Difference between revisions of "Concave and convex operators"
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Non-linear operators in semi-ordered spaces that are analogues of concave and convex functions of a real variable. | Non-linear operators in semi-ordered spaces that are analogues of concave and convex functions of a real variable. | ||
| − | A non-linear operator | + | 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 | + | 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 | + | where $ u _ {0} $ |
| + | is some fixed non-zero element of $ K $ | ||
| + | and $ \alpha ( x) $ | ||
| + | and $ \beta ( x) $ | ||
| + | are positive scalar functions; | ||
| − | 2) for each | + | 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: | the following relations are valid: | ||
| − | + | $$ \tag{* } | |
| + | A ( tx) \geq ( 1 + \eta ( x, t)) tA ( x),\ 0 < t < 1, | ||
| + | $$ | ||
| − | where | + | where $ \eta ( x, t) > 0 $. |
| − | In a similar manner, an operator | + | 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 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 | + | 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==== | ====References==== | ||
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> M.A. Krasnosel'skii, P.P. Zabreiko, "Geometric methods of non-linear analysis" , Springer (1983) (Translated from Russian)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> M.A. Krasnosel'skii, P.P. Zabreiko, "Geometric methods of non-linear analysis" , Springer (1983) (Translated from Russian)</TD></TR></table> | ||
Latest revision as of 17:46, 4 June 2020
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) |
Concave and convex operators. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Concave_and_convex_operators&oldid=16007