An Attension And Application Of Krasnoselski Theorem

In this paper,we mainly discuss the bifurcation problem of nonlinear equationF(λ,u) =λu-G(u) =θ, where F : R×Xâ†'X is a nonlinear di?erential mapping,X is a Banach space.The Krasnoselski's classical bifurcation proved using Morse theorem which pmultiplicity eigenvalues of A = G (θ) are bifurcation points for F(λ,u) =θwith thecase in which G∈C1(X,X) is a compact variational operator.Moreover,he provedvia topological theorem that the eigenvalue of A with odd algebraic multiplicityis a bifurcation point of F(λ,u) =θin the case that A = G (θ) and G are com-pact. We weak the condition for G (θ) is compact,and 0 < dim(N(λ*I - G (θ)) =p <∞,N(λ*I - G (θ)) \ R(λ*I - G (θ)) = {θ}, prove the Krasnoselski's bifur-cation theorem via Lyapunov-Schmidt reduction and the implicit function theo-rem.Furthermore,we compute the direction of bifurcation. At last,to understandthe abstract theorem more easily,we applying the main theorem to the semilinearelliptic equation and systems.We have the local construct of the solution sets nearthe bifurcation points from the abstract theorem.