Contributions on the degree theory for perturbation of maximal monotone maps Let X be a real reflexive separable locally uniformly convex Banach space with locally uniformly convex dual space X^*. 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(\|x\|), 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 convex 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+\epsilon J)^{-1}:X^*\rightarrow X is compact for every $\epsilon >0. We present a relevant result in chapter 2 that says there exists an open ball around zero in the image of a relatively open set by a continuous and bounded perturbation of a maximal monotone operator with compact resolvents. The generalized degree function for compact perturbations of m-accretive operators established by Y.-Z Chen in [7] is extended to the case of a multivalued compact perturbations of maximal monotone maps by appealing to the topological degree for set-valued compact fields in locally convex spaces introduced by Tsoy Wo-Ma in [25]. Such is the content of the third chapter. A unified eigenvalue theory is developed for the pair(T,S), where T:X\supset D(T)\rightarrow 2^{X^*} is a quasimonotone-type operator which belong to the so-called A_G(QM) class introduced by Arto Kittila in [23] and S is a bounded demicontinuous mapping of class (S)_+. Conditions are given for the existence of a pair (x,\lambda)\in (0,\infty)\times (D(T+S)\cap\partial G)$ such that Tx+\lambda Sx\ni 0$. This is the content of Chapter 4. English
Contributions on the degree theory for perturbation of maximal monotone maps
Let X be a real reflexive separable locally uniformly convex Banach space with locally uniformly convex dual space X^*. 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(\|x\|), 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 convex 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+\epsilon J)^{-1}:X^*\rightarrow X is compact for every $\epsilon >0. We present a relevant result in chapter 2 that says there exists an open ball around zero in the image of a relatively open set by a continuous and bounded perturbation of a maximal monotone operator with compact resolvents. The generalized degree function for compact perturbations of m-accretive operators established by Y.-Z Chen in [7] is extended to the case of a multivalued compact perturbations of maximal monotone maps by appealing to the topological degree for set-valued compact fields in locally convex spaces introduced by Tsoy Wo-Ma in [25]. Such is the content of the third chapter. A unified eigenvalue theory is developed for the pair(T,S), where T:X\supset D(T)\rightarrow 2^{X^*} is a quasimonotone-type operator which belong to the so-called A_G(QM) class introduced by Arto Kittila in [23] and S is a bounded demicontinuous mapping of class (S)_+. Conditions are given for the existence of a pair (x,\lambda)\in (0,\infty)\times (D(T+S)\cap\partial G)$ such that Tx+\lambda Sx\ni 0$. This is the content of Chapter 4.
English