Studies On Solving Generalized Equations

Let f:Ω(?) X→Y be Frechet differentiable on an open setΩin X and G:X→2Y be a set-valued mapping with closed graph, where X and Y are Banach spaces. This thesis is concerned with the problem of approximating a solution of the generalized equation of the form find x∈Ω, such that 0∈f(x)+G(x). This kind of inclusion is an abstract model of a wide variety of variational problems in-cluding linear and nonlinear complementarity problems, systems of nonlinear equations, variational inequalities, first-order necessary conditions for nonlinear programming etc. There are also plenty of applications in engineering (analysis of elastoplastic structures, traffic equilibrium problems) and economics (Walrasian equilibrium, Nash equilibrium). Many researchers have been conducted due to its practical background, see for example [16,17,27,28,36,37,47,48]. However, the sequences generated are not unique. There-fore, from the viewpoint of practical computations, we establish a semilocal convergence of sequences generated by Gauss-Newton's method by using majorizing function technique. Moreover, noting that nonlinear equation is an important special case of generalized equation, which is widely studied in both theoretical and applied areas of mathematics, we establish a semilocal as well as a local convergence analyses of Newton-Steffensen's method to solve nonlinear operator equations in Banach spaces. Some numerical examples are shown. This thesis is divided into four parts as follows:In Chapter 1, we give some necessary notations, definitions and preliminary results and summarize the progress of research on generalized equations.In Chapter 2, we define the set-valued mapping as follows:Qx(·):= f(x)+f'(x)(·x)+G(·). Under the assumptions that f'satisfies the L-average Lipschitz condition and Qx0-1(·) is Lipchitz-like (x0 is an initial point), we establish a semilocal convergence analysis for Gauss-Newton's method and verify that the sequence generated by Gauss-Newton's method converges to some solution of the generalized equation. Moreover, we specialize the function L(u) appearing in L-average Lipschitz condition to two important cases of the function:L= constant and L=2γ/(1-γu)3, to get Kantorovich type and Smale type theorems. In particular, mimicking Smale'sγ-theory about the approximation zeros for Newton's method in solving nonlinear equations, we introduce a new kind of approximation zero for Gauss-Newton's method in solving generalized equations and prove that the initial point of the sequence generated by Gauss-Newton's method is anη-approximate solution of the generalized equation.In Chapter 3, we study a special case of the generalized equation-a nonlinear equa- tion: F(x)= 0, where D (?) X is an open set and F:D (?) X→Y is a nonlinear operator. Under the assumption that the second derivative of the operator satisfies Lipschitz condition, the convergence criterion and convergence ball for Newton-Steffensen's method are es-tablished. Applying the obtained semilocal convergence result to a nonlinear boundary value problem, we can see that Newton-Steffensen's method is very valid and feasible.