Representation Theory Of Measures Of Noncompactness And Countable Determination Of The Kuratowski Measure Of Noncompactness

The study of measures of noncompactness has continued for 90 years.It has been shown that the theory of measures of noncompactness was used in a wide variety of topics in nonlinear analysis,such as fixed point theory,spectral theory,integral equations,ordinary differential equations,partial differential equations,geometry of Banach spaces,fractional differential equations,and optimization theory etc.On the other hand,the development of theory of measures of noncompactness is not so well because of the lack of theoretical tools,and it cannot meet the needs of the applications in both depth and breadth.The reason is that the link between measures of noncompactness and other branches of mathematics has not been built yet in theory,so the theories and methods of other branches of mathematics cannot be used for reference.Because there is no good representation method for measures of noncompactness,we have to go back to the original definition when in application.There are three goals of this thesis:(1)we show a representation theorem of convex measures of noncompactness(convex MNCs,for simplicity)and of their generalizations.(2)We prove that the Kuratowski MNC has property of countable determination,which gives an affirmative answer to a long-standing question.(3)As an application of the representation theorem,we establish a number of classes of basic integral inequalities related to an initial-value problem in Banach spaces.The main results of this thesis are as follows:1.Theorem 1(A representation theorem of convex MNCs).For every Banach space X,there exist a Banach function space C(K)for some compact Hausdorff space K,and a 1-Lipschitzian three-time order preserving mapping T=TQJ from B(X)of all nonempty bounded subsets of X to the positive cone C(K)+of C(K)such that for every convex MNC μ on X,there is a monotone continuous convex function F defined on the cone V=T(B(X))? C(K)+such that F(T(B))=μB(B),? B∈B(X),and F is cr-Lipschitzian on V ∩(rBC(K))for each r≥0,where cr=μ[(1+r)BX].2.Theorem 2(A class of basic integral inequalities).Let μ be a convex MNC on a Banach space X.For every nonempty subset G ?L1([0,a],X)with ψ(t)≡supg∈G ‖g(t)‖ integrable on[0,a]such that the mapping JG:[0,a]→Cb(Ω)defined for t∈[0,a]by JG(t)(ω)=(?)<ω,g(t)>≡σG(t)(ω),ω∈Ω≡BX*,is strongly(Lebesgue-Bochner)measurable,then we haveμ{∫0τG(s)ds}≤1/τ∫0τ∫μ{τG(s)}ds,?0<τ≤a;in particular,if μ is a sublinear MNC,or,τ≤1,thenμ{∫0τG(s)ds}≤∫0τG(s)}ds.3.Theorem 3(Countable determination of the Kuratowski MNC).For every bounded subset B of a metric space,there exists a countable subset B0?B such that a(B0)=α(B).4.As an application of Theorem 1 and Theorem 2,we extend some classical results of solvability of the initial-value problem in Banach spaces x’(t)=f(t,x),x(0)=x0.5.We show that the representation theorem(Theorem 1)is still true for convex measures of non generalized compactness(convex MNGCs,for simplicity),including convex measures of non-weak compactness,of non-super weak compactness,of non-Radon-Nikodym property and of non-Asplundness etc.And the basic integral inequalities in Theorem 2 still hold for convex MNGCs under the same conditions.In this paper,a)with the aid of a "three-time order preserving embedding theorem" introduced by L.Cheng et al in 2018 and by arguments in convex analysis,especially,in subdifferentiability and Gateaux differentiability of convex functions,we show the representation theorem of convex MNCs and convex MNGCs.b)In the proof of representation theorem,the significant difficulty is to prove that the convex function determined by a convex MNC or a convex MNGC is locally Lipschitzian,which defined on a metric cone with empty interiors.There is no classical method for reference in convex analysis.c)With the help of the localized setting of finite representability developed in the last decade,we introduce stronger finite representability,which is called "strongly finite representability",then show for every nonempty subset A of a Banach space,there is a countable subset A0 ? A such that A is strongly finitely representable in A0.