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.

In this work, we demonstrate that the Leray-Schauder topological degree theory can be used for the development of a topological degree theory for maximal monotone perturbations of demicontinuous operators of type (S+) in separable reflexive Banach spaces. This is an extension of Berkovits’ degree development for operators as the perturbations above. Berkovits has developed a topological degree for demicontinuous mappings of type (S+), and has shown that the degree mapping is unique under the assumption that it satisfies certain general properties. He proved that if f is a bounded demicontinous mapping of type (S+), G is an open bounded subset of X, and 0 ∈/ f(∂G), then there exists ε0 > 0 such that for every ε ∈ (0, ε0) we have 0 ∈/ (I+ (1/ε)QQ∗ (f))(∂G). Here, Q is a compact linear injection from a Hilbert space H into X, such that Q(H) is dense in X, and Q∗ its adjoint. The map I+ 1 εQQ∗ (f) is a compact displacement of the identity, for which the Leray-Schauder degree is well defined. The Berkovits degree is obtained as the limit of this Leray-Schauder degree as ε tends to zero. We utilize a demicontinuous (S+)-approximation of the form Tt + f, where Tt is the Yosida approximant of T. Namely, we show that if G is an open bounded set in X and 0 ∈/ (T + f)(∂G), then there exist ε0 > 0, t0 > 0, such that for every ε ∈ (0, ε0), t ∈ (0, t0), we have 0 ∈/ (I + (1/ε)QQ∗ (Tt + f))(∂G). Our degree is the limit of the Leray-Schauder degree of the compact displacement of the identity I + (1/ε)QQ∗ (Tt + f) as ε, t → 0. Various extension of the degree has been considered. Finally some properties and applications in invariance of domain, eigenvalue and surjectivity results have also been discussed.

Let X be a real reflexive locally uniformly convex Banach space with locally uniformly convex dual space X* . Let G be a bounded open subset of X. Let T:X⊃ D(T)⇒ 2X* be maximal monotone and S: X ⇒ 2X* be bounded pseudomonotone and such that 0 notin cl((T+S)(D(T)∩partG)). Chapter 1 gives general introduction and mathematical prerequisites. In Chapter 2 we develop a homotopy invariance and uniqueness results for the degree theory constructed by Zhang and Chen for multivalued (S+) perturbations of maximal monotone operators. Chapter 3 is devoted to the construction of a new topological degree theory for the sum T+S with the degree mapping d(T+S,G,0) defined by d(T+S,G,0)=limepsilondarr 0+ dS+(T+S+ J,G,0), where dS+ is the degree for bounded (S+)-perturbations of maximal monotone operators. The uniqueness and homotopy invariance result of this degree mapping are also included herein. As applications of the theory, we give associated mapping theorems as well as degree theoretic proofs of known results by Figueiredo, Kenmochi and Le. In chapter 4, we consider T:X D(T)⇒ 2X* to be maximal monotone and S:D(S)=K⇒ 2X* at least pseudomonotone, where K is a nonempty, closed and convex subset of X with 0isinKordm. Let Phi:X⇒ ( infin, infin] be a proper, convex and lower-semicontinuous function. Let f* isin X* be fixed. New results are given concerning the solvability of perturbed variational inequalities for operators of the type T+S associated with the function f. The associated range results for nonlinear operators are also given, as well as extensions and/or improvements of known results by Kenmochi, Le, Browder, Browder and Hess, Figueiredo, Zhou, and others.