On dominated multivalued operators involving nonlinear contractions and applications

Tahair Rasham; Najma Noor; Muhammad Safeer; Ravi Prakash Agarwal; Hassen Aydi; Manuel De La Sen; Tahair Rasham; Najma Noor; Muhammad Safeer; Ravi Prakash Agarwal; Hassen Aydi; Manuel De La Sen

doi:10.3934/math.2024001

AIMS Mathematics

2024, Volume 9, Issue 1: 1-21. doi: 10.3934/math.2024001

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Research article

On dominated multivalued operators involving nonlinear contractions and applications

1.
Department of Mathematics, University of Poonch Rawalakot, Azad Kashmir, Pakistan
2.
Department of Mathematics Texas A& M University-Kingsville 700 University Blvd., MSC 172 Kingsville, Texas 78363-8202 USA
3.
Universit'e de Sousse, Institut Superieur d'Informatique et des Techniques de Communication, H. Sousse 4000, Tunisia
4.
China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
5.
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Ga-Rankuwa, South Africa
6.
Institute of Research and Development of Processes IIDP, University of the Basque Country, Campus of Leioa, PO Box 48940, Leioa, Bizkaia 48940, Spain

Received: 28 July 2023 Revised: 06 November 2023 Accepted: 07 November 2023 Published: 24 November 2023
MSC : 05C12, 39Bxx, 47H04, 47H10, 54H25

The objective of this research is to establish new results for set-valued dominated mappings that meet the criteria of advanced locally contractions in a complete extended b-metric space. Additionally, we intend to establish new fixed point outcomes for a couple of dominated multi-functions on a closed ball that satisfy generalized local contractions. In this study, we present novel findings for dominated maps in an ordered complete extended b-metric space. Additionally, we introduce a new concept of multi-graph dominated mappings on a closed ball within these spaces and demonstrate some original results for graphic contractions equipped with a graphic structure. To demonstrate the uniqueness of our new discoveries, we verify their applicability in obtaining a joint solution of integral and functional equations. Our findings have also led to modifications of numerous classical and contemporary results in existing research literature.
- fixed point,
- advanced locally contraction,
- ordered extended b-metric space,
- dominated multi-functions,
- graph theory,
- integral equation,
- functional equation
Citation: Tahair Rasham, Najma Noor, Muhammad Safeer, Ravi Prakash Agarwal, Hassen Aydi, Manuel De La Sen. On dominated multivalued operators involving nonlinear contractions and applications[J]. AIMS Mathematics, 2024, 9(1): 1-21. doi: 10.3934/math.2024001

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Abstract

The objective of this research is to establish new results for set-valued dominated mappings that meet the criteria of advanced locally contractions in a complete extended b-metric space. Additionally, we intend to establish new fixed point outcomes for a couple of dominated multi-functions on a closed ball that satisfy generalized local contractions. In this study, we present novel findings for dominated maps in an ordered complete extended b-metric space. Additionally, we introduce a new concept of multi-graph dominated mappings on a closed ball within these spaces and demonstrate some original results for graphic contractions equipped with a graphic structure. To demonstrate the uniqueness of our new discoveries, we verify their applicability in obtaining a joint solution of integral and functional equations. Our findings have also led to modifications of numerous classical and contemporary results in existing research literature.

References

[1]	Ö. Acar, G. Durmaz, G. Minak, Generalized multivalued F-contractions on complete metric spaces, Bull. Iranian Math. Soc., 40 (2014), 1469–1478.
[2]	R. P. Agarwal, U. Aksoy, E. Karapınar, I. M. Erhan, F-contraction mappings on metric-like spaces in connection with integral equations on time scales, RACSAM, 114 (2020), 147. https://doi.org/10.1007/s13398-020-00877-5 doi: 10.1007/s13398-020-00877-5
[3]	J. Ahmad, A. Al-Rawashdeh, A. Azam, Some new fixed point theorems for generalized contractions in complete metric spaces. Fixed Point Theory Appl., 2015 (2015), 80. https://doi.org/10.1186/s13663-015-0333-2 doi: 10.1186/s13663-015-0333-2
[4]	B. Alqahtani, H. Aydi, E. Karapınar, V. Rakočević, A Solution for Volterra fractional integral equations by hybrid contractions, Mathematics, 7 (2019), 694. https://doi.org/10.3390/math7080694 doi: 10.3390/math7080694
[5]	M. U. Ali, T. Kamran, E. Karapınar, Further discussion on modified multivalued α^-ψ-contractive type mapping, Filomat*, 29 (2015), 1893–1900. https://doi.org/10.2298/FIL1508893A doi: 10.2298/FIL1508893A
[6]	H. H. Alsulami, E., Karapinar, H. Piri, Fixed points of modified F-contractive mappings in complete metric-like spaces, J. Funct. Space., 2015 (2015), 270971. https://doi.org/10.1155/2015/270971 doi: 10.1155/2015/270971
[7]	S. Al-Sadi, M. Bibi, M. Muddassar, S. Kermausuor, Generalized m-preinvexity on fractal set and related local fractional integral inequalities with applications, J. Math. Comput. Sci., 30 (2023), 352–371. https://doi.org/10.22436/jmcs.030.04.05 doi: 10.22436/jmcs.030.04.05
[8]	I. Altun, G. Mınak, M. Olgun, Fixed points of multivalued nonlinear F-contractions on complete metric spaces, Nonlinear Anal. Model., 21 (2016), 201–210. https://doi.org/10.15388/NA.2016.2.4 doi: 10.15388/NA.2016.2.4
[9]	E. Ameer, M. Arshad, Two new generalization for F-contraction on closed ball and fixed point theorem with applications, J. Math. Ext., 11 (2017), 43–67.
[10]	H. Aydi, E. Karapinar, H. Yazidi, Modified F-contractions via α-admissible mappings and application to integral equations, Filomat, 31 (2017), 1141–1148. https://doi.org/10.2298/FIL1705141A doi: 10.2298/FIL1705141A
[11]	S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133–181. https://doi.org/10.4064/FM-3-1-133-181 doi: 10.4064/FM-3-1-133-181
[12]	I. A. Bakhtin, The contraction mapping principle in almost quasispaces, Funkts. Anal., 30 (1989), 26–37.
[13]	L. B. Ciric, Fixed point for generalized multivalued contractions, Mat. Vesnik., 9 (1972), 265–272.
[14]	M. Cosentino, M. Jleli, B. Samet, C. Vetro, Solvability of integrodifferential problems via fixed point theory in b-metric spaces, Fixed Point Theory Appl., 2015 (2015), 70. https://doi.org/10.1186/s13663-015-0317-2 doi: 10.1186/s13663-015-0317-2
[15]	S. Czerwik, Contraction mappings in b-metric spaces, Acta Math. Inform. Univ. Ostrav., 1 (1993), 5–11.
[16]	S. Czerwik, Nonlinear set-valued contraction mappings in b-metric spaces, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), 263–276.
[17]	G. J. de Cabral-García, K. Baquero-Mariaca, J. Villa-Morales, A fixed point theorem in the space of integrable functions and applications, Rend. Circ. Mat. Palerm., 72 (2023), 655–672. https://doi.org/10.1007/s12215-021-00714-7 doi: 10.1007/s12215-021-00714-7
[18]	M. Demma, R. Saadati, P. Vetro, Fixed point results on b-metric space via Picard sequences and b-simulation functions, Iran. J. Math. Sci. Inform., 11 (2016), 123–136. https://doi.org/10.7508/ijmsi.2016.01.011 doi: 10.7508/ijmsi.2016.01.011
[19]	G. M. A. Elhamed, Fixed point results for $ (\beta, \alpha)$-implicit contractions in two generalized b-metric spaces, J. Nonlinear Sci. Appl., 14 (2021), 39–47. https://doi.org/10.22436/jnsa.014.01.05 doi: 10.22436/jnsa.014.01.05
[20]	Y. Feng, S. Liu, Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings, J. Math. Anal. Appl., 317 (2006), 103–112. https://doi.org/10.1016/j.jmaa.2005.12.004 doi: 10.1016/j.jmaa.2005.12.004
[21]	J. Jachymski, The contraction principle for mappings on a metric space with a graph, Proc. Amer. Math. Soc., 136 (2008), 1359–1373. https://doi.org/10.1090/S0002-9939-07-09110-1 doi: 10.1090/S0002-9939-07-09110-1
[22]	M. Jleli, B. Samet, C. Vetro, F. Vetro, Fixed points for multivalued mappings in b-metric spaces, Abstr. Appl. Anal., 2015 (2015), 718074. https://doi.org/10.1155/2015/718074 doi: 10.1155/2015/718074
[23]	T. Kamran, M. Samreen, Q. Ain, A generalization of b-metric space and some fixed point theorems, Mathematics, 5 (2017), 19. https://doi.org/10.3390/math5020019 doi: 10.3390/math5020019
[24]	E. Karapınar, M. A. Kutbi, H. Piri, D. O'Regan, Fixed points of conditionally F-contractions in complete metric-like spaces, Fixed Point Theory Appl., 2015 (2015), 126. https://doi.org/10.1186/s13663-015-0377-3 doi: 10.1186/s13663-015-0377-3
[25]	E. Karapınar, A. Fulga, M. Rashid, L. Shahid, H. Aydi, Large contractions on quasi-metric spaces with an application to nonlinear fractional differential equations, Mathematics, 7 (2019), 444. https://doi.org/10.3390/math7050444 doi: 10.3390/math7050444
[26]	Q. Kiran, N. Alamgir, N. Mlaiki, H.Aydi, On some new fixed point results in complete extended b-metric space. Mathematics, 7 (2019), 476. https://doi.org/10.3390/math7050476 doi: 10.3390/math7050476
[27]	G. Minak, M. Olgun, I. Altun, A new approach to fixed point theorems for multivalued contractive maps, Carpathian J. Math., 31 (2015), 241–248.
[28]	S. B. Nadler, Multivalued contraction mappinġs, Pac. J. Math., 30 (1969), 475–488. https://doi.org/10.2140/pjm.1969.30.475 doi: 10.2140/pjm.1969.30.475
[29]	L. N. Mishra, V. Dewangan, V. N. Mishra, S. Karateke, Best proximity points of admissible almost generalized weakly contractive mappings with rational expressions on b-metric spaces, J. Math. Comput. Sci., 22 (2021); 97–109. https://doi.org/10.22436/jmcs.022.02.01 doi: 10.22436/jmcs.022.02.01
[30]	H. K. Nashine, L. K. Dey, R. W. Ibrahim, S. Radenovic, Feng-Liu-type fixed point result in orbital b-metric spaces and application to fractal integral equation, Nonlinear Anal. Model., 26 (2021), 522–533. https://doi.org/10.15388/namc.2021.26.22497 doi: 10.15388/namc.2021.26.22497
[31]	H. K. Nashine, Z. Kadelburg, Cyclic generalized ϕ-contractions in b-metric spaces and an application to integral equations, Filomat, 28 (2014), 2047–2057. https://doi.org/10.2298/FIL140047N doi: 10.2298/FIL140047N
[32]	M. Nazam, C. Park, M. Arshad, Fixed point problems for generalized contractions with applications, Adv. Difffer. Equ., 2021 (2021), 247. https://doi.org/10.1186/s13662-021-03405-w doi: 10.1186/s13662-021-03405-w
[33]	J. J. Nieto, R. Rodríguez-López, Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order, 22 (2005), 223–239. https://doi.org/10.1007/s11083-005-9018-5 doi: 10.1007/s11083-005-9018-5
[34]	A. Padcharoen, D. Gopal, P. Chaipunya, P. Kumam, Fixed point and periodic point results for α-type F-contractions in modular metric spaces, Fixed Point Theory Appl., 2016 (2016), 39. https://doi.org/10.1186/s13663-016-0525-4 doi: 10.1186/s13663-016-0525-4
[35]	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. https://doi.org/10.1007/s11784-018-0525-6 doi: 10.1007/s11784-018-0525-6
[36]	T. Rasham, M. S. Shabbir, P. Agarwal, S. Momani, On a pair of fuzzy dominated mappings on closed ball in the multipli-cative metric space with applications, Fuzzy Set Syst., 437 (2022), 81–96. https://doi.org/10.1016/j.fss.2021.09.002 doi: 10.1016/j.fss.2021.09.002
[37]	T. Rasham, M. D. La Sen, A novel study for hybrid pair of multivalued dominated mappings in b-multiplicative metric space with applications, J. Inequal. Appl., 2022 (2022), 107. https://doi.org/10.1186/s13660-022-02845-6 doi: 10.1186/s13660-022-02845-6
[38]	T. Rasham, A. Shoaib, G. Marino, B. A. S. Alamri, M. Arshad, Sufficient conditions to solve two systems of integral equations via fixed point results, J. Inequal. Appl., 2019 (2019), 182. https://doi.org/10.1186/s13660-019-2130-7 doi: 10.1186/s13660-019-2130-7
[39]	T. Rasham, G. Marino, A. Shahzad, C. Park, A. Shoaib, Fixed point results for a pair of fuzzy mappings and related applications in b-metric like spaces, Adv. Differ. Equ., 2021 (2021), 259. https://doi.org/10.1186/s13662-021-03418-5 doi: 10.1186/s13662-021-03418-5
[40]	N. A. Secelean, Iterated function systems consisting of F-contractions, Fixed Point Theory Appl., 2013 (2013), 277. https://doi.org/10.1186/1687-1812-2013-277 doi: 10.1186/1687-1812-2013-277
[41]	M. Sgroi, C. Vetro, Multi-valued F-contractions and the solution of certain functional and integral equations, Filomat, 27 (2013), 1259–1268. https://doi.org/10.2298/FIL1307259S doi: 10.2298/FIL1307259S
[42]	D. Wardowski, Fixed point theory of a new type of contractive mappings in complete metric spaces, Fixed Point Theory Appl., 2012 (2012), 94. https://doi.org/10.1186/1687-1812-2012-94 doi: 10.1186/1687-1812-2012-94
[43]	K. Zhang, J. Xu, Solvability for a system of Hadamard-type hybrid fractional differential inclusions, Demonstr. Math., 56 (2023), 20220226. https://doi.org/10.1515/dema-2022-0226 doi: 10.1515/dema-2022-0226

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