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Article

Cayley Inclusion Problem Involving XOR-Operation

1
Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
2
Center for Fundamental Science, and Research Center for Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung 80708, Taiwan
3
Department of Medical Research, Kaohsiung Medical University Hospital, Kaohsiung 80708, Taiwan
*
Author to whom correspondence should be addressed.
Submission received: 9 February 2019 / Revised: 14 March 2019 / Accepted: 21 March 2019 / Published: 25 March 2019
(This article belongs to the Special Issue Fixed Point, Optimization, and Applications)

Abstract

:
In this paper, we study an absolutely new problem, namely, the Cayley inclusion problem which involves the Cayley operator and a multi-valued mapping with XOR-operation. We have shown that the Cayley operator is a single-valued comparison and it is Lipschitz-type-continuous. A fixed point formulation of the Cayley inclusion problem is shown by using the concept of a resolvent operator as well as the Yosida approximation operator. Finally, an existence and convergence result is proved. An example is constructed for some of the concepts used in this work.
MSC:
47H05; 49H10; 47J25

1. Introduction

It is well known that inclusion problems were introduced and studied as a generalization of equilibrium problems, which include a vast range of problems in analysis such as variational inequalities, vector optimization, game theory, fixed point problems, the Nash equilibrium problem, complementary problems, traffic equilibrium problems, economics, etc., see [1,2,3]. It is interesting to note that the term “Variational inclusion”, is understood with different aspects in several works. That is, it means simply multi-valued variational inequalities in [4,5] and the problem of finding the zeros of maximal monotone mappings in [6,7,8], etc. Variational inclusions involving different kinds of operators are useful and have a wide range of applications in industry, mathematical finance, decision sciences, ecology, engineering sciences, etc., see [9,10,11,12,13,14,15].
Due to the fact that the projection methods cannot be used to solve variational inclusion problems, the resolvent operator methods came into the picture to solve them efficiently. It is also known that the monotone operators in abstract spaces can be regularized into single-valued Lipschitzian monotone operator through a process known as Yosida approximation, see [16,17,18,19].
The XOR-operation ⊕ is a binary operation and behaves like ADD operation, which is associative as well as commutative. XOR-operation depicts interesting facts and observations and forms various real time applications, i.e., data encryption, error detection in digital communication, parity check and helps to implement multi-layer perception in neural networks.
Many problems related to ordered variational inequalities and ordered equations were studied by H.G.Li together with his co-authors, see [20,21,22,23,24] and I.Ahmad with his co-authors, see [25,26]. Considering all the facts mentioned above, in this paper, we introduce and study a quite new and interesting problem which we call Cayley inclusion problem involving XOR-operation. The Cayley inclusion problem involves a Cayley operator and a multi-valued mapping. We have shown some properties of the Cayley operator, that is, it is single-valued, comparison as well as Lipschitz-type-continuous. A fixed point formulation of the Cayley inclusion problem is given by using the concept of resolvent operator and Yosida approximation operator. An iterative algorithm is established and finally an existence and convergence result is proved for the Cayley inclusion problem involving XOR-operation. An example is constructed to illustrate some of the concepts used in this paper.

2. Preliminaries

Throughout this paper, we suppose that H is a real ordered Hilbert space endowed with a norm . and an inner product · , · , d is the metric induced by the norm . and 2 H is the family of all nonempty subsets of H .
Now, we recall some known concepts are results which we need to prove the main result of this paper and can be found in [22,23,24,27,28].
Definition 1.
A nonempty closed convex subset C of H is said to be a cone, if
(i) 
for any x C and λ > 0 , then λ x C ,
(ii) 
for any x , C and x C , then x = 0 .
Definition 2.
A cone C is said to be normal if and only if, there exist a constant λ N > 0 such that 0 x y implies x λ N y , where λ N is the normal constant of C.
Definition 3.
Let C be a cone. For arbitrary element x , y H , x y holds if and only if, x y C . The relation in H is called partial ordered relation.
Definition 4.
For arbitrary elements x , y H , if x y (or y x ) holds, then x and y are said to be comparable to each other (denoted by x y ).
Definition 5.
For arbitrary elements x , y H , l u b { x , y } and g l b { x , y } means least upper bound and greatest upper bound of the set { x , y } . Suppose l u b { x , y } and g l b { x , y } exist, then some binary operations are defined as follows:
(i) 
x y = l u b { x , y } ,
(ii) 
x y = g l b { x , y } ,
(iii) 
x y = ( x y ) ( y x ) ,
(iv) 
x y = ( x y ) ( y x ) .
The operations , , and are called OR, AND, XOR and XNOR operations, respectively.
Lemma 1.
If x y , then lub { x , y } and glb { x , y } exist, x y y x and 0 ( x y ) ( y x ) .
Lemma 2.
For any natural number n , x y n and y n y as n , then x y .
Proposition 1.
Let be an XOR-operation and be an XNOR-operation. Then the following relations hold:
(i) 
x x = 0 , x y = y x = ( x y ) = ( y x ) ,
(ii) 
if x 0 , then x 0 x x 0 ,
(iii) 
( λ x ) ( λ y ) = | λ | ( x y ) ,
(iv) 
0 x y , if x y ,
(v) 
if x y , then x y = 0 if and only if x = y ,
(vi) 
( x + y ) ( u + v ) ( x u ) + ( y v ) ,
(vii) 
( x + y ) ( u + v ) ( x v ) + ( y u ) ,
(viii) 
if x , y and w are comparable to each other, then ( x y ) x w + w y ,
(ix) 
α x β x = | α β | x = ( α β ) x , if x 0 , x , y , u , v H and α , β , λ R .
Proposition 2.
Let C be a normal cone in H with normal constant λ N , then for each x , y H , the following relations hold:
(i) 
0 0 = 0 = 0 ,
(ii) 
x y x y x + y ,
(iii) 
x y x y λ N | x y ,
(iv) 
if x y , then x y = x y .
Definition 6.
Let A : H H be a single-valued mapping.
(i) 
A is said to be a comparison mapping if x y then A ( x ) A ( y ) , x A ( x ) and y A ( y ) , f o r a l l x , y H ,
(ii) 
A is said to be strongly comparison mapping, if A is a comparison mapping and A ( x ) A ( y ) if and only if x y , f o r a l l x , y H .
Definition 7.
A mapping A : H H is said to be β-ordered comparison mapping, if A is comparison mapping and
A ( x ) A ( y ) β ( x y ) , f o r 0 β 1 , f o r a l l x , y H .
Definition 8.
Let M : H 2 H be a multi-valued mapping. Then
(i) 
M is said to be a comparison mapping, if for any v x M ( x ) , x v x , and if x y , then for any v x M ( x ) and any v y M ( y ) , v x v y , f o r a l l x , y H ,
(ii) 
a comparison mapping M is said to be α-non-ordinary difference mapping, if for each x , y H , v x M ( x ) and v y M ( y ) such that
( v x v y ) α ( x y ) = 0 ,
(iii) 
a comparison mapping M is said to be γ-ordered rectangular, if there exists a constant γ > 0 , and for any x , y H , there exist v x M ( x ) and v y M ( y ) such that
v x v y , ( x y ) γ x y 2 ,
holds.
(iv) 
M is said to be weak comparison mapping, if for any x , y H , x y , then there exist v x M ( x ) and v y M ( y ) such that x v x , y v y and v x v y .
(v) 
M is said to be λ-weak ordered different comparison mapping, if there exist a constant λ > 0 such that for any x , y H , there exist v x M ( x ) , v y M ( y ) , λ ( v x v y ) ( x y ) holds.
(vi) 
a weak comparison mapping M : H 2 H is said to be a ( γ , λ ) -weak ordered rectangular different multi-valued mapping, if M is a γ-ordered rectangular and λ-weak ordered different comparison mapping and [ I + λ M ] ( H ) = H , for λ > 0 .
Definition 9.
Let M : H 2 H be a multi-valued mapping. The operator R I , λ M : H H defined by
R I , λ M ( x ) = [ I + λ M ] 1 ( x ) , f o r a l l x H ,
is called the resolvent operator associated with M, where λ > 0 is a constant.
It is well known that the resolvent operator associated with M is single-valued.
Definition 10.
The Yosida approximation operator J I , λ M associated with M is defined by
J I , λ M ( x ) = 1 λ [ I R I , λ M ] ( x ) , f o r a l l x H ,
where λ > 0 is a constant.
Now we define the Cayley operator based on resolvent operator (1)
Definition 11.
The Cayley operator C I , λ M of M is defined as:
C I , λ M ( x ) = [ 2 R I λ M I ] ( x ) , f o r a l l x H ,
where I is the identity operator.
Proposition 3.
Let M : H 2 H is a γ-ordered rectangular multi-valued mapping. Then, the Yosida approximation operator J I , λ M is single-valued, for λ > 0 .
Proof. 
For the proof we refer to [25]. □
Proposition 4.
Let M : H 2 H is a γ-ordered rectangular multi-valued mapping. Then, the Cayley operator C I , λ M associated with M is single valued, for γ λ > 1 .
Proof. 
Let x , y C I , λ M ( u ) . Then
x C I , λ M ( u ) = ( 2 R I , λ M I ) ( u ) , 1 2 ( x + u ) R I , λ M ( u ) = [ I + λ M ] 1 ( u ) , i . e . , u 1 2 ( x + u ) [ I + λ M ] , u 1 2 ( x + u ) + 1 2 λ M ( x + u ) , 2 u ( x + u ) + λ M ( x + u ) , u x λ M ( x + u ) .
Thus 1 λ ( u x ) M ( z 1 ) , where z 1 = x + u . Let 1 λ ( u x ) = v z 1 , then v z 1 M ( z 1 ) .
Similarly, for y C I , λ M ( u ) , we have v z 2 M ( z 2 ) , where
v z 2 = 1 λ ( u y ) a n d z 2 = y + u .
Now, we evaluate v z 1 v z 2 by using the values of v z 1 and v z 2 calculated above and using Proposition 1.
v z 1 v z 2 = [ 1 λ ( u x ) 1 λ ( u y ) ] = 1 λ [ ( u x ) ( u y ) ] 1 λ ( x y ) .
Since M is γ -ordered rectangular multi-valued mapping and using (4), we have
γ ( x + u ) ( y + u ) 2 v z 1 v z 2 , [ ( x + u ) ( y + u ) ] = v z 1 v z 2 , x y 1 λ x y , x y ,
which implies that
γ x y 2 1 λ x y 2 ( γ λ 1 ) x y 2 0 i . e . , x y = 0 w h i c h i m p l i e s t h a t x = y .
Therefore, the Cayley operator C I , λ M associated with M is single-valued. □
Proposition 5.
For any x , y H , let x y and R I , λ M ( x ) R I , λ M ( y ) .Then, the Cayley operator C I , λ M associated with M is a comparison mapping.
Proof. 
For any x , y H , let x y then obviously I ( x ) I ( y ) . As R I , λ M is a comparison mapping, we have R I , λ M ( x ) R I , λ M ( y ) . Thus, we have
[ 2 R I , λ M I ] ( x ) [ 2 R I , λ M I ] ( y ) .
i.e., we have
C I , λ M ( x ) C I , λ M ( y ) .
Therefore, the Cayley operator C I , λ M associated with M is a comparison mapping. □
Lemma 3.
Let M : H 2 H be a γ-ordered rectangular multi-valued mapping with respect to R I , λ M , for λ > 1 γ . Then the following condition holds:
R I , λ M ( x ) R I , λ M ( y ) θ x y , f o r a l l x , y H , w h e r e θ = 1 γ λ 1 .
That is, the resolvent operator R I , λ M is Lipschitz-type-continuous.
Proof. 
For the proof we refer to [25]. □
Lemma 4.
Let M : H 2 H be a ( γ , λ ) -weak ordered rectangular different multi-valued mapping with respect to R I , λ M and the resolvent operator R I , λ M defined by (1) is θ-Lipschitz-type-continuous. Then, the Yosida approximation operator J I , λ M defined by (2) is θ -Lipschitz-type-continuous. i.e.,
J I , λ M ( x ) J I , λ M ( y ) θ x y , f o r a l l x , y H , w h e r e θ = 1 γ λ 1 , θ = γ γ λ 1 a n d γ λ > 1 .
That is, the Yosida approximation operator J I , λ M is Lipschitz-type-continuous.
Proof. 
For the proof we refer to [25]. □
Lemma 5.
Let M : H 2 H be ( γ , λ ) -weak ordered rectangular different multi-valued mapping with respect to R I , λ M and the resolvent operator R I , λ M is θ-Lipschitz-type-continuous. Then, the Cayley operator C I , λ M defined by (3) is ( 2 θ + 1 ) -Lipschitz-type-continuous. That is,
C I , λ M ( x ) C I , λ M ( y ) ( 2 θ + 1 ) x y , f o r a l l x , y H , w h e r e θ = 1 γ λ 1 a n d γ λ > 1 .
Proof. 
Using Cauchy-Schwartz inequality and Proposition 1, we have
C I , λ M ( x ) C I , λ M ( y ) 2 = C I , λ M ( x ) C I , λ M ( y ) , C I , λ M ( x ) C I , λ M ( y ) = ( 2 R I , λ M ( x ) I ( x ) ) ( 2 R I , λ M ( y ) I ( y ) ) , ( 2 R I , λ M ( x ) I ( x ) ) ( 2 R I , λ M ( y ) I ( y ) ) = ( 2 R I , λ M ( x ) 2 R I , λ M ( y ) ) + ( I ( x ) I ( y ) ) , ( 2 R I , λ M ( x ) 2 R I , λ M ( y ) ) + ( I ( x ) I ( y ) ) = 2 R I , λ M ( x ) 2 R I , λ M ( y ) , 2 R I , λ M ( x ) 2 R I , λ M ( y ) + 2 R I , λ M ( x ) 2 R I , λ M ( y ) , I ( x ) I ( y ) + I ( x ) I ( y ) , 2 R I , λ M ( x ) 2 R I , λ M ( y ) + I ( x ) I ( y ) , I ( x ) I ( y ) 4 R I , λ M ( x ) R I , λ M ( y ) 2 + 2 R I , λ M ( x ) R I , λ M ( y ) x y + 2 R I , λ M ( x ) R I , λ M ( y ) x y + x y 2 = 2 R I , λ M ( x ) R I , λ M ( y ) + x y 2 i . e . , C I , λ M ( x ) C I , λ M ( y ) 2 R I , λ M ( x ) R I , λ M ( y ) + x y .
Using the Lipschitz-type-continuity of the resolvent operator R I , λ M , we have
C I , λ M ( x ) C I , λ M ( y ) 2 θ x y + x y , i . e . , C I , λ M ( x ) C I , λ M ( y ) ( 2 θ + 1 ) x y , w h e r e θ = 1 γ λ 1 a n d γ λ > 1 .
i.e., the Cayley operator C I , λ M is Lipschitz-type-continuous. □
We construct the following example in support of some of the concepts used in this paper.
Example 1.
Let C H be a normal cone with constant λ N . Let M : H 2 H be the multi-valued mapping defined by M ( x ) = { x + 1 : x H } and x y .
A s x y , c l e a r l y M ( x ) M ( y )
That is, M is a comparison mapping.
L e t v x = x + 1 M ( x ) a n d v y = y + 1 M ( y ) a n d v x v y , ( x y ) = v x v y , x y = ( x + 1 ) ( y + 1 ) , x y = x y , x y = x y 2 4 5 x y 2 , x , y H .
Thus, M is 4 5 -ordered rectangular mapping. Also it is easy to see that for λ = 2 , M is 2-weak ordered different comparison mapping. Hence, M is ( 4 5 , 2 ) -weak ordered rectangular different multi-valued mapping.
The resolvent operator defined by (1) is given by
R I , λ M ( x ) = ( x 2 ) 3 , f o r a l l x H .
Also,
R I , λ M ( x ) R I , λ M ( y ) = ( x 2 ) 3 ( y 2 ) 3 = 1 3 x y 5 3 x y .
That is, the resolvent operator R I , λ M is 5 3 -Lipschitz-type-continuous.
In view of the above, the Cayley operator C I , λ M defined by (3) is of the form:
C I , λ M ( x ) = ( x 4 ) 3 , f o r a l l x H .
It is easy to see that that the Cayley operator defined above is a comparison and single-valued mapping. Also,
C I , λ M ( x ) C I , λ M ( y ) = ( x 4 ) 3 ( y 4 ) 3 = 1 3 x y 13 3 x y , C I , λ M ( x ) C I , λ M ( y ) 13 3 x y , f o r a l l x , y H .
That is, the Cayley operator C I , λ M is 13 3 -Lipschitz-type-continuous.

3. Formulation of The Problem and Existence of Solution

Let H be a real ordered Hilbert space. Let M : H 2 H be the multi-valued mapping and C I , λ M be the Cayley operator. We consider the following problem:
Find x H such that
0 C I , λ M ( x ) M ( x ) .
We call Problem (5) a Cayley inclusion problem involving XOR-operation.
If C = 0 , then the Problem (5) reduces to the problem of finding x H such that
0 M ( x ) .
Problem (6) is a fundamental problem of inclusions in analysis and studied by Li et al. [22] and others.
The following Lemma is a fixed point formulation of Cayley inclusion Problem involving XOR-operation (5).
Lemma 6.
The Cayley inclusion Problem (5) involving XOR-operation has a solution x H if and only if, it satisfies the following equation:
x = R I , λ M λ ( J I , λ M ( x ) C I , λ M ( x ) ) + R I , λ M ( x ) .
Proof. 
From Equation (7), we have
x = R I , λ M λ ( J I , λ M ( x ) C I , λ M ( x ) ) + R I , λ M ( x ) .
Using the definition of resolvent operator and Yosida approximation operator, we obtain
x = [ I + λ M ] 1 λ ( J I , λ M ( x ) C I , λ M ( x ) ) + R I , λ M ( x ) , x + λ M ( x ) = λ . 1 λ ( I R I , λ M ) ( x ) λ C I , λ M ( x ) + R I , λ M ( x ) = x R I , λ M ( x ) λ C I , λ M ( x ) + R I , λ M ( x ) , λ M ( x ) = λ C I , λ M ( x ) ,
which implies that
0 C I , λ M ( x ) M ( x ) ,
i.e., the required Cayley inclusion Problem involving XOR-operation (5). □
Based on Lemma 6, we define the following iterative algorithm for finding the solution of the Cayley inclusion problem involving XOR-operation (5).
Iterative Algorithm 1.
For initial element x 0 H , compute the sequence { x n } by the following iterative scheme:
x n + 1 = ( 1 α ) x n + α R I , λ M { λ ( J I , λ M ( x n ) C I , λ M ( x n ) ) + R I , λ M ( x n ) } ,
where α [ 0 , 1 ] , λ > 0 is a constant and I is the identity operator.
Theorem 1.
Let H be a real ordered Hilbert space and C be a normal cone with normal constant λ N with ordering . Let M : H 2 H be γ-ordered rectangular, ( γ , λ ) -weak ordered rectangular different multi-valued mapping. Let J I , λ M be the Yosida approximation operator defined by (2) and C I , λ M be the Cayley operator defined by (3) such that both the operators are Lipschitz-type-continuous with constant θ and ( 2 θ + 1 ) , respectively. Let x n + 1 x n and J I , λ M ( x ) J I , λ M ( y ) C I , λ M ( x ) C I , λ M ( y ) , for all x , y H , n = 0 , 1 , 2 , 3 , such that the following condition is satisfied:
θ + 2 θ < [ 1 θ ( 1 + λ ) ] λ θ , w h e r e θ = 1 γ λ 1 , θ = γ γ λ 1 a n d γ λ > 1 .
Then the sequence { x n } generated by the Algorithm 1 strongly converges to x , the solution of the Cayley inclusion Problem involving XOR-operation (5). In addition, for any x 0 H , the following condition holds:
x x 0 1 + ( λ N 1 ) [ ( 1 α ) + α θ [ 1 + λ θ + λ ( 2 θ + 1 ) ] 1 [ ( 1 α ) + α θ [ 1 + λ θ 1 + λ ( 2 θ + 1 ) ] × α x 0 + R I , λ M { λ ( J I , λ M ( x 0 ) C I , λ M ( x 0 ) ) + R I , λ M ( x 0 ) } .
Proof. 
By using Algorithm 1 and Proposition 1, we have
0 x n + 1 x n = ( 1 α ) x n + α R I , λ M { λ ( J I , λ M ( x n ) C I , λ M ( x n ) ) + R I , λ M ( x n ) } ( 1 α ) x n 1 + α R I , λ M { λ ( J I , λ M ( x n 1 ) C I , λ M ( x n 1 ) ) + R I , λ M ( x n 1 ) } = ( 1 α ) ( x n x n 1 ) + α ( R I , λ M { λ ( J I , λ M ( x n ) C I , λ M ( x n ) ) + R I , λ M ( x n ) } R I , λ M { λ ( J I , λ M ( x n 1 ) C I , λ M ( x n 1 ) ) + R I , λ M ( x n 1 ) } ) .
Using Proposition 2, we calculate
x n + 1 x n λ N ( 1 α ) ( x n x n 1 ) + α ( R I , λ M { λ ( J I , λ M ( x n ) C I , λ M ( x n ) ) + R I , λ M ( x n ) } R I , λ M { λ ( J I , λ M ( x n 1 ) C I , λ M ( x n 1 ) ) + R I , λ M ( x n 1 ) } ) λ N ( 1 α ) x n x n 1 + λ N α R I , λ M { λ ( J I , λ M ( x n ) C I , λ M ( x n ) ) } R I , λ M { λ ( J I , λ M ( x n 1 ) C I , λ M ( x n 1 ) ) } + λ N α R I , λ M ( x n ) R I , λ M ( x n 1 ) .
As R I , λ M is Lipschitz-type-continuous, we have
x n + 1 x n λ N ( 1 α ) x n x n 1 + λ N α λ θ [ J I , λ M ( x n ) C I , λ M ( x n ) ] [ J I , λ M ( x n 1 ) C I , λ M ( x n 1 ) ] + λ N α θ x n x n 1 λ N ( 1 α ) + λ N α θ x n x n 1 + λ N α λ θ J I , λ M ( x n ) J I , λ M ( x n 1 C I , λ M ( x n ) C I , λ M ( x n 1 ) λ N ( 1 α ) + λ N α θ x n x n 1 + λ N α λ θ ( J I , λ M ( x n ) J I , λ M ( x n 1 ) ) ( C I , λ M ( x n ) C I , λ M ( x n 1 ) ) .
That is,
x n + 1 x n λ N ( 1 α ) + λ N α θ x n x n 1 + λ N α λ θ J I , λ M ( x n ) J I , λ M ( x n 1 ) + λ N α λ θ C I , λ M ( x n ) C I , λ M ( x n 1 ) , w h e r e θ = 1 γ λ 1 a n d γ λ > 1 .
Using the Lipschitz-type-continuity of Yosida approximation operator J I , λ M and Cayley operator C I , λ M , we have
x n + 1 x n λ N ( 1 α ) + λ N α θ x n x n 1 + λ N α λ θ θ x n x n 1 + λ N α λ θ ( 2 θ + 1 ) x n x n 1 = λ N ( 1 α ) + α θ + α λ θ θ + α λ θ ( 2 θ + 1 ) x n x n 1 , i . e . , x n + 1 x n λ N ( 1 α ) + α θ ( 1 + λ θ + λ ( 2 θ + 1 ) ) x n x n 1 ,
where θ = 1 γ λ 1 , θ = γ γ λ 1 a n d γ λ > 1 .
Since x n + 1 x n , we have
x n + 1 x n = x n + 1 x n λ N ( 1 α ) + α θ [ 1 + λ θ + λ ( 2 θ + 1 ) ] x n x n 1 .
Thus, we have
x n + 1 x n λ N ν n x 1 x 0 , w h e r e ν = ( 1 α ) + α θ [ 1 + λ θ + λ ( 2 θ + 1 ) ] .
Hence, for m > n > 0 , we have
x m x n i = n m 1 x i + 1 x i λ N x 1 x 0 i = n m 1 ν i .
It follows from condition (8) that 0 < ν < 1 , and thus x m x n 0 , a s n and so { x n } is a Cauchy sequence in H . Since H is complete x n x H , a s n . Thus, we can write
x = lim n x n + 1 = lim n ( 1 α ) x n + α [ R I , λ M { λ ( J I , λ M ( x n ) C I , λ M ( x n ) ) + R I , λ M ( x n ) } ] = ( 1 α ) lim n x n + α lim n R I , λ M { λ ( J I , λ M ( x n ) C I , λ M ( x n ) ) + R I , λ M ( x n ) } = ( 1 α ) x + α R I , λ M { λ ( J I , λ M ( lim n x n ) C I , λ M ( lim n x n ) ) + R I , λ M ( lim n x n ) } = ( 1 α ) x + α R I , λ M { λ ( J I , λ M ( x ) C I , λ M ( x ) ) + R I , λ M ( x ) } .
It follows that x satisfies the Equation (7),
i . e . , x = R I , λ M λ ( J I , λ M ( x ) C I , λ M ( x ) ) + R I , λ M ( x ) .
By Lemma 6, x is a solution of Cayley inclusion problem involving XOR-operation (5). On the other hand, it follows that R I , λ M λ ( J I , λ M ( x ) C I , λ M ( x ) ) + R I , λ M ( x ) x , n = 0 , 1 , 2 , .
Using Lemma 1 and (12), we have
x x 0 = lim n x n x lim n i = 1 n x i + 1 x i lim n λ N i = 2 n ν i 1 x 1 x 0 + x 1 x 0 1 + ( λ N 1 ) [ ( 1 α ) + α θ [ 1 + λ θ + λ ( 2 θ + 1 ) ] 1 [ ( 1 α ) + α θ [ 1 + λ θ + λ ( 2 θ + 1 ) ] × α x 0 + R I , λ M { λ J I , λ M ( x 0 ) C I , λ M ( x 0 ) + R I , λ M ( x 0 ) } .
This complete the proof. □

4. Conclusions

We have introduced and studied a new problem which involves a Cayley operator and a multi-valued mapping with XOR-operation in real ordered Hilbert space, called the Cayley Inclusion problem involving XOR-operation. A fixed point formulation of the Cayley inclusion problem involving XOR-operation is given by using the Yosida approximation operator and resolvent operator. Finally, an existence and convergence result is proved with some extra condition. An example is constructed to illustrate some of the concepts used in this paper.
We remark that our results may be extended in ordered Banach spaces and other higher dimensional spaces.

Author Contributions

All the authors have contributed equally to this paper. All the authors read and approved the final manuscript.

Funding

This work was supported by the Taiwan MOST [grant number 107-2115-M-037-001].

Acknowledgments

The authors of this paper are thankful to the referees for their valuable suggestions, which improve the paper a lot.

Conflicts of Interest

The authors declare no conflict of interest.

References

  1. Luc, D.T.; Tan, N.X. Existence conditions in variational inclusions with constraints. Optimization 2004, 53, 505–515. [Google Scholar] [CrossRef]
  2. Tan, N.X. On the existence of solutions of quasivariational inclusion problem. J. Optim. Theory Appl. 2004, 123, 619–638. [Google Scholar] [CrossRef]
  3. Tuan, L.A.; Sach, P.H. Existence of solutions of generalized quasivariational inequalities with set-valued maps. Acta Math. Vietnam 2004, 29, 309–316. [Google Scholar]
  4. Ding, X.P.; Xia, F.Q. A new class of completely generalized quasivariational inclusions in Banach spaces. J. Copmut. Appl. Math. 2002, 147, 369–383. [Google Scholar]
  5. Kristály, A.; Vagra, C. Set-valued versions of Ky Fan’s inequality with applications to variational inclusion theory. J. Math. Anal. Appl. 2003, 282, 8–20. [Google Scholar]
  6. Agarwal, R.P.; Huang, N.J.; Tan, M.Y. Sensitivity analysis for a new system of generalized nonlinear mixed quasi-variational inclusions. Appl. Math. Lett. 2004, 17, 345–352. [Google Scholar] [CrossRef]
  7. Chang, S.S. Set-valued variational inclusions in Banach spaces. J. Math. Anal. Appl. 2000, 248, 438–454. [Google Scholar] [CrossRef]
  8. Chang, S.S.; Kim, J.K.; Kim, K.H. On the existence and the iterative approximation problems of solutions for set-valued variational inclusions in Banach spaces. J. Math. Anal. Appl. 2002, 268, 89–108. [Google Scholar] [CrossRef]
  9. Ahmad, R.; Dilshad, M.; Wong, M.M.; Yao, J.C. H(.,.)-cocoercive operator and an application for solving generalized variational inclusions. Abs. Appl. Anal. 2011, 2011, 261534. [Google Scholar] [CrossRef]
  10. Ansari, Q.H.; Yao, J.C. Iterative schemes for solving mixed variational-like inequalities. J. Optim. Theory Appl. 2001, 108, 527–541. [Google Scholar] [CrossRef]
  11. Hassouni, A.; Moudafi, A. A Perturbed algorithm for variational inclusions. J. Math. Anal. Appl. 1994, 185, 706–712. [Google Scholar] [CrossRef]
  12. Petrot, N.; Balooee, J. General nonconvex variational inclusions and fixed point problems. J. Ineqal. Appl. 2013, 2013, 377. [Google Scholar] [CrossRef] [Green Version]
  13. Shelmas, E.A. Resolvent operator method for general variational inclusions. J. Math. Ineqal. 2009, 3, 455–462. [Google Scholar] [CrossRef]
  14. Xu, Z.; Wang, Z. A generalized mixed variational inclusion involving (H(.,.),η)-monotone operators in Banach spaces. J. Math. Res. 2010, 2, 47–56. [Google Scholar] [CrossRef]
  15. Zou, Y.Z.; Huang, N.J. H(.,.)-Accretive operator with an application for solving variational inclusions in Banach spaces. Appl. Math. Comput. 2008, 204, 809–816. [Google Scholar] [CrossRef]
  16. Attouch, H. Variational Convergence for Functions and Operators; Pitman: Boston, MA, USA, 1984. [Google Scholar]
  17. Attouch, H.; Moudafi, A.; Riahi, H. Quantitative stability analysis for maximal monotone operators and semi-groups of contractions. Sémin. Anal. Convexe Montp. 1991, 21, 1–38. [Google Scholar] [CrossRef]
  18. Barbu, V. Nonlinear Semigroups and Differential Equations in Banach Spaces; Noordhoff: Groningen, The Netherlands, 1976. [Google Scholar]
  19. Pazy, A. Semigroups of Nonlinear Contractions in Hilbert Spaces. In Problems in Nonlinear Analysis; Springer: Berlin/Heidelberg, Germany, 2010; pp. 343–430. [Google Scholar]
  20. Li, H.G.; Li, L.P.; Jin, M.M. A class of nonlinear mixed ordered inclusion problems for ordered (αA,λ)-ANODM set valued mappings with strong comparison mapping A. Fixed Point Theory Appl. 2014, 1, 79. [Google Scholar] [CrossRef]
  21. Li, H.G.; Qiu, D.; Zou, Y. Characterizations of weak-ANODD set-valued mappings with applications to approximate solution of GNMOQV inclusions involving ⊕ operator in ordered Banach spaces. Fixed Point Theory Appl. 2013. [Google Scholar] [CrossRef]
  22. Li, H.G. A nonlinear inclusion problem involving (α,λ)-NODM set-valued mappings in ordered Hilbert space. Appl. Math. Lett. 2012, 25, 1384–1388. [Google Scholar] [CrossRef]
  23. Li, H.G. Nonlinear inclusion problems for ordered RME set-valued mappings in ordered Hilbert spaces. Nonlinear Funct. Anal. Appl. 2001, 16, 1–8. [Google Scholar]
  24. Li, H.G.; Pan, X.B.; Deng, Z.Y.; Wang, C.Y. Solving frameworks involving (γG,λ)-weak-GRD set-valued mappings in positive Hilbert spaces. Fixed Point Theory Appl. 2014. [Google Scholar] [CrossRef]
  25. Ahmad, I.; Pang, C.T.; Ahmad, R.; Ishtyak, M. System of Yosida inclusions involving XOR-operation. J. Nonlinear Convex Anal. 2017, 18, 831–845. [Google Scholar]
  26. Ahmad, I.; Ahmad, R.; Iqbal, J. A resolvent operator for solving a set-valued variational inclusion problem using weak-RRD mapping. Korean J. Math. 2016, 24, 199–213. [Google Scholar] [CrossRef]
  27. Du, Y.H. Fixed points of increasing operators in ordered Banach spaces and applications. Appl. Anal. 1990, 38, 1–20. [Google Scholar] [CrossRef]
  28. Schaefer, H.H. Banach Lattices and Positive Operators; Springer: Berlin/Heidelberg, Germany; New York, NY, USA, 1974. [Google Scholar]

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MDPI and ACS Style

Ali, I.; Ahmad, R.; Wen, C.-F. Cayley Inclusion Problem Involving XOR-Operation. Mathematics 2019, 7, 302. https://0-doi-org.brum.beds.ac.uk/10.3390/math7030302

AMA Style

Ali I, Ahmad R, Wen C-F. Cayley Inclusion Problem Involving XOR-Operation. Mathematics. 2019; 7(3):302. https://0-doi-org.brum.beds.ac.uk/10.3390/math7030302

Chicago/Turabian Style

Ali, Imran, Rais Ahmad, and Ching-Feng Wen. 2019. "Cayley Inclusion Problem Involving XOR-Operation" Mathematics 7, no. 3: 302. https://0-doi-org.brum.beds.ac.uk/10.3390/math7030302

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