Let x be a real reflexive separable locally uniformly convex banach space with locally uniformly convex dual spacex *. Let t:x\supset d(t)\rightarrow 2 {x *} be maximal monotone with 0\in t(0), 0\in intd(t) and c:x\supset d(c)\rightarrow x *. Assume that $l\subset d(c)$ is a dense linear subspace of x, c is of class (s_+)_l and \langle cx,x\rangle\geq-\psi(\lx\l), x\in d(c), where \psi:\mathbb{r} +\rightarrow\mathbb{r} + is nondecreasing. a new topological degree is developed for the sum t+c in chapter one. This theory extends the recent degree theory for the operators c of type (s_+)_{0,l} in [15]. unlike such a recent extension to multivalued (s_+)_{0,l}-type operators, the current approach utilizes the approximate degree d(t_t+c,g,0), t\downarrow 0, where t_t = (T {-1}+tJ {-1}) {-1}and G is an open bounded subset of X and is such that $0\in G$, for the single-valued mapping $T_t+C$. The subdifferential\partial\varphi, for \varphi belonging to a large class of proper c onvex lower semicontinuous functions, gives rise to operators T to which this degree theory applies. Theoretical applications to problems of Nonlinear Analysis are included, as well as applications from the field of partial differential equations. Let T:X\supset D(T)\rightarrow 2 {X *} be maximal monotone with compact resolvents, i.e, the operator $(T+\epsilonJ) {-1}:X *\rightarrow X is compact for every \in 0. We present a relevant result in chapter 2 that says there exists an open ball around zero in the image of a relatively open set by a continuous and bounded perturbation of a maximal monotone operator with compact resolvents. The generalized degree function for compact perturbations of m-accretive operators established by Y. -Z Chen in [7] isextended to the case of a multivalued compact perturbations of maximal monotone maps by appealing to the topological degree forset-valued compact fields in locally convex spaces introduced by Tsoy Wo-Ma in [25]. Such is the content of the thi rd chapter. A unified eigen value theory is developed for the pair(T,S), where T:X\supset D(T)\rightarrow 2 {X *} is aquasimonotone-type operator which belong to the so-called A_G(QM) class introduced by Arto Kittila in [23] and S is abounded demicontinuous mapping of class (S)_+. Conditions are given for the existence of a pair (x,\lambda)\in (0,\infty)\times(D(T+S)\cap\partial G)$ such that Tx+\lambda Sx\ni 0$. This is the content of Chapter 4.