Positive solutions of boundary value problem for singular positone and semi-positone third-order difference equations

<p>Abstract</p> <p>This article studies the boundary value problems for the third-order nonlinear singular difference equations</p> <p><display-formula><m:math name="1687-1847-2011-38-i1" xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mrow> <m:msup> <m:mrow> <m:mi>&#916;</m:mi> </m:mrow> <m:mrow> <m:mstyle class="text"> <m:mtext class="textsf" mathvariant="sans-serif">3</m:mtext> </m:mstyle> </m:mrow> </m:msup> <m:mi>u</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo class="MathClass-bin">-</m:mo> <m:mstyle class="text"> <m:mtext class="textsf" mathvariant="sans-serif">2</m:mtext> </m:mstyle> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-bin">+</m:mo> <m:mi>&#955;</m:mi> <m:mi>a</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>i</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mi>f</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo class="MathClass-punc">,</m:mo> <m:mi>u</m:mi> <m:mrow> <m:mo class="MathClass-open">(</m:mo> <m:mrow> <m:mi>i</m:mi> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> </m:mrow> <m:mo class="MathClass-close">)</m:mo> </m:mrow> <m:mo class="MathClass-rel">=</m:mo> <m:mn>0</m:mn> <m:mo class="MathClass-punc">,</m:mo> <m:mspace width="1em" class="quad"/> <m:mi>i</m:mi> <m:mo class="MathClass-rel">&#8712;</m:mo> <m:mrow> <m:mo class="MathClass-open">[</m:mo> <m:mrow> <m:mstyle class="text"> <m:mtext class="textsf" mathvariant="sans-serif">2</m:mtext> </m:mstyle> <m:mo class="MathClass-punc">,</m:mo> <m:mi>T</m:mi> <m:mo class="MathClass-bin">+</m:mo> <m:mstyle class="text"> <m:mtext class="textsf" mathvariant="sans-serif">2</m:mtext> </m:mstyle> </m:mrow> <m:mo class="MathClass-close">]</m:mo> </m:mrow> <m:mo class="MathClass-punc">,</m:mo> </m:mrow> </m:math> </display-formula></p> <p>satisfying five kinds of different boundary value conditions. This article shows the existence of positive solutions for positone and semi-positone type. The nonlinear term may be singular. Two examples are also given to illustrate the main results. The arguments are based upon fixed point theorems in a cone.</p> <p><b>M</b>SC [2008]: 34B15; 39A10.</p