We show some results about local homology modules when they are in a Serre subcategory of the category of R-modules. For an ideal a of R, we also define the concept of the condition C^a on a Serre category, which seems dual to the condition Ca in Melkersson [1]. As a main result we show that for an Artinian R-module M and any Serre subcategory S of the category of R-modules and a non-negative integer s, HomR(R/a,H^a_s (M)) ∈ S if H^a_i (M) ∈ S for all i > s.

2010 Mathematics Subject Classification. 13D45, 13M10; 13E15.

M. A. Selby [8-10] and, independently, N. Sibony [11] proved that on the complex plane c-completeness is equivalent to c-finitely compactness. Their proofs are quite similar and are based on [4]. We give more refined equivalent conditions and, along the way, simplify the proofs.

2010 Mathematics Subject Classification. 30H05, 30H50.

In this paper, we study several fractional variational problems with functionals that contain n unknown functions with their higher order Caputo derivatives and Riemann–Liouville integrals. We prove generalized fractional Euler–Lagrange equations. We also study an isoperimetric problem with multiple constraints, and we find the optimality conditions. Some examples are provided to illustrate the applications of the results.

2000 Mathematics Subject Classification. 49K10, 26A33, 26B20.