This paper provides a direct equivalence proof for minimax solutions of A.I. Subbotin and generalized weak solutions in the sense of idempotent analysis. It is shown that the Hamilton-Jacobi equation Vt+H(t, x,DxV ) = 0 (with the Hamiltonian H(t, x, s) concave in s), considered in the context of minimax generalized solutions, is linear w.r.t. ⊕ = min and ̄ = +. This leads to a representation fo...

The Cauchy problem for the Euler-Poisson-Darboux equation on the unit sphere Sn gives rise to a family of fractional integrals M cos f(x) which integrate f over the spherical cap of radius centered at the point x 2 Sn. These fractional integrals are called the generalized Minkowski-Funk transforms because various transforms of integral geometry (including those of Minkowski and Funk) are partic...

Several symmetry problems are discussed. These include the Pompeiu problem and similar conjectures for the heat and wave equations.

In this paper, we prove the existence and uniqueness of solutions of the β-Cauchy problem of second order β-difference equations [Formula: see text] [Formula: see text], in a neighborhood of the unique fixed point [Formula: see text] of the strictly increasing continuous function β, defined on an interval [Formula: see text]. These equations are based on the general quantum difference operator ...