| Let H be a real Hilbert space with inner product〈?, ?〉and norm‖?‖. Suppose that C is closed convex subset of H and F :H→H is an operater. In chapter 2 ,we study thefollowing classical variational inequality problem: find u*∈C ,such that, Let T1,T2…TN:H→H be finite nonexpansive mappings. From an arbitrary point x0∈H, an explicit iteration scheme is defined as follows:Where, (?), F is a nonlinear operatoron a real Hilert space H which isη-strongly monotone and k -Lipschizian on anonempty convex subset of H. Under some suitable conditions, the sequence {xn} is shown to converge weakly to a point of C . necessary sufficient and conditions that{xn} converges strongly to a point of C are obtained. In chapter 3, we study the following variational inequality problem: find u*∈C such that g(u*)∈C and GVI(F,g,C) :〈F(u*),v-g(u*)〉≥0,v∈C,where F is a nonlinearoperator on a real Hilert space H which isη-strongly monotone and k -Lipschizian on a nonempty convex subset of H, Assume also that g: H→H si a continuous mapping andσ-Lipschizian,δ-strongly monotone. By using Hybrid Steepest Descent Method, we divise an iterative algorithm which generates a sequence {un} from an arbitrary initial pointu0∈H,under suitable assumptions imposed on the algorithm parameters .The sequence {un} isshown to converge in norm to the unique solution u* of the generalized variational inequality GVI(F, g, C) .In chapter 4, we show that theorem of chapter 3 can be applied to k -constrained generalized psceu-doinverse problems. |
