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In this paper we consider a generalized polynomial f: ℝ → ℝ of degree two that satisfies the additional equation f(x)f(y) = 0 for the pairs (x, y) ∈ D, where D ⊆ ℝ2 is given by some algebraic condition. In the particular cases when there exists a positive rational m fulfilling D = { (x , y) ∈ ℝ 2 | x 2 - m y 2 = 1 } , we prove that f(x) = 0 for all x ∈ ℝ. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Hyers–Ulam stability of the coefficient multipliers on the Hardy space H 2 and the Bergman space A 2 , meanwhile, we also investigate the Hyers–Ulam stability of the coefficient multipliers between the Bergman space A 2 and the Hardy space H 2 . We give the necessary and sufficient condition for the coefficient multipliers to have the Hyers–Ulam stability on the Hardy space H 2 , on the Bergman space A 2 and between the Bergman space A 2 and the Hardy space H 2 , respectively. We also show that the best constant of Hyers–Ulam stability exists under different circumstances. Some results generalized the results of MacGregor and Zhu when p = 2 in MacGregor and Zhu article (Mathematika 42:413–426, 1995). Moreover, some illustrative examples are also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we study a coupled system of beam deflection type that involves nonlinear equations with sequential Caputo fractional derivatives. Under flexible/fixed end-conditions, two main theorems on the existence and uniqueness of solutions are proved by using two fixed point theorems. Some examples are discussed to illustrate the applications of the existence and uniqueness of solution results. Another main result on the Ulam–Hyers stability of solutions for the introduced system is also discussed. Some examples of stability are discussed. New travelling wave solutions are obtained for another conformable coupled system of beam type that has a connection with the first considered system. A conclusion follows at the end. [ABSTRACT FROM AUTHOR]

In this paper, we solve the system of functional equations { f (x + y) + g (y − x) = 2 f (x) , g (x + y) − f (y − x) = 2 g (y) and we investigate the stability of g-derivations in Banach algebras. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of Hadamard 픇-hom-der on Banach algebras. Further, we apply the fixed point method to prove the stability of the Hadamard 픇-hom-der on Banach algebras. [ABSTRACT FROM AUTHOR]

In this paper, a computer program developed in the computer algebra system Maple is presented, which investigates alienness and strong alienness of linear functional equations. [ABSTRACT FROM AUTHOR]

In this paper, the discrete parametrix method is adopted to investigate the estimation of transition kernel for Euler-Maruyama scheme SDEs driven by α -stable noise, which implies Krylov's estimate and Khasminskii's estimate. As an application, the convergence rate of Euler-Maruyama scheme for a class of multidimensional SDEs with singular drift (by the use of Zvonkin's transformation) is obtained. [ABSTRACT FROM AUTHOR]

This paper is aimed to discuss the existence and exponential stability of mild solutions for second-order impulsive neutral stochastic differential equations (INSDEs). Firstly, we derive the existence of mild solutions for INSDEs by using the Banach fixed point theorem. Then, the exponential stability in the pth moment of mild solution for the considered INSDEs is proved by means of the generalized impulsive-integral inequality, which can effectively improve some known existing ones. Finally, as an application, an example is given to illustrate the efficiency of the obtained theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we continue the discussion about relations between exponential polynomials and generalized moment functions on a commutative hypergroup. We are interested in the following problem: is it true that every finite-dimensional variety is spanned by moment functions? Let m be an exponential on X. In our former paper, we have proved that if the linear space of all m-sine functions in the variety of an m-exponential monomial is (at most) one-dimensional, then this variety is spanned by moment functions generated by m. In this paper, we show that this may happen also in cases where the m-sine functions span a more than one-dimensional subspace in the variety. We recall the notion of a polynomial hypergroup in d variables, describe exponentials on it and give the characterization of the so-called m-sine functions. Next we show that the Fourier algebra of a polynomial hypergroup in d variables is the polynomial ring in d variables. Finally, using the Ehrenpreis–Palamodov Theorem, we show that every exponential polynomial on the polynomial hypergroup in d variables is a linear combination of moment functions contained in its variety. [ABSTRACT FROM AUTHOR]

In this paper, we establish some Hardy inequalities for the p-Laplace operator associated with Robin boundary conditions in general domains. Moreover, by using the new Hardy inequality, we derive the classical Hardy inequality with the weight of mean distance function. [ABSTRACT FROM AUTHOR]

In this current paper, using q-fractional calculus, we study the Duffing–Rayleigh type problem with sequential fractional q-derivative of the Caputo type. We investigate the existence and uniqueness of solutions by applying some classical fixed point theorems. Also we define and study the Ulam–Hyers and the Ulam–Hyers–Rassias stabilities of solutions for our problem. An example is presented to illustrate the main results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to prove that if on a commutative hypergroup an exponential monomial has the property that the linear subspace of all sine functions in its variety is one dimensional, then this exponential monomial is a linear combination of generalized moment functions. [ABSTRACT FROM AUTHOR]

In this paper generalized moment functions are considered. They are closely related to the well-known functions of binomial type which have been investigated on various abstract structures. The main purpose of this work is to prove characterization theorems for generalized moment functions on commutative groups. At the beginning a multivariate characterization of moment functions defined on a commutative group is given. Next the notion of generalized moment functions of higher rank is introduced and some basic properties on groups are listed. The characterization of exponential polynomials by means of complete (exponential) Bell polynomials is given. The main result is the description of generalized moment functions of higher rank defined on a commutative group as the product of an exponential and composition of multivariate Bell polynomial and an additive function. Furthermore, corollaries for generalized moment function of rank one are also stated. At the end of the paper some possible directions of further research are discussed. [ABSTRACT FROM AUTHOR]

In this paper, we deal with perturbations of two general functional equations in several variables. Namely, we prove the generalized, in the spirit of Bourgin, Ulam stability of these equations in Banach spaces. In order to do this, we use the fixed point method. Moreover, as corollaries from our main results, we get several outcomes on approximate solutions of a few important classic equations. They include, among others, the functional equations which characterize multi-additive and multi-quadratic mappings. In consequence, the perturbation of homomorphisms of Banach spaces and quadratic mappings is also treated. [ABSTRACT FROM AUTHOR]

In this paper, versions of the famous Jensen inequality for sublinear isotonic functionals are proved. The obtained results generalize classic Jessen's and McShane's inequalities. Applications to generalized means and to Hölder's and Minkowski's inequalities are also given. [ABSTRACT FROM AUTHOR]

The aim of this sequence of work is to investigate polynomial equations satisfied by additive functions. As a result of this, new characterization theorems for homomorphisms and derivations can be given. More exactly, in this paper the following type of equation is considered ∑ i = 1 n f i (x p i) g i (x q i) = 0 x ∈ F , where n is a positive integer, F ⊂ C is a field, f i , g i : F → C are additive functions and p i , q i are positive integers for all i = 1 , ... , n . [ABSTRACT FROM AUTHOR]

We prove that the first eigenvalue of the fractional Dirichlet–Laplacian of order s on a simply connected set of the plane can be bounded from below in terms of its inradius only. This is valid for 1 / 2 < s < 1 and we show that this condition is sharp, i.e., for 0 < s ≤ 1 / 2 such a lower bound is not possible. The constant appearing in the estimate has the correct asymptotic behavior with respect to s, as it permits to recover a classical result by Makai and Hayman in the limit s ↗ 1 . The paper is as self-contained as possible. [ABSTRACT FROM AUTHOR]

In this paper we employ a Gaussian-type heat kernel estimate to establish Krylov's estimate and Khasminskii's estimate for the Euler–Maruyama (EM) algorithm. For applications, by taking Zvonkin's transformation into account, we investigate the convergence rate of the EM algorithm for a class of multidimensional stochastic differential equations (SDEs) with low regular drifts, which need not be piecewise Lipschitz. [ABSTRACT FROM AUTHOR]

In this study, we discuss the existence of solutions for a coupled system of two- sequential random fractional differential equations involving nonlocal periodic boundary conditions. Using a version of Leray-Schauder alternative theorem, we present recent results on the existence of coupled solutions for the problem. The obtained solutions as well as their Caputo derivatives satisfy some stochastic processes properties. An illustrative example is also discussed at the end of this paper. [ABSTRACT FROM AUTHOR]

This paper is concerned with the study of a system of q-fractional differential equations. By applying the Leray-Schauder alternative, we show the existence of multiple solutions to our system. Then, the uniqueness of the solution is shown using the Banach fixed point theorem. On the other hand, we study stability in the sense of Ulam-Hyers then that of Ulam-Hyers Rassias. Finally, we give some examples to illustrate our results. [ABSTRACT FROM AUTHOR]

Let S be a semigroup and K a field. A function f : S → K is additive if f (x y) = f (x) + f (y) for all x , y ∈ S , and functions g , h : S → K form a sine pair if they satisfy the sine addition law g (x y) = g (x) h (y) + h (x) g (y) for all x , y ∈ S . Adding these two equations we arrive at the functional equation (*) f (x y) + g (x y) = f (x) + f (y) + g (x) h (y) + h (x) g (y) . The alienation question for additivity and sine additivity asks whether (*) implies that f is additive and (g, h) is a sine pair. To fully answer this question we find the general solution of (*) for unknown functions f , g , h : S → C . The solution illustrates a significant amount of interdependence between additivity and sine additivity. [ABSTRACT FROM AUTHOR]

By introducing some appropriate definitions for Ulam type stabilities, in this paper, we discuss the Ulam and Hyers and the generalized Ulam and Hyers stabilities for two different classes of nonlinear singular fractional integro-differential problems of Lane and Emden type. The existence and uniqueness results for our two classes have already been published respectively in Math. Method. Appl. Sci., 2020, and Moroc. J. Pure Appl. Anal., 2020. [ABSTRACT FROM AUTHOR]

The present article is devoted to generalized Salem functions, the generalized shift operator, and certain related problems. A description of further investigations of the author of this article is given. These investigations (in terms of various representations of real numbers) include generalized Salem functions and generalizations of the Gauss–Kuzmin problem. [ABSTRACT FROM AUTHOR]

This paper deals with a nonlinear random differential equation involving mean square Caputo fractional derivative. We study the existence and uniqueness of solutions for the considered problem. Then, by introducing new concepts on the continuous and fractional derivative dependance, we prove new result of the stability on some random/deterministic data dependance. [ABSTRACT FROM AUTHOR]