\r\n\tWith this rapid transformation of computing and communication world, information-system security has moved from a largely self-contained bounded environment interacting with a generally known and disciplined user community to one of worldwide scope with a body of users that may not be known and are not necessarily trusted. Importantly, security control now must deal with circumstances over which there is largely no control or expectation of avoiding their impact. Computer security, as it has evolved, shares a similarity with liability assurance; they each face a threat environment that is known in a very general way and can face attacks over a broad spectrum of sources; however, the exact details or even time or certainty of an attack is unknown until an incident actually occurs.
\r\n\tThe purpose of this book is to discuss some of the critical security challenges in today’s computing world and to discuss mechanisms for defending against those attacks by using classical and modern approaches of cryptography and other security solutions. With this objective, the book invites contributions from researchers in the field of cryptography and its applications in network security. Some illustrative topics of interest (but not limited to) are: cryptography algorithms, authentication, authorization, integrity, confidentiality, privacy, security in wireless networks, security in wireless local area networks, wireless sensor networks, wireless ad hoc networks, vehicular ad hoc networks, security and privacy in the Internet of Things.
Let
where
where
which is
where
At the same time, integral equations have been developed to solve boundary value problems for both ordinary and partial differential equations and play a very important role in nonlinear analysis. Many problems of mathematical physics, theory of elasticity, viscodynamics fluid and mixed problems of mechanics of continuous media reduce to the Fredholm integral Eq. A rich literature on existence of solutions for nonlinear integral equations, which contain particular cases of important integral and functional equations can be found, for example, see [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. An important technique to solve integral equations is to construct an iterative procedure to generate approximate solutions and find their limit, a host of attractive methods have been proposed for the approximate solutions of Fredholm integral equations of the second kind, see [15, 16, 17, 18, 19]. We consider a non-homogeneous Fredholm integral equation of second kind of the form
where
An advancement in this direction is to find the solution of such mathematical models by using fixed point theorems. In this technique, we generate a sequence by iterative procedure for some self-map T and then look for a fixed point of T, that is actually the solution of given mathematical model. The simplest case is when T is a contraction mapping, that is a self-mapping satisfying
where
In this chapter, we prove the existence of solution for matrix Eq. (1) and integral Eq. (2) by using newly developed fixed point theorems.
Let
for all
Denote by
where
If
Definition 2.1 [22] Let
where
Further, Definition 2.1 is generalized in the following way.
Definition 2.2 [23] Let
If
Consistent with Du and Khojasteh [24], we denote by
Example 3.1 [24] Let
Example 3.2 Let
where
so,
Definition 3.3 Let
Now we prove first result of this section.
Theorem 3.4 Let
there exists
for a sequence
Then
Proof. Let
If
By
Also note that if
So, from (5) and (7), we get
Let
Since
This implies that there exists
By induction, we form a sequence
and
by taking
By using (7), (8), (13), and (15), we get for each
this implies that
Which means that
Now, if
This contradiction shows that
then from Eq. (9), we have
(19) implies that
For any
Put
Since
Since
Letting limit
Let
Definition 3.5 [27] Let
Theorem 3.6 [26] Let
Definition 3.7 ([27]). Let
Theorem 3.8 ([27]). Let
Theorem 3.9 ([27]). Let
For more in this direction, see, [28, 29, 30, 31]. Here, we give the concept of multivalued
Definition 3.10 Let
where,
and
Note that
Definition 3.11 Let
Theorem 3.12 Let
the map
there exists
Then
Proof. Let
where
Since
Since
Suppose that
consequently,
or,
On continuing recursively, we get a sequence
Since
and
Combining (39) and (40) gives
Let
Let
From (38), we also get
Now consider the sequence
Case 1. For each
Case 2. There is
From
Letting
since
Thus,
In the following theorem we take
Theorem 3.13 Let
Proof. Let
where
The rest of the proof can be completed as in the proof of Theorem 3.12 by considering the closedness of
Theorem 3.14 Let
there exists
there exists
and for all
where
Then
Proof. By following the steps in the proof of Theorem 3.12, we get the required result.□
Note that Theorem 3.14 cannot be obtained from Theorem 3.12, because in Theorem 3.12,
Theorem 3.15 Let
From Theorems 3.14 and 3.15, we get the following fixed point result for single valued mappings:
Theorem 3.16 Let
there exists
there exists
and for all
where
Then
Now, let
Theorem 3.17 Let
there exists
for a sequence
Then
Proof. Define
then for
In case of single valued mapping Theorem 3.17 reduced to the following:
Theorem 3.18 Let
there exists
for a sequence
Definition 3.19 Let
where,
Theorem 3.20 Let
the map
there exists
Then
Proof. By using the similar arguments as in the proof of Theorem 3.17 and using Theorem 3.12, we get the result.□
Theorem 3.21 Let
Theorem 3.22 Let
there exists
there exists
and for all
where
Then
Proof. By defining
Theorem 3.23 Let
From Theorems 3.22 and 3.23, we get the following fixed point result for single valued mapping.
Theorem 3.24 Let
there exists
there exists
and for all
where
Then
In this section, by using the fixed point results proved in the previous section, we obtain the existence of the solution of integral Eq. (2) and matrix Eq. (1).
Let
Theorem 4.1 Let
there exists a continuous function
for each
there exists
for a sequence
Then the integral Eq. (2) has a solution in
Proof. Let
Then
This implies that
So
Theorem 4.2 Let
where
Then (1) has a solution in
Proof. Define
and
and so,
This implies that
Consequently,
Also, from
"I work with IntechOpen for a number of reasons: their professionalism, their mission in support of Open Access publishing, and the quality of their peer-reviewed publications, but also because they believe in equality. Throughout the world, we are seeing progress in attracting, retaining, and promoting women in STEMM. IntechOpen are certainly supporting this work globally by empowering all scientists and ensuring that women are encouraged and enabled to publish and take leading roles within the scientific community." Dr. Catrin Rutland, University of Nottingham, UK
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