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)