CONTROLLABILITY IN A CLASS OF CANCER THERAPY MODELS WITH CO-EVOLVING RESISTANCE

The paper’s Theorem 2 is proved via a nonautonomous Poincaré–Bendixson theorem for differential inclusions (Theorem A) together with a nested family of extremal-flow loops Ω_δ built by slightly enlarging the input range; this yields convergence of any trajectory to one enc Ω_j without asserting forward invariance except in the node-type case j=1 (Appendix A and Theorem 1.1) . By contrast, the candidate solution incorrectly claims strong forward invariance for every enc Ω_j using a Nagumo tangent-cone argument. The paper explicitly states that saddle- and mixed-type sets (enc Ω2, enc Ω3) are not forward invariant and can be exited along stable-manifold boundary segments, see Figure 6(a) and Remark 3 . The candidate’s global attraction step based on extremal nullclines (Γ1±, Γ2±) also asserts a uniform negative Dini derivative of distance to K without proof; the paper instead uses the ω-limit set alternative and upper semicontinuity of enc Ω_δ to conclude convergence . Hence the paper’s argument is correct, while the model’s proof contains a critical error (forward invariance for j=2,3) and missing hypotheses.