# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Existence results for fractional order functional differential equations with infinite delay. (English) Zbl 1209.34096
The paper deals with the existence of solutions for initial value problems for fractional-order functional differential equations with infinite delay $$D^\alpha y(t)=f(t,y_t),\quad t\in J=[0,b],\quad 0<\alpha<1,$$ $$y(t)=\varphi(t),\quad t\in (-\infty,0]$$ and $$D^\alpha[y(t)-g(t,y_t)]=f(t,y_t),\quad t\in J=[0,b],$$ $$y(t)=\varphi(t),\quad t\in (-\infty,0],$$ where $D^\alpha$ is the standard Riemann-Liouville fractional derivative. The Banach fixed point theorem and a nonlinear alternative of Leray-Schauder type are used to investigate the given initial value problems.
Reviewer: Zdeněk Šmarda (MR2386501)

##### MSC:
 34K37 Functional-differential equations with fractional derivatives 34K40 Neutral functional-differential equations 47N20 Applications of operator theory to differential and integral equations
Full Text:
##### References:
 [1] Corduneanu, C.; Lakshmikantham, V.: Equations with unbounded delay, Nonlinear anal. 4, 831-877 (1980) · Zbl 0661.34068 [2] Diethelm, K.; Freed, A. D.: On the solution of nonlinear fractional order differential equations used in the modeling of viscoplasticity, Scientific computing in chemical engineering II -- computational fluid dynamics, reaction engineering and molecular properties, 217-224 (1999) [3] Delboso, D.; Rodino, L.: Existence and uniqueness for a nonlinear fractional differential equation, J. math. Anal. appl. 204, 609-625 (1996) · Zbl 0881.34005 · doi:10.1006/jmaa.1996.0456 [4] Diethelm, K.; Ford, N. J.: Analysis of fractional differential equations, J. math. Anal. appl. 265, 229-248 (2002) · Zbl 1014.34003 [5] Diethelm, K.; Walz, G.: Numerical solution of fractional order differential equations by extrapolation, Numer. algorithms 16, 231-253 (1997) · Zbl 0926.65070 · doi:10.1023/A:1019147432240 [6] Gaul, L.; Klein, P.; Kempfle, S.: Damping description involving fractional operators, Mech. systems signal processing 5, 81-88 (1991) [7] Glockle, W. G.; Nonnenmacher, T. F.: A fractional calculus approach of self-similar protein dynamics, Biophys. J. 68, 46-53 (1995) [8] Granas, A.; Dugundji, J.: Fixed point theory, (2003) · Zbl 1025.47002 [9] Hale, J.; Kato, J.: Phase space for retarded equations with infinite delay, Funkcial. ekvac. 21, 11-41 (1978) · Zbl 0383.34055 [10] Henry, D.: Geometric theory of semilinear parabolic partial differential equations, (1989) [11] Hino, Y.; Murakami, S.; Naito, T.: Functional differential equations with infinite delay, Lecture notes in math. 1473 (1991) · Zbl 0732.34051 [12] Kappel, F.; Schappacher, W.: Some considerations to the fundamental theory of infinite delay equations, J. differential equations 37, 141-183 (1980) · Zbl 0466.34036 [13] Lakshmikantham, V.; Wen, L.; Zhang, B.: Theory of differential equations with unbounded delay, Math. appl. (1994) · Zbl 0823.34069 [14] Mainardi, F.: Fractional calculus: some basic problems in continuum and statistical mechanics, Fractals and fractional calculus in continuum mechanics, 291-348 (1997) · Zbl 0917.73004 [15] Metzler, F.; Schick, W.; Kilian, H. G.; Nonnenmacher, T. F.: Relaxation in filled polymers: A fractional calculus approach, J. chem. Phys. 103, 7180-7186 (1995) [16] Miller, K. S.; Ross, B.: An introduction to the fractional calculus and differential equations, (1993) · Zbl 0789.26002 [17] Momani, S. M.; Hadid, S. B.: Some comparison results for integro-fractional differential inequalities, J. fract. Calc. 24, 37-44 (2003) · Zbl 1057.45003 [18] Momani, S. M.; Hadid, S. B.; Alawenh, Z. M.: Some analytical properties of solutions of differential equations of noninteger order, Int. J. Math. math. Sci. 2004, 697-701 (2004) · Zbl 1069.34002 · doi:10.1155/S0161171204302231 [19] Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008 [20] Podlubny, I.; Petraš, I.; Vinagre, B. M.; O’leary, P.; Dorčk, L.: Analogue realizations of fractional-order controllers, Nonlinear dynam. 29, 281-296 (2002) · Zbl 1041.93022 · doi:10.1023/A:1016556604320 [21] Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives. Theory and applications, (1993) · Zbl 0818.26003 [22] Schumacher, K.: Existence and continuous dependence for differential equations with unbounded delay, Arch. ration. Mech. anal. 64, 315-335 (1978) · Zbl 0383.34052 [23] Shin, J. S.: An existence theorem of functional differential equations with infinite delay in a Banach space, Funkcial. ekvac. 30, 19-29 (1987) · Zbl 0647.34066 [24] Yu, C.; Gao, G.: Existence of fractional differential equations, J. math. Anal. appl. 310, 26-29 (2005) · Zbl 1088.34501 · doi:10.1016/j.jmaa.2004.12.015