Boundary Value Problems For Second Order Differential Inclusions

As a intersection of second order differential equation theory and set-valued analysis, the new subject second order differential inclusions has wide application in mechanics, engineering, optimization and control theory. Here is an illustrative example.Consider a particle m moving under the rule of an external force f.By Newton's second law of motion, we know that the displacement x(t) of m conforms the following equationwhere the force f imposed on m varies about the time t,the displacement x and the velocity v = x'.If we place the particle m into a central (gravitational or electro-magnetic) field, then the force f can be represented by the gradient of a convex potential, namely,Sometimes, the external force can be decomposed into two factors, one comes from a central field, which changes smoothly, while the other one is a perturbation, denoted by F, which comes from many directions and sources, and changed discontinuously and uncertainly. To describe the motion of m precisely in this case, we can only appeal to the differential inclusionIf the perturbation is controllable with the form F(t, x(t),x'(t),u(t)), where the data u(t) can be controlled by a set-valued function U(t,x(t),x'(t)),then we get a feed back control system In the last three decades, the new subject has been received an increasing interest. A lot of scholars, such as V. Barbu (Romania), N. H. Pavel (U.S.A.), N. C. Apreutesei (Romania), N. S. Papageorgiou (Greece), S. Hu (P.R.C.), F. Papalini (Italy), D. Motreanu (France) and M. Frigon (Canada) etc studied the second order differential inclusions with various boundary conditionswhere A is a maximal monotone map on R^N(especially, it is equal to the gradient or subdifferential operator of a convex function), F is a multivalued perturbation satisfying some conditions, and BC denotes a boundary condition having the following forms:In dealing with the boundary value problems, there is an important task, that is to seek for a solution lying in a given set C, which contains the following forms:To complete this task, we always use the method of truncations and penalization and obtain an auxiliary problem, which solutions can be proved existing by means of fixed point theory and approximation. And from an priori estimate, we can find that all the solutions lie in the given set C,and then solve the original problem automatically. Recently, many authors paid their attentions to the boundary value problems driven by the p-Laplacian (or p-Laplacian-like) operators. Based on the works of J. Mawhin (U.S.A), D. Man(?)sevich (Chile) and M. Zhang (P.R.C), a series of papers with high quality appeared. In these works, authors' discussions were concentrated on the Fucik spectrum, the resonance and unresonance conditions, and the constant sign and nodal solutions, etc.This thesis is devoted to study the boundary value problems for second order inclusions, which contents can be divided into two parts.In the first part, we investigate the multivalued boundary problems with monotone terms. Our approach relies on the theory of fixed points of set-valued maps and Yosida approximation of maxiaml monotone operators. This part contains three chapters.In Chapter 1,we study the following modelwhere a :R^N→R^N is a classical p-Laplacian-like operator, and the multivalued perturbation satisfies the generalized Hartman's condition (see Chapter 1H(F)_iv)In Chapter 2, we consider the semilinear problemwith a pair of solution tube existing.(see Section 1, Chapter 2)And Chapter 3 is devoted to the scalar nonlinear problemcoupled with a pair of upper-lower solutions.To solve the above three problems, we employ truncated functions (see (1.2.1), (1.2.2), (2.1.6), (2.1.7), (3.2.1), (3.2.2)) and get a transformationτdefined on a function space. By a careful analysis on r, we find the relations between the differentials of x and (τx)' (see (1.2.5), (1.4.7), (2.1.11), (2.3.11), (3.3.8)), which make us perform dual products (or inner products) on the approximate equations and get the boundedness of the approximate solutions. Based on the reformed method, we improve and extend some results of those in the works of Papalini, Papageorgiou, Mawhin and Frigon etc.The second part also has three chapters. In Chapter 4, we continue to discuss the multivalued boundary problemwhere g(t,·) is a family of lower semicotinuous and convex proper functions, and the boundary conditions arc described by the subdifferentials of two convex functions respectively. This model has not been studied ever before, since the domain Dg(t,·) varies about t,and thus the approximate equations could not be used any longer. To dear with the model, our discussion is concentrated on the family g(t,·).Under some reasonable conditions, we prove that the functional (?)derived from g(t,·)is also lsc,convex and proper, which subdifferential is equal to the Nemytskij operator of (?)g(t,·).Thus, using fixed point theory again, a class of problems have been proved correspondingly. As a by-product, an abstract result about the embedding of a maximal monotone map from a Banach space into one of its dense subspaces has been got simultaneously (see theorem 4.15, Chapter 4). This result has important value in theory and application.In the last two chapters, we turn to investigate the periodic differential systems.By introducing a suitable functional defined on some function space, the task to search for a solution for the boundary problem turns to be one to seek for a critical point of the functional.There are two models we deal with, both of them are concerning periodic solutions.In Chapter 5, there is model 1, For this model, we introduce the functionalΦ=φ+ψ,whereφis a convex function derived from g(t,·) andψis a locally Lipschitz one associated with f(t.·).Using the generalized nonsmooth PS-condition and the least action principle, we prove the existence of solutions of the equationwhich are also called the critical points ofΦ.This model has wide application in boundary problems, and its abstract form can be viewed as a useful supplement of critical point theory (see Theorem 5.10, Chapter 5).In the last chapter, we discuss model 2,Our discussion focusses on the weak AR condition (see H(f)(iv)). By means of nonsmooth C-condition and the Mountain Pass Lemma of nonsmooth type, we prove the existence of positive solutions of the model. This is the first time to investigate the periodic problem using the weak A.R. condition.At the end of Chapter 6, the multiplicity of solutions and the homoclinic solutions are all taken into account.To sum up, in this thesis, we study several types of second order differential inclusions with boundary conditions. Employing the tools of set-valued analysis, convex and nonsmooth analysis, and critical point theory, we discuss the existence and multiplicity of solutions for these models. This work improves and extends some results in the works followed by us, discussed two new models (see Chapter 4 and Chapter 5) and provides a useful method for other authors in this field.