American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms
  • DCDS-S Home
  • This Issue
  • Next Article
    Dimension reduction of thermistor models for large-area organic light-emitting diodes
doi: 10.3934/dcdss.2020420

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

Erdal Karapınar 1,, , Abdon Atangana 2, and  Andreea Fulga 2,

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.

Keywords: Pata type contraction, rational expression, $ C $-condition, orbital admissible mapping.
Mathematics Subject Classification: Primary: 54H25, 47H10; Secondary: 54E50.
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:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

Z. Kadelburg and S. Radenović, A note on Pata-type cyclic contractions, Sarajevo J. Math., 11 (2015), 235-245.   Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15] 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
[16] 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
[17]

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

[18]

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

[19]

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

[20]

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

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

Z. Kadelburg and S. Radenović, A note on Pata-type cyclic contractions, Sarajevo J. Math., 11 (2015), 235-245.   Google Scholar

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15] 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
[16] 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
[17]

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

[18]

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

[19]

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

[20]

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

[1]

Beom-Seok Han, Kyeong-Hun Kim, Daehan Park. A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3415-3445. doi: 10.3934/dcds.2021002
[2]

Siqi Chen, Yong-Kui Chang, Yanyan Wei. Pseudo $ S $-asymptotically Bloch type periodic solutions to a damped evolution equation. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021017
[3]

Jong Yoon Hyun, Yoonjin Lee, Yansheng Wu. Connection of $ p $-ary $ t $-weight linear codes to Ramanujan Cayley graphs with $ t+1 $ eigenvalues. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020133
[4]

Jennifer D. Key, Bernardo G. Rodrigues. Binary codes from $ m $-ary $ n $-cubes $ Q^m_n $. Advances in Mathematics of Communications, 2021, 15 (3) : 507-524. doi: 10.3934/amc.2020079
[5]

Chaoqian Li, Yajun Liu, Yaotang Li. Note on $ Z $-eigenvalue inclusion theorems for tensors. Journal of Industrial & Management Optimization, 2021, 17 (2) : 687-693. doi: 10.3934/jimo.2019129
[6]

Ruonan Liu, Run Xu. Hermite-Hadamard type inequalities for harmonical $ (h1,h2)- $convex interval-valued functions. Mathematical Foundations of Computing, 2021  doi: 10.3934/mfc.2021005
[7]

Dean Crnković, Nina Mostarac, Bernardo G. Rodrigues, Leo Storme. $ s $-PD-sets for codes from projective planes $ \mathrm{PG}(2,2^h) $, $ 5 \leq h\leq 9 $. Advances in Mathematics of Communications, 2021, 15 (3) : 423-440. doi: 10.3934/amc.2020075
[8]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2635-3652. doi: 10.3934/dcds.2020378
[9]

Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113
[10]

Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195
[11]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1631-1648. doi: 10.3934/dcdss.2020447
[12]

Raj Kumar, Maheshanand Bhaintwal. Duadic codes over $ \mathbb{Z}_4+u\mathbb{Z}_4 $. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020135
[13]

Brian Ryals, Robert J. Sacker. Bifurcation in the almost periodic $ 2 $D Ricker map. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021089
[14]

Alfonso Castro, Jorge Cossio, Sigifredo Herrón, Carlos Vélez. Infinitely many radial solutions for a $ p $-Laplacian problem with indefinite weight. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021058
[15]

Zhaoxia Wang, Hebai Chen. A nonsmooth van der Pol-Duffing oscillator (I): The sum of indices of equilibria is $ -1 $. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021096
[16]

Yishui Wang, Dongmei Zhang, Peng Zhang, Yong Zhang. Local search algorithm for the squared metric $ k $-facility location problem with linear penalties. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2013-2030. doi: 10.3934/jimo.2020056
[17]

Zhaoxia Wang, Hebai Chen. A nonsmooth van der Pol-Duffing oscillator (II): The sum of indices of equilibria is $ 1 $. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021101
[18]

Said Taarabti. Positive solutions for the $ p(x)- $Laplacian : Application of the Nehari method. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021029
[19]

Khalid Latrach, Hssaine Oummi, Ahmed Zeghal. Existence results for nonlinear mono-energetic singular transport equations: $ L^p $-spaces. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021028
[20]

Yao Nie, Jia Yuan. The Littlewood-Paley $ pth $-order moments in three-dimensional MHD turbulence. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3045-3062. doi: 10.3934/dcds.2020397

2019 Impact Factor: 1.233

Tools

Metrics

  • PDF downloads (87)
  • HTML views (266)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences