On a generalisation of monotonic sequences

E. T. Copson

doi:10.1017/S0013091500009433

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A bounded monotonic sequence is convergent. Dr J. M. Whittaker recently suggested to me a generalisation of this result, that, if a bounded sequence {an} of real numbers satisfies the inequality

then it is convergent. This I was able to prove by considering the corresponding difference equation

Crossref Citations

This article has been cited by the following publications. This list is generated based on data provided by Crossref.

Borwein, D. 1972. Convergence criteria for bounded sequences. Proceedings of the Edinburgh Mathematical Society, Vol. 18, Issue. 2, p. 99.

Russell, Dennis C. 1974. On bounded sequences satisfying a linear inequality. Proceedings of the Edinburgh Mathematical Society, Vol. 19, Issue. 1, p. 11.

Bibby, John 1974. Axiomatisations of the average and a further generalisation of monotonic sequences. Glasgow Mathematical Journal, Vol. 15, Issue. 1, p. 63.

Russell, Dennis C. 1987. General Inequalities 5. p. 103.

Essén, Matts 1987. General Inequalities 5. p. 19.

Rödler, Richard 1998. A convergence theorem for a special class of stochastic processes. Stochastic Analysis and Applications, Vol. 16, Issue. 1, p. 153.

Karakostas, G.L. and Stević†, S. 2004. Slowly Varying Solutions of the Difference Equation. Journal of Difference Equations and Applications, Vol. 10, Issue. 3, p. 249.

Stevic, Stevo 2006. Asymptotic behavior of a class of nonlinear difference equations. Discrete Dynamics in Nature and Society, Vol. 2006, Issue. , p. 1.

Iricanin, Bratislav D. 2007. A Global Convergence Result for a Higher Order Difference Equation. Discrete Dynamics in Nature and Society, Vol. 2007, Issue. , p. 1.

Stevic, Stevo 2007. Asymptotic Periodicity of a Higher-Order Difference Equation. Discrete Dynamics in Nature and Society, Vol. 2007, Issue. , p. 1.

Xi, Hongjian Sun, Taixiang Yu, Weiyong and Zhao, Jinfeng 2009. On Boundedness of Solutions of the Difference Equation xn+1=(pxn+qxn−1)/(1+xn) for q>1+p>1. Advances in Difference Equations, Vol. 2009, Issue. , p. 1.

Sun, Taixiang Qin, Bin Xi, Hongjian and Han, Caihong 2009. Global Behavior of the Max-Type Difference Equationxn+1=max⁡{1/xn,An/xn−1}. Abstract and Applied Analysis, Vol. 2009, Issue. , p. 1.

Iričanin, Bratislav D. 2010. Global stability of some classes of higher-order nonlinear difference equations. Applied Mathematics and Computation, Vol. 216, Issue. 4, p. 1325.

Berg, Lothar and Stević, Stevo 2011. On some systems of difference equations. Applied Mathematics and Computation, Vol. 218, Issue. 5, p. 1713.

Altun, Muhammed 2011. On the Characterization of a Class of Difference Equations. Discrete Dynamics in Nature and Society, Vol. 2011, Issue. , p. 1.

Stević, Stevo 2011. On the difference equation xn=xn−2/(bn+cnxn−1xn−2). Applied Mathematics and Computation, Vol. 218, Issue. 8, p. 4507.

Sun, Taixiang Xi, Hongjian and He, Qiuli 2011. On boundedness of the difference equation xn+1=pn+xn-3s+1xn-s+1 with period-k coefficients. Applied Mathematics and Computation, Vol. 217, Issue. 12, p. 5994.

Stević, Stevo 2012. On the difference equation. Applied Mathematics and Computation, Vol. 218, Issue. 11, p. 6291.

Stević, Stevo 2012. On some solvable systems of difference equations. Applied Mathematics and Computation, Vol. 218, Issue. 9, p. 5010.

Stević, Stevo 2012. On some linear difference equations and inequalities with continuous argument. Applied Mathematics and Computation, Vol. 218, Issue. 19, p. 9831.

Download full list

Article contents

On a generalisation of monotonic sequences

Extract

Save article to Kindle

Save article to Dropbox

Save article to Google Drive

Reply to: Submit a response

Your details

You have entered the maximum number of contributors

Conflicting interests