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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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Fixed points, eigenvalues and surjectivity for (ws)-compact operators on unbounded convex sets

100%
Amar A.
Open Mathematics
|
2013
|
tom 11
|
nr 1
85-93
EN
The paper studies the existence of fixed points for some nonlinear (ws)-compact, weakly condensing and strictly quasibounded operators defined on an unbounded closed convex subset of a Banach space. Applications of the newly developed fixed point theorems are also discussed for proving the existence of positive eigenvalues and surjectivity of quasibounded operators in similar situations. The main condition in our results is formulated in terms of axiomatic measures of weak noncompactness.
3
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Some global results for nonlinear fourth order eigenvalue problems

68%
Aliyev Z.
Open Mathematics
|
2014
|
tom 12
|
nr 12
1811-1828
EN
In this paper, we consider the nonlinear fourth order eigenvalue problem. We show the existence of family of unbounded continua of nontrivial solutions bifurcating from the line of trivial solutions. These global continua have properties similar to those found in Rabinowitz and Berestycki well-known global bifurcation theorems.
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