Class Of Nonlinear Operator, Fixed Point Theorem And Its Applications

Nonlinear functional analysis is an important branch of mordern analysismathematics.more and more mathematicians are devoting their time to it which can explain many kinds of natural phenomena. Among them,the positivefixed point theory for nonlinear operators plays a very important role in nonlinear integral and differential equations, especially in the study of positive solutions of nonlinear integral equations in nuclear physics.In this paper,we discuss the existence and uniqueness of fixed point theoremfor a class nolinear operators without continuity or compactness in Banach space with cone theory,iterative method and so on. The thesis is divided into four sections according to contents.Chapter 1 is the introduction of this paper, which introduces development and research significance of nonlinear functional analysis.In Chapter 2,with the partially method,cone theory and iterative method,we extend the existence and uniqueness of fixed point of the concave operator.It corresponds with the conclusion in document[6]. Also, we apply this theoremto the two-point boundary value problem for a second-order differential equationIn Chapter 3,we discuss the existence and uniqueness of fixed point of(?)(t),(?)(t,u, v)mixed monotone models operators which have no continuous or compactconditions.It extends the conclusion in document[21]. Also we consider the continuous positive solutions of the nonlinear integral equation as followswhere x∈C[0,1].In Chapter 4 ,with partially method,cone theory and iterative method,we et the existence and uniqueness of fixed point of the operator C = A + B in Banach space.It extends the conclusion in document[29], where A is decreasing operator,B is sublinear operator, and we don't require the continuity or compactnessof the operator.Also,we apply the theory to the following nonlinear integral equation and we get the unique positive solution∫_α1^α2 G(t,s)f(x(s))ds=G₁(t)x(t)-(1-G₂(t))x(t+τ),t∈R,whereτis real numbers.