Demiclosedness and weak convergence of supper hybrid mappings in Banach spaces

 Co esponding au ho : Ikechukwu Godwin Ezugo ie
Copy igh © 2025 Au ho (s) e ain he copy igh o his a icle. This a icle is published unde he e ms o he C ea i e Commons A ibu ion License 4.0.
Demiclosedness and weak con e gence o suppe hyb id mappings in Banach spaces
Ikechukwu Godwin Ezugo ie *
Depa men o Ma hema ics, Enugu S a e Uni e si y o Science and Technology, Enugu, Enugu S a e, Nige ia.
Wo ld Jou nal o Ad anced Resea ch and Re iews, 2025, 27(02), 1564-1570
Publica ion his o y: Recei ed on 09 July 2025; e ised on 19 Augus ; accep ed on 22 Augus 2025
A icle DOI: h ps://doi.o g/10.30574/wja .2025.27.2.2987
Abs ac
We in oduce and s udy a new class o mapping in Banach Spaces, e med (𝛼 , 𝛽,𝛾) - suppe hyb id mappings, which
gene alize he well – known ( 𝛼 , 𝛽 ) - gene alized hyb id mappings. This ex ended amewo k encompasses a b oade
spec um o nonlinea o nonlinea ope a o s and allows o e ined con ol ia an addi ional pa ame e 𝛾 ≥ 0. We
es ablish se e al ounda ional p ope ies o suppe hyb id mappings, including quasi – nonexpansi enes and he
demiclosedness p inciple a ze o. Fu he mo e, we p o e a nonlinea e godic heo em o Baillon’s ype in Hilbe spaces
o suppe hyb id mappings, demons a ed weak con e gence o he Cesà o means o a ixed poin . Ou app oach
le e ages me ic p ojec ions and echniques inspi ed by Takahashi, he eby ex ending classical ixed poin heo y o
his new ope a o class.
Keywo ds: Suppe hyb id mapping; Nonlinea e godic heo em; Quasi – nonexpansi e mapping; Fixed poin ; Banach
space; Demiclosedness p inciple; Cesà o mean; Weak con e gence
1. In oduc ion
Fixed poin heo y o nonlinea mappings in Banach and Hilbe spaces has seen ex ensi e de elopmen , pa icula ly
h ough he s udy o nonexpensi e, quasi – nonexpansi e, hyb id mappings. Among hese gene alized hyb id mappings,
in oduced o in e pola e be ween con ac i e and nonexpansi e beha iou s. Ha e p o en ins umen al in analyzing
i e a i e algo i hms and a ia ional inequali ies.
In his pape , we p opose a new class o mappings, e med (𝛼 ,𝛽 ,𝛾)- suppe hyb id mappings, which ex end he classical
(𝛼 ,𝛽) - gene alized hyb id mappings by inco po a ing an addi ional nonnega i e pa ame e s 𝛾. This ex ension allows
o g ea e lexibili y in modeling nonlinea phenomenon and uni ies se e al ope a o classes unde one single
amewo k.
Ou p ima y con ibu ions a e h ee old. Fi s , we show ha suppe hyb id mappings wi h ixed poin s a e quasi –
nonexpansi e, he eby inhe i ing a key s abili y p ope y. Second, we es ablish he demiclosednes p inciple o (𝐼−𝑇)
a ze o unde mild assump ions on he duali y mapping, bo h o suppe hyb id and gene alized hyb id mappings. Thi d,
we p o e a nonlinea e godic heo em o Baillon’s ype o suppe hyb id mappings in Hilbe spaces, demons a ing
weak con e gence o he Cesà o means o a ixed poin .
The echniques employed d aw inspi a ion om Takahashi’s wo k on on e godic heo ems and ixed poin
app oxima ions, and ou esul s con ibu e o he ongoing e o o gene alize and e ine con e gence p inciple in
nonlinea analysis. The s uc u e o he pape is as ollows. In sec ion 2, we p esen he de ini ion o suppe hyb id
mappings and es ablish hei basic p ope ies. Sec ion 3 con ains he main esul s, including he demiclosedness
p inciple and he e godic heo em. We conclude wi h ema ks on po en ial ex ensions and applica ions.
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2. P elimina ies
Le 𝐸 be a eal Banach space wi h dual space 𝐸∗ and le 〈 .,.〉 deno e he duali y pai ing be ween 𝐸 and 𝐸∗. A subse 𝐶 ⊂
𝐸 is said o be con ex i o all 𝑥 ,𝑦 ∈ 𝐶 and ∈ [ 0 ,1 ] , he poin
𝑡𝑥 + (1 − 𝑡)𝑦 ∈ 𝐶 . A mapping 𝐽 ∶ 𝐸 ⟶ 2𝐸∗ is called a duali y mapping i
𝐽(𝑥):={ 𝑥∗∈ 𝐸∗∶ 〈 𝑥 ,𝑥∗ 〉 = ‖𝑥∗‖2 = ‖𝑥‖2 },∀ 𝑥 ∈ 𝐸 .
We say ha 𝐽 is weakly con inuous i 𝑥𝑛 ⇀ 𝑥 in 𝐸 implies 𝐽(𝑥𝑛) ⟶ 𝐽(𝑥) in he weak opology o 𝐸∗
Le 𝐻 be a eal Hilbe space. The me ic p ojec ion 𝑃𝐶 : 𝐻 ⟶ 𝐶 on o a nonemp y closed con ex subse 𝐶 ⊂ 𝐸 is
de ined by
𝑃𝐶 𝑥 ∶ = a gmin
𝑦 ∈ 𝐶‖𝑥−𝑦‖ ,∀ 𝑥 ∈ 𝐻
I is well known ha 𝑃𝐶 is nonexpansi e and sa is ies he a ia ional inequali y
〈 𝑥 − 𝑃𝐶 𝑥 ,𝑦 − 𝑃𝐶 𝑥 〉 ≤0 , ∀ 𝑦 ∈ 𝐶 .
A mapping 𝑇 ∶ 𝐶 ⟶ 𝐶 is called: - nonexpansi e i ‖𝑇𝑥−𝑇𝑦‖ ≤ ‖𝑥−𝑦‖ o all 𝑥 ,𝑦 ∈ 𝐶 ;
quasi nonexpansi e i ‖𝑇𝑥−𝑝‖ ≤‖𝑥−𝑝‖ , o all 𝑥 ∈ 𝐶 and 𝑝 ∈𝐹 (𝑇) , whe e
𝐹 (𝑇)∶={ 𝑥 ∈ 𝐶 ∶ 𝑇𝑥 = 𝑥 }, deno es he se o ixed poin s o 𝑇.
We ecall he demiclosedness p inciple, which plays a cen al ole in ixed poin heo y:
2.1. Lemma 2.1 (Demiclosedness P inciple)
Le 𝐸 be a Banach space wi h a weakly con inuous duali y mapping, and le 𝑇 ∶ 𝐶 ⟶ 𝐸 be a mapping. I 𝑇 is quasi –
nonexpansi e and 𝑥𝑛⇀ 𝑥 in 𝐸 wi h (𝐼−𝑇)𝑥𝑛 ⟶ 0 , hen 𝑥 ∈ 𝐹(𝑇).
We also ecall he classical Cesà o means used in e godic heo y. Fo a mapping ∶ 𝐶 ⟶ 𝐶 , he sequence { 𝑆𝑛 𝑥 }
de ined by
𝑆𝑛 𝑥 ∶ = 1
𝑛 ∑𝑇𝑘
𝑛−1
𝑘=0 𝑥
is called he Cesà o mean o he i e a es o 𝑇. In Hilbe space, such sequence o en con e ge weakly o a ixed poin
unde sui able condi ions.
Th oughou his pape , we use he no a ion 𝑇𝑛𝑥 o deno e he 𝑛− old composi ion o 𝑇 applied o , and we assume
ha all mappings ac on nonemp y closed con ex subse s o Banach o Hilbe spaces unless s a ed o he wise.
2.2. Lemma 2.2 ([2])
Assuming ha E is a Banach space has a weakly con inuous duali y mapping wi h guage 𝜑. Then o any sequences {𝑥𝑛}
ha con e ges weakly o 𝑥 , we ha e o any 𝑦 ∈ 𝐸,
lim sup
𝑛→∞ Φ (‖𝑥𝑛−𝑦‖) = lim sup
𝑛→∞ Φ (‖𝑥𝑛−𝑥‖) + lim sup
𝑛→∞ Φ (‖𝑥−𝑦‖)
De ini ion 2.2 Le 𝐾 be a nonemp y closed subse o a Banach space . A mapping 𝑇 ∶ 𝐾 ⟶ 𝐸 is called suppe hyb id i
he e a e 𝛼,𝛽,𝛾 ∈ ℝ wi h 𝛾 ≥0 such ha o all 𝑥 ,𝑦 ∈𝐾, we ha e
𝛼‖𝑇𝑥−𝑇𝑦‖2 + (1−𝛼+𝛾)‖𝑥−𝑇𝑦‖2≤ (𝛽+(𝛽−𝛼)𝛾)‖𝑇𝑥−𝑦‖2
Wo ld Jou nal o Ad anced Resea ch and Re iews, 2025, 27(02), 1564-1570
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+ (1 − 𝛽− (𝛽− 𝛼− 1)𝛾) ‖𝑥−𝑦‖2+(𝛼−𝛽)𝛾)‖𝑥−𝑇𝑦‖2+ 𝛾 ‖𝑦−𝑇𝑦‖2, ….. (1)
We call such a mapping an (𝛼,𝛽,𝛾) - suppe hyb id mapping (see [3]) . No ice ha an (𝛼,𝛽,0)- suppe hyb id mapping
is (𝛼,𝛽)- gene alized hyb id mapping, ha is
𝛼‖𝑇𝑥−𝑇𝑦‖2 + (1−𝛼)‖𝑥−𝑇𝑦‖2≤ 𝛽‖𝑇𝑥−𝑦‖2 + (1 − 𝛽) ‖𝑥−𝑦‖2 (2)
So, he class o suppe hyb id mappings con ains he class o gene alized hyb id mappings.
3. Main Resul s
3.1. P oposi ion 3.1
Le 𝐸 be a Banach space, le 𝐶 be a nonemp y subse o 𝐸, hen a suppe hyb id mappings wi h a ixed poin is quasi –
nonexpansi e.
P oo : Since 𝑇 ∶ 𝐶 ⟶ 𝐶 is a suppe hyb id mapping o 𝛼,𝛽,𝛾 ∈ ℝ wi h 𝛾 ≥0 and 𝑥 ,𝑦 ∈𝐶, as in (1). Le 𝑣 ∈ 𝐹 (𝑇),
hen we ha e ha o any 𝑥 ∈𝐶, om (1), ha
𝛼‖𝑇𝑥−𝑣‖2 ≤ (𝛽+(𝛽−𝛼)𝛾)‖𝑇𝑥−𝑣‖2+ (1−𝛽−(𝛽−𝛼−1)𝛾)‖𝑥−𝑣‖2
+(𝛼−𝛽)𝛾)‖𝑥−𝑣‖2+ 𝛾 ‖𝑣−𝑣‖2−(1−𝛼+𝛾)‖𝑥−𝑣‖2
Which implies ha
[𝛼−𝛽−(𝛽−𝛼)𝛾]‖𝑇𝑥−𝑣‖2 ≤[𝛼− 𝛽+ (𝛼−𝛽)𝛾)‖𝑥−𝑣‖2
and hence ‖𝑇𝑥−𝑣‖2≤ ‖𝑥−𝑣‖2. This implies ha 𝑇 is quasi–nonexpansi e.
3.2. P oposi ion 3.2
Le 𝐶 be a nonemp y closed con ex subse o a eal Banach space 𝐸, wi h a weakly con inuous duali y mapping and le 𝑇 ∶
𝐶 ⟶ 𝐸 be (𝛼,𝛽,𝛾) –suppe hyb id mappings wi h 𝛼,𝛽,𝛾 ∈ ℝ wi h 𝛾 ≥0. Then wi h (𝐼−𝑇) is demiclosed a wi h 0.
P oo : Le {𝑥𝑛}𝑛=1
∞ C be a sequence in 𝐶 which con e ges weakly o 𝑝 and {𝑥𝑛− 𝑇𝑥𝑛}𝑛=1
∞
Con e ges s ongly o 0. We show ha 𝑝 is a ixed poin o 𝑇. Since {𝑥𝑛}𝑛=1
∞ con e ges weakly, i is bounded. Clea ly,
{𝑇𝑥𝑛}𝑛=1
∞ is also bounded sequence. Since 𝑇 ∶ 𝐶 ⟶ 𝐸 is suppe –hyb id mapping, implies ha om (1), since {𝑥𝑛}𝑛=1
∞
con e ges weakly, i is bounded.
Fo each 𝑥 ∈𝐸
De ine by 𝑓 ∶ 𝐸 ⟶[0,∞) by
𝑓(𝑥)∶= lim sup
𝑛→∞ ‖𝑥𝑛−𝑥‖2
Then om Lemma 2.2, aking Φ(‖𝑥‖) = 1
2‖𝑥‖2, we ob ain,
𝑓(𝑥) = lim sup
𝑛→∞ ‖𝑥𝑛−𝑝‖2+‖𝑝−𝑥‖2, ∀ 𝑥 ∈𝐸
Thus,
𝑓(𝑥) = 𝑓(𝑝) + ‖𝑝−𝑥‖2 , ∀ 𝑥 ∈𝐸
and
𝑓(𝑇𝑝) = 𝑓(𝑝) + ‖𝑝−𝑇𝑝‖2 ……… (3)
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Obse e also ha om (1) and (3)
𝛼𝑓(𝑇𝑝) = 𝛼 lim sup
𝑛→∞ ‖𝑥𝑛−𝑇𝑝‖2
= 𝛼 lim sup
𝑛→∞ ‖𝑥𝑛−𝑇𝑥𝑛+𝑇𝑥𝑛−𝑇𝑝‖2
= 𝛼limsup
𝑛→∞ ‖𝑇𝑥𝑛− 𝑇𝑝‖2
≤ limsup
𝑛→∞ [ (𝛽+(𝛽−𝛼)𝛾)‖𝑇𝑥𝑛−𝑝‖2+ (1 −𝛽− (𝛽−𝛼−1)) ‖𝑥𝑛−𝑝‖2
+ (𝛼−𝛽)‖𝑥𝑛−𝑇𝑥𝑛‖2+ 𝛾 ‖𝑝−𝑇𝑝‖2− (1 − 𝛼+ 𝛾)‖𝑥𝑛−𝑇𝑝‖2]
= (𝛽+(𝛽−𝛼)𝛾) 𝑓(𝑝)+(1−𝛽−(𝛽−𝛼−1) 𝛾 )𝑓(𝑝)
+ 𝛾 ‖𝑝−𝑇𝑝‖2− (1 − 𝛼+ 𝛾) 𝑓(𝑇𝑝)
= 𝑓(𝑝)+ 𝛾 [ 𝑓(𝑝) + ‖𝑝−𝑇𝑝‖2] − (1 − 𝛼+ 𝛾) 𝑓(𝑇𝑝)
= 𝑓(𝑝)+ 𝛾 𝑓(𝑝) − (1 − 𝛼+ 𝛾) 𝑓(𝑇𝑝)
= 𝑓(𝑝) − (1−𝛼) 𝑓(𝑇𝑝)
The e o e,
𝑓(𝑇𝑝) ≤ 𝑓(𝑝) …………. (4)
Hence i ollows om (3) and (4) ha ‖𝑝−𝑇𝑝‖ = 0.
3.3. P oposi ion 3.3
Le 𝐶 be a nonemp y closed con ex subse o a eal Banach space 𝐸 wi h a weakly con inuous duali y mapping, and le 𝑇 ∶
𝐶 ⟶ 𝐶 be (𝛼,𝛽)–gene alized hyb id mappings wi h 𝛼,𝛽 ∈ ℝ. Then (𝐼−𝑇) is demiclosed a 0.
P oo : F om Lemma 3.2 i hen we ob ain he desi ed esul . We now p o e he ollowing Nonlinea e godic heo em o
Baillon’s ype [1] by using he echnique de eloped by Takahashi [4].
Theo em 3.1 Le 𝐻 be a Hilbe space and le 𝐶 be closed con ex subse o 𝐻, le 𝑇∶ 𝐶 ⟶ 𝐶 be a suppe hyb id mapping,
wi h 𝐹(𝑇) ≠ 0 and le 𝐶 be a me ic p ojec ion o 𝐻 on o 𝐹(𝑇). Then o 𝑥 ∈𝐶,
𝑆𝑛 𝑥 ∶ = 1
𝑛 ∑𝑇𝑘
𝑛−1
𝑘=0 𝑥………(5)
con e ges weakly o an elemen 𝑝 o 𝐹(𝑇), whe e 𝑝= 𝑙𝑖𝑚𝑃𝑇𝑛𝑥
𝑛→∞ .
P oo : Le 𝑇 ∶ 𝐶 ⟶ 𝐶 be (𝛼,𝛽,𝛾) suppe -hyb id wi h wi h 𝛾 ≥0, hen om P oposi ion 3.1 𝑇 is quasi-nonexpansi e,
we ha e ha 𝐹(𝑇) is closed and con ex. Le 𝑥 ∈𝐶 and le 𝑃 be he me ic p ojec ion o 𝐻 on o 𝐹(𝑇). Then, we ha e
‖𝑃𝑇𝑛𝑥−𝑇𝑛𝑥‖ ≤ ‖𝑃𝑇𝑛−1𝑥−𝑇𝑛𝑥‖ …….. (6)
≤ ‖𝑃𝑇𝑛−1𝑥−𝑇𝑛−1𝑥‖ ……….. (7)
This implies ha {‖𝑃𝑇𝑛𝑥−𝑇𝑛𝑥‖} non inc easing. We also know ha o any 𝑣 ∈𝐶 and 𝑢 ∈𝐹(𝑇)
〈 𝑣 − 𝑃𝑣 ,𝑃𝑣 − 𝑢 〉 ≥ 0
and hence
‖𝑣−𝑃𝑣‖2 ≤ 〈 𝑣 − 𝑃𝑣 ,𝑃𝑣 − 𝑢 〉.
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So, we ge
‖𝑃𝑣−𝑢‖2= ‖𝑃𝑣−𝑣+𝑣−𝑢‖2
= ‖𝑃𝑣−𝑣‖2−2〈 𝑃𝑣 − 𝑣 ,𝑢 − 𝑣 〉 + ‖𝑣−𝑢‖2
= ‖𝑣−𝑢‖2−‖𝑃𝑣−𝑣‖2
Le 𝑚 ,𝑛 ∈ℕ wi h 𝑚 ≥ 𝑛. Pu ing 𝑣 = 𝑇𝑚𝑥 and 𝑢 = 𝑇𝑛𝑥, we ha e
‖𝑃𝑇𝑚𝑥−𝑃𝑇𝑛𝑥‖2 ≤ ‖𝑇𝑚𝑥−𝑃𝑇𝑛𝑥‖2− ‖𝑃𝑇𝑚𝑥−𝑇𝑚𝑥‖2
≤ ‖𝑇𝑛𝑥−𝑃𝑇𝑛𝑥‖2− ‖𝑃𝑇𝑚𝑥−𝑇𝑚𝑥‖2
So, {𝑃𝑇𝑛𝑥} is a Cauchy sequence. Since 𝐹(𝑇), is closed, {𝑃𝑇𝑛𝑥} con e ges s ongly o an elemen 𝑝 o 𝐹(𝑇). Then we
ob ain, o any 𝑛 ∈ℕ,
‖𝑆𝑛 𝑥 − 𝑢‖ ≤ 1
𝑛 ∑‖ 𝑇𝑘𝑥−𝑢‖
𝑛−1
𝑘=0 ≤ ‖𝑥−𝑢‖
So, {𝑆𝑛𝑥} is bounded and hence he e exis s a weakly con e gen subsequence {𝑆𝑛𝑖𝑥} o 𝑆𝑛𝑥}.
I 𝑆𝑛𝑖𝑥 ⇀𝑣, hen we ha e 𝑣 ∈𝐹(𝑇). In ac , o any 𝑦 ∈𝐶 and 𝑘 ∈ ℕ ⋃ {0}, we ha e
0 ≤ (𝛽+(𝛽−𝛼)𝛾)‖𝑇𝑘+1𝑥−𝑦‖2+ 1 − 𝛽− (𝛽−𝛼−1)𝛾)‖𝑇𝑘𝑥−𝑦‖2
+ (𝛼−𝛽)𝛾‖𝑇𝑘𝑥−𝑇𝑦‖2+ 𝛾 ‖𝑦−𝑇𝑦‖2
−𝛼‖𝑇𝑘+1𝑥−𝑇𝑦‖2 − (1 − 𝛼+ 𝛾)‖𝑇𝑘𝑥−𝑇𝑦‖2]
= (𝛽+(𝛽−𝛼)𝛾)‖𝑇𝑘+1𝑥−𝑦‖2+ (1 − 𝛽−(𝛽−𝛼−1)𝛾)‖𝑇𝑘𝑥−𝑦‖2
+ (𝛼−𝛽)𝛾‖𝑇𝑘𝑥−𝑇𝑦‖2+ 𝛾 ‖𝑦−𝑇𝑦‖2−𝛼[‖𝑇𝑘+1𝑥−𝑦‖2
+ ‖𝑦−𝑇𝑦‖2+2 〈 𝑇𝑘+1𝑥− 𝑦 ,𝑦− 𝑇𝑦 〉]
− (1−𝛼+𝛾)[‖𝑇𝑘𝑥−𝑦‖2+‖𝑦−𝑇𝑦‖2+ 2 〈 𝑇𝑘𝑥− 𝑦 ,𝑦− 𝑇𝑦 〉]
= (𝛽+(𝛽−𝛼)𝛾)‖𝑇𝑘+1𝑥−𝑦‖2+ (1 − 𝛽−(𝛽−𝛼−1)𝛾)‖𝑇𝑘𝑥−𝑦‖2
+ (𝛼−𝛽)𝛾‖𝑇𝑘𝑥−𝑇𝑦‖2−‖𝑦−𝑇𝑦‖2
− 𝛼[‖𝑇𝑘+1𝑥−𝑦‖2+ 2 〈 𝑇𝑘+1𝑥− 𝑦 ,𝑦 − 𝑇𝑦 〉]−(1−𝛼+𝛾)[‖𝑇𝑘𝑥−𝑦‖2
+ 2 〈 𝑇𝑘𝑥− 𝑦 ,𝑦− 𝑇𝑦 〉] ……… (8)
Summing up he inequali y (8) wi h espec o 𝑘 =0,1,2,3,...,𝑛−1, we ge
0 ≤ (𝛽+(𝛽−𝛼)𝛾)‖𝑇𝑛𝑥−𝑦‖2+ (1 − 𝛽− (𝛽−𝛼−1)𝛾)‖𝑥−𝑦‖2
+ (𝛼−𝛽)𝛾‖𝑥−𝑇𝑦‖2− 𝑛 ‖𝑦−𝑇𝑦‖2−𝛼[‖𝑇𝑛𝑥−𝑦‖2
+ 2 〈 (𝑛+1)𝑆(𝑛+1)𝑥−𝑥−𝑛𝑦 ,𝑦− 𝑇𝑦 〉]
− (1−𝛼+𝛾)[‖𝑥−𝑦‖2+ 2 〈 𝑥−𝑦 ,𝑦− 𝑇𝑦 〉 ] ………… (9)
Di iding he inequali y (9) by 𝑛, we ha e

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0 ≤ (𝛽+(𝛽−𝛼)𝛾)
𝑛‖𝑇𝑛𝑥−𝑦‖2+ (1−𝛽−(𝛽−𝛼−1)𝛾)
𝑛‖𝑥−𝑦‖2
+ (𝛼−𝛽)𝛾
𝑛‖𝑥−𝑇𝑦‖2− ‖𝑦−𝑇𝑦‖2−𝛼[1
𝑛‖𝑇𝑛𝑥−𝑦‖2 ……… (10)
+ 2 〈 (𝑛+1)
𝑛𝑆(𝑛+1)𝑥−𝑥
𝑛− 𝑦 ,𝑦− 𝑇𝑦 〉]
− (1−𝛼+𝛾)[1
𝑛‖𝑥−𝑦‖2+ 2
𝑛 〈 𝑥− 𝑦 ,𝑦− 𝑇𝑦 〉 ] , ………… (11)
whe e ∑𝑇𝑘+1
𝑛
𝑘=0 𝑥 =(𝑛+1)𝑆(𝑛+1)𝑥−𝑥 om (5). Replacing 𝑛 by 𝑛𝑖 and le ing by 𝑛𝑖→∞, we ob ain om 𝑆(𝑛𝑖+1)𝑥 ⇀
𝑣 ha
0 ≤ − ‖𝑦−𝑇𝑦‖2 …………. (12)
Pu ing 𝑦= 𝑣 in (12) we ge
0 ≤ − ‖𝑣−𝑇𝑣‖2,
ha is
‖𝑣−𝑇𝑣‖2 ≤0
Hence, 𝑇𝑣 =𝑣. To comple e he p oo , i is su icien o show ha i 𝑆(𝑛𝑖+1)𝑥 ⇀ 𝑣 hen 𝑣 =𝑝. We ha e ha
〈 𝑇𝑘𝑥−𝑃𝑇𝑘𝑥 ,𝑃𝑇𝑘𝑥− 𝑢 〉 ≥0
o all 𝑢 ∈𝐹(𝑇). Since {‖𝑇𝑘𝑥−𝑃𝑇𝑘𝑥‖} is noninc easing, we ha e
〈 𝑢− 𝑝 ,𝑇𝑘𝑥−𝑃𝑇𝑘𝑥 〉 ≤ 〈 𝑃𝑇𝑘𝑥−𝑝 ,𝑇𝑘𝑥−𝑃𝑇𝑘𝑥 〉
≤ ‖𝑃𝑇𝑘𝑥−𝑝‖ .‖𝑇𝑘𝑥−𝑃𝑇𝑘𝑥‖
≤ ‖𝑃𝑇𝑘𝑥−𝑝‖ .‖𝑥−𝑃𝑥‖
Adding hese inequali ies om 𝑘 =0 o 𝑘 =𝑛−1 and di iding by 𝑛, we ha e
〈 𝑢− 𝑝 ,𝑆𝑛 𝑥− 1
𝑛 ∑ 𝑃𝑇𝑘𝑥
𝑛−1
𝑘=0 〉 ≤‖𝑥−𝑃𝑥‖
𝑛∑‖𝑃𝑇𝑘𝑥−𝑝‖.
𝑛−1
𝑘=0
Since, 𝑆(𝑛𝑖+1)𝑥 ⇀ 𝑣 and 𝑃𝑇𝑘𝑥 →𝑝, we ha e
〈 𝑢− 𝑝 ,𝑣− 𝑝 〉≤ 0.
We know 𝑣 ∈ 𝐹(𝑇). So, pu ing 𝑢 =𝑣, we ha e 〈 𝑣− 𝑝 ,𝑣− 𝑝 〉≤0 and hence ‖𝑣−𝑝‖2≤0. So, we ob ain 𝑣 =𝑝. This
comple es he p oo .
Co olla y 3.2 Le 𝐶 be a nonemp y closed con ex subse o a eal Hilbe space 𝐻, and le 𝑇 ∶ 𝐶 ⟶ 𝐶 be a suppe hyb id
mapping, wi h nonemp y ixed poin se 𝐹(𝑇). Then, o any 𝑥 ∈𝐶, he Cesa o means
𝑆𝑛 𝑥 ∶ = 1
𝑛 ∑𝑇𝑘
𝑛−1
𝑘=0 𝑥
con e ges weakly o a poin 𝑝∈𝐹(𝑇).
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P oo : This ollows di ec ly om he nonlinea e godic heo em es ablished in Theo em 3.1, oge he wi h he
demiclosedness p inciple and he weak compac ness o closed con ex subse s in Hilbe spaces.
Applica ions
Va ia ional Inequali ies: The con e gence o Cesa o means o suppe hyb id mappings can be used o app oxima e
solu ions o a ia ional inequali y p oblems o he o m: ind 𝑥∗∈𝐶 such ha
〈 𝐴𝑥∗ ,𝑦 − 𝑥∗ 〉≥0, ∀ 𝑦 ∈𝐶,
whe e 𝐴 ∶ 𝐶 ⟶ 𝐻 is a mono one ope a o . By cons uc ing sui able suppe hyb id mappings associa ed wi h he
esol en o 𝐴, one can apply he e godic heo em o ob ain weak con e gence o a solu ion. is
Con ex Feasibili y P oblems: In he con ex o inding a poin in he in e sec ion o con ex se s 𝐶1 ,𝐶2 , 𝐶3 ,..., 𝐶𝑚 ⊂
𝐻 suppe hyb id mapping can be designe o encode p ojec ion – based i e a i e schemes. The e godic con e gence o
Cesà o means hen p o ides a mechanism o app oxima ing easible poin s when di ec p ojec ion is compu a ionally
expensi e o in easible.
4. Conclusion
In his pape , we ha e in oduced and analyzed a nonlinea e godic heo em o a new class o mappings e med suppe
hyb id mappings in Banach and Hilbe spaces. By employing he demiclosedness p inciple and p ope ies o quasi-
nonexpansi e mappings, we es ablished he weak con e gence o Cesà o means o ixed poin s unde mild assump ions.
A key co olla y demons a es ha such con e gence holds o any ini ial poin in he domain, he eby ex ending classical
e godic esul s o a b oade class o nonlinea ope a o s.
Beyond i s heo e ical signi icance, he main esul admi s applica ions o a ia ional inequali y p oblems and con ex
easibili y o mula ions, whe e suppe hyb id mappings can be used o cons uc i e i e a i e schemes wi h gua an eed
con e gence. These indings o e a uni ied amewo k o analyzing nonlinea i e a i e p ocess in in ini e-dimensional
se ings.
Fu he esea ch may ocus on quan i a i e con e gence a es, s abili y unde pe u ba ions, and ex ensions o mo e
gene al classes o mappings. Applica ions o mono one inclusion p oblems and ope a o spli ing me hods also p esen
p omising di ec ions.
Re e ences
[1] J.-B. Baillon, Un heo em de ype e godique pou les con ac ions non lineai e dans un espace de Hilbe , C. R.
Acad. Sci. Pa is Se . A-B, 280 (1975), 1511-1514.
[2] T. C. Lim, H. K. Xu, Fixed Poin heo ems o asymp o ically nonexpansi e mappings, Nonlinea Anal. 22 (1994)
1345-1355.
[3] P. Kocou ek, W. Takahashi, and J.-C. Yao, Fixed Poin heo ems and weak con e gence heo ems o gene alized
hyb id mappings in Hilbe spaces, Taiwanese Jou nal o Ma hema ics, ol. 14, no. 6, 2497-2511, 2010.
[4] W. Takahashi, A nonlinea e godic heo em o an amenable semig oup o nonexpansi e mappings in a Hilbe
space, P oc. Ame . Ma h. Soc., 81 (1981), 253-256.