On existence theorems for coupled systems of quadratic Hammerstein-Urysohn integral equations in Orlicz spaces

Ateq Alsaadi; Mohamed M. A. Metwali; Ateq Alsaadi; Mohamed M. A. Metwali

doi:10.3934/math.2022889

AIMS Mathematics

2022, Volume 7, Issue 9: 16278-16295. doi: 10.3934/math.2022889

Previous Article Next Article

Research article

On existence theorems for coupled systems of quadratic Hammerstein-Urysohn integral equations in Orlicz spaces

Ateq Alsaadi ¹,
Mohamed M. A. Metwali ^{2
,
,}

1.
Department of Mathematics and Statistics, College of Science, Taif University, Saudi Arabia
2.
Department of Mathematics, Faculty of Science, Damanhour University, Damanhour, Egypt

Received: 22 March 2022 Revised: 24 May 2022 Accepted: 29 May 2022 Published: 04 July 2022
MSC : 45G10, 47H30, 47N20

We present two existence theorems for a general system of functional quadratic Hammerstein-Urysohn integral equations in arbitrary Orlicz spaces $ L_\varphi $, namely when the generating $ N $-functions fulfill $ \Delta' $ and $ \Delta_3 $-conditions. The studied system contains many integral equations as special cases such as the Chandrasekhar equations, which have significant applications in technology and different disciplines of science. Our analysis is concerned with the fixed point approach and a measure of noncompactness.
- measure of noncompactness,
- Orlicz spaces,
- coupled systems of integral equations,
- compact in measure,
- $ \Delta'\; \Delta_3 $-conditions
Citation: Ateq Alsaadi, Mohamed M. A. Metwali. On existence theorems for coupled systems of quadratic Hammerstein-Urysohn integral equations in Orlicz spaces[J]. AIMS Mathematics, 2022, 7(9): 16278-16295. doi: 10.3934/math.2022889

Related Papers:

Abstract

We present two existence theorems for a general system of functional quadratic Hammerstein-Urysohn integral equations in arbitrary Orlicz spaces $ L_\varphi $, namely when the generating $ N $-functions fulfill $ \Delta' $ and $ \Delta_3 $-conditions. The studied system contains many integral equations as special cases such as the Chandrasekhar equations, which have significant applications in technology and different disciplines of science. Our analysis is concerned with the fixed point approach and a measure of noncompactness.

References

[1]	M. A. Polo-Labarrios, S. Q. Garcia, G. E. Paredes, L. F. Perez, J. O. Villafuerta, Novel numerical solution to the fractional neutron point kinetic equation in nuclear reactor dynamics, Ann. Nucl. Energy, 137 (2020), 10717. https://doi.org/10.1016/j.anucene.2019.107173 doi: 10.1016/j.anucene.2019.107173
[2]	H. Chen, J. I. Frankel, M. Keyhani, Two-probe calibration integral equation method for nonlinear inverse heat conduction problem of surface heat flux estimation, Int. J. Heat Mass Tran., 121 (2018), 246–264. https://doi.org/10.1016/j.ijheatmasstransfer.2017.12.072 doi: 10.1016/j.ijheatmasstransfer.2017.12.072
[3]	T. E. Roth, W. C. Chew, Stability analysis and discretization of A-$\phi$ time domain integral equations for multiscale electromagnetic, J. Comput. Phys., 408 (2020), 109102. https://doi.org/10.1016/j.jcp.2019.109102 doi: 10.1016/j.jcp.2019.109102
[4]	V. Gafiychuk, B. Datsko, V. Meleshko, D. Blackmore, Analysis of the solutions of coupled nonlinear fractional reaction diffusion equations, Chaos Soliton. Fract., 41 (2009), 1095–1104. https://doi.org/10.1016/j.chaos.2008.04.039 doi: 10.1016/j.chaos.2008.04.039
[5]	E. Cuesta, M. Kirance, S. A. Malik, Image structure preserving denoising using generalized fractional time integrals, Signal Process., 92 (2012), 553–563.
[6]	R. Arab, Application of measure of noncompactness for the system of functional integral equations, Filomat, 30 (2016), 3063–3073. https://doi.org/10.2298/FIL1611063A doi: 10.2298/FIL1611063A
[7]	A. Deep, Deepmala, J. Roshan, T. Abdeljawad, An extension of Darbo's fixed point theorem for a class of system of nonlinear integral equations, Adv. Differ. Equ., 2020 (2020), 483. https://doi.org/10.1186/s13662-020-02936-y doi: 10.1186/s13662-020-02936-y
[8]	M.I. Youssef, On the solvability of a general class of a coupled system of stochastic functional integral equations, Arab J. Basic Appl. Sci., 27 (2020), 142–148. https://doi.org/10.1080/25765299.2020.1744071 doi: 10.1080/25765299.2020.1744071
[9]	X. W. Su, Boundary value problem for a coupled system of nonlinear fractional differential equations, Appl. Math. Lett., 22 (2009), 64–69. https://doi.org/10.1016/j.aml.2008.03.001 doi: 10.1016/j.aml.2008.03.001
[10]	B. D. Karande, S. N. Kondekar, Existence the solution of coupled system of quadratic hybrid functional integral equation in Banach algebras, J. Mech. Continua Math. Sci., 15 (2020), 243–255. https://doi.org/10.26782/jmcms.2020.09.00020 doi: 10.26782/jmcms.2020.09.00020
[11]	S. Baghdad, Existence and stability of solutions for a system of quadratic integral equations in Banach algebras, Ann. Univ. Paedagog. Crac. Stud. Math., 19 (2020), 203–218. https://doi.org/10.2478/aupcsm-2020-0015 doi: 10.2478/aupcsm-2020-0015
[12]	H.A. Hammad, H. Aydi, C. Park, Fixed point approach for solving a system of Volterra integral equations and Lebesgue integral concept in $F_CM$-spaces, AIMS Mathematics, 7 (2022), 9003–9022. https://doi.org/10.3934/math.2022501 doi: 10.3934/math.2022501
[13]	A. El-Sayed, S. Abd El-Salam, Coupled system of a fractional order differential equations with weighted initial conditions, Open Math., 17 (2019), 1737–1749. https://doi.org/10.1515/math-2019-0120 doi: 10.1515/math-2019-0120
[14]	M. Cichoń, M. Metwali, On a fixed point theorem for the product of operators, J. Fixed Point Theory Appl., 18 (2016), 753–770. https://doi.org/10.1007/s11784-016-0319-7 doi: 10.1007/s11784-016-0319-7
[15]	J. Berger, J. Robert, Strongly nonlinear equations of Hammerstein type, J. Lond. Math. Soc., 15 (1977), 277–287. https://doi.org/10.1112/jlms/s2-15.2.277 doi: 10.1112/jlms/s2-15.2.277
[16]	M. A. Krasnosel'skii, Y. B. Rutickii, Convex functions and Orlicz spaces, Gröningen: Noordhoff, 1961.
[17]	I. Y. S. Cheng, J. J. Kozak, Application of the theory of Orlicz spaces to statistical mechanics. I. Integral equations, J. Math. Phys., 13 (1972), 51–58. https://doi.org/10.1063/1.1665850 doi: 10.1063/1.1665850
[18]	R. P. Agarwal, D. O'Regan, P. J. Y. Wong, Constant-sign solutions of a system of Volterra integral equations in Orlicz spaces, J. Integr. Equations Appl., 20 (2008), 337–378.
[19]	D. O'Regan, Solutions in Orlicz spaces to Urysohn integral equations, Proc. R. Ir. Acad., Sect. A, 96 (1996), 67–78.
[20]	A. Sołtysiak, S. Szufla, Existence theorems for $L_\varphi$-solutions of the Hammerstein integral equation in Banach spaces, Commentat. Math., 30 (1990), 177–190.
[21]	C. Bardaro, J. Musielak, G. Vinti, Nonlinear integral operators and applications, Berlin: Walter de Gruyter, 2003.
[22]	R. Płuciennik, S. Szufla, Nonlinear Volterra integral equations in Orlicz spaces, Demonstr. Math., 17 (1984), 515–532. https://doi.org/10.1515/dema-1984-0221 doi: 10.1515/dema-1984-0221
[23]	M. Cichoń, M. Metwali, On quadratic integral equations in Orlicz spaces, J. Math. Anal. Appl., 387 (2012), 419–432. https://doi.org/10.1016/j.jmaa.2011.09.013 doi: 10.1016/j.jmaa.2011.09.013
[24]	M. Cichoń, M. Metwali, On solutions of quadratic integral equations in Orlicz spaces, Mediterr. J. Math., 12 (2015), 901–920. https://doi.org/10.1007/s00009-014-0450-x doi: 10.1007/s00009-014-0450-x
[25]	M. Metwali, Nonlinear quadratic Volterra-Urysohn functional-integral equations in Orlicz spaces, Filomat, 35 (2021), 2963–2972. https://doi.org/10.2298/FIL2109963M doi: 10.2298/FIL2109963M
[26]	M. Metwali, On perturbed quadratic integral equations and initial value problem with nonlocal conditions in Orlicz spaces, Demonstr. Math., 53 (2020), 86–94. https://doi.org/10.1515/dema-2020-0052 doi: 10.1515/dema-2020-0052
[27]	A. Deep, Deepmala, R. Ezzati, Application of Petryshyn's fixed point theorem to solvability for functional integral equations, Appl. Math. Comput., 395 (2021), 125878. https://doi.org/10.1016/j.amc.2020.125878 doi: 10.1016/j.amc.2020.125878
[28]	K. Kavitha, V. Vijayakumar, R. Udhayakumar, C. Ravichandran, Results on controllability of Hilfer fractional differential equations with infinite delay via measures of noncompactness, Asian J. Control., 24 (2022), 1406–1415. https://doi.org/10.1002/asjc.2549 doi: 10.1002/asjc.2549
[29]	S. Singh, S. Kumar, M. Metwali, S. Aldosary, K. Nisar, An existence theorem for nonlinear functional Volterra integral equations via Petryshyn's fixed point theorem, AIMS Mathematics, 7 (2022), 5594–5604. https://doi.org/10.3934/math.2022309 doi: 10.3934/math.2022309
[30]	H. Hashem, A. El-Sayed, Stabilization of coupled systems of quadratic integral equations of Chandrasekhar type, Math. Nachr., 290 (2017), 341–348. https://doi.org/10.1002/mana.201400348 doi: 10.1002/mana.201400348
[31]	A.Fahem, A. Jeribi, N. Kaddachi, Existence of solutions for a system of Chandrasekhar's equations in Banach algebras under weak topology, Filomat, 33 (2019), 5949–5957. https://doi.org/10.2298/FIL1918949F doi: 10.2298/FIL1918949F
[32]	T. Nabil, Existence results for nonlinear coupled system of integral equations of Urysohn Volterra-Chandrasekhar mixed type, Demonstr. Math., 53 (2020), 236–248. https://doi.org/10.1515/dema-2020-0017 doi: 10.1515/dema-2020-0017
[33]	A. Jeribi, N. Naddachi, B. Krichen, Fixed-point theorems for multivalued operator matrix under weak topology with an applications, Bull. Malays. Math. Sci. Soc., 43 (2020), 1047–1067. https://doi.org/10.1007/s40840-019-00724-w doi: 10.1007/s40840-019-00724-w
[34]	F. Haq, K. Shah, G. UR-Rahmanc, M. Shahzada, Existence results for a coupled systems of Chandrasekhar quadratic integral equations, Commun. Nonlinear Anal., 3 (2017), 15–22.
[35]	A. Bellour, D. O'Regan, M. A. Taoudi, On the existence of integrable solutions for a nonlinear quadratic integral equation, J. Appl. Math. Comput., 46 (2014), 67–77. https://doi.org/10.1007/s12190-013-0737-2 doi: 10.1007/s12190-013-0737-2
[36]	J. Caballero, A.B. Mingarelli, K. Sadarangani, Existence of solutions of an integral equation of Chandrasekhar type in the theory of radiative transfer, Electron. J. Differ. Eq., 2006 (2006), 57. http://eudml.org/doc/127518
[37]	S. Chandrasekhar, Radiative transfer, New York: Dover Publications, 1960.
[38]	L. Maligranda, Orlicz spaces and interpolation, Seminars in mathematics, 1989.
[39]	M. Väth, Volterra and integral equations of vector functions, New York: Marcel Dekker, 2000.
[40]	J. Banaś, On the superposition operator and integrable solutions of some functional equations, Nonlinear Anal. Theor., 12 (1988), 777–784. https://doi.org/10.1016/0362-546X(88)90038-7 doi: 10.1016/0362-546X(88)90038-7
[41]	J. Banaś, K. Goebel, Measures of noncompactness in Banach spaces, New York: Marcel Dekker, 1980.
[42]	N. Erzakova, Compactness in measure and measure of noncompactness, Sib. Math. J., 38 (1997), 926–928. https://doi.org/10.1007/BF02673034 doi: 10.1007/BF02673034

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)