Fixed point in Banach *-algebras with an application to functional integral equation of fractional order
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Abstract
In this paper, we investigate the solvability of an operator equation involving four operators in the setting of Banach *-algebras using Schauder's fixed point theorem. Moreover, we have given an application of our result to the following functional integral equation of fractional order:
$$
\xi(t)=g(t,\xi(\psi_1(t)))I^{\alpha} f_1(t,I^{\beta}u(t,\xi(\psi_2(t))))+h(t,\xi(\psi_3(t)))I^{\gamma}f_2(t,I^{\delta}v(t,\xi^*(\psi_4(t))))
$$
for proving the existence as well as the uniqueness of the solution in Banach *-algebras under some generalized conditions.
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References
[1] H. Afshari, H. Shojaat and A. Fulga, Common new fixed point results on b-cone Banach spaces over Banach algebras, Appl. Gen. Topol. 23 (1) (2022), 145–156. https://dx.doi.org/10.4995/agt.2022.15571 Google Scholar
[2] S.M. Al-Issa and N.M. Mawed, Results on solvability of nonlinear quadratic integral equations of fractional orders in Banach algebra, J. Nonlinear Sci. Appl. 14 (4) (2021), 181–195. https://dx.doi.org/10.22436/jnsa.014.04.01 Google Scholar
[3] I.A. Bhat, L.N. Mishra, V.N. Mishra, C. Tunc and O. Tunc, Precision and efficiency of an interpolation approach to weakly singular integral equations, Int. J. Numer. Methods Heat Fluid Flow 34 (3) (2024), 1479–1499. https://dx.doi.org/10.1108/HFF-09-2023-0553 Google Scholar
[4] F.F. Bonsall and J. Duncan, Complete normed Algebras, Springer-Verlag, 1973. Google Scholar
[5] R.K. Bose, Some Random Fixed Point Theorems Concerning Three Random Operators on a Banach Algebra, Int. J. Pure Appl. Math. 45 (3) (2008), 453–462. Google Scholar
[6] Y.M. Chu, S. Rashid, F. Jarad, M.A. Noor and 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. https://dx.doi.org/10.3934/dcdss.2021063 Google Scholar
[7] B.C. Dhage, On some variants of Schauder’s fixed point principle and applications to nonlinear integral equations, J. Math. Phys. 25 (1988), 603–611. Google Scholar
[8] B.C. Dhage, Remarks on two fixed point theorems involving the sum and product of two operators, Comput. Math. Appl. 46 (12) (2003), 1779–1785. https://dx.doi.org/10.1016/S0898-1221(03)90236-7 Google Scholar
[9] B.C. Dhage, A fixed point theorem in Banach algebras involving three operators with applications, Kyungpook Math. J. 44 (1) (2004), 145–145. Google Scholar
[10] B.C. Dhage, On a fixed point theorem in Banach algebras with applications, Appl. Math. Lett. 18 (3) (2005), 273–280. https://dx.doi.org/10.1016/j.aml.2003.10.014 Google Scholar
[11] B.C. Dhage, Some nonlinear alternatives in Banach algebras with applications II, Kyungpook Math. J. 45 (2) (2005), 281–292. Google Scholar
[12] B.C. Dhage, Coupled hybrid fixed point theory involving the sum and product of three coupled operators in a partially ordered Banach algebra with applications, J. Fixed Point Theory Appl. 19 (2017), 3231–3264. https://dx.doi.org/10.1007/s11784-017-0471-8 Google Scholar
[13] B.C. Dhage, Some variants of two basic hybrid fixed point theorems of Krasnoselskii and Dhage with applications, Nonlinear Stud. 25 (3) (2018), 559–573. Google Scholar
[14] B.C. Dhage, A coupled hybrid fixed point theorem involving the sum of two coupled operators in a partially ordered Banach space with applications, Tamkang J. Math. 50 (1) (2019), 1–36. https://dx.doi.org/10.5556/j.tkjm.50.2019.2502 Google Scholar
[15] A.M.A. El-Sayed and H. Hashem, Existence results for nonlinear quadratic integral equations of fractional order in Banach algebra, Fract. Calc. Appl. Anal. 16 (4) (2013), 816–826. https://dx.doi.org/10.2478/s13540-013-0051-6 Google Scholar
[16] A.M.A. El-Sayed and S.M. Al-Issa, Monotonic integrable solution for a mixed type integral and differential inclusion of fractional orders, Int. J. Differ. Equ. Appl. 18 (1) (2019), 1–9. Google Scholar
[17] A.M.A. El-Sayed and S.M. Al-Issa, Monotonic solutions for a quadratic integral equation of fractional order, AIMS Math. 4 (3) (2019), 821–830. https://dx.doi.org/10.3934/math.2019.3.821 Google Scholar
[18] J. Fernandez, N. Malviya, Z.D. Mitrovi ́c, A. Hussain and V. Parvaneh, Some fixed point results on N b-cone metric spaces over Banach algebra, Adv. Differential Equations 2020 (1) (2020), 529. https://dx.doi.org/10.1186/s13662-020-02991-5 Google Scholar
[19] J. Fernandez, N. Malviya, S. Radenoviˇc and K. Saxena, F-cone metric spaces over Banach algebra, Fixed Point Theory Appl. 2017 (1) (2016), 7. https://dx.doi.org/10.1186/s13663-017-0600-5 Google Scholar
[20] K.S. Miller and B. Ross, An Introduction to the fractional calculus and fractional differential equations, John Wiley & Sons, New York, (1993). Google Scholar
[21] M.M. Metwali, On a class of quadratic Urysohn–Hammerstein integral equations of mixed type and initial value problem of fractional order, Mediterr. J. Math. 13 (2016), 2691–2707. https://dx.doi.org/10.1007/s00009-015-0647-7 Google Scholar
[22] M.M.A. Metwali and V.N. Mishra, On the measure of noncompactness in Lp(R+) and applications to a product of n-integral equations, Turkish J. Math. 47 (1) (2023), 372–386. https://dx.doi.org/10.55730/1300-0098.3365 Google Scholar
[23] S. Mubeen, S. Habib and M.N. Naeem, The Minkowski inequality involving generalized k-fractional conformable integral, J. Inequal. Appl. 2019 (81) (2019). https://dx.doi.org/10.1186/s13660-019-2040-8 Google Scholar
[24] S.K. Paul, L.N. Mishra, V.N. Mishra and D. Baleanu, An effective method for solving nonlinear integral equations involving the Riemann-Liouville fractional operator, AIMS Math. 8 (8) (2023), 17448–17469. https://dx.doi.org/10.3934/math.2023891 Google Scholar
[25] S.K. Paul, L.N. Mishra, V.N. Mishra and D. Baleanu, Analysis of mixed type nonlinear Volterra–Fredholm integral equations involving the Erdélyi–Kober fractional operator, J. King Saud Univ. Sci. 35 (10) (2023), 102949. https://dx.doi.org/10.1016/j.jksus.2023.102949 Google Scholar
[26] V.K. Pathak, L.N. Mishra, V.N. Mishra and D. Baleanu, On the Solvability of Mixed-Type Fractional-Order Non-Linear Functional Integral Equations in the Banach Space C(I), Fractal fract. 6 (12) (2022), 744. https://dx.doi.org/10.3390/fractalfract6120744 Google Scholar
[27] V.K. Pathak, L.N. Mishra and V.N. Mishra, On the solvability of a class of nonlinear functional integral equations involving Erdélyi-Kober fractional operator, Math. Methods Appl. Sci. 46 (13)(2023), 14340–14352. https://dx.doi.org/10.1002/mma.9322 Google Scholar
[28] H.K. Pathak, Remarks on some fixed point theorems of Dhage, Appl. Math. Lett. 25 (11) (2012), 1969–1975. https://dx.doi.org/10.1016/j.aml.2012.03.011 Google Scholar
[29] A.G. Sanatee, L. Rathour, V.N. Mishra and V. Dewangan, Some fixed point theorems in regular modular metric spaces and application to Caratheodory’s type anti-periodic boundary value problem, The J. Anal. 31 (2023), 619–632. https://dx.doi.org/10.1007/s41478-022-00469-z Google Scholar
[30] D.R. Smart, Fixed point theorems, Cambridge University Press, 1974. Google Scholar
[31] B.C. Tripathy, P. Sudipta and R.D. Nanda, Banach’s and Kannan’s fixed point results in fuzzy 2-metric spaces, Proyecciones 32 (4) (2013), 359–375. https://dx.doi.org/10.4067/S0716-09172013000400005 Google Scholar
[32] B.C. Tripathy, S. Paul and R.N. Das, A fixed point theorem in a generalized fuzzy metric space, Bol. Soc. Parana. Mat. 32 (2) (2014), 221–227. Google Scholar
[33] P. Thongin and W. Fupinwong, The fixed point property of a Banach algebra generated by an element with infinite spectrum, J. Funct. Spaces 2018 (2018). https://dx.doi.org/10.1155/2018/9045790 Google Scholar