Caristi, Nadler and
H
+
-Type Contractive Mappings and Their Fixed Points in θ-Metric Spaces
Abstract
:1. Introduction
2. Preliminaries
- (i)
- and for all ,
- (ii)
- (iii)
- for each and for each , there exists such that ,
- (iv)
- , for all .
- (i)
- , where ,
- (ii)
- , where ,
- (iii)
- ,
- (iv)
- .
- (a1)
- .
- (a2)
- and .
- (a3)
- η is continuous with respect to the first variable.
- (a4)
- If , then .
- (i)
- if and only if ,
- (ii)
- , for all ,
- (iii)
- , for all .
- (i)
- a sequence in X is said to be converged to if as ,
- (ii)
- a sequence is Cauchy if, for each , there exists such that, for all , .
- (iii)
- is complete if every Cauchy sequence is convergent in X.
- (iv)
- A self-mapping T on is said to be θ-continuous if whenever as .
- ()
- if and , then ;
- ()
- there exist a mapping which is nondecreasing with respect to each variable such that .
- (E1)
- there exists such that is bounded below and lower semicontinuous, and is upper semicontinuous for each ,
- (E2)
- if and only if ,
- (E3)
- for each
- (i)
- for each ,
- (ii)
- ν is nondecreasing map,
- (iii)
- ν is continuous,
- (iv)
- if and only if .
- (a)
- Let ; thus, . Now, let be defined by
- (b)
- Let ; thus, . Now, let be defined by
- (c)
- Let , ; thus, . Now, let be defined by
- A point is called a fixed point of a mapping if and only if .
- A point is called a periodic point of a mapping if and only if there exists such that .
3. Caristi Type Fixed Point Theorems
- (i)
- For every we have
- (ii)
- Since is nondecreasing for , for is too.
- (iii)
- Continuity of log function implies continuity of ν.
- (iv)
- Let .
4. and Metrics
5. Fixed Point Results for Set-Valued Mappings
5.1. Lim–Nadler Type Fixed Point Theorems
- (i)
- for each , there exists such that implies ,
- (ii)
- for all ,
- (iii)
- if and only if .
- (i)
- for every ,
- (ii)
- for every sequence such that as , , we have
- (i)
- for , we consider ; then, . For , we have .
- (ii)
- For every we have
- (iii)
- .
5.2. Pathak and Sahzad Type Fixed Point Theorem
- (i)
- there exists such that for every
- (ii)
- for every , and , there exists such that
- If , then and . Thus, is satisfied for ,
- If , then and . Thus, is satisfied for
- If , then and . Thus, is satisfied for
- If , , , there exists such that .
- If , ,
- (i)
- let , , there exists such that ,
- (ii)
- let , , there exists say such that .
- If , ,
- (i)
- let , , there exists such that ,
- (ii)
- let , , there exists say such that .
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
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Patle, P.; Vujaković, J.; Patel, D.; Radenović, S.
Caristi, Nadler and
Patle P, Vujaković J, Patel D, Radenović S.
Caristi, Nadler and
Patle, Pradip, Jelena Vujaković, Deepesh Patel, and Stojan Radenović.
2019. "Caristi, Nadler and