Interval‐Valued β‐Volterra Dynamics Under Generalized Hukuhara Calculus

Source abstract

ABSTRACT We study interval‐valued Volterra dynamics under the ‐calculus, which combines the generalized Hukuhara difference with the general quantum operator on a continuous real interval; a thermal model with hereditary heat exchange is the main application. The coefficients and kernel are real‐valued, while the initial state and forcing may be interval‐valued. Under explicit boundedness and contraction assumptions, all bounded solutions share one scalar midpoint trajectory, and their radii solve a scalar ‐Volterra equation driven by a binary sign selection . We prove that every solution‐admissible selection generates a bounded solution and, conversely, that every such solution arises from at least one such selection. Under a sufficient uniform width condition, the constant positive selection yields an outer interval envelope. For the thermal model with nonnegative exchange coefficients, the type‐ band also encloses every admissible crisp trajectory, whereas the type‐ band need not contain them. Further results include an orbit form of the second fundamental theorem, fixed‐type well‐posedness and Ulam‐Hyers stability, a classical Jackson limit, and a truncated ‐orbit Picard method whose a priori error bound separates iteration and orbit‐truncation errors. Dimensionless experiments with constant and fading‐memory kernels illustrate the common midpoint, switching points and orbit‐induced jumps, the geometric truncation rate, and the convergence of fixed‐type solutions to the classical hereditary model.