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Báo cáo hoa học: " Research Article Summation Characterization of the Recessive Solution for Half-Linear Difference Equations"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Summation Characterization of the Recessive Solution for Half-Linear Difference Equations | Hindawi Publishing Corporation Advances in Difference Equations Volume 2009 Article ID 52i058 16 pages doi 10.1155 2009 521058 Research Article Summation Characterization of the Recessive Solution for Half-Linear Difference Equations Ondrej Dosly1 and Simona Fisnarova2 1 Department of Mathematics and Statistics Masaryk University Kotlarska 2 611 37 Brno Czech Republic 2 Department of Mathematics Mendel University of Agriculture and Forestry in Brno Zemẽdẽlskú 1 613 00 Brno Czech Republic Correspondence should be addressed to Ondrej Dosly dosly@math.muni.cz Received 24 June 2009 Accepted 24 August 2009 Recommended by Martin J. Bohner We show that the recessive solution of the second-order half-linear difference equation A rk Axk Ck xfc 1 0 x x p-2x p 1 where r c are real-valued sequences is closely related to the divergence of the infinite series 2 rkXfcXfc 1 Axfc p-2 . Copyright 2009 O. Dosly and S. Fisnarova. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction We consider the second-order half-linear difference equation A rfcO Axk CkO xfc 1 0 O x x p 2x p 1 1.1 where r c are real-valued sequences and rk 0 and we investigate properties of its recessive solution. Qualitative theory of 1.1 was established in the series of the papers of Rehak 1-5 and it is summarized in 6 Chapter 3 . It was shown there that the oscillation theory of 1.1 is very similar to that of the linear equation A rk Axk CkXk 1 0 1.2 which is the special case p 2 in 1.1 . We will recall basic facts of the oscillation theory of 1.1 in the following section. 2 Advances in Difference Equations The concept of the recessive solution of 1.1 has been introduced in 7 . There are several attempts in literature to find a summation characterization of this solution see 8 and also related references 9 10 which are based on the asymptotic