On the stability of the generalized Cauchy–Jensen set-valued functional equations

In this paper, we prove the Hyers–Ulam–Rassias stability of the generalized Cauchy–Jensen set-valued functional equation defined by

$$\begin{aligned} \alpha f\left( \frac{x+y}{\alpha } + z\right) = f(x) \oplus f(y)\oplus \alpha f(z) \end{aligned}$$

for all \(x,y,z \in X\) and \(\alpha \ge 2\) on a Banach space by using the fixed point alternative theorem.

     

On the stability of Hosszú’s functional equation

Two stability results are proved. The first one states that Hosszú’s functional equation $$f(x+y-xy)+f(xy)=f(x)-f(y)=0\ \ \ \ \ (x,y \in \rm R)$$ is stable. The second is a local stability theorem for additive functions in a Banach space setting.     

On Hyers–Ulam stability of two functional equations in non-Archimedean spaces

In this paper we prove, using the fixed point method, the generalized Hyers–Ulam stability of two functional equations in complete non-Archimedean normed spaces. One of these equations characterizes multi-Cauchy–Jensen mappings, and the other gives a characterization of multi-additive-quadratic mappings.     

Optimal control approach to stability criteria on Hill’s equations

This paper gives a tutorial on how to prove Lyapunov type criteria by optimal control methods. Firstly, we consider stability criteria on Hill's equations with nonnegative potential. By optimal control methods developed in 1990s, we obtain several stability criteria including Lyapunov's criterion, Neǐgauz and Lidskiǐ's criterion. Secondly, we present stability criteria on Hill's equations with sign-changing potential in which Brog's criterion and Krein's criterion are included.     

On the asymptotic stability ofθ-methods for delay differential equations

Stability regions of -methods for the linear delay differential test equations

On regularity for de Rham’s functional equations

We consider regularity for solutions of a class of de Rham’s functional equations. Under some smoothness conditions of functions making up the equation, we improve some results in Hata (Japan J Appl Math 2:381–414, 1985). Our results are applicable to some cases when the functions making up the equation are non-linear functions on an interval, specifically, polynomials and linear fractional transformations. Our results imply the singularity of some well-known singular functions, in particular, Minkowski’s question-mark function, and, some small perturbed functions of the singular functions.     

On restricting Cauchy–Pexider functional equations to submanifolds

Sufficient geometric conditions are given which determine when the Cauchy–Pexider functional equation f(x)g(y) = h(x + y) restricted to x, y lying on a hypersurface in ${\mathbb{R}^d}$ has only solutions which extend uniquely to exponential affine functions ${\mathbb{R}^d \to \mathbb{C}}$ (when f, g, h are assumed to be measurable and non-trivial). The Cauchy–Pexider-type functional equations ${\prod_{j=0}^df_j(x_j)=F(\sum_{j=0}^dx_j)}$ for ${x_0, \ldots,x_d}$ lying on a curve and ${f_1(x_1)f_2(x_2)f_3(x_3)=F(x_1+x_2+x_3)}$ for x ₁, x ₂, x ₃ lying on a hypersurface are also considered.     

Application of Lyapunov’s direct method to the study of the stability of solutions to systems of impulsive differential equations

Classical theorems on the stability of the solutions of impulsive differential equations are further developed.     

On the functional differential equations with “maximums”

The paper deals with an existence - uniqueness problem of the solution for functional differential equations with "maximums", Which arise in the theory of automatic control     

On the functional differential equations with “maximums”

The paper deals with the existence-uniqueness problem of the solution of functional differential equations with maximums, which arises in the theory of automatic control. Making use of some previous results of the authors, sufficient conditions are obtained for the existence and uniqueness of the global solution of the initial value problem for these equations.     

On the regularity of the pressure field of Brenier’s weak solutions to incompressible Euler equations

In this paper we improve the regularity in time of the gradient of the pressure field arising in Brenier’s variational weak solutions (Comm Pure Appl Math 52:411–452, 1999) to incompressible Euler equations. This improvement is necessary to obtain that the pressure field is not only a measure, but a function in . In turn, this is a fundamental ingredient in the analysis made by Ambrosio and Figalli (2007, preprint) of the necessary and sufficient optimality conditions for the variational problem by Brenier (J Am Mat Soc 2:225–255, 1989; Comm Pure Appl Math 52:411–452, 1999).     

On some functional equations of Goląb-Schinzel type

Summary LetE be a real Hausdorff topological vector space. We consider the following binary law * on ·E:(, ) * (, ) = (, ^k + ) for(, ), (, ) × E where is a nonnegative real number,k andl are integers.In order to find all subgroupoids of ( ·E, *) which depend faithfully on a set of parameters, we have to solve the following functional equation:f(f(y) ^k x +f(x) ^l y) =f(x)f(y) (x, y E). (1)In this paper, all solutionsf: of (1) which are in the Baire class I and have the Darboux property are obtained. We obtain also all continuous solutionsf: E of (1). The subgroupoids of (^* ·E, *) which dapend faithfully and continuously on a set of parameters are then determined in different cases. We also deduce from this that the only subsemigroup ofL _n ¹ of the form {(F(x ₂,x ₃, ,x _n ),x ₂,x ₃, ,x _n ); (x ₂, ,x _n ) ^{n – 1} }, where the mappingF: ^{n – 1 *} has some regularity property, is {1} × ^{n – 1} .We may noitice that the Gob-Schinzel functional equation is a particular case of equation (1)(k = 0, l = 1, = 1). So we can say that (1) is of Gob—Schinzel type. More generally, whenE is a real algebra, we shall say that a functional equation is of Gob—Schinzel type if it is of the form:f(f(y) ^k x +f(x) ^l y) =F(x,y,f(x),f(y),f(xy)) wherek andl are integers andF is a given function in five variables. In this category of functional equations, we study here the equation:f(f(y) ^k x +f(x) ^l y) =f(xy) (x, y f: ). (4)This paper extends the results obtained by N. Brillouët and J. Dhombres in [3] and completes some results obtained by P. Urban in his Ph.D. thesis [11] (this work has not yet been published).Dedicated to the memory of Alexander M. Ostrowski on the occasion of the 100th anniversary of his birth     

On the stability of self-similar solutions of 1D cubic Schrödinger equations

In this paper we will study the stability properties of self-similar solutions of $1$ D cubic NLS equations with time-dependent coefficients of the form 0.1 $$\begin{aligned} \displaystyle { iu_t+u_{xx}+\frac{u}{2} \left(|u|^2-\frac{A}{t}\right)=0, \quad A\in \mathbb{R }. } \end{aligned}$$ The study of the stability of these self-similar solutions is related, through the Hasimoto transformation, to the stability of some singular vortex dynamics in the setting of the Localized Induction Equation (LIE), an equation modeling the self-induced motion of vortex filaments in ideal fluids and superfluids. We follow the approach used by Banica and Vega that is based on the so-called pseudo-conformal transformation, which reduces the problem to the construction of modified wave operators for solutions of the equation $$\begin{aligned} iv_t+ v_{xx} +\frac{v}{2t}(|v|^2-A)=0. \end{aligned}$$ As a by-product of our results we prove that Eq. (0.1) is well-posed in appropriate function spaces when the initial datum is given by $u(0,x)= z_0 \mathrm p.v \frac{1}{x}$ for some values of $z_0\in \mathbb{C }\setminus \{ 0\}$ , and $A$ is adequately chosen. This is in deep contrast with the case when the initial datum is the Dirac-delta distribution.