Generalized Darbo-Type F -Contraction and F -Expansion and Its Applications to a Nonlinear Fractional-Order Differential Equation

In this work, we introduce various Darbo-type F £ -contractions, and utilizing these contractions, we present some fixed point theorems. Moreover, we introduce a Darbo-type F £ -expanding mapping and prove fixed point theorems under the Darbo-type F £ -expanding mapping. Employing our results, we check the existence of a solution to the nonlinear fractional-order differential equation under the integral type boundary conditions. For its validity, an appropriate example is given.

CONVERGENCE OF INVARIANT MEASURES FOR DEGENERATING SEQUENCES OF HYPERBOLIC MAPPINGS WITH SINGULARITIES

This paper is devoted to presenting and giving a sketch of the proof of the theorem which states that, if the sequence of hyperbolic mappings with singularities converges to degenerating piecewise expanding mapping, then the corresponding sequence of measures of a Sinai-Bowen-Ruelle type converges to an absolutely continuous invariant measure.

The Hausdorff dimension of certain solenoids

For the solid torus V = S1 × and a C1 embedding f: V → V given by with dϕ/dt > 1, 0 < λi(t) < 1 the attractor Λ = ∩i = 0∞fi(V) is a solenoid, and for each disk D(t) = {t} × (t ∈ S1) the intersection Λ(t) = Λ ∩ D(t) is a Cantor set. It is the aim of the paper to find conditions under which the Hausdorff dimension of Λ(t) is independent of t and determined by where the real numbers pi are characterized by the condition that the pressure of the function log : S1 → ℝ with respect to the expanding mapping ϕ: S1 → S1 becomes zero. (There are two further characterizations of these numbers.)It is proved that (0.1) holds provided λ1, λ2 are sufficiently small and Λ satisfies a condition called intrinsic transverseness. Then it is shown that in the C1 space of all embeddings f with sup λi > Θ−2 (Θ the mapping degree of ϕ: S1 → S1) all those f which have an intrinsically transverse attractor Λ form an open and dense subset.

Invariant subsets of expanding mappings of the circle

The continuity of Hausdorff dimension of closed invariant subsetsKof aC2-expanding mappinggof the circle is investigated. Ifg/Ksatisfies the specification property then the equilibrium states of Hölder continuous functions are studied. It is proved that iffis a piecewise monotone continuous mapping of a compact interval and φ a continuous function withP(f,φ)> sup(φ), then the pressureP(f,φ) is attained on one-dimensional ‘Smale's horseshoes’, and some results of Misiurewicz and Szlenk [M−Sz] are extended to the case of pressure.

On Hausdorff dimension of invariant sets for expanding maps of a circle

Given an orientation preserving C2 expanding mapping g: S1 → Sl of a circle we consider the family of closed invariant sets Kg(ε) defined as those points whose forward trajectory avoids the interval (0, ε). We prove that topological entropy of g|Kg(ε) is a Cantor function of ε. If we consider the map g(z) = zq then the Hausdorff dimension of the corresponding Cantor set around a parameter ε in the space of parameters is equal to the Hausdorff dimension of Kg(ε). In § 3 we establish some relationships between the mappings g|Kg(ε) and the theory of β-transformations, and in the last section we consider DE-bifurcations related to the sets Kg(ε).