二阶微分系统奇异正定超线性周期边值问题的多重正解

主要研究了二阶微分系统具有奇异正定超线性周期边值问题多重正解的存在性问题,利用Leray-Schauder抉择定理和锥不动点定理给出了奇异正定超线性周期边值问题-(p(t)x′)′+q1(t)x=f1(t,x,y),t∈I=0,1]-(p(t)y′)′+q2(t)y=f2(t,x,y)x(0)=x(1),x1](0)=x1](1)y(0)=y(1),y1](0)=y1](1)(1.1)的多重正解的存在性,其中非线性项fi(t,x,y)(i=1,2)在x=∞,y=∞点处超线性,在(x,y)=(0,0)处具有奇性.这里定义x1](t)=p(t)x′(t),y1](t)=p(t)y′(t)为准导数,其中系数p(t),qi(t)(i=1,2)是定义在0,1]上的可测函数,且p(t)>0,qi(t)>0(i=1,2),a.e0,1],fi(t,x,y)∈C(I×R×R,R+),R+=(0,+∞).

Multiplicity of Positive Solutions to Singular Positone Superlinear Second-order Periodic Boundary Value Problems for Second-order Differential Systems

We are devoted to establish the multiplicity of positive solutions to positone superlinear singular equations for second-order differential systems with periodic boundary conditions {- (p(t)x')' + q(t)x = f_n(t,x(t)) +q(t)/n, 0≤t≤1 x(0) = x(l),x~(1])(0) = x~(1])(l) It is proved that such a problem has at least two positive solutions under our reasonable conditions. Our nonlinearity f_i(t,x,y)(i = 1,2) may be singular in (x,y) = (0,0) and superlinear at x =∞,y = ∞, where x~(1])(t) = p(t)x' (t) ,y~(1])(t) = p(t)y'(t) are quasi-derivative and p(t),q_i(t)(i=1,2) in 0,1] measurable functions, p(t) >0,q_i(t) > 0(i = 1,2), a.e 0,l],f_i(t,x,y)∈ C(I × R × R,R~+),R~+= (0, + ∞). The proof relies on a nonlinear alternative of Leray-Schauder type and on Krasnoselskii fixed point theorem on compression and expansion of cones.