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Scattered homoclinics to a class of time-recurrent Hamiltonian systems

Gregory S. Spradlin (2007)

ESAIM: Control, Optimisation and Calculus of Variations

A second-order Hamiltonian system with time recurrence is studied. The recurrence condition is weaker than almost periodicity. The existence is proven of an infinite family of solutions homoclinic to zero whose support is spread out over the real line.

Semilinear perturbations of Hille-Yosida operators

Horst R. Thieme, Hauke Vosseler (2003)

Banach Center Publications

The semilinear Cauchy problem (1) u’(t) = Au(t) + G(u(t)), u ( 0 ) = x D ( A ) ¯ , with a Hille-Yosida operator A and a nonlinear operator G: D(A) → X is considered under the assumption that ||G(x) - G(y)|| ≤ ||B(x -y )|| ∀x,y ∈ D(A) with some linear B: D(A) → X, B ( λ - A ) - 1 x = λ 0 e - λ t V ( s ) x d s , where V is of suitable small strong variation on some interval [0,ε). We will prove the existence of a semiflow on [ 0 , ) × D ( A ) ¯ that provides Friedrichs solutions in L₁ for (1). If X is a Banach lattice, we replace the condition above by |G(x) - G(y)| ≤ Bv whenever...

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,...

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.

Smooth bifurcation for a Signorini problem on a rectangle

Jan Eisner, Milan Kučera, Lutz Recke (2012)

Mathematica Bohemica

We study a parameter depending semilinear elliptic PDE on a rectangle with Signorini boundary conditions on a part of one edge and mixed (zero Dirichlet and Neumann) boundary conditions on the rest of the boundary. We describe smooth branches of smooth nontrivial solutions bifurcating from the trivial solution branch in eigenvalues of the linearized problem. In particular, the contact sets of these nontrivial solutions are intervals which change smoothly along the branch. The main tools of the proof...

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