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Eigenvalue results for pseudomonotone perturbations of maximal monotone operators

100%
Kim I.-S., Bae J.-H.
Open Mathematics
|
2013
|
tom 11
|
nr 5
851-864
EN
Let X be an infinite-dimensional real reflexive Banach space such that X and its dual X* are locally uniformly convex. Suppose that T: X⊃D(T) → 2X* is a maximal monotone multi-valued operator and C: X⊃D(C) → X* is a generalized pseudomonotone quasibounded operator with L ⊂ D(C), where L is a dense subspace of X. Applying a recent degree theory of Kartsatos and Skrypnik, we establish the existence of an eigensolution to the nonlinear inclusion 0 ∈ T x + λ C x, with a regularization method by means of the duality operator. Moreover, possible branches of eigensolutions to the above inclusion are discussed. Furthermore, we give a surjectivity result about the operator λT + C when λ is not an eigenvalue for the pair (T, C), T being single-valued and densely defined.
2
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Equivariant degree of convex-valued maps applied to set-valued BVP

100%
Dzedzej Z.
Open Mathematics
|
2012
|
tom 10
|
nr 6
2173-2186
EN
An equivariant degree is defined for equivariant completely continuous multivalued vector fields with compact convex values. Then it is applied to obtain a result on existence of solutions to a second order BVP for differential inclusions carrying some symmetries.
3
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Equivariant Morse equation

100%
Styborski M.
Open Mathematics
|
2012
|
tom 10
|
nr 6
2138-2159
EN
The paper is concerned with the Morse equation for flows in a representation of a compact Lie group. As a consequence of this equation we give a relationship between the equivariant Conley index of an isolated invariant set of the flow given by .x = −∇f(x) and the gradient equivariant degree of ∇f. Some multiplicity results are also presented.
4
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Semilinear systems with a multi-valued nonlinear term

68%
Kim I.-S., Hong S.-J.
Open Mathematics
|
2017
|
tom 15
|
nr 1
628-644
EN
Introducing a topological degree theory, we first establish some existence results for the inclusion h ∈ Lu − Nu in the nonresonance and resonance cases, where L is a closed densely defined linear operator on a Hilbert space with a compact resolvent and N is a nonlinear multi-valued operator of monotone type. Using the nonresonance result, we next show that abstract semilinear system has a solution under certain conditions on N = (N1, N2), provided that L = (L1, L2) satisfies dim Ker L1 = ∞ and dim Ker L2 < ∞. As an application, periodic Dirichlet problems for the system involving the wave operator and a discontinuous nonlinear term are discussed.
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