Existence Of Positive Solutions Of Several Kind Of Differential Equations

This paper is concerned with the existence of positive solutions for bvp of several kinds of differential equations. The main results are obtained by using the fixed point index theory on cone and fixed point theorem in cones.These results extend some of the existing literature.This thesis of Master is composed of four chapters. Chapter 1 introduce issues arising from the historical background and main tasks of this article.Chapter 2 discuss the existence of positive periodic solutions for the functional differential equation with parameterwhere a(t, x(t))∈C(R, (0,∞));Ï„_i(t)∈C(R, R),0≤i≤n;f∈C[R×[0,∞)ⁿ⁺¹, [0,∞)); R = (-∞,+∞);λ＞0 is a parameter,Ï„_i(t),a(t,x(t)),f(t,·) are continuousandω-periodic function in t, By using the fixed point index theory in cones.We extend some known results,and obtain some existence and nonexistence results.In Chapter 3, we study the second-order three-point boundary value problemwhereη∈(0,1) is a constant, f∈C([0,1]×[0,∞), [0,∞)).α≥0β≥0, 0＜k＜(?). The existence criteria for multiple positive solutions isobtained by using Krasnoselskii fixed point theorem and Leggett-Williams fixedpointtheorem.This result extends some of the existing literature.Chapter 4 investigates the existence of positive solutions for second order Neumann singular boundary value problem where M∈(0, (?)) is a constant,f(t, u) is required to be nonnegetive and continuous, and f(t, u) may be singular at t = 0, t = 1 mostand x = 0.By constructing a special cone, using the fixed point index theory on cone and method of approximation,we get the existence of at least two positive solutions.This result extends some of the existing literature.