Abstract
This paper deals with some existence and Ulam stability results for a class of partial integral equations via Hadamard’s fractional integral, by applying Schauder’s fixed-point theorem.
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Abbas S., Benchohra M.: Partial hyperbolic differential equations with finite delay involving the Caputo fractional derivative. Commun. Math. Anal. 7, 62–72 (2009)
Abbas S., Benchohra M.: Fractional order integral equations of two independent variables. Appl. Math. Comput. 227, 755–761 (2014)
Abbas S., Benchohra M., N’Guérékata G.M.: Topics in Fractional Differential Equations. Springer, New York (2012)
Abbas S., Benchohra M., N’Guérékata G.M.: Advanced Fractional Differential and Integral Equations. Nova Science Publishers, New York (2015)
Abbas S., Benchohra M., Vityuk A.N.: On fractional order derivatives and Darboux problem for implicit differential equations. Fract. Calc. Appl. Anal. 15, 168–182 (2012)
Benchohra M., Henderson J., Ntouyas S.K., Ouahab A.: Existence results for functional differential equations of fractional order. J. Math. Anal. Appl. 338, 1340–1350 (2008)
Bota-Boriceanu M.F., Petrusel A.: Ulam–Hyers stability for operatorial equations and inclusions. Analele Univ. I. Cuza Iasi. 57, 65–74 (2011)
Butzer P.L., Kilbas A.A., Trujillo J.J.: Fractional calculus in the Mellin setting and Hadamard-type fractional integrals. J. Math. Anal. Appl. 269, 1–27 (2002)
Butzer P.L., Kilbas A.A., Trujillo J.J.: Mellin transform analysis and integration by parts for Hadamard-type fractional integrals. J. Math. Anal. Appl. 270, 1–15 (2002)
Castro L.P., Ramos A.: Hyers–Ulam–Rassias stability for a class of Volterra integral equations. Banach J. Math. Anal. 3, 36–43 (2009)
Granas A., Dugundji J.: Fixed Point Theory. Springer-Verlag, New York (2003)
Hadamard J.: Essai sur l’étude des fonctions données par leur développment de Taylor. J. Pure Appl. Math. 4(8), 101–186 (1892)
Hilfer R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)
Hyers D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. 27, 222–224 (1941)
Hyers, D.H.; Isac, G.; Rassias, Th.M.: Stability of Functional Equations in Several Variables. Birkhuser (1998)
Jung S.-M.: Hyers–Ulam–Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor (2001)
Jung S.-M.: Hyers–Ulam–Rassias Stability of Functional Equations in Nonlinear Analysis. Springer, New York (2011)
Jung, S.-M.: A fixed point approach to the stability of a Volterra integral equation. Fixed Point Theory Appl. 2007 (2007), Article ID 57064, 9 pages
Kilbas, A.A.; Srivastava, H.M.; Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. Elsevier Science B.V.; Amsterdam (2006)
Miller K.S., Ross B.: An Introduction to the Fractional Calculus and Differential Equations. John Wiley, New York (1993)
Petru T.P., Petrusel A., Yao J.-C.: Ulam–Hyers stability for operatorial equations and inclusions via nonself operators. Taiwan. J. Math. 15, 2169–2193 (2011)
Pooseh S., Almeida R., Torres D.: Expansion formulas in terms of integer-order derivatives for the Hadamard fractional integral and derivative. Numer. Funct. Anal. Optim. 33(3), 301–319 (2012)
Rassias Th.M.: On the stability of linear mappings in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)
Rus I.A.: Ulam stability of ordinary differential equations. Studia Univ. Babes-Bolyai Math. LIV(4, 125–133 (2009)
Rus I.A.: Remarks on Ulam stability of the operatorial equations. Fixed Point Theory 10, 305–320 (2009)
Samko S.G., Kilbas A.A., Marichev O.I.: Fractional Integrals and Derivatives. Theory and Applications. Gordon and Breach, Yverdon (1993)
Tarasov, V.E.: Fractional Dynamics: Application of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, Heidelberg; Higher Education Press, Beijing (2010)
Ulam S.M.: A Collection of Mathematical Problems. Interscience Publishers, New York (1968)
Vityuk, A.N.: On solutions of hyperbolic differential inclusions with a nonconvex right-hand side. (Russian) Ukran. Mat. Zh. 47(4), 531-534 (1995); translation in Ukrainian Math. J. 47 (1995), no. 4, 617–621 (1996)
Vityuk A.N., Golushkov A.V.: Existence of solutions of systems of partial differential equations of fractional order. Nonlinear Oscil. 7, 318–325 (2004)
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Abbas, S., Albarakati, W., Benchohra, M. et al. Ulam stabilities for partial Hadamard fractional integral equations. Arab. J. Math. 5, 1–7 (2016). https://doi.org/10.1007/s40065-015-0142-9
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DOI: https://doi.org/10.1007/s40065-015-0142-9