1. Introduction
This work was motivated by some recent works on Caristi’s fixed point theorem for mappings defined on metric spaces with a partial order or graph. It seems that the terminology of graph theory instead of partial ordering gives clearer pictures and yields generalized fixed point theorems. The Caristi fixed point theorem is known as one of the very interesting and useful generalizations of the Banach fixed point theorem for self-mappings on a complete normed space. The Caristi’s fixed point theorem is a modification of the ε-variational principle of Ekeland ( [1] [2] ), which is a crucial tool in nonlinear analysis like optimization, variational inequalities, differential equations, and control theory. Furthermore, in 1977, Western [3] proved that the conclusion of Caristi’s theorem is equivalent to norm completeness. In the last decades, Caristi’s fixed point theorem has been generalized and extended in several directions (i.e. [4] [5] and the related references therein). Here at present, we discussed Caristi’s fixed point theorem in normed spaces where Caristi’s fixed point theorem was in matric space [6].
2. Preliminaries
We will discuss some applications of Caristi’s fixed point theorem in complete normed space.
Theorem-1 [7]
Let be a complete normed space, and let be a mapping such that for
.
where is a lower semi-continuous mapping. The T has at least a fixed point.
We denote by N the set of positive integers and by R the set of real numbers.
Let be a complete normed space.
We denote by CB(X) the family of all no-empty closed and bounded subsets of X.
A function .
Defined by is said to be Housdorff norm on CB(X) induced by the norm on X.
A point v in X is a fixed point of a map T if (when is a single-valued map), or (when is a multi-valued map).
Let be a complete normed space and a map . Suppose there exists a function satisfying , for and suppose that is right upper semi-continuous such that . Then T has a unique fixed point.
Problem-1 [8] [9] [10]
Let be a complete normed space and let be a multi-valued mapping such that .
For all , where is continuous and increasing map such that . Does T have a fixed point?
In these works, the authors consider additional conditions on the mapping to find a fixed point.
1) Daffer et al. assumed that
i) is upper right semi-continuous;
ii) and;
iii) where on some interval , .
2) Jachymski assumed that
i) is super additive i.e. and;
ii) is non-decreasing.
Problem-2 [11]
Let be a complete normed space and let be a mapping such that .
For all , where is a mapping such that .
For all . Does T have a fixed point?
Theorem-2 [12]
Let be a complete normed space and let be a mapping such that .
For all , where is a mapping such that .
For all . Then T has a fixed point.
Normed Spaces [13]
A normed on X is a real function defined on X such that for any and for all .
i) . ii) if and only if . iii) .
iv) (Triangle inequality).
A norm on X defines a metric d on X which is given by
and is called the metric induced by the norm.
The normed space is denoted by or simply by X.
3. Further Extension
In this section, many of the known fixed point results can be deduced from light version of Caristi’s theorem in normed space.
Theorem-3
Let be a complete normed space, and let be a mapping such that for
.
where is lower semi-continuous with respect to the first variable.
Then T has a unique fixed point.
Proof:
For each , let and . Then for each .
(1)
and is a lower semi-continuous mapping.
If X is a normed space and is a mapping, then
(2)
where is a lower semi-continuous mapping? And T has at least a fixed point.
Comparing Equations (1) and (2), then we get is a lower semi-continuous mapping and T has a fixed point.
Let u, v be two distinct fixed points for T.
Now we will prove that the uniqueness of the fixed point i.e. u = v
Hence T has a unique fixed point.
3.1. Banach Contraction Principle
Theorem-4 [14]
Let be a complete normed space, and let be a mapping such that for some , .
Then T has a unique fixed point.
Proof: We define . Then we have
It means that
Hence T has a unique fixed point.
Theorem-5
Let be a complete normed space, and let be a mapping such that
where is lower semi-continuous mapping such that
, for each and is a non-decreasing map. Then T has a unique fixed point.
Proof:
Define if and otherwise . Then we
have
Since is non-decreasing and
Hence T has a unique fixed point.
Theorem-6
Let be a complete normed space, and let be a non-expansive mapping such that, for each , and for all , there exists such that where is lower semi-continuous mapping w. r. to the first variable. Then T has a fixed point.
Proof:
Let and . If , then is a fixed.
Now we will prove that T has a fixed point.
If such that
(3)
Alternatively, we choose such that and we find such that
(4)
which means that is a non-increasing sequence and bounded below. So it is converges to .
From Equation (4), then we get
(5)
Taking the limit in Equation (5), then we get,
(6)
Also for all with
(7)
Taking the limit sup on both sides in Equation (7), then we get,
It means that is a Cauchy sequence and so it converges to
Now we will prove that x is a fixed point of T.
[Using (6)]
(8)
Taking the limit on both sides in Equation (8), then we get ,
Hence T has a unique fixed point.
Theorem-7
Let be a complete normed space, and let be a multi-valued function such that
where is a lower semi-continuous map such that
and is a non-decreasing. Then T has a fixed point.
Proof:
Let and . If , then T has a fixed and the proof is complete.
We suppose that . Then we define
Since
(9)
Thus there exists such that
(10)
So there exists such that
[Using (10)]
[Using (9)]
Again, we suppose that , then we have
(11)
[Since is also a non-decreasing function and ].
Define if , otherwise 0 for all .
From Equation (3), then we get,
Hence T has a unique fixed point.
3.2. Mizoguchi-Takahashi’s Type
Theorem-8
Let be a complete normed space, and let be a multi-valued mapping such that
where is a lower semi-continuous and non-decreasing mapping? Then T has a fixed point.
Proof:
Let and . If , then T has a fixed and the proof is complete.
Let and is a non-decreasing
mapping. Since,
(12)
Thus there exists such that
(13)
So there exists such that
[Using (13)]
[Using (12)]
(14)
Again, we suppose that , then we have
(15)
[Since is also a non-decreasing function and ].
Define if , otherwise 0 for all .
From Equation (15), then we get,
Hence T has a unique fixed point.
Theorem-9
Let be a complete normed space, and let be a multi-valued functions such that
where is an upper semi-continuous map such that
and is a non-increasing. Then T has a fixed point.
Proof:
Let and . If , then T has a fixed and the proof is complete.
Let , for each . Then , for each and
is a non-decreasing. Since
(16)
Thus there exists such that
(17)
So there exists such that
[Using (17)]
[Using (16)]
(18)
Again, we suppose that , then we have
(19)
[Since is non-increasing function and ].
Define if , otherwise 0 for all .
From Equation (19), then we get,
Hence T has a unique fixed point.
4. Remark
1) If is sub-additive i.e. , for each , then it is a non-decreasing continuous map such that .
2) Here we show that many of the known Banach contraction generalizations can be deduced and generalized by Caristi’s fixed point theorem in normed space and its consequences.
5. Conclusion
Our aim is to discuss the Caristi’s fixed point theorem on normed spaces. We hope that this work will be useful for functional analysis related to normed spaces and fixed point theory. All expected results in this paper will help us to understand better solution of complicated theorem. We give an important application and use the fixed point theory related to different branches of mathematics for the solution of physical problems. In future, we will discuss Caristi’s fixed point properties related to physical problems.
Acknowledgements
I would like to thank my respectable teacher Prof. Dr. Nurul Alam Khan for encouragement and valuable suggestions.
Author Contributions
Authors have made equal contributions for paper.
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