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J. Nonlinear Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"452","DOI":"10.1006\/jmaa.1997.5875","article-title":"Existence results for partial neutral functional differential equations with unbounded delay","volume":"221","year":"1998","journal-title":"J. Math. Anal. Appl."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1006\/jmaa.1997.5899","article-title":"Existence of periodic solutions of partial neutral functional differential equations with unbounded delay","volume":"221","year":"1998","journal-title":"J. Math. Anal. Appl."},{"key":"ref_4","first-page":"360","article-title":"Existence of solutions for double perturbed neutral functional evolution equation","volume":"8","author":"Ye","year":"2009","journal-title":"Int. J. Nonlinear Sci."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Hale, J., and Lunel, S.M.V. (1993). Introduction to Functional Differential Equations. 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Anal."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1340","DOI":"10.1016\/j.jmaa.2007.06.021","article-title":"Existence results for fractional order functional differential equations with infinite delay","volume":"338","author":"Benchohra","year":"2008","journal-title":"J. Math. Anal. Appl."},{"key":"ref_10","first-page":"47","article-title":"Existence results for a fractional functional differential equation of mixed type","volume":"15","author":"Darwish","year":"2008","journal-title":"Commun. Appl. Nonlinear Anal."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2091","DOI":"10.1016\/j.na.2008.02.111","article-title":"Fractional functional differential inclusions with finite delay","volume":"70","author":"Henderson","year":"2009","journal-title":"Nonlinear Anal."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"3337","DOI":"10.1016\/j.na.2007.09.025","article-title":"Theory of fractional functional differential equations","volume":"69","author":"Lakshmikantham","year":"2008","journal-title":"Nonlinear Anal."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1007\/s12591-009-0027-5","article-title":"On the solutions for impulsive fractional functional differential equations","volume":"17","author":"Chen","year":"2009","journal-title":"Differ. Equ. Dynam. Syst."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1359","DOI":"10.1016\/j.camwa.2011.03.062","article-title":"Attractivity of fractional functional differential equations","volume":"62","author":"Chen","year":"2011","journal-title":"Comput. Math. Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/S0362-546X(97)00525-7","article-title":"Nonlinear functional differential equations of arbitrary orders","volume":"33","year":"1998","journal-title":"Nonlinear Anal."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3414","DOI":"10.1016\/j.camwa.2011.12.054","article-title":"Impulsive fractional functional differential equations","volume":"64","author":"Guo","year":"2012","journal-title":"Comput. Math. Appl."},{"key":"ref_17","first-page":"1","article-title":"Initial value problems of fractional order Hadamard-type functional differential equations","volume":"77","author":"Ahmad","year":"2015","journal-title":"Electron. J. Differ. Equ."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1186\/s13662-016-0810-x","article-title":"Hadamard-type fractional functional differential equations and inclusions with retarded and advanced arguments","volume":"2016","author":"Agarwal","year":"2016","journal-title":"Adv. Differ. Equ."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"460","DOI":"10.1016\/j.cnsns.2016.09.006","article-title":"A Caputo fractional derivative of a function with respect to another function","volume":"44","author":"Almeida","year":"2017","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Gambo, Y.Y., Jarad, F., Baleanu, D., and Abdeljawad, T. (2014). On Caputo modification of the Hadamard fractional derivatives. Adv. Differ. Equ., 10.","DOI":"10.1186\/1687-1847-2014-10"},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Jarad, F., Abdeljawad, T., and Baleanu, D. (2012). Caputo-type modification of the Hadamard fractional derivatives. Adv. Differ. Equ., 142.","DOI":"10.1186\/1687-1847-2012-142"},{"key":"ref_22","unstructured":"Kilbas, A.A., Srivastava, H.M., and Trujillo, J.J. (2006). Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, Elsevier Science B.V."},{"key":"ref_23","first-page":"249","article-title":"Caputo-type modification of the Erd\u00e9lyi\u2013Kober fractional derivative","volume":"10","author":"Luchko","year":"2007","journal-title":"Fract. Calc. Appl. Anal."},{"key":"ref_24","unstructured":"Samko, S.G., Kilbas, A.A., and Marichev, O.I. (1993). Fractional Integrals and Derivatives, Translated from the 1987 Russian Original, Gordon and Breach."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"336","DOI":"10.1002\/mma.4617","article-title":"Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications","volume":"41","author":"Almeida","year":"2018","journal-title":"Math. Meth. Appl. Sci."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"358","DOI":"10.2298\/AADM170428002A","article-title":"What is the best fractional derivative to fit data?","volume":"11","author":"Almeida","year":"2017","journal-title":"Appl. Anal. Discrete Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"160","DOI":"10.1007\/s40314-019-0941-2","article-title":"Fractional differential equations and Volterra\u2013Stieltjes integral equations of the second kind","volume":"38","author":"Asanov","year":"2019","journal-title":"Comput. 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