α-β-ψ-φ Contraction in Digital Metric Spaces

Ibtisam A. Masmali¹, Ghaliah Y. Alhamzi² and Sumitra Dalal^1,

¹Jazan University, Jazan, K.S.A.

²Al Imam Mohammad Ibn Saud Islamic University, Riyadh, K.S.A.

Cite this paper:
Ibtisam A. Masmali, Ghaliah Y. Alhamzi and Sumitra Dalal. α-β-ψ-φ Contraction in Digital Metric Spaces. American Journal of Mathematical Analysis. 2018; 6(1):5-15. doi: 10.12691/ajma-6-1-2

Abstract

Samet et al. (Nonlinear Anal. 75, 2012, 2154-2165) introduced a new, simple and unified approach by using the concepts of α-ψ-contractive type mappings and α-admissible mappings in metric spaces and presented some nice fixed point results. Recently, Sridevi et.al (International Journal of Mathematics Trends and Technology, Volume 48, Number 3 August 2017) proposed the concept of α-ψ-φ contraction and generalized α-ψ-φ for self map in digital metric spaces. The purpose of this paper is to present a new class of contractive pair of mappings called α-β-ψ-φ contraction and generalized α-β-ψ-φ contractive pair of mappings and study various fixed point theorems for such mappings in digital metric spaces. For this, we introduce a new notion of α-β-admissible w.r.t T mapping which in turn generalizes the concept of g-monotone mapping recently given by “Ciric et al. (Fixed Point Theory Appl. 2008 (2008), Article ID 131294, 11 pages)”. Also, we give some fixed point theorems for cyclic contractive mapping in such spaces. The presented theorems hold without using completeness of the space and without the assumption of continuity of the given mappings. Our results extend, generalize and subsumes digital version of various known comparable results [[1-4,8,13,16,18-22], worth to mention here]. Some illustrative examples are quoted to demonstrate the main results.

Keywords:
Digital image Digital metric space Banach contractive principle -admissible maps contraction and generalized contraction

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References:

[1]	Aydi, H., Bota, M-F., Karapinar, E and Moradi, S.,A Common Fixed Point For Weak –Phi Contractions on b-Metric Spaces, Fixed Point Theory, 13( 2012),no:2, 337-346.
[2]	Banach, S., Surles operations dans les ensembles abstraits et leur application aux equations itegrales, Fundamenta Mathematicae 3, 133-181 (1922).
[3]	Bilgili, N., Karapinar, E. and Samet, B., Generalized α - ψ contractive mappings in quasimetric spaces and related fixed-point theorems, Journal of Inequalities and Applicationss, (2014).
[4]	Bota, M., Chifu, C and Karapinar, E., Fixed point theorems for generalized (alpha-psi)-Ciric-type contractive multivalued operators in b-metric spaces J. Nonlinear Sci. Appl. 9 (2016), Issue: 3 pages 1165-1177.
[5]	Boxer L., Digitally Continuous Functions, Pattern Recognition Letters, 15 (1994), 833-839.
[6]	Boxer L., A Classical Constructions for The Digital Fundamental Group, J. Math. Imaging Vis., 10(1999), 51-62.
[7]	Boxer L., Continuous Maps on Digital Simple Closed Curves, Appl. Math., 1(2010), 377-386.
[8]	Ciric, L., Cakic, N., Rajovic, M., Ume, J.S., Monotone generalized nonlinear contractions in partially ordered metric spaces, Fixed Point Theory Appl. 2008(2008), Article ID 131294, 11 pages.
[9]	Ege O, K., Some Results on Simplicial Homology Groups of 2D Digital Images, Int. J. Inform. Computer Sci., 1(2012), 198-203.
[10]	Ege O, K., Lefschetz Fixed Point Theorem for Digital Images, Fixed Point Theory Appl., 2013(2013), 13 pages.
[11]	Ege O, K., Applications of The Lefschetz Number to Digital Images, Bull. Belg. Math. Soc. Simon Stevin, 21(2014), 823-839.
[12]	Ege O, K., Banach Fixed Point Theorem for Digital Images, J. Nonlinear Sci. Appl., 8(2015), 237-245.
[13]	Gulyaz, O,S., On some alpha-admissible contraction mappings on Branciari b-metric spaces, Advances in the Theory of Nonlinear Analysis and its Applications 1(1),1-13 (2017), Article Id: 2017:1:1
[14]	Herman GT, Oriented Surfaces in Digital Spaces, CVGIP: Graphical Models and Image Processing, 55(1993), 381-396.
[15]	Kong TY, A Digital Fundamental Group, Computers and Graphics, 13(1989), 159-166.
[16]	Karapinar, E., Samet, B.:Generalized α-ψ-contractive type mappings and related fixed point theorems withapplications, Abstract and Applied Analysis 2012 Article ID 793486, 17 pages.
[17]	Rosenfeld A, Digital Topology, Amer. Math. Monthly, 86(1979), 76-87.
[18]	Rani A, Jyoti K and Rani A. 2016. Common fixed point theorems in digital metric spaces, International Journal of Scientific & Engineering Research, Volume 7, Issue 12, December-2016 ISSN 2229-5518
[19]	Samet, B., Vetro, C., Vetro, P.: Fixed point theorem for α-ψ contractive type mappings, Nonlinear Anal. 75, 2154-2165 (2012).
[20]	Sumitra Dalal, Common Fixed Point Results for Weakly Compatible Map in Digital Metric Spaces, Scholars Journal of Physics, Mathematics and Statistics, Sch. J. Phys. Math. Stat. 2017; 4(4):196-201.
[21]	Sumitra Dalal, Common Fixed Point Results for Compatible Map in Digital Metric Spaces, Journal of Advances in Mathematics, Volume13, Number.
[22]	Sridevi, K., Kameswari, M. V. R. and Kiran, D. M. K., Fixed Point Theorems for Digital Contractive Type Mappings in Digital Metric Spaces, International Journal of Mathematics Trends and Technology (IJMTT) – Volume 48 Number 3 August 2017.