Strong convergence of an inertial Halpern type algorithm in Banach spaces
In this article, we obtain the strong convergence of the new modified Halpern iteration process x n + 1 = α n u + ( 1 - α n ) T n P ( x n + θ n ( x n - x n - 1 ) ) , n = 1 , 2 , 3 , … , to a common fixed point of { T n } , where { T n } n = 1 ∞ is a family of nonexpansive mappings on the closed and...
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| Vydáno v: | Rendiconti del Circolo matematico di Palermo Ročník 72; číslo 2; s. 1517 - 1526 |
|---|---|
| Hlavní autor: | |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Cham
Springer International Publishing
01.03.2023
|
| Témata: | |
| ISSN: | 0009-725X, 1973-4409 |
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| Abstract | In this article, we obtain the strong convergence of the new modified Halpern iteration process
x
n
+
1
=
α
n
u
+
(
1
-
α
n
)
T
n
P
(
x
n
+
θ
n
(
x
n
-
x
n
-
1
)
)
,
n
=
1
,
2
,
3
,
…
,
to a common fixed point of
{
T
n
}
, where
{
T
n
}
n
=
1
∞
is a family of nonexpansive mappings on the closed and convex subset
C
of a Banach space
X
,
P
:
X
⟶
C
is a nonexpansive retraction,
{
α
n
}
⊂
[
0
,
1
]
and
{
θ
n
}
⊂
R
+
. Some applications of this result are also presented. |
|---|---|
| AbstractList | In this article, we obtain the strong convergence of the new modified Halpern iteration process
x
n
+
1
=
α
n
u
+
(
1
-
α
n
)
T
n
P
(
x
n
+
θ
n
(
x
n
-
x
n
-
1
)
)
,
n
=
1
,
2
,
3
,
…
,
to a common fixed point of
{
T
n
}
, where
{
T
n
}
n
=
1
∞
is a family of nonexpansive mappings on the closed and convex subset
C
of a Banach space
X
,
P
:
X
⟶
C
is a nonexpansive retraction,
{
α
n
}
⊂
[
0
,
1
]
and
{
θ
n
}
⊂
R
+
. Some applications of this result are also presented. |
| Author | Ranjbar, Sajad |
| Author_xml | – sequence: 1 givenname: Sajad surname: Ranjbar fullname: Ranjbar, Sajad email: sranjbar@eghlid.ac.ir, sranjbar74@yahoo.com organization: Department of Mathematics, Higher Education Center of Eghlid |
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| Cites_doi | 10.1016/j.na.2006.08.032 10.1090/S0002-9939-1974-0341038-7 10.1016/0022-247X(73)90024-3 10.1512/iumj.1973.22.22041 10.1090/S0002-9904-1970-12486-7 10.1007/BF02761184 10.1016/0022-247X(81)90013-5 10.1016/0022-247X(79)90024-6 10.1073/pnas.54.4.1041 10.2140/pjm.1973.47.341 10.1016/0022-1236(77)90022-2 10.1023/A:1011253113155 10.1016/0022-247X(80)90323-6 10.4064/bc77-0-12 10.1016/j.jmaa.2008.03.028 10.1007/s11228-006-0027-3 10.1016/B978-0-12-434180-7.50033-4 10.1002/mana.19650300312 10.1016/j.cam.2007.07.021 10.1016/j.na.2010.07.031 |
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| Keywords | Strong convergence 47H10 Halpern iteration 47H09 Iterative methods Accretive operator Fixed point |
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| References | Aoyama, Kimura, Takahashi, Toyoda (CR2) 2007; 67 Reich (CR18) 1977; 26 Khatibzadeh, Rezaei, Ranjbar (CR10) 2021; 5 CR3 Bruck (CR7) 1974; 43 Browder (CR4) 1965; 54 Reich (CR22) 1981; 79 CR8 CR19 Bruck (CR5) 1973; 47 CR14 Kopecka, Reich (CR11) 2007; 77 CR13 CR24 CR12 Reich (CR21) 1980; 75 Reich, Zaslavski (CR23) 2000; 15 Nakajo (CR15) 2006; 7 Bruck (CR6) 1970; 76 Karlovitz (CR9) 1972; 22 Nevanlinna, Reich (CR16) 1979; 32 Reich (CR17) 1973; 44 Reich (CR20) 1979; 67 Alvarez, Attouch (CR1) 2001; 9 S Reich (749_CR17) 1973; 44 749_CR3 749_CR8 RE Bruck (749_CR5) 1973; 47 S Reich (749_CR22) 1981; 79 RE Bruck (749_CR6) 1970; 76 LA Karlovitz (749_CR9) 1972; 22 S Reich (749_CR21) 1980; 75 H Khatibzadeh (749_CR10) 2021; 5 749_CR12 K Nakajo (749_CR15) 2006; 7 E Kopecka (749_CR11) 2007; 77 RE Bruck (749_CR7) 1974; 43 K Aoyama (749_CR2) 2007; 67 F Alvarez (749_CR1) 2001; 9 749_CR14 749_CR13 749_CR24 FE Browder (749_CR4) 1965; 54 749_CR19 S Reich (749_CR18) 1977; 26 S Reich (749_CR23) 2000; 15 S Reich (749_CR20) 1979; 67 O Nevanlinna (749_CR16) 1979; 32 |
| References_xml | – volume: 67 start-page: 2350 year: 2007 end-page: 2360 ident: CR2 article-title: Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space publication-title: Nonlinear Anal. doi: 10.1016/j.na.2006.08.032 – volume: 43 start-page: 173 year: 1974 end-page: 175 ident: CR7 article-title: A characterization of Hilbert space publication-title: Proc. Am. Math. Soc. doi: 10.1090/S0002-9939-1974-0341038-7 – volume: 44 start-page: 57 year: 1973 end-page: 70 ident: CR17 article-title: Asymptotic behavior of contractions in Banach spaces publication-title: J. Math. Anal. Appl. doi: 10.1016/0022-247X(73)90024-3 – ident: CR19 – volume: 22 start-page: 473 year: 1972 end-page: 481 ident: CR9 article-title: The construction and application of contractive retractions in two-dimensional normed linear spaces publication-title: Indiana Univ. Math. 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Anal. – volume: 77 start-page: 161 year: 2007 ident: 749_CR11 publication-title: Banach Center Publ. doi: 10.4064/bc77-0-12 – volume: 76 start-page: 384 year: 1970 ident: 749_CR6 publication-title: Bull. Am. Math. Soc. doi: 10.1090/S0002-9904-1970-12486-7 – ident: 749_CR14 doi: 10.1016/j.jmaa.2008.03.028 – volume: 47 start-page: 341 year: 1973 ident: 749_CR5 publication-title: Pac. J. Math. doi: 10.2140/pjm.1973.47.341 – volume: 54 start-page: 1041 year: 1965 ident: 749_CR4 publication-title: Proc. Nat. Acad. Sci. USA doi: 10.1073/pnas.54.4.1041 – volume: 32 start-page: 44 year: 1979 ident: 749_CR16 publication-title: Israel J. Math. doi: 10.1007/BF02761184 – volume: 79 start-page: 113 year: 1981 ident: 749_CR22 publication-title: J. Math. Anal. Appl. doi: 10.1016/0022-247X(81)90013-5 – volume: 26 start-page: 378 year: 1977 ident: 749_CR18 publication-title: J. Funct. Anal. doi: 10.1016/0022-1236(77)90022-2 – volume: 67 start-page: 2350 year: 2007 ident: 749_CR2 publication-title: Nonlinear Anal. doi: 10.1016/j.na.2006.08.032 – volume: 67 start-page: 274 year: 1979 ident: 749_CR20 publication-title: J. Math. Anal. Appl. doi: 10.1016/0022-247X(79)90024-6 – volume: 22 start-page: 473 year: 1972 ident: 749_CR9 publication-title: Indiana Univ. Math. J. doi: 10.1512/iumj.1973.22.22041 – ident: 749_CR12 doi: 10.1007/s11228-006-0027-3 – ident: 749_CR19 doi: 10.1016/B978-0-12-434180-7.50033-4 – ident: 749_CR8 doi: 10.1002/mana.19650300312 – ident: 749_CR13 doi: 10.1016/j.cam.2007.07.021 – ident: 749_CR24 doi: 10.1016/j.na.2010.07.031 – volume: 7 start-page: 71 year: 2006 ident: 749_CR15 publication-title: J. Nonlinear Convex Anal. |
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| Snippet | In this article, we obtain the strong convergence of the new modified Halpern iteration process
x
n
+
1
=
α
n
u
+
(
1
-
α
n
)
T
n
P
(
x
n
+
θ
n
(
x
n
-
x
n
-
1... |
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| SubjectTerms | Algebra Analysis Applications of Mathematics Geometry Mathematics Mathematics and Statistics |
| Title | Strong convergence of an inertial Halpern type algorithm in Banach spaces |
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