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 $A$ that is positive on a cone $K$ in a Banach space is said to be concave (more exactly, $u _ {0}$- concave on $K$) if

1) the following inequalities are valid for any non-zero $x \in K$:

$$\alpha ( x) u _ {0} \leq Ax \leq \beta ( x) u _ {0} ,$$

where $u _ {0}$ is some fixed non-zero element of $K$ and $\alpha ( x)$ and $\beta ( x)$ are positive scalar functions;

2) for each $x \in K$ such that

$$\alpha _ {1} ( x) u _ {0} \leq x \leq \beta _ {1} ( x) u _ {0} ,\ \ \alpha _ {1} , \beta _ {1} > 0,$$

the following relations are valid:

$$\tag{* } A ( tx) \geq ( 1 + \eta ( x, t)) tA ( x),\ 0 < t < 1,$$

where $\eta ( x, t) > 0$.

In a similar manner, an operator $A$ is said to be convex (more exactly, $u _ {0}$- convex on $K$) if conditions 1) and 2) are met but the inequality (*) is replaced by the opposite inequality, with a function $\eta ( x, t) < 0$.

A typical example is Urysohn's integral operator

$$A [ x ( t)] = \ \int\limits _ { G } k ( t, s, x ( s)) ds,$$

the concavity and convexity of which is ensured by, respectively, the concavity and convexity of the scalar function $k( t, s, u)$ with respect to the variable $u$. 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)