Journal of Mathematical Analysis and Applications

Abstract: We discuss existence, uniqueness, and structural stability of solutions of nonlinear differential equations of fractional order. The differential operators are taken in the Riemann–Liouville sense and the initial conditions are specified according to Caputo's suggestion, thus allowing for interpretation in a physically meaningful way. We investigate in particular the dependence of the solution on the order of the differential equation and on the initial condition, and we relate our results to the selection of appropriate numerical schemes for the solution of fractional differential equations.

Abstract: In probability theory [I], an event, A, is a member of a a-field, CY, of subsets of a sample space ~2. A probability measure, P, is a normed measure over a measurable space (Q, GY); that is, P is a real-valued function which assigns to every A in Gk’ a probability, P(A), such that (a) P(A) > 0 for all A E a, (b) P(Q) = 1; and (c) P is countably additive, i.e., if {Ai} is any collection of disjoint events, then

Abstract: The variational principle states that if a differentiable functional F attains its minimum at some point zi, then F’(C) = 0; it has proved a valuable tool for studying partial differential equations. This paper shows that if a differentiable function F has a finite lower bound (although it need not attain it), then, for every E > 0, there exists some point u( where 11 F’(uJj* < l , i.e., its derivative can be made arbitrarily small. Applications are given to Plateau’s problem, to partial differential equations, to nonlinear eigenvalues, to geodesics on infinite-dimensional manifolds, and to control theory.