SciELO - Scientific Electronic Library Online

vol.23 issue1Idempotents in an ultrametric Banach algebra author indexsubject indexarticles search
Home Pagealphabetic serial listing  

Services on Demand




Related links

  • On index processCited by Google
  • Have no similar articlesSimilars in SciELO
  • On index processSimilars in Google


Cubo (Temuco)

On-line version ISSN 0719-0646

Cubo vol.23 no.1 Temuco  2021 


Existence, well-posedness of coupled fixed points and application to nonlinear integral equations

1 Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah-711103, West Bengal, India.

2 Department of Mathematics, Sovarani Memorial College, Jagatballavpur, Howrah-711408, West Bengal, India.

3 Department of Mathematics, Government General Degree College, Salboni, Paschim Mednipur-721516, West Bengal, India.


We investigate a fixed point problem for coupled Geraghty type contraction in a metric space with a binary relation. The role of the binary relation is to restrict the scope of the contraction to smaller number of ordered pairs. Such possibilities have been explored for different types of contractions in recent times which has led to the emergence of relational fixed point theory. Geraghty type contractions arose in the literatures as a part of research seeking the replacement contraction constants by appropriate functions. Also coupled fixed point problems have evoked much interest in recent times. Combining the above trends we formulate and solve the fixed point problem mentioned above. Further we show that with some additional conditions such solution is unique. Well-posedness of the problem is investigated. An illustrative example is discussed. The consequences of the results are discussed considering α-dominated mappings and graphs on the metric space. Finally we apply our result to show the existence of solution of some system of nonlinear integral equations.

Keywords and Phrases: Metric space; coupled fixed point; well-posedness; application


Investigamos un problema de punto fijo para contracciones acopladas de tipo Geraghty en un espacio métrico con una relación binaria. El rol de la relación binaria es restringir el alcance de la contracción a un número menor de pares ordenados. Tales posibilidades han sido exploradas para diferentes tipos de contracciones recientemente, lo que ha conllevado el nacimiento de la teoría de punto fijo relacional. Las contracciones de tipo Geraghty aparecen en la literatura como parte de la investigación buscando reemplazar las constantes de contracción por funciones apropiadas. También problemas de puntos fijos acoplados han sido de mucho interés recientemente. Combinando las ideas anteriores, formulamos y resolvemos el problema de punto fijo mencionado anteriormente. Más aún, mostramos que bajo condiciones adicionales tal solución es única. Se investiga la bien-definición del problema. Se discute un ejemplo ilustrativo. Las consecuencias de los resultados se discuten considerando aplicaciones α-dominadas y grafos en espacios métricos. Finalmente aplicamos nuestros resultados para mostrar la existencia de soluciones de algunos sistemas de ecuaciones integrales no lineales.

Texto completo disponible sólo en PDF

Full text available only in PDF format.


The suggestions of the learned referee are gratefully acknowledged.


[1] A. Alam, and M. Imad, “Relation-theoretic contraction principle", J. Fixed Point Theory Appl., vol. 17, pp. 693-702, 2015. [ Links ]

[2] M. R. Alfuraidan, and M. A. Khamsi, “Caristi fixed point theorem in metric spaces with a graph", Abstr. Appl. Anal., Article ID 303484, 5 pages, 2014. [ Links ]

[3] M. S. Asgari, and B. Mousavi, “Coupled fixed point theorems with respect to binary relations in metric spaces", J. Nonlinear Sci. Appl., vol. 8, pp. 153-162, 2015. [ Links ]

[4] I. Beg, A. R. Butt, and S. Radojević, “The contraction principle for set valued mappings on a metric space with a graph", Comput. Math. Appl., vol. 60, pp. 1214-1219, 2010. [ Links ]

[5] D. W. Boyd, and T. S. W. Wong, “On nonlinear contractions", Proc. Amer. Math. Soc., vol. 20, pp. 458-464, 1969. [ Links ]

[6] C. Chifu, and G. Petruşel, “Coupled fixed point results for (ϕ; G)-contractions of type (b) in b-metric spaces endowed with a graph", J. Nonlinear Sci. Appl. , vol. 10. pp. 671-683, 2017. [ Links ]

[7] B. S. Choudhury, and A. Kundu, “A coupled coincidence point result in partially ordered metric spaces for compatible mappings", Nonlinear Anal., vol. 73, pp. 2524-2531, 2010. [ Links ]

[8] B. S. Choudhury , and A. Kundu, “On coupled generalised Banach and Kannan type contractions", J. Nonlinear Sci. Appl. , vol. 5, pp. 259-270, 2012. [ Links ]

[9] B. S. Choudhury , N. Metiya, and M. Postolache, “A generalized weak contraction principle with applications to coupled coincidence point problems", Fixed Point Theory Appl., 152(2013), 2013. [ Links ]

[10] B. S. Choudhury , N. Metiya , and S. Kundu, “Existence and stability results for coincidence points of nonlinear contractions", Facta Universitatis (NÎS) Ser. Math. Inform., vol. 32, no. 4, pp. 469-483, 2017. [ Links ]

[11] B. S. Choudhury , N. Metiya , and S. Kundu, “Fixed point sets of multivalued contractions and stability analysis", Commun. Math. Sci., vol. 2, pp. 163-171, 2018. [ Links ]

[12] M. Dinarvand, “Fixed point results for (ϕ − ѱ) contractions in metric spaces endowed with a graph and applications", Matematichki Vesnik, vol. 69, no. 1, pp. 23-38, 2017. [ Links ]

[13] D. Dorić, “Common fixed point for generalized (ѱ ; ϕ)-weak contractions", Appl. Math. Lett., vol. 22, pp. 1896-1900, 2009. [ Links ]

[14] P. N. Dutta, andB. S. Choudhury , “A generalisation of contraction principle in metric spaces", Fixed Point Theory Appl. , Article ID 406368, 2008. [ Links ]

[15] T. Gnana Bhaskar, and V. Lakshmikantham, “Fixed point theorems in partially ordered metric spaces and applications", Nonlinear Anal. , vol. 65, pp. 1379-1393, 2006. [ Links ]

[16] M. Geraghty, “On contractive mappings", Proc. Amer. Math. Soc. , vol. 40, pp. 604-608, 1973. [ Links ]

[17] D. Guo, and V. Lakshmikantham, “Coupled fixed points of nonlinear operators with applications", Nonlinear Anal. , vol. 11, pp. 623-632, 1987. [ Links ]

[18] J. Harjani, B. López, and K. Sadarangani, “Fixed point theorems for mixed monotone operators and applications to integral equations", Nonlinear Anal. , vol. 74, pp. 1749-1760, 2011. [ Links ]

[19] N. Hussain, E. Karapinar, P. Salimi, and F. Akbar, “α-admissible mappings and related fixed point theorems", J. Inequal. Appl., 114(2013), 2013. [ Links ]

[20] Z. Kadelburg, P. Kumam, S. Radenović, and W. Sintunavarat, “Common coupled fixed point theorems for Geraghty-type contraction mappings using monotone property", Fixed Point Theory Appl. , 27(2015), 2015. [ Links ]

[21] E. Karapinar , “Couple fixed point theorems for nonlinear contractions in cone metric spaces", Comput. Math. Appl. , vol. 59, pp. 3656-3668, 2010. [ Links ]

[22] M. S. Khan, M. Swaleh, and S. Sessa, “Fixed points theorems by altering distances between the points", Bull. Aust. Math. Soc., vol. 30, pp. 1-9, 1984. [ Links ]

[23] M. S. Khan , M. Berzig, and S. Chandok, “Fixed point theorems in bimetric space endowed with a binary relation and application", Miskolc Mathematical Notes, vol. 16, no. 2, pp. 939-951, 2015. [ Links ]

[24] M. A. Kutbi, and W. Sintunavarat, “Ulam-Hyers stability and well-posedness of fixed point problems for α − λ-contraction mapping in metric spaces", Abstr. Appl. Anal. , Article ID 268230, vol. 2014, 6 pages, 2014. [ Links ]

[25] B. K. Lahiri, and P. Das, “Well-posedness and porosity of a certain class of operators", Demonstratio Math., vol. 1, pp. 170-176, 2005. [ Links ]

[26] X. L Liu, M. Zhou, and B. Damjanović, “Common coupled fixed point theorem for Geraghtytype contraction in partially ordered metric spaces", Journal of Function Spaces, vol. 2018, Article ID 9063267, 11 pages, 2018. [ Links ]

[27] S. Phiangsungnoen, andP. Kumam , “Generalized Ulam-Hyers stability and well-posedness for fixed point equation via α-admissibility", J. Inequal. Appl. , 418(2014), 2014. [ Links ]

[28] V. Popa, “Well-posedness of fixed point problem in orbitally complete metric spaces", Stud. Cercet. Stiint., Ser. Mat., vol. 16, pp 209-214, 2006. [ Links ]

[29] P. Salimi , A. Latif, andN. Hussain , “Modified α−ѱ -contractive mappings with applications", Fixed Point Theory Appl. , 151(2013), 2013. [ Links ]

[30] B. Samet, and C. Vetro, “Coupled fixed point theorems for multi-valued nonlinear contraction mappings in partially ordered metric spaces", Nonlinear Anal. , vol. 74, no. 12, pp. 4260-4268, 2011. [ Links ]

[31] B. Samet , C. Vetro, and P. Vetro, “Fixed point theorem for α− -contractive type mappings", Nonlinear Anal. , vol. 75, pp. 2154-2165, 2012. [ Links ]

[32] K. P. R. Sastry, G. V. R. Babu, P. S. Kumar, and B. R. Naidu, “Fixed point theorems for α-Geraghty contraction type maps in Generalized metric spaces", MAYFEB Journal of Mathematics, vol. 3, pp. 28-44, 2017. [ Links ]

[33] E. Yolacan, and M. Kir, “New results for α−Geraghty type contractive maps with some applications", GU J Sci, vol. 29, no. 3, pp. 651-658, 2016. [ Links ]

Accepted: March 29, 2021; Received: June 14, 2020

Creative Commons License This is an open-access article distributed under the terms of the Creative Commons Attribution License