Relation theoretic contractions and their applications in b-metric like spaces

• Autorzy: Tomar Anita , Joshi Meena , Padaliya S. K.

Identyfikatory

DOI

10.1515/jaa-2021-2083

• Języki publikacji: EN

Abstrakty

EN

We introduce a relational generalized Meir-Keeler contraction and a relational generalized Meir-Keeler contraction with rational terms in non-complete relational b-metric like spaces in order to establish non-unique fixed point results for a discontinuous single-valued map. Also, we provide an illustrative example to demonstrate that a relational generalized Meir-Keeler contraction with rational terms in a relational b-metric like space admits discontinuity at the fixed point. Thereby,we provide a novel explanation via a binary relation to the question of the existence of a contractive map admitting a fixed point at the point of discontinuity. Finally, we give applications to solve an initial value problem and a non-linear matrix equation which demonstrate the usability and effectiveness of our results.

Słowa kluczowe

EN

binary relation; fixed point; R-completeness; R-continuity; transitivity

• Wydawca: De Gruyter
• Czasopismo: Journal of Applied Analysis
• Rocznik: 2022
• Tom: Vol. 28, nr 2
• Strony: 295--309
• Opis fizyczny: Bibliogr. 44 poz., 1 rys.

Twórcy

autor

Tomar Anita

• Department of Mathematics, Sri Dev Suman Uttarakhand University, Pt. L. M. S. Campus, Rishikesh, Uttarakhand 249201, India

• https://orcid.org/0000-0001-8033-856X

autor

Joshi Meena

• Department of Mathematics, Shri Guru Ram Rai (P. G.) College, Dehradun, India

• https://orcid.org/0000-0002-1562-0988

autor

Padaliya S. K.

• Department of Mathematics, Shri Guru Ram Rai (P. G.) College, Dehradun, India

Bibliografia

• [1] A. Alam and M. Imdad, Relation-theoretic contraction principle, J. Fixed Point Theory Appl. 17 (2015), no. 4, 693-702.
• [2] A. Alam and M. Imdad, Relation-theoretic metrical coincidence theorems, Filomat 31 (2017), no. 14, 4421-4439.
• [3] M. A. Alghamdi, N. Hussain and P. Salimi, Fixed point and coupled fixed point theorems on b-metric-like spaces, J. Inequal. Appl. 2013 (2013), Paper No. 402.
• [4] A. Amini-Harandi, Metric-like spaces, partial metric spaces and fixed points, Fixed Point Theory Appl. 2012 (2012), Paper No. 204.
• [5] I. A. Bakhtin, The contraction mapping principle in almost metric space, Funct. Anal. 30 (1989), 26-37.
• [6] S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math. 3 (1922), 133-181.
• [7] D. W. Boyd and J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969), 458-464.
• [8] F. E. Browder, On the convergence of successive approximations for nonlinear functional equations, Nederl. Akad. Wetensch. Proc. Ser. A 71 30 (1968), 27-35.
• [9] T. A. Burton, Integral equations, implicit functions, and fixed points, Proc. Amer. Math. Soc. 124 (1996), no. 8, 2383-2390.
• [10] L. Ćirić, A new fixed-point theorem for contractive mappings, Publ. Inst. Math. (Beograd) (N. S.) 30(44) (1981), 25-27.
• [11] S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostraviensis 1 (1993), 5-11.
• [12] W. L. de Koning, Infinite horizon optimal control of linear discrete time systems with stochastic parameters, Automatica J. IFAC 18 (1982), no. 4, 443-453.
• [13] J. Dugundji, Positive definite functions and coincidences, Fund. Math. 90 (1975/76), no. 2, 131-142.
• [14] J. Dugundji and A. Granas, Weakly contractive maps and elementary domain invariance theorem, Bull. Soc. Math. Grèce (N. S.) 19 (1978), no. 1, 141-151.
• [15] M. Fréchet, Sur quelques points du calcul fonctionnel, Rend. Circ. Mat. Palermo 22 (1906), 1-72.
• [16] M. A. Geraghty, On contractive mappings, Proc. Amer. Math. Soc. 40 (1973), 604-608.
• [17] M. A. Geraghty, An improved criterion for fixed points of contraction mappings, J. Math. Anal. Appl. 48 (1974), 811-817.
• [18] N. Gholamian and M. Khanehgir, Fixed points of generalized Meir-Keeler contraction mappings in b-metric-like spaces, Fixed Point Theory Appl. 2016 (2016), Paper No. 34.
• [19] J. A. Górnicki, Remarks on contractive type mappings, Fixed Point Theory Appl. 2017 (2017), Paper No. 8.
• [20] H. A. Hammad and M. De la Sen, Generalized contractive mappings and related results in b−metric like spaces with an application, Symmetry 11 (2019), DOI 10.3390/sym11050667.
• [21] J. Jachymski, Equivalent conditions and the Meir-Keeler type theorems, J. Math. Anal. Appl. 194 (1995), no. 1, 293-303.
• [22] M. Joshi, A. Tomar and SK Padaliya, Fixed Point to Fixed Ellipse in Metric Spaces and Discontinuous Activation Function, Appl. Math. E-Notes 21 (2021), 225-237.
• [23] B. Kolman, R. C. Busby and S. Ross, Discrete Mathematical Structures, 3rd ed., PHI Pvt., New Delhi, 2000.
• [24] M. A. Kutbi, A. Roldán, W. Sintunavarat, J. Martínez-Moreno and C. Roldán, F -closed sets and coupled fixed point theorems without the mixed monotone property, Fixed Point Theory Appl. 2013 (2013), Paper No. 330.
• [25] J. Liouville, Second Mémoire sur le développement des functions ou parties de functions en series dont divers terms sont assujettis á satisfaire a une meme equation différentielle du second ordre contenant unparamétre variable, J. Math. Pure Appl. 2 (1837), 16-35.
• [26] R. D. Maddux, Relation Algebras, Stud. Logic Found. Math. 150, Elsevier, Amsterdam, 2006.
• [27] S. Manro and A. Tomar, Common fixed point theorems using property (EA) and its variants involving quadratic terms, Annals of Fuzzy Mathematics and Informatics 7 (2014), no. 3, 473-484.
• [28] J. Matkowski, Fixed point theorems for contractive mappings in metric spaces, Časopis Pěst. Mat. 105 (1980), no. 4, 341-344, 409.
• [29] S. G. Matthews, Partial metric topology, in: Papers on General Topology and Applications (Flushing 1992), Ann. New York Acad. Sci. 728, New York Academy of Sciences, New York (1994), 183-197.
• [30] A. Meir and E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl. 28 (1969), 326-329.
• [31] S. Lipschutz, Schaum’s Outlines of Theory and Problems of Set Theory and Related Topics, McGraw-Hill Book, New York, 1964.
• [32] A. Mukherjea, Contractions and completely continuous mappings, Nonlinear Anal. 1 (1976/77), no. 3, 235-247.
• [33] E. Picard, Mémoire sur la théorie des équations aux dérivées partielles et la méthode des approximations successives, J. Math. Pures Appl. 6 (1890), 145-210.
• [34] B. E. Rhoades, Contractive definitions and continuity, in: Fixed Point Theory and its Applications (Berkeley 1986), Contemp. Math. 72, American Mathematical Society, Providence (1988), 233-245.
• [35] A.-F. Roldán-López-de Hierro, A unified version of Ran and Reuring’s theorem and Nieto and Rodríguez-López’s theorem and low-dimensional generalizations, Appl. Math. Inf. Sci. 10 (2016), no. 2, 383-393.
• [36] B. Samet and M. Turinici, Fixed point theorems on a metric space endowed with an arbitrary binary relation and applications, Commun. Math. Anal. 13 (2012), no. 2, 82-97.
• [37] B. Samet, C. Vetro and P. Vetro, Fixed point theorems for α-ψ-contractive type mappings, Nonlinear Anal. 75 (2012), no. 4, 2154-2165.
• [38] B. Samet, C. Vetro and H. Yazidi, A fixed point theorem for a Meir-Keeler type contraction through rational expression, J. Nonlinear Sci. Appl. 6 (2013), no. 3, 162-169.
• [39] S. Shukla, Partial b-metric spaces and fixed point theorems, Mediterr. J. Math. 11 (2014), no. 2, 703-711.
• [40] M. R. Tasković, A generalization of Banach’s contraction principle, Publ. Inst. Math. (Beograd) (N. S.) 23(37) (1978), 179-191.
• [41] A. Tomar and M. Joshi, Relation-theoretic nonlinear contractions in an F-metric space and applications, Rend. Circ. Mat. Palermo (2) 70 (2021), no. 2, 835-852.
• [42] A. Tomar and R. Sharma, Some coincidence and common fixed point theorems concerning F-contraction and applications, J. Internat. Math. Virtual Institute 8 (2018), 181-198.
• [43] A. Tomar, Giniswamy, C. Jeyanthi and PG. Maheshwari, Coincidence and common fixed point of F-contractions via CLRST property, Surveys in Mathematics and its Applications 11 (2016), 21-31.
• [44] A. Tomar, M. Joshi, S. K. Padaliya, B. Joshi and A. Diwedi, Fixed point under set-valued relation-theoretic nonlinear contractions and application, Filomat 33 (2019), no. 14, 4655-4664.

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2024).