Caristi fixed-point theorem: Difference between revisions

Content deleted Content added

Inline

Revision as of 16:46, 29 September 2020

In mathematics, the Caristi fixed-point theorem (also known as the Caristi–Kirk fixed-point theorem) generalizes the Banach fixed-point theorem for maps of a complete metric space into itself. Caristi's fixed-point theorem modifies the ε-variational principle of Ekeland (1974, 1979).^[1]^[2] The conclusion of Caristi's theorem is equivalent to metric completeness, as proved by Weston (1977).^[3] The original result is due to the mathematicians James Caristi and William Arthur Kirk.^[4]

Caristi fixed-point theorem can be applied to derive other classical fixed-point results, and also to prove the existence of bounded solutions of a functional equation.^[5]

Statement of the theorem

Let (X, d) be a complete metric space. Let T : X → X and f : X → [0, +∞) be a lower semicontinuous function from X into the non-negative real numbers. Suppose that, for all points x in X,

d{\big (}x,T(x){\big )}\leq f(x)-f{\big (}T(x){\big )}.

Then T has a fixed point in X, i.e. a point x₀ such that T(x₀) = x₀. The proof of this result utilizes Zorn's lemma to guarantee the existence of a minimal element which turns out to be a desired fixed point.^[6]

References

^ Ekeland, Ivar (1974). "On the variational principle". J. Math. Anal. Appl. 47 (2): 324–353. doi:10.1016/0022-247X(74)90025-0. ISSN 0022-247X.
^ Ekeland, Ivar (1979). "Nonconvex minimization problems". Bull. Amer. Math. Soc. (N.S.). 1 (3): 443–474. doi:10.1090/S0273-0979-1979-14595-6. ISSN 0002-9904.
^ Weston, J. D. (1977). "A characterization of metric completeness". Proc. Amer. Math. Soc. 64 (1): 186–188. doi:10.2307/2041008. ISSN 0002-9939. JSTOR 2041008.
^ Caristi, James (1976). "Fixed point theorems for mappings satisfying inwardness conditions". Trans. Amer. Math. Soc. 215: 241–251. doi:10.2307/1999724. ISSN 0002-9947. JSTOR 1999724.
^ Khojasteh, Farshid; Karapinar, Erdal; Khandani, Hassan (27 January 2016). "Some applications of Caristi's fixed point theorem in metric spaces". Fixed Point Theory and Applications. doi:10.1186/s13663-016-0501-z.
^ Dhompongsa, S.; Kumam, P. (2021). "A Remark on the Caristi's Fixed Point Theorem and the Brouwer Fixed Point Theorem". In Kreinovich, V. (ed.). Statistical and Fuzzy Approaches to Data Processing, with Applications to Econometrics and Other Areas. Berlin: Springer. pp. 93–99. doi:10.1007/978-3-030-45619-1_7. ISBN 978-3-030-45618-4.

[1] Ekeland, Ivar (1974). "On the variational principle". J. Math. Anal. Appl. 47 (2): 324–353. doi:10.1016/0022-247X(74)90025-0. ISSN 0022-247X.

[2] Ekeland, Ivar (1979). "Nonconvex minimization problems". Bull. Amer. Math. Soc. (N.S.). 1 (3): 443–474. doi:10.1090/S0273-0979-1979-14595-6. ISSN 0002-9904.

[3] Weston, J. D. (1977). "A characterization of metric completeness". Proc. Amer. Math. Soc. 64 (1): 186–188. doi:10.2307/2041008. ISSN 0002-9939. JSTOR 2041008.

[4] Caristi, James (1976). "Fixed point theorems for mappings satisfying inwardness conditions". Trans. Amer. Math. Soc. 215: 241–251. doi:10.2307/1999724. ISSN 0002-9947. JSTOR 1999724.

[5] Khojasteh, Farshid; Karapinar, Erdal; Khandani, Hassan (27 January 2016). "Some applications of Caristi's fixed point theorem in metric spaces". Fixed Point Theory and Applications. doi:10.1186/s13663-016-0501-z.

[6] Dhompongsa, S.; Kumam, P. (2021). "A Remark on the Caristi's Fixed Point Theorem and the Brouwer Fixed Point Theorem". In Kreinovich, V. (ed.). Statistical and Fuzzy Approaches to Data Processing, with Applications to Econometrics and Other Areas. Berlin: Springer. pp. 93–99. doi:10.1007/978-3-030-45619-1_7. ISBN 978-3-030-45618-4.

[1]

[2]

[3]

[4]

[5]

[6]

@@ Line 1: / Line 1: @@
-In [[mathematics]], the '''Caristi fixed-point theorem''' (also known as the '''Caristi–Kirk fixed-point theorem''') generalizes the [[Banach fixed point theorem]] for maps of a [[complete space|complete]] [[metric space]] into itself. Caristi's fixed-point theorem modifies  the ''ε''-[[Ekeland's variational principle|variational principle of]] [[Ivar Ekeland|Ekeland]] (1974, 1979). The conclusion of Caristi's theorem is equivalent to metric completeness, as proved by Weston (1977). The original result is due to the mathematicians [[James Caristi]] and [[William Arthur Kirk]].
+In [[mathematics]], the '''Caristi fixed-point theorem''' (also known as the '''Caristi–Kirk fixed-point theorem''') generalizes the [[Banach fixed-point theorem]] for maps of a [[complete space|complete]] [[metric space]] into itself. Caristi's fixed-point theorem modifies  the ''ε''-[[Ekeland's variational principle|variational principle of]] [[Ivar Ekeland|Ekeland]] (1974, 1979).<ref>{{cite journal
-Caristi fixed-point theorem can be applied to derive other classical fixed-point results, and also to prove the existence of bounded solutions of a [[functional equation]].<ref>{{cite journal|last1=Khojasteh|first1=Farshid|last2=Karapinar|first2=Erdal|last3=Khandani|first3=Hassan|title=Some applications of Caristi’s fixed point theorem in metric spaces|journal=Fixed Point Theory and Applications|date=27 January 2016|doi=10.1186/s13663-016-0501-z|doi-access=free}}</ref>
-==Statement of the theorem==
-Let (''X'',&nbsp;''d'') be a complete metric space. Let ''T''&nbsp;:&nbsp;''X''&nbsp;→&nbsp;''X'' and ''f''&nbsp;:&nbsp;''X''&nbsp;→&nbsp;[0,&nbsp;+∞) be a [[lower semicontinuous]] function from ''X'' into the non-negative [[real numbers]]. Suppose that, for all points ''x'' in ''X'',
-:<math>d \big( x, T(x) \big) \leq f(x) - f \big( T(x) \big).</math>
-Then ''T'' has a fixed point in ''X'', i.e. a point ''x''<sub>0</sub> such that ''T''(''x''<sub>0</sub>)&nbsp;=&nbsp;''x''<sub>0</sub>.
-==Notes==
-{{reflist}}
-==References==
-* {{cite journal
-|     last = Caristi
-|    first = James
-|    title = Fixed point theorems for mappings satisfying inwardness conditions
-|  journal = [[Transactions of the American Mathematical Society|Trans. Amer. Math. Soc.]]
-|   volume = 215
-|     year = 1976
-|    pages = 241&ndash;251
-|     issn = 0002-9947
-|      doi = 10.2307/1999724
-|      jstor = 1999724
-|doi-access = free
-}}
-* {{cite journal
 |     doi = 10.1016/0022-247X(74)90025-0
 |     last = Ekeland
@@ Line 39: / Line 10: @@
 |     issn = 0022-247X
 |     issue = 2
+}}</ref><ref>{{cite journal
-}}
-* {{cite journal
 |     last = Ekeland
 |    first = Ivar
@@ Line 52: / Line 22: @@
 |      doi = 10.1090/S0273-0979-1979-14595-6
 |doi-access = free
+}}</ref> The conclusion of Caristi's theorem is equivalent to metric completeness, as proved by Weston (1977).<ref>{{cite journal
-}}
-* {{cite journal
 |     last = Weston
 |    first = J. D.
@@ Line 65: / Line 34: @@
 |      doi = 10.2307/2041008
 |      jstor = 2041008
+}}</ref> The original result is due to the mathematicians [[James Caristi]] and [[William Arthur Kirk]].<ref>{{cite journal
-}}
+|     last = Caristi
+|    first = James
+|    title = Fixed point theorems for mappings satisfying inwardness conditions
+|  journal = [[Transactions of the American Mathematical Society|Trans. Amer. Math. Soc.]]
+|   volume = 215
+|     year = 1976
+|    pages = 241&ndash;251
+|     issn = 0002-9947
+|      doi = 10.2307/1999724
+|      jstor = 1999724
+|doi-access = free
+}}</ref>
+Caristi fixed-point theorem can be applied to derive other classical fixed-point results, and also to prove the existence of bounded solutions of a [[functional equation]].<ref>{{cite journal|last1=Khojasteh|first1=Farshid|last2=Karapinar|first2=Erdal|last3=Khandani|first3=Hassan|title=Some applications of Caristi’s fixed point theorem in metric spaces|journal=Fixed Point Theory and Applications|date=27 January 2016|doi=10.1186/s13663-016-0501-z|doi-access=free}}</ref>
+==Statement of the theorem==
+Let (''X'',&nbsp;''d'') be a complete metric space. Let ''T''&nbsp;:&nbsp;''X''&nbsp;→&nbsp;''X'' and ''f''&nbsp;:&nbsp;''X''&nbsp;→&nbsp;[0,&nbsp;+∞) be a [[lower semicontinuous]] function from ''X'' into the non-negative [[real numbers]]. Suppose that, for all points ''x'' in ''X'',
+:<math>d \big( x, T(x) \big) \leq f(x) - f \big( T(x) \big).</math>
+Then ''T'' has a fixed point in ''X'', i.e. a point ''x''<sub>0</sub> such that ''T''(''x''<sub>0</sub>)&nbsp;=&nbsp;''x''<sub>0</sub>. The proof of this result utilizes [[Zorn's lemma]] to guarantee the existence of a [[Maximal and minimal elements|minimal element]] which turns out to be a desired fixed point.<ref>{{cite book |first=S. |last=Dhompongsa |first2=P. |last2=Kumam |chapter=A Remark on the Caristi’s Fixed Point Theorem and the Brouwer Fixed Point Theorem |editor-last=Kreinovich |editor-first=V. |title=Statistical and Fuzzy Approaches to Data Processing, with Applications to Econometrics and Other Areas |location=Berlin |publisher=Springer |year=2021 |pages=93-99 |isbn=978-3-030-45618-4 |doi=10.1007/978-3-030-45619-1_7 }}</ref>
+==References==
+{{reflist}}
 [[Category:Fixed-point theorems]]