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In this article, we apply the concept of fuzzy soft δ -open sets to define new types of compactness in fuzzy soft topological spaces, namely, δ -compactness and δ ∗ -compactness. We explore some of their basic properties and reveal the relationships between these types and that which are defined by other authors. Also, we show that δ ∗ -compactness is more general than that which is presented in other papers. To clarify the obtained results and relationships, some illustrative examples are given.

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In this paper, we study a class of initial value problems for a nonlinear implicit fractional differential equation with nonlocal conditions involving the Atangana–Baleanu–Caputo fractional derivative. The applied fractional operator is based on a nonsingular and nonlocal kernel. Then we derive a formula for the solution through the equivalent fractional functional integral equations to the proposed problem. The existence and uniqueness are obtained by means of Schauder’s and Banach’s fixed point theorems. Moreover, two types of the continuous dependence of solutions to such equations are discussed. Finally, the paper includes two examples to substantiate the validity of the main results.