Decreasing solutions of cyclic second-order difference systems

The existence and asymptotic behavior of positive decreasing solutions to the cyclic second-order nonlinear difference system <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="block"> <a:mi mathvariant="normal">Δ</a:mi> <a:mo stretchy="false">(</a:mo> <a:msub> <a:mi>p</a:mi> <a:mi>i</a:mi> </a:msub> <a:mo stretchy="false">(</a:mo> <a:mi>n</a:mi> <a:mo stretchy="false">)</a:mo> <a:mrow class="MJX-TeXAtom-ORD"> <a:mo stretchy="false">|</a:mo> </a:mrow> <a:mi mathvariant="normal">Δ</a:mi> <a:msub> <a:mi>x</a:mi> <a:mi>i</a:mi> </a:msub> <a:mo stretchy="false">(</a:mo> <a:mi>n</a:mi> <a:mo stretchy="false">)</a:mo> <a:msup> <a:mrow class="MJX-TeXAtom-ORD"> <a:mo stretchy="false">|</a:mo> </a:mrow> <a:mrow class="MJX-TeXAtom-ORD"> <a:msub> <a:mi>α</a:mi> <a:mi>i</a:mi> </a:msub> <a:mo>−</a:mo> <a:mn>1</a:mn> </a:mrow> </a:msup> <a:mi mathvariant="normal">Δ</a:mi> <a:msub> <a:mi>x</a:mi> <a:mi>i</a:mi> </a:msub> <a:mo stretchy="false">(</a:mo> <a:mi>n</a:mi> <a:mo stretchy="false">)</a:mo> <a:mo stretchy="false">)</a:mo> <a:mo>=</a:mo> <a:msub> <a:mi>q</a:mi> <a:mi>i</a:mi> </a:msub> <a:mo stretchy="false">(</a:mo> <a:mi>n</a:mi> <a:mo stretchy="false">)</a:mo> <a:mrow class="MJX-TeXAtom-ORD"> <a:mo stretchy="false">|</a:mo> </a:mrow> <a:msub> <a:mi>x</a:mi> <a:mrow class="MJX-TeXAtom-ORD"> <a:mi>i</a:mi> <a:mo>+</a:mo> <a:mn>1</a:mn> </a:mrow> </a:msub> <a:mo stretchy="false">(</a:mo> <a:mi>n</a:mi> <a:mo>+</a:mo> <a:mn>1</a:mn> <a:mo stretchy="false">)</a:mo> <a:msup> <a:mrow class="MJX-TeXAtom-ORD"> <a:mo stretchy="false">|</a:mo> </a:mrow> <a:mrow class="MJX-TeXAtom-ORD"> <a:msub> <a:mi>β</a:mi> <a:mi>i</a:mi> </a:msub> <a:mo>−</a:mo> <a:mn>1</a:mn> </a:mrow> </a:msup> <a:msub> <a:mi>x</a:mi> <a:mrow class="MJX-TeXAtom-ORD"> <a:mi>i</a:mi> <a:mo>+</a:mo> <a:mn>1</a:mn> </a:mrow> </a:msub> <a:mo stretchy="false">(</a:mo> <a:mi>n</a:mi> <a:mo>+</a:mo> <a:mn>1</a:mn> <a:mo stretchy="false">)</a:mo> <a:mo>,</a:mo> <a:mspace width="1em"/> <a:mi>i</a:mi> <a:mo>=</a:mo> <a:mover> <a:mrow> <a:mn>1</a:mn> <a:mo>,</a:mo> <a:mi>N</a:mi> </a:mrow> <a:mo accent="false">¯</a:mo> </a:mover> <a:mo>,</a:mo> </a:math> are studied, where <hb:math xmlns:hb="http://www.w3.org/1998/Math/MathML"> <hb:mspace width="thinmathspace"/> <hb:mspace width="thinmathspace"/> <hb:msub> <hb:mi>x</hb:mi> <hb:mrow class="MJX-TeXAtom-ORD"> <hb:mi>N</hb:mi> <hb:mo>+</hb:mo> <hb:mn>1</hb:mn> </hb:mrow> </hb:msub> <hb:mo>=</hb:mo> <hb:msub> <hb:mi>x</hb:mi> <hb:mn>1</hb:mn> </hb:msub> <hb:mo>,</hb:mo> </hb:math> <lb:math xmlns:lb="http://www.w3.org/1998/Math/MathML"> <lb:mi>i</lb:mi> <lb:mo>=</lb:mo> <lb:mover> <lb:mrow> <lb:mn>1</lb:mn> <lb:mo>,</lb:mo> <lb:mi>N</lb:mi> </lb:mrow> <lb:mo accent="false">¯</lb:mo> </lb:mover> </lb:math> and <nb:math xmlns:nb="http://www.w3.org/1998/Math/MathML"> <nb:msub> <nb:mi>q</nb:mi> <nb:mi>i</nb:mi> </nb:msub> <nb:mo>=</nb:mo> <nb:mo fence="false" stretchy="false">{</nb:mo> <nb:msub> <nb:mi>q</nb:mi> <nb:mi>i</nb:mi> </nb:msub> <nb:mo stretchy="false">(</nb:mo> <nb:mi>n</nb:mi> <nb:mo stretchy="false">)</nb:mo> <nb:mo fence="false" stretchy="false">}</nb:mo> </nb:math> are positive real sequences, and the constants <ub:math xmlns:ub="http://www.w3.org/1998/Math/MathML"> <ub:msub> <ub:mi>α</ub:mi> <ub:mi>i</ub:mi> </ub:msub> </ub:math> and <vb:math xmlns:vb="http://www.w3.org/1998/Math/MathML"> <vb:msub> <vb:mi>β</vb:mi> <vb:mi>i</vb:mi> </vb:msub> <vb:mo>,</vb:mo> </vb:math> <wb:math xmlns:wb="http://www.w3.org/1998/Math/MathML"> <wb:mi>i</wb:mi> <wb:mo>=</wb:mo> <wb:mover> <wb:mrow> <wb:mn>1</wb:mn> <wb:mo>,</wb:mo> <wb:mi>N</wb:mi> </wb:mrow> <wb:mo accent="false">¯</wb:mo> </wb:mover> </wb:math> are positive and satisfy the sublinear condition <yb:math xmlns:yb="http://www.w3.org/1998/Math/MathML"> <yb:msub> <yb:mi>α</yb:mi> <yb:mn>1</yb:mn> </yb:msub> <yb:msub> <yb:mi>α</yb:mi> <yb:mn>2</yb:mn> </yb:msub> <yb:mo>⋅</yb:mo> <yb:mo>…</yb:mo> <yb:mo>⋅</yb:mo> <yb:msub> <yb:mi>α</yb:mi> <yb:mi>N</yb:mi> </yb:msub> <yb:mo>></yb:mo> <yb:msub> <yb:mi>β</yb:mi> <yb:mn>1</yb:mn> </yb:msub> <yb:msub> <yb:mi>β</yb:mi> <yb:mn>2</yb:mn> </yb:msub> <yb:mo>⋅</yb:mo> <yb:mo>…</yb:mo> <yb:mo>⋅</yb:mo> <yb:msub> <yb:mi>β</yb:mi> <yb:mi>N</yb:mi> </yb:msub> <yb:mo>.</yb:mo> </yb:math> Two distinct types of positive decreasing solutions are considered, depending on whether the series <zb:math xmlns:zb="http://www.w3.org/1998/Math/MathML"> <zb:munderover> <zb:mo>∑</zb:mo> <zb:mrow class="MJX-TeXAtom-ORD"> <zb:mi>n</zb:mi> <zb:mo>=</zb:mo> <zb:mn>1</zb:mn> </zb:mrow> <zb:mi mathvariant="normal">∞</zb:mi> </zb:munderover> <zb:mrow class="MJX-TeXAtom-ORD"> <zb:msub> <zb:mi>p</zb:mi> <zb:mi>i</zb:mi> </zb:msub> <zb:mo stretchy="false">(</zb:mo> <zb:mi>n</zb:mi> <zb:msup> <zb:mo stretchy="false">)</zb:mo> <zb:mrow class="MJX-TeXAtom-ORD"> <zb:mo>−</zb:mo> <zb:mn>1</zb:mn> <zb:mrow class="MJX-TeXAtom-ORD"> <zb:mo>/</zb:mo> </zb:mrow> <zb:msub> <zb:mi>α</zb:mi> <zb:mi>i</zb:mi> </zb:msub> </zb:mrow> </zb:msup> </zb:mrow> </zb:math> is divergent or convergent. In the first case, necessary and sufficient conditions for the existence of solutions tending to a positive constant as well as solutions tending to zero, while their associated quasi-differences approach a nonzero limit, are rigorously derived using fixed point techniques. In the second case, the analysis is focused on solutions whose components and quasi-differences both tend to zero. Under the additional assumption that the coefficient sequences are regularly varying, necessary and sufficient conditions for the existence of such solutions are obtained, and their precise asymptotic behavior is determined using the theory of discrete regular variation.