Comptes Rendus
Complex analysis/Functional analysis
Complex variable approach to the analysis of a fractional differential equation in the real line
[Approche par variable complexe de l'analyse d'une équation différentielle fractionnaire sur la droite réelle]
Comptes Rendus. Mathématique, Volume 356 (2018) no. 3, pp. 293-300.

L'objectif principal de ce travail est d'établir un théorème d'existence de type Peano pour un problème aux valeurs initiales faisant intervenir une dérivée fractionnaire, puis, comme conséquence, de donner une réponse partielle à l'existence locale d'une solution continue du problème aux valeurs initiales suivant :

{Dxqu(x)=f(x,u(x)),u(0)=b,(b0).
De plus, nous étudions les propriétés géométriques des solutions pour quelques cas particuliers.

The first aim of this work is to establish a Peano-type existence theorem for an initial value problem involving a complex fractional derivative, and then, as a consequence of this theorem, to give a partial answer for the local existence of the continuous solution to the initial value problem:

{Dxqu(x)=f(x,u(x)),u(0)=b,(b0).
Moreover, for some special cases of the problem, we investigate the corresponding geometric properties of the solutions.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2018.01.008

Müfit Şan 1

1 Department of Mathematics, Faculty of Science, Çankırı Karatekin University, TR-18100, Çankırı, Turkey
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Müfit Şan. Complex variable approach to the analysis of a fractional differential equation in the real line. Comptes Rendus. Mathématique, Volume 356 (2018) no. 3, pp. 293-300. doi : 10.1016/j.crma.2018.01.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.01.008/

[1] D. Baleanu; O.G. Mustafa On the global existence of solutions to a class of fractional differential equations, Comput. Math. Appl., Volume 59 (2010) no. 5, pp. 1835-1841

[2] M. Chen; H. Irmak; H.M. Srivastava Some families of multivalently analytic functions with negative coefficients, J. Math. Anal. Appl., Volume 214 (1997) no. 2, pp. 674-690

[3] D. Delboso; L. Rodino Existence and uniqueness for a nonlinear fractional differential equation, J. Math. Anal. Appl., Volume 204 (1996), pp. 609-625

[4] J.W. Dettman Applied Complex Variables, Dover Publications Inc., New York, 1965

[5] K. Diethelm Smoothness properties of solutions of Caputo-type fractional differential equations, Fract. Calc. Appl. Anal., Volume 10 (2007) no. 2, pp. 151-160

[6] M.J. Garrido-Atienza; K. Lu; B. Schmalfuß Compensated fractional derivatives and stochastic evolution equations, C. R. Acad. Sci., Ser. I Math., Volume 350 (2012) no. 23–24, pp. 1037-1042

[7] A.W. Goodman Univalent Functions, vol. II, Mariner Publishing Co., Inc., Tampa, FL, 1983

[8] H. Irmak; M. Şan Some relations between certain inequalities concerning analytic and univalent functions, Appl. Math. Lett., Volume 23 (2010) no. 8, pp. 897-901

[9] A.A.A. Kilbas; H.M. Srivastava; J.J. Trujillo Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier Science Limited, 2006

[10] V. Lakshmikanthan; A.S. Vatsala Basic theory of fractional differential equations, Nonlinear Anal., Volume 69 (2008) no. 8, pp. 2677-2682

[11] P.T. Mocanu Some starlikeness conditions for analytic functions, Rev. Roum. Math. Pures Appl., Volume 33 (1988), pp. 117-124

[12] T. Ohsawa Analysis of Several Complex Variables, Transl. Math. Monogr., vol. 211, American Mathematical Society, 2002

[13] M.D. Ortigueira; L. Rodríguez-Germá; J.J. Trujillo Complex Grünwald-Letnikov, Liouville, Riemann–Liouville, and Caputo derivatives for analytic functions, Commun. Nonlinear Sci. Numer. Simul., Volume 16 (2011) no. 11, pp. 4174-4182

[14] S. Owa; H. Saitoh; H.M. Srivastava; R. Yamakawa Geometric properties of solutions of a class of differential equations, Comput. Math. Appl., Volume 47 (2004) no. 10, pp. 1689-1696

[15] I. Podlubny Fractional Differential Equations, Academic Press, San Diego, 1999

[16] S. Ponnusamy; H. Silverman Complex Variables with Applications, Springer Science & Business, Media, 2007

[17] H. Saitoh Univalence and starlikeness of solutions W+aW+bW=0, Ann. Univ. Mariae Curie-Skłodowska, Sect. A, Volume 53 (1999), pp. 209-216

[18] S.G. Samko; A.A. Kilbas; O.I. Marichev Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993

[19] M. Şan; H. Irmak Some novel applications of certain higher order ordinary complex differential equations to normalized analytic functions, J. Appl. Anal. Comput., Volume 5 (2015) no. 3, pp. 479-484

[20] M. Şan; K.N. Soltanov The New Existence and Uniqueness Results for Complex Nonlinear Fractional Differential Equation, 2015 (preprint) | arXiv

[21] H.M. Srivastava; S. Owa Univalent Functions, Fractional Calculus, and Their Applications, Ellis Horwood/Halsted Press, New York/Toronto, 1989

[22] C. Yu; G. Gao Existence of fractional differential equations, J. Math. Anal. Appl., Volume 310 (2005), pp. 26-29

[23] E. Zeidler Nonlinear Functional Analysis and its Applications, I: Fixed-Point Theorems, Springer-Verlag, New York, 1985

[24] S. Zhang Monotone iterative method for initial value problem involving Riemann–Liouville fractional derivatives, Nonlinear Anal., Theory Methods Appl., Volume 71 (2009) no. 5, pp. 2087-2093

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