Let a, b, c be fixed positive integers satisfying a2 + ab + b2 = c with gcd(a, b) = 1. We show that the Diophantine equation a2x+axby+b2y = cz has only the positive integer solution (x, y, z) = (1, 1, 1) under some conditions. The proof is based on elementary methods and Cohn’s ones concerning the Diophantine equation x2 + C = yn. Mathematics Subject Classification: 11D61

A mathematical model of an adaptive SamuelMarshall type single component market described by quasilinear functional differential equations with dependent on phase coordinates and frequently switched an ergodic Markov process is presented. The proposed method is based on an averaging procedure with respect to time along the critical solutions of the generative average linear equation and with re...

The fractional Zener constitutive law is frequently used as a model of solid-like viscoelastic behavior. In this work, class linear models type, which generalize the model, studied by use Bernstein functions technique. We prove that corresponding relaxation moduli are completely monotone under appropriate thermodynamic restrictions on parameters. Based property, we study propagation function an...

the main purpose of this paper is to establish some new results onthe superstability and stability via a fixed point approach forthe pexiderized exponential equation, i.e.,$$|f(x+y)-g(x)h(y)|leq psi(x,y),$$where $f$, $g$ and $h$ are three functions from an arbitrarycommutative semigroup $s$ to an arbitrary unitary complex banachalgebra and also $psi: s^{2}rightarrow [0,infty)$ is afunction. fur...

In this paper, we have characterized the nature and form of solutions following nonlinear delay-differential equation: $$f^{n}(z)+\sum_{i=1}^{n-1}b_{i}f^{i}(z)+q(z)e^{Q(z)}L(z,f)=P(z),$$ where $b_i\in\mathbb{C}$, $L(z,f)$ are a linear polynomial $f$; $n$ is positive integers; $q$, $Q$ $P$ respectively nonzero, nonconstant any polynomials. Different special cases our result will accommodate all ...

In this paper a Constitutive equation to dcscribe the mechanical behaviour of materials, reinforced with unidirectional fibres, is presented. The material behaviour of both matrix and fibres may be viscoelastic. The constitutive equation is a linear relation between the second Piola-Kirchhoff stress tensor and the Green-Lagrange strain tensor. The effective relaxation functions in the constitut...

In this paper, a nonlinear constitutive law and a curve fitting, two relationships between the stress-strain and the shear stress-strain for sandstone material were used to obtain a second-order polynomial constitutive equation. Based on the established polynomial constitutive equations and Newton’s second law, a mathematical model of the non-homogeneous nonlinear wave equation under an externa...

In this paper, we study the existence and regularity of positive solution for an elliptic system on a bounded and regular domain. The non linearities in this equation are functions of Caratheodory type satisfying some exponential growth conditions.