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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 the asymptotic boundary of traveling waves (first-component) in a discrete diffusive epidemic model, whose existence is well-known. Meanwhile, we derive the nonexistence of traveling wave solutions with non-positive wave speed. We solve two open problems left in two papers [Fu et al., Nonlinear Convex A. 17 (2016), pp. 1739–1751; Wu, J. Differential Equations 262 (2017), pp. 272–282]. [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 investigate the stability in the sense of Hyers–Ulam–Rassias for the two types of generalized fractional differential systems by the ρ -Laplace transform method. Several examples are given to illustrate the theoretical results. [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 the present paper, a general class of linear functional equations is considered and a computer program is described, which determines the exact solutions of systems of equations belonging to this class. [ABSTRACT FROM AUTHOR]

Let k ≥ 2 be an integer. The purpose of this paper is first to introduce the notation of Felbin's type fuzzy normed linear spaces, and then by virtue of this notation to study some stability results concerning the more general cubic functional equation of the form f (x + ky) + f (x - ky) + f (kx) = k 2 f (x + y) + k 2 f (x - y) + (k 3 - 2 k 2 + 2) f (x) in the setting of Felbin's type fuzzy normed linear spaces by employing the direct and fixed point methods. Then some applications of our results for the stability of the cubic functional equation from a real normed space to a Banach space will be demonstrated. Furthermore, the interdisciplinary relation between the theory of Felbin's type fuzzy spaces and the theory of functional equations are also presented in this paper. [ABSTRACT FROM AUTHOR]

In an earlier paper we discussed the composition of derivations of order 1 on a commutative ring R, showing that (i) the composition of n derivations of order 1 yields a derivation of order at most n, and (ii) under additional conditions on R the composition of n derivations of order exactly 1 forms a derivation of order exactly n. In the present paper we consider the composition of derivations of any orders on rings. We show that on any commutative ring R the composition of a derivation of order at most n with a derivation of order at most m results in a derivation of order at most n + m . If R is an integral domain of sufficiently large characteristic, then the composition of a derivation of order exactly n with a derivation of order exactly m results in a derivation of order exactly n + m . As in the previous paper, the results are proved using Leibniz difference operators. [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 paper, we introduce a new quadratic functional equation and, motivated by this equation, we investigate n-variables mappings which are quadratic in each variable. We show that such mappings can be unified as an equation, namely, multi-quadratic functional equation. We also apply a fixed point technique to study the stability for the multi-quadratic functional equations. Furthermore, we present an example and a few corollaries corresponding to the stability and hyperstability outcomes. [ABSTRACT FROM AUTHOR]

The existence and uniqueness of the mild solutions for a class of degenerate functional stochastic partial differential equations (SPDEs) are obtained, where the drift is assumed to be Hölder–Dini continuous. Moreover, the non-explosion of the solution is proved under some reasonable conditions. In addition, the Harnack inequality is derived by the method of coupling by change of measure. Finally, the shift Harnack inequality is obtained for the equations without delay, which is new even in the non-degenerate case. An example is presented in the final part of the paper. [ABSTRACT FROM AUTHOR]

The aim of this paper is to prove the superstability of the following functional equations: f (P (x , y)) = g (x) h (y) , f (x + y) = g (x) h (y). [ABSTRACT FROM AUTHOR]

This research paper deals with general solution and the Hyers–Ulam stability of a new generalized n-variable mixed type of additive and quadratic functional equations in fuzzy modular spaces by using the fixed point method. [ABSTRACT FROM AUTHOR]

Certain Hardy inequalities in domains of Euclidean space contain sharp but unreachable constants. V. G. Maz'ya and other authors used this fact to improve the corresponding inequalities by adding new integral terms. In this paper, a survey of results in this direction initiated by H. Brezis and M. Marcus is presented. Also, we give some generalizations of Brezis–Marcus-type inequalities to the case of Rellich-type inequalities with weights that are powers of the distance from a point to the boundary of the domain. Generalizations to the case of conformally invariant integral inequalities in simply connected and doubly connected planar hyperbolic domains are discussed. [ABSTRACT FROM AUTHOR]

In the present paper, inspired by methods contained in Gajda and Kominek (Stud Math 100:25-38, 1991) we generalize the well known sandwich theorem for subadditive and superadditive functionals to the case of delta-subadditive and delta-superadditive mappings. As a consequence we obtain the classical Hyers-Ulam stability result for the Cauchy functional equation. We also consider the problem of supporting delta-subadditive maps by additive ones. [ABSTRACT FROM AUTHOR]

In this paper, we consider random multi-normed spaces introduced by Dales and Polyakov (Multi-Normed Spaces, 2012). Next, by the fixed point method, we approximate the multiplicatives on these spaces. [ABSTRACT FROM AUTHOR]

This article has two aims. First, we provide the solution to a problem posed by the author in a previous paper. Second, we consider a problem posed by Kannappan and Kurepa (Aequat Math 4: 163-175, 1970). Our results show that additive functions linked by certain types of functional equations are combinations of linear functions and derivations of various orders. We show that this is not generally the case for the problem of Kannappan and Kurepa, and we modify their problem accordingly. [ABSTRACT FROM AUTHOR]

In this paper, we unify the system of functional equations defining a multi-Jensen-quadratic mapping to obtain a single equation. We also prove, using the fixed point method, the generalized Hyers-Ulam stability of this equation both in Banach spaces and in complete non-Archimedean normed spaces. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence of nontrivial solutions to the following fractional p-Laplacian system with homogeneous nonlinearities of critical Sobolev growth: { (- Δ p) s ⁢ u = Q u ⁢ (u , v) + H u ⁢ (u , v) in ⁢ Ω , (- Δ p) s ⁢ v = Q v ⁢ (u , v) + H v ⁢ (u , v) in ⁢ Ω , u = v = 0 in ⁢ ℝ N ∖ Ω , u , v ≥ 0 , u , v ≠ 0 in ⁢ Ω , \left\{\begin{aligned} \displaystyle{}(-\Delta_{p})^{s}u&\displaystyle=Q_{u}(u% ,v)+H_{u}(u,v)&&\displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle(-\Delta_{p})^{s}v&\displaystyle=Q_{v}(u,v)+H_{v}(u,v)&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u=v&\displaystyle=0&&\displaystyle\phantom{}\text{in }\mathbb{R}% ^{N}\setminus\Omega,\\ \displaystyle u,v&\displaystyle\geq 0,\quad u,v\neq 0&&\displaystyle\phantom{}% \text{in }\Omega,\end{aligned}\right. where (- Δ p) s {(-\Delta_{p})^{s}} denotes the fractional p-Laplacian operator, p > 1 {p>1} , s ∈ (0 , 1) {s\in(0,1)} , p ⁢ s < N {ps

Let k ≥ 2 be an integer. The purpose of this paper is first to introduce the notation of Felbin's type fuzzy normed linear spaces, and then by virtue of this notation to study some stability results concerning the more general cubic functional equation of the form f (x + ky) + f (x - ky) + f (kx) = k 2 f (x + y) + k 2 f (x - y) + (k 3 - 2 k 2 + 2) f (x) in the setting of Felbin's type fuzzy normed linear spaces by employing the direct and fixed point methods. Then some applications of our results for the stability of the cubic functional equation from a real normed space to a Banach space will be demonstrated. Furthermore, the interdisciplinary relation between the theory of Felbin's type fuzzy spaces and the theory of functional equations are also presented in this paper. [ABSTRACT FROM AUTHOR]