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Semigroups and generators on convex domains with the hyperbolic metric

Simeon Reich, David Shoikhet (1997)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Let D be domain in a complex Banach space X , and let ρ be a pseudometric assigned to D by a Schwarz-Pick system. In the first section of the paper we establish several criteria for a mapping f : D → X to be a generator of a ρ -nonexpansive semigroup on D in terms of its nonlinear resolvent. In the second section we let X = H be a complex Hilbert space, D = B the open unit ball of H , and ρ the hyperbolic metric on B . We introduce the notion of a ρ -monotone mapping and obtain simple characterizations of generators...

Separable solutions of quasilinear Lane–Emden equations

Alessio Porretta, Laurent Véron (2013)

Journal of the European Mathematical Society

For 0 < p - 1 < q and either ϵ = 1 or ϵ = - 1 , we prove the existence of solutions of - Δ p u = ϵ u q in a cone C S , with vertex 0 and opening S , vanishing on ∂ C S , of the form u ( x ) = x - β ω ( x / x ) . The problem reduces to a quasilinear elliptic equation on S and the existence proof is based upon degree theory and homotopy methods. We also obtain a nonexistence result in some critical case by making use of an integral type identity.

Sets in the ranges of nonlinear accretive operators in Banach spaces

Athanassios Kartsatos (1995)

Studia Mathematica

Let X be a real Banach space and G ⊂ X open and bounded. Assume that one of the following conditions is satisfied: (i) X* is uniformly convex and T:Ḡ→ X is demicontinuous and accretive; (ii) T:Ḡ→ X is continuous and accretive; (iii) T:X ⊃ D(T)→ X is m-accretive and Ḡ ⊂ D(T). Assume, further, that M ⊂ X is pathwise connected and such that M ∩ TG ≠ ∅ and M ∩ T ( ∂ G ) ¯ = ∅ . Then M ⊂ T G ¯ . If, moreover, Case (i) or (ii) holds and T is of type ( S 1 ) , or Case (iii) holds and T is of type ( S 2 ) , then M ⊂ TG. Various results of Morales,...

Singlevaluedness of monotone operators on subspaces of GSG spaces

Martin Heisler (1996)

Commentationes Mathematicae Universitatis Carolinae

We extend Zajíček’s theorem from [Za] about points of singlevaluedness of monotone operators on Asplund spaces. Namely we prove that every monotone operator on a subspace of a Banach space containing densely a continuous image of an Asplund space (these spaces are called GSG spaces) is singlevalued on the whole space except a σ -cone supported set.

Singular φ -Laplacian third-order BVPs with derivative dependance

Smaïl Djebali, Ouiza Saifi (2016)

Archivum Mathematicum

This work is devoted to the existence of solutions for a class of singular third-order boundary value problem associated with a φ -Laplacian operator and posed on the positive half-line; the nonlinearity also depends on the first derivative. The upper and lower solution method combined with the fixed point theory guarantee the existence of positive solutions when the nonlinearity is monotonic with respect to its arguments and may have a space singularity; however no Nagumo type condition is assumed....

Small functions and iterative methods

Michal Fečkan (1992)

Commentationes Mathematicae Universitatis Carolinae

Iterative methods based on small functions are used both to show local surjectivity of certain operators and a fixed point property of mappings on scales of complete metric spaces.

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