Fixed point theorems for nonlinear contractions with applications to iterated function systems

Rajendra Pant

Abstract

We introduce a new type of nonlinear contraction and present some fixed point results without using continuity or semi-continuity. Our result complement, extend and generalize a number of fixed point theorems including the the well-known Boyd and Wong theorem [On nonlinear contractions, Proc. Amer. Math. Soc. 20(1969)]. Also we discuss an  application to  iterated function systems.

Keywords

Suzuki type contraction; self-similarity; iterated function systems; fractals

Subject classification

47H10; 54H25

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References

M. Barnsley, Fractals Everywhere, Academic Press, Inc., Boston, MA, 1988. xii+396 pp.

M. Barnsley and A. Vince, The eigenvalue problem for linear and affine iterated function systems, Linear Algebra Appl. 435, no. 12 (2011), 3124-3138. https://doi.org/10.1016/j.laa.2011.05.011

L. Block and J. Keesling, Iterated function systems and the code space, Topology Appl. 122, no. 1-2 (2002), 65-75. https://doi.org/10.1016/S0166-8641(01)00134-1

J. Bohnstengel and M. Kessebohmer, Wavelets for iterated function systems, J. Funct. Anal. 259, no. 3 (2010), 583-601. https://doi.org/10.1016/j.jfa.2010.04.014

J. Bousch and J. Mairesse, Asymptotic height optimization for topical IFS, Tetris heaps, and the finiteness conjecture, J. Amer. Math. Soc. 15, no. 1 (2002), 77-111. https://doi.org/10.1090/S0894-0347-01-00378-2

D. W. Boyd and J. S. W. Wong, On nonlinear contractions, Proc. Amer. Math. Soc. 20 (1969), 458-464. https://doi.org/10.1090/S0002-9939-1969-0239559-9

B. S. Daya, G. Rangarajan and D. Veneziano, Preface- Fractals in geophysics, Chaos Solitons Fractals 19 (2004), 237-239. https://doi.org/10.1016/S0960-0779(03)00037-7

S. Dhompongsa and H. Yingtaweesittikul, Fixed points for multivalued mappings and the metric completeness, Fixed Point Theory Appl. 2009, Art. ID 972395, 15 pp.

D. Doric, Z. Kadelburg and S. Radenovic, Edelstein-Suzuki-type fixed point results in metric and abstract metric spaces, Nonlinear Anal. 75, no. 4 (2012), 1927-1932. https://doi.org/10.1016/j.na.2011.09.046

D. Dumitru, L. Ioana, R. C. Sfetcu and F. Strobin, Topological version of generalized (infinite) iterated function systems, Chaos Solitons Fractals 71 (2015), 78-90. https://doi.org/10.1016/j.chaos.2014.12.005

U. Freiberg, D. L. Torre and F. Mendivil, Iterated function systems and stability of variational problems on self-similar objects, Nonlinear Anal. Real World Appl. 12, no. 2 (2011), 1123-1129. https://doi.org/10.1016/j.nonrwa.2010.09.006

J. E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30, no. 5 (1981), 713-747. https://doi.org/10.1512/iumj.1981.30.30055

J. R. Jachymski, Equivalence of some contractivity properties over metrical structures, Proc. Amer. Math. Soc. 125, no. 8 (1997), 2327-2335. https://doi.org/10.1090/S0002-9939-97-03853-7

S. K. Kashyap, B. K. Sharma, A. Banerjee and S. C. Shrivastava, On Krasnoselskii fixed point theorem and fractal, Chaos Solitons Fractals 61 (2014), 44-45. https://doi.org/10.1016/j.chaos.2014.02.003

M. Kikkawa and T. Suzuki, Three fixed point theorems for generalized contractions with constants in complete metric spaces, Nonlinear Anal. 69, no. 9 (2008), 2942-2949. https://doi.org/10.1016/j.na.2007.08.064

M. Kikkawa and T. Suzuki, Some similarity between contractions and Kannan mappings, Fixed Point Theory Appl. 2008, Art. ID 649749, 8 pp.

G. Mot and A. Petrusel, Fixed point theory for a new type of contractive multivalued operators, Nonlinear Anal. 70, no. 9 (2009), 3371-3377. https://doi.org/10.1016/j.na.2008.05.005

M. S. Naschie, Iterated function systems and the two-slit experiment of quantum mechanics, Chaos Solitons Fractals 4 (1994), no. 10, 1965–-1968.

R. Pant, S. L. Singh and S. N. Mishra, A coincidence and fixed point theorems for semi-quasi contractions, Fixed Point Theory 17 (2016), no. 2, 449-456.

O. Popescu, Two generalizations of some fixed point theorems, Comput. Math. Appl. 62, no. 10 (2011), 3912-3919. https://doi.org/10.1016/j.camwa.2011.09.044

B. E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math. Soc. 226 (1977), 257-290. https://doi.org/10.1090/S0002-9947-1977-0433430-4

S. Ri, A new fixed point theorem in the fractal space, Indag. Math. (N.S.) 27, no. 1 (2016), 85-93.

D. R. Sahu, A. Chakraborty and R. P. Dubey, $K$-iterated function system, Fractals 18, no. 1 (2010), 139-144. https://doi.org/10.1142/S0218348X10004713

N. Secelean, Generalized iterated function systems on the space $l^infty(X),$ J. Math. Anal. Appl. 410, no. 2 (2014), 847-858. https://doi.org/10.1016/j.jmaa.2013.09.007

S. L. Singh, B. Prasad and A. Kumar, Fractals via iterated functions and multifunctions, Chaos Solitons Fractals 39, no. 3 (2009), 1224-1231. https://doi.org/10.1016/j.chaos.2007.06.014

S. L. Singh, S. N. Mishra, R. Chugh and R. Kamal, General common fixed point theorems and applications, J. Appl. Math. 2012, Art. ID 902312, 14 pp.

W. Slomczynski, From quantum entropy to iterated function systems, Chaos Solitons Fractals 8, no. 11 (1997), 1861-1864. https://doi.org/10.1016/S0960-0779(97)00073-8

F. Strobin, Attractors of generalized IFSs that are not attractors of IFSs, J. Math. Anal. Appl. 422, no. 1 (2015), 99-108. https://doi.org/10.1016/j.jmaa.2014.08.029

F. Strobin and J. Swaczyna, On a certain generalisation of the iterated function system, Bull. Aust. Math. Soc. 87, no. 1 (2013), 37-54. https://doi.org/10.1017/S0004972712000500

T. Suzuki, A generalized Banach contraction principle that characterizes metric completeness, Proc. Amer. Math. Soc. 136, no. 5 (2008), 1861-1869. https://doi.org/10.1090/S0002-9939-07-09055-7

X. Y. Wang and F. P. Li, A class of nonlinear iterated function system attractors, Nonlinear Anal. 70, no. 2 (2009), 830-838. https://doi.org/10.1016/j.na.2008.01.013

S. Xu, S. Cheng and Z. Zhou, Reich's iterated function systems and well-posedness via fixed point theory, Fixed Point Theory Appl. 2015, 2015:71, 11 pp.

Y. Y. Yao, Generating iterated function systems of some planar self-similar sets, J. Math. Anal. Appl. 421, no. 1 (2015), 938-949. https://doi.org/10.1016/j.jmaa.2014.07.051

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