<?xml version="1.0" encoding="UTF-8"?><xml><records><record><source-app name="Biblio" version="7.x">Drupal-Biblio</source-app><ref-type>17</ref-type><contributors><authors><author><style face="normal" font="default" size="100%">PUSHP R. AWASTHI</style></author><author><style face="normal" font="default" size="100%">LYNN H. ERBE</style></author><author><style face="normal" font="default" size="100%">ALLAN C. PETERSON</style></author></authors></contributors><titles><title><style face="normal" font="default" size="100%">EXISTENCE AND UNIQUENESS RESULTS FOR POSITIVE SOLUTIONS OF A NONLINEAR FRACTIONAL DIFFERENCE EQUATION</style></title></titles><keywords><keyword><style  face="normal" font="default" size="100%">39A10</style></keyword></keywords><dates><year><style  face="normal" font="default" size="100%">2015</style></year><pub-dates><date><style  face="normal" font="default" size="100%">2015</style></date></pub-dates></dates><urls><web-urls><url><style face="normal" font="default" size="100%">http://www.acadsol.eu/en/articles/19/1/5.pdf</style></url></web-urls></urls><volume><style face="normal" font="default" size="100%">19</style></volume><pages><style face="normal" font="default" size="100%">18</style></pages><language><style face="normal" font="default" size="100%">eng</style></language><abstract><style face="normal" font="default" size="100%">&lt;p&gt;In this paper we are concerned with the fractional self-adjoint equation $${∆^µ_{µ−1} (p∆x)(t) + q(t + µ − 1)x(t + µ − 1) = h(t), \ \ \ t ∈ \mathbb{N}_0, \ \ \ \&amp;nbsp; (0.1) }$$ where ${ 0 &amp;lt; µ ≤ 1, \ p : \mathbb{N}_{µ−1} → (0, ∞), \ q : \mathbb{N}_{µ−1} → [0, ∞), \&amp;nbsp; h : \mathbb{N}_0 → \mathbb{R} }$. Our sole reason for calling this equation self-adjoint is that when ${µ = 1}$&amp;nbsp; we get the well-studied second order self-adjoint difference equation [16, Chapters 6-9]. We will prove various results concerning the existence and uniqueness of positive solutions of the nonlinear fractional equation$${ ∆^µ_{µ−1} (p∆x)(t) + F(t, x(t + µ − 1)) = 0, \ t ∈ \mathbb{N}_0, \ \ \ \ \ \ &amp;nbsp;\ (0.2) }$$ where ${ 0 &amp;lt; µ ≤ 1, \ p : \mathbb{N}_{µ−1} → (0, ∞)}$, and ${F : \mathbb{N}_0 × \mathbb{R} → [0, ∞) }$ by applying the Contraction Mapping Theorem. We also give examples illustrating our main results.&lt;/p&gt;
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