An ecological generalized model with r-Kirchhoff-type problem

Abstract In the present paper we investigate the existence of solutions for an ecological generalized model with r -Kirchhoff problem via the method of sub-supersolutions in $$\zeta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ζ</mml:mi> </mml:math> -generalized space $$\mathcal {H}^{\alpha , \eta ;\zeta }_{r}(\Lambda )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msubsup> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mi>r</mml:mi> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> <mml:mi>η</mml:mi> <mml:mo>;</mml:mo> <mml:mi>ζ</mml:mi> </mml:mrow> </mml:msubsup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Λ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . We also discuss some special cases.