Some Discussion For Nonlinear Operators And Nonlinear Equations

The purpose of this thesis is to discuss problems of fixed points for somenonlinear operators in partial ordered linear space. And we obtain the conclusionthat such operators under certain conditions will have unique fixed point.Moreover, this thesis is also to study the existence of the solutions for someclasses nonlinear differential equations by using the fixed point theorem fornonlinear operators.This thesis includes four chapters:In chapterâ… , by using partial order theory, we study a class of mixedmonotone mappings with convexity and concavity in partial ordered linear space.That is A(x,y): D×D→E, which is increasing withx, decreasing withy, andsatisfies A(tx, t^-1y)≥t[1+η(t, x, y)]A(x, y)(where x,y∈D,t∈(0,1)). In particular, we give conditions, both necessary andsufficient, for the existence and uniqueness of fixed points. The relevant resultsare generalized and improved.In chapterâ…¡, by using three-solution theorems in paper[12] and fixed-pointindex theory, we investigate the second-order three-point boundary valueproblem where 0＜η＜1,αis positive constants, when the following conditions aresatisfied, (H₁) f∈C([0,∞),[0,∞)); (H₂) h∈C([0,1],[0,∞)), there exists x₀∈(0,1) such that h(x₀)＞0; (H₃) a∈C[0,1], b∈C([0,1],(-∞, 0)).and the nonlinear term f also satisfies f₀,f_∞(?){0,∞}(where f₀=(?)f(u)/u,f_∞=(?)f(u)/u), then the second-order three-point boundary value problem abovewill have at least three non-negative solutions, and we also give the interval forthese solutions.In chapterâ…¢, we discuss the existence of multiple solutions for the secondorder impulsive differential equation.Where J=[0,1], 0＜t₁＜t₂＜......＜t_m＜1, f∈C[J×R⁺,R⁺],I∈C[R⁺,R⁺]and R⁺=[0,+∞),Δx'|_t=tk=x'(t_k⁺)-x'(t_k^-) and x'(t_k⁺, x'(t_k^-) are the left and right limit of x'(t) att=t_k respectively, and we get the conclusion that the second-order impulsivedifferential equations above may have multiple solutions when the nonlinearterm f and impulsive term satisfy certain conditions.In chapterâ…£, through discussing the fixed points theorems for a class ofincreasing operators, we investigate following nonlinear evolution equation where f: R×X→X,f(t,u) isω-periodic with respect to t, non-decreasingwith u(f is increasing), and f is also e- concave, (that is(?)0＜λ＜1, (?)0＜α=α(λ)＜1 such that f(t,λv)≥λ^α(λ)f(t,v), 0≤t≤ω, v∈P), byusing super-solutions and sub-solution method in ordered Banach space, also byconsidering the theory and important characteristics of positive operatorssemigroups. And the sufficient and necessary conditions for the existence ofperiodic solution for the nonlinear evolution equations above are given.