Quasi-Linear Generalized Inverse And Imperfect Bifurcation Theory And Their Applications

Nonlinear mathematics, as a branch of nonlinear science, is the theoretical foundationof nonlinear science and it is one of the most studied directions in modern mathematics.The content of our paper belongs to nonlinear mathematics, which is connected to manynonlinear problems in physics, chemistry, engineering and biology by their mathematicalmodels.In general, the mechanism of real world problems and modern science technology canbe described by state equations like algebraic equations, ordinary di?erential equations,di?erence equations, partial di?erential equations and other types of equations. If thestate space and input space are taken to be Banach spaces X and Y , we can express thestate equation as operator equation F(x) = y or F(λ,x) = y, where F and F(·,·) aremappings from X to Y and R×X to Y respectively. For nonlinear systems, the equationmentioned above is general nonlinear equation:F(λ,x) = y. (0.0.2)If F is independent ofλand linear about x, then (0.0.2) becomesTx = y,where T∈L(X,Y ) is a linear operator. If the operator equation satisfies the followingconditions: (i) the existence of solution; (ii) the uniqueness of solution (iii) continuity anddependence of solution with respect to y, then the operator equation is well-posed. Ifany of the three conditions failed, then the operator equation is ill-posed. In the last fiftyyears, ill-posed mathematics has attracted lots of interests, which has been an importantnew research branch. The research subjects of ill-posed and nonlinear operator equationsinclude bifurcation, catastrophe and chaos etc. And the research subjects of ill-posed andlinear operator equations include the generalized inverse theory of the linear operator etc. In this paper, we will discuss quasi-linear generalized inverse and the application ofimperfect bifurcation theory in some partial di?erential equations.First, we show that the best generalized inverse among all bounded quasi-lineargeneralized inverse set in linear operator is the Moore-Penrose generalized inverse. In thecase X and Y are finite dimension spaces Rn and Rm respectively, this result deduces themain theorem by G.R.Goldstein and J.A.Goldstein in 2000.Second, we study the generalized bifurcation theorem from a non-simple eigenvalue,which will be applied to perturbed nonlinear equations and it provides new ideas to theresearch in bifurcation theory.Finally, we study analytic bifurcation theory on Banach spaces. By using Morselemma instead of implicit function theorem, we obtain the two crossing solution curvesfrom conditions on F at the bifurcation point; We prove that the celebrated bifurcationtheorem of Crandall and Rabinowitz is a special case of our result. We study imperfectbifurcation under small perturbations by using this new bifurcation theorem and givedi?erent transversality conditions, and obtain the classification of degenerate solutions.We give applications in semilinear elliptic equations. In particular, we apply the imperfectbifurcation theory in Banach spaces to study the exact multiplicity of solutions to aperturbed logistic type equation on a symmetric spatial domain and obtain the precisebifurcation diagrams.