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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

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.

Semilinear systems with a multi-valued nonlinear term

100%

Kim I.-S., Hong S.-J.

Open Mathematics

2017

tom 15

nr 1

628-644

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.

Ograniczanie wyników

2 Open Mathematics

2 Kim I.-S.

1 Bae J.-H.

1 Hong S.-J.

1 2017

1 2013

Wyniki wyszukiwania

Eigenvalue results for pseudomonotone perturbations of maximal monotone operators

Semilinear systems with a multi-valued nonlinear term