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Cubo (Temuco)

On-line version ISSN 0719-0646

Cubo vol.23 no.1 Temuco  2021

http://dx.doi.org/10.4067/S0719-06462021000100171 

Articles

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. binayak12@yahoo.co.in

2 Department of Mathematics, Sovarani Memorial College, Jagatballavpur, Howrah-711408, West Bengal, India. metiya.nikhilesh@gmail.com

3 Department of Mathematics, Government General Degree College, Salboni, Paschim Mednipur-721516, West Bengal, India. sunirmalkundu2009@rediffmail.com

Abstract

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

Resumen

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.

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Acknowledgement

The suggestions of the learned referee are gratefully acknowledged.

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

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