Nonlocal Cauchy problems for semilinear differential inclusions with fractional order in Banach spaces

• Nonlocal Cauchy problems for fractional differential inclusions is studied.
• Measure of noncompactness and fixed point theorem for multifunctions are used.
• Fractional lattice inclusions with global neighborhood interactions are investigated.

In this paper we present two existence results of nonlocal Cauchy problems for semilinear differential inclusions with fractional order in Banach spaces. The first result relies on a growth condition on the whole time interval. Our second result relies on a revised growth condition which is divided into two parts, one for the subintervals containing the points associated with the nonlocal conditions, and the other for the rest of the interval. The used technique is based on fractional calculus, the properties of the measure of noncompactness and a powerful fixed point theorem for multifunctions due to O’Regan–Precup. Finally, we apply the theoretical results to fractional differential inclusions on lattices with global neighborhood interactions.