In this paper, we investigate a new perturbation theorem for the Moore-Penrose metric generalized inverses of a bounded linear operator in Banach space. The main tool in this paper is ＂the generalized Neumann lemma＂...In this paper, we investigate a new perturbation theorem for the Moore-Penrose metric generalized inverses of a bounded linear operator in Banach space. The main tool in this paper is ＂the generalized Neumann lemma＂ which is quite different from the method in [12] where ＂the generalized Banach lemma＂ was used. By the method of the perturba- tion analysis of bounded linear operators, we obtain an explicit perturbation theorem and three inequalities about error estimates for the Moore-Penrose metric generalized inverse of bounded linear operator under the generalized Neumann lemma and the concept of stable perturbations in Banach spaces.展开更多
Let B（R^n） be the set of all n x n real matrices, Sr the set of all matrices with rank r, 0 ≤ r ≤ n, and Sr^# the number of arcwise connected components of Sr. It is well-known that Sn =GL（R^n） is a Lie group an...Let B（R^n） be the set of all n x n real matrices, Sr the set of all matrices with rank r, 0 ≤ r ≤ n, and Sr^# the number of arcwise connected components of Sr. It is well-known that Sn =GL（R^n） is a Lie group and also a smooth hypersurface in B（R^n） with the dimension n × n.展开更多
In this paper, we study a semilinear elliptic equation defined on a bounded smooth domain. This type of problem arises from the study of spatial ecology model, and the growth function in the equation has a strong Alle...In this paper, we study a semilinear elliptic equation defined on a bounded smooth domain. This type of problem arises from the study of spatial ecology model, and the growth function in the equation has a strong Allee effect and is inhomogeneous. We use variational methods to prove that the equation has at least two positive solutions for a large parameter if it satisfies some appropriate conditions. We also prove some nonexistence results.展开更多
Twenty-five years ago America's small hydroelectric power generations were reactivated by private, municipal, and local governments to utilize the renewable energy from the small hydro sties available 10,000 exist...Twenty-five years ago America's small hydroelectric power generations were reactivated by private, municipal, and local governments to utilize the renewable energy from the small hydro sties available 10,000 existing dams, small rivers and/or streams, and energy recovery facilities at many water transmission and delivery systems.展开更多
Let E, F be two Banach spaces, B(E, F),B +(E, F), Φ(E, F), SΦ(E, F) and R(E, F) be bounded linear, double splitting, Fredholm, semi-Frdholm and finite rank operators from E into F, respectively. Let Σ be any one of...Let E, F be two Banach spaces, B(E, F),B +(E, F), Φ(E, F), SΦ(E, F) and R(E, F) be bounded linear, double splitting, Fredholm, semi-Frdholm and finite rank operators from E into F, respectively. Let Σ be any one of the following sets: {T ∈ Φ(E, F): Index T = constant and dim N(T) = constant}, {T ∈ SΦ(E, F): either dim N(T) =constant< ∞ or codim R(T) =constant< ∞} and {T ∈ R(E, F): Rank T =constant< ∞}. Then it is known that gS is a smooth submanifold of B(E, F) with the tangent space T A Σ = {B ∈ B(E, F): BN(A) ? R(A)} for any A ∈ Σ. However, for B*(E, F) = {T ∈ B +(E, F): dimN(T) = codimR(T) = ∞} without the characteristic numbers, dimN(A), codimR(A), index(A) and Rank(A) of the equivalent classes above, it is very difficult to find which class of operators in B*(E, E) forms a smooth submanifold of B(E, F). Fortunately, we find that B*(E, F) is just a smooth submanifold of B(E, F) with the tangent space T A B*(E, F) = {T ∈ B(E, F): TN(A) ? R(A)} for each A ∈ B*(E, F). Thus the geometric construction of B +(E, F) is obtained, i.e., B +(E, F) is a smooth Banach submanifold of B(E, F), which is the union of the previous smooth submanifolds disjoint from each other. Meanwhile we give a smooth submanifold S(A) of B(E, F), modeled on a fixed Banach space and containing A for any A ∈ B +(E, F). To end these, results on the generalized inverse perturbation analysis are generalized. Specially, in the case E = F = ? n , it is obtained that the set Σ r of all n × n matrices A with Rank(A) = r < n is an arcwise connected and smooth hypersurface (submanifold) in B(? n ) with dimΣ r = 2nr × r 2. Then a new geometrical construction of B(? n ), analogous to B +(E, F), is given besides its analysis and algebra constructions known well.展开更多
In this paper, continuous homogeneous selections for the set-valued metric generalized inverses T^ of linear operators T in Banach spaces are investigated by means of the methods of geometry of Banach spaces. Necessar...In this paper, continuous homogeneous selections for the set-valued metric generalized inverses T^ of linear operators T in Banach spaces are investigated by means of the methods of geometry of Banach spaces. Necessary and sufficient conditions for bounded linear operators T to have continuous homogeneous selections for the set-valued metric generalized inverses T~ are given. The results are an answer to the problem posed by Nashed and Votruba.展开更多
Given two Banach spaces E,F,let B(E,F) be the set of all bounded linear operators from E into F,and R(E,F) the set of all operators in B(E,F) with finite rank.It is well-known that B(Rn) is a Banach space as well as a...Given two Banach spaces E,F,let B(E,F) be the set of all bounded linear operators from E into F,and R(E,F) the set of all operators in B(E,F) with finite rank.It is well-known that B(Rn) is a Banach space as well as an algebra,while B(Rn,Rm) for m = n,is a Banach space but not an algebra;meanwhile,it is clear that R(E,F) is neither a Banach space nor an algebra.However,in this paper,it is proved that all of them have a common property in geometry and topology,i.e.,they are all a union of mutual disjoint path-connected and smooth submanifolds (or hypersurfaces).Let Σr be the set of all operators of finite rank r in B(E,F) (or B(Rn,Rm)).In fact,we have that 1) suppose Σr∈ B(Rn,Rm),and then Σr is a smooth and path-connected submanifold of B(Rn,Rm) and dimΣr = (n + m)r-r2,for each r ∈ [0,min{n,m});if m = n,the same conclusion for Σr and its dimension is valid for each r ∈ [0,min{n,m}];2) suppose Σr∈ B(E,F),and dimF = ∞,and then Σr is a smooth and path-connected submanifold of B(E,F) with the tangent space TAΣr = {B ∈ B(E,F) : BN(A)-R(A)} at each A ∈Σr for 0 r 【 ∞.The routine methods for seeking a path to connect two operators can hardly apply here.A new method and some fundamental theorems are introduced in this paper,which is development of elementary transformation of matrices in B(Rn),and more adapted and simple than the elementary transformation method.In addition to tensor analysis and application of Thom’s famous result for transversility,these will benefit the study of infinite geometry.展开更多
In this paper, we investigate the perturbation problem for the Moore-Penrose bounded quasi-linear projection generalized inverses of a closed linear operaters in Banach space. By the method of the perturbation analysi...In this paper, we investigate the perturbation problem for the Moore-Penrose bounded quasi-linear projection generalized inverses of a closed linear operaters in Banach space. By the method of the perturbation analysis of bounded quasi-linear operators, we obtain an explicit perturbation theorem and error estimates for the Moore-Penrose bounded quasi-linear generalized inverse of closed linear operator under the T-bounded perturbation, which not only extend some known results on the perturbation of the oblique projection generalized inverse of closed linear operators, but also extend some known results on the perturbation of the Moore-Penrose metric generalized inverse of bounded linear operators in Banach spaces.展开更多
Let X, Y be Banach spaces and M be a linear subspace in X x Y = {{x,y}lx E X,y C Y}. We may view M as a multi-valued linear operator from X to Y by taking M（x） = {yl（x,y} C M}. In this paper, we give several criter...Let X, Y be Banach spaces and M be a linear subspace in X x Y = {{x,y}lx E X,y C Y}. We may view M as a multi-valued linear operator from X to Y by taking M（x） = {yl（x,y} C M}. In this paper, we give several criteria for a single-valued operator from Y to X to be the metric generalized inverse of the multi-valued linear operator M. The principal tool in this paper is also the generalized orthogonal decomposition theorem in Banach spaces.展开更多
The authors prove the uniqueness and existence of positive solutions for the semilinear elliptic system which involves nonlinearities with sublinear growth conditions.
In this paper, we introduce the concepts of generalized regular points and narrow spectrum points of bounded linear operators on Hilbert spaces. The concept of generalized regular points is an extension of the concept...In this paper, we introduce the concepts of generalized regular points and narrow spectrum points of bounded linear operators on Hilbert spaces. The concept of generalized regular points is an extension of the concept regular points, and so, the set of all spectrum points is reduced to the narrow spectrum. We present not only the same and different properties of spectrum and of narrow spectrum but also show the relationship between them. Finally, the well known problem about the invariant subspaces of bounded linear operators on separable Hilbert spaces is simplified to the problem of the operator with narrow spectrum only.展开更多
We apply the imperfect bifurcation theory in Banach spaces to study the exact multiplicity of solutions to a perturbed logistic type equations on a symmetric spatial domain. We obtain the precise bifurcation diagrams.
基金Supported by the Nature Science Foundation of China(11471091 and 11401143)
文摘In this paper, we investigate a new perturbation theorem for the Moore-Penrose metric generalized inverses of a bounded linear operator in Banach space. The main tool in this paper is ＂the generalized Neumann lemma＂ which is quite different from the method in [12] where ＂the generalized Banach lemma＂ was used. By the method of the perturba- tion analysis of bounded linear operators, we obtain an explicit perturbation theorem and three inequalities about error estimates for the Moore-Penrose metric generalized inverse of bounded linear operator under the generalized Neumann lemma and the concept of stable perturbations in Banach spaces.
文摘Let B（R^n） be the set of all n x n real matrices, Sr the set of all matrices with rank r, 0 ≤ r ≤ n, and Sr^# the number of arcwise connected components of Sr. It is well-known that Sn =GL（R^n） is a Lie group and also a smooth hypersurface in B（R^n） with the dimension n × n.
基金supported by the National Natural Science Foundation of China (No.10671049)the US-NSF grants (No.DMS-0314736)
文摘In this paper, we study a semilinear elliptic equation defined on a bounded smooth domain. This type of problem arises from the study of spatial ecology model, and the growth function in the equation has a strong Allee effect and is inhomogeneous. We use variational methods to prove that the equation has at least two positive solutions for a large parameter if it satisfies some appropriate conditions. We also prove some nonexistence results.
文摘Twenty-five years ago America's small hydroelectric power generations were reactivated by private, municipal, and local governments to utilize the renewable energy from the small hydro sties available 10,000 existing dams, small rivers and/or streams, and energy recovery facilities at many water transmission and delivery systems.
基金supported by National Natural Science Foundation of China (Grant No.10771101,10671049)
文摘Let E, F be two Banach spaces, B(E, F),B +(E, F), Φ(E, F), SΦ(E, F) and R(E, F) be bounded linear, double splitting, Fredholm, semi-Frdholm and finite rank operators from E into F, respectively. Let Σ be any one of the following sets: {T ∈ Φ(E, F): Index T = constant and dim N(T) = constant}, {T ∈ SΦ(E, F): either dim N(T) =constant< ∞ or codim R(T) =constant< ∞} and {T ∈ R(E, F): Rank T =constant< ∞}. Then it is known that gS is a smooth submanifold of B(E, F) with the tangent space T A Σ = {B ∈ B(E, F): BN(A) ? R(A)} for any A ∈ Σ. However, for B*(E, F) = {T ∈ B +(E, F): dimN(T) = codimR(T) = ∞} without the characteristic numbers, dimN(A), codimR(A), index(A) and Rank(A) of the equivalent classes above, it is very difficult to find which class of operators in B*(E, E) forms a smooth submanifold of B(E, F). Fortunately, we find that B*(E, F) is just a smooth submanifold of B(E, F) with the tangent space T A B*(E, F) = {T ∈ B(E, F): TN(A) ? R(A)} for each A ∈ B*(E, F). Thus the geometric construction of B +(E, F) is obtained, i.e., B +(E, F) is a smooth Banach submanifold of B(E, F), which is the union of the previous smooth submanifolds disjoint from each other. Meanwhile we give a smooth submanifold S(A) of B(E, F), modeled on a fixed Banach space and containing A for any A ∈ B +(E, F). To end these, results on the generalized inverse perturbation analysis are generalized. Specially, in the case E = F = ? n , it is obtained that the set Σ r of all n × n matrices A with Rank(A) = r < n is an arcwise connected and smooth hypersurface (submanifold) in B(? n ) with dimΣ r = 2nr × r 2. Then a new geometrical construction of B(? n ), analogous to B +(E, F), is given besides its analysis and algebra constructions known well.
基金supported by National Science Foundation of China (Grant No.11071051)Youth Science Foundation of Heilongjiang Province of China (Grant No.QC2009C73)+1 种基金the second author is supported by the State Committee for Scientific Research of Poland (Grant No.N N201 362236)the third author is supported by National Science Foundation of China (Grant No.11071051)
文摘In this paper, continuous homogeneous selections for the set-valued metric generalized inverses T^ of linear operators T in Banach spaces are investigated by means of the methods of geometry of Banach spaces. Necessary and sufficient conditions for bounded linear operators T to have continuous homogeneous selections for the set-valued metric generalized inverses T~ are given. The results are an answer to the problem posed by Nashed and Votruba.
基金supported by National Natural Science Foundation of China (Grant Nos.10671049,10771101)
文摘Given two Banach spaces E,F,let B(E,F) be the set of all bounded linear operators from E into F,and R(E,F) the set of all operators in B(E,F) with finite rank.It is well-known that B(Rn) is a Banach space as well as an algebra,while B(Rn,Rm) for m = n,is a Banach space but not an algebra;meanwhile,it is clear that R(E,F) is neither a Banach space nor an algebra.However,in this paper,it is proved that all of them have a common property in geometry and topology,i.e.,they are all a union of mutual disjoint path-connected and smooth submanifolds (or hypersurfaces).Let Σr be the set of all operators of finite rank r in B(E,F) (or B(Rn,Rm)).In fact,we have that 1) suppose Σr∈ B(Rn,Rm),and then Σr is a smooth and path-connected submanifold of B(Rn,Rm) and dimΣr = (n + m)r-r2,for each r ∈ [0,min{n,m});if m = n,the same conclusion for Σr and its dimension is valid for each r ∈ [0,min{n,m}];2) suppose Σr∈ B(E,F),and dimF = ∞,and then Σr is a smooth and path-connected submanifold of B(E,F) with the tangent space TAΣr = {B ∈ B(E,F) : BN(A)-R(A)} at each A ∈Σr for 0 r 【 ∞.The routine methods for seeking a path to connect two operators can hardly apply here.A new method and some fundamental theorems are introduced in this paper,which is development of elementary transformation of matrices in B(Rn),and more adapted and simple than the elementary transformation method.In addition to tensor analysis and application of Thom’s famous result for transversility,these will benefit the study of infinite geometry.
基金Supported by National Nature Science Foundation of China(Grant No.11471091)
文摘In this paper, we investigate the perturbation problem for the Moore-Penrose bounded quasi-linear projection generalized inverses of a closed linear operaters in Banach space. By the method of the perturbation analysis of bounded quasi-linear operators, we obtain an explicit perturbation theorem and error estimates for the Moore-Penrose bounded quasi-linear generalized inverse of closed linear operator under the T-bounded perturbation, which not only extend some known results on the perturbation of the oblique projection generalized inverse of closed linear operators, but also extend some known results on the perturbation of the Moore-Penrose metric generalized inverse of bounded linear operators in Banach spaces.
基金Supported by National Natural Science Foundation of China (Grant No. 11071051)
文摘Let X, Y be Banach spaces and M be a linear subspace in X x Y = {{x,y}lx E X,y C Y}. We may view M as a multi-valued linear operator from X to Y by taking M（x） = {yl（x,y} C M}. In this paper, we give several criteria for a single-valued operator from Y to X to be the metric generalized inverse of the multi-valued linear operator M. The principal tool in this paper is also the generalized orthogonal decomposition theorem in Banach spaces.
基金Partially supported by National Natural Science Foundation of China （Grant No. 11071051）, Tianyuan Foundation for Mathematics of National Natural Science Foundation of China （Grant No. 10926060）, Youth Science Foundation of Heilongjiang Province （Grant No. QC2009C73）
文摘The authors prove the uniqueness and existence of positive solutions for the semilinear elliptic system which involves nonlinearities with sublinear growth conditions.
基金Supported by National Natural Science Foundation of China (Grant No. 11071051)Youth Science Foundation of Heilongjiang Province (Grant No. QC2009C73)the State Committee for Scientific Research of Poland (Grant No. N N201 362236)
文摘In this paper, we introduce the concepts of generalized regular points and narrow spectrum points of bounded linear operators on Hilbert spaces. The concept of generalized regular points is an extension of the concept regular points, and so, the set of all spectrum points is reduced to the narrow spectrum. We present not only the same and different properties of spectrum and of narrow spectrum but also show the relationship between them. Finally, the well known problem about the invariant subspaces of bounded linear operators on separable Hilbert spaces is simplified to the problem of the operator with narrow spectrum only.
基金the National Natural Science Foundation of China (Grant No. 10671049), Longjiang Scholar GrantScience Research Fund of the Education Department of Heilongjiang Province (Grant No.11531246)Harbin Normal University Academic Backbone of Youth Project
文摘We apply the imperfect bifurcation theory in Banach spaces to study the exact multiplicity of solutions to a perturbed logistic type equations on a symmetric spatial domain. We obtain the precise bifurcation diagrams.
