Abstract
In this paper, we study some properties of Fredholm-type fractional integro-differential equations. For obtaining the results, \(\psi\)-Hilfer fractional derivative and \(\psi\) Riemann–Liouville integral operator are used. The existence and uniqueness of solution are studied using fixed point theorem and \(\psi\)-fractional Bielecki-type norm, and the properties such as estimates and continuous dependence of solution are studied using \(\psi\)-fractional Gronwall type of inequalities.
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Almeida, R.: A Caputo fractional derivative of a function with respect to another function. Commun. Nonlinear Sci. Numer. Simul. 44, 460–481 (2017)
Baleanu, D., Lopes, A.M.: Handbook of Fractional Calculus with Applications, Applications in Engineering, Life and Social Sciences, vol. 7. De Gruyter, Berlin (2019)
Furati, K.M., Kassim, M.D., Tatar, N.E.: Existence and uniqueness for a problem involving Hilfer fractional derivative. Comput. Math. Appl. 64, 1616–1626 (2012)
Harikrishnan, S., Shah, K., Kanagarajan, K.: Study of a boundary value problem for fractional order $\psi $-Hilfer fractional derivative. Arab. J. Math. (2019)
Harikrishnan, S., Kanagarajan, K., Vivek, D.: Existence and stability results for boundary value problem for differential equation with $\psi $-Hilfer fractional derivative. J. Appl. Nonlinear Dyn. 8(2), 251–259 (2019)
Kilbas, A.A., Srivastava, H.M., Trujilio J.J.: Theory and Applications of Fractional Differential Equations, North Holland Mathematics Studies, vol. 207 (2006)
Kucche, K.D., Mali, A.D., Sousa, J.C.: On the nonlinear $\psi $-Hilfer fractional differential equations. Comp. Appl. Math. 38–73 (2018)
Petras, I.: Handbook of Fractional Calculus with Applications, Applications in Control, vol. 6. De Gruyter, Berlin (2019)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science Publishers, London (1993)
Sousa, J.V.D.C., De Oliveira, E.C.: On the $\psi $-Hilfer fractional derivative. Commun Nonlinear Sci. Numer. Simul. 60, 72–91 (2018)
Sousa, J.V.D.C., Rodrigues, F.G., De Oliveira, E.C.: Stability of the fractional Volterra integrodifferential equation by means of $\psi $-Hilfer operator. Math. Methods Appl. Sci. 42, 3033–3043 (2019)
Sousa, J.V.D.C., De Oliveira, E.C.: On the stability of a hyperbolic fractional partial differential equation. Differ. Equ. Dyn. Syst. (2018)
Sousa, J.V.D.C., De Oliveira, E.C.: A Gronwall inequality and the Cauchy type problem by means of $\psi $-Hilfer Operator. Differ. Equ. Appl. 11(1), 87–106 (2019)
Sousa, J.V.D.C., De Oliveira, E.C., Kucche, K.D.: On the fractional functional differential equation with abstract volterra operator. Bull. Braz. Math. Soc. (2019)
Sousa, J.V.D.C., de Oliveira, E.C.: Leibniz type rule: $\psi $-Hilfer operator. Commun. Nonlinear Sci. Numer. Simul. 77, 305–311 (2019)
Sousa, J.V.D.C., Frederico, G.F., De Oliveira, E.C.: $\psi $-Hilfer pseudo-fractional operator: new results about fractional calculus. Comput. Appl. Math. 39, 254 (2020)
Thabet, S., Ahmad, B., Agrwal, R.P.: On abstract Hilfer fractional integrodifferential equations with boundary conditions. Arab J. Math. Sci. (2019)
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Communicated by Carlos Lizama.
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Pachpatte, D.B. Properties of some \(\psi\)-Hilfer fractional Fredholm-type integro-differential equations. Adv. Oper. Theory 6, 7 (2021). https://doi.org/10.1007/s43036-020-00114-1
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DOI: https://doi.org/10.1007/s43036-020-00114-1