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

We studed 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.这里， （V，―）是一个可伸缩的Banach空间，其中有一个密集嵌入到Hilber空间（H，<·，·>）中， A μ·H×H是一个最大单元操作员满意度<Ax-Ay，x-y>[大于或等于等于] ― x-y ― ^p （p>1）， K是一个封闭的，H的卷积子集，f ： [0，T]×K→H是一个日历映射。由于f（t，·）的连续性条件是V的拓扑，我们的结果是两者的延伸。我们也假定了f.的亚切线条件。Moreover，我们假定计量投影P ： H→K饱和度P（V）〔V and P ： （V，―） →（V，―）是连续性与f的连续性和H的拓扑结构相关的。我们展示了这些假设在应用中是自然的，我们展示了二阶普通方程式u（t）+G'（u（t））=f（t）， t的存在 ... More R in R^N.这里， N是一个自然数字， f ∈C（R，R^N）是一个T周期函数满意度（1/T）≤ ^T_0 f（t）dt=0，G∈C^2（R^N，R）是一个没有舒适度的函数。在使用莫尔斯的不平等的情况下，当f的正常值足够小时，我们还展示了最少两个解决方案，其中包括κ T-周期性，但不是每个T-周期性都足够大的素数。在同源的Dirichlet边界条件下，我们研究了单一椭圆等式的多个积极解决方案的存在-γ u= γ u^<-q>+u^p。这里， Ω是R^N中的边界域， λ>0，q>0和p>1。在0<-q <1的情况下，我们的结果与Ambrosetti-Brezis-Cerami有关。由于相关功能的行为是类似的，我们自然可以期望得到类似的结果;因此，在这种情况下，当λ>0是相当小的，我们展示了这个问题至少有两个积极的解决方案。我们还展示了所获得的积极解决方案在某些假设下是经典的。假设是较小的限制性，即Ω是C^2。Less（低）

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