ON WELL-POSEDNESS OF A GENERAL FRACTIONAL INTEGRAL EQUATION IN ORLICZ SPACES

Hala H. Taha; Mohamed M. A. Metwali; Kottakkaran S. Nisar; Fatemah Mofarreh

doi:10.11948/20250268

2026 Volume 16 Issue 6

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Hala H. Taha, Mohamed M. A. Metwali, Kottakkaran S. Nisar, Fatemah Mofarreh. ON WELL-POSEDNESS OF A GENERAL FRACTIONAL INTEGRAL EQUATION IN ORLICZ SPACES[J]. Journal of Applied Analysis & Computation, 2026, 16(6): 3029-3045. doi: 10.11948/20250268

Citation:

Hala H. Taha, Mohamed M. A. Metwali, Kottakkaran S. Nisar, Fatemah Mofarreh. ON WELL-POSEDNESS OF A GENERAL FRACTIONAL INTEGRAL EQUATION IN ORLICZ SPACES[J]. Journal of Applied Analysis & Computation, 2026, 16(6): 3029-3045. doi: 10.11948/20250268

ON WELL-POSEDNESS OF A GENERAL FRACTIONAL INTEGRAL EQUATION IN ORLICZ SPACES

1.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia
2.
Department of Mathematics, College of Science and Humanities in AlKharj, Prince Sattam Bin Abdulaziz University, AlKharj 11942, Saudi Arabia
3.
Department of Mathematics and Computer Science, Faculty of Science, Damanhour University, 22511 Damanhour, Egypt
4.
Hourani Center for Applied Scientific Research, Al-Ahliyya Amman University, Amman, Jordan

Author Bio: Email: hhtaha@pnu.edu.sa(H. H. Taha); Email: n.sooppy@psau.edu.sa(K. S. Nisar); Email: fyalmofarrah@pnu.edu.sa(F. Mofarreh)
Corresponding author: Email: metwali@sci.dmu.edu.eg; m.metwali@spau.edu.sa(M. M. A. Metwali)

Abstract

Abstract

This article has two objectives. The first objective is to utilize Darbo's fixed-point theorem ($ \mathbf{\mathcal{FPT}} $) and the measure of noncompactness ($ \mathbf{\mathcal{MNC}} $) technique to study the well-posedness (i.e., the existence and the uniqueness in addition to continuous dependence on the data) of a generalized fractional quadratic integral equation in Orlicz spaces $ L_\psi $. Our second objective is to prove and explain the novel properties of $ g $-fractional operators, such as boundedness, continuity, acting, and monotonicity within $ L_\psi $. As a result of our work, several fractional operators are generalized, unified, and extended, including Riemann-Liouville, Hadamard, and Erdélyi-Kober, as well as related classical and quadratic fractional problems. Our theories are supported by several examples.
- Orlicz spaces L_ψ /
- ($\mathbf{\mathcal{MNC}}$) measure of noncompactness /
- g-fractional operator /
- ($\mathbf{\mathcal{FPT}}$) fixed point theorem
MSC: 47N20, 45G10, 47H30, 47H10

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References

[1]	R. P. Agarwal, D. O'Regan and P. Wong, Solutions of a system of integral equations in Orlicz spaces, J. Integral Equations Appl., 2009, 21, 469–498. Google Scholar
[2]	B. Ahmad and J. J. Nieto, Existence of solutions for anti-periodic boundary value problems involving fractional differential equations via Leray-Schauder degree theory, Topol. Methods Nonlinear Anal., 2010, 35, 295–304. Google Scholar
[3]	S. F. Aldosary and M. Metwali, Solvability of product of $n$-quadratic Hadamard-type fractional integral equations in Orlicz spaces, AIMS Math., 2024, 9(5), 11039–11050. DOI: 10.3934/math.2024541. CrossRef $n$-quadratic Hadamard-type fractional integral equations in Orlicz spaces" target="_blank">Google Scholar
[4]	A. M. Alotaibi, M. Metwali, H. Taha and R. P. Agarwal, Existence, uniqueness, continuous dependence on the data for the product of $n$-fractional integral equations in Orlicz spaces, AIMS Math., 2025, 10(4), 8382–8397. doi: 10.3934/math.2025386 CrossRef $n$-fractional integral equations in Orlicz spaces" target="_blank">Google Scholar
[5]	M. A. Abdel-Aty and M. A. Abdou, Analytical and numerical discussion for the quadratic integral equations, Filomat, 2023, 37, 8095–8111. doi: 10.2298/FIL2324095A CrossRef Google Scholar
[6]	J. Banaś, On measures of noncompactness in Banach spaces, Comment. Math. Univ. Carolin., 1980, 21, 131–143. Google Scholar
[7]	J. Banaś, A. Chlebowicz and B. Rzepka, Infinite systems of differential and integral equations: Current state and some open problems, Symmetry, 2025, 17(4), 575. DOI: 10.3390/sym17040575. CrossRef Google Scholar
[8]	J. Banaś and K. Goebel, Measures of noncompactness in Banach spaces, Comment. Math. Univ. Carolin., 1980, 21, 131–143. Google Scholar
[9]	M. Benchohra, J. Henderson, S. K. Ntouyas and A. Ouahab, Existence results for fractional order functional differential equations with infinite delay, J. Math. Anal. Appl., 2008, 338, 1340–1350. doi: 10.1016/j.jmaa.2007.06.021 CrossRef Google Scholar
[10]	K. G. Binmore, Mathematical Analysis: A Straightforward Approach, Cambridge University Press, Cambridge, 1977. Google Scholar
[11]	P. Bora and B. Hazarika, On solvability of quadratic Erdélyi-Kober fractional integral equations in Orlicz space, Bol. Soc. Paran. Mat., 2025, 43, 1–10. DOI: 10.5269/bspm.70062. CrossRef Google Scholar
[12]	S. Chandrasekhar, Radiative Transfer, Dover Publications, New York, 1960. Google Scholar
[13]	N. M. Chuong, N. T. Hong and H. D. Hung, Multilinear Hardy-Cesáro operator and commutator on the product of Morrey-Herz spaces, Anal. Math., 2017, 43, 547–565. DOI: 10.1007/s10476-017-0502-0. CrossRef Google Scholar
[14]	M. Cichoń and M. Metwali, On solutions of quadratic integral equations in Orlicz spaces, Mediterr. J. Math., 2015, 12, 901–920. doi: 10.1007/s00009-014-0450-x CrossRef Google Scholar
[15]	M. Cichoń and M. Metwali, Existence of monotonic $L_\phi$-solutions for quadratic Volterra functional integral equations, Electron. J. Qual. Theory Differ. Equ., 2015, 13, 1–16. $L_\phi$-solutions for quadratic Volterra functional integral equations" target="_blank">Google Scholar
[16]	R. M. Corless, G. H. Gonnet, D. E. G. Hare, D. J. Jeffrey and D. E. Knuth, On the Lambert W function, Adv. Comput. Math., 1996, 5, 329–359. doi: 10.1007/BF02124750 CrossRef Google Scholar
[17]	N. Erzakova, Compactness in measure and measure of noncompactness, Sib. Math. J., 1997, 38, 926–928. doi: 10.1007/BF02673034 CrossRef Google Scholar
[18]	H. H. Hashem and M. S. Zaki, Carathéodory theorem for quadratic integral equations of Erdélyi-Kober type, J. Fract. Calc. Appl., 2013, 4, 56–72. Google Scholar
[19]	S. Hu, M. Khavanin and W. Zhuang, Integral equations arising in the kinetic theory of gases, Appl. Anal., 1989, 34, 261–266. doi: 10.1080/00036818908839899 CrossRef Google Scholar
[20]	C. T. Kelly, Approximation of solutions of some quadratic integral equations in transport theory, J. Integral Equ., 1982, 4, 221–237. Google Scholar
[21]	A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, 2006. Google Scholar
[22]	M. A. Krasnosel'skii and Y. Rutitskii, Convex Functions and Orlicz Spaces, Nordhoff, Groningen, 1961. Google Scholar
[23]	V. Lakshmikantham, S. Leela and J. Vasundara Devi, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, Cambridge, 2009. Google Scholar
[24]	W. A. Majewski and L. E. Labuschagne, On applications of Orlicz spaces to statistical physics, Ann. Henri Poincaré, 2014, 15, 1197–1221. DOI: 10.1007/s00023-013-0267-3. CrossRef Google Scholar
[25]	L. Maligranda, Orlicz Spaces and Interpolation, Universidade Estadual de Campinas, Campinas, 1989. Google Scholar
[26]	M. Metwali, On some properties of Riemann-Liouville fractional operator in Orlicz spaces and applications to quadratic integral equations, Filomat, 2022, 36, 6009–6020. doi: 10.2298/FIL2217009M CrossRef Google Scholar
[27]	M. Metwali, On $L_\varphi$-Solutions for $n$-Product of Fractional Integral Operators, in: M. Younis, L. Chen and D. Singh (eds.), Recent Developments in Fixed-Point Theory, Springer, Singapore, 2024. DOI: 10.1007/978-981-99-9546-2_7. Google Scholar
[28]	M. Metwali, On perturbed quadratic integral equations and initial value problem with nonlocal conditions in Orlicz spaces, Demonstr. Math., 2020, 53, 86–94. doi: 10.1515/dema-2020-0052 CrossRef Google Scholar
[29]	M. Metwali, Solvability of quadratic Hadamard-type fractional integral equations in Orlicz spaces, Rocky Mountain J. Math., 2023, 53, 531–540. DOI: 10.1216/rmj.2023.53.531. CrossRef Google Scholar
[30]	M. Metwali and S. A. M. Alsallami, On qualitative properties of $L_\Theta$-solutions for coupled systems of Hadamard-type fractional integral equations in Banach spaces, Eur. J. Pure Appl. Math., 2025, 18(2), 6157. DOI: 10.29020/nybg.ejpam.v18i2.6157. CrossRef $L_\Theta$-solutions for coupled systems of Hadamard-type fractional integral equations in Banach spaces" target="_blank">Google Scholar
[31]	M. Metwali and S. A. M. Alsallami, On Erdélyi–Kober fractional operator and quadratic integral equations in Orlicz spaces, Mathematics, 2023, 11, 3901. DOI: 10.3390/math11183901. CrossRef Google Scholar
[32]	K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, 1993. Google Scholar
[33]	E. Nakai, On generalized fractional integrals, Taiwanese J. Math., 2001, 5, 587–602. Google Scholar
[34]	R. O'Neil, Fractional integration in Orlicz spaces. I, Trans. Amer. Math. Soc., 1965, 115, 300–328. doi: 10.1090/S0002-9947-1965-0194881-0 CrossRef Google Scholar
[35]	H. Salem, M. Cichoń and W. Shammakh, Generalized fractional calculus in Banach spaces and applications to existence results for boundary value problems, Bound. Value Probl., 2023, 57. DOI: 10.1186/s13661-023-01745-y. CrossRef Google Scholar
[36]	S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, Amsterdam, 1993. Google Scholar
[37]	I. N. Sneddon, The use in mathematical analysis of Erdélyi-Kober operators and some of their applications, in: Fractional Calculus and Its Applications, Lecture Notes in Math., Vol. 457, Springer-Verlag, New York, 1975, 37–79. Google Scholar
[38]	H. M. Srivastava and R. K. Saxena, Operators of fractional integration and their applications, Appl. Math. Comput., 2001, 118, 1–52. Google Scholar
[39]	M. Väth, Volterra and Integral Equations of Vector Functions, Marcel Dekker, New York, 2000. Google Scholar
[40]	J. Wang, X. Dong and Y. Zhou, Analysis of nonlinear integral equations with Erdélyi-Kober fractional operator, Commun. Nonlinear Sci. Numer. Simul., 2012, 17, 3129–3139. doi: 10.1016/j.cnsns.2011.12.002 CrossRef Google Scholar
[41]	J. Xin, C. Zhu, J. Wang and F. Chen, Nondecreasing solutions of fractional quadratic integral equations involving Erdélyi-Kober singular kernels, Topol. Methods Nonlinear Anal., 2014, 44, 73–88. DOI: 10.12775/TMNA.2014.036. CrossRef Google Scholar
[42]	S. Zhang, Existence of positive solution for some class of nonlinear fractional differential equations, J. Math. Anal. Appl., 2003, 278, 136–148. doi: 10.1016/S0022-247X(02)00583-8 CrossRef Google Scholar