Krasnoselskii-type results for equiexpansive and equicontractive operators with application in radiative transfer equations

Niaz Ahmad; Nayyar Mehmood; Thabet Abdeljawad; Aiman Mukheimer; Niaz Ahmad; Nayyar Mehmood; Thabet Abdeljawad; Aiman Mukheimer

doi:10.3934/math.2023746

AIMS Mathematics

2023, Volume 8, Issue 6: 14592-14608. doi: 10.3934/math.2023746

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Krasnoselskii-type results for equiexpansive and equicontractive operators with application in radiative transfer equations

1.
Department of Mathematics and Statistics, International Islamic University H-10, Islamabad, Pakistan
2.
Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
3.
Department of Medical Research, China Medical University, Taichung 40402, Taiwan
4.
Department of Mathematics Kyung Hee University 26 Kyungheedae-ro, Dongdaemun-gu Seoul 02447, Republic of Korea

Received: 08 October 2022 Revised: 17 March 2023 Accepted: 06 April 2023 Published: 20 April 2023
MSC : 47H10, 54H25

In this paper, we study the results of fixed points for the operator equations of type $ x = H(\digamma x, x) $ using the idea of measure of noncompactness and assuming that the operator $ \digamma $ is $ k $ -set contractive (strictly $ k $ -set contractive, or a continuous) and the family $ \{H(u, .):u\} $ is equiexpansive or equicontractive. The obtained results are generalization of Krasnoselskii type fixed point results. Some examples are given to elaborate new concepts. We use the main result to find the existence of solutions for the stationary radiative transfer equation in a channel. We demonstrate our theory with an example by comparison of an approximate solution with the exact solution.
- Krasnoselskii,
- equiexpansive,
- equicontractive,
- fixed points,
- radiative transfer
Citation: Niaz Ahmad, Nayyar Mehmood, Thabet Abdeljawad, Aiman Mukheimer. Krasnoselskii-type results for equiexpansive and equicontractive operators with application in radiative transfer equations[J]. AIMS Mathematics, 2023, 8(6): 14592-14608. doi: 10.3934/math.2023746

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Abstract

In this paper, we study the results of fixed points for the operator equations of type $ x = H(\digamma x, x) $ using the idea of measure of noncompactness and assuming that the operator $ \digamma $ is $ k $ -set contractive (strictly $ k $ -set contractive, or a continuous) and the family $ \{H(u, .):u\} $ is equiexpansive or equicontractive. The obtained results are generalization of Krasnoselskii type fixed point results. Some examples are given to elaborate new concepts. We use the main result to find the existence of solutions for the stationary radiative transfer equation in a channel. We demonstrate our theory with an example by comparison of an approximate solution with the exact solution.

References

[1]	N. Dunford, J. T. Schwartz, Linear operators, part 1: General theory. John Wiley & Sons, 1988 Feb 23.
[2]	E. Kreyszig, Introductory functional analysis with applications, New York: wiley, 1978 Jan.
[3]	E. Malkowsky, V. Rakočević, Advanced functional analysis, CRC Press, 2019 Feb 25.
[4]	W. Rudin, Principles of mathematical analysis, New York: McGraw-hill, 1964 Jan.
[5]	D. Wardowski, Family of mappings with an equicontractive-type condition, J. Fix. Point Theory A., 22 (2020), 55. https://doi.org/10.1007/s11784-020-00789-2 doi: 10.1007/s11784-020-00789-2
[6]	D. R. Smart, Fixed point theorems, Cup Archive, 1980 Feb 14.
[7]	T. A. Burton, A fixed-point theorem of Krasnoselskii, Appl. Math. Lett., 11 (1998), 85–89. https://doi.org/10.1016/S0893-9659(98)00016-0 doi: 10.1016/S0893-9659(98)00016-0
[8]	Y. Z. Chen, Krasnoselskii-type fixed point theorems using $ \alpha $-concave operators, J. Fix.Point Theory A., 22 (2020), 52. https://doi.org/10.1007/s11784-020-00792-7 doi: 10.1007/s11784-020-00792-7
[9]	S. Park, Generalizations of the Krasnoselskii fixed point theorem, Nonlinear Anal-Theory,, 67 (2007), 3401–3410. https://doi.org/10.1016/j.na.2006.10.024 doi: 10.1016/j.na.2006.10.024
[10]	E. Pourhadi, R. Saadati, Z. Kadelburg, Some Krasnosel'skii-type fixed point theorems for Meir–Keeler-type mappings, Nonlinear Anal. Model., 25 (2020), 257–265. https://orcid.org/0000-0003-3056-4299
[11]	G. Bal, Diffusion approximation of radiative transfer equations in a channel, Transport Theor. Stat., 30 (2001), 269–293. https://doi.org/10.1081/TT-100105370 doi: 10.1081/TT-100105370
[12]	A. Peraiah, An introduction to radiative transfer: Methods and applications in astrophysics, Cambridge University Press, 2002.
[13]	E. Zeidler, P. R. Wadsack, Nonlinear functional analysis and Its Applications: Fixed-point Theorems/Transl, by Peter R. Wadsack. Springer-Verlag, 1993.
[14]	F. F Bonsall, K. B Vedak, Lectures on some fixed point theorems of functional analysis, Bombay: Tata Institute of Fundamental Research, 1962.
[15]	G. L. Karakostas, An extension of Krasnoselskiĭ's fixed point theorem for contractions and compact mappings, Topol. Method. Nonl. An., 22 (2003), 181–191. https://doi.org/10.18261/ISSN1500-1571-2004-03-12 doi: 10.18261/ISSN1500-1571-2004-03-12
[16]	W. R. Melvin, Some extensions of the Krasnoselskii fixed point theorems, J. Differ. Equations., 11 (1972), 335–348. https://doi.org/10.1016/0022-0396(72)90049-6 doi: 10.1016/0022-0396(72)90049-6
[17]	M. Z. Nashed, J. S. Wong, Some variants of a fixed point theorem of Krasnoselskii and applications to nonlinear integral equations, J. Math. Mec., 18 (1969), 767–777. Available from: https://www.jstor.org/stable/24893136
[18]	T. Xiang, R. Yuan, Critical type of Krasnosel'skii fixed point theorem, P. Am. Math. Soc., 139 (2011), 1033–1044. https://doi.org/10.1090/S0002-9939-2010-10517-8 doi: 10.1090/S0002-9939-2010-10517-8
[19]	T. Xiang, R. Yuan, A class of expansive-type Krasnosel'skii fixed point theorems, Nonlinear Anal. Theor., 71 (2009), 3229–3339. https://doi.org/10.1016/j.na.2009.01.197 doi: 10.1016/j.na.2009.01.197
[20]	T. Xiang, Notes on expansive mappings and a partial answer to Nirenberg's problem, Electron. J. Differ. Eq., 2013 (2013), 1–6. Available from: http://ejde.math.txstate.edu or http://ejde.math.unt.eduftpejde.math.txstate.edu
[21]	O. Diekmann, S. A. Van Gils, S. M. V. Lunel, H. O. Walther, Delay equations: Functional-, complex-, and nonlinear analysis, Springer Science & Business Media, 2012 Dec 6.
[22]	N. Ahmad, N. Mehmood, A. Al-Rawashdeh, Some variants of Krasnoselskii-Type fixed point results for equiexpansive mappings with applications, J. Function Space., 2021 (2021), Article ID 2648057. https://doi.org/10.1155/2021/2648057 doi: 10.1155/2021/2648057
[23]	J. M. A Toledano, T. D. Benavides, G. L. Acedo, Measures of noncompactness in metric fixed point theory, Oper. Theory Adv. Appl., 99 (1997).
[24]	R. R Akhmerov, M. I Kamenskii, A. S. Potapov, A. E. Rodkina, B. N. Sadovskii, Measures of noncompactness and condensing operators, Basel: Birkhä user, 1992.
[25]	J. Banaś, On measures of noncompactness in Banach spaces, Comment. Math. Univ. Ca., 21 (1980), 131–143. Available from: http://dml.cz/dmlcz/105982
[26]	J. Banaś, M. Jleli, M. Mursaleen, B. Samet, C. Vetro, Eds, Advances in nonlinear analysis via the concept of measure of noncompactness, Singapore: Springer Singapore; 2017 Apr 25.
[27]	V. I. Istratesecu, On a measure of noncompactness, B. Math. Soc. Sci. Math., 16 (1972), 195–197.
[28]	V. Rakočević, Measures of noncompactness and some applications, Filomat, 1 (1998), 87–120. Available from: https://www.jstor.org/stable/43999286
[29]	T. Xiang, S. G. Georgiev, Noncompact-type Krasnoselskii fixed-point theorems and their applications, Math. Method. Appl. Sci., 39 (2016), 833–863. https://doi.org/10.1002/mma.3525 doi: 10.1002/mma.3525

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