Common Fixed Point Theorem via Cyclic (α,β)–(ψ,φ)S-Contraction with Applications
Abstract
:1. Introduction
2. Preliminaries
- 1.
- θ is continuous and nondecreasing;
- 2.
- if and only if .
- 1.
- for some
- 2.
- for some
- 1.
- 2.
- ;
- 3.
- .
- (a)
- b-convergent if there exists such that as .
- (b)
- b-Cauchy if as .
3. Results
- (i)
- with are closed subspaces of ;
- (ii)
- there exists with and ;
- (iii)
- if is a sequence in X with for all n and , then ;
- (iv)
- and whenever and .
- (i)
- with are closed subspaces of X;
- (ii)
- there exists with and ;
- (iii)
- if is a sequence in X with for all n and , then
- (iv)
- and whenever and .
- (i)
- there exists with and ;
- (ii)
- if is a sequence in X with for all n and , then
- (iii)
- and whenever and ;
- (i)
- with are closed subspaces of X;
- (ii)
- there exists with and ;
- (iii)
- if is a sequence in X with for all n and , then
- (iv)
- and whenever and ;
4. Applications in Dynamic Programming
- C0
- such that are closed subspaces of ;
- C1
- there exists such that and ;
- C2
- is a sequence in such that and for all n, then .
- C3
- if and , for all , then for all we have:
- C4
- C5
- and whenever and ;
- C6
- for some , , whenever ;
- C7
- for , and are bounded.
Author Contributions
Funding
Conflicts of Interest
References
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Zada, M.B.; Sarwar, M.; Jarad, F.; Abdeljawad, T. Common Fixed Point Theorem via Cyclic (α,β)–(ψ,φ)S-Contraction with Applications. Symmetry 2019, 11, 198. https://doi.org/10.3390/sym11020198
Zada MB, Sarwar M, Jarad F, Abdeljawad T. Common Fixed Point Theorem via Cyclic (α,β)–(ψ,φ)S-Contraction with Applications. Symmetry. 2019; 11(2):198. https://doi.org/10.3390/sym11020198
Chicago/Turabian StyleZada, Mian Bahadur, Muhammad Sarwar, Fahd Jarad, and Thabet Abdeljawad. 2019. "Common Fixed Point Theorem via Cyclic (α,β)–(ψ,φ)S-Contraction with Applications" Symmetry 11, no. 2: 198. https://doi.org/10.3390/sym11020198
APA StyleZada, M. B., Sarwar, M., Jarad, F., & Abdeljawad, T. (2019). Common Fixed Point Theorem via Cyclic (α,β)–(ψ,φ)S-Contraction with Applications. Symmetry, 11(2), 198. https://doi.org/10.3390/sym11020198