Fixed point results for a new contraction mapping with integral and fractional applications

Hasanen A. Hammad; Hassen Aydi; Choonkil Park; Hasanen A. Hammad; Hassen Aydi; Choonkil Park

doi:10.3934/math.2022765

AIMS Mathematics

2022, Volume 7, Issue 8: 13856-13873. doi: 10.3934/math.2022765

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Research article

Fixed point results for a new contraction mapping with integral and fractional applications

1.
Department of Mathematics, Unaizah College of Sciences and Arts, Qassim University, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Sohag University, Sohag 82524, Egypt
3.
Université de Sousse, Institut Supérieur 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.
Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea

Received: 18 February 2022 Revised: 17 May 2022 Accepted: 18 May 2022 Published: 24 May 2022
MSC : 47H10, 47H09, 54E50, 55M20

The purpose of this manuscript is to present some fixed point results for a $ \Lambda $-Ćirić mapping in the setting of non-triangular metric spaces. Also, two numerical examples are given to support the theoretical study. Finally, under suitable conditions, the existence and uniqueness of a solution to a general Fredholm integral equation, a Riemann-Liouville fractional differential equation and a Caputo non-linear fractional differential equation are discussed as applications.
- fixed point technique,
- non-triangular metric space,
- Fredholm integral equation,
- fractional differential equation
Citation: Hasanen A. Hammad, Hassen Aydi, Choonkil Park. Fixed point results for a new contraction mapping with integral and fractional applications[J]. AIMS Mathematics, 2022, 7(8): 13856-13873. doi: 10.3934/math.2022765

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Abstract

The purpose of this manuscript is to present some fixed point results for a $ \Lambda $-Ćirić mapping in the setting of non-triangular metric spaces. Also, two numerical examples are given to support the theoretical study. Finally, under suitable conditions, the existence and uniqueness of a solution to a general Fredholm integral equation, a Riemann-Liouville fractional differential equation and a Caputo non-linear fractional differential equation are discussed as applications.

References

[1]	S. Banach, Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales, Fund. Math., 3 (1922), 133–181.
[2]	P. Saipara, P. Kumam, Y. Cho, Random fixed point theorems for Hardy-Rogers self-random operators with applications to random integral equations, Stochastics, 90 (2017), 297–311. https://doi.org/10.1080/17442508.2017.1346655 doi: 10.1080/17442508.2017.1346655
[3]	M. Sgroi, C. Vetro, Multi-valued $F$-contractions and the solution of certain functional and integral equations, Filomat, 27 (2013), 1259–1268.
[4]	T. Kamran, M. Postolache, M. U. Ali, Q. Kiran, Feng and Liu type $F$-contraction in $b$-metric spaces with application to integral equations, J. Math. Anal., 7 (2016), 18–27.
[5]	V. Joshi, D. Singh, A. Petrusel, Existence results for integral equations and boundary value problems via fixed point theorems for generalized $F$-contractions in $b$-metric-like spaces, J. Funct. Spaces, 2017 (2017), Article ID 1649864. https://doi.org/10.1155/20174/1649864 doi: 10.1155/20174/1649864
[6]	H. A. Hammad, M. De la Sen, Fixed-point results for a generalized almost $(s, q)$-Jaggi $F$-contraction-type on $b$-metric-like spaces, Math., 8 (2020), Paper No. 63. https://doi.org/10.3390/math8010063
[7]	S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Amsterdam: Gordon and Breach, 1993.
[8]	I. Podlubny, Fractional Differential Equations, New York: Academic Press, 1999.
[9]	C. Li, W. Deng, Remarks on fractional derivatives, Appl. Math. Comput., 187 (2007), 777–784. https://doi.org/10.1016/j.amc.2006.08.163 doi: 10.1016/j.amc.2006.08.163
[10]	F. Li, Mild solutions for fractional differential equations with nonlocal conditions, Adv. Difference Equ., 2010 (2010), Article ID 287861. https://doi.org/10.1155/2010/287861 doi: 10.1155/2010/287861
[11]	B. Ahmad, A. Alsaedi, Existence and uniqueness of solutions for coupled systems of higher order nonlinear fractional differential equations, Fixed Point Theory Appl., 2010 (2010), Article ID 364560. https://doi.org/10.1155/2010/364560 doi: 10.1155/2010/364560
[12]	H. A. Hammad, H. Aydi, M. De la Sen, Solutions of fractional differential type equations by fixed point techniques for multivalued contractions, Complexity, 2021 (2021), Article ID 5730853. https://doi.org/10.1155/2021/5730853 doi: 10.1155/2021/5730853
[13]	Y. Cui, Existence results for singular boundary value problem of nonlinear fractional differential equation, Abstr. Appl. Anal., 2011 (2011), Article ID 605614. https://doi.org/10.1155/2011/605614 doi: 10.1155/2011/605614
[14]	J. Mao, Z. Zhao, N. Xu, The existence and uniqueness of positive solutions for integral boundary value problems, Bull. Malays. Math. Sci. Soc., 34 (2011), 153–164.
[15]	J. Liu, F. Li, L. Lu, Fixed point and applications of mixed monotone operator with superlinear nonlinearity, Acta Math. Scientia. Ser. A (Chinses Ed.), 23 (2003), 19—24.
[16]	S. Xu, B. Jia, Fixed point theorems of $\phi$ concave-(-$\psi$) convex mixed monotone operators and applications, J. Math. Anal. Appl., 295 (2004), 645–657. https://doi.org/10.1016/j.jmaa.2004.03.049 doi: 10.1016/j.jmaa.2004.03.049
[17]	S. Czerwik, Contraction mappings in $b$-metric spaces, Acta Math. Inform. Univ. Ostraviensis, 1 (1993), 5–11.
[18]	S. G. Matthews, Partial metric topology, Ann. New York Acad. Sci., 728 (1994), 183–197. https://doi.org/10.1111/j.1749-6632.1994.tb44144.x doi: 10.1111/j.1749-6632.1994.tb44144.x
[19]	A. Amini-Harandi, Metric-like spaces, partial metric spaces and fixed points, Fixed Point Theory Appl., 2012 (2012), Paper No. 204. https://doi.org/10.1186/1687-1812-2012-204
[20]	Z. Mustafa, B. Sims, A new approach to generalized metric spaces, J. Nonlinear Convex Anal., 7 (2006), 289–297.
[21]	T. Kamran, M. Samreen, Q. U. Ain, A generalization of $b$-metric space and some fixed point theorems, Math., 5 (2017), Paper No. 19. https://doi.org/10.3390/math5020019
[22]	N. Mlaiki, H. Aydi, N. Souayah, T. Abdeljawad, Controlled metric type spaces and the related contraction principle, Math., 6 (2018), Paper No. 194. https://doi.org/10.3390/math6100194
[23]	T. Abdeljawad, N. Mlaiki, H. Aydi, N. Souayah, Double controlled metric type spaces and some fixed point results, Math., 6 (2018), Paper No. 320. https://doi.org/10.3390/math6120320
[24]	M. Jleli, B. Samet, A generalized metric space and related fixed point theorems, Fixed Point Theory Appl., 2015 (2015), Paper No. 61. https://doi.org/10.1186/s13663-015-0312-7
[25]	F. Khojasteh, H. Khandani, Scrutiny of some fixed point results by $S$-operator without triangle inequality, Math. Slovaca, 70 (2020), 467–476. https://doi.org/10.1515/ms-2017-0364 doi: 10.1515/ms-2017-0364
[26]	S. Pourrazi, F. Khojasteh, M. Javahernia, H. Khandani, On non-triangular metric and $JS$-metric spaces and related consequences via $\widehat {Man}(\mathbb{R})$-contractions, J. Math. Anal., 10 (2019), 31–39.
[27]	L. B. Ćirić, Generalized contractions and fixed-point theorems, Publ. Inst. Math., 12 (1971), 19–26.
[28]	D. Wardowski, Fixed points of a new type of contractive mapping in complete metric spaces, Fixed Point Theory Appl., 2012 (2012), Paper No. 94. https://doi.org/10.1186/1687-1812-2012-94
[29]	R. A. Rashwan, H. A. Hammad, A common random fixed point theorem of rational inequality in polish spaces with application, Facta Univ. Ser. Math. Inform., 32 (2017), 703–714. https://doi.org/10.22190/FUMI1705703R doi: 10.22190/FUMI1705703R
[30]	El-S. El-Hady, E. Agrekci, On Hyers-Ulam-Rassias stability of fractional differential equations with Caputo derivative, J. Nonlinear Sci. Appl., 22 (2022), 325–332.
[31]	K. Maazouz, R. Rodriguez-Lopez, Differential equations of arbitrary order under Caputo-Fabrizio derivative: Some existence results and study of stability, Math. Biosci. Eng., 19 (2022), 6234–6251. https://doi.org/10.3934/mbe.2022291 doi: 10.3934/mbe.2022291
[32]	El-S. El-Hady, E. Agrekci, On Hyers-Ulam-Rassias stability of fractional differential equations with Caputo derivative, J. Math. Comput. Sci., 22 (2021), 325–332. https://doi.org/10.22436/jmcs.022.04.02 doi: 10.22436/jmcs.022.04.02
[33]	M. Alghamdi, A. Aljehani, A. E. Hamza, Hyers-Ulam-Rassias stability of abstract second-order linear dynamic equations on time scales, J. Math. Comput. Sci., 24 (2022), 110–118. https://doi.org/10.22436/jmcs.024.02.02 doi: 10.22436/jmcs.024.02.02
[34]	A. Bartwal, R. C. Dimri, G. Prasad, Some fixed point theorems in fuzzy bipolar metric spaces, J. Nonlinear Sci. Appl., 13 (2020), 196–204. https://doi.org/10.22436/jnsa.013.04.04 doi: 10.22436/jnsa.013.04.04

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