Some existence and uniqueness results for a solution of a system of equations

Authors

DOI:

https://doi.org/10.4995/agt.2024.19798

Keywords:

Matkowski's contraction, system of equations, control function, metric space

Abstract

This paper presents some existence and uniqueness results for a solution of a system of equations. Our results extend and generalize the well-known and celebrated results of Boyd and Wong [Proc. Amer. Math. Soc. 20 (1969)], Matkowski [Dissertations Math.(Rozprawy Mat.) 127 (1975)], Proinov [Nonlinear Anal. 64 (2006)], Ri [Indag. Math. (N. S.) 27 (2016)] and many others. We also present some illustrative examples to validate our results.

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Author Biographies

Deepak Khantwal, University of Johannesburg

Department of Mathematics & Applied Mathematics

Rajendra Pant, University of Johannesburg

Department of Mathematics & Applied Mathematics

References

J. B. Baillon and S. L. Singh, Nonlinear hybrid contractions on product spaces, Far East J. Math. Sci. 1, no. 2 (1993), 117-127.

R. K. Bisht, R. P. Pant, and V. Rakočević , Proinov contractions and discontinuity at fixed point, Miskolc Math. Notes 20, no. 1 (2019), 131-137. https://doi.org/10.18514/MMN.2019.2277

D. W. Boyd and J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969), 458-464. https://doi.org/10.1090/S0002-9939-1969-0239559-9

F. E. Browder and W. V. Petryshyn, The solution by iteration of nonlinear functional equations in Banach spaces, Bull. Amer. Math. Soc. 72 (1966), 571-575. https://doi.org/10.1090/S0002-9904-1966-11544-6

Lj. B. Ćirić , A generalization of Banach's contraction principle, Proc. Amer. Math. Soc. 45 (1974), 267-273. https://doi.org/10.1090/S0002-9939-1974-0356011-2

S. Czerwik, A fixed point theorem for a system of multivalued transformations, Proc. Amer. Math. Soc. 55, no. 1 (1976), 136-139. https://doi.org/10.1090/S0002-9939-1976-0394619-0

S. Czerwik, Generalization of Edelstein's fixed point theorem, Demonstratio Math. 9, no. 2 (1976), 281-285. https://doi.org/10.1515/dema-1976-0215

M. Edelstein, On fixed and periodic points under contractive mappings, J. London Math. Soc. 37 (1962), 74-79. https://doi.org/10.1112/jlms/s1-37.1.74

U. C. Gairola and P. S. Jangwan, Co-ordinatewise R-weakly commuting maps and fixed point theorem on product spaces, Demonstratio Math. 36, no. 4 (2003), 939-949. https://doi.org/10.1515/dema-2003-0418

U. C. Gairola and P. S. Jangwan, Coincidence theorem for multi-valued and single-valued systems of transformations, Demonstratio Math. 41, no. 1 (2008), 129-136. https://doi.org/10.1515/dema-2008-0113

U. C. Gairola, S. N. Mishra, and S. L. Singh, Coincidence and fixed point theorems on product spaces, Demonstratio Math. 30, no. 1 (1997), 15-24. https://doi.org/10.1515/dema-1997-0104

U. C. Gairola, S. L. Singh, and J. H. M. Whitfield, Fixed point theorems on product of compact metric spaces, Demonstratio Math. 28, no. 3 (1995), 541-548. https://doi.org/10.1515/dema-1995-0305

F. R. Gantmakher, The theory of matrices, vol. 2, American Mathematical Soc., 2000.

J. Jachymski, Equivalent conditions and the Meir-Keeler type theorems, J. Math. Anal. Appl. 194, no. 1 (1995), 293-303. https://doi.org/10.1006/jmaa.1995.1299

D. Khantwal, S. Aneja, and U. C. Gairola, A generalization of Matkowski's and Suzuki's fixed point theorems, Asian-Eur. J. Math. 15, no. 9 (2022), Article ID 2250169, 12. https://doi.org/10.1142/S1793557122501698

D. Khantwal and U. C. Gairola, An extension of Matkowski's and Wardowski's fixed point theorems with applications to functional equations, Aequationes Math. 93, no. 2 (2019), 433-443. https://doi.org/10.1007/s00010-018-0562-7

D. Khantwal, I. K. Letlhage, and R. Pant, Fixed point results for Suzuki type contractions in relational metric spaces with applications, Indian J. Math. 64, no. 3 (2022), 279-304.

M. A. Krasnoselski, Two remarks on the method of successive approximations, Acad. R. P. Romîne. An. Romîno-Soviet. Ser. Mat.-Fiz. (3) 10 (1956), no. 2 (17), 55-59.

J. Matkowski, Some inequalities and a generalization of Banach's principle, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21 (1973), 323-324.

J. Matkowski, Integrable solutions of functional equations, Dissertationes Math. (Rozprawy Mat.) 127 (1975), 68.

J. Matkowski, Fixed point theorems for contractive mappings in metric spaces, Časopis Pĕst. Mat. 105, no. 4 (1980), 341-344. https://doi.org/10.21136/CPM.1980.108246

J. Matkowski, and S. L. Singh, Banach type fixed point theorems on product of spaces, Indian J. Math. 38, no. 1 (1996), 73-80.

A. Meir and E. Keeler, A theorem on contraction mappings, J. Math. Anal. Appl. 28 (1969), 326-329. https://doi.org/10.1016/0022-247X(69)90031-6

R. Pant and R. Shukla, New fixed point results for Proinov-Suzuki type contractions in metric spaces, Rend. Circ. Mat. Palermo (2) 71, no. 2 (2022), 633-645. https://doi.org/10.1007/s12215-021-00649-z

P. D. Proinov, Fixed point theorems in metric spaces, Nonlinear Anal. 64, no. 3 (2006), 546-557. https://doi.org/10.1016/j.na.2005.04.044

K. B. Reddy, and P. V. Subrahmanyam, Extensions of Krasnoselskii 's and Matkowski's fixed point theorems, Funkcial. Ekvac. 24, no. 1 (1981), 67-83.

K. B. Reddy and P. V. Subrahmanyam, Altman's contractors and fixed points of multivalued mappings, Pacific J. Math. 99, no. 1 (1982), 127-136. https://doi.org/10.2140/pjm.1982.99.127

S. Ri, A new fixed point theorem in the fractal space, Indag. Math. (N.S.) 27, no. 1 (2016), 85-93. https://doi.org/10.1016/j.indag.2015.07.006

S. L. Singh, and U. C. Gairola, Coordinatewise commuting and weakly commuting maps, and extension of Jungck and Matkowski contraction principles, J. Math. Phys. Sci. 25, no. 4 (1991), 305-318.

S. L. Singh, and U. C. Gairola, A general fixed point theorem, Math. Japon. 36, no. 4 (1991), 791-801.

S. L. Singh, S. N. Mishra, and V. Chadha, Round-off stability of iterations on product spaces, C. R. Math. Rep. Acad. Sci. Canada 16, no. 2-3 (1994), 105-109.

S. L. Singh, S. N. Mishra, and R. Pant, New fixed point theorems for asymptotically regular multi-valued maps, Nonlinear Anal. 71, no. 7-8 (2009), 3299-3304. https://doi.org/10.1016/j.na.2009.01.212

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Published

2024-04-02

How to Cite

[1]
D. Khantwal and R. Pant, “Some existence and uniqueness results for a solution of a system of equations”, Appl. Gen. Topol., vol. 25, no. 1, pp. 159–174, Apr. 2024.

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Regular Articles