Topological degree theories and nonlinear operator equations in Banach spaces

Adhikari, Dhruba R.; Kartsatos, Athanassios G.

doi:10.1016/j.na.2007.06.026

Cited by 15 publications

(14 citation statements)

References 13 publications

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“…However, if A is an S-contractive map, i.e., I − A is of S + -type which have been widely studied, for example by Skrypnik [39,40], Browder [6], Adhikari and Kartsatos [2], then r A may be neither a compact map nor a condensing map and if A is demicontinuous, then r A need not be continuous. Therefore, these fixed point index theories mentioned above cannot be used to treat the variational inequality (1.1) when A is a demicontinuous S-contractive map.…”

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confidence: 99%

“…In addition, it seems to be difficult to employ the degree theories for maps of S + -type [6,39,40] to derive an index theory for (1.1) due to the fact that I − Ar may not be of S + -type although the classic fixed point index theory for compact maps was developed by using the degree theory of I − Ar [3]. We refer to [2] for the study of the existence of nonzero solutions for demicontinuous maps of S + -type, where the degree theories of Skrypnik [39,40] and Browder [6] are employed.…”

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confidence: 99%

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A variational inequality theory for demicontinuous S-contractive maps with applications to semilinear elliptic inequalities

Lan

2009

Journal of Differential Equations

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A variational inequality theory for demicontinuous S-contractive maps in Hilbert spaces is established by employing the ideas of Granas' topological transversality. Such a variational inequality theory has many properties similar to those of fixed point theory for demicontinuous weakly inward S-contractive maps and to those of fixed point index for condensing maps. The variational inequality theory will be applied to study the existence of positive weak solutions and eigenvalue problems for semilinear secondorder elliptic inequalities with nonlinearities which satisfy suitable lower bound conditions involving the critical Sobolev exponent. There has been little discussion for such elliptic inequalities involving the critical Sobolev exponent in the literature.

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confidence: 99%

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A variational inequality theory for demicontinuous S-contractive maps with applications to semilinear elliptic inequalities

Lan

2009

Journal of Differential Equations

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“…A is said to be S-contractive on D if I − A is of S + -type, that is, if {y n } ⊂ D with y n y ∈ H and lim sup( y n − Ay n , y n − y) 0 together imply y n → y (see [7,26,27]). We refer to [2,7,40] for the degree theories for maps of S + -type and applications. A is said to be compact if A is continuous and…”

Section: Principles Of Variational Inequalitiesmentioning

confidence: 99%

Positive weak solutions of semilinear second order elliptic inequalities via variational inequalities

Lan¹

2011

Journal of Mathematical Analysis and Applications

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Keywords:Variational inequality Demicontinuous pseudo-contractive map Convergence of approximants Semilinear elliptic inequalities Critical Sobolev exponent Existence, uniqueness and convergence of approximants of positive weak solutions for semilinear second order elliptic inequalities are obtained. The nonlinearities involved in these inequalities satisfy suitable upper or lower bound conditions or monotonicity conditions. The lower bound conditions are allowed to contain the critical Sobolev exponents. The methodology is to establish variational inequality principles for demicontinuous pseudo-contractive maps in Hilbert spaces by considering convergence of approximants and apply them to the corresponding variational inequalities arising from the semilinear second order elliptic inequalities. Examples on the existence, uniqueness and convergence of approximants of positive weak solutions of the semilinear second order elliptic inequalities are given.

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“…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, pp.…”

Section: Introduction Preliminariesmentioning

confidence: 99%

“…In Boubakari and Kartsatos [6] one may find the development of a degree theory, in a separable space X , which extends the degree theory of Berkovits in [4] for demicontinuous bounded (S + )-mappings f to operators T + f with T strongly quasibounded and maximal monotone. Applications of degree theories to the existence of zeros of nonlinear operator equations may be found in the authors' paper [2].…”

Section: Introduction Preliminariesmentioning

confidence: 99%

Strongly quasibounded maximal monotone perturbations for the Berkovits–Mustonen topological degree theory

Adhikari

Kartsatos

2008

Journal of Mathematical Analysis and Applications

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Keywords:Berkovits-Mustonen degree theory Browder degree theory Maximal monotone operator Bounded demicontinuous operator of type (S + ) 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.

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Topological degree theories and nonlinear operator equations in Banach spaces

Cited by 15 publications

References 13 publications

A variational inequality theory for demicontinuous S-contractive maps with applications to semilinear elliptic inequalities

A variational inequality theory for demicontinuous S-contractive maps with applications to semilinear elliptic inequalities

Positive weak solutions of semilinear second order elliptic inequalities via variational inequalities

Strongly quasibounded maximal monotone perturbations for the Berkovits–Mustonen topological degree theory

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