A Relation Theoretic Approach for φ-Fixed Point Result in Metric Space with an Application to an Integral Equation

Document Type : Original Article


1 Department of Applied Mathematics & Humanities, S. V. National Institute of Technology, Surat-395007, Gujarat, India

2 Poornima College of Engineering, Jaipur-302022, Rajasthan, India


In this paper, we prove the existence and uniqueness of φ- fixed point for (F,φ,θ)-contraction mapping in a complete metric space with a binary relation. Here the contractive condition is required to hold only for
those elements that are related under the binary relation and not for the whole space. An application is
given to show the φusability of our result obtained.


[1]A. Alam and M. Imdad, Relation-theoretic contraction principle. Journal of Fixed Point Theory and Applications, 17(4) (2015), 693-702.
[2]A. Sahin, Z. Kalkan, H. Arisoy, On the solution of a nonlinear Volterra integral equation with delay, Sakarya University Journal of Science, 21 (6), 2017, 1367-1376, doi: 10.16984/saufenbilder.305632.
[3]S. Banach, Sur les operations dans les ensembles abstraits et leur application aux equations integrales. Fund.Math. 3 (1922), 13 3181.1
[4]M. Jleli, B. Samet, C. Vetro, Fixed point theory in partial metric spaces via φ- fixed point's concept in metric spaces, Journal of Inequalities and Applications, 2014 (2014) 426.
[5]P. Kumrod, and W. Sintunavarat, A new contractive condition approach to φ- fixed point results in metric spaces and its applications. Journal of Computational and Applied Mathematics 311 (2017) , 194-204.
[6]M. A. Kutbi, A. Roldan, W. Sintunavarat, J. Martinez-Moreno and C. Roldan, F-closed sets and coupled fixed point theorems without the mixed monotone property. Fixed Point Theory and its Applications, 2013 (2013),doi:10.1186/1687-1812-2013- 330, 11 pp.1.4 
[7]S. Lipschutz, Schaums Outlines of Theory and Problems of Set Theory and Related Topics. McGrawHill, New York, 1964.1
[8]A.M. Rismani, H. Monfared, Numerical solution of singular ivps of laneemden type using a modiff ed legendrespectral
method. Applied Mathematical Modelling, 36 (2012), 4830-4836.
[9]Y. Khan, Z. Svoboda, Z. Smarda, Solving certain classes of lane-emden type equations using the differential transformation method. Advances in Di erence Equations, 2012(1) (2012), 174. doi:10.1186/1687-1847-2012-174.
[10]S. Liao, A new analytic algorithm of laneemden type equations. Applied Mathematics and Computation, 142(1)(2003), 1-16. doi:10.1016/S0096-3003(02)00943-8.
[11]A. Yldrm, T.
OZiS. , Solutions of singular ivps of laneemden type by homotopy perturbation method. Physics Letters
A, 369(1) (2007), 70{76. doi:10.1016/j.physleta.2007.04.072.
[12]C.M. Bender, K.A. Milton, S.S. Pinsky, L.M. Simmons, A new perturbative approach to nonlinear problems. Journal of Mathematical Physics, 30(7), (1989) 1447-1455.