The Extension Of Three-Solution Therom And The Duality Of Generalized Orlicz Space

The thesis mainly consists of two parts as follow:Content of the first part is related to the steepest descent method in calculus of variations. calculus of variations is an math area dealt with the function of function, contrasting to the differential and integral calculus which deals with the function of number, and such function can be constructed by some unknown function's integrate and differential. The final aim of calculus of variations is to find the polar function which can enable the function to get the maximum or minimum. It can be explained by such example of the steepest descent curve along which a particle can arrive at point B from point A which is indirectly above of point B in possible short time under the effect of gravity. We must minimize the expression standing for the descent time among all of curves from A to B. Calculus of variations is to obtain the solution of nonlinear operator equations by means of the polar points of certain functions. This thesis extends the Three-solution theorem on nonlinear operator equations from Hilbert spaces to Banach spaces by using the pseudo gradient and the idea of steepest descent method, weakens some conditions of primary theorem, for example, it weakens the locally Lipschits conditions to the boundedness of gradient operator and so on.Content of the second part is about the norm of Orlicz spaces. Orlicz spaces are the extension of Lp spaces. Orlicz spaces theory is an important branch of Banach spaces theory. Its generating functions are very beautiful and have wide variation, which enables Orlicz spaces to supply abstract Banach spaces with plenty of examples and counter examples, and even supplies lots of methods and skills to solve nonlinear problems of Banach spaces theory. Besides of Banach space theory, Orlicz spaces theory has a good application in every walk of life, and it already has a wide applied foreground especially in approximation theory,control theory,fixed point theory,prediction operator theory and probability theory and so on.As we known, the dual spaces of Banach spaces play a very import role in researching the gradient of operators, so, in order to research nonlinear problems of generalized Orlicz space, we must discuss its duality.There are mainly two norms in Orlicz spaces: Orlicz norm and Luxemburg norm. These two norms were respectively introduced by Orlicz in 1932 and Luxemburg in 1955. In this paper we introduce two new norms respectively named as generalized Orlicz norm and generalized Luxemburg norm, and then we proves the equivalence between the new norms with Orlicz norm and Luxemburg norm. At last, we give another expression of generalized Orlicz norm.