Generalized Hyers-Ulam Stability of Trigonometric Functional Equations

We will investigate the superstability of the trigonometric functional equation from the following Pexider type functional equation: f ( x + y ) − f ( x − y ) = λ · g ( x ) h ( y ) , λ is constant, which is a trigonometric functional equation mixed by the sine and cosine function. More- over, the equation can be considered by the mixed functional equation of the hyperbolic trigonometric functions, several exponential type functions, and Jensen type equation.

We generalize the super-stability result for cosine functional equation mappings with values in the field of complex numbers to the case of an arbitrary Hilbert space with the Hadamard product. On the other hand, by an example we prove that this generalization is not true for sine functional equation. Mathematics subject classification (2010): Primary: 39B72, 39B82; secondary: 46E40.

In the field of functional equations and their solutions, Ulam's stability is an essential concept. This theory examines whether the function approximating a certain functional equation is close to the function that exactly satisfies it. A broader extension of the stability concept is generalized Hyers–Ulam stability. Classifying, analysing and solving functional equations across multiple spaces are made easier by using this. In this investigation, we examine the generalized Hyers–Ulam stability of generalized cubic- and quartic-type functional equations of the form: m { f ( m ϑ + μ ) + f ( m ϑ − μ ) } + f ( ϑ + m μ ) + f ( ϑ − m μ ) = 2 m 2 { f ( ϑ + μ ) + f ( ϑ − μ ) } + 2 ( m 4 − 2 m 2 + 1 ) f ( ϑ ) and f ( m ϑ + μ ) + f ( m ϑ − μ ) + f ( ϑ + m μ ) + f ( ϑ − m μ ) = 2 m 2 + { f ( ϑ + μ ) + f ( ϑ − μ ) } + 2 ( m 4 − 2 m 2 + 1 ) { f ( ϑ ) + f ( μ ) } , in setting of non-Archimedean (n-A) normed spaces by using distinguished Hyers direct method. Also, we provide a graphical representation of an approximate solution and how it differs from an exact solution for both equations. Furthermore, we present counterexamples that demonstrate the failure case of stability.

In this paper, we discuss the existence and uniqueness of a solution for the implicit two-order fractional integro-differential equation with m-point boundary conditions by applying the Banach fixed point theorem. Moreover, in the paper we establish the four different varieties of Ulam stability (Hyers–Ulam stability, generalized Hyers–Ulam stability, Hyers–Ulam-Rassias stability, and generalized Hyers–Ulam–Rassias stability) for the given problem.

In this paper, we are introducing pertinent Euler–Lagrange–Jensen type k‐quintic functional equations and investigate the ‘Ulam stability’ of these new k‐quintic functional mappings f:X→Y, where X is a real normed linear space and Y a real complete normed linear space. We also solve the Ulam stability problem for Euler–Lagrange–Jensen alternative k‐quintic mappings. Copyright © 2016 John Wiley & Sons, Ltd.

The purpose of this paper is to study the Hyers–Ulam stability and Generalized Hyers–Ulam stability of the general Linear Differential Equations of first order and second order with constant coefficients using Fourier Transform method. Moreover, the Hyers–Ulam stability constants of these differential equations are obtained. Some examples are given to illustrate the main results.

In this paper, we study a system governed by impulsive semilinear nonautonomous differential equations. We present the β –Ulam stability, β –Hyers–Ulam stability and β –Hyers–Ulam–Rassias stability for the said system on a compact interval and then extended it to an unbounded interval. We use Grönwall type inequality and evolution family as a basic tool for our results. We present an example to demonstrate the application of the main result.

In this paper, we use direct and fixed-point techniques to examine the generalised Ulam–Hyers stability results of the general Euler–Lagrange quadratic mapping in non-Archimedean IFN spaces (briefly, non-Archimedean Intuitionistic Fuzzy Normed spaces) over a field.

Let X, Y be Banach modules over a -algebra and let be given. We prove the generalized Hyers–Ulam stability of the following functional equation in Banach modules over a unital -algebra: We show that if and an odd mapping satisfies the functional equation (0.1) then the odd mapping is Cauchy additive. As an application, we show that every almost linear bijection of a unital -algebra A onto a unital -algebra B is a -algebra isomorphism when for all unitaries , all , and all . The concept of generalized Hyers–Ulam stability originated from Th.M. Rassias' stability Theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297–300.

This paper studies the existence theorems and Ulam stability results of solutions for implicit (p, q)-fractional difference equations. By applying Banach and Schauder fixed-point principles, we derive results related to the existence and uniqueness of solutions. Additionally, we analyze generalized Ulam-Hyers stability under (p, q)-Gronwall inequality. Key results are supported with illustrative examples, demonstrating the applicability of the proposed framework. Compared to previous studies restricted to the standard q-calculus, the present work introduces the (p, q)-Caputo fractional difference setting, which offers a more flexible and generalized approach. This novelty extends existing results and provides new perspectives for the analysis of stability and solvability of fractional systems.

In this paper, we discuss existence and stability results for a new class of impulsive fractional boundary value problems with non-separated boundary conditions containing the Caputo proportional fractional derivative of a function with respect to another function. The uniqueness result is discussed via Banach's contraction mapping principle, and the existence of solutions is proved by using Schaefer's fixed point theorem. Furthermore, we utilize the theory of stability for presenting different kinds of Ulam's stability results of the proposed problem. Finally, an example is also constructed to demonstrate the application of the main results.

In the present paper we characterize, in terms of characters, additive functions and matrix elements of irreducible two-dimensional representations, the continuous solutions of some trigonometric functional equations for mappings defined on an arbitrary group and taking their values in a complex Hilbert space with the Hadamard product. In addition, we investigate a superstability result for these equations.

Superstability of a Pexider-type trigonometric functional equation in normed algebras

Corrigendum to "On trigonometric functional equations of rectangular type" in Aequationes Math. 53 (1997), pp. 36-53

On trigonometric functional equations of rectangular type