We establish five theorems giving lists of nonlinear contractive conditions which turn out to be mutually equivalent. We derive them from some general lemmas concerning subsets of the plane which may be applied both in the single- or set-valued case as well as for a family of mappings. A separation theorem for concave functions is proved as an auxiliary result. Also, we discuss briefly the following problems for several classes of contractions: stability of procedure of successive approximations,...

This paper is mainly concerned with existence of mild solutions and optimal controls for nonlinear delay integrodifferential systems with Caputo fractional derivative in infinite-dimensional spaces. We do not assume that the relevant strongly continuous semigroup is compact.

We consider a nonlinear fractional differential inclusion with nonlocal fractional integro-differential boundary conditions in a Banach space. The existence of at least one solution is proved by using the set-valued analog of Mönch fixed point theorem associated with the technique of measures of noncompactness.

For the nonlinear heat equation with a fractional Laplacian $u_t+{(-\Delta )}^{\alpha /2}u=u^2$, 1 < α ≤ 2, the first initial-boundary value problem in a disk is considered. Small initial conditions, homogeneous boundary conditions, and periodicity conditions in the angular coordinate are imposed. Existence and uniqueness of a global-in-time solution is proved, and the solution is constructed in the form of a series of eigenfunctions of the Laplace operator in the disk. First-order long-time asymptotics of the solution is obtained....

In this paper, we establish sufficient conditions for the existence of solutions for nonlinear Hadamard-type implicit fractional differential equations with finite delay. The proof of the main results is based on the measure of noncompactness and the Darbo’s and Mönch’s fixed point theorems. An example is included to show the applicability of our results.

This paper aims at discussing asymptotic behaviour of nonoscillatory solutions of the forced fractional difference equations of the form $$\begin{array}{cc}\hfill {\Delta }^{\gamma }u\left(\kappa \right)& +\Theta [\kappa +\gamma ,w(\kappa +\gamma \left)\right]\hfill \\ \hfill =& \Phi (\kappa +\gamma )+{\rm Y}(\kappa +\gamma )w^{\nu }(\kappa +\gamma )+\Psi [\kappa +\gamma ,w(\kappa +\gamma )],\kappa \in {\mathbb{N}}_{1-\gamma },\hfill \\ \hfill u_0=& c_0,\hfill \end{array}$$ where ${\mathbb{N}}_{1-\gamma }=\{1-\gamma ,2-\gamma ,3-\gamma ,\cdots \}$, $0<\gamma \le 1$, ${\Delta }^{\gamma }$ is a Caputo-like fractional difference operator. Three cases are investigated by using some salient features of discrete fractional calculus and mathematical inequalities. Examples are presented to illustrate the validity of the theoretical results.

In this work we introduce a nonparametric recursive aggregation process called Multilayer Aggregation (MLA). The name refers to the fact that at each step the results from the previous one are aggregated and thus, before the final result is derived, the initial values are subjected to several layers of aggregation. Most of the conventional aggregation operators, as for instance weighted mean, combine numerical values according to a vector of weights (parameters). Alternatively, the MLA operators...

We establish that for a fairly general class of topologically transitive dynamical systems, the set of non-transitive points is very small when the rate of transitivity is very high. The notion of smallness that we consider here is that of σ-porosity, and in particular we show that the set of non-transitive points is σ-porous for any subshift that is a factor of a transitive subshift of finite type, and for the tent map of [0,1]. The result extends to some finite-to-one factor systems. We also show...

We revisit the properties of Bessel-Riesz operators and present a different proof of the boundedness of these operators on generalized Morrey spaces. We also obtain an estimate for the norm of these operators on generalized Morrey spaces in terms of the norm of their kernels on an associated Morrey space. As a consequence of our results, we reprove the boundedness of fractional integral operators on generalized Morrey spaces, especially of exponent $1$, and obtain a new estimate for their norm.

For X ⊆ [0,1], let $D_X$ denote the collection of subsets of ℕ whose densities lie in X. Given the exact location of X in the Borel or difference hierarchy, we exhibit the exact location of $D_X$. For α ≥ 3, X is properly $D_{\xi }\left({\Pi }_{\alpha }^0\right)$ iff $D_X$ is properly $D_{\xi }\left({\Pi }_{1+\alpha }^0\right)$. We also show that for every nonempty set X ⊆[0,1], $D_X$ is ${\Pi }_3^0$-hard. For each nonempty ${\Pi }_2^0$ set X ⊆ [0,1], in particular for X = x, $D_X$ is ${\Pi }_3^0$-complete. For each n ≥ 2, the collection of real numbers that are normal or simply normal to base n is ${\Pi }_3^0$-complete. Moreover, $D_{\mathbb{Q}}$, the...

We study the relation between the Lusin-Menchoff property and the $F_{\sigma }$-“semiseparation” property of a fine topology in normal spaces. Three examples of normal topological spaces having the $F_{\sigma }$-“semiseparation” property without the Lusin-Menchoff property are given. A positive result is obtained in the countable compact space.

The paper discusses basics of calculus of backward fractional differences and sums. We state their definitions, basic properties and consider a special two-term linear fractional difference equation. We construct a family of functions to obtain its solution.