View source for Darbo fixed-point theorem

The notion of  "measure of non-compactness"  was first introduced by C. Kuratowski [[#References|[a1]]]. For any bounded set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130020/d1300201.png" /> in a [[Metric space|metric space]] its measure of non-compactness, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130020/d1300202.png" />, is defined to be the infimum of the positive numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130020/d1300203.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130020/d1300204.png" /> can be covered by a finite number of sets of diameter less than or equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130020/d1300205.png" />.

Another measure of non-compactness is the ball measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130020/d1300206.png" />, or Hausdorff measure, which is defined as the infimum of the positive numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130020/d1300207.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130020/d1300208.png" /> can be covered by a finite number of balls of radii smaller than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130020/d1300209.png" />. See also [[Hausdorff measure|Hausdorff measure]].

Roughly speaking, a measure of non-compactness is some function defined on the family of all non-empty bounded subsets of a given metric space such that it is equal to zero on the whole family of relatively compact sets.

G. Darbo used a measure of non-compactness to investigate operators whose properties can be characterized as being intermediate between those of contraction and compact mappings (cf. also [[Compact mapping|Compact mapping]]; [[Compact operator|Compact operator]]; [[Contraction(2)|Contraction]]). He was the first to use the index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130020/d13002010.png" /> in the theory of fixed points [[#References|[a2]]]. Darbo's fixed-point theorem is a generalization of the well-known Schauder fixed-point theorem (cf. also [[Schauder theorem|Schauder theorem]]). It states that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130020/d13002011.png" /> is a non-empty bounded closed convex subset of a [[Banach space|Banach space]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130020/d13002012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130020/d13002013.png" /> is a [[Continuous mapping|continuous mapping]] such that for any set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130020/d13002014.png" />,

<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/d/d130/d130020/d13002015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130020/d13002016.png" /> is a constant, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130020/d13002017.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130020/d13002018.png" /> has a [[Fixed point|fixed point]]. This theorem is true for the measure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130020/d13002019.png" /> also.

Note that every completely-continuous mapping (or [[Compact mapping|compact mapping]]; cf. also [[Completely-continuous operator|Completely-continuous operator]]) satisfies (a1) with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130020/d13002020.png" />, while all Lipschitz mappings with constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130020/d13002021.png" /> (cf. [[Lipschitz condition|Lipschitz condition]]) also satisfy (a1). Further, mappings that are not completely continuous but satisfy the condition (a1) are of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130020/d13002022.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130020/d13002023.png" /> is completely continuous and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130020/d13002024.png" /> satisfies the Lipschitz condition with constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130020/d13002025.png" />. The significance of this type of mapping is due to the fact that compactness of either the domain or the range is not required.

Methods for determining the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130020/d13002026.png" /> for a given set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d130/d130020/d13002027.png" /> in a Banach space are given in [[#References|[a3]]].

Darbo's fixed-point theorem is useful in establishing the existence of solutions of various classes of differential equations, especially for implicit differential equations, integral equations and integro-differential equations, see [[#References|[a3]]]. It is also used to study the controllability problem for dynamical systems represented by implicit differential equations [[#References|[a4]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  C. Kuratowski,   "Sur les espaces complets"  ''Fundam. Math.'' , '''15'''  (1930)  pp. 301–309</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Darbo,   "Punti uniti in transformazioni a condominio non compacto"  ''Rend. Sem. Mat. Univ. Padova'' , '''24'''  (1955)  pp. 84–92</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J. Banas,   K. Goebel,   "Measure of noncompactness in Banach spaces" , M. Dekker  (1980)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  K. Balachandran,   J.P. Dauer,   "Controllability of nonlinear systems via fixed point theorems"  ''J. Optim. Th. Appl.'' , '''53'''  (1987)  pp. 345–352</TD></TR></table>