research article fixed point theorems on nonlinear binary ...fixed point theorems on nonlinear...
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Research ArticleFixed Point Theorems on Nonlinear Binary OperatorEquations with Applications
Baomin Qiao
Department of Mathematics, Shangqiu Normal College, Shangqiu 476000, China
Correspondence should be addressed to Baomin Qiao; [email protected]
Received 18 April 2014; Revised 11 June 2014; Accepted 11 June 2014; Published 19 June 2014
Academic Editor: Krzysztof CieplinΜski
Copyright Β© 2014 Baomin Qiao.This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The existence and uniqueness for solution of systems of some binary nonlinear operator equations are discussed by using cone andpartial order theory and monotone iteration theory. Furthermore, error estimates for iterative sequences and some correspondingresults are obtained. Finally, the applications of our results are given.
1. Introduction
In recent years, more and more scholars have studied binaryoperator equations and have obtained many conclusions; see[1β6]. In this paper, we will discuss solutions for these equa-tionswhich associatedwith an ordinal symmetric contractionoperator and obtain some results which generalized andimproved those of [3β6]. Finally, we apply our conclusions totwo-point boundary value problem with two-degree super-linear ordinary differential equations.
In the following, letπΈ always be a realBanach spacewhichis partially ordered by a cone π, let π be a normal cone of πΈ,π is normal constant of π, partial order β€ is determined byπ and π denotes zero element of πΈ. Let π’, V β πΈ, π’ < V, π· =[π’, V] = {π₯ β πΈ : π’ β€ π₯ β€ V} denote an ordering interval of πΈ.
For the concepts of normal cone and partially order,mixed monotone operator, coupled solutions of operatorequations, and so forth see [1, 5].
Definition 1. Let π΄ : π· Γ π· β πΈ be a binary operator. π΄ issaid to be πΏ-ordering symmetric contraction operator if thereexists a bounded linear and positive operator πΏ : πΈ β πΈ,where spectral radius π(πΏ) < 1 such that π΄(π¦, π₯) β π΄(π₯, π¦) β€πΏ(π¦ β π₯) for any π₯, π¦ β π·, π₯ β€ π¦, where πΏ is called acontraction operator of π΄.
2. Main Results
Theorem 2. Let π΄ : π· Γ π· β πΈ be πΏ-ordering symmetriccontraction operator, and there exists a πΌ β [0, 1) such that
π΄ (π₯2, π¦2) β π΄ (π₯
1, π¦1) β₯ βπΌ (π₯
2β π₯1) ,
π’ β€ π₯1β€ π₯2β€ V, π’ β€ π¦
2β€ π¦1β€ V.
(1)
If condition (H1) π’ β€ π΄(π’, V), π΄(V, π’) β€ V β πΌ(V β π’) or (H
2)
π’ + πΌ(V β π’) β€ π΄(π’, V), π΄(V, π’) β€ V holds, then the followingstatements hold.
(C1) π΄(π₯, π₯) = π₯ has a unique solution π₯β β π·, and for
any coupled solutions π₯, π¦ β π·, π₯ = π¦ = π₯β.(C2) For any π₯
0, π¦0β π·, we construct symmetric iterative
sequences:
π₯π=
1
πΌ + 1[π΄ (π₯πβ1
, π¦πβ1
) + πΌπ₯πβ1
] ,
π¦π=
1
πΌ + 1[π΄ (π¦πβ1
, π₯πβ1
) + πΌπ¦πβ1
] ,
π = 1, 2, 3, . . . .
(2)
Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2014, Article ID 241942, 4 pageshttp://dx.doi.org/10.1155/2014/241942
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2 Discrete Dynamics in Nature and Society
Then π₯π
β π₯β, π¦π
β π₯β(π β β), and for any π½ β
(π(πΏ), 1), there exists a natural number π; and if π β₯ π, weget error estimates for iterative sequences (2):
π₯π (π¦π) β π₯β β€ 2π(
πΌ + π½
πΌ + 1)
π
βπ’ β Vβ . (3)
Proof. Setπ΅(π₯, π¦) = (1/(πΌ+1))[π΄(π₯, π¦)+πΌπ₯], and if condition(H1) or (H
2) holds, then it is obvious that
π’ β€ π΅ (π’, V) , π΅ (V, π’) β€ V. (4)
By (1), we easily prove thatπ΅ : π·Γπ· β πΈ ismixedmonotoneoperator, and for any π₯, π¦ β π·, π’ β€ π₯ β€ π¦ β€ V,
π β€ π΅ (π¦, π₯) β π΅ (π₯, π¦) β€ π» (π¦ β π₯) , (5)
whereπ» = (1/(πΌ+1))(πΏ+πΌπΌ) is a bounded linear and positiveoperator and πΌ, is identical operator.
By the mathematical induction, we easily prove that
π β€ π΅π(π¦, π₯) β π΅
π(π₯, π¦) β€ π»
π(π¦ β π₯) , π’ β€ π₯ β€ π¦ β€ V,
(6)
where π΅π(π₯, π¦) = π΅(π΅πβ1(π₯, π¦), π΅πβ1(π¦, π₯)), π₯, π¦ β π·, π β₯ 2.By the character of normal cone π, it is shown thatπ΅π(π¦, π₯) β π΅
π(π₯, π¦)
β€ ππ»π
π¦ β π₯ , π’ β€ π₯ β€ π¦ β€ V.
(7)
For any π½ β (π(πΏ), 1), since limπββ
||π»π||1/π
= π(π») β€
(πΌ + π(πΏ))/(πΌ + 1) < (πΌ +π½)/(πΌ + 1) < 1, there exists a naturalnumberπ, and if π β₯ π, we have ||π»π|| < ((πΌ + π½)/(πΌ + 1))π,andπ||π»π|| < 1. Considering mixed monotone operator π΅πand constantπ||π»π||, π΅π(π₯, π₯) = π₯ has a unique solution π₯βand for any coupled solution π₯, π¦ β π·, such that π₯ = π¦ = π₯βbyTheorem 3 in [3].
From π΅π(π΅(π₯β, π₯β), π΅(π₯β, π₯β)) = π΅(π΅π(π₯β, π₯β), π΅π(π₯β,π₯β)) = π΅(π₯
β, π₯β), and the uniqueness of solution with
π΅π(π₯, π₯) = π₯, then we have π΅(π₯β, π₯β) = π₯β and π΄(π₯β, π₯β) =
π₯β.We take note of that π΄(π₯, π₯) = π₯ and π΅(π₯, π₯) = π₯ have
the same coupled solution; therefore, a coupled solution forπ΅(π₯, π₯) = π₯ must be a coupled solution for π΅π(π₯, π₯) = π₯;consequently, (C
1) has been proved.
Considering iterative sequence (2), we construct iterativesequences:
π’π= π΅ (π’
πβ1, Vπβ1
) , Vπ= π΅ (V
πβ1, π’πβ1
) , (8)
where π’0= π’, V
0= V, it is obvious that
π₯π= π΅ (π₯
πβ1, π¦πβ1
) , π¦π= π΅ (π¦
πβ1, π₯πβ1
) ,
π β€ Vπβ π’πβ€ π»π(V β π’) ,
(9)
by themathematical induction and characterization ofmixedmonotone of π΅; then
π’πβ€ π₯ββ€ Vπ, π’
πβ€ π₯πβ€ Vπ, π’
πβ€ π¦πβ€ Vπ. (10)
Hence,π₯π (π¦π) β π’π
β€ πVπ β π’π
,
π₯ββ π’π
β€ πVπ β π’π
,
π = 1, 2, 3, . . . .
(11)
Moreover, if π β₯ π, we getπ₯π (π¦π) β π₯
β β€ 2πVπ β π’π
β€ 2ππ»π βV β π’β β€ 2π(
πΌ + π½
πΌ + 1)
π
βπ’ β Vβ .(12)
Consequently, π₯πβ π₯β, π¦πβ π₯β(π β β).
Remark 3. When πΌ = 0, Theorem 1 in [4] is a special case ofthis paper Theorem 2 under condition (H
1) or (H
2).
Corollary 4. Let π΄ : π· Γ π· β πΈ be πΏ-ordering symmetriccontraction operator; if there exists a πΌ β [0, 1) such that π΄satisfies condition of Theorem 2, the following statement holds.
(C3) For any π½ β (π(πΏ), 1) and πΌ+π½ < 1, we make iterative
sequences:
π’π= π΄ (π’
πβ1, Vπβ1
) ,
Vπ= π΄ (V
πβ1, π’πβ1
) + πΌ (Vπβ1
β π’πβ1
) ,
π = 1, 2, 3, . . . ,
(13)
orπ’π= π΄ (π’
πβ1, Vπβ1
) β πΌ (Vπβ1
β π’πβ1
) ,
Vπ= π΄ (V
πβ1, π’πβ1
) ,
π = 1, 2, 3, . . . ,
(14)
where π’0= π’, V
0= V.
Thus, π’π
β π₯β, Vπ
β π₯β(π β β), and there exists a
natural number π, and if π β₯ π, we have error estimates foriterative sequences (13) or (14):
π’π (Vπ) β π₯β β€ π(πΌ + π½)
π
βπ’ β Vβ . (15)
Proof. By the character of mixedmonotone ofπ΄, then (1) and(C1), (C2) [in (1), (C
2) where πΌ = 0] hold.
In the following, we will prove (C3).
Consider iterative sequence (13); since π’ β€ π₯β β€ V, we get
π’1= π΄ (π’, V) β€ π΄ (π₯β, π₯β) = π₯β β€ π΄ (V, π’)
= V1β πΌ (V β π’) β€ V
1.
(16)
By themathematical induction, we easily prove π’πβ€ π₯ββ€ Vπ,
π β₯ 1, hence
π β€ π₯ββ π’πβ€ Vπβ π’π, π β€ V
πβ π₯ββ€ Vπβ π’π. (17)
It is clear thatπ β€ Vπβ π’πβ€ (πΏ + πΌπΌ) (V
πβ1β π’πβ1
)
= (πΏ + πΌπΌ)π(V β π’) , π β₯ 1.
(18)
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Discrete Dynamics in Nature and Society 3
For any π½ β (π(πΏ), 1), πΌ + π½ < 1, since
limπββ
(πΏ + πΌπΌ)π
1/π
= π (πΏ + πΌπΌ) β€ π (πΏ) + πΌ < πΌ + π½ < 1,
(19)
there exists a natural numberπ, if π β₯ π, such that(πΏ + πΌπΌ)
π < (πΌ + π½)π
. (20)
Moreover,π’π (Vπ) β π₯
β β€ π(πΏ + πΌπΌ)
π βπ’ β Vβ
β€ π(πΌ + π½)π
βπ’ β Vβ , (π β₯ π) .(21)
Consequently, π’πβ π₯β, Vπβ π₯β, (π β β).
Similarly, we can prove (14).
Theorem 5. Let π΄ : π· Γ π· β πΈ be a πΏ-ordering symmetriccontraction operator; if there exists a πΌ β [0, 1) such that (1 βπΌ)π’ β€ π΄(π’, V), π΄(V, π’) β€ (1βπΌ)V, then the following statementshold.
(C4) Operator equation π΄(π₯, π₯) = (1 β πΌ)π₯ has a unique
solution π₯β β π·, and for any coupled solutions π₯, π¦ β π·, π₯ =π¦ = π₯
β.(C5) For any π₯
0, π¦0, π€0, π§0
β π·, we make symmetriciterative sequences
π₯π=
1
1 β πΌπ΄ (π₯πβ1
, π¦πβ1
) ,
π¦π=
1
1 β πΌπ΄ (π¦πβ1
, π₯πβ1
) ,
π = 1, 2, 3, . . . ,
(22)
π€π= π΄ (π€
πβ1, π§πβ1
) + πΌπ€πβ1
,
π§π= π΄ (π§
πβ1, π€πβ1
) + πΌπ§πβ1
,
π = 1, 2, 3, . . . .
(23)
Then π₯π
β π₯β, π¦π
β π₯β, π€π
β π₯β, π§π
β π₯β(π β β),
and for any π½ β (π(πΏ), 1), πΌ + π½ < 1, there exists a naturalnumber π, and if π β₯ π, then we have error estimates foriterative sequences (22) and (23), respectively,
π₯π (π¦π) β π₯β β€ 2π(
π½
1 β πΌ)
π
βπ’ β Vβ ,
π€π (π§π) β π₯β β€ 2π(πΌ + π½)
π
βπ’ β Vβ .
(24)
Proof. Set π΅(π₯, π¦) = (1/(1 β πΌ))π΄(π₯, π¦) or πΆ(π₯, π¦) =π΄(π₯, π¦) + πΌπ₯; we can prove that this theorem imitates proofof Theorem 2.
Similarly, we can prove the following theorems.
Theorem 6. Let π΄ : π· Γ π· β πΈ be πΏ-ordering symmetriccontraction operator; if there exists a πΌ β [0, 1) such that π’ +πΌV β€ π΄(π’, V), π΄(V, π’) β€ V + πΌπ’, then the following statementshold.
(C6) Equation π΄(π₯, π₯) = (1 + πΌ)π₯ has a unique solution
π₯ββ π·, and for any coupled solutions π₯, π¦ β π·π₯ = π¦ = π₯β.(C7) For any π₯
0, π¦0
β π·, we make symmetric iterativesequence:
π₯π=
1
1 + πΌπ΄ (π₯πβ1
, π¦πβ1
) ,
π¦π=
1
1 + πΌπ΄ (π¦πβ1
, π₯πβ1
) ,
π = 1, 2, 3, . . . .
(25)
Thenπ₯πβ π₯β,π¦πβ π₯β(π β β); moreover,π½ β (π(πΏ), 1),
and there exists natural numberπ, and if π β₯ π, then we haveerror estimates for iterative sequence (25):
π₯π (π¦π) β π₯β β€ 2π(
π½
πΌ + 1)
π
βπ’ β Vβ , (26)
(C8) For any π½ β (π(πΏ), 1)(πΌ+π½ < 1), π€
0, π§0β π·, we make
symmetry iterative sequence π€π= π΄(π€
πβ1, π§πβ1
) β πΌπ§πβ1
, π§π=
π΄(π§πβ1
, π€πβ1
) β πΌπ€πβ1
, π β₯ 1; then π€π
β π₯β, π§π
β
π₯β(π β β), and there exists a natural number π, and if
π β₯ π, we have error estimates for iterative sequence (24).
Remark 7. When πΌ = 0, Corollary 2 in [4] is a special case ofthis paper Theorems 2β6.
Remark 8. The contraction constant of operator in [5] isexpand into the contraction operator of this paper.
Remark 9. Operator π΄ of this paper does not need characterof mixed monotone as operator in [6].
3. Application
We consider that two-point boundary value problem for two-degree super linear ordinary differential equations:
π₯+ π (π‘) π₯
π+
1
1 + π (π‘) π₯= 0, π‘ β [0, 1] , (π β₯ 2)
π₯ (0) = π₯(1) = 0.
(27)
Let π(π‘, π ) be Green function with boundary value prob-lem (23); that is,
π (π‘, π ) = min {π‘, π } = {π‘, π‘ β€ π
π , π < π‘.(28)
Then the solution with boundary value problem (23) andsolution for nonlinear integral equation with type of Ham-merstein
π₯ (π‘) = β«
1
0
π (π‘, π ) {π (π ) [π₯ (π )]π+
1
1 + π (π ) π₯ (π )} ππ (29)
are equivalent, where maxπ‘β[0,1]
β«1
0π(π‘, π )ππ = 1/2.
Theorem10. Let π(π‘), π(π‘) be nonnegative continuous functionin [0, 1], π = max
π‘β[0,1]π(π‘), π = max
π‘β[0,1]π(π‘). If π < 1,ππ +
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4 Discrete Dynamics in Nature and Society
π < 2, then boundary value problem (23) has a unique solutionπ₯β(π‘) such that 0 β€ π₯β(π‘) β€ 1 (π‘ β [0, 1]). Moreover, for any
initial function π₯0(π‘), π¦0(π‘), such that
0 β€ π₯0(π‘) β€ 1, 0 β€ π¦
0(π‘) β€ 1 (π‘ β [0, 1]) , (30)
we make iterative sequence:
π₯π(π‘) = β«
1
0
π (π‘, π ) {π (π ) [π₯πβ1
(π )]π
+1
1 + π (π ) π¦πβ1
(π )} ππ ,
π¦π(π‘) = β«
1
0
π (π‘, π ) {π (π ) [π¦πβ1
(π )]π
+1
1 + π (π ) π₯πβ1
(π )} ππ ,
π = 1, 2, 3, . . . .
(31)
Then π₯π(π‘) and π¦
π(π‘) are all uniformly converge to π₯β(π‘) on
[0, 1], and we have error estimates:
π₯π (π‘) (π¦π (π‘)) β π₯β(π‘)
β€ 2(ππ + π
2)
π
,
π‘ β [0, 1] , π = 1, 2, 3, . . . .
(32)
Proof. Let πΈ = πΆ[0, 1], π = {π₯ β πΈ | π₯(π‘) β₯ 0, π‘ β[0, 1]}, βπ₯β = max
π‘β[0,1]|π₯(π‘)| denote norm of; then πΈ has
become π΅ππππβ space, π is normal cone of πΈ, and its normalconstant π = 1. It is obvious that integral Equation (24)transforms to operator equation π΄(π₯, π₯) = π₯, where
π΄ (π₯, π¦) (π‘)
= β«
1
0
π (π‘, π ) {π (π ) [π₯ (π )]π+
1
1 + π (π ) π¦ (π )} ππ , π‘ β [0, 1] .
(33)
Set π’ = π’(π‘) β‘ 0, V = V(π‘) β‘ 1; then π· = [0, 1] denoteordering interval of πΈ, π΄ : π· Γ π· β πΈ is mixed monotoneoperator, and 0 β€ π΄(0, 1), π΄(1, 0) β€ (1 + π)/2 < 1.
Set
πΏπ₯ (π‘) = β«
1
0
π (π , π‘) [ππ (π ) + π (π )] π₯ (π ) ππ , π‘ β [0, 1] .
(34)
Then πΏ : πΈ β πΈ is bounded linear operator, its spectralradius π(πΏ) β€ (ππ+π)/2 < 1, and for anyπ₯, π¦ β πΈ, 0 β€ π₯(π‘) β€π¦(π‘) β€ 1 such that 0 β€ π΄(π¦, π₯)(π‘)βπ΄(π₯, π¦)(π‘) β€ πΏ(π¦βπ₯)(π‘),π΄is πΏ-ordering symmetric contraction operator, by Theorem 2(where πΌ = 0); thenTheorem 10 has been proved.
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper.
Acknowledgments
This work is supported by the NSF of Henan EducationBureau (2000110019) and by the NSF of Shangqiu(200211125).
References
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[3] Q. Zhang, βContraction mapping principle of mixed monotonemapping and applications,βHenan Science, vol. 18, no. 2, pp. 121β125, 2000.
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