Fixed points of nonlinear contractions with applications

Mohammed Shehu Shagari; Qiu-Hong Shi; Saima Rashid; Usamot Idayat Foluke; Khadijah M. Abualnaja; Mohammed Shehu Shagari; Qiu-Hong Shi; Saima Rashid; Usamot Idayat Foluke; Khadijah M. Abualnaja

doi:10.3934/math.2021545

AIMS Mathematics

2021, Volume 6, Issue 9: 9378-9396. doi: 10.3934/math.2021545

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Research article

Fixed points of nonlinear contractions with applications

1.
Department of Mathematics, Faculty of Physical Sciences, Ahmadu Bello University, Zaria, Nigeria
2.
Department of Mathematics, Huzhou University, Huzhou 313000, China
3.
Department of Mathematics, Government College University, Faisalabad, Pakistan
4.
Department of Mathematics, Faculty of Physical Sciences, University of Ilorin, Ilorin, Nigeria
5.
Department of Mathematics, Faculty of Science, Taif University, P. O. Box 11099, Taif 21944, Saudi Arabia

Received: 13 December 2020 Accepted: 10 June 2021 Published: 23 June 2021
MSC : Primary 47H09, 47H10, 90C39, 45D05, 34A12; Secondary 54H25

The aim of this paper is to initiate a new concept of nonlinear contraction under the name $ r $-hybrid $ \psi $-contraction and establish some fixed point results for such mappings in the setting of complete metric spaces. The presented ideas herein unify and extend a number of well-known results in the corresponding literature. A few of these special cases are pointed out and analysed. From application point of view, we investigate the existence and uniqueness criteria of solutions to certain functional equation arising in dynamic programming and integral equation of Volterra type. Nontrivial illustrative examples are provided to show the generality and validity of our obtained results.
- fixed point,
- $ \psi $-contraction,
- $ r $-hybrid $ \psi $-contraction,
- dynamic programming,
- integral equation
Citation: Mohammed Shehu Shagari, Qiu-Hong Shi, Saima Rashid, Usamot Idayat Foluke, Khadijah M. Abualnaja. Fixed points of nonlinear contractions with applications[J]. AIMS Mathematics, 2021, 6(9): 9378-9396. doi: 10.3934/math.2021545

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Abstract

The aim of this paper is to initiate a new concept of nonlinear contraction under the name $ r $-hybrid $ \psi $-contraction and establish some fixed point results for such mappings in the setting of complete metric spaces. The presented ideas herein unify and extend a number of well-known results in the corresponding literature. A few of these special cases are pointed out and analysed. From application point of view, we investigate the existence and uniqueness criteria of solutions to certain functional equation arising in dynamic programming and integral equation of Volterra type. Nontrivial illustrative examples are provided to show the generality and validity of our obtained results.

References

[1]	Q. L. Dong, Y. C. Tang, Y. J. Cho, T. M. Rassias, Optimal choice of the step length of the projection and contraction methods for solving the split feasibility problem, J. Global Optim., 71 (2018), 341–360. doi: 10.1007/s10898-018-0628-z
[2]	Y. Yao, R. P. Agarwal, M. Postolache, Y. C. Liou, Algorithms with strong convergence for the split common solution of the feasibility problem and fixed point problem, Fixed Point Theory Appl., 2014 (2014), 183. doi: 10.1186/1687-1812-2014-183
[3]	M. A. Noor, On nonlinear variational inequalities, Int. J. Math. Math. Sci., 14 (1991), 399–402. doi: 10.1155/S0161171291000479
[4]	Y. M. Chu, S. Rashid, F. Jarad, M. A. Noor, H. Kalsoom, More new results on integral inequalities for generalized K-fractional conformable integral operators, Discrete Contin. Dyn. Syst. Ser. S, 2021. DOI: 10.3934/dcdss.2021063.
[5]	S. S. Zhou, S. Rashid, A. Rauf, F. Jarad, Y. S. Hamed, K. M. Abualnaja, Efficient computations for weighted generalized proportional fractional operators with respect to a monotone function, AIMS Math., 6 (2021), 8001–8029. doi: 10.3934/math.2021465
[6]	S. Rashid, S. Sultana, F. Jarad, H. Jafari, Y. S. Hamed, More efficient estimates via h-discrete fractional calculus theory and applications, Chaos Solitons Fract., 147 (2021), 110981. doi: 10.1016/j.chaos.2021.110981
[7]	H. G. Jile, S. Rashid, F. B. Farooq, S. Sultana, Some inequalities for a new class of convex functions with applications via local fractional integral, J. Funct. Spaces Appl., 2021 (2021), 1–17.
[8]	S. Rashid, S. Parveen, H. Ahmad, Y. M. Chu, New quantum integral inequalities for some new classes of generalized $\psi$-convex functions and their scope in physical systems, Open Phys., 19 (2021). DOI: 10.1515/phys-2021-0001.
[9]	D. W. Barnes, L. A. Lambe, A fixed point approach to homological perturbation theory, Proc. Amer. Math. Soc., 112 (1991), 881–892. doi: 10.1090/S0002-9939-1991-1057939-0
[10]	B. Hazarika, H. M. Srivastava, R. Arab, M. Rabbani, Existence of solution for an infinite system of nonlinear integral equations via measure of noncompactness and homotopy perturbation method to solve it, J. Comput. Appl. Math., 343 (2018), 341–352. doi: 10.1016/j.cam.2018.05.011
[11]	M. Abukhaled, S. A. Khuri, A semi-analytical solution of amperometric enzymatic reactions based on Green's functions and fixed point iterative schemes, J. Electroanal. Chem., 792 (2017), 66–71. doi: 10.1016/j.jelechem.2017.03.015
[12]	A. A. El-Deeb, S. Rashid, On some new double dynamic inequalities associated with Leibniz integral rule on time scales, Adv. Differ. Equ., 2021 (2021), 1–22. doi: 10.1186/s13662-020-03162-2
[13]	S. S. Zhou, S. Rashid, S. Parveen, A. O. Akdemir, Z. Hammouch, New computations for extended weighted functionals within the Hilfer generalized proportional fractional integral operators, AIMS Math., 6 (2021), 4507–4525. doi: 10.3934/math.2021267
[14]	M. Al-Qurashi, S. Rashid, S. Sultana, H. Ahmad, K. A. Gepreel, New formulation for discrete dynamical type inequalities via $ h $-discrete fractional operator pertaining to nonsingular kernel, Math. Biosci. Eng., 18 (2021), 1794–1812. doi: 10.3934/mbe.2021093
[15]	Y. M. Chu, S. Rashid, J. Singh, A novel comprehensive analysis on generalized harmonically $\Psi$-convex with respect to Raina's function on fractal set with applications, Math. Methods Appl. Sci., 2021. DOI: 10.1002/mma.7346.
[16]	S. Rashid, F. Jarad, Z. Hammouch, Some new bounds analogous to generalized proportional fractional integral operator with respect to another function, Discrete. Contin. Dyn. Syst. Ser. S, 2021. DOI: 10.3934/DCDSS.2021020.
[17]	H. Iiduka, Fixed point optimization algorithm and its application to network bandwidth allocation, J. Comput. Appl. Math., 236 (2012), 1733–1742. doi: 10.1016/j.cam.2011.10.004
[18]	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
[19]	E. Karapinar, Revisiting the Kannan type contractions via interpolation, Adv. Theory Nonlinear Anal. Appl., 2 (2018), 85–87.
[20]	E. Karapınar, O. Alqahtani, H. Aydi, On interpolative Hardy-Rogers type contractions, Symmetry, 11 (2019), 1–7.
[21]	S. Reich, Some remarks concerning contraction mappings, Can. Math. Bull., 14 (1971), 121–124. doi: 10.4153/CMB-1971-024-9
[22]	D. Wardowski, Fixed points of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl., 2012 (2012), 94. doi: 10.1186/1687-1812-2012-94
[23]	D. Wardowski, N. Van Dung, Fixed points of $F$-weak contractions on complete metric spaces, Demonstratio Math., 47 (2014), 146–155.
[24]	N. A. Secelean, Iterated function systems consisting of F-contractions, Fixed Point Theory Appl., 2013 (2013), 1–13. doi: 10.1186/1687-1812-2013-1
[25]	H. Piri, P. Kumam, Some fixed point theorems concerning $F$-contraction in complete metric spaces, Fixed Point Theory Appl., 2014 (2014), 210. doi: 10.1186/1687-1812-2014-210
[26]	M. Cosentino, P. Vetro, Fixed point results for $F$-contractive mappings of Hardy-Rogers-type, Filomat, 28 (2014), 715–722. doi: 10.2298/FIL1404715C
[27]	M. Alansari, S. S. Mohammed, A. Azam, N. Hussain, On multivalued hybrid contractions with applications, J. Funct. Spaces Appl., 2020 (2020), 1–12.
[28]	E. Ameer, H. Aydi, M. Arshad, H. Alsamir, M. S. Noorani, Hybrid multivalued type contraction mappings in $\alpha K$-complete partial $b$-metric spaces and applications, Symmetry, 11 (2019). DOI: 10.3390/sym11010086.
[29]	E. Karapinar, A. Fulga, New hybrid contractions on $b$-metric spaces, Mathematics, 7 (2019), 578. doi: 10.3390/math7070578
[30]	E. Karapinar, H. Aydi, A. Fulga, On $p$-hybrid Wardowski contractions, J. Math., 2020 (2020), 1–7.
[31]	J. Ahmad, A. Al-Rawashdeh, A. Azam, New fixed point theorems for generalized $F$-contractions in complete metric spaces, Fixed Point Theory Appl., 2015 (2015), 80. doi: 10.1186/s13663-015-0333-2
[32]	H. Aydi, E. Karapinar, A. F. Roldán López de Hierro, $\omega$-interpolative Ćirić-Reich-Rus-type contractions, Mathematics, 7 (2019), 57. doi: 10.3390/math7010057
[33]	H. Aydi, C. M. Chen, E. Karapınar, Interpolative Ćirić-Reich-Rus type contractions via the Branciari distance, Mathematics, 7 (2019), 84. doi: 10.3390/math7010084
[34]	D. Derouiche, H. Ramoul, New fixed point results for $F$-contractions of Hardy-Rogers type in $b$-metric spaces with applications, J. Fixed Point Theory Appl., 22 (2020), 1–44. doi: 10.1007/s11784-019-0746-3
[35]	E. Karapinar, R. Agarwal, H. Aydi, Interpolative Reich-Rus-Ćirić type contractions on partial metric spaces, Mathematics, 6 (2018), 256. doi: 10.3390/math6110256
[36]	T. Rasham, A. Shoaib, N. Hussain, M. Arshad, S. U. Khan, Common fixed point results for new Ciric-type rational multivalued $F$-contraction with an application, J. Fixed Point Theory Appl., 20 (2018), 45. doi: 10.1007/s11784-018-0525-6
[37]	R. Bellman, E. S. Lee, Functional equations in dynamic programming, Aequationes Math., 17 (1978), 1–18. doi: 10.1007/BF01818535

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