{"title":"Oscillation of Operator Differential Equations in Hilbert Space","authors":"H. Onose","doi":"10.5036/BFSIU1968.16.49","DOIUrl":null,"url":null,"abstract":"where P: R+→L(H,H) is continuous and self-adjoint for each t∈R+=[0,∞). By a solution of (I), we mean any function X: R+→L(H,H) which satisfies (I) on R+. Here X\" denotes the second strong derivative of X. Any solution X(t) of (I) satisfies X*(t)X'(t)=X*'(t)X(t), t∈R+ is called to be prepared. A prepared solution of (I) is nonoscillatory if there exists t0≧0 such that, for every t≧t0 X(t) is an isomorphism, i.e., X(t) is one-to-one and onto. Otherwise X(t) is said to be oscillatory. Equation (I) is oscillatory if every prepared solution of (I) is oscillatory. A bounded linear functional f: L(H,H)→R is said to be positive if f(T)≧0 for every positive operator T∈L(H,H), i.e., any T with <Tx,x>≧0 for all x∈H, and f(A*B)=f(B*A) for every A, B∈L(H,H). Here <,> denotes the inner product of H. Every positive linear functional f satisfies ‖f‖=f(I) and if f is not the zero functional, it satisfies","PeriodicalId":141145,"journal":{"name":"Bulletin of The Faculty of Science, Ibaraki University. Series A, Mathematics","volume":"9 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"1984-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of The Faculty of Science, Ibaraki University. Series A, Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.5036/BFSIU1968.16.49","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
where P: R+→L(H,H) is continuous and self-adjoint for each t∈R+=[0,∞). By a solution of (I), we mean any function X: R+→L(H,H) which satisfies (I) on R+. Here X" denotes the second strong derivative of X. Any solution X(t) of (I) satisfies X*(t)X'(t)=X*'(t)X(t), t∈R+ is called to be prepared. A prepared solution of (I) is nonoscillatory if there exists t0≧0 such that, for every t≧t0 X(t) is an isomorphism, i.e., X(t) is one-to-one and onto. Otherwise X(t) is said to be oscillatory. Equation (I) is oscillatory if every prepared solution of (I) is oscillatory. A bounded linear functional f: L(H,H)→R is said to be positive if f(T)≧0 for every positive operator T∈L(H,H), i.e., any T with ≧0 for all x∈H, and f(A*B)=f(B*A) for every A, B∈L(H,H). Here <,> denotes the inner product of H. Every positive linear functional f satisfies ‖f‖=f(I) and if f is not the zero functional, it satisfies