The Best Approximation and Related Nonlinear Analysis with Applications Under the Framework of P-Vector Spaces

The goal of this paper is develop a number of new analysis tools for nonlinear analysis under the general framework of p-vector spaces for p iin (0, 1], which would be very useful for the study of nonlinear problems from optimization, nonlinear programming, variational inequality, complementarity, game theory, mathematical economics, social science, applied science and related areas. In particular, we first establish one best approximation, acting as a tool to establish the principle of nonlinear alternative, which then allows us to give general principle of nonlinear alternative under the general framework of p-vector spaces for p in (0, 1]. The results given in thispaper on the principle of nonlinear alterative and Leray - Schauder type not only include the corresponding results in the existing literature as special cases, but also would become very useful new tools for the study of nonlinear problems from social science, engineering, applied mathematics and related topics and areas