Open Access

On the System of Nonlinear Mixed Implicit Equilibrium Problems in Hilbert Spaces

Journal of Inequalities and Applications20102010:437976

DOI: 10.1155/2010/437976

Received: 22 December 2009

Accepted: 10 January 2010

Published: 17 January 2010

Abstract

We use the Wiener-Hopf equations and the Yosida approximation notions to prove the existence theorem of a system of nonlinear mixed implicit equilibrium problems (SMIE) in Hilbert spaces. The algorithm for finding a solution of the problem (SMIE) is suggested; the convergence criteria and stability of the iterative algorithm are discussed. The results presented in this paper are more general and are viewed as an extension, refinement, and improvement of the previously known results in the literature.

1. Introduction and Preliminaries

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq1_HTML.gif be a real Hilbert space whose inner product and norm are denoted by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq2_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq3_HTML.gif respectively. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq4_HTML.gif be given two bi-functions satisfying https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq5_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq6_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq7_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq8_HTML.gif be a nonlinear mapping. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq9_HTML.gif be a nonempty closed convex subset of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq10_HTML.gif . In this paper, we consider the following problem.

Find https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq11_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ1_HTML.gif
(1.1)

The problem of type (1.1) is called the system of nonlinear mixed implicit equilibrium problems.

We denote by SMIE https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq12_HTML.gif the set of all solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq13_HTML.gif of the problem (1.1).

Some examples of the problem (1.1) are as follows.
  1. (I)

    If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq14_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq15_HTML.gif is a maximal monotone mapping for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq16_HTML.gif then the problem (1.1) becomes the following problem.

     

Find https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq17_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ2_HTML.gif
(1.2)

which is called the system of variational inclusion problems. In particular, when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq18_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq19_HTML.gif the problem (1.2) is reduced to the problem, so-called the generalized variational inclusion problem, which was studied by Kazmi and Bhat [1].

It is worth noting that the variational inclusions and related problems are being studied extensively by many authors and have important applications in operations research, optimization, mathematical finance, decision sciences, and other several branches of pure and applied sciences.
  1. (II)

    If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq20_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq21_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq22_HTML.gif is a real valued function for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq23_HTML.gif Then the problem (1.1) reduces to the following problem.

     

Find https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq24_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ3_HTML.gif
(1.3)
Some corresponding results to the problem (1.3) were considered by Kassay and Kolumbán [2] when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq25_HTML.gif .
  1. (III)

    For each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq26_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq27_HTML.gif be a nonlinear mapping and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq28_HTML.gif fixed positive real numbers. If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq29_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq30_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq31_HTML.gif , then the problem (1.3) reduces to the following problem.

     

Find https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq32_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ4_HTML.gif
(1.4)
which is called the system of nonlinear mixed variational inequalities problems. A special case of the problem (1.4), when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq33_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq34_HTML.gif , has been studied by He and Gu [3].
  1. (IV)

    If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq35_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq36_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq37_HTML.gif is the indicator function of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq38_HTML.gif defined by

     
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ5_HTML.gif
(1.5)

then the problem (1.4) reduces to the following problem.

Find https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq39_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ6_HTML.gif
(1.6)

which is called the system of nonlinear variational inequalities problems. Some corresponding results to the problem (1.6) were studied by Agarwal et al. [4], Chang et al. [5], Cho et al. [6], J. K. Kim and D. S. Kim, [7] and Verma [8, 9].

For the recent trends and developments in the problem (1.6) and its special cases, see [3, 811] and the references therein, for examples.
  1. (V)

    If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq40_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq41_HTML.gif is a univariate mapping, then the problem (1.6) reduces to the following problem.

     

Find https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq42_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ7_HTML.gif
(1.7)

which is known as the classical variational inequality introduced and studied by Stampacchia [12] in 1964. This shows that a number of classes of variational inequalities and related optimization problems can be obtained as special cases of the system (1.1) of mixed equilibrium problems.

Inspired and motivated by the recent research going on in this area, in this paper, we use the Wiener-Hopf equations and the Yosida approximation notion to suggest and prove the existence and uniqueness of solutions for the problem (1.1). We also discuss the convergence criteria and stability of the iterative algorithm. The results presented in this paper improve and generalize many known results in the literature.

In the sequel, we need the following basic concepts and lemmas.

Definition 1.1 (Blum and Oettli [13]).

A real valued bifunction https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq43_HTML.gif is said to be:

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq44_HTML.gif
monotone if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ8_HTML.gif
(1.8)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq45_HTML.gif
strictly monotone if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ9_HTML.gif
(1.9)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq46_HTML.gif
upper-hemicontinuous if
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ10_HTML.gif
(1.10)

Definition 1.2.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq47_HTML.gif
A function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq48_HTML.gif is said to be lower semicontinuous at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq49_HTML.gif if, for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq50_HTML.gif there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq51_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ11_HTML.gif
(1.11)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq52_HTML.gif denotes the ball with center https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq53_HTML.gif and radius https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq54_HTML.gif , that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ12_HTML.gif
(1.12)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq55_HTML.gif
The function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq56_HTML.gif is said to be lower semicontinuous on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq57_HTML.gif if it is lower semicontinuous at every point of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq58_HTML.gif .

Lemma 1.3 (Combettes and Hirstoaga [14]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq59_HTML.gif be a nonempty closed convex subset of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq60_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq61_HTML.gif be a bifunction of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq62_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq63_HTML.gif satisfying the following conditions:

(C1) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq64_HTML.gif is monotone and upper hemicontinuous;

(C2) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq65_HTML.gif is convex and lower semi-continuous for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq66_HTML.gif .

For all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq67_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq68_HTML.gif , define a mapping https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq69_HTML.gif as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ13_HTML.gif
(1.13)

Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq70_HTML.gif is a single-valued mapping.

Definition 1.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq71_HTML.gif be a positive number. For any bi-function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq72_HTML.gif the associated Yosida approximation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq73_HTML.gif over https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq74_HTML.gif and the corresponding regularized operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq75_HTML.gif are defined as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ14_HTML.gif
(1.14)
in which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq76_HTML.gif is the unique solution of the following problem:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ15_HTML.gif
(1.15)

Remark 1.5.

Definition 1.4 is an extension of the Yosida approximation notion introduced in [15]. The existence and uniqueness of the solution of the problem (1.15) follow from Lemma 1.3.

Definition 1.6.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq77_HTML.gif be a set-valued mapping.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq78_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq79_HTML.gif is said to be monotone if, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq80_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ16_HTML.gif
(1.16)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq81_HTML.gif
A monotone operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq82_HTML.gif is said to bemaximal if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq83_HTML.gif is not properly contained in any other monotone operators.

Example 1.7 (Huang et al. [16]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq84_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq85_HTML.gif is a maximal monotone mapping. Then it directly follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ17_HTML.gif
(1.17)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq86_HTML.gif is the Yosida approximation of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq87_HTML.gif and we recover the classical concepts.

Using the idea as in Huang et al. [16], we have the following result.

Lemma 1.8.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq88_HTML.gif is a monotone function, then the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq89_HTML.gif is a nonexpansive mapping, that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ18_HTML.gif
(1.18)

Proof.

From (1.15), for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq90_HTML.gif , we can obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ19_HTML.gif
(1.19)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ20_HTML.gif
(1.20)
By adding (1.19) with (1.20) and using the monotonicity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq91_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ21_HTML.gif
(1.21)
and so
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ22_HTML.gif
(1.22)

This implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq92_HTML.gif is a nonexpansive mapping. This completes the proof.

Now, for solving the problem (1.1), we consider the following equation: let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq93_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq94_HTML.gif be fixed positive real numbers. Find https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq95_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ23_HTML.gif
(1.23)

Lemma 1.9.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq96_HTML.gif
is a solution of the problem (1.1) if and only if the problem (1.23) has a solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq97_HTML.gif where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ24_HTML.gif
(1.24)
that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ25_HTML.gif
(1.25)

Proof.

The proof directly follows from the definitions of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq98_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq99_HTML.gif .

In this paper, we are interested in the following class of nonlinear mappings.

Definition 1.10.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq100_HTML.gif
A mapping https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq101_HTML.gif is said to be https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq102_HTML.gif -strongly monotone if there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq103_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ26_HTML.gif
(1.26)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq104_HTML.gif
A mapping https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq105_HTML.gif is said to be https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq106_HTML.gif -Lipschitz if there exist constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq107_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ27_HTML.gif
(1.27)

2. Existence of Solutions of the Problem (1.1)

In this section, we give an existence theorem of solutions for the problem (1.1). Firstly, in view of Lemma 1.9, we can obtain the following, which is an important tool, immediately.

Lemma 2.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq108_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq109_HTML.gif if and only if there exist positive real numbers https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq110_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq111_HTML.gif is a fixed point of the mapping https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq112_HTML.gif defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ28_HTML.gif
(2.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq113_HTML.gif are defined, respectively, by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ29_HTML.gif
(2.2)

Now, we are in position to prove the existence theorem of solutions for the problem (1.1).

Theorem 2.2.

For each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq114_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq115_HTML.gif be a monotone bi-function. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq116_HTML.gif be a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq117_HTML.gif -strongly monotone with respect to the first argument and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq118_HTML.gif -Lipschitz mapping and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq119_HTML.gif be a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq120_HTML.gif -strongly monotone with respect to the second argument and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq121_HTML.gif -Lipschitz mapping. Suppose that there are positive real numbers https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq122_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ30_HTML.gif
(2.3)

Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq123_HTML.gif is a singleton.

Proof.

Notice that, in view of Lemma 2.1, it is sufficient to show that the mapping https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq124_HTML.gif defined in Lemma 2.1 has the unique fixed point. Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq125_HTML.gif is nonexpansive, we have the following estimate:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ31_HTML.gif
(2.4)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq126_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq127_HTML.gif -Lipschitz mapping and, for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq128_HTML.gif , the mapping https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq129_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq130_HTML.gif -strongly monotone, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ32_HTML.gif
(2.5)
Consequently, from (2.4)-(2.5), it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ33_HTML.gif
(2.6)
Next, we have the following estimate:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ34_HTML.gif
(2.7)
From (2.6) and (2.7), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ35_HTML.gif
(2.8)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ36_HTML.gif
(2.9)
Now, define the norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq131_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq132_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ37_HTML.gif
(2.10)
Notice that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq133_HTML.gif is a Banach space and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ38_HTML.gif
(2.11)

By the condition (2.3), we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq134_HTML.gif , which implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq135_HTML.gif is a contraction mapping. Hence, by Banach contraction principle, there exists a unique https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq136_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq137_HTML.gif This completes the proof.

3. Convergence and Stability Analysis

In view of Lemma 2.1, for the fixed point formulation of the problem (2.1), we suggest the following iterative algorithm.

3.1. Mann Type Perturbed Iterative Algorithm (MTA)

For any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq138_HTML.gif , compute approximate solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq139_HTML.gif given by the iterative schemes:

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ39_HTML.gif
(3.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq140_HTML.gif is a sequence of real numbers such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq141_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq142_HTML.gif

In order to consider the convergence theorem of the sequences generated by the algorithm (MTA), we need the following lemma.

Lemma 3.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq143_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq144_HTML.gif be two nonnegative real sequences satisfying the following conditions. There exists a positive integer https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq145_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ40_HTML.gif
(3.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq146_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq147_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq148_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq149_HTML.gif

Now, we prove the convergence theorem for a solution for the problem (1.1).

Theorem 3.2.

If all the conditions of the Theorem 2.2 hold, then the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq150_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq151_HTML.gif generated by the algorithm (3.1) converges strongly to the unique solution for the problem (1.1).

Proof.

It follows from Theorem 2.2 that there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq152_HTML.gif which is the unique solution for the problem (1.1). Moreover, in view of Lemma 2.1, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ41_HTML.gif
(3.3)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq153_HTML.gif is nonexpansive, from the iterative sequences (3.1) and (3.3), it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ42_HTML.gif
(3.4)
Next, we have the following estimate:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ43_HTML.gif
(3.5)
Substituting (3.5) into (3.4) yields that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ44_HTML.gif
(3.6)
Similarly, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ45_HTML.gif
(3.7)
Thus, from (3.6) and (3.7), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ46_HTML.gif
(3.8)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq154_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq155_HTML.gif are givenin (2.9). Setting
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ47_HTML.gif
(3.9)
From the condition (2.3), it follows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq156_HTML.gif and so https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq157_HTML.gif . Moreover, since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq158_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq159_HTML.gif . Hence all the conditions of Lemma 3.1 are satisfied and so https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq160_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq161_HTML.gif that is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ48_HTML.gif
(3.10)

Thus the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq162_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq163_HTML.gif converges strongly to a solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq164_HTML.gif for the problem (1.1). This completes the proof.

3.2. Stability of the Algorithm (MTA)

Consider the following definition as an extension of the concept of stability of the iterative procedure given by Harder and Hicks [17].

Definition 3.3 (Kazmi and Khan [18]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq165_HTML.gif be a Hilbert space and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq166_HTML.gif be nonlinear mappings. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq167_HTML.gif be defined as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq168_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq169_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq170_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq171_HTML.gif defines an iterative procedure which yields a sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq172_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq173_HTML.gif . Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq174_HTML.gif and the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq175_HTML.gif converges to some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq176_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq177_HTML.gif be an arbitrary sequence in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq178_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ49_HTML.gif
(3.11)

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq179_HTML.gif implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq180_HTML.gif , then the iterative procedure https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq181_HTML.gif is said to be https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq182_HTML.gif -stable or stable with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq183_HTML.gif .

Theorem 3.4.

Assume that all the conditions of Theorem 2.2 hold. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq184_HTML.gif be an arbitrary sequence in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq185_HTML.gif and define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq186_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ50_HTML.gif
(3.12)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ51_HTML.gif
(3.13)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq187_HTML.gif is a sequence defined in (3.1). If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq188_HTML.gif is defined as in (2.1), then the iterative procedure generated by (3.1) is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq189_HTML.gif -stable.

Proof.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq190_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq191_HTML.gif be the unique fixed point of the mapping https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq192_HTML.gif This means that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ52_HTML.gif
(3.14)
Now, from (3.12) and (3.13), it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ53_HTML.gif
(3.15)
Notice that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq193_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq194_HTML.gif , which implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ54_HTML.gif
(3.16)
Using (3.16) and the assumption https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_IEq195_HTML.gif , it follows from (3.15) that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F437976/MediaObjects/13660_2009_Article_2151_Equ55_HTML.gif
(3.17)

This completes the proof.

Declarations

Acknowledgments

The first author was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050). The second author was supported by the Commission on Higher Education and the Thailand Research Fund (project no. MRG5180178).

Authors’ Affiliations

(1)
Department of Mathematics Education and the RINS, Gyeongsang National University
(2)
Department of Mathematics, Faculty of Science, Naresuan University

References

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© Y. J. Cho and N. Petrot. 2010

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