    ## Pata type contractions involving rational expressions with an application to integral equations

 1 ETSI Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam, Department of Mathematics, Çankaya University, 06790, Etimesgut, Ankara, Turkey, Department of Medical Research, China Medical University Hospital, China Medical University, 40402, Taichung, Taiwan 2 University of the Free State, Bloemfontein, South Africa, Department of Mathematics and Computer Sciences, Transilvania University of Brasov, Romania

* Corresponding author: Erdal Karapınar

Received  October 2019 Revised  January 2020 Published  August 2020

In this paper, we introduce the notion of rational Pata type contraction in the complete metric space. After discussing the existence and uniqueness of a fixed point for such contraction, we consider a solution for integral equations.

Citation: Erdal Karapınar, Abdon Atangana, Andreea Fulga. Pata type contractions involving rational expressions with an application to integral equations. Discrete & Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2020420
##### References:
  T. Abdeljawad, R. P. Agarwal, E. Karapinar and P. Sumati Kumari, Solutions of the nonlinear integral equation and fractional differential equation using the technique of a fixed point with a numerical experiment in extended b-metric space, Symmetry, 11 (2019), Article Number 686. doi: 10.3390/sym11050686. Google Scholar  A. Ali, K. Shah, F. Jarad, V. Gupta and T. Abdeljawad, Existence and stability analysis to a coupled system of implicit type impulsive boundary value problems of fractional-order differential equations, Adv. Difference Equ., (2019), Article Number 101, 21 pp. doi: 10.1186/s13662-019-2047-y.  Google Scholar  A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Phys. A, 505 (2018), 688-706.  doi: 10.1016/j.physa.2018.03.056.  Google Scholar  A. Atangana and T. Mekkaoui, Trinition the complex number with two imaginary parts: Fractal, chaos and fractional calculus, Chaos Solitons Fractals, 128 (2019), 366-381.  doi: 10.1016/j.chaos.2019.08.018.  Google Scholar  S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133-181.  doi: 10.4064/fm-3-1-133-181.  Google Scholar  R. I. Batt, T. Abdeljawad, M. A.Alqudah and Mujeeb ur Rehman, Ulam stability of Caputo q-fractional delay difference equation: q-fractional Gronwall inequality approach, J. Inequal. Appl., 2019 (2019), 305. doi: 10.1186/s13660-019-2257-6.  Google Scholar  F. Jarad, T. Abdeljawad and Z. Hammouch, On a class of ordinary differential equations in the frame of Atangana-Baleanu fractional derivative, Chaos Solitons Fractals, 117 (2018), 16-20.  doi: 10.1016/j.chaos.2018.10.006.  Google Scholar  Z. Kadelburg and S. Radenović, Fixed point theorems under Pata-type conditions in metric spaces, J. Egyptian Math. Soc., 24 (2016), 77-82.  doi: 10.1016/j.joems.2014.09.001.  Google Scholar  Z. Kadelburg and S. Radenović, A note on Pata-type cyclic contractions, Sarajevo J. Math., 11 (2015), 235-245. Google Scholar  Z. Kadelburg and S. Radenović, Pata-type common fixed point results in b-metric and $b$-rectangular metric spaces, J. Nonlinear Sci. Appl., 8 (2015), 944-954.  doi: 10.22436/jnsa.008.06.05.  Google Scholar  Z. Kadelburg and S. Radenovic, Fixed point and tripled fixed point theprems under Pata-type conditions in ordered metric spaces, International Journal of Analysis and Applications, 6, (2014), 113–122. Google Scholar  E. Karapinar, T. Abdeljawad and F. Jarad, Applying new fixed point theorems on fractional and ordinary differential equations, Adv. Difference Equ., 2019 (2019), Paper No. 421, 25 pp. doi: 10.1186/s13662-019-2354-3.  Google Scholar  E. Karapinar, I. M. Erhan and Ü. Aksoy, Weak $\psi$-contractions on partially ordered metric spaces and applications to boundary value problems, Bound. Value Probl., 2014 (2014), 149, 15 pp. doi: 10.1186/s13661-014-0149-8.  Google Scholar  J. Liouville, Second mémoire sur le développement des fonctions ou parties de fonctions en séries dont divers termes sont assujettis á satisfaire a une m eme équation différentielle du second ordre contenant un paramétre variable, J. Math. Pure et Appi., 2 (1837), 16-35.   Google Scholar  S. K. Panda, T. Abdeljawad and C. Ravichandran, Novel fixed point approach to Atangana-Baleanu fractional and -Fredholm integral equations, Alexandria Engineering Journal, in press, 2020.  doi: 10.1016/j.aej.2019.12.027. Google Scholar  S. K. Panda, T. Abdeljawad and K. K. Swamy, New numerical scheme for solving integral equations via fixed point method using distinct $\omega-F$-contractions, Alexandria Engineering Journal, in press, 2020.  doi: 10.1016/j.aej.2019.12.034. Google Scholar  V. Pata, A fixed point theorem in metric spaces, J. Fixed Point Theory Appl., 10 (2011), 299-305.  doi: 10.1007/s11784-011-0060-1.  Google Scholar  O. Popescu, Some new fixed point theorems for $\alpha$-Geraghty contractive type maps in metric spaces, Fixed Point Theory Appl., 2014 (2014), 12 pp. doi: 10.1186/1687-1812-2014-190.  Google Scholar  T. Suzuki, A new type of fixed point theorem in metric spaces, Nonlinear Anal., 71 (2009), 5313-5317.  doi: 10.1016/j.na.2009.04.017.  Google Scholar  T. Suzuki, A generalized Banach contraction principle which characterizes metric completeness, Proc. Amer. Math. Soc., 136 (2008), 1861-1869.  doi: 10.1090/S0002-9939-07-09055-7.  Google Scholar

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##### References:
  T. Abdeljawad, R. P. Agarwal, E. Karapinar and P. Sumati Kumari, Solutions of the nonlinear integral equation and fractional differential equation using the technique of a fixed point with a numerical experiment in extended b-metric space, Symmetry, 11 (2019), Article Number 686. doi: 10.3390/sym11050686. Google Scholar  A. Ali, K. Shah, F. Jarad, V. Gupta and T. Abdeljawad, Existence and stability analysis to a coupled system of implicit type impulsive boundary value problems of fractional-order differential equations, Adv. Difference Equ., (2019), Article Number 101, 21 pp. doi: 10.1186/s13662-019-2047-y.  Google Scholar  A. Atangana, Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties, Phys. A, 505 (2018), 688-706.  doi: 10.1016/j.physa.2018.03.056.  Google Scholar  A. Atangana and T. Mekkaoui, Trinition the complex number with two imaginary parts: Fractal, chaos and fractional calculus, Chaos Solitons Fractals, 128 (2019), 366-381.  doi: 10.1016/j.chaos.2019.08.018.  Google Scholar  S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133-181.  doi: 10.4064/fm-3-1-133-181.  Google Scholar  R. I. Batt, T. Abdeljawad, M. A.Alqudah and Mujeeb ur Rehman, Ulam stability of Caputo q-fractional delay difference equation: q-fractional Gronwall inequality approach, J. Inequal. Appl., 2019 (2019), 305. doi: 10.1186/s13660-019-2257-6.  Google Scholar  F. Jarad, T. Abdeljawad and Z. Hammouch, On a class of ordinary differential equations in the frame of Atangana-Baleanu fractional derivative, Chaos Solitons Fractals, 117 (2018), 16-20.  doi: 10.1016/j.chaos.2018.10.006.  Google Scholar  Z. Kadelburg and S. Radenović, Fixed point theorems under Pata-type conditions in metric spaces, J. Egyptian Math. Soc., 24 (2016), 77-82.  doi: 10.1016/j.joems.2014.09.001.  Google Scholar  Z. Kadelburg and S. Radenović, A note on Pata-type cyclic contractions, Sarajevo J. Math., 11 (2015), 235-245. Google Scholar  Z. Kadelburg and S. Radenović, Pata-type common fixed point results in b-metric and $b$-rectangular metric spaces, J. Nonlinear Sci. Appl., 8 (2015), 944-954.  doi: 10.22436/jnsa.008.06.05.  Google Scholar  Z. Kadelburg and S. Radenovic, Fixed point and tripled fixed point theprems under Pata-type conditions in ordered metric spaces, International Journal of Analysis and Applications, 6, (2014), 113–122. Google Scholar  E. Karapinar, T. Abdeljawad and F. Jarad, Applying new fixed point theorems on fractional and ordinary differential equations, Adv. Difference Equ., 2019 (2019), Paper No. 421, 25 pp. doi: 10.1186/s13662-019-2354-3.  Google Scholar  E. Karapinar, I. M. Erhan and Ü. Aksoy, Weak $\psi$-contractions on partially ordered metric spaces and applications to boundary value problems, Bound. Value Probl., 2014 (2014), 149, 15 pp. doi: 10.1186/s13661-014-0149-8.  Google Scholar  J. Liouville, Second mémoire sur le développement des fonctions ou parties de fonctions en séries dont divers termes sont assujettis á satisfaire a une m eme équation différentielle du second ordre contenant un paramétre variable, J. Math. Pure et Appi., 2 (1837), 16-35.   Google Scholar  S. K. Panda, T. Abdeljawad and C. Ravichandran, Novel fixed point approach to Atangana-Baleanu fractional and -Fredholm integral equations, Alexandria Engineering Journal, in press, 2020.  doi: 10.1016/j.aej.2019.12.027. Google Scholar  S. K. Panda, T. Abdeljawad and K. K. Swamy, New numerical scheme for solving integral equations via fixed point method using distinct $\omega-F$-contractions, Alexandria Engineering Journal, in press, 2020.  doi: 10.1016/j.aej.2019.12.034. Google Scholar  V. Pata, A fixed point theorem in metric spaces, J. Fixed Point Theory Appl., 10 (2011), 299-305.  doi: 10.1007/s11784-011-0060-1.  Google Scholar  O. Popescu, Some new fixed point theorems for $\alpha$-Geraghty contractive type maps in metric spaces, Fixed Point Theory Appl., 2014 (2014), 12 pp. doi: 10.1186/1687-1812-2014-190.  Google Scholar  T. Suzuki, A new type of fixed point theorem in metric spaces, Nonlinear Anal., 71 (2009), 5313-5317.  doi: 10.1016/j.na.2009.04.017.  Google Scholar  T. Suzuki, A generalized Banach contraction principle which characterizes metric completeness, Proc. Amer. Math. Soc., 136 (2008), 1861-1869.  doi: 10.1090/S0002-9939-07-09055-7.  Google Scholar
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