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In mathematics, the '''Banach [[fixed-point theorem]]''' (also known as the '''contraction mapping theorem''' or '''contraction mapping principle''') is an important tool in the theory of [[metric space]]s; it guarantees the existence and uniqueness of [[fixed point (mathematics)|fixed points]] of certain self-maps of metric spaces, and provides a constructive method to find those fixed points. The theorem is named after [[Stefan Banach]] (1892–1945), and was first stated by him in 1922.<ref>http://www.emis.de/journals/BJMA/tex_v1_n1_a1.pdf</ref>
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== The Statement==
Ingesting oat meal in the morning can provide you with a great start to the day. The grain in oat meal helps keep you satisfied and whole for prolonged.<br><br>A good digestive tract is essential to accomplish and maintain weight loss and good health. Ensure you are consuming a lot of drinking water and ingesting your day-to-day consumption of fibers. A helping or a couple of probiotics located in yogurt and the like is another wise decision.<br><br>Incorporating Quinoa in what you eat will maintain a proteins absorption which happens to be healthy, whilst lowering the steak amount you consume. There are a variety of proteins inside it. Quinoa is also a great, gluten-totally free option. Quinoa has a really understated, nutty taste that many discover to become a great differ from meats.<br><br>Selfmade fruits drinks are wonderful to need to start off your day. Beware of drinks from junk food areas that happen to be loaded with added sugar and brief on real fruit. When you allow it to be on your own, you management the ingredients. You can be employed in essentially anything at all. Attempt wholesome elements like fresh fruit, skim dairy and Ancient greek low fat yogurt being a basic.<br><br>A fantastic guideline to remember is always to make the plate as vibrant as you possibly can whenever you fill it up. Each of the hues of your spectrum will be the idea to locating fresh vegatables and fruits that help keep you nurtured without the need of unnecessary energy. Make an attempt to get one or more providing of fruit and veggies with each meal. Having the delicious skin area of fruits is nutritionally valuable, since they consist of plenty of anti-oxidants.<br><br>Generate fresh fruits drinks. Store-purchased options are greasy and sugary. Do you know what vitamins and minerals are getting in a smoothie if you ensure it is on your own. 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This well-known species of fish brings each omega-3's and niacin into your diet plan. Omega-3 has proven to lessen heart disease, cancer, and oppression. Niacin might stop Alzheimer's illness. To be able to lessen being exposed to hazardous toxins, decide on outdoors salmon as an alternative to farmed.<br><br>Weight a pizza with vegetables to get your loved ones to consume them. When you coating the dairy products and pepperoni, add on wholesome options such as olives, tomatoes, onions and peppers. Be sure no person selections them off, both.<br><br>Enjoy yourself when attempting new food items by motivating your children to tell you what they think about foods these are trying. Permitting them to know what the feel is like, for instance, will make them curious sufficient that they'll want to go on a nibble.<br><br>Being familiar with greens is a great way to improve the diet you possess. Salads are far a lot more than ranch dressing up and lettuce. An incredible salad might include from prepared lean meats to crunch greens, cereals and fruit. Your creative thinking will be your only reduce. Salads can either be cool or hot, and they may be the primary meal that can help you stay satiated. You may definitely include some selection for your salads with selfmade salad dressings. Check out some components you've by no means found in your salads just before. Concepts may incorporate fresh herbal remedies, dried up fruits, tahini and in many cases peanut butter!<br><br>If you are looking for the merely wholesome goody, then look no further than natural veggies. These assist to curtail your hunger when offering you great nutrients and vitamins. These are rarely more complex to make and shop than artificial unhealthy foods. Moreover, they don't result in as much of a clutter for your comfort food products. Uncooked greens come up with a great connection between your normal meals.<br><br>Multivitamins are ideal for nutritional supplements but they are not alternatives for healthy food items. It's important that you get nutrients and vitamins from meals too. Unless bought by your medical doctor, your daily amount of your multi-nutritional should not be in excess of 1. Using lots of [http://nasebook.com/index.php?do=/blog/256339/the-best-way-to-train-for-maximum-outcomes/add-comment/ vitamins] could have a unfavorable result on the body.<br><br>A great way to rejuvenate your diet plan-shifting advancement is to remember the standards that first determined you to definitely begin having more healthy. Everybody has some other purpose as to why, but if you get returning to your preliminary motives, it will be simpler to acquire your determination back.<br><br>A colleague can be a wonderful assist method whilst you get used to a huge change in your daily life, like your diet program. Anyone you decide on for help can be quite a coach, or simply another person looking for the same final results. It's essential to have somebody to communicate to about it.<br><br>Remember that healthy foods can flavor excellent. If lasagna is one of your chosen food products, all you need to do is change the recipe a bit. Prepare with fat-free cheddar cheese and whole-grain noodles. Use floor poultry as opposed to terrain beef and give a small green spinach. These couple of alterations is not going to modify the preference of the meal and you will be more healthy.<br><br>Consuming a much healthier meals are some thing you are doing in little amounts, not large actions. This article will reveal to you how to get started out. These methods are not required, and you could uncover much more. Bare in mind that every step movements you even closer to your nutritional targets.
''Definition.'' Let (''X'', ''d'') be a metric space.  Then a map ''T'' : ''X'' → ''X'' is called a [[contraction mapping]] on ''X'' if there exists ''q'' ∈ [0, 1) such that
:<math>d(T(x),T(y)) \le q d(x,y)</math>
for all ''x'', ''y'' in ''X''.
 
<blockquote>'''Banach Fixed Point Theorem.''' Let (''X'', ''d'') be a non-empty [[complete metric space]] with a contraction mapping ''T'' : ''X'' → ''X''. Then ''T'' admits a unique fixed-point ''x*'' in ''X'' (i.e. ''T''(''x*'') = ''x*''). Furthermore, ''x*'' can be found as follows: start with an arbitrary element ''x''<sub>0</sub> in ''X'' and define a sequence {''x<sub>n</sub>''} by ''x<sub>n</sub>'' = ''T''(''x''<sub>''n''−1</sub>), then ''x<sub>n</sub>'' → ''x*''.</blockquote>
 
''Remark 1.'' The following inequalities are equivalent and describes the [[Rate of convergence|speed of convergence]]:
 
:<math>\begin{align}
d(x^*, x_n) &\leq \frac{q^n}{1-q} d(x_1,x_0), \\
d(x^*, x_{n+1}) &\leq \frac{q}{1-q} d(x_{n+1},x_n), \\
d(x^*, x_{n+1}) &\leq q d(x^*,x_n).
\end{align}</math>
 
Any such value of ''q'' is called a ''[[Lipschitz constant]]'' for ''T'', and the smallest one is sometimes called "the best Lipschitz constant" of ''T''.
 
''Remark 2.'' ''d''(''T''(''x''),&nbsp;''T''(''y''))&nbsp;<&nbsp;''d''(''x'',&nbsp;''y'') for all ''x'' ≠ ''y'' is in general not enough to ensure the existence of a fixed point, as is shown by the map ''T'' : [1, ∞) → [1, ∞), ''T''(''x'')&nbsp;=&nbsp;''x''&nbsp;+&nbsp;1/''x'', which lacks a fixed point. However, if ''X'' is [[Compact space|compact]], then this weaker assumption does imply the existence and uniqueness of a fixed point, that can be easily found as a minimizer of ''d''(''x'',&nbsp;''T''(''x'')), indeed, a minimizer exists by compactness, and has to be a fixed point of ''T''. It then easily follows that the fixed point is the limit of any sequence of iterations of ''T''.
 
''Remark 3.'' When using the theorem in practice, the most difficult part is typically to define ''X'' properly so that ''T''(''X'') ⊆ ''X''.
 
==Proofs==
===Banach's Original Proof===
Let ''x''<sub>0</sub> ∈ (''X'', ''d'') be arbitrary and define a sequence {''x<sub>n</sub>''} by setting: ''x<sub>n</sub>'' = ''T''(''x''<sub>''n''−1</sub>). Banach's original proof can be broken off into several Lemmas:
 
<blockquote>'''Lemma 1.''' For all ''n'' ∈ '''N''', ''d''(''x''<sub>''n''+1</sub>, ''x<sub>n</sub>'') ≤ ''q<sup>n</sup>d''(''x''<sub>1</sub>, ''x''<sub>0</sub>).</blockquote>
 
''Proof.'' We will proceed using induction, the base of the induction (''n'' = 1) holds:
 
:<math>d(x_{1+1}, x_1) = d(x_2, x_1) = d(T(x_1), T(x_0)) \leq qd(x_1, x_0).</math>
 
Suppose the statement holds for some ''k'' ∈ '''N'''. Then we have
 
: <math>\begin{align}
d(x_{(k + 1) + 1}, x_{k + 1}) & = d(x_{k + 2}, x_{k + 1}) \\
& = d(T(x_{k + 1}), T(x_k)) \\
& \leq q d(x_{k + 1}, x_k) \\
& \leq q q^kd(x_1, x_0) && \text{Induction Hypothesis}\\
& = q^{k + 1}d(x_1, x_0).
\end{align}</math>
 
By the [[principle of mathematical induction]], for all ''n'' ∈ '''N''', the Lemma is proven.
 
<blockquote>'''Lemma 2.''' {''x<sub>n</sub>''} is a [[Cauchy sequence]] in (''X'', ''d'') and hence converges to a limit ''x*'' in ''X''.</blockquote>
 
''Proof.'' Let ''m'', ''n'' ∈ '''N''' such that ''m'' > ''n''.
 
: <math>\begin{align}
d(x_m, x_n) & \leq d(x_m, x_{m-1}) + d(x_{m-1}, x_{m-2}) + \cdots + d(x_{n+1}, x_n) && \text{Triangle Inequality} \\
& \leq q^{m-1}d(x_1, x_0) + q^{m-2}d(x_1, x_0) + \cdots + q^nd(x_1, x_0)  && \text{Lemma 1}\\
& = q^n d(x_1, x_0) \sum_{k=0}^{m-n-1} q^k \\
& \leq q^n d(x_1, x_0) \sum_{k=0}^\infty q^k \\
& = q^n d(x_1, x_0) \left ( \frac{1}{1-q} \right ) && \text{Geometric Series}
\end{align}</math>
 
Let ε > 0 be arbitrary, since ''q'' ∈ [0, 1), we can find a large ''N'' ∈ '''N''' so that
 
:<math>q^N < \frac{\varepsilon(1-q)}{d(x_1, x_0)}.</math>
 
Therefore by choosing ''m'', ''n'' large enough we may write:
 
:<math>d(x_m, x_n)  \leq q^n d(x_1, x_0) \left ( \frac{1}{1-q} \right ) < \left (\frac{\varepsilon(1-q)}{d(x_1, x_0)} \right ) d(x_1, x_0) \left ( \frac{1}{1-q} \right ) = \varepsilon.</math>
 
Since ε > 0 was arbitrary this proves that sequence is Cauchy.
 
<blockquote>'''Lemma 3.''' ''x*'' is a [[Fixed point (mathematics)|fixed point]] of ''T''.</blockquote>
 
''Proof.'' Take the limit of both sides of the recurrence ''x<sub>n</sub>'' = ''T''(''x''<sub>''n''−1</sub>''),
 
:<math> \lim_{n\to\infty} x_n = \lim_{n\to\infty} T(x_{n-1})</math>
 
Since ''T'' is a contraction mapping, it is continuous, so we may take the limit inside:
 
:<math>\lim_{n\to\infty} x_n = T\left(\lim_{n\to\infty} x_{n-1} \right). </math>
 
Thus, ''x*'' = ''T''(''x*'').
 
<blockquote>'''Lemma 4.''' ''x*'' is the only fixed point of ''T'' in (''X'', ''d'').</blockquote>
 
''Proof.'' Suppose ''y'' also satisfies ''T''(''y'') = ''y''. Then
 
:<math>0 \leq d(x^*, y) = d(T(x^*), T(y)) \leq q d(x^*, y).</math>
 
Remembering that ''q'' ∈ [0, 1), the above implies that 0 ≤ (1−''q'')''d''(''x*'', ''y'') ≤ 0, which shows that ''d''(''x*'', ''y'') = 0, whence by [[Metric (mathematics)#Definition|positive definiteness]], ''x*'' = ''y''.
 
===Shorter Proof===
Now we present a simpler proof that appeared recently in the Journal of Fixed Point Theory and its Application (see reference).  
 
By the triangle inequality, for all ''x'', ''y'' in ''X'',
:<math>\begin{align}
d(x,y) &\le d(x,T(x)) + d(T(x),T(y)) + d(T(y),y) \\
&\le  d(x,T(x)) + q d(x,y) + d(T(y),y)
\end{align}</math>
solving for ''d''(''x'', ''y'') we get the ``Fundamental  Contraction Inequality":
:<math>d(x,y) \le \frac{d(T(x), x) + d(T(y),y)}{1-q},</math>
and we note that if ''x'' and ''y'' are both fixed points then this implies that ''d''(''x'', ''y'') = 0, so ''x'' = ''y'', proving that ''T'' has at most one fixed point. Now define the mapping ''T<sup>n</sup>'' by composing ''T'' with itself ''n'' times and note by induction that it satisfies a Lipschitz condition with constant ''q<sup>n</sup>''. It remains to show that for any ''x''<sub>0</sub> in ''X'', the sequence {''T<sup>n</sup>''(''x''<sub>0</sub>)} is Cauchy and so converges to a point ''x*'' of ''X'', which as noted above is clearly a fixed point of ''T''. If in the Fundamental Inequality we replace ''x'' and ''y'' by ''T<sup>n</sup>''(''x''<sub>0</sub>) and ''T<sup>m</sup>''(''x''<sub>0</sub>), we find that  
:<math>\begin{align}
d(T^n(x_0), T^m(x_0)) &\le \frac{d(T(T^n(x_0)), T^n(x_0)) + d(T(T^m(x_0)),T^m(x_0))}{1-q}, \\
&= \frac{d(T^n(T(x_0)), T^n(x_0)) + d(T^m(T(x_0)), T^m(x_0))}{1-q} \\
&\le \frac{q^n d(T(x_0), x_0) + q^m d(T(x_0), x_0)}{1-q} \\
&= \frac {q^n + q^m} {1-q} d(T(x_0) ,x_0)
\end{align}</math>
since ''q'' < 1, the last expression converges to zero as ''n'', ''m'' → ∞, proving that {''T<sup>n</sup>''(''x''<sub>0</sub>)} is Cauchy. Note also that as ''m'' → ∞ gives us
:<math> d(T^n(x_0), x^*) \le \frac {q^n} {1-q} d(T(x_0) ,x_0) </math>
derived in the first proof that gives the rate at which {''T<sup>n</sup>''(''x''<sub>0</sub>)} converges to ''x*''.
 
==Applications==
*A standard application is the proof of the [[Picard–Lindelöf theorem]] about the existence and uniqueness of solutions to certain [[ordinary differential equation]]s. The sought solution of the differential equation is expressed as a fixed point of a suitable integral operator which transforms continuous functions into continuous functions. The Banach fixed-point theorem is then used to show that this integral operator has a unique fixed point.
 
*One consequence of the Banach fixed-point theorem is that small Lipschitz perturbation of the identity are bi-lipschitz homeomorphisms. Let Ω be an open set of a Banach space ''E''; let ''I'' : Ω → ''E'' denote the identity (inclusion) map and let ''g'' : Ω → ''E'' be a Lipschitz map of constant ''k'' < 1. Then
#Ω′ := (''I''+''g'')(Ω) is an open subset of ''E'': precisely, for any ''x'' in Ω such that ''B''(''x'', ''r'') ⊂ Ω one has ''B''((''I''+''g'')(''x''), ''r''(1−''k'')) ⊂ Ω′;
#''I''+''g'' : Ω → Ω′ is a bi-lipschitz homeomorphism;
:precisely, (''I''+''g'')<sup>−1</sup> is still of the form ''I'' + ''h'' : Ω → Ω′ with ''h'' a Lipschitz map of constant ''k''/(1−''k''). A direct consequence  of this result yields the proof of the [[inverse function theorem]].
 
==Converses==
Several converses of the Banach contraction principle exist. The following is due to [[Czesław Bessaga]], from 1959:
 
Let ''f'' : ''X'' → ''X'' be a map of an abstract [[set (mathematics)|set]] such that each [[iterated function|iterate]] ''f<sup>n</sup>'' has a unique fixed point. Let ''q'' ∈ (0, 1), then there exists a complete metric on ''X'' such that ''f'' is contractive, and ''q'' is the contraction constant.
 
Indeed, very weak assumptions suffice to obtain such a kind of converse. For example if ''f'' : ''X'' → ''X'' is a map on a [[T1 space|''T''<sub>1</sub> topological space]] with a unique [[fixed point (mathematics)|fixed point]] ''a'', such that for each ''x'' in ''X'' we have ''f<sup>n</sup>''(''x'') → ''a'', then there already exists a metric on ''X'' with respect to which ''f'' satisfies the conditions of the Banach contraction principle with contraction constant 1/2.<ref>Pascal Hitzler, Anthony K. Seda, A "Converse" of the Banach Contraction Mapping Theorem. Journal of Electrical Engineering 52 (10/s), 2001, 3-6.</ref> In this case the metric is in fact an [[ultrametric]].
 
==Generalizations==
There are a number of generalizations as immediate corollaries, which are of some interest for the sake of applications. Let ''T'' : ''X'' → ''X'' be a map on a complete non-empty metric space.
*Assume that some iterate ''T<sup>n</sup>'' of ''T'' is a contraction. Then ''T'' has a unique fixed point.
*Assume that T is a continuous function, and for all ''x'' and ''y'' in ''X'',  
::<math>\sum\nolimits_n d(T^n(x),T^n(y))<\infty.</math>  
:Then ''T'' has a unique fixed point.{{Citation needed|date=July 2013}}
However, in most applications the existence and unicity of a fixed point can be shown directly with the standard Banach fixed point theorem, by a suitable choice of the metric that makes the map ''T'' a contraction. Indeed, the above result by Bessaga strongly suggests to look for such a metric. See also the article on [[fixed point theorems in infinite-dimensional spaces]] for generalizations.
 
A different class of generalizations arise from suitable generalizations of the notion of [[metric space]], e.g. by weakening the defining axioms for the notion of metric.<ref>Pascal Hitzler and Anthony Seda, Mathematical Aspects of Logic Programming Semantics. Chapman and Hall/CRC, 2010.</ref> Some of these have applications, e.g., in the theory of programming semantics in theoretical computer science.<ref>Anthony Karel Seda and Pascal Hitzler, Generalized Distance Functions in the Theory of Computation. The Computer Journal 53(4), 443-464, 2010.</ref>
 
==See also==
* [[Fixed-point theorem]]s
* [[Brouwer fixed-point theorem]]
* [[Infinite compositions of analytic functions]]
* [[Caristi fixed-point theorem]]
 
==Notes==
<references/>
 
==References==
* Banach, S. "Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales." ''Fund. Math.'' 3(1922), 133–181. [http://matwbn.icm.edu.pl/ksiazki/or/or2/or215.pdf]
* Vasile I. Istratescu, ''Fixed Point Theory, An Introduction'', D.Reidel, the Netherlands (1981). ISBN 90-277-1224-7 See chapter 7.
* Andrzej Granas and [[James Dugundji]], ''Fixed Point Theory'' (2003) Springer-Verlag, New York, ISBN 0-387-00173-5.
*{{cite book|author = Kirk, William A.; Khamsi, Mohamed A.|title = An Introduction to Metric Spaces and Fixed Point Theory|year= 2001| publisher= John Wiley, New York | isbn  = 978-0-471-41825-2}}
* William A. Kirk and Brailey Sims, ''Handbook of Metric Fixed Point Theory'' (2001), Kluwer Academic, London ISBN 0-7923-7073-2.
* Palais, R. "A simple proof of the Banach contraction principle." ''J. fixed point theory appl. 2 (2007), 221–223''
----
An earlier version of this article was posted on [http://planetmath.org/encyclopedia/BanachFixedPointTheorem.html Planet Math]. This article is [[open content]].
 
{{DEFAULTSORT:Banach Fixed-Point Theorem}}
[[Category:Topology]]
[[Category:Fixed-point theorems]]
[[Category:Articles containing proofs]]

Revision as of 02:36, 5 March 2014

Changing your diet program to get more far healthier and have more nutritious food items can certainly make a massive various on your own life. A dietician will offer excellent guidance in relation to consuming a much more proper diet.

Ingesting oat meal in the morning can provide you with a great start to the day. The grain in oat meal helps keep you satisfied and whole for prolonged.

A good digestive tract is essential to accomplish and maintain weight loss and good health. Ensure you are consuming a lot of drinking water and ingesting your day-to-day consumption of fibers. A helping or a couple of probiotics located in yogurt and the like is another wise decision.

Incorporating Quinoa in what you eat will maintain a proteins absorption which happens to be healthy, whilst lowering the steak amount you consume. There are a variety of proteins inside it. Quinoa is also a great, gluten-totally free option. Quinoa has a really understated, nutty taste that many discover to become a great differ from meats.

Selfmade fruits drinks are wonderful to need to start off your day. Beware of drinks from junk food areas that happen to be loaded with added sugar and brief on real fruit. When you allow it to be on your own, you management the ingredients. You can be employed in essentially anything at all. Attempt wholesome elements like fresh fruit, skim dairy and Ancient greek low fat yogurt being a basic.

A fantastic guideline to remember is always to make the plate as vibrant as you possibly can whenever you fill it up. Each of the hues of your spectrum will be the idea to locating fresh vegatables and fruits that help keep you nurtured without the need of unnecessary energy. Make an attempt to get one or more providing of fruit and veggies with each meal. Having the delicious skin area of fruits is nutritionally valuable, since they consist of plenty of anti-oxidants.

Generate fresh fruits drinks. Store-purchased options are greasy and sugary. Do you know what vitamins and minerals are getting in a smoothie if you ensure it is on your own. Additionally, you will be able making it satisfy your own having requirements. Low fat yogurt, clean and iced fruits, and tofu are all great additions.

You are able to truly feel and check better by rehearsing better diet. Decreasing the foods and drinks you consume which contain refined sweets is a good thing to do yourself. When you have virtually any questions concerning where along with the best way to make use of Best weight gainer Protein Mix, you are able to contact us at our web-site. Be suspicious of items like fresh fruit juices and fizzy drinks. For you to do the best to stay away from foods using a substantial blood sugar articles. If you remove sugars through your meals program you will notice an immediate distinction. You will also look and feel, a lot better.

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If you are looking for the merely wholesome goody, then look no further than natural veggies. These assist to curtail your hunger when offering you great nutrients and vitamins. These are rarely more complex to make and shop than artificial unhealthy foods. Moreover, they don't result in as much of a clutter for your comfort food products. Uncooked greens come up with a great connection between your normal meals.

Multivitamins are ideal for nutritional supplements but they are not alternatives for healthy food items. It's important that you get nutrients and vitamins from meals too. Unless bought by your medical doctor, your daily amount of your multi-nutritional should not be in excess of 1. Using lots of vitamins could have a unfavorable result on the body.

A great way to rejuvenate your diet plan-shifting advancement is to remember the standards that first determined you to definitely begin having more healthy. Everybody has some other purpose as to why, but if you get returning to your preliminary motives, it will be simpler to acquire your determination back.

A colleague can be a wonderful assist method whilst you get used to a huge change in your daily life, like your diet program. Anyone you decide on for help can be quite a coach, or simply another person looking for the same final results. It's essential to have somebody to communicate to about it.

Remember that healthy foods can flavor excellent. If lasagna is one of your chosen food products, all you need to do is change the recipe a bit. Prepare with fat-free cheddar cheese and whole-grain noodles. Use floor poultry as opposed to terrain beef and give a small green spinach. These couple of alterations is not going to modify the preference of the meal and you will be more healthy.

Consuming a much healthier meals are some thing you are doing in little amounts, not large actions. This article will reveal to you how to get started out. These methods are not required, and you could uncover much more. Bare in mind that every step movements you even closer to your nutritional targets.

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