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Nonlinear functional analysis is a new course that develop just inthe last several decades,it is developed when people study the medicalscience,biology,classicality and modern physics,economics etc. Nonlinearproblem is one of modern science foundation knowledge, Nonlinear functionanalysis provided the theories tools for studying nonlinear problems.Thebasic methods are topological method,variational method,analytic method,half ordering method and monotony method etc. Professor Dajun Guo sumup some important problems and its application in [1].Such as some typi-cal nonlinear operator, Hammerstein integral equation,constant or partialdi?erential equation,transfer equation,cone theory and positive solution fornonlinear operator equation, topology and fixed point theory,etc.The constant di?erential equation theory in abstract space is a newmathematics ramus that develop just in the thirty years ago,it put to-gether di?erential equation theories and function analysis theories,we canusing the method of the functional analysis to study the di?erential equa-tion of the abstract space. When we need solve a di?erential equation inreal space,we usually change it into a integral equation,and from it ab-stract a operator, then discuss completely continuous of the operator,atlast,applying fixed point theory, we can find the fixed point of the opera-tor, thus gained the solution of the original equation.Singular problem is put forward to studying atmosphere convection,sph-ere evolvement and some ?ow mechanics problems.We can translate theminto the following Boundary Value problem???????u (t) + t?λu?m(t) = 0, t ∈ (0,1),u ∈ B(0,1), λ,m > 0Afterwards,the research begin around the following di?erential equation???????u (t) + f(t,u(t),u (t)) = 0, t ∈ (0,1),u ∈ B(0,1),What is called singular is that the function f is infinity at some points.In the first chapter, Applying fixed point index theory, this paperdeals with the existence of multiple positive solutions for a class of singu-lar boundary value problem of impulsive di?erential equations in abstractBanach space. This consideration is resulted from the boundary valueproblem in [4],which is Profession Yansheng Liu considered. This disserta-tion refers to the function f(t,x) instead of the function f(t,x,x ),refers tothe solution space PC[J,E] instead of the solution space PC1[J,E]. Us-ing cone fixed point theorem we get the existence of one or two positivesolutions.In the second chapter, using fixed point theory of cone expansionand compression, this paper investigate the existence of multiple positivesolutions for singular boundary value problems of a coupled system of non-linear ordinary di?erential equations.This consideration is resulted fromthe system of nonlinear ordinary di?erential equations in [27], which isProfession Agarwal and O’Regan considered.Using Leray-Schauder theo-rem they investigated the existence of solution for a coupled system ofnonlinear ordinary di?erential equations. We investigate the existence oftwo positive solutions with some special conditions.In the third chapter, This paper establishes the exact multiplicities ofproperties of positive solutions for some singular boundary value problems.This consideration is resulted from a problem in [12],what is put forwardby Agarwal and O’Regan,they proved that the following equation have anonnegative solution for su?cient small δ > 0,????????y (t) + δ(y?α(t) + yβ(t) + 1) = 0, t ∈ (0,1),y(0) = y(1) = 0,Where 0 ≤ α < 1 < β,δ > 0 is a parameter.this chapter educe the exactnumber and property of solution based on paper [12].
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