Solvability of Quadratic Integral Equations of Urysohn Type Involving Hadamard Variable-Order Operator

Year 2024, Volume: 7 Issue: 2, 108 - 117, 30.06.2024

Zoubida Bouazza , Mohammed Said Souıd , Amar Benkerrouche , Ali Yakar

https://doi.org/10.33401/fujma.1405875

Cited By: 1

Abstract

This study investigates the existence of solutions to integral equations in the form of quadratic Urysohn type with Hadamard fractional variable order integral operator. Due to the lack of semigroup properties in variable-order fractional integrals, it becomes challenging to get the existence and uniqueness of corresponding integral equations, hence the problem is examined by employing the concepts of piecewise constant functions and generalized intervals to address this issue. In this context, the problem is reformulated as integral equations with constant orders to obtain the main results. Both the Schauder and Banach fixed point theorems are employed to prove the uniqueness results. In addition, an illustration is included in order to verify those results in the final step.

Keywords

Fixed-point theorem, Fractional variable order, Volterra integral equations

References

• [1] A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, Elsevier Science Tech, (2006). $ \href{https://doi.org/10.1016/S0304-0208(06)80002-2}{\mbox{[CrossRef]}}$
• [2] V. Lakshmikantham, S. Leela and J. V. Devi, Theory of Fractional Dynamic Systems, Cambridge Scientfic Publishers, Cambridge, (2009). $\href{https://cambridgescientificpublishers.com/product/theory-of-fractional-dynamic-systems}{\mbox{[Web]}} $
• [3] R.K. Saxena and S.L. Kalla, On a fractional generalization of the free electron laser equation, Appl. Math. Comput., 143(1) (2003), 89-97. $ \href{https://doi.org/10.1016/S0096-3003(02)00348-X}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-0037809667&origin=resultslist&sort=plf-f&src=s&sid=553cbe7692df2307735295a6a031ea28&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+a+fractional+generalization+of+the+free+electron+laser+equation%22%29&sl=77&sessionSearchId=553cbe7692df2307735295a6a031ea28&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000183349300006}{\mbox{[Web of Science]}}$
• [4] A. Tepljakov, Fractional-order Modeling and Control of Dynamic Systems, Springer, (2017). $\href{https://doi.org/10.1007/978-3-319-52950-9}{\mbox{[CrossRef]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000416339100010}{\mbox{[Web of Science]}} $
• [5] S. Chakraverty, R.M. Jena and S.K. Jena, Time-Fractional Order Biological Systems with Uncertain Parameters, Synthesis Lectures on Mathematics & Statistics, Springer International Publishing, (2022). $\href{https://doi.org/10.1007/978-3-031-02423-8}{\mbox{[CrossRef]}} $
• [6] H. Singh, D. Kumar, D. Baleanu, Methods of Mathematical Modelling: Fractional Differential Equations, Mathematics and its Applications, CRC Press, (2019). $ \href{https://doi.org/10.1201/9780429274114}{\mbox{[CrossRef]}}$
• [7] J. Singh, J.Y. Hristov, Z. Hammouch, New Trends in Fractional Differential Equations with Real-World Applications in Physics, Frontiers Research Topics, Frontiers Media SA, 2020. $ \href{https://www.frontiersin.org/research-topics/12142/new-trends-in-fractional-differential-equations-with-real-world-applications-in-physics}{\mbox{[Web]}}$
• [8] V.E. Tarasov, Mathematical economics: application of fractional calculus, Mathematics, 8(5) 660 (2020), 1-3. $\href{https://doi.org/10.3390/math8050660}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85085550694&origin=resultslist&sort=plf-f&src=s&sid=553cbe7692df2307735295a6a031ea28&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Mathematical+economics%3A+application+of+fractional+calculus%22%29&sl=77&sessionSearchId=553cbe7692df2307735295a6a031ea28&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000542738100164}{\mbox{[Web of Science]}} $
• [9] M. Yağmurlu, Y. Uçar and A. Esen, Collocation method for the KdV-Burgers-Kuramoto equation with Caputo fractional derivative, Fundam. Contemp. Math. Sci., 1(1) (2020), 1-13. $\href{https://dergipark.org.tr/en/pub/fcmathsci/issue/52208/663140}{\mbox{[Web]}} $
• [10] S.G. Samko and B. Ross, Integration and differentiation to a variable fractional order, Integral Transforms Spec. Funct., 1(4) (1993), 277-300. $\href{https://doi.org/10.1080/10652469308819027}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84948882036&origin=resultslist&sort=plf-f&src=s&sid=553cbe7692df2307735295a6a031ea28&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Integration+and+differentiation+to+a+variable+fractional+order%22%29&sl=77&sessionSearchId=553cbe7692df2307735295a6a031ea28&relpos=0}{\mbox{[Scopus]}} $
• [11] R. Almeida, D. Tavares and D.F.M. Torres, The Variable-Order Fractional Calculus of Variations, Springer, (2018). $\href{https://doi.org/10.1007/978-3-319-94006-9}{\mbox{[CrossRef]}} $
• [12] H.G. Sun, A. Chang, Y. Zhang and W. Chen, A review on variable-order fractional differential equations: Mathematical foundations, physical models, numerical methods and applications, Fract. Calc. Appl. Anal., 22(1) (2019), 27-59. $ \href{http://dx.doi.org/10.1515/fca-2019-0003}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85063769565&origin=resultslist&sort=plf-f&src=s&sid=553cbe7692df2307735295a6a031ea28&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22A+review+on+variable-order+fractional+differential+equations%3A+Mathematical+foundations%2C+physical+models%2C+numerical+methods+and+applications%22%29&sl=77&sessionSearchId=553cbe7692df2307735295a6a031ea28&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000462023200003}{\mbox{[Web of Science]}}$
• [13] D. Sierociuk,W. Malesza and M. Macias, Derivation, interpretation, and analog modelling of fractional variable order derivative definition, Appl. Math. Model., 39(13) (2015), 3876-3888. $ \href{https://doi.org/10.1016/j.apm.2014.12.009}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84930042560&origin=resultslist&sort=plf-f&src=s&sid=553cbe7692df2307735295a6a031ea28&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Derivation%2C+interpretation%2C+and+analog+modelling+of+fractional+variable+order+derivative+definition%22%29&sl=77&sessionSearchId=553cbe7692df2307735295a6a031ea28&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000356544600021}{\mbox{[Web of Science]}} $
• [14] C.J. Zuniga-Aguilar, H.M. Romero-Ugalde, J.F. Gomez-Aguilar, R.F. Escobar-Jimenez and M. Valtierra-Rodrıguez, Solving fractional differential equations of variable-order involving operators with Mittag-Leffler kernel using artificial neural networks, Chaos Solitons Fract., 103 (2017), 382-403. $\href{https://doi.org/10.1016/j.chaos.2017.06.030}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85021751772&origin=resultslist&sort=plf-f&src=s&sid=553cbe7692df2307735295a6a031ea28&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Solving+fractional+differential+equations+of+variable-order+involving+operators+with+Mittag-Leffler+kernel+using+artificial+neural+networks%22%29&sl=77&sessionSearchId=553cbe7692df2307735295a6a031ea28&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000410680700042}{\mbox{[Web of Science]}} $
• [15] J.F. Gomez-Aguilar, Analytical and numerical solutions of a nonlinear alcoholism model via variable-order fractional differential equations, Physica A Stat. Mech. Appl., 494 (2018), 52-75. $ \href{https://doi.org/10.1016/j.physa.2017.12.007}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85037976084&origin=resultslist&sort=plf-f&src=s&sid=553cbe7692df2307735295a6a031ea28&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Analytical+and+numerical+solutions+of+a+nonlinear+alcoholism+model+via+variable-order+fractional+differential+equations%22%29&sl=77&sessionSearchId=553cbe7692df2307735295a6a031ea28&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000424176800006}{\mbox{[Web of Science]}} $
• [16] J. Yang, H. Yao, and B. Wu, An efficient numerical method for variable order fractional functional differential equation, Appl. Math. Lett., 76 (2018), 221-226. $\href{https://doi.org/10.1016/j.aml.2017.08.020}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85037976084&origin=resultslist&sort=plf-f&src=s&sid=553cbe7692df2307735295a6a031ea28&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Analytical+and+numerical+solutions+of+a+nonlinear+alcoholism+model+via+variable-order+fractional+differential+equations%22%29&sl=77&sessionSearchId=553cbe7692df2307735295a6a031ea28&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000414889200034}{\mbox{[Web of Science]}} $
• [17] D. Valerio and J.S. Da Costa, Variable-order fractional derivatives and their numerical approximations, Signal Process., 91(3) (2011), 470-483. $\href{https://doi.org/10.1016/j.sigpro.2010.04.006}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-78049352592&origin=resultslist&sort=plf-f&src=s&sid=553cbe7692df2307735295a6a031ea28&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Variable-order+fractional+derivatives+and+their+numerical+approximations%22%29&sl=77&sessionSearchId=553cbe7692df2307735295a6a031ea28&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000288038800011}{\mbox{[Web of Science]}} $
• [18] K. Benia, M.S. Souid, F. Jarad, M.A. Alqudah and T. Abdeljawad, Boundary value problem of weighted fractional derivative of a function with a respect to another function of variable order, J. Inequal. Appl., 2023(1) (2023), 127. $\href{http://dx.doi.org/10.1186/s13660-023-03042-9}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85173710659&origin=resultslist&sort=plf-f&src=s&sid=553cbe7692df2307735295a6a031ea28&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Boundary+value+problem+of+weighted+fractional+derivative+of+a+function+with+a+respect+to+another+function+of+variable+order%22%29&sl=77&sessionSearchId=553cbe7692df2307735295a6a031ea28&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001078085800001}{\mbox{[Web of Science]}} $
• [19] D. O’Regan, S. Hristova and R.P. Agarwal, Ulam-type stability results for variable order Y-tempered Caputo fractional differential equations, Fractal Fract., 8(1) 11 (2024), 1-15. $\href{https://doi.org/10.3390/fractalfract8010011}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85183202419&origin=resultslist&sort=plf-f&src=s&sid=553cbe7692df2307735295a6a031ea28&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Ulam-type+stability+results+for+variable+order+%24%5CPsi%24-tempered+Caputo+fractional+differential+equations%22%29&sl=77&sessionSearchId=553cbe7692df2307735295a6a031ea28&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:001151879400001}{\mbox{[Web of Science]}} $
• [20] B. Telli, M.S. Souid and I. Stamova, Boundary-value problem for nonlinear fractional differential equations of variable order with finite delay via Kuratowski measure of noncompactness, Axioms, 12(1) 80 (2023), 1-23. $\href{https://doi.org/10.3390/axioms12010080}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85146797555&origin=resultslist&sort=plf-f&src=s&sid=553cbe7692df2307735295a6a031ea28&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Boundary-value+problem+for+nonlinear+fractional+differential+%09equations+of+variable+order+with+finite+delay+via+Kuratowski+measure+of+%09noncompactness%22%29&sl=77&sessionSearchId=553cbe7692df2307735295a6a031ea28&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000914427000001}{\mbox{[Web of Science]}} $
• [21] A. Benkerrouche, S. Etemad, M.S. Souid, S. Rezapour, H. Ahmad and T. Botmart, Fractional variable order differential equations with impulses: A study on the stability and existence properties, AIMS Math., 8(1) (2022), 775-791. $\href{https://doi.org/10.3934/math.2023038}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85139511805&origin=resultslist&sort=plf-f&src=s&sid=553cbe7692df2307735295a6a031ea28&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Fractional+variable+order+differential+equations+with+impulses%3A+A+study+on+the+stability+and+existence+properties%22%29&sl=77&sessionSearchId=553cbe7692df2307735295a6a031ea28&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000875234100001}{\mbox{[Web of Science]}} $
• [22] S. Zhang, S. Li and L. Hu, The existeness and uniqueness result of solutions to initial value problems of nonlinear diffusion equations involving with the conformable variable derivative, Rev. Real Acad. Cienc. Exactas Fis. Nat. - A: Mat., 113(2) (2019), 1601-1623. $\href{http://dx.doi.org/10.1007/s13398-018-0572-2}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85064907608&origin=resultslist&sort=plf-f&src=s&sid=553cbe7692df2307735295a6a031ea28&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22The+existeness+and+uniqueness+result+of+solutions+to+initial+value+problems+of+nonlinear+diffusion+equations+involving+with+the+conformable+variable+derivative%22%29&sl=77&sessionSearchId=553cbe7692df2307735295a6a031ea28&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000467148800079}{\mbox{[Web of Science]}} $
• [23] A. Benkerrouche, M.S. Souid, S. Etemad, A. Hakem, P. Agarwal, S. Rezapour, S.K. Ntouyas and J. Tariboon, Qualitative study on solutions of a Hadamard variable order boundary problem via the Ulam–Hyers–Rassias stability, Fractal Fract., 5(3) 108 (2021), 1-20. $\href{https://doi.org/10.3390/fractalfract5030108}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85114765366&origin=resultslist&sort=plf-f&src=s&sid=553cbe7692df2307735295a6a031ea28&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Qualitative+study+on+solutions+of+a+Hadamard+variable+order+boundary+problem+via+the+Ulam%E2%80%93Hyers%E2%80%93Rassias+stability%22%29&sl=77&sessionSearchId=553cbe7692df2307735295a6a031ea28&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000699638400001}{\mbox{[Web of Science]}} $
• [24] M.S. Souid, Z. Bouazza and A. Yakar, Existence, uniqueness, and stability of solutions to variable fractional order boundary value problems, J. New Theo., 41 (2022), 82-93. $\href{https://doi.org/10.53570/jnt.1182795}{\mbox{[CrossRef]}} $
• [25] S. Zhang and L. Hu, The existence of solutions and generalized Lyapunov-type inequalities to boundary value problems of differential equations of variable order, AIMS Math., 5(4) (2020), 2923-2943. $ \href{https://doi.org/10.3934/math.2020189}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85082711880&origin=resultslist&sort=plf-f&src=s&sid=553cbe7692df2307735295a6a031ea28&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22The+existence+of+solutions+and+generalized+Lyapunov-type+inequalities+to+boundary+value+problems+of+differential+equations+of+variable+order%22%29&sl=77&sessionSearchId=553cbe7692df2307735295a6a031ea28&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000532484000009}{\mbox{[Web of Science]}} $
• [26] A. Refice, M.S. Souid and A. Yakar, Some qualitative properties of nonlinear fractional integro-differential equations of variable order, Int. J. Optim. Control: Theor. Appl., 11(3) (2021), 68-78. $ \href{https://doi.org/10.11121/ijocta.2021.1198}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85128641586&origin=resultslist&sort=plf-f&src=s&sid=553cbe7692df2307735295a6a031ea28&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Some+qualitative+properties+of+nonlinear+fractional+integro-differential+equations+of+variable+order%22%29&sl=77&sessionSearchId=553cbe7692df2307735295a6a031ea28&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000873914800005}{\mbox{[Web of Science]}}$
• [27] S. Hristova, A. Benkerrouche, M.S. Souid and A. Hakem, Boundary value problems of Hadamard fractional differential equations of variable order, Symmetry 13(5) 896 (2021), 1-16. $\href{https://doi.org/10.3390/sym13050896}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85106977558&origin=resultslist&sort=plf-f&src=s&sid=553cbe7692df2307735295a6a031ea28&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Boundary+value+problems+of+Hadamard+fractional+differential+equations+of+variable+order%22%29&sl=77&sessionSearchId=553cbe7692df2307735295a6a031ea28&relpos=3}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000654623500001}{\mbox{[Web of Science]}} $
• [28] A. Refice, M.S. Souid and I. Stamova, On the boundary value problems of Hadamard fractional differential equations of variable order via Kuratowski MNC technique, Mathematics, 9(10) 1134 (2021), 1-16. $ \href{https://doi.org/10.3390/math9101134}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85106948665&origin=resultslist&sort=plf-f&src=s&sid=553cbe7692df2307735295a6a031ea28&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+the+boundary+value+problems+of+Hadamard+fractional+differential+equations+of+variable+order+via+Kuratowski+MNC+technique%22%29&sl=77&sessionSearchId=553cbe7692df2307735295a6a031ea28&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000655042400001}{\mbox{[Web of Science]}} $
• [29] S. Zhang, S. Sun and L. Hu, Approximate solutions to initial value problem for differential equation of variable order, J. Fract. Calc. Appl, 9(2) (2018), 93-112. $ \href{https://doi.org/10.21608/jfca.2018.308469}{\mbox{[CrossRef]}}$
• [30] A. Benkerrouche, D. Baleanu, M.S. Souid, A. Hakem and M. Inc, Boundary value problem for nonlinear fractional differential equations of variable order via Kuratowski MNC technique, Adv. Differ. Equ., 2021(1) 365 (2021). $\href{https://doi.org/10.1186/s13662-021-03520-8}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85111983571&origin=resultslist&sort=plf-f&src=s&sid=553cbe7692df2307735295a6a031ea28&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Boundary+value+problem+for+nonlinear+fractional+differential+equations+of+variable+order+via+Kuratowski+MNC+technique%22%29&sl=77&sessionSearchId=553cbe7692df2307735295a6a031ea28&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000683734000004}{\mbox{[Web of Science]}} $
• [31] T.A. Burton, Volterra Integral and Differential Equations, Elsevier Science, (2005). $\href{https://doi.org/10.1007/978-3-642-21449-3_5}{\mbox{[CrossRef]}} $
• [32] C. Corduneanu, Integral Equations and Applications, Cambridge University Press, (2008). $ \href{https://doi.org/10.1017/CBO9780511569395}{\mbox{[CrossRef]}}$
• [33] S. Abbas, M. Benchohra, J.R. Graef and J. Henderson, Implicit Fractional Differential and Integral Equations: Existence and Stability, De Gruyter Series in Nonlinear Analysis and Applications. De Gruyter, (2018). $ \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000480763700013}{\mbox{[Web of Science]}}$
• [34] E.H. Doha, M.A. Abdelkawy, A.Z.M. Amin and A.M. Lopes, On spectral methods for solving variable-order fractional integro-differential equations, Comput. Appl. Math., 37(3) (2018), 3937–3950. $\href{https://doi.org/10.1007/s40314-017-0551-9}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85044324919&origin=resultslist&sort=plf-f&src=s&sid=553cbe7692df2307735295a6a031ea28&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+spectral+methods+for+solving+variable-order+fractional+integro-differential+equations%22%29&sl=77&sessionSearchId=553cbe7692df2307735295a6a031ea28&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000438312300078}{\mbox{[Web of Science]}} $
• [35] R.M. Ganji, H. Jafari and S. Nemati, A new approach for solving integro-differential equations of variable order, J. Comput. Appl. Math., 379 (2020) 112946. $ \href{https://doi.org/10.1016/j.cam.2020.112946}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85083899738&origin=resultslist&sort=plf-f&src=s&sid=553cbe7692df2307735295a6a031ea28&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22A+new+approach+for+solving+integro-differential+equations+of+variable+order%22%29&sl=77&sessionSearchId=553cbe7692df2307735295a6a031ea28&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000536751200003}{\mbox{[Web of Science]}} $
• [36] A. Benkerrouche, M.S. Souid, G. Stamov and I. Stamova, On the solutions of a quadratic integral equation of the Urysohn type of fractional variable order, Entropy, 24(7) 886 (2022), 1-14. $\href{https://doi.org/10.3390/e24070886}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85133330171&origin=resultslist&sort=plf-f&src=s&sid=553cbe7692df2307735295a6a031ea28&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22On+the+solutions+of+a+quadratic+integral+equation+of+the+Urysohn+type+of+fractional+variable+order%22%29&sl=77&sessionSearchId=553cbe7692df2307735295a6a031ea28&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000832286200001}{\mbox{[Web of Science]}} $
• [37] M.H. Heydari, Chebyshev cardinal wavelets for nonlinear variable-order fractional quadratic integral equations, Appl. Numer. Math., 144 (2019), 190-203. $\href{https://doi.org/10.1016/j.apnum.2019.04.019}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85065171544&origin=resultslist&sort=plf-f&src=s&sid=553cbe7692df2307735295a6a031ea28&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Chebyshev+cardinal+wavelets+for+nonlinear+variable-order+fractional+quadratic+integral+equations%22%29&sl=77&sessionSearchId=553cbe7692df2307735295a6a031ea28&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000474328000011}{\mbox{[Web of Science]}} $
• [38] M.H. Heydari, A computational method for a class of systems of nonlinear variable-order fractional quadratic integral equations, Appl. Numer. Math., 153 (2020), 164-178. $\href{https://doi.org/10.1016/j.apnum.2020.02.011}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85079614589&origin=resultslist&sort=plf-f&src=s&sid=553cbe7692df2307735295a6a031ea28&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22A+computational+method+for+a+class+of+systems+of+nonlinear+variable-order+fractional+quadratic+integral+equations%22%29&sl=77&sessionSearchId=553cbe7692df2307735295a6a031ea28&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000527660200011}{\mbox{[Web of Science]}} $
• [39] A. Wazwaz, First Course In Integral Equations, (A Second Edition), World Scientific Publishing Company, (2015). $\href{http://dx.doi.org/10.1142/9570}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84967622869&origin=resultslist&sort=plf-f&src=s&sid=553cbe7692df2307735295a6a031ea28&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22First+Course+In+Integral+Equations%22%29&sl=77&sessionSearchId=553cbe7692df2307735295a6a031ea28&relpos=0}{\mbox{[Scopus]}} $
• [40] S. Samko, Fractional integration and differentiation of variable order: an overview, Nonlinear Dyn., 71(4) (2013) 653-662. $\href{https://doi.org/10.1007/s11071-012-0485-0}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84881547849&origin=resultslist&sort=plf-f&src=s&sid=553cbe7692df2307735295a6a031ea28&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Fractional+integration+and+differentiation+of+variable+order%3A+an+overview%22%29&sl=77&sessionSearchId=553cbe7692df2307735295a6a031ea28&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000315437600007}{\mbox{[Web of Science]}} $
• [41] S. Zhang, Existence of solutions for two point boundary value problems with singular differential equations of variable order, Electron. J. Differ. Equ., 245 (2013), 1-16. $\href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84887743065&origin=resultslist&sort=plf-f&src=s&sid=553cbe7692df2307735295a6a031ea28&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Existence+of+solutions+for+two+point+boundary+value+problems+with+singular+differential+equations+of+variable+order%22%29&sl=77&sessionSearchId=553cbe7692df2307735295a6a031ea28&relpos=0}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000328075400001}{\mbox{[Web of Science]}} $
• [42] D.R. Smart, Fixed Point Theorems, Cambridge Tracts in Mathematics, Cambridge University Press, (1980). $\href{https://www.cambridge.org/tr/universitypress/subjects/mathematics/abstract-analysis/fixed-point-theorems?format=PB&isbn=9780521298339}{\mbox{[Web]}} $