Fixed Point Theorems And Applications Of Mixed Monotone Operators

Recently, the nonlinear analysis problem has become an important research direction in modern mathematics. And then, the hotspot issue of the nonlinear analysis is mixed monotone operator and its application. The mixed monotone operator was put forward ?rstly by D. J.Guo professor and Lakshmikantham in 1987. Later, many authors have investigated all kinds of theories about mixed monotone operators, and these theories are also widely used in the physics or the applied mathematics. In this paper, with the help of the monotone iterative methods and cone theory, we mainly investigate the sum of mixed monotone operator in Banach space. In the end, we can conclude the existence and uniqueness of solutions for the operator equations. Meanwhile, we apply the results to differential equations.The thesis is divided into four sections according to contents.Chapter 1 Preference, we introduce the main contents of this paper.Chapter 2 We use the ?xed point theory of mixed monotone operator to investigate existence and uniqueness of the solution for the following operator equation C(x, x) + D(x, x) = x where C, D are mixed monotone operators, and C is α-concave, D is sub-homogeneous. As applications, we use the results obtained to study a fractional differential equation.Chapter 3 Using the mixed monotone theories, we discuss the following two equations where Dv0+u(t) is Riemann-Liouville fractional derivative(n > 3, n ∈ N).The nonlinear term f, g have the property of mixed monotonicity. In the second equation, the equation has derivative term.Chapter 4 We use the ?xed point theory of mixed monotone operator to investigate existence and uniqueness of the solution for the following operator equation A(x, x) + B(x, x) = x,where A, B are mixed monotone operators, and A is e-concave(convex)operator, B is sub-homogeneous.