In this paper, we study multivalued nonlocal elliptic problems driven by the fractional double phase operator with variable exponents and \(\omega \) -logarithmic perturbation formulated by \(\begin{aligned} {\left\{ \begin{array}{ll} \left( -\Delta \right) ^s_{\mathcal {H}} u \in \mathcal {F}(x,u) \quad & \text {in } \Omega ,\\ u=0& \text {on } \mathbb {R}^N\setminus \Omega . \end{array}\right. } \end{aligned}\) We are going to establish maximum principles for the fractional perturbed double phase operator and show the boundedness of weak solutions to the above problem. Finally, under appropriate assumptions we discuss the existence of infinitely many small (non-negative) weak solutions to a single-valued nonlocal double phase problem.
