Positive Solutions Of Nonlinear Differential Equation Boundary Value Problem

?Nonlinear functional analysis is not only profound theory but also widelyused research discipline in mathematics, in the context of nonlinear problemsof mathematics and natural sciences, a number of theories and methods of dil-ing with nonlinear problems have been established. Because great importanceof nonlinear problems has been concerned in mathematics field and naturalsciences field at home and abroad, researches of the nonlinear functional andapplication are undoubtedly important application of theoretical significanceand value, so it has good prospects for researches and applications.?Nonlinear differential equations problems arise in a variety of areas ofmathematics, physics, cybernetics and other applied sciences, which is an im-portant branch of modern analysis mathmatics and is one of the most activeareas of mathematics presently. Because it can explain all kinds of natural phe-nomena, more and more mathematicans are devoting their time to it. Amongof them, the singular muliti-point boundary value problems come from allkinds of areas of applied mathematics and physics model, it has wide applica-tion prospects. Theories and methods related to the direction of mathematicswill have greater impact on development, at the same time, it has a positive im-pact on development in physical, chemical, biological sciences, astronomy, andother related disciplines, so it has great theoretical significance and the valueof application. The present paper employs the cone theory, fixed point indextheory, krasnoselskii fixed point theory and leggett-williams fixed point theoryand so on to investigate the existence of positove solutions of several classes ofnonlinear differential equations boundary value problem. The results obtainedare either new or essentially generalize and improve the previous relevant ones.The thesis is divided into three chapters according to contents.?In Chapter 1, we mainly use the krasnoselskii fixde point index thorem as well as the leggett-williams fixed point theorem, in view of boundary valueproblemwe will discuss existence of at least one positive solution, at least two positivesolutions and three positive solutions, where f,α,βsatisfies:(H₁) f : [0,1] x [0,∞)→[0,∞)continuously;(H₂)α,β∈R, satisfiesβ<2Ï€²,α≥β²/4,α/Ï€⁴+β/Ï€²<1,L=Ï€⁴-βπ²-α.It improves and generalize some results of articles [3], [6] and [7] essentially.In Chapter 2, we inspect fourth-order singular nonlinear deferential equa-tion problem of value condition more generalby using krasnoselskii's fixed point theorem of cone expansion and compressionand under certain supposition conditions, we obtain existence of at least onepositive solution, at least two positive solutions and infinitely many positivesolutions of the boundary value problem. whereλ> 0 is a positive parameter,a_i≥0,b_i> 0, c_i > 0, d_i > 0, i=1, 2.ρ_i=b_ic_i+a_ic_i+a_id_i>0, i = 1, 2. f∈C([0,∞), [0,∞),α∈C((0,1), [0, +∞)), namely permit a has strange in t=0and t=1 place.In Chapter 3, we exploit the fixed point theory of cone expansion and com-pression to study existence of positive solutions of a class three-order singularnonlinear differential equation with three-point boundary value problems: where 0<η< 1,1<α< 1/η,f∈C([0,∞), [0,∞)),α∞∈C((0,l),[0,+∞)),and is not identically zero on [η/α,η], a is allowed to be singular at t= 0 ort = 1. The article preceding part is that the existence of positive solutionto the boundary value problem is discussed, under condisions of f satisfingf₀=f_∞=0 or f₀=f_∞=∞; the latter part is that the existence ofpositive solution to the boundary value problem is discussed, under condisionsof f satisfing f₀=f_∞(?) {0, +∞}, where f₀=(?)f(u)/u, f_∞=(?)f(u)/u. Itimproves and includes some known results of [6].