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Quotients of continuous convex functions on nonreflexive Banach spaces

Holicky P., Kalenda O. F., Vesely L., Zajícek L.

Bulletin of the Polish Academy of Sciences. Mathematics

2007

Vol. 55, no 3

211-217

On each nonreflexive Banach space X there exists a positive continuous convex function ƒ such that 1/ƒ is not a d.c. function (i.e., a difference of two continuous convex functions). This result together with known ones implies that X is reflexive if and only if each everywhere defined quotient of two continuous convex functions is a d.c. function. Our construction also gives a stronger version of Klee's result concerning renormings of nonreflexive spaces and non-norm-attaining functionals.

Convergence of Toland's critical points for sequences of D.C. functions and application to the resolution of semilinear elliptic problems

Yassine A., Alaa N., Elhilali Alaoui A.

Control and Cybernetics

2001

Vol. 30, no 4

405-417

We prove that if a sequence (fn)n of D.C. functions (Difference of two Convex functions) converges to a D.C. function f in some appropriate way and if un is a critical point of fn, in the sense described by Toland, and is such that (un)n converges to u, then a is a critical point of f, still in Toland's sense. We also build a new algorithm which searches for this critical point u and then apply it in order to compute the solution of a semilinear elliptic equation.