The Structure Of The Extremal Solutions Set Of Ill-posed Equations In Banach Spaces And Its Applications In A System Of An Autocatalytic Chemical Reaction Model

In the feld of modern applied science, the system is formulated by a mathematicalmodelling to be a state equation bellowT (λ, x)=y,(1)where X, Y are Banach spaces(or manifolds), T: Λ×X→Y is an operator, and λ∈Λ Rnis viewed as a parameter. If the equation (1) and the parameter λ are independent ofeach other, then the state equation (1) can be simplifed to beT x=y.(2)The research on the ill-posedness of the equation (1) covers many aspects, includingbifurcations, catastrophes and chaos etc. While the true solutions of (2) are more concrete,the extremal solutions, the optimal approximation solutions and its structure of solutionset are also important.In Chapter2, we study the discriminant conditions of the existence of the ill-posedness, where the range of the operator T in the equation (2) is co-dimension1.And a description for the structure of the set of the extremal solution is given, too. Weuse the generalization of operators and the geometry of Banach spaces as the the maintools and methods in this chapter. For the nonlinear equation (1), we analyze a specialform ofu T (λ, u)=θin chapter3. Utilizing the well-kown topological methods(Leray-Schauder degree), weobtain an abstract global bifurcation theorem. Still, a description for the structure of thesolution set is given in Chapter3.The last chapter of this thesis deals with the existence of steady-state solutions andthe dynamics behavior of the model coming from an autocatalytic chemical reaction with some decay, which can be viewed as an application of the global bifurcation theoremobtained in Chapter2. Besides the global bifurcation theorem, another efective ways wedepend on is the sub-supper solution method in Mixed Quasi-monotone Theory. In thecase of strong decay, we get the asymptotically stability of the dynamical system, and wealso describe the structure of the solution set.