A strong maximum principle in quantum calculus

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Abstract

In this paper, we investigate qualitative properties of solutions to second-order q-difference inequalities within the framework of quantum calculus. We establish a strong maximum principle on a nonuniform q-grid and show that an interior maximum forces the solution to be constant. As a consequence, we derive a comparison principle and a uniqueness result for linear boundary value problems. In addition, we construct explicit barriers in the constant-coefficient case and obtain pointwise bounds along the grid. The resulting bounds are used to prove a right-hand Hermite–Hadamard-type inequality.

Keywords

quantum calculus / q-difference inequalities / maximum principle / comparison principle / Hermite–Hadamard-type inequality

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