This paper develops a unified qualitative framework for four classes of Liouville–Caputo fractional difference equations arising from different combinations of fractional sums and integer-order differences. Based on the equivalence between initial/final value problems and Volterra-type summation equations, sufficient conditions for the existence and uniqueness of solutions are established by applying the Banach contraction mapping principle together with refined combinatorial estimates. Furthermore, the continuous dependence of solutions on prescribed initial or final data is investigated. By deriving explicit error estimates through a discrete fractional Gronwall-type inequality, we prove that Lipschitz solutions depend continuously on perturbations of boundary data. Numerical experiments for a representative case are presented to verify the theoretical results, including the influence of the fractional order and the sensitivity with respect to boundary data, while additional examples illustrate the applicability of the framework. The obtained results extend the unified discrete fractional calculus framework by providing a rigorous well-posedness analysis and offering a theoretical foundation for further applications of discrete fractional models with memory effects and diverse boundary conditions.

This study addresses multistage task assignment and recovery for heterogeneous unmanned aerial vehicle (UAV) teams under progressively revealed disruptions and epistemic parameter uncertainty. A finite scenario tree is used to represent the sequential revelation of service demand and persistent UAV unavailability. Assignment and routing decisions are indexed by information nodes and linked through nonanticipativity, allowing a policy to adapt after new information becomes available without using future observations. Unit-distance fuel consumption is represented by normal uncertain variables, whereas service effectiveness is driven by node-specific lognormal uncertain factors. The resulting formulation incorporates heterogeneous capabilities, task completion, cross-stage collaboration, route continuity, flight-range limits, and consumable-payload capacities. A K–$\beta$ risk model is developed to minimize the fuel-consumption and weighted service-deficit thresholds at prescribed belief levels. Inverse and generalized inverse uncertainty distributions yield a deterministic multiobjective reformulation while preserving the efficient policy set. A branch-risk-aware multiobjective evolutionary algorithm based on decomposition (BRA–MOEA/D) is designed as a structure-compatible solution method for the resulting combinatorial problem. On the tested medium and large benchmark instances, computational experiments produce feasible nondominated policies with distinct fuel–service tradeoffs; these results are evidence at the investigated scales rather than a general claim of algorithmic superiority or scalability. Mechanism comparisons show that adaptive branching and event-triggered recovery materially change the attainable risk-threshold front, while a sensitivity analysis confirms that the two belief levels separately regulate fuel-risk and service-risk conservatism. The model thereby links sequential adaptation and recovery decisions with belief-based risk control in heterogeneous UAV service operations.

Fractional calculus has emerged as a powerful tool for characterizing non-classical dynamic phenomena, yet its discretization remains fragmented, with existing studies primarily focusing on single combinations of variable bounds and difference directions. To address this gap, this paper proposes a unified theoretical framework for discrete fractional calculus by systematically introducing four novel Caputo fractional difference definitions, which integrate variable upper/lower-limit sums with forward/backward difference operations. First, we rigorously derive the fundamental properties of these four definitions, including the commutativity of fractional sums and differences, and their consistency with integer-order difference operations. Second, we construct fractional difference equations for each definition, establish their equivalence to Volterra sum equations, and provide explicit solutions and strict proofs for their corresponding initial and final value problems. To validate the theoretical results, we design four targeted computational cases and numerical simulations, confirm the consistency between theoretical solutions and numerical results, and intuitively demonstrate the long-memory effect of fractional-order discrete systems. Furthermore, we present a concise comparison of the four definitions, clarifying their suitability for discrete systems with distinct boundary conditions and dynamic characteristics. This work not only completes the theoretical system of discrete fractional calculus with variable bounds but also provides standardized and targeted mathematical tools for modeling complex discrete dynamic processes, laying a solid foundation for the practical application of discrete fractional calculus in fields such as engineering control, infectious disease modeling, and economic dynamics.