Existence of tangent lines to Carnot–Carathéodory geodesics

We show that length minimizing curves in Carnot–Carathéodory spaces possess at any point at least one tangent curve (i.e., a blow-up in the nilpotent approximation) equal to a straight horizontal line.     

Radon and Carathéodory type theorems for staircase convexity

Aequationes mathematicae - We establish the following analogues of the familiar Radon and Carathéodory theorems for staircase convex sets in $$mathbb {R}^d$$ : Various “best...     

Carathéodory approximate solutions for a class of perturbed stochastic differential equations with reflecting boundary

Abstract

In this work, we study the Carathéodory approximate solution for a class of one-dimensional perturbed stochastic differential equations with reflecting boundary (PSDERB). Based on the Carathéodory approximation procedure, we prove that PSDERB have a unique solution and show that the Carathéodory approximate solution converges to the solution of PSDERB whose both drift and diffusion coefficients are non-Lipschitz. After that, we establish an explicit rate of convergence in the case of PSDERB with Lipschitz coefficients.     

Solvability of a class of elastic beam equations with strong Carathéodory nonlinearity

We study the existence of a solution to the nonlinear fourth-order elastic beam equation with nonhomogeneous boundary conditions

Continuity and boundary behavior of the Carathéodory metric

The boundary behavior of the higher-order Carathéodory metrics, the singular Carathéodory metric, and the Azukawa metric near anh-extendable boundary point of a bounded smooth pseudoconvex domain in ℂⁿ are studied. Translated fromMathematicheskie Zametski, Vol. 67, No. 2, pp. 230–240, February, 2000.     

Successively Iterative Technique of a Classical Elastic Beam Equation with Carathéodory Nonlinearity

In this paper, we consider an elastic beam equation where the nonlinear term is a Carathéodory function and the boundary condition is nonhomogeneous. We construct an iterative sequence by the help of monotonic technique and prove that the sequence approximates successively to the solution of the equation under suitable assumptions.     

Carathéodory’s Theorem and moduli of local connectivity

We give a quantitative proof of the Carathéodory Theorem by means of the concept of a modulus of local connectivity and the extremal distance of the separating curves of an annulus.     

Solvability of Third-order Three-point Boundary Value Problems with Carathéodory Nonlinearity

A class of third-order three-point boundary value problems is considered,where the nonlinear term is a Carathéodory function. By introducing a height function and considering the integration of this height function, an existence theorem ofsolution is proved when the limit growth function exists. The main tools are theLebesgue dominated convergence theorem and the Schauder fixed point theorem.     

Solvability of Third-order Three-point Boundary Value Problems with Carathéodory Nonlinearity

A class of third-order three-point boundary value problems is considered,where the nonlinear term is a Carathéodory function.By introducing a height function and considering the integration of this hei...     

Dichotomies for linear differential equations with delays: the Carathéodory case

Summary This work is concerned with differential equations with delays, on [0, ∞[, of the form +Mu=r, and the corresponding homogeneous equation +Mu=0, where u and r take values in a Banach space E, and M is a ? memory ?. The main results relate the existence of dichotomies or exponential dichotomies of the solutions of the homogeneous equation (a kind of conditional stability) to the admissibility of certain pairs of function spaces for the inhomogeneous equation. The ? memory ? M is quite general: it is a linear mapping that takes continuous E-valued functions on [−1, ∞[ into locally integrable functions on [0, ∞[ in such a way that Mu on [a, b] ⊂ [0, ∞[ depends only on u on [a −1, b], and satisfies a reasonable boundedness condition. Entrata in Redazione il 10 settembre 1973. This work was partially supported by NSF Grant GP-33364X.     

Carathéodory balls and norm balls in a class of convex bounded Reinhardt domains

A family of the spherical fractional integrals on the unit sphere Σ _n in ℝ ⁿ⁺¹ is investigated. This family includes the spherical Radon transform (α = 0) and the Blaschke-Levy representation (α>1). Explicit inversion formulas and a characterization ofT ^αƒ are obtained for ƒ belonging to the spacesC ^∞,C, L^p and for the case when ƒ is replaced by a finite Borel measure. All admissiblen ≥ 2,α ε ℂ, andp are considered. As a tool we use spherical wavelet transforms associated withT ^α. Wavelet type representations are obtained forT ^α ƒ, ƒ εL ^p, in the case Reα ≤ 0, provided thatT ^α is a linear bounded operator inL ^p. Partially supported by the Edmund Landau Center for Research in Mathematical Analysis, sponsored by the Minerva Foundation (Germany).     

Existence and global stability of periodic solution for impulsive predator-prey model with diffusion and distributed delay

In this paper, the nonlinear nonautonomous predator-prey model with impulsive effect is considered. By using the method of coincidence degree theorem, a set of easily verifiable sufficient conditions are obtained for the existence of at least one strictly positive periodic solution, and by the means of a suitable Lyapunov function, the uniqueness and global attractivity of positive periodic solution are presented. An example shows the feasibility of our main results.     

On the Carathéodory superposition of multifunctions and an existence theorem

Let I ? ? be an interval and Y a reflexive Banach space. We introduce the (H) property of a multifunction F from I × Y to Y and prove that the Carathéodory superposition of F with each continuous function f from I to Y is a derivative provided that F has the (H) property. Some application of this theorem to the existence of solutions of differential inclusions f′(x) ∈ F(x, f(x)) is given.     

A topological principle for retarded functional differential equations of Carathéodory type

The paper presents a Razumikhin-type version of Wa?ewski's principle for RFDEs of Carathéodory type, thus generalizing the author's previous results (J. Differential Equations36, 117–138 (1980)). A few facts from the theory of RFDEs are reviewed, and the concept of a regular polyfacial set with respect to an RFDE of Carathéodory type is introduced and the results are state and proved. Most of the examples given in the above work can easily be adapted to the more general setting of Carathéodory systems. An indication how to do it is given in only one case. Throughout the paper, standard notation is used.     

The Grunsky function and Carathéodory metric of Teichmüller spaces

A new proof of the equality of all non-expanding invariant metrics on the universal Teichmüller space is given. The arguments extend to Teichmüller space of the punctured disc.