This paper deals with the problem of approximate controllability of infinite dimensional linear systems in nonreflexive state spaces. A necessary and sufficient condition for approximate controllability via L^p([0, T...This paper deals with the problem of approximate controllability of infinite dimensional linear systems in nonreflexive state spaces. A necessary and sufficient condition for approximate controllability via L^p([0, T], U), 1≤p〈∞ is obtained,where L^p( [0, T], U) is the control function space.展开更多
In this paper, the author studies a class of mixed nonlinear variational-like inequalities in reflexive Banach space. By applying a minimax inequality obtained by the author, some existence uniqueness theorems of solu...In this paper, the author studies a class of mixed nonlinear variational-like inequalities in reflexive Banach space. By applying a minimax inequality obtained by the author, some existence uniqueness theorems of solutions for the mixed nonlinear variational-like inequalities are proved. Next, by applying the auxiliary problem technique, rite author suggests an innovative iterative algorithm to compute the approximate solutions of the mixed nonlinear variational-like inequalities. Finally, the convergence criteria is also discussed.展开更多
A new bilevel generalized mixed equilibrium problem (BGMEP) involving generalized mixed variational-like inequality problems (GMVLIPs) is introduced and studied in the reflexive Banach spaces. First, an auxiliary ...A new bilevel generalized mixed equilibrium problem (BGMEP) involving generalized mixed variational-like inequality problems (GMVLIPs) is introduced and studied in the reflexive Banach spaces. First, an auxiliary generalized mixed equilibrium problem (AGMEP) is introduced to compute the approximate solutions of the BGMEP involving the GMVLIPs. By using a minimax inequality, the existence and the unique- ness of solutions of the AGMEP are proved under mild conditions without any coercive assumptions. By using an auxiliary principle technique, the new iterative algorithms are proposed and analyzed, with which the approximate solutions of the BGMEP are computed. The strong convergence of the iterative sequence generated by the algorithms is shown under mild conditions without any coercive assumptions. These new results can generalize some recent results in this field.展开更多
A new system of generalized mixed equilibrium problems (SGMEPs) involving generalized mixed variational-like inequality problems is introduced and studied in reflexive Banach spaces. A system of auxiliary generalize...A new system of generalized mixed equilibrium problems (SGMEPs) involving generalized mixed variational-like inequality problems is introduced and studied in reflexive Banach spaces. A system of auxiliary generalized mixed equilibrium problems (SAGMEPs) for solving the SGMEPs is first introduced. Existence and uniqueness of the solutions to the SAGMEPs is proved under quite mild assumptions without any coercive conditions in reflexive Banach spaces. Using the auxiliary principle technique, a new iterative algorithm for solving the SGMEPs is proposed and analyzed. Strong convergence of the iterative sequences generated by the algorithm is also proved under quite mild assumptions without any coercive conditions. These results improve, unify, and generalize some recent results in this field.展开更多
The aim of our work is to formulate and demonstrate the results of the normality, the Lipschitz continuity, of a nonlinear feedback system described by the monotone maximal operators and hemicontinuous, defined on rea...The aim of our work is to formulate and demonstrate the results of the normality, the Lipschitz continuity, of a nonlinear feedback system described by the monotone maximal operators and hemicontinuous, defined on real reflexive Banach spaces, as well as the approximation in a neighborhood of zero, of solutions of a feedback system [A,B] assumed to be non-linear, by solutions of another linear, This approximation allows us to obtain appropriate estimates of the solutions. These estimates have a significant effect on the study of the robust stability and sensitivity of such a system see <a href="#ref1">[1]</a> <a href="#ref2">[2]</a> <a href="#ref3">[3]</a>. We then consider a linear FS <img src="Edit_4629d4d0-bbb2-478d-adde-391efde3d1e0.bmp" alt="" />, and prove that, if <img src="Edit_435aae08-e821-4b4d-99d2-e2a2b47609c1.bmp" alt="" />;<img src="Edit_4fa030bc-1f97-4726-8257-ca8d00657aac.bmp" alt="" /> , with <img src="Edit_63ab4faa-ba40-45fe-8b8a-7a6caef91794.bmp" alt="" />the respective solutions of FS’s [A,B] and <img src="Edit_e78e2e6d-8934-4011-93eb-8b7eb52fa856.bmp" alt="" /> corresponding to the given (u,v) in <img src="Edit_0e18433c-8c7a-454f-8eec-6eb9fb69469a.bmp" alt="" /> . There exists,<img src="Edit_3dcd8afc-8cea-4c06-a920-e4148a5f793e.bmp" alt="" />, positive real constants such that, <img src="Edit_edb88446-3e39-4fe0-865a-114de701e78e.bmp" alt="" />. These results are the subject of theorems 3.1, <span style="font-size:10.0pt;font-family:;" "="">... </span>, 3.3. The proofs of these theorems are based on our lemmas 3.2, <span style="font-size:10.0pt;font-family:;" "="">... </span>, 3.5, devoted according to the hypotheses on A and B, to the existence of the inverse of the operator <em>I+BA</em> and <img src="Edit_2db1326b-cb5b-44cf-8d1f-df22bd6da45f.bmp" alt="" />. The results obtained and demonstrated along this document, present an extension in general Banach space of those in <a href="#ref4">[4]</a> on a Hilbert space <em>H</em> and those in <a href="#ref5">[5]</a> on a extended Hilbert space <img src="Edit_b70ce337-1812-4d4b-ae7d-a24da7e5b3cf.bmp" alt="" />.展开更多
Let K be a nonempty, closed and convex subset of a real reflexive Banach space E which has a uniformly Gateaux differentiable norm. Assume that every nonempty closed con- vex and bounded subset of K has the fixed poin...Let K be a nonempty, closed and convex subset of a real reflexive Banach space E which has a uniformly Gateaux differentiable norm. Assume that every nonempty closed con- vex and bounded subset of K has the fixed point property for nonexpansive mappings. Strong convergence theorems for approximation of a fixed point of Lipschitz pseudo-contractive map- pings which is also a unique solution to variational inequality problem involving φ-strongly pseudo-contractive mappings are proved. The results presented in this article can be applied to the study of fixed points of nonexpansive mappings, variational inequality problems, con- vex optimization problems, and split feasibility problems. Our result extends many recent important results.展开更多
In this paper, we propose a new hybrid iterative scheme for finding a common solution of an equilibrium problem and fixed point of Bregman totally quasi-asymptotically nonexpansive mapping in reflexive Banach spaces. ...In this paper, we propose a new hybrid iterative scheme for finding a common solution of an equilibrium problem and fixed point of Bregman totally quasi-asymptotically nonexpansive mapping in reflexive Banach spaces. Moreover, we prove some strong convergence theorems under suitable control conditions. Finally, the application to zero point problem of maximal monotone operators is given by the result.展开更多
Let X, Y be reflexive strictly convex Banach spaces, let T, δT : X → Y be bounded linear operators with closed range R(T). Put T= T + δT. In this paper, by using the concept of quasiadditivity and the so called...Let X, Y be reflexive strictly convex Banach spaces, let T, δT : X → Y be bounded linear operators with closed range R(T). Put T= T + δT. In this paper, by using the concept of quasiadditivity and the so called generalized Neumman lemma, we will give some error estimates of the bounds of ||T^M||. By using a relation between the concepts of the reduced minimum module and the gap of two subspaces, some new existence characterization of the Moore-Penrose metric generalized inverse T^M of the perturbed operator T will be also given.展开更多
It is our purpose in this paper to show that some results obtained in uniformly convex real Banach space with uniformly Gateaux differentiable norm are extendable to more general reflexive and strictly convex real Ban...It is our purpose in this paper to show that some results obtained in uniformly convex real Banach space with uniformly Gateaux differentiable norm are extendable to more general reflexive and strictly convex real Banach space with uniformly G&teaux differentiable norm. Demicompactness condition imposed in such results is dispensed with. Furthermore, Applications of our theorems to approximation of common fixed point of countable infinite family of continuous pseudocontractive mappings and approximation of common solution of countable infinite family of generalized mixed equilibrium problems are also discussed. Our theorems improve, generalize, unify and extend several recently announced results.展开更多
A new system of generalized mixed implicit equilibrium problems (SGMIEP) involving nonmonotone set-valued mappings is introduced and studied in real reflexive Banach spaces. First, an auxiliary mixed equilibrium pro...A new system of generalized mixed implicit equilibrium problems (SGMIEP) involving nonmonotone set-valued mappings is introduced and studied in real reflexive Banach spaces. First, an auxiliary mixed equilibrium problem (AMEP) is introduced. The existence and the uniqueness of the solutions to the AMEP are proved under quite mild assumptions without any coercive conditions. Next, by using the solution mapping of the AMEP, a system of generalized equation problems (SGEP) is considered, and its equivalence with the SGMIEP is shown. By using the SGEP, a new iterative algorithm for solving the SGMIEP is proposed and analyzed. The strong convergence of the iterative sequences generated by the algorithm is proved under suitable conditions. These results are new, which unify and generalize some recent results in this field.展开更多
Let E be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E^*, and C be a nonempty closed convex subset of E. Let {T(t) : t ≥ 0} be a nonexpansive semigroup on...Let E be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E^*, and C be a nonempty closed convex subset of E. Let {T(t) : t ≥ 0} be a nonexpansive semigroup on C such that F :=∩t≥0 Fix(T(t)) ≠ 0, and f : C → C be a fixed contractive mapping. If {αn}, {βn}, {an}, {bn}, {tn} satisfy certain appropriate conditions, then we suggest and analyze the two modified iterative processes as:{yn=αnxn+(1-αn)T(tn)xn,xn=βnf(xn)+(1-βn)yn{u0∈C,vn=anun+(1-an)T(tn)un,un+1=bnf(un)+(1-bn)vnWe prove that the approximate solutions obtained from these methods converge strongly to q ∈∩t≥0 Fix(T(t)), which is a unique solution in F to the following variational inequality:〈(I-f)q,j(q-u)〉≤0 u∈F Our results extend and improve the corresponding ones of Suzuki [Proc. Amer. Math. Soc., 131, 2133-2136 (2002)], and Kim and XU [Nonlear Analysis, 61, 51-60 (2005)] and Chen and He [Appl. Math. Lett., 20, 751-757 (2007)].展开更多
In this paper, we propose a new hybrid iterative scheme for finding a common solution to a finite of equilibrium problems and fixed point of Bregman totally quasi- asymptotically nonexpansive mapping in reflexive Bana...In this paper, we propose a new hybrid iterative scheme for finding a common solution to a finite of equilibrium problems and fixed point of Bregman totally quasi- asymptotically nonexpansive mapping in reflexive Banach spaces. Moreover, we prove some strong convergence theorems under suitable control conditions.展开更多
文摘This paper deals with the problem of approximate controllability of infinite dimensional linear systems in nonreflexive state spaces. A necessary and sufficient condition for approximate controllability via L^p([0, T], U), 1≤p〈∞ is obtained,where L^p( [0, T], U) is the control function space.
文摘In this paper, the author studies a class of mixed nonlinear variational-like inequalities in reflexive Banach space. By applying a minimax inequality obtained by the author, some existence uniqueness theorems of solutions for the mixed nonlinear variational-like inequalities are proved. Next, by applying the auxiliary problem technique, rite author suggests an innovative iterative algorithm to compute the approximate solutions of the mixed nonlinear variational-like inequalities. Finally, the convergence criteria is also discussed.
基金Project supported by the Scientific Research Fund of Sichuan Normal University(No.09ZDL04)the Leading Academic Discipline Project of Sichuan Province of China(No.SZD0406)
文摘A new bilevel generalized mixed equilibrium problem (BGMEP) involving generalized mixed variational-like inequality problems (GMVLIPs) is introduced and studied in the reflexive Banach spaces. First, an auxiliary generalized mixed equilibrium problem (AGMEP) is introduced to compute the approximate solutions of the BGMEP involving the GMVLIPs. By using a minimax inequality, the existence and the unique- ness of solutions of the AGMEP are proved under mild conditions without any coercive assumptions. By using an auxiliary principle technique, the new iterative algorithms are proposed and analyzed, with which the approximate solutions of the BGMEP are computed. The strong convergence of the iterative sequence generated by the algorithms is shown under mild conditions without any coercive assumptions. These new results can generalize some recent results in this field.
基金supported by the Scientific Research Fund of Sichuan Normal University (No.09ZDL04)the Sichuan Province Leading Academic Discipline Project (No.SZD0406)
文摘A new system of generalized mixed equilibrium problems (SGMEPs) involving generalized mixed variational-like inequality problems is introduced and studied in reflexive Banach spaces. A system of auxiliary generalized mixed equilibrium problems (SAGMEPs) for solving the SGMEPs is first introduced. Existence and uniqueness of the solutions to the SAGMEPs is proved under quite mild assumptions without any coercive conditions in reflexive Banach spaces. Using the auxiliary principle technique, a new iterative algorithm for solving the SGMEPs is proposed and analyzed. Strong convergence of the iterative sequences generated by the algorithm is also proved under quite mild assumptions without any coercive conditions. These results improve, unify, and generalize some recent results in this field.
文摘The aim of our work is to formulate and demonstrate the results of the normality, the Lipschitz continuity, of a nonlinear feedback system described by the monotone maximal operators and hemicontinuous, defined on real reflexive Banach spaces, as well as the approximation in a neighborhood of zero, of solutions of a feedback system [A,B] assumed to be non-linear, by solutions of another linear, This approximation allows us to obtain appropriate estimates of the solutions. These estimates have a significant effect on the study of the robust stability and sensitivity of such a system see <a href="#ref1">[1]</a> <a href="#ref2">[2]</a> <a href="#ref3">[3]</a>. We then consider a linear FS <img src="Edit_4629d4d0-bbb2-478d-adde-391efde3d1e0.bmp" alt="" />, and prove that, if <img src="Edit_435aae08-e821-4b4d-99d2-e2a2b47609c1.bmp" alt="" />;<img src="Edit_4fa030bc-1f97-4726-8257-ca8d00657aac.bmp" alt="" /> , with <img src="Edit_63ab4faa-ba40-45fe-8b8a-7a6caef91794.bmp" alt="" />the respective solutions of FS’s [A,B] and <img src="Edit_e78e2e6d-8934-4011-93eb-8b7eb52fa856.bmp" alt="" /> corresponding to the given (u,v) in <img src="Edit_0e18433c-8c7a-454f-8eec-6eb9fb69469a.bmp" alt="" /> . There exists,<img src="Edit_3dcd8afc-8cea-4c06-a920-e4148a5f793e.bmp" alt="" />, positive real constants such that, <img src="Edit_edb88446-3e39-4fe0-865a-114de701e78e.bmp" alt="" />. These results are the subject of theorems 3.1, <span style="font-size:10.0pt;font-family:;" "="">... </span>, 3.3. The proofs of these theorems are based on our lemmas 3.2, <span style="font-size:10.0pt;font-family:;" "="">... </span>, 3.5, devoted according to the hypotheses on A and B, to the existence of the inverse of the operator <em>I+BA</em> and <img src="Edit_2db1326b-cb5b-44cf-8d1f-df22bd6da45f.bmp" alt="" />. The results obtained and demonstrated along this document, present an extension in general Banach space of those in <a href="#ref4">[4]</a> on a Hilbert space <em>H</em> and those in <a href="#ref5">[5]</a> on a extended Hilbert space <img src="Edit_b70ce337-1812-4d4b-ae7d-a24da7e5b3cf.bmp" alt="" />.
文摘Let K be a nonempty, closed and convex subset of a real reflexive Banach space E which has a uniformly Gateaux differentiable norm. Assume that every nonempty closed con- vex and bounded subset of K has the fixed point property for nonexpansive mappings. Strong convergence theorems for approximation of a fixed point of Lipschitz pseudo-contractive map- pings which is also a unique solution to variational inequality problem involving φ-strongly pseudo-contractive mappings are proved. The results presented in this article can be applied to the study of fixed points of nonexpansive mappings, variational inequality problems, con- vex optimization problems, and split feasibility problems. Our result extends many recent important results.
基金supported by the Province Natural Science Foundation of China(2014J01008)
文摘In this paper, we propose a new hybrid iterative scheme for finding a common solution of an equilibrium problem and fixed point of Bregman totally quasi-asymptotically nonexpansive mapping in reflexive Banach spaces. Moreover, we prove some strong convergence theorems under suitable control conditions. Finally, the application to zero point problem of maximal monotone operators is given by the result.
基金supported by China Postdoctoral Science Foundation(Grant No.2015M582186)He’nan Institute of Science and Technology Postdoctoral Science Foundation(Grant No.5201029470209)
文摘Let X, Y be reflexive strictly convex Banach spaces, let T, δT : X → Y be bounded linear operators with closed range R(T). Put T= T + δT. In this paper, by using the concept of quasiadditivity and the so called generalized Neumman lemma, we will give some error estimates of the bounds of ||T^M||. By using a relation between the concepts of the reduced minimum module and the gap of two subspaces, some new existence characterization of the Moore-Penrose metric generalized inverse T^M of the perturbed operator T will be also given.
文摘It is our purpose in this paper to show that some results obtained in uniformly convex real Banach space with uniformly Gateaux differentiable norm are extendable to more general reflexive and strictly convex real Banach space with uniformly G&teaux differentiable norm. Demicompactness condition imposed in such results is dispensed with. Furthermore, Applications of our theorems to approximation of common fixed point of countable infinite family of continuous pseudocontractive mappings and approximation of common solution of countable infinite family of generalized mixed equilibrium problems are also discussed. Our theorems improve, generalize, unify and extend several recently announced results.
基金Project supported by the Sichuan Province Leading Academic Discipline Project(No.SZD0406)the Scientific Research Fund of Sichuan Normal University(No.11ZDL01)
文摘A new system of generalized mixed implicit equilibrium problems (SGMIEP) involving nonmonotone set-valued mappings is introduced and studied in real reflexive Banach spaces. First, an auxiliary mixed equilibrium problem (AMEP) is introduced. The existence and the uniqueness of the solutions to the AMEP are proved under quite mild assumptions without any coercive conditions. Next, by using the solution mapping of the AMEP, a system of generalized equation problems (SGEP) is considered, and its equivalence with the SGMIEP is shown. By using the SGEP, a new iterative algorithm for solving the SGMIEP is proposed and analyzed. The strong convergence of the iterative sequences generated by the algorithm is proved under suitable conditions. These results are new, which unify and generalize some recent results in this field.
基金Supported by the National Natural Science Foundation of China (Grant No. 10771050)supported by the Higher Education Commission, Pakistan, through Research Grant No. 1-29/HEC/HRD/2005/90
文摘Let E be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E^*, and C be a nonempty closed convex subset of E. Let {T(t) : t ≥ 0} be a nonexpansive semigroup on C such that F :=∩t≥0 Fix(T(t)) ≠ 0, and f : C → C be a fixed contractive mapping. If {αn}, {βn}, {an}, {bn}, {tn} satisfy certain appropriate conditions, then we suggest and analyze the two modified iterative processes as:{yn=αnxn+(1-αn)T(tn)xn,xn=βnf(xn)+(1-βn)yn{u0∈C,vn=anun+(1-an)T(tn)un,un+1=bnf(un)+(1-bn)vnWe prove that the approximate solutions obtained from these methods converge strongly to q ∈∩t≥0 Fix(T(t)), which is a unique solution in F to the following variational inequality:〈(I-f)q,j(q-u)〉≤0 u∈F Our results extend and improve the corresponding ones of Suzuki [Proc. Amer. Math. Soc., 131, 2133-2136 (2002)], and Kim and XU [Nonlear Analysis, 61, 51-60 (2005)] and Chen and He [Appl. Math. Lett., 20, 751-757 (2007)].
基金supported by the Natural Science Foundation of Fujian Province(Grant No.2014J01008)
文摘In this paper, we propose a new hybrid iterative scheme for finding a common solution to a finite of equilibrium problems and fixed point of Bregman totally quasi- asymptotically nonexpansive mapping in reflexive Banach spaces. Moreover, we prove some strong convergence theorems under suitable control conditions.