Existence Uniqueness And Multiplicity Of Solutions Of Boundary Value Problems For The Differential Equation -u"=λ～2u+f(u,u')

In this paper ,by using the shooting method,Sturm separation principle and some special techniques, we study second-order Dirichlet boundary value problemswhere λ > 0, α ≥ 0 are parameters andwhere λ > 0 is parameter.In Chapter 2,we proved the following theorem:then if 0 < λ < π,α ≥ 0, (*) has at least one solution; if λ ≥ π, (*) has no positive solution.In Chapter 3, the following theorem was established:then if 0 < A < tt, a > 0, (*) has a unique positive solution; if A > tt, (*) has no positive solution.In Chapter 4,we obtained the following result:Theorem 4.1 Let / G C(R2, R) .satisfying: (F) (i) u/(u,i;)>0lV u?0,v^0, (ii) there exists fi, h,9i,92 € C(R+, R+), R+ = [0, oo)with the propertyMM)9i(\v\)<\f(u,v)\0,(F2) lim^ = O,3-+0 Sds—TT < °°?(Gj.i) lii= oo,Va>0.y a + g2{s) Then(1) for all A > 0, (**) has infinitely many solutions;(2) if 0 < A < 7T, (**) has at least one positive solution and one negative solution^ A > 7T, (**) has neither positive solution nor negative solution;(3) if kn < A < (k + l)7r,the solution of (**) has at least k zeros in (0,1) , and for every n> k, (**) has at least two solutions each of which has exactly n zeros in (0,1)-...