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.     

The Hyers-Ulam stability of a functional equation deriving from quadratic and cubic functions in quasi-��-normed spaces

In this paper, we investigate the Hyers-Ulam stability of the following function equation 2f（2x ＋ y） ＋ 2f（2x - y） = 4f（x ＋ y） ＋ 4f（x - y） ＋ 4f（2x） ＋ f（2y） - Sf（x） - 8f（y） in quasi-β-normed spaces.     

Generalized Leray–Schauder principles for general classes of maps in completely regular topological spaces

In this paper, we present new continuation theorems for maps in a very general class.     

Uniqueness results for fractional functional differential equations with infinite delay in Fréchet spaces

In this article, a recent nonlinear alternative for contraction maps in Fréchet spaces due to Frigon and Granas [1998, Résultats de type Leray-Schauder pour des contractions sur des espaces de Fréchet, Ann. Sci. Math. Québec 22, 161–168] is used to investigate the existence and uniqueness of solutions for fractional order functional differential equations with infinite delay.     

Families of Liapunov-Krasovskiį functionals and stability for functional differential equations

This paper discusses the stability of solutions of nonautonomous functional differential equations with infinite delay with respect to a parr of admissible phase spaces of Hale and Kato. A one-parameter family of Liapunov-Krasovskiį functional, together with some additional analysis, is used to prove new sufficient conditions of asymptotic and uniform asymptotic stability for such equations. It is also shown that the so-called Razumikhin condition is unessential when families of Liapunov-Krasovskiį functionals are used. Entrata in Redazione il 25 settembre 1997. Invited address at the Second Marrakesh International Conference on Differential Equations, Marrakesh, Morocco, June 1995.     

Asymptotic stability for the Navier–Stokes equations in L n

Considering the Navier–Stokes equations in W ì \mathbbRⁿ\Omega \subset {\mathbb{R}}^n, we prove the asymptotic stability for weak solutions in the marginal class u ∈ C _B (0, ∞; L ⁿ ) with arbitrary initial and external perturbations.     

On the stability of a modified Nyström method for Mellin convolution equations in weighted spaces

This paper deals with the numerical solution of second kind integral equations with fixed singularities of Mellin convolution type. The main difficulty in solving such equations is the proof of the stability of the chosen numerical method, being the noncompactness of the Mellin integral operator the chief theoretical barrier. Here, we propose a Nyström method suitably modified in order to achieve the theoretical stability under proper assumptions on the Mellin kernel. We also provide an error estimate in weighted uniform norm and prove the well-conditioning of the involved linear systems. Some numerical tests which confirm the efficiency of the method are shown.     

Nearly (k, s)-Fibonacci functional equations in β-normed spaces

In this paper, we solve the (k, s)-Fibonacci functional equation $$f_{k,s}(x)=kf_{k,s}(x-1)+sf_{k,s}(x-2).$$ Moreover, we prove the Hyers–Ulam stability of the (k, s)-Fibonacci functional equation in β-normed spaces.     

One dimensional φ-Laplacian functional equations

In this paper we study functional φ-Laplacian equations with functional boundary conditions. The non-linearity belongs to a class of completely continuous operators, which includes Carathéodory ones, while the functional boundary conditions are given by compactly-fixed operators. We extend, in particular, a solvability result of C. Bereanu and J. Mawhin.     

Generalized Krasnoselskii–Mann-type iterations for nonexpansive mappings in Hilbert spaces

The Krasnoselskii–Mann iteration plays an important role in the approximation of fixed points of nonexpansive operators; it is known to be weakly convergent in the infinite dimensional setting. In this present paper, we provide a new inexact Krasnoselskii–Mann iteration and prove weak convergence under certain accuracy criteria on the error resulting from the inexactness. We also show strong convergence for a modified inexact Krasnoselskii–Mann iteration under suitable assumptions. The convergence results generalize existing ones from the literature. Applications are given to the Douglas–Rachford splitting method, the Fermat–Weber location problem as well as the alternating projection method by John von Neumann.     

Generalized Hyers-Ulam Stability for a General Mixed Functional Equation in Quasi-��-normed Spaces

In this paper, we establish the general solution and investigate the generalized Hyers-Ulam stability of the following mixed additive and quadratic functional equation

Generalized φ-contraction for a pair of mappings on cone metric spaces

Using the setting of cone metric space, a fixed point theorem is proved for two maps, and several corollaries are obtained. In these cases, the cone does not need to be normal. These results generalize several well known compatible recent and classical results in the literature. As an application, the existence of solution of an integral equation is presented.     

On ω-limit sets of ordinary differential equations in Banach spaces

Let X be an infinite-dimensional real Banach space. We classify ω-limit sets of autonomous ordinary differential equations x^′=f(x), x(0)=x₀, where f:X→X is Lipschitz, as being of three types I-III. We denote by SX the class of all sets in X which are ω-limit sets of a solution to (1), for some Lipschitz vector field f and some initial condition x₀∈X. We say that S∈SX is of type I if there exists a Lipschitz function f and a solution x such that S=Ω(x) and . We say that S∈SX is of type II if it has non-empty interior. We say that S∈SX is of type III if it has empty interior and for every solution x (of Eq. (1) where f is Lipschitz) such that S=Ω(x) it holds . Our main results are the following: S is a type I set in SX if and only if S is a closed and separable subset of the topological boundary of an open and connected set U⊂X. Suppose that there exists an open separable and connected set U⊂X such that , then S is a type II set in SX. Every separable Banach space with a Schauder basis contains a type III set. Moreover, in all these results we show that in addition f may be chosen Ck-smooth whenever the underlying Banach space is Ck-smooth.     

A new type of approximation for the radical quintic functional equation in non-Archimedean (2,β)-Banach spaces

Let $\mathbb{R}$ be the set of real numbers. In this paper, we first introduce the notions of non-Archimedean $(2,\beta )$-normed spaces $(X,{6?,?6}_{?,\beta })$ and we will reformulate the fixed point theorem [10, Theorem 1] in this space, after it, we introduce and solve the radical quintic functional equation

$f\left(\sqrt[5]{x^5+y^5}\right)=f(x)+f(y),x,y\in \mathbb{R}.$

Also, under some weak natural assumptions on the function $\gamma :\mathbb{R}\times \mathbb{R}\times X\to [0,\infty )$, we show that this theorem is a very efficient and convenient tool for proving the hyperstability results when $f:\mathbb{R}\to X$ satisfy the following radical quintic inequality

${6f\left(\sqrt[5]{x^5+y^5}\right)?f(x)?f(y),z6}_{?,\beta }\le \gamma (x,y,z),x,y\in \mathbb{R}?\{0\},z\in X,$

with $x\ne ?y$.     

Schauder estimates for elliptic equations in Banach spaces associated with stochastic reaction–diffusion equations

We consider some reaction–diffusion equations perturbed by white noise and prove Schauder estimates for the elliptic problem associated with the generator of the corresponding transition semigroup, defined in the Banach space of continuous functions. This requires the proof of some new interpolation result.     

Entropy and renormalized solutions to the general nonlinear elliptic equations in Musielak–Orlicz spaces

In this paper we mainly prove the existence and uniqueness of entropy solutions and the uniqueness of renormalized solutions to the general nonlinear elliptic equations in Musielak-Orlicz spaces. Moreover, we also obtain the equivalence of entropy solutions and renormalized solutions in the present conditions.     

Asymptotic stability for the Navier–Stokes equations in L n

Considering the Navier–Stokes equations in , we prove the asymptotic stability for weak solutions in the marginal class u ∈ C _B (0, ∞; L ⁿ ) with arbitrary initial and external perturbations.