Related papers: A General Version of Carath\'{e}odory's Existence …
In the paper, we improve our earlier results concerning the existence, uniqueness and differentiability of a global implicit function. Some application to a Cauchy problem for an integro-differential Volterra system of nonconvolution type,…
We prove, using a fixed point theorem in a Banach algebra, an existence result for a fractional functional differential equation in the Riemann-Liouville sense. Dependence of solutions with respect to initial data and an uniqueness result…
The Carath\'eodory theorem on the construction of a measure is generalized by replacing the outer measure with an approximation of it and generalizing the Carath\'eodory measurability. The new theorem is applied to obtain dynamically…
We study some already introduced and some new strong and weak topologies of integral type to provide continuous dependence on continuous initial data for the solutions of non-autonomous Carath\'eodory delay differential equations. As a…
Given $l>2\nu>2d\geq 4$, we prove the persistence of a Cantor--family of KAM tori of measure $O(\varepsilon^{1/2-\nu/l})$ for any non--degenerate nearly integrable Hamiltonian system of class $C^l(\mathscr D\times\mathbb{T}^d)$, where…
This paper is in concern with Cauchy problems involving the fractional derivatives with respect to another function. Results of existence, uniqueness, and Taylor series among others are established in appropriate functional spaces. We prove…
In this article, we extend the foundations of the theory of differential inclusions in the space of compactly supported probability measures, introduced recently by the authors, to the setting of general Wasserstein spaces. In this context,…
We extend in this paper the definition of Caputo derivatives of order in $(0,1)$ to a certain class of locally integrable functions using a convolution group. Our strategy is to define a fractional calculus for a certain class of…
A variational principle for Lagrangian densities containing derivatives of real order is formulated and the invariance of this principle is studied in two characteristic cases. Necessary and sufficient conditions for an infinitesimal…
We generalize the classical mean value theorem of differential calculus by allowing the use of a Caputo-type fractional derivative instead of the commonly used first-order derivative. Similarly, we generalize the classical mean value…
In this paper, we show that higher-order optimality conditions can be obtain for arbitrary nonsmooth function. We introduce a new higher-order directional derivative and higher-order subdifferential of Hadamard type of a given proper…
By means of two fractional order integral inequalities we investigate the existence and uniqueness of the solutions of the fractional nonlinear Volterra integral equation and a fractional nonlinear integrodifferential equation in Banach…
Metric topological vector spaces of Carath\'eodory functions and topologies of $L^p_{loc}$ type are introduced, depending on a suitable set of moduli of continuity. Theorems of continuous dependence on initial data for the solutions of…
We study two variations of the classical one-delta problem for entire functions of exponential type, known also as the Carath\'eodory--Fej\'er--Tur\'an problem. The first variation imposes the additional requirement that the function is…
Motivated by works on extension sets in standard domains we introduce a notion of the Carath\'eodory set that seems better suited for the methods used in proofs of results on characterization of extension sets. A special stress is put on a…
We prove the uniqueness of the infinite length axisymmetric solution to the capillary equation. We observe that capillary equation can be viewed, at large depth, as a perturbation of an integrable two-dimensional differential system.…
The Carath\'eodory's Extension Theorem is a powerful tool that allows us to generate a measure, over a sigma-algebra, from a pre-measure defined over an algebra of sets. However, although this result reduces our work to define a measure by…
In this paper, we introduce a new higher-order directional derivative and higher-order subdifferential of Hadamard type of a given proper extended real function. This derivative is harmonized with the classical higher-order Fr\'echet…
We prove existence and uniqueness of the solution for a class of mixed fractional stochastic differential equations with discontinuous drift driven by both standard and fractional Brownian motion. Additionally, we establish a generalized…
Numerical solving differential equations with fractional derivatives requires elimination of the singularity which is inherent in the standard definition of fractional derivatives. The method of integration by parts to eliminate this…