On the eigenvalue problem for perturbed nonlinear maximal monotone operators in reflexive Banach spaces

Athanassios G. Kartsatos, Igor V. Skrypnik
2005 Transactions of the American Mathematical Society  
Let X be a real reflexive Banach space with dual X * and G ⊂ X open and bounded and such that 0 ∈ G. Let T : X ⊃ D(T ) → 2 X * be maximal monotone with 0 ∈ D(T ) and 0 ∈ T (0), and C : X ⊃ D(C) → X * with 0 ∈ D(C) and C(0) = 0. A general and more unified eigenvalue theory is developed for the pair of operators (T, C). Further conditions are given for the existence of a pair (λ, The "implicit" eigenvalue problem, with C(λ, x) in place of λCx, is also considered. The existence of continuous
more » ... of continuous branches of eigenvectors of infinite length is investigated, and a Fredholm alternative in the spirit of Necas is given for a pair of homogeneous operators T, C. No compactness assumptions have been made in most of the results. The degree theories of Browder and Skrypnik are used, as well as the degree theories of the authors involving densely defined perturbations of maximal monotone operators. Applications to nonlinear partial differential equations are included. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ATHANASSIOS G. KARTSATOS AND IGOR V. SKRYPNIK T is maximal if and only if R(T + λJ) = X * for all λ ∈ (0, ∞). If T is maximal monotone, then the operator T t ≡ (T −1 + tJ −1 ) −1 : X → X * is bounded, demicontinuous, maximal monotone and such that T t x T {0} x as t → 0 + for every x ∈ D(T ), where T {0} x denotes the element y * ∈ T x of minimum norm, i.e. T {0} x = inf{ y * : y * ∈ T x}. In our setting, this infimum is always attained and D(T {0} ) = D(T ). Also, T t x ∈ T J t x, where J t ≡ I − tJ −1 T t : X → X and satisfies lim t→0 J t x = x for all x ∈ coD(T ), where coA denotes the convex hull of the set A. In addition, x ∈ D(T ) and t 0 > 0 imply lim t→t 0 T t x = T t 0 x. The operators T t , J t were introduced by Brézis, Crandall and Pazy in [2] . For their basic properties, we refer the reader to [2] as well as Pascali and Sburlan [18,. In our setting, the duality mapping J is single-valued and bicontinuous. An operator T : X ⊃ D(T ) → Y, with Y another real Banach space, is "bounded" if it maps bounded subsets of D(T ) onto bounded sets. It is "compact" if it is continuous and maps bounded subsets of D(T ) onto relatively compact subsets of Y. It is "demicontinuous" ("completely continuous") if it is strong-weak (weak-strong) continuous on D(T ). Given an operator T : X ⊃ D(T ) → 2 X * , we say that T has the property P "locally" on G ⊂ X if for every x 0 ∈ D(T )∩G there exists a closed ball B r (x 0 ) ⊂ G such that T has the property P on D(T ) ∩ B r (x 0 ). If G = X, then we simply say that T has "locally" the property P. We say that an operator T
doi:10.1090/s0002-9947-05-03761-x fatcat:dcckal2wgndxhegug6l3mzltem