Minimization Of The First Positive Eigenvalue For Some Second-order Linear Boundary Value Problem With Indefinite Weight

It is well known that the principal eigenvalue of the corresponding linear problem plays a crucial role in study of many nonlinear problems.In this paper,by using the method of G raphical analysis and the Implicit function theorem,we study the first positive eigenvalue of some seeond-order linear boundary value problem with given indefinite weight.The main works are:1.By using the Implicit function theorem,we study the first positive eigenvalue of following boundary value problem and when the weight function m(x)take the form where α∈[0,1].We get the results:if m(x)=m1(x),then the value of first positive eigenvalue of problem(1)and(2)are decrease with respect to α；if m(x)=m2(x), then the value of first positive eigenvalue of problem(1)and(2)are increase with respect to α.2.By using the method of Graphical analysis,we study the first positive eigen-value of following boundary value problem when the weight function m(x)take the form and where in m3(x)and m4(x),T is a fixed constant and0<T<1/2,a belong to the interval of[0,T]and[0,1-T]；in m5(x)and m6(x),T is a variable.We get the results:If m(x)=m3(x)or m(x)=m4(x),then the value of first positive eigenvalue of problem(3)is independent with a.If m(x)=m5(x),T∈(0,1/2),then the value of first positive eigenvalue of problem(3)is decrease with respect to T； If m(x)=m6(x),T∈(1/2,1),then the value of first positive eigenvalue of problem (3)is increase with respect to T.