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Let X be an infinite dimensional real reflexive Banach space with dual space X∗ and G⊂X open and bounded. Let T:X⊃D(T)→2X∗ be a maximal monotone operator with 0∈D(T) and 0∈T(0), and let C:X⊃D(C)→2X∗ be densely defined strongly quasibounded and of type (S~+). A new topological degree theory is introduced for the sum T+C with a degree mapping d(T+C,G,0) defined eventually in terms of the Ma degree for multivalued compact operators. Unlike single-valued operators considered by Kartsatos and Skrypnik, the operator C here is multivalued so that the multivalued generalized pseudomonotone operators considered by Browder and Hess include such C and even T+C. Consequently, the main existence results of Browder and Hess are obtained via the new degree theory and some of their existence results are extended. An application of the theory to elliptic partial differential inclusions in divergence form is included.

Let X be a real reflexive Banach space with X ⁎ its dual space and G be a nonempty and open subset of X. Let A : X ⊇ D ( A ) → 2 X ⁎ be a strongly quasibounded maximal monotone operator and T : X ⊇ D ( T ) → 2 X ⁎ be an operator of class A G ( S + ) introduced by Kittila. We develop a topological degree theory for the operator A + T . The theory generalizes the Browder degree theory for operators of type ( S + ) and extends the Kittila degree theory for operators of class A G ( S + ) . New existence results are established. The existence results give generalizations of similar known results for operators of type ( S + ) . Applications to strongly nonlinear problems are included.

Preface: Thematic Issue in Mathematical Biology and Applied Evolutionary Equations

Let be a real reflexive Banach space with dual . Let be densely defined linear maximal monotone. Let be maximal monotone with and and bounded, demicontinuous and of type w.r.t. . An invariance of domain result is established for the sum . An eigenvalue problem of the type is also solved, where is now maximal monotone and strongly quasibounded with and is like above. The recent topological degree theory of the authors is used, utilizing the graph norm topology on along with the methodology of Berkovits and Mustonen and recent invariance of domain and eigenvalue results by Kartsatos and Skrypnik. The results are original even in the case Possible applications to time-dependent problems are also included.

Let X be an infinite dimensional real reflexive Banach space with dual space X ∗ and G ⊂ X , open and bounded. Assume that X and X ∗ are locally uniformly convex. Let T : X ⊃ D ( T ) → 2 X ∗ be maximal monotone and strongly quasibounded, S : X ⊃ D ( S ) → X ∗ maximal monotone, and C : X ⊃ D ( C ) → X ∗ strongly quasibounded w.r.t. S and such that it satisfies a generalized ( S + ) -condition w.r.t. S . Assume that D ( S ) = L ⊂ D ( T ) ∩ D ( C ) , where L is a dense subspace of X , and 0 ∈ T ( 0 ) , S ( 0 ) = 0 . A new topological degree theory is introduced for the sum T + S + C , with degree mapping d ( T + S + C , G , 0 ) . The reason for this development is the creation of a useful tool for the study of a class of time-dependent problems involving three operators. This degree theory is based on a degree theory that was recently developed by Kartsatos and Skrypnik just for the single-valued sum S + C , as above.

Let X be a real reflexive Banach space with dual X ∗ . Let L : X ⊃ D ( L ) → X ∗ be densely defined, linear and maximal monotone. Let T : X ⊃ D ( T ) → 2 X ∗ , with 0 ∈ D ( T ) and 0 ∈ T ( 0 ) , be strongly quasibounded and maximal monotone, and C : X ⊃ D ( C ) → X ∗ bounded, demicontinuous and of type ( S + ) w.r.t. D ( L ) . A new topological degree theory has been developed for the sum L + T + C . This degree theory is an extension of the Berkovits–Mustonen theory (for T = 0 ) and an improvement of the work of Addou and Mermri (for T : X → 2 X ∗ bounded). Unbounded maximal monotone operators with 0 ∈ D ˚ ( T ) are strongly quasibounded and may be used with the new degree theory.

Let X be a real Banach space and G 1 , G 2 two nonempty, open and bounded subsets of X such that 0 ∈ G 2 and G 2 ¯ ⊂ G 1 . The problem ( ∗ ) T x + C x = 0 is considered, where T : X ⊃ D ( T ) → X (or X ∗ ) is an accretive (or monotone) operator with 0 ∈ D ( T ) and T ( 0 ) = 0 , while C : X ⊃ D ( C ) → X (or X ∗ ) can be, e.g. one of the following types: (a) compact; (b) continuous and bounded with the resolvents of T compact; (c) demicontinuous, bounded and of type ( S + ) with T positively homogeneous of degree one; (d) quasi-bounded and satisfying a generalized ( S + ) -condition w.r.t. the operator T , while T is positively homogeneous of degree one. Solutions are sought for the problem (∗) lying in the set D ( T + C ) ∩ ( G 1 ∖ G 2 ) . These solutions are nontrivial even when C ( 0 ) = 0 . The degree theories of Leray and Schauder, Browder, and Skrypnik are used, as well as the degree theory by Kartsatos and Skrypnik for densely defined operators T , C . The last three degree theories do not assume any compactness conditions. The excision and additivity properties of these degree theories are employed, and the main results are significant extensions or generalizations of previous results by Krasnoselskii, Guo, Ding and Kartsatos, and other authors, involving the relaxation of compactness conditions and/or conditions on the boundedness of the operator T . An application in the field of partial differential equations is also included.