Compution Of Topological Degree And Fixed Point Index In Product Spaces And Applications

Nonlinear functional analysis has been one of the most significant branches in modern mathematics, which mainly includes topological degree theory,Cone expansion and cone fixed point theory,the variational method, monotone operator theory, cone theory.etc People at home and abroad had profound works in the research of nonlinear functional analysis.L.E.J.Brouwer had established the concept of topological degree for infinite dimensional space(Broawer degree) in 1912. J.Leray and J.schauder had extended the completely continuous field of Banach space and established the Leray schauder degrees in 1934. Many famous mathematicians in China, say Zhang Gongqing,Guo Dajun,Zhang Chengkui,Ge Weigao etc.,played an important role in nonlinear functional analysis.The boundary value problems of nonlinear differential equations is among the most im-portant parts in the theory of differential equations They can be found in applied mathematics physics and other applied sciences. Because of the importance in both theory and applications, boundary value problems for ordinary differential equations have been studied widely, with many interesting and in-depth results established in recent years.In this thesis, we mainly use fixed point index theory to study the existence of positive solution and nontrivial solutions of boundary value problems. It divides into four chapters.In Chapter One,we compute, by using cone theory, the fixed point index of a class of completely continuous fields. Our results generalize the existing ones in [68]. Finally, we use our abstract results to establish the existence of positive solutions for the system of nonlinear differential Sturm-Liouville boundary value problems.In Chapter Two, we study the existence of positive solutions for a fourth-order boundary value problem: where n≥ 2;f ∈ C([0,1] ×Rn+,R+)(R+:= [0, ∞)), αi,bi,ci,di≥0(i= 0,1,...,n-1),且Δi= aidi+bici+aici> O.we use the fixed point index theory to prove the existence of positive for the problem. Under more general conditions,namely the coefficient of the boundary condition is different, Our main results improve the results in [79].In Chapter Three, we study the existence of positive solutions for the system of higher- order quasilinear boundary value problemswhere n≥2, m≥2,φ:R+ â†'R+,is either a convex or concave homeomorphism ,and f,g ∈ C([0,1] ×R+×R+,R+),(R+:= [0,∞).Based on a priori estimates achieved by utilizing Jensen’s inequalities for concave and convex functions,we use fixed point index theory to establish our main resultsIn Chapter Four,we compute, by using cone theory, the topological degree of a class of completely continuous fields. Our results generalize the existing ones in [39]. Finally, we use our abstract results to establish the existence of nontrivial solutions for the system of superlinear Hammerstein integral equations.