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<p>This paper introduces a novel class of discrete boundary value problems governed by variable-order nabla-type fractional difference operators subject to nonlocal linear constraints. We generalize the constant-order framework by allowing the fractional order to vary spatially and by replacing classical Dirichlet conditions with linear hyperplane constraints that represent cumulative or sectoral dependencies. Using variational methods, we establish the existence of nontrivial solutions via the Ricceri variational principle and prove multiplicity through genus theory under symmetry assumptions. We characterize bifurcation thresholds analytically and verify them numerically. The results reveal threshold effects, equilibrium multiplicity, and unbounded continuation branches as the bifurcation parameter tends to zero. Economic interpretations emphasize systemic responsiveness to external stimuli and multistable regimes. A numerical scheme based on discrete convolution is developed and tested, confirming the theoretical predictions.</p>

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Existence and multiplicity of solutions for discrete variable-order fractional boundary value problems with nonlocal constraints

  • 期刊论文

  • 发表日期：

    2026年4月20日

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  • David Barilla,
  • Maria Frasca,
  • Luca Guerrini

摘要

This paper introduces a novel class of discrete boundary value problems governed by variable-order nabla-type fractional difference operators subject to nonlocal linear constraints. We generalize the constant-order framework by allowing the fractional order to vary spatially and by replacing classical Dirichlet conditions with linear hyperplane constraints that represent cumulative or sectoral dependencies. Using variational methods, we establish the existence of nontrivial solutions via the Ricceri variational principle and prove multiplicity through genus theory under symmetry assumptions. We characterize bifurcation thresholds analytically and verify them numerically. The results reveal threshold effects, equilibrium multiplicity, and unbounded continuation branches as the bifurcation parameter tends to zero. Economic interpretations emphasize systemic responsiveness to external stimuli and multistable regimes. A numerical scheme based on discrete convolution is developed and tested, confirming the theoretical predictions.