Integral type contractions and best proximity points

Subject Areas : Fixed point theory

K. Fallahi ¹ , F. Esmaeilnia ² , A. Pourmoslemi ³

1 - Department of Mathematics, Payame Noor University, Tehran, Iran
2 - Department of Mathematics, Payame Noor University, Tehran, Iran
3 - Department of Mathematics, Payame Noor University, Tehran, Iran

Received: 2023-02-26 Accepted : 2023-03-29 Published : 2023-03-30

Keywords: Integral type contraction, Best proximity point, Lebesgue-integrable function, graph proximal,

Abstract :

‎In the present work‎, ‎Banach and Kannan integral type contractions in metric spaces endowed with a graph are considered and‎ ‎the existence and uniqueness of best proximity points for mappings satisfying in these contractions are proved‎.

References:

[1] A. Aghanians, K. Nourouzi, Fixed points of integral type contractions in uniform spaces, Filomat. 29 (7) (2015), 1613-1621.

[2] S. Banach, Sur les operations dans les ensembles abstraits et leur application aux quations intgrales, Fund. Math. J. 3 (1922), 133-181.

[3] A. Branciari, A fixed point theorem for mappings satisfying a general contractive condition of integral type, Int. J. Math. Sci. 29 (9) (2002), 531-536.

[4] K. Fallahi, A. Aghanians, On quasi-contractions in metric spaces with a graph, Hacettepe J. Math. Statis. 45 (4) (2016), 1033-1047.

[5] K. Fallahi, H. Ghahramani, G. Soleimani Rad, Integral type contractions in partially ordered metric spaces and best proximity point, Iranian J. Sci. Tech. Transactions A. Sci. 44 (2020), 177-183.

[6] K. Fallahi, G. Soleimani Rad, A. Fulga, Best proximity points for (φ − ψ)-weak contractions and some applications, Filomat. 37 (6) (2023), 1835-1842.

[7] R. Kannan, Some results on fixed points-II, Amer. Math. Monthly. 76 (1969), 405-408.

[8] J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc. 136 (4) (2008), 1359-1373.

[9] J. J. Nieto, R. Rodriguez-Lopez, Contractive mappings theorems in partially ordered sets and applications to ordinary differential equations, Order. 22 (3) (2005), 223-239.

[10] A. Petrusel, I. A. Rus, Fixed point theory in terms of a metric and of an order relation, Fixed Point Theory. 20 (2) (2019), 601-622.

[11] VS. Raj, A best proximity point theorem for weakly contractive nonself-mappings, Nonlinear Anal. 74 (2011), 4804-4808.

[12] A. C. M. Ran, M. C. B. Reurings, A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc. 132 (5) (2004), 1435-1443.

[13] S. Sadiq Basha, Best proximity point theorems in the frameworks of fairly and proximally complete spaces, J. Fixed Point Theory Appl. 19 (3) (2017), 1939-1951.

[14] S. Sadiq Basha, Discrete optimization in partially ordered sets, J. Glob. Optim. 54 (2012), 511-517.