*10.1. Stating Problem*

Note that the sinusoidal response to fractional oscillators attracts research interests but it is ye<sup>t</sup> a problem that has not been solved satisfactorily. In fact, the existence of the sinusoidal response to fractional oscillators remains a problem. In mathematics, it is regarded as a problem of periodic solution to fractional oscillators. Kaslik and Sivasundaram stated that the exact periodic solution does not exist ([81], p. 1495, Remark 5). The view of Kaslik and Sivasundaram's in [81] is also implied in other works of researchers. Taking fractional oscillators in Class I as an example, Mainardi noticed that the solution to fractional oscillators for 1 < *α* < 2, when driven by sinusoidal function, does not exhibit permanent oscillations but asymptotically algebraic decayed ([25], p. 1469), also see Achar et al. ([33], lines above Equation (14)), Duan et al. ([39], p. 49).

As a matter of fact, when considering a fractional oscillator of Class I type for 1 < *α* < 2 without the case of *α* = 2 in the form

$$m\frac{d^\alpha y\_1(t)}{dt^\alpha} + ky\_1(t) = \cos\omega t, 1 < \alpha < 2,\tag{258}$$

it is obvious that *y*1(*t*) must contain steady-state component that is not equal to 0 for *t* → ∞ no matter what value of *α* ∈ (1, 2) is. Otherwise, the conservation law of energy would be violated. The problem is what the complete solution of *y*1(*t*) should be.

The actual solution *y*1(*t*) should, in reality, consist of two parts. One is the steady-state part, denoted by *y*<sup>1</sup>*s*(*t*), where the subscript *s* stands for steady-state, which is not equal to 0 for *t* → ∞ and for any value of *α* ∈ (1, 2). The other is the transient part, denoted by *y*<sup>1</sup>*tr*(*t*), where the subscript *tr* means transient. Thus, the complete solution should, qualitatively, be in the form

$$y\_1(t) = y\_{1tr}(t) + y\_{1s}(t). \tag{259}$$

We contribute the complete solutions to three classes of fractional oscillators regarding their sinusoidal responses in this section. Our results will show that there exist steady-state components for fractional oscillators in either class with any value of *α* ∈ (1, 2) for those in Class I, or *β* ∈ (0, 1) in Class II, or any combination of *α* ∈ (1, 2) with *β* ∈ (0, 1) for those in Class III.
