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Some observations on generalized non-expansive mappings with an application

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Abstract

This study aimed at showing that the classes of generalized non-expansive mappings due to Hardy and Rogers and the mappings satisfying Suzuki’s condition (C) are independent and study some basic properties of generalized non-expansive mappings. Also, we introduce a new iterative scheme, called JF iterative scheme, and prove convergence results for generalized non-expansive mappings due to Hardy and Rogers in uniformly convex Banach spaces. Moreover, we show numerically that JF iterative scheme converges to a fixed point of generalized non-expansive mappings faster than some known and leading iterative schemes. As an application, we utilize newly defined iterative scheme to approximate the solution of a delay differential equation. Also, we present some nontrivial illustrative numerical examples to support main results. Our results are new and extend several relevant results in the existing literature.

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References

  • Agrawal RP, O’Regan D, Sahu DR (2007) Iterative construction of fixed points of nearly asymptotically non-expansive mappings. J Nonlinear Convex Anal 8(1):61–79

    MathSciNet  MATH  Google Scholar 

  • Ali J, Ali F, Kumar P (2019) Approximation of fixed points for Suzuki’s generalized non-expansive mappings. Mathematics 7(6):522

    Article  Google Scholar 

  • Bae JS (1984) Fixed point theorems of generalized non-expansive maps. J Korean Math Soc 21(2):233–248

    MathSciNet  MATH  Google Scholar 

  • Bogin J (1976) A generalization of a fixed point theorem of Goebel, Kirk and Shimi. Can Math Bull 19:7–12

    Article  MathSciNet  Google Scholar 

  • Bose RK, Mukherjee RN (1981) Approximating fixed points of some mappings. Proc Am Math Soc 82:603–606

    Article  MathSciNet  Google Scholar 

  • Coman GH, Pavel G, Rus I, Rus IA (1976) Introduction in the theory of operational equation, Ed. Dacia, Cluj-Napoca

  • Cooke KL, van den Driessche P, Zou X (1999) Interaction of maturation delay and nonlinear birth in population and epidemic models. J Math Biol 39:332–352

    Article  MathSciNet  Google Scholar 

  • Dhomphongsa S, Inthakon W, Kaewkhao A (2009) Edelstein’s method and fixed point theorems for some generalized non-expansive mappings. J Math Anal Appl 350(1):12–17

    Article  MathSciNet  Google Scholar 

  • Fukhar-ud-din H, Saleh K (2018) One-step iterations for a finite family of generalized non-expansive mappings in CAT(0) spaces. Bull Malays Math Sci Soc 41(2):597–608

    MathSciNet  MATH  Google Scholar 

  • Fuster EL, Gàlvez EM (2011) The fixed point theory for some generalized non-expansive mappings, Abstract and Applied Analysis 2011, p 15s

  • Geobel K, Kirk WA (1990) Topic in metric fixed point theory. Cambridge University Press, Cambridge

    Book  Google Scholar 

  • Goebel K, Kirk WA, Shimi TN (1973) A fixed point theorem in uniformly convex spaces. Boll Un Mat Ital. 7:67–75

    MathSciNet  MATH  Google Scholar 

  • Gürsoy F (2016) A Picard-S iterative method for approximating fixed point of weak-contraction mappings. Filomat 30(10):2829–2845

    Article  MathSciNet  Google Scholar 

  • Gürsoy F, Khan AR, Ertürk M, Karakaya V (2018) Convergence and data dependency of normal-S iterative method for discontinuous operators on Banach space. Numer Funct Anal Optim 39(3):322–345

    Article  MathSciNet  Google Scholar 

  • Gürsoy F, Eksteen JJA, Khan AR, Karakaya V (2019) An iterative method and its application to stable inversion. Soft Comput 23(16):7393–7406

    Article  Google Scholar 

  • Gursoy F, Karakaya V (2014) A Picard-S hybrid type iteration method for solving a differential equation with retarded argument. arXiv:1403.2546v2

  • Hardy GF, Rogers TD (1973) A generalization of a fixed point theorem of Reich. Can Math Bull 16:201–206

    Article  MathSciNet  Google Scholar 

  • Ishikawa S (1974) Fixed points by a new iteration method. Proc Am Math Soc 44:147–150

    Article  MathSciNet  Google Scholar 

  • Maiti M, Ghosh MK (1989) Approximating fixed points by Ishikawa iterates. Bull Austral Math Soc 40:113–117

    Article  MathSciNet  Google Scholar 

  • Mann WR (1953) Mean value methods in iteration. Proc Am Math Soc 4:506–510

    Article  MathSciNet  Google Scholar 

  • Noor MA (2000) New approximation schemes for general variational inequalities. J Math Anal Appl 251(1):217–229

    Article  MathSciNet  Google Scholar 

  • Opial Z (1967) Weak convergence of the sequence of successive approximations for non-expansive mappings. Bull Am Math Soc 73:595–597

    Article  Google Scholar 

  • Park JY, Jeong JU (1994) Weak convergence to a fixed point of the sequence of Mann type iterates. J Math Anal Appl 184(1):75–81

    Article  MathSciNet  Google Scholar 

  • Picard E (1890) Memoire sur la theorie des equations aux derivees partielles et la methode des approximations successives. J Math Pures Appl 6:145–210

    MATH  Google Scholar 

  • Schu J (1991) Weak and strong convergence to fixed points of asymptotically non-expansive mappings. Bull Aust Math Soc 43(1):153–159

    Article  MathSciNet  Google Scholar 

  • Senter HF, Dotson WG (1974) Approximating fixed points of non-expansive mappings. Proc Am Math Soc 44(2):375–380

    Article  Google Scholar 

  • Soltuz SM, Otrocol D (2007) Classical results via Mann–Ishikawa iteration. Revue d’Analyse Numérique et de Théorie de l’Approximation 36(2):195–199

    MathSciNet  MATH  Google Scholar 

  • Suzuki T (2008) Fixed point theorems and convergence theorems for some generalized non-expansive mappings. J Math Anal Appl 340(2):1088–1095

    Article  MathSciNet  Google Scholar 

  • Tan KK, Xu HK (1993) Approximating fixed points of non-expansive mappings by the Ishikawa iteration process. J Math Anal Appl 178:301–308

    Article  MathSciNet  Google Scholar 

  • Thakur BS, Thakur D, Postolache M (2016) A new iterative scheme for numerical reckoning fixed points of Suzuki’s generalized non-expansive mappings. Appl Math Comput 275:147–155

    MathSciNet  MATH  Google Scholar 

  • Turchin P (1990) Rarity of density dependence or population regulation with lags. Nature 344:660–663

    Article  Google Scholar 

  • Uddin I, Imdad M (2015) Some convergence theorems for a hybrid pair of generalized nonexpansive mappings in CAT(0) spaces. J Nonlinear Convex Anal 16(3):447–457

    MathSciNet  MATH  Google Scholar 

  • Uddin I, Imdad M (2015) On certain convergence of S-iteration scheme in CAT(0) spaces. Kuwait J Sci 42(2):93–106

    MathSciNet  Google Scholar 

  • Uddin I, Imdad M (2018) Convergence of SP-iteration for generalized nonexpansive mapping in Hadamard spaces. Hacet J Math Stat 47(6):1595–1604

    MathSciNet  Google Scholar 

  • Villasana M, Radunskaya A (2003) A delay differential equation model for tumor growth. J Math Biol 47(3):270–294

    Article  MathSciNet  Google Scholar 

  • Wong CS (1974) Generalized contractions and fixed point theorems. Proc Am Math Soc 42:409–417

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors are grateful to the anonymous referees for their valuable comments and suggestions which improve the paper. The first author would like to thank Council of Scientific and Industrial Research, Government of India for SRF (09/112(0536)/2016-EMR-I). The work of J. J. Nieto has been partially supported by Agencia Estatal de Investigacin (AEI) of Spain under grant MTM2016-75140-P co-financed by the European Community fund FEDER, and XUNTA de Galicia under grants GRC2015-004 and R2016-022.

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Authors and Affiliations

  1. Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India

    Faeem Ali & Javid Ali

  2. Department of Statistics, Mathematical Analysis and Optimization, Faculty of Mathematics, University of Santiago de Compostela, Santiago de Compostela, Spain

    Juan J. Nieto

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  1. Faeem Ali
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  2. Javid Ali
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  3. Juan J. Nieto
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Correspondence to Javid Ali.

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Communicated by Carlos Conca.

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Ali, F., Ali, J. & Nieto, J.J. Some observations on generalized non-expansive mappings with an application. Comp. Appl. Math. 39, 74 (2020). https://doi.org/10.1007/s40314-020-1101-4

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