Fixed Point Analysis of Variable-Order Fractional Problems with Nonlocal Conditions

In this talk, we investigate a class of generalized time-variable fractional integro-differential equations involving multi-term Caputo derivatives of variable order and nonlocal initial conditions. The model consists of a second-order classical derivative coupled with several variable-order fractional operators and a nonlinear source term depending on a fractional derivative of the unknown function. The initial condition is given in a nonlocal integral form, reflecting global memory effects.

Variable-order fractional operators arise naturally in applications where the memory intensity evolves over time, but their analysis is significantly more challenging than in the constant-order case. In particular, fundamental properties such as the semigroup law and simple inverse relations between fractional integrals and derivatives no longer hold. Using the definition of variable-order Caputo derivatives and without relying on invalid composition rules, we derive an equivalent Volterra-type integral formulation of the problem. This formulation provides a suitable framework for analysis and allows us to establish the existence of mild solutions via fixed point techniques, such as the Leray–Schauder alternative, under natural continuity and growth assumptions.

Finally, a numerical example is presented to illustrate the theoretical results and to demonstrate the influence of the time-dependent fractional order and the nonlocal initial condition on the qualitative behavior of solutions.