Existence,Nonexistence And Other Properties For Solutions In Several Kinds Of Nonlinear Equations

There are many problems in modern applied science which are needed to be solved urgently,we can develop mathematical models to establish differential equation or equa-tions to describe these problems.To analyze and solve the equation or equations,we should choose proper state spaces and define appropriate operators in order to transfor-m them to suitable operator equation in abstract spaces.The existence for solutions of these kinds of nonlinear equations and nonlinear equations with random disturbance can be considered by methods of nonlinear analysis and stochastic analysis.In the past few decades,the research is on the rise.Since linear mathematic can be solved by general theory uniformly,we should say for nonlinear systems,each case is unique,they must be discussed separately.Thus there are many different research fields in nonlinear analysis.Nonlinear system and stochastic system are two most important aspects to reflect the complexity of the system.First,by using the bounded quasi-linear generalized inverse of a closed linear opera-tor,we give a necessary and sufficient condition for a closed linear subspace in a Banach space to be complemented.And as an application in bifurcation,Saddle-node,transcriti-cal and pitchfork bifurcations theorems from multiple eigenvalue is given under a weaker and easier to validate condition.Next,we study an important nonlinear chemical model-Schnakenberg model.For a coupled two-cell Schnakenberg model with homogenous Neumann boundary condition,instead of transforming to abstract equation,but by using comparing theorem,we give a priori estimate to the positive solution,then we obtain the non-existence and existence of positive non-constant solution as parameters changes.We find that if the diffusion coefficient of one of the chemicals product is sufficient large or another chemical sources is sufficient small,the steady state system has no non-constant solution.Thus we get when the domain is small,there is non non-constant solution.By using Leray-Schauder degree,we prove under some conditions on the parameters,the steady state equation has at least one non-constant solution.On this basis,we can transform the equations to abstract operator equation to discuss its bifurcation properties.Finally,we study a nonlinear stochastic spatiotemporal diffusive predator-prey sys-tem with Holling-III.It is proved that under some existence condition,the system can be turn into stochastic abstract operator equation.Therefor there is a unique mild global solution for the system,and the system can not be persistent if the noises are large and the system can be stability in mean square if the noises are small.Finally,we proved that the global mild solution is a Markov process,and an unique invariant measure for the system is obtained under a more stringent conditions than the existential conditions,i.e.,the system is ergodic.The existence of the solution is the foundation of discussing a bifurcation problem.After finishing the existence of the solution for these specific equations,then we can consider their bifurcation problem by choosing proper operators and spaces.