Existence Results for Fractional Integral Equations in Frechet Spaces

Year 2022, Volume: 5 Issue: 3, 105 - 112, 01.12.2022

Said Baghdad

https://doi.org/10.33187/jmsm.1059716

Abstract

The objective of this paper is to present results on the existence of solutions for a class of fractional integral equations in Fr\'{e}chet spaces of Banach space-valued functions on the unbounded interval. Our main tool is the technique of measures of noncompactness and fixed points theorems.

Keywords

Fixed-point theorems, Frechet spaces, Hadamard--Volterra--Stieltjes integral equations, Measure of noncompactness

References

• [1] J. Banas, K. Geobel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Appl. Math., 60, Marcel Dekker, New York and Basel, 1980.
• [2] M. Mursaleen, Syed M. H. Rizvi, B. Samet, Measures of Noncompactness and their Applications, Advances in Nonlinear Analysis via the Concept of Measure of Noncompactness, 59-125, Springer, Singapore, 2017.
• [3] 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.
• [4] S. Baghdad, M. Benchohra, Global existence and stability results for Hadamard-Volterra-Stieltjes integral equation, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 68(2) (2019), 1387-1400.
• [5] M. Benchohra, M. A. Darwish, On quadratic integral equations of Urysohn type in Fr´echet spaces, Acta Math. Univ. Comenian. (N.S.), 79(1) (2010), 105-110.
• [6] L. Olszowy, Fixed point theorems in the Fr´echet space C(R+) and functional integral equations on an unbounded interval, Appl. Math. Comput. 218(18) (2012), 9066-9074.
• [7] K. D. Bierstedt, J. Bonet, Some aspects of the modern theory of Fr´echet spaces, RACSAM. Rev. R. Acad. Cienc. Exactas F´ıs. Nat. Ser. A Mat., 97(2) (2003), 159-188.
• [8] V. Dietmar, Lectures on Fr´echet Spaces, Bergische Universit¨at Wuppertal Sommersemester, 2000.
• [9] D. Baleanu, A. Jajarmi, H. Mohammadi, S. Rezapour, A new study on the mathematical modelling of human liver with Caputo-Fabrizio fractional derivative, Chaos Solitons Fractals, 134 (2020), 109705.
• [10] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.
• [11] H. Mohammadi, S. Kumar, S. Rezapour; S. Etemad, A theoretical study of the Caputo-Fabrizio fractional modeling for hearing loss due to Mumps virus with optimal control, Chaos Solitons Fractals, 144 (2021), 110668.
• [12] Y.-M. Chu, S. Rashid, F. Jarad, M. A. Noor, H. Kalsoom, More new results on integral inequalities for generalized K -fractional conformable integral operators, Discrete Contin. Dyn. Syst. Ser. S 14(7) (2021), 2119-2135.
• [13] S. Abbas, M. Benchohra, G. M. N’Gu´er´ekata Advanced Fractional Differential and Integral Equations, Nova Science Publishers, New York, 2015.
• [14] D. Baleanu, S. Etemad, S. Rezapour, A hybrid Caputo fractional modeling for thermostat with hybrid boundary value conditions, Bound. Value Probl., 2020(1) (2020), Article number: 64, 16 pages.
• [15] P. O. Mohammed, T. Abdeljawad, F. Jarad, Y. M. Chu, Existence and uniqueness of uncertain fractional backward difference equations of Riemann-Liouville type, Math. Probl. Eng., 2020 (2020), Article ID: 6598682, 8 pages.
• [16] J. Bana´s; T. Zaja¸c, A new approach to the theory of functional integral equations of fractional order, J. Math. Anal. Appl., 375(2) (2011), 375-387.
• [17] S. Abbas, M. Benchohra, J. Henderson, Asymptotic behavior of solutions of nonlinear fractional order Riemann-Liouville Volterra-Stieltjes quadratic integral equations, Int. Elect. J. Pure Appl. Math., 4(3) (2012), 195-209.
• [18] S. Abbas, M. Benchohra, J. J. Nieto, Global attractivity of solutions for nonlinear fractional order Riemann-Liouville Volterra-Stieltjes partial integral equations, Electron. J. Qual. Theory Differ. Equ., 81 (2012), 1-15.
• [19] S. Samko, A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives (Theorie and Applications), Gordon and Breach Science Publishers, Yverdon, 1993.
• [20] I. P. Natanson, Theory of Functions of a Real Variable, Ungar, New York, 1960.
• [21] B. G. Pachpatte, Inequalities for Differential and Integral Equations, William F. Ames, Georgia Institute of Technology, 1998.
• [22] L. Olszowy, S. Dudek, On generalization of Darbo-Sadovskii type fixed point theorems for iterated mappings in Fr´echet spaces, J. Fixed Point Theory Appl., 20(4) (2018), Article number: 146, 12 pages.
• [23] J. Daneˇs, Some fixed point theorems, Comment. Math. Univ. Carolinae, 9 (1968), 223-235.
• [24] F. Wang, H. Zhou, Fixed point theorems in locally convex spaces and a nonlinear integral equation of mixed type, Fixed Point Theory Appl., 2015(1)(2015), Article number: 228228, 11 pages.