The Solution of Binary Nonlinear Operator Equations with Applications ()
1. Introduction
In recent years, more and more scholars have studied binary operator equations and have obtained many conclusions, such as references [1-3] etc. In this paper, we will discuss solutions for ordinal symmetric contraction operator and obtain some general conclusions; some corresponding results of references [4,5] are improved and generalized. Finally, we apply our conclusions to two point boundary value problems with two degree superlinear ordinary differential equations.
In the following, let E always be a real Banach space which is partially ordered by a cone P, P be normal cone of E, N is normal constant of P, partial order ≤ is determined by P, denotes zero element of E. For and, let
denotes an ordering interval of E.
The concepts of normal cone and partially order, mixed monotone operator, coupled solutions of operator equations etc. see [6].
Definition 1.1. Let be binary operator, A is said to be L-ordering symmetric contraction operator if there exists a bounded linear operator, which its spectral radius such that
for any, where L is called contraction operator of A.
2. Main Results
Theorem 2.1. Let be L-ordering symmetric contraction operator, and there exists a, for any, such that
. (1)
If condition
(H1) ;
or
(H2)
holds, then the following statements hold:
(C1) has a unique solution, and for any coupled solution such that;
(C2) For any, we make up symmetric iterative sequences
(2)
then
and for any, there exists a natural numbers m, if, we get error estimates for iterative sequences (2):
.
Proof. Set
if condition (H1) or (H2) holds, then it is obvious
by (1), we easily prove that is mixed monotone operator, and for any such that
where
is a bounded linear operator, I is identical operator.
By the mathematical induction, we easily prove that
where
.
By the character of normal cone P, we implies
For any, since
so there exists a natural numbers m, if, such that
and constant.
Considering mixed monotone operator and constant, by Theorem 3 in reference [3], then we know has an unique solution, and for any coupled solution such that
.
From
and uniqueness of solutions with, then we have and.
We take note of that and have same coupled solution, therefore coupled solution for must be coupled solution for x, consequently, (C1) has been proved.
Considering that iterative sequence (2) and set iterative sequences:
where it is obvious that
by the mathematical induction and character of mixed monotone of B, then
hence
moreover, if, we get
consequently,.
Remark 1. When, Theorem 1 in [4] is a special case of this paper Theorem 2.1 under condition (H1) or (H2).
Corollary 2.1. Let be L-ordering symmetric contraction operator, if there exists a such that A satisfies condition of Theorem 2.1, then (C1), (C2) hold and the following statements holds:
(C3) For any and, we make up iterative sequences
(3)
or
(4)
where thus, and there exists a natural numbers m, if, we have error estimates for iterative sequences (3) or (4):
. (5)
Proof. By the character of mixed monotone of A, then (1) and (C1), (C2) [in (1), (C2) where] hold. In the following, we will prove (C3).
Consider iterative sequence (3), since
so we get
by the mathematical induction, we easily prove
hence
It is clear
For any, , since
thus there exists a natural numbers m, if, such that
Moreover,
consequently, ,.
Similarly, we can prove (4).
Theorem 2.2. Let be L-ordering symmetric contraction operator, if there exists a such that
then the following statements holds:
(C4) Operator equation
has an unique of solution, and for its any coupled solution, such that;
(C5) For any, we make up symmetric iterative sequence
(6)
(7)
then
and that for any and, there exists a natural numbers m, if, then we have error estimates for iterative sequences (6) and (7) respectively:
(8)
Proof. Set
or
we can prove this theorem imitate proof of Theorem 2.1, over.
Similarly, we can prove following theorems.
Theorem 2.3. Let be L-ordering symmetric contraction operator, if there exists a such that
then the following statements holds:
(C6) Equation
has an unique solution, and for any coupled solution such that;
(C7) For any, we make up symmetric iterative sequence
(9)
then that, moreover, , there exist natural number m, if, then we have error estimates for iterative sequence (9):
;
(C8) For any , , we make up symmetry iterative sequence
Then
and there exists a natural numbers m, if, we have error estimates for iterative sequence (8).
Remark 2. When, Corollary 2 in [4] is a special case of this paper Theorem 2.1 - 2.3.
Remark 3. The contraction constant of operator in [5] is expand into the contraction operator of this paper.
Remark 4. Operator A of this paper does not need character of mixed monotone as operator in [6].
3. Application
We consider that two point boundary value problems for two degree super linear ordinary differential equations
(10)
Let be Green function with boundary value problem (7), that is
then that the solution with boundary value problem (7) and solution for nonlinear integral equation with type of Hammerstein
(11)
is equivalent, where
.
Theorem 3.1. Let are nonnegative continuous function in
.
If, then boundary value problem (7) have an unique solution such that
;
Moreover, for any initial function such that
we make up iterative sequence
Then, uniform convergence to on, and we have error estimates
Proof. Let
,
denote norm of E, then that E has become Banach space, P is normal cone of E and its normal constant N = 1. It is obvious that integral Equation (8) is transformed to operator equation, where
Set
then denote ordering interval of E, is mixed monotone operator ,and
.
Set
then is bounded linear operator, its spectral radius and for any, such that that is, A is L-ordering symmetric contraction operator, by Theorem 2.1 (where), then Theorem 3.1 has be proved.
4. Acknowledgements
Supported by the Natural Science Foundation of Henan under Grant 122300410425; the NSF of Henan Education Bureau (2000110019); Supported by the NSF of Shangqiu (200211125).