Oscillatory behavior of solutions of fractional differential equations with two different derivatives

This study investigates the asymptotic and oscillatory behavior of solutions to a class of forced nonlinear fractional differential equations (FDEs) characterized by two distinct Caputo fractional derivatives of the form ${}^C{D}_c^{\alpha }Z\left(\ell \right)+a{}^C{D}_c^{\beta }Z\left(\ell \right)-g_1(\ell ,Z\left(\ell \right))=e\left(\ell \right)+g_2(\ell ,Z\left(\ell \right)),c>1,$where $0<\beta <1<1+\beta \le \alpha <2$. While multi-term FDEs are known to effectively model complex multi-scale phenomena like viscoelasticity, the qualitative theory for equations where the derivatives span both the sub-diffusive ($\beta <1$) and super-diffusive ($\alpha >1$) regimes remains underdeveloped. This study aims to bridge this gap. Under specific growth conditions on the nonlinear functions $g_1$ and $g_2$, novel sufficient criteria are established to ensure that all solutions of the equation are oscillatory. The proof technique employs a unified framework integrating the semi-group properties of integer-order calculus with fractional operators and proceeds via a counterfactual argument by assuming the existence of a non-oscillatory solution and deriving a contradiction. The results presented here significantly extend the existing oscillation theory for fractional differential equations with multiple derivatives. The theoretical findings are further illustrated with some examples.