The main goal of this paper, is to obtain the forms of the solutions of the following nonlinear fifteenth-order difference equations \[ x_{n+1}=\frac{x_{n-14}}{\pm 1\pm x_{n-2}x_{n-5}x_{n-8}x_{n-11}x_{n-14}},\ \ \ \ n=0,1,2,\ldots,\] where the initial conditions \(x_{-14},x_{-13},\ldots,x_{0}\) are arbitrary real numbers. Moreover, we investigate stability, boundedness, oscillation and the periodic character of these solutions. Finally, we confirm the results with some numerical examples and graphs by using Matlab program.

In this paper, we investigate the behavior of solutions of the difference equation \[ x_{n+1}=\frac{\alpha \left( x_{n-2}+x_{n-3}\right) +\left( \alpha -1\right) x_{n-2}x_{n-3}}{x_{n-2}x_{n-3}+\alpha },\;\ \ \ n=0,1,2,\ldots, \] where the initial conditions \(x_{-3},x_{-2},x_{-1},x_{0}\) are arbitrary non-negative real numbers and the parameter \(\alpha \in \lbrack 1,\infty ).\) More precisely, we study the boundedness, stability, and oscillation of the solutions of this equation.

Institute of Mathematics of the Czech Academy of Sciences

In this paper, we investigate the behavior of solutions of the difference equation \[ x_{n+1}=\frac{\alpha \left( x_{n-1}+x_{n-2}\right) +\left( \alpha -1\right) x_{n-1}x_{n-2}}{x_{n-1}x_{n-2}+\alpha },\;\ \ \ n=0,1,2,\ldots, \] where the initial conditions \(x_{-2},x_{-1},x_{0}\) are arbitrary non-negative real numbers and the parameter \(\alpha \in \lbrack 1,\infty ).\) More precisely, we study the boundedness, stability, and oscillation of the solutions of this equation.

The aim of this work is to obtain the forms of the solutions of the following nonlinear eighteenth-order difference equations \[ x_{n+1}=\frac{x_{n-17}}{\pm 1\pm x_{n-2}x_{n-5}x_{n-8}x_{n-11}x_{n-14}x_{n-17}},\ \ \ \ n=0,1,2,\ldots, \] where the initial conditions \(x_{-17},x_{-16},\ldots,x_{0}\) are arbitrary real numbers. Moreover, we investigate stability, boundedness, oscillation, and the periodic character of these solutions. Finally, we confirm the results with some numerical examples and graphs by using Matlab program.

The purpose of the paper is to introduce and study a new class of operators on semi-Hilbertian spaces i.e.; spaces generated by positive semidefinite sesquilinear forms. Let H be a Hilbert space and let A be a positive bounded operator on H .