a proper metric that is related to the elliptic operator L_λ,we prove the differentiability on the boundary when some well-posed boundary conditions are satisfied.The main difficulty is the construction of new barrier functions in this article.
文摘The purpose of this paper is to verify the Smulyan lemma for the support function, and also the Gateaux differentiability of the support function is studied on its domain. Moreover, we provide a characterization of Frechet differentiability of the support function on the extremal points.
文摘This paper studies the smoothness of solutions of the higher dimensional polynomial-like iterative equation. The methods given by Zhang Weinian([7]) and by Kulczvcki M, Tabor j.([3]) are improved by constructing a new operator for the structure of the equation in order to apply fixed point theorems. Existence, uniqueness and stability of continuously differentiable solutions are given.
基金supported by Young Scholar Award for Doctoral Students of the Ministry of Education of Chinathe Marie Curie Initial Training Network(Grant No. PITN-GA-2008-213841)
文摘In this paper,we study the differentiability of the solutions of stochastic differential equations driven by the G-Brownian motion with respect to the initial data and the parameter.
基金Project supported by the National Natural Science Foundation of China (No.10471114)the Pujian Provincial Natural Science Foundation of China (No.F00021)the Tianyuan Foundation of Mathematics (No,A0324618).
文摘This paper presents a type of variational tinuous functions on certain subsets in duals principles for real valued ω^＊ lower semicon- of locally convex spaces, and resolve a problem concerning differentiability of convex functions on general Banach spaces. They are done through discussing differentiability of convex functions on nonlinear topological spaces and convexification of nonconvex functions on topological linear spaces.
基金supported by National Natural Science Foundation of China (Grant No.10871101)the Research Fund for the Doctoral Program of Higher Education (Grant No. 20060055010)
文摘Let X and Y be real Banach spaces.Suppose that the subset sm[S1(X)] of the smooth points of the unit sphere [S1(X)] is dense in S1(X).If T0 is a surjective 1-Lipschitz mapping between two unit spheres,then,under some condition,T0 can be extended to a linear isometry on the whole space.
基金Supported by the National Natural Science Foundation of China(Grant Nos.12071229,12101452)Tianjin Normal University Doctoral Research Project(Grant No.52XB2110)。
文摘This paper focuses on the higher order fractional differentiability of weak solution pairs to the following nonlinear stationary Stokes system{div A(x-Du)-■π=divF,inΩdivu=0,inΩ.In terms of the difference quotient method,our first result reveals that if F∈B_(p,q.loc)^(β)(Ω,R^(n))for p=2 and 1≤q≤2n/n-2β,then such extra Besov regularity can transfer to the symmetric gradient Du and its pressureπwith no losses under a suitable fractional differentiability assumption on x■A(x,ξ).Furthermore,when the vector field A(x,Du)is simplified to the full gradient■u,we improve the aforementioned Besov regularity for all integrability exponents p and q by establishing a new Campanato-type decay estimates for(■u,π).
