Complex analysis/Functional analysis
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.

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:

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

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 :

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

Accepted:
Published online:
DOI: 10.1016/j.crma.2018.01.008
Şan, Müfit 1

1 Department of Mathematics, Faculty of Science, Çankırı Karatekin University, TR-18100, Çankırı, Turkey
@article{CRMATH_2018__356_3_293_0,
author = {\c{S}an, M\"ufit},
title = {Complex variable approach to the analysis of a fractional differential equation in the real line},
journal = {Comptes Rendus. Math\'ematique},
pages = {293--300},
publisher = {Elsevier},
volume = {356},
number = {3},
year = {2018},
doi = {10.1016/j.crma.2018.01.008},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2018.01.008/}
}
TY  - JOUR
AU  - Şan, Müfit
TI  - Complex variable approach to the analysis of a fractional differential equation in the real line
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 293
EP  - 300
VL  - 356
IS  - 3
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2018.01.008/
DO  - 10.1016/j.crma.2018.01.008
LA  - en
ID  - CRMATH_2018__356_3_293_0
ER  - 
%0 Journal Article
%A Şan, Müfit
%T Complex variable approach to the analysis of a fractional differential equation in the real line
%J Comptes Rendus. Mathématique
%D 2018
%P 293-300
%V 356
%N 3
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2018.01.008/
%R 10.1016/j.crma.2018.01.008
%G en
%F CRMATH_2018__356_3_293_0
Şan, Müfit. 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. http://www.numdam.org/articles/10.1016/j.crma.2018.01.008/

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

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

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

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

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

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

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

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

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

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

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

[13] Ortigueira, M.D.; Rodríguez-Germá, L.; Trujillo, J.J. 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] Owa, S.; Saitoh, H.; Srivastava, H.M.; Yamakawa, R. Geometric properties of solutions of a class of differential equations, Comput. Math. Appl., Volume 47 (2004) no. 10, pp. 1689-1696

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

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

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

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

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

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

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

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

[24] Zhang, S. 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

Cited by Sources: