This paper investigates the Hyers-Ulam stability and superstability of the functional equation f(x2 + yf(z)) = xf(x) + zf(y) for real-valued functions f : R -> R on some restricted subsets of R.

In this paper, we introduce and solve the following additive (ρ 1, ρ 2)-functional inequalities: 1 $$\displaystyle \begin{aligned} \begin{array}{rcl} \left\|f\left(x-y\right) - f(x )+ f(y)\right\| &\displaystyle \ge &\displaystyle \|\rho_1 (f(x+y)-f(x)-f(y))\| \\ &\displaystyle + &\displaystyle \left\|\rho_2 \left( f(y-x)-f(y)+f(x)\right)\right\|, {} \end{array} \end{aligned} $$ where ρ 1 and ρ 2 are fixed complex numbers with |ρ 1| + |ρ 2| > 1, and 2 $$\displaystyle \begin{aligned} \begin{array}{rcl} \left\|f\left(x+y\right) - f(x )- f(y)\right\|&\displaystyle \ge &\displaystyle \|\rho_1 (f(x-y)-f(x)+f(y))\| \\ &\displaystyle + &\displaystyle \left\|\rho_2 \left( f(y-x)-f(y)+f(x)\right)\right\| ,{} \end{array} \end{aligned} $$ where ρ 1 and ρ 2 are fixed complex numbers with 1 + |ρ 1| > |ρ 2| > 1. Using the fixed point method and the direct method, we prove the Hyers–Ulam stability of the additive (ρ 1, ρ 2)-functional inequalities (2) and (1) in complex Banach spaces.

In (Park et al., Rocky Mountain J Math 49:593–607, 2019), Park introduced the following bi-additive s-functional inequality 1 $$\displaystyle \begin{aligned} & \| f(x+y, z-w) + f(x-y, z+w) -2f(x,z)+2 f(y, w)\| \\ & \quad \le \left \|s \left (2f\left (\frac {x+y}{2}, z-w\right ) + 2f\left (\frac {x-y}{2}, z+w\right ) - 2f(x,z )+ 2 f(y, w)\right )\right \|,{} \end{aligned} $$ where s is a fixed nonzero complex number with |s| < 1. Using the fixed point method, we prove the Hyers–Ulam stability of ternary biderivations and ternary bihomomorphism in C∗-ternary algebras, associated with the bi-additive s-functional inequality (1).

In this paper, we apply the well-known aggregation mappings on Mittag-Leffler-type functions to investigating new approximation error estimates of a W-Hilfer fractional differential equation, by a different concept of Ulam-type stability in both bounded and unbounded domains.

In this paper, we apply some special functions to introduce a new class of control functions that help us define the concept of multi-stability. Further, we investigate the multi-stability of homomorphisms in $C^{*}$ -algebras and Lie $C^{*}$ -algebras, multi-stability of derivations in $C^{*}$ -algebras, and Lie $C^{*}$ -algebras for the following random operator equation via fixed point methods: $$ \mu f \biggl(\eth , \frac{x+y}{2} \biggr) + \mu f \biggl(\eth , \frac{x-y}{2} \biggr) = f(\eth , \mu x) . $$ In particular, for $\mu = 1$ , the above equation turns out to be Jensen’s random operator equation.

It is Euler who started a new field in mathematics, now known as variational analysis. Euler in 1744 deduced the first general rule, now known as Euler's differential equation, for the characterization of the maximizing or minimizing arcs. The years 1929-1936 proved to be very essential for the development of the calculus of variations in the large of integrals depending on a curve. One of the most important questions concerning the problem of Plateau, and especially the existence and non-uniqueness theory of the problem is the one studied by Morse and Tompkins and Shiffman. Their main theorem is stated as: If r is a Jordan curve in R3 which bounds two minimal surfaces which are proper relative minima, it bounds at least one minimal surface which is not a proper relative minimum. This chapter applies the Morse-Smale Index Theorem to the problem of a global analysis of variational problems in several variables.

In the present paper we establish a few equivalent conditions of a Hilbert-type integral inequality with a non-homogeneous kernel in the whole plane. A few equivalent conditions of a Hilbert-type integral inequality with the homogeneous kernel in the whole plane are deduced, in the form of applications. We additionally consider operator expressions and several interesting particular cases.

In this paper we will study Hyers-Ulam stability for a general linear partial differential equation of first order in a Banach space.

Series on Computers and Operations ResearchAnalysis, Geometry, Nonlinear Optimization and Applications, pp. 625-639 (2023) No AccessChapter 23: Stability of Some Functional Equations on Restricted DomainsAbbas Najati, Mohammad B. Moghimi, Batool Noori, and Themistocles M. RassiasAbbas NajatiDepartment of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil, Iran, Mohammad B. MoghimiDepartment of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil, Iran, Batool NooriDepartment of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil, Iran, and Themistocles M. RassiasDepartment of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greecehttps://doi.org/10.1142/9789811261572_0023Cited by:0 PreviousNext AboutSectionsPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail Abstract: In this work, we investigate the Hyers–Ulam stability for some functional equations on restricted domains. As a consequence, we obtain asymptotic behaviors of these functional equations. FiguresReferencesRelatedDetails Analysis, Geometry, Nonlinear Optimization and ApplicationsMetrics History PDF download

In this paper we first introduce a new concept of a functional equation called multi-Drygas equation . We deal with the generalized hyperstability results of the multi-Drygas functional equation on a restricted domain by applying the Brzdȩk’s fixed point theorem (Brzdȩk et al. in Nonlinear Anal. 74 : 6728–6732, 2011, Theorem 1). Our main results improve and generalize results obtained in Aiemsombonn and Sintunavarat (Bull Aust Math Soc 92: 269–280, 2016), El-Fassi(J Fixed Point Theory Appl 9: 2529–2540, 2017), Piszczek, Szczawińska(J Funct Spaces Appl 2013: 912718, 2013) . Some applications of our results are also provided.