In this paper, we prove the generalized Hyers-Ulam-Rassias stability for the Drygas functional equation$$f(x+y)+f(x-y)=2f(x)+f(y)+f(-y)$$ in Banach spaces by using the Brzc{d}ek's fixed point theorem. Moreover, we give a general result on the hyperstability of this equation. Our results are improvements and generalizations of the main result of M. Piszczek and J. Szczawi'{n}ska [21].

Winding numbers are fundamental objects arising in algebraic topology, with many applications in non-constructive complex analysis. We present a formalization in Coq of the winding numbers and their main properties. As an application of this development, we also give non-constructive proofs of the following theorems: the Fundamental Theorem of Algebra, the 2-dimensional Brouwer Fixed-Point theo...

In this paper we study Hyers-Ulam stability of Euler’s equation in the calculus variations two special cases: when F=F(x,y′) and F=F(y,y′). For first case use direct method for second Laplace transform. Theorem Example corresponding estimations Jyx−Jy0x are given. We mention that it is time problem Ulam-stability extremals functionals represented integral form studied.

In this paper, we study the maximal ideals in a commutative ring of bicomplex numbers and then describe algebra. We found that kernel nonzero multiplicative BC-linear functional Banach algebra need not be ideal. Finally, introduce notion division generalize Gelfand–Mazur theorem for

The purpose of this paper is to present some coupled fixed point results on a metric space endowed with two $b$-metrics. We shall apply a fixed point theorem for an appropriate operator on the Cartesian product of the given spaces endowed with directed graphs. Data dependence, well-posedness and Ulam-Hyers stability are also studied. The results obtained here will be applied to prove the existe...

The fixed point method, which is the second most popular technique of proving the Hyers–Ulam stability of functional equations, was used for the first time in 1991 by J.A. Baker who applied a variant of Banach’s fixed point theorem to obtain the stability of a functional equation in a single variable. However, most authors follow Radu’s approach and make use of a theorem of Diaz and Margolis. T...

The stability problem of functional equations started with the question concerning stability of group homomorphisms proposed by Ulam 1 during a talk before a Mathematical Colloquium at the University of Wisconsin, Madison. In 1941, Hyers 2 gave a partial solution of Ulam’s problem for the case of approximate additive mappings in the context of Banach spaces. In 1978, Rassias 3 generalized the t...

In 1940, Ulam [1] proposed the famous Ulam stability problem of linear mappings. In 1941, Hyers [2] considered the case of approximately additive mappings f : E→ E′, where E and E′ are Banach spaces and f satisfies Hyers inequality ‖ f (x+ y)− f (x)− f (y)‖ ≤ ε for all x, y ∈ E. It was shown that the limit L(x) = limn→∞ 2−n f (2nx) exists for all x ∈ E and that L : E→ E′ is the unique additive ...

The existence of non-trivial solutions X to matrix equations of the form F (X,A1,A2, · · · ,As) = G(X,A1,A2, · · · ,As) over the real numbers is investigated. Here F and G denote monomials in the (n × n)-matrix X = (xij) of variables together with (n × n)-matrices A1,A2, · · · ,As for s ≥ 1 and n ≥ 2 such that F and G have different total positive degrees in X. An example with s = 1 is given by...