Concave and convex operators

Non-linear operators in semi-ordered spaces that are analogues of concave and convex functions of a real variable.

A non-linear operator that is positive on a cone in a Banach space is said to be concave (more exactly, -concave on ) if

1) the following inequalities are valid for any non-zero :

where is some fixed non-zero element of and and are positive scalar functions;

2) for each such that

the following relations are valid:

(*)

where .

In a similar manner, an operator is said to be convex (more exactly, -convex on ) if conditions 1) and 2) are met but the inequality (*) is replaced by the opposite inequality, with a function .

A typical example is Urysohn's integral operator

the concavity and convexity of which is ensured by, respectively, the concavity and convexity of the scalar function with respect to the variable . Concavity of an operator means that it contains only "weak" non-linearities — the values of the operator on the elements of the cone increase "slowly" with the increase in the norms of the elements. Convexity of an operator means, as a rule, that it contains "strong" non-linearities. For this reason equations involving concave operators differ in many respects from equations involving convex operators; the properties of the former resemble the corresponding scalar equations, unlike the latter for which the theorem on the uniqueness of a positive solution is not valid.

References

[1]	M.A. Krasnosel'skii, P.P. Zabreiko, "Geometric methods of non-linear analysis" , Springer (1983) (Translated from Russian)