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In this paper we consider the question of under which conditions multi-valued dynamical systems admit invariant measures. We give results on the existence of invariant measures with full support on orbit spaces of multi-valued dynamical systems. We use these measures on the orbit space to induce measures on the original dynamical system. We focus on the question of when a non-atomic invariant measure on the orbit space induces an atomic invariant measure on the multi-valued dynamical system. This phenomenon is an indicator of complicated multi-periodic behaviour.

For an ideal I on ω, we introduce the notions of I -invariant and bi- I -invariant injections from ω to ω. We study injections that are invariant with respect to selected classes of ideals. We show some applications to ideal convergence.

In this article we prove a classification theorem (Main Theorem) of real planar cubic vector fields which possess four distinct infinite singularities and eight invariant straight lines, including the line at infinity and including their multiplicities. This classification, which is taken modulo the action of the group of real affine transformations and time rescaling, is given in terms of invariant polynomials. The algebraic invariants and comitants allow one to verify for any given real cubic system with four infinite distinct singularities whether or not it has invariant lines of total multiplicity eight, and to specify its configuration of lines endowed with their corresponding real singularities of this system. The calculations can be implemented on computer.

In this note, we show that if T ∈ B ( H ) is such that p ( T ) satisfies the ( G 1 ) -condition for every polynomial p and if the outer boundary of the spectrum of T is a finite union of Jordan curves, then T has a nontrivial invariant subspace. In addition, we give a connection between operator-valued Poisson kernels and invariant subspaces.

The set of sampling in a shift invariant space plays an important role in signal processing and has many applications. This paper addresses the problem when some randomly chosen samples X = { x j : j ∈ J } form a set of sampling in a shift invariant space. That is, when the inequality of the form c p ‖ f ‖ L p ( R d ) p ≤ ∑ x j ∈ X | f ( x j ) | p ≤ C p ‖ f ‖ L p ( R d ) p holds uniformly for all functions f in a shift invariant space, where c p and C p are positive constants ( 1 ≤ p ≤ ∞ ) . We prove that with overwhelming probability, the above sampling inequality holds for certain compact subsets of the shift invariant space when the sampling size is sufficiently large.

Recently, the authors provided an example of an integrable Liouvillian planar polynomial differential system that has no finite invariant algebraic curves; see Gine and Llibre (2012) [8] . In this note, we prove that, if a complex differential equation of the form y ′ = a 0 ( x ) + a 1 ( x ) y + ⋯ + a n ( x ) y n , with a i ( x ) polynomials for i = 0 , 1 , … , n , a n ( x ) ≠ 0 , and n ≥ 2 , has a Liouvillian first integral, then it has a finite invariant algebraic curve. So, this result applies to Riccati and Abel polynomial differential equations. We shall prove that in general this result is not true when n = 1 , i.e., for linear polynomial differential equations.

We study the notion of linear invariance on the unit ball of a J B ⁎ -triple X , and we obtain some connection between the norm-order of a linear invariant family and the starlikeness of order 1/2. Also, we give some result concerning the radius of univalence of some linear invariant families. Finally, if the dimension of X is finite and if the norm-order of a linear invariant family is finite, then we prove the normality of the linear invariant family and we also obtain upper bounds on the distortion and the growth of mappings in a linear invariant family with specified norm-order. In particular, our results are valid for the classical Cartan domains and the unit polydisc.

The problem of invariant curves for continuous mean-type mappings is considered. The obtained results are applied in solving some functional equations.

We investigate shift invariant subspaces of L 2 ( G ) , where G is a locally compact abelian group. We show, among other things, that every shift invariant space can be decomposed as an orthogonal sum of spaces each of which is generated by a single function whose shifts form a Parseval frame.

An example proves that the Uniqueness theorem for non-locally finite invariant measures on a topological group does not hold, In this paper a uniqueness theorem for locally σ-finite invariant measures is given and it is proved that two invariant measures with the same sets of finite measure on a non-compact group are mutually absolutely continuous. This result is improved in the case of Hausdorff measures.

We show how, for nonlinear equations satisfying certain symmetry conditions, the determining equations reduce for solutions taken in some invariant subspace.

In the first section of this note we prove an approximate selection theorem for upper semicontinuous mappings defined on paracompact subsets of a Hausdorff topological vector space. In Section 2 we study continuity properties of setvalued metric projections. In the third section we combine the results obtained in the previous sections and establish some new fixed point theorems. In [36] we presented a set-valued version of Fan’s fixed point theorem for inward singlevalued mappings [ 171. We assumed there that the set-valued mapping in question was continuous. Now we are able to show that the same result is true for upper semicontinuous mappings. We also improve recent theorems due to Fitzpatrick and Petryshyn [19] and extend a theorem of Lim’s [27]. In the last section we consider invariance criteria that are similar to the inwardness conditions used in Section 3. In particular, we relate Nagumo’s subtangency condition [31] with Browder’s local support cones [6].

The existence of an infinite number of symmetric two-particle interactions is shown. These interactions are translation invariant, Lorentz invariant, and lead to second-order differential equations. Examples are analytic.

In this paper we introduce and study the notions of isotropic mapping and essential kernel. In addition some theorems on the Borel graph and Baire mapping for polynomial operators are proved. It is shown that a polynomial functional from an infinite dimensional complex linear space into the field of complex numbers vanishes on some infinite dimensional affine subspace.

We classify all centers of a planar weight-homogeneous polynomial vector field of weight degree 1, 2, 3 and 4.