In this paper, we study the reciprocal sums of products of Fibonacci and Lucas numbers. Some identities are obtained related to the numbers ??,k=n 1/FkLk+m and ??,k=n 1/LkFk+m, m ? 0.

In this paper, we will apply the operator method to prove the generalized Hyers-Ulam stability of the wave equation, u t t ( x , t ) − c 2 ▵ u ( x , t ) = f ( x , t ) , for a class of real-valued functions with continuous second partial derivatives. Finally, we will discuss the stability more explicitly by giving examples.

We will prove the generalized Hyers–Ulam stability of isometries, with a focus on the stability for restricted domains. More precisely, we prove the generalized Hyers–Ulam stability of the orthogonality equation and we use this result to prove the stability of the equations $\parallel f\left(x\right)-f\left(y\right)\parallel =\parallel x-y\parallel$ and ${\parallel f\left(x\right)-f\left(y\right)\parallel }^2={\parallel x-y\parallel }^2$ on the restricted domains. As we can easily see, these functional equations are symmetric in the sense that they become the same equations even if the roles of variables x and y are exchanged.

The question of whether there is a true isometry approximating the ε -isometry defined in the bounded subset of the n -dimensional Euclidean space has long been considered an interesting question. In 1982, Fickett published the first article on this topic, and in early 2000, Alestalo et al. and Väisälä improved Fickett’s result significantly. Recently, the second author of this article published a paper improving the previous results. The main purpose of this article is to significantly improve all of the aforementioned results by applying a basic and intuitive method.

The question of whether there is a true isometry that approximates the $\epsilon$-isometry defined on a bounded set has long interested mathematicians. The first paper on this topic was published by Fickett, whose result was subsequently greatly improved by Alestalo et al., Väisälä and Vestfrid. Recently, the authors published some papers improving the previous results. The main purpose of this paper is to improve all of the abovementioned results by utilizing the properties of the norm and inner product for Euclidean space.

In Applied Mathematics Letters 74 (2017), 147-153, the Hyers-Ulam stability of the one-dimensional time-independent Schrodinger equation was investigated when the relevant system has a potential well of finite depth. As a continuous work, we prove in this paper a type of Hyers-Ulam stability of the one-dimensional time-independent Schrodinger equation with a rectangular potential barrier of V-0 in height and 2c in width, where V-0 is assumed to be greater than the energy E of the particle under consideration.