By representing random conjugate spaces a general representation theorem on classical duals is proved. For application, we unify and improve many known important representation theorems of the dual of Lebesgue-Bochner...By representing random conjugate spaces a general representation theorem on classical duals is proved. For application, we unify and improve many known important representation theorems of the dual of Lebesgue-Bochner function spaces.展开更多
In general, Banach space-valued Riemann integrable functions defined on [0, 1] （equipped with the Lebesgue measure） need not be weakly continuous almost everywhere. A Banach space is said to have the weak Lebesgue p...In general, Banach space-valued Riemann integrable functions defined on [0, 1] （equipped with the Lebesgue measure） need not be weakly continuous almost everywhere. A Banach space is said to have the weak Lebesgue property if every Riemann integrable function taking values in it is weakly continuous almost everywhere. In this paper we discuss this property for the Banach space LX^1 of all Bochner integrable functions from [0, 1] to the Banach space X. We show that LX^1 has the weak Lebesgue property whenever X has the Radon-Nikodym property and X＊ is separable. This generalizes the result by Chonghu Wang and Kang Wan [Rocky Mountain J. Math., 31（2）, 697-703 （2001）] that L^1[0, 1] has the weak Lebesgue property.展开更多
Let X be a complex Banach space and let B and C be two closed linear operators on X satisfying the condition D(B)?D(C),and let d∈L^(1)(R_(+))and 0≤β<α≤2.We characterize the well-posedness of the fractional int...Let X be a complex Banach space and let B and C be two closed linear operators on X satisfying the condition D(B)?D(C),and let d∈L^(1)(R_(+))and 0≤β<α≤2.We characterize the well-posedness of the fractional integro-differential equations D^(α)u(t)+CD^(β)u(t)=Bu(t)+∫_(-∞)td(t-s)Bu(s)ds+f(t),(0≤t≤2π)on periodic Lebesgue-Bochner spaces L^(p)(T;X)and periodic Besov spaces B_(p,q)^(s)(T;X).展开更多
This paper is devoted to studying the representation of measures of non-generalized compactness,in particular,measures of noncompactness,of non-weak compactness and of non-super weak compactness,defined on Banach spac...This paper is devoted to studying the representation of measures of non-generalized compactness,in particular,measures of noncompactness,of non-weak compactness and of non-super weak compactness,defined on Banach spaces and its applications.With the aid of a three-time order-preserving embedding theorem,we show that for every Banach space X,there exist a Banach function space C(K)for some compact Hausdorff space K and an order-preserving affine mapping T from the super space B of all the nonempty bounded subsets of X endowed with the Hausdorff metric to the positive cone C(K)^(+) of C(K),such that for every convex measure,in particular,the regular measure,the homogeneous measure and the sublinear measure of non-generalized compactnessμon X,there is a convex function F on the cone V=T(B)which is Lipschitzian on each bounded set of V such that F(T(B))=μ(B),■B∈B.As its applications,we show a class of basic integral inequalities related to an initial value problem in Banach spaces,and prove a solvability result of the initial value problem,which is an extension of some classical results due to Bana′s and Goebel(1980),Goebel and Rzymowski(1970)and Rzymowski(1971).展开更多
We give necessary and sufficient conditions of Lp-maximal regularity(resp.B sp ,q-maximal regularity or F sp ,q-maximal regularity) for the second order delay equations:u″(t)=Au(t) + Gu't + F u t + f(t), t ∈ [0...We give necessary and sufficient conditions of Lp-maximal regularity(resp.B sp ,q-maximal regularity or F sp ,q-maximal regularity) for the second order delay equations:u″(t)=Au(t) + Gu't + F u t + f(t), t ∈ [0, 2π] with periodic boundary conditions u(0)=u(2π), u′(0)=u′(2π), where A is a closed operator in a Banach space X,F and G are delay operators on Lp([-2π, 0];X)(resp.Bsp ,q([2π, 0];X) or Fsp,q([-2π, 0;X])).展开更多
In this paper, we study the well-posedness of the third-order differential equation with finite delay(P3): αu’"(t) + u"(t) = Au(t) + Bu’(t) + Fut +f(t)(t ∈ T := [0,2π]) with periodic boundary conditions...In this paper, we study the well-posedness of the third-order differential equation with finite delay(P3): αu’"(t) + u"(t) = Au(t) + Bu’(t) + Fut +f(t)(t ∈ T := [0,2π]) with periodic boundary conditions u(0) = u(2π), u’(0) = u"(2π),u"(0)=u"(2π) in periodic Lebesgue-Bochner spaces Lp(T;X) and periodic Besov spaces Bp,qs(T;X), where A and B are closed linear operators on a Banach space X satisfying D(A) ∩ D(B) ≠ {0}, α≠ 0 is a fixed constant and F is a bounded linear operator from Lp([-2π, 0];X)(resp. Bp,qs([-2π, 0];X)) into X, ut is given by ut(s) = u(t + s) when s ∈ [-2π,0]. Necessary and sufficient conditions for the Lp-well-posedness(resp. Bp,qs-well-posedness)of(P3) are given in the above two function spaces. We also give concrete examples that our abstract results may be applied.展开更多
文摘By representing random conjugate spaces a general representation theorem on classical duals is proved. For application, we unify and improve many known important representation theorems of the dual of Lebesgue-Bochner function spaces.
基金supported by MEC and FEDER (Project MTM2005-08350-C03-03)Generalitat Valenciana (Project GV/2007/191)+3 种基金supported by MEC and FEDER (Project MTM2005-08379)Fundacion Seneca (Project 00690/PI/04) the "Juan de la Cierva" Programme (MEC and FSE)supported by MEC and FEDER (Project MTM2006-11690-C02-01)
文摘In general, Banach space-valued Riemann integrable functions defined on [0, 1] （equipped with the Lebesgue measure） need not be weakly continuous almost everywhere. A Banach space is said to have the weak Lebesgue property if every Riemann integrable function taking values in it is weakly continuous almost everywhere. In this paper we discuss this property for the Banach space LX^1 of all Bochner integrable functions from [0, 1] to the Banach space X. We show that LX^1 has the weak Lebesgue property whenever X has the Radon-Nikodym property and X＊ is separable. This generalizes the result by Chonghu Wang and Kang Wan [Rocky Mountain J. Math., 31（2）, 697-703 （2001）] that L^1[0, 1] has the weak Lebesgue property.
基金the NSF of China(12171266,12171062)the NSF of Chongqing(CSTB2022NSCQ-JQX0004)。
文摘Let X be a complex Banach space and let B and C be two closed linear operators on X satisfying the condition D(B)?D(C),and let d∈L^(1)(R_(+))and 0≤β<α≤2.We characterize the well-posedness of the fractional integro-differential equations D^(α)u(t)+CD^(β)u(t)=Bu(t)+∫_(-∞)td(t-s)Bu(s)ds+f(t),(0≤t≤2π)on periodic Lebesgue-Bochner spaces L^(p)(T;X)and periodic Besov spaces B_(p,q)^(s)(T;X).
基金supported by National Natural Science Foundation of China(Grant No.11731010)。
文摘This paper is devoted to studying the representation of measures of non-generalized compactness,in particular,measures of noncompactness,of non-weak compactness and of non-super weak compactness,defined on Banach spaces and its applications.With the aid of a three-time order-preserving embedding theorem,we show that for every Banach space X,there exist a Banach function space C(K)for some compact Hausdorff space K and an order-preserving affine mapping T from the super space B of all the nonempty bounded subsets of X endowed with the Hausdorff metric to the positive cone C(K)^(+) of C(K),such that for every convex measure,in particular,the regular measure,the homogeneous measure and the sublinear measure of non-generalized compactnessμon X,there is a convex function F on the cone V=T(B)which is Lipschitzian on each bounded set of V such that F(T(B))=μ(B),■B∈B.As its applications,we show a class of basic integral inequalities related to an initial value problem in Banach spaces,and prove a solvability result of the initial value problem,which is an extension of some classical results due to Bana′s and Goebel(1980),Goebel and Rzymowski(1970)and Rzymowski(1971).
文摘We give necessary and sufficient conditions of Lp-maximal regularity(resp.B sp ,q-maximal regularity or F sp ,q-maximal regularity) for the second order delay equations:u″(t)=Au(t) + Gu't + F u t + f(t), t ∈ [0, 2π] with periodic boundary conditions u(0)=u(2π), u′(0)=u′(2π), where A is a closed operator in a Banach space X,F and G are delay operators on Lp([-2π, 0];X)(resp.Bsp ,q([2π, 0];X) or Fsp,q([-2π, 0;X])).
基金Supported by the NSF of China(Grant Nos.11571194,11731010 and 11771063)the Natural Science Foundation of Chongqing(Grant No.cstc2017jcyjAX0006)+2 种基金Science and Technology Project of Chongqing Education Committee(Grant No.KJ1703041)the University Young Core Teacher Foundation of Chongqing(Grant No.020603011714)Talent Project of Chongqing Normal University(Grant No.02030307-00024)
文摘In this paper, we study the well-posedness of the third-order differential equation with finite delay(P3): αu’"(t) + u"(t) = Au(t) + Bu’(t) + Fut +f(t)(t ∈ T := [0,2π]) with periodic boundary conditions u(0) = u(2π), u’(0) = u"(2π),u"(0)=u"(2π) in periodic Lebesgue-Bochner spaces Lp(T;X) and periodic Besov spaces Bp,qs(T;X), where A and B are closed linear operators on a Banach space X satisfying D(A) ∩ D(B) ≠ {0}, α≠ 0 is a fixed constant and F is a bounded linear operator from Lp([-2π, 0];X)(resp. Bp,qs([-2π, 0];X)) into X, ut is given by ut(s) = u(t + s) when s ∈ [-2π,0]. Necessary and sufficient conditions for the Lp-well-posedness(resp. Bp,qs-well-posedness)of(P3) are given in the above two function spaces. We also give concrete examples that our abstract results may be applied.
