GENERALIZED BOCHNER THEOREM ON LOCALLY CONVEX SPACES

ThisworkissupportedinpartbythePostdoctoralScienceFoundationofChina.1.IntroductionTheclassicalBochnertheoremstatesthatafunctionCiR"-CisthecharacteristicfunctionofaprobabilitymeasureonR"iffCispositivedefinite,C(0)=1andcontinuous.FOrvariousgeneralizationsofthetheoremtoinfinitedimensionspaces,thereaderisreferredto{1]andT3I.Inthemonographl'],MinlostheoremgeneralizestheclassicalBochnertheoremtonuclearspaces,whichcharacterizestheclassofcharacteristicfunctionalsofRadonprobabilitymeasuresonstron…     

Diestel-Faires定理在局部凸空间中的推广

通过Banach 空间与局部凸空间的对比,将Banach 空间上的Diestel-Faires 定理在局部凸空间上进行推广。进一步给出了局部凸空间上的Orlicz-Pettis定理与推论。     

On a fixed-point theorem on locally convex linear topological spaces

Iff is a self mapping on a closed convex subsetK of a separated quasicomplete locally convex linear topological spaceE such that (i)E is strictly convex, (ii)f (K) is contained in a compact subset ofK and (iii)f satisfies a contraction condition, then it is shown that for eachxK, the sequence of {U ⁿ (x)} _n =1 of iterates, whereU KK is defined byU (y)=f(y)+(1-) y, yK, converges to a fixed point off.     

On an extension of Rolle's theorem to locally convex spaces

This work concerns the extension of a weak form of the Rolle's theorem to locally convex spaces that satisfy an axiom of separation. The result provides a condition for asserting the uniqueness of a solution to nonlinear functional equations, including nonlinear integro-differential equations. We use the extended Rolle's theorem to prove the uniqueness of a solution to a nonlinear, fractional differential equation.     

On a fixed-point theorem on locally convex linear topological spaces

Iff is a self mapping on a closed convex subsetK of a separated quasicomplete locally convex linear topological spaceE such that (i)E is strictly convex, (ii)f (K) is contained in a compact subset ofK and (iii)f satisfies a contraction condition, then it is shown that for eachxK, the sequence of {U ⁿ (x)} _n =1 of iterates, whereU KK is defined byU (y)=f(y)+(1-) y, yK, converges to a fixed point off.     

A spectral mapping theorem for scalar-type spectral operators in locally convex spaces

LetT be a continuous scalar-type spectral operator defined on a quasi-complete locally convex spaceX, that is,T=fdP whereP is an equicontinuous spectral measure inX andf is aP-integrable function. It is shown that (T) is precisely the closedP-essential range of the functionf or equivalently, that (T) is equal to the support of the (unique) equicontinuous spectral measureQ ^* defined on the Borel sets of the extended complex plane ^* such thatQ ^*({})=0 andT=zdQ _*(z). This result is then used to prove a spectral mapping theorem; namely, thatg((T))=(g(T)) for anyQ ^*-integrable functiong: ^{* *} which is continuous on (T). This is an improvement on previous results of this type since it covers the case wheng((T))/{} is an unbounded set in a phenomenon which occurs often for continuous operatorsT defined in non-normable spacesX.     

Phelps' lemma, Danes' drop theorem and Ekeland's principle in locally convex spaces

A generalization of Phelps' lemma to locally convex spaces is proven, applying its well-known Banach space version. We show the equivalence of this theorem, Ekeland's principle and Danes' drop theorem in locally convex spaces to their Banach space counterparts and to a Pareto efficiency theorem due to Isac. This solves a problem, concerning the drop theorem, proposed by G. Isac in 1997.

We show that a different formulation of Ekeland's principle in locally convex spaces, using a family of topology generating seminorms as perturbation functions rather than a single (in general discontinuous) Minkowski functional, turns out to be equivalent to the original version.

     

On limitedness in locally convex spaces

Archiv der Mathematik -     

Nonlinear programming in locally convex spaces

Farkas' lemma is generalized both to nonlinear functions and to infinite-dimensional spaces; the version for linear maps is less restricted than Hurwicz's result. A generalization of F. John's necessary condition for constrained minima is deduced for infinite dimension and cone constraints. Some theorems on converse and symmetric duality in nonlinear programming are obtained, which extend the known results, even in the finite-dimensional case.Communicated by P. P. Varayia