Solutions for Strongly Monotone Operator Equations in Riesz Spaces

Abstract

This paper is devoted to the study of solutions for a class of operator equations governed by strongly monotone operators on real Riesz spaces continuously embedded into Banach lattices. By exploiting the intrinsic lattice structure of Banach lattices, we establish refined growth assumptions on nonlinear terms to guarantee that suitable neighbourhoods of positive and negative cones are invariant under the descending flow. Combining the descending flow invariant set technique with the theory of strongly monotone operators, we derive abstract existence theorems: the operator equation possesses at least one positive solution, one negative solution and one sign-changing solution. These abstract results are further applied to \((p,q)\)-Laplacian boundary value problems, yielding corresponding multiplicity conclusions on positive, negative and sign-changing solutions.