文摘
Due to its numerous applications, existence of positive solutions for the algebraic system <span id="mmlsi1" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302543&_mathId=si1.gif&_user=111111111&_pii=S0893965916302543&_rdoc=1&_issn=08939659&md5=8ff080ccaac3476d94a60a7e1d05fb0a" title="Click to view the MathML source">x=GF(x)span><span class="mathContainer hidden"><span class="mathCode">span>span>span> has been extensively studied, where <span id="mmlsi2" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302543&_mathId=si2.gif&_user=111111111&_pii=S0893965916302543&_rdoc=1&_issn=08939659&md5=b4f31805feb4ab99ef94d72f85c0087c" title="Click to view the MathML source">Gspan><span class="mathContainer hidden"><span class="mathCode">span>span>span> is the coefficient matrix and <span id="mmlsi3" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302543&_mathId=si3.gif&_user=111111111&_pii=S0893965916302543&_rdoc=1&_issn=08939659&md5=584188725b09b2ed9056d64638f5f32d" title="Click to view the MathML source">F:R<sup>nsup>→R<sup>nsup>span><span class="mathContainer hidden"><span class="mathCode">span>span>span> is nonlinear. However, all results require the matrix <span id="mmlsi2" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302543&_mathId=si2.gif&_user=111111111&_pii=S0893965916302543&_rdoc=1&_issn=08939659&md5=b4f31805feb4ab99ef94d72f85c0087c" title="Click to view the MathML source">Gspan><span class="mathContainer hidden"><span class="mathCode">span>span>span> to be positive. When <span id="mmlsi2" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302543&_mathId=si2.gif&_user=111111111&_pii=S0893965916302543&_rdoc=1&_issn=08939659&md5=b4f31805feb4ab99ef94d72f85c0087c" title="Click to view the MathML source">Gspan><span class="mathContainer hidden"><span class="mathCode">span>span>span> contains a zero-element, positive solutions have not been proved because of the difficulties of cone construction. In the present paper, an existence result is obtained for nonnegative <span id="mmlsi2" class="mathmlsrc"><span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302543&_mathId=si2.gif&_user=111111111&_pii=S0893965916302543&_rdoc=1&_issn=08939659&md5=b4f31805feb4ab99ef94d72f85c0087c" title="Click to view the MathML source">Gspan><span class="mathContainer hidden"><span class="mathCode">span>span>span> by introducing a new cone. To show applications of the theorem, two explanatory examples are given. The new result can be naturally extended to some more general systems. In particular, the system can be transformed into an operator equation on a Banach space. Thus, the new method also provides a novel idea for operator equations.