On existence of solutions for differential equations and their properties via functional analysis

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We studied the existence of solutions of both initial value problem and periodic problem for the evolution equation u'(t)+Au(t)∋f(t,u(t)), 0【less than or equal】t【less than or equal】T under the condition u(t)∈K. Here, (V,‖・‖) is a reflexive Banach space which is densely imbedded into a Hilber space (H,<・,・>), A⊂・H×H is a maximal monotone operator satisfying <Ax-Ay,x-y>【greater than or equal】‖x-y‖^p (p>1), K is a closed, convex subset of H, and f : [0,T]×K→H is a Caratheodory mapping. Since the condition for continuity of f(t,・) is on the topology of V, our result is an extension of Bothe's. We also assumed a subtangential condition for f. Moreover, we assumed that the metric projection P : H→K satisfies P(V)⊂V and P : (V,‖・‖) →(V,‖・‖) is continuous in order to associate the continuity of f and the topology of H. We showed that these assumptions are natural in applications.We showed existence of subharmonic solutions for the sysytem of second order ordinary equation u(t)+G'(u(t))=f(t), t … More ∈R in R^N. Here, N is a natural number, f ∈C(R,R^N) is a T-periodic function satisfying (1/T) ∫^T_0 f(t)dt=0,and G∈C^2(R^N,R) a function without convexity. Using Morse's inequality, in the case when the norm of f is sufficiently small, we also showed that there exist at least two solutions which are κT-periodic but not T-periodic for each sufficiently large prime number.Under the homogeneous Dirichlet boundary condition, we studied existence of multiplicity of positive solutions for a singular elliptic equation -Δu=λu^<-q>+u^p in Ω. Here, Ω is a bounded domain in R^N, λ>0,q>0 and p>1. In the case of 0<-q<1,our result is related to Ambrosetti-Brezis-Cerami's. Since behaviors for associated functionals are similar, we can naturally expect that a similar result holds ; so in the case when λ>0 is sufficiently small, we showed that the problem has at least two positive solutions. We also showed that the obtained positive solutions are clasical under some assumption. The assumption is less restrictive the condition that ∂Ω is C^2. Less