A strong maximum principle in quantum calculus

Bessem Samet; Bessem Samet

doi:10.3934/era.2026232

Electronic Research Archive

2026, Volume 34, Issue 8: 5217-5235. doi: 10.3934/era.2026232

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A strong maximum principle in quantum calculus

Bessem Samet ^,

Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia

Received: 25 March 2026 Revised: 18 May 2026 Accepted: 28 May 2026 Published: 17 June 2026

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.
- quantum calculus,
- $ q $-difference inequalities,
- maximum principle,
- comparison principle,
- Hermite–Hadamard-type inequality
Citation: Bessem Samet. A strong maximum principle in quantum calculus[J]. Electronic Research Archive, 2026, 34(8): 5217-5235. doi: 10.3934/era.2026232

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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.

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