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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024390/c0243901.png" /> that is positive on a cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024390/c0243902.png" /> in a Banach space is said to be concave (more exactly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024390/c0243904.png" />-concave on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024390/c0243905.png" />) if
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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024390/c0243906.png" />:
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1) the following inequalities are valid for any non-zero $  x \in K $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024390/c0243907.png" /></td> </tr></table>
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$$
 +
\alpha ( x) u _ {0}  \leq  Ax  \leq  \beta ( x) u _ {0} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024390/c0243908.png" /> is some fixed non-zero element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024390/c0243909.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024390/c02439010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024390/c02439011.png" /> are positive scalar functions;
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where $  u _ {0} $
 +
is some fixed non-zero element of $  K $
 +
and $  \alpha ( x) $
 +
and $  \beta ( x) $
 +
are positive scalar functions;
  
2) for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024390/c02439012.png" /> such that
+
2) for each $  x \in K $
 +
such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024390/c02439013.png" /></td> </tr></table>
+
$$
 +
\alpha _ {1} ( x) u _ {0}  \leq  x  \leq  \beta _ {1} ( x) u _ {0} ,\ \
 +
\alpha _ {1} , \beta _ {1} > 0,
 +
$$
  
 
the following relations are valid:
 
the following relations are valid:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024390/c02439014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$ \tag{* }
 +
A ( tx)  \geq  ( 1 + \eta ( x, t)) tA ( x),\  0 < t < 1,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024390/c02439015.png" />.
+
where $  \eta ( x, t) > 0 $.
  
In a similar manner, an operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024390/c02439016.png" /> is said to be convex (more exactly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024390/c02439018.png" />-convex on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024390/c02439019.png" />) if conditions 1) and 2) are met but the inequality (*) is replaced by the opposite inequality, with a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024390/c02439020.png" />.
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024390/c02439021.png" /></td> </tr></table>
+
$$
 +
A [ x ( t)]  = \
 +
\int\limits _ { G }
 +
k ( t, s, x ( s))  ds,
 +
$$
  
the concavity and convexity of which is ensured by, respectively, the concavity and convexity of the scalar function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024390/c02439022.png" /> with respect to the variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c024/c024390/c02439023.png" />. 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.
+
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)
How to Cite This Entry:
Concave and convex operators. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Concave_and_convex_operators&oldid=16007
This article was adapted from an original article by M.I. Voitsekhovskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article