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 of solutions for multi point boundary value problems for fractional differential equations. (English) Zbl 1247.35190

Summary: By employing the Leggett-Williams fixed point theorem, we study the existence of three solutions in the multi point fractional boundary value problem

D 0 + α u(t)=f(t,u(t),u ' (t)),t[0,1],u(0)=u ' (0)=0,u(1)- i=1 m a i u(ξ i )=λ,

where 2<α3 and m1 are integers, 0<ξ 1 <ξ 2 <<ξ n <1 are constants, λ(0,) is a parameter, α i >0 for 1im and i=1 m a i ξ i ξ i α-1 <1, fC([0,1]×[0,)×[0,);[0,)).


MSC:
35R11Fractional partial differential equations
35A01Existence problems for PDE: global existence, local existence, non-existence
References:
[1]Agarwal, R. P.; Benchohra, M.; Hamani, S.: A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions, Acta appl. Math. 109, 973-1033 (2010) · Zbl 1198.26004 · doi:10.1007/s10440-008-9356-6
[2]Cang, J.; Tan, Y.; Xu, H.; Liao, S. J.: Series solutions of non-linear Riccati differential equations with fractional order, Chaos solitons fractals 40, No. 1, 1-9 (2009) · Zbl 1197.34006 · doi:10.1016/j.chaos.2007.04.018
[3]Diethelm, K.; Ford, N.; Freed, A.: A predictor-corrector approach for the numerical solution of fractional differential equations, Nonlinear dynam. 29, 3-22 (2002) · Zbl 1009.65049 · doi:10.1023/A:1016592219341
[4]Diethelm, K.; Ford, N.; Freed, A.: Detailed error analysis for a fractional Adams method, Numer. algorithms 36, 31-52 (2004) · Zbl 1055.65098 · doi:10.1023/B:NUMA.0000027736.85078.be
[5]El-Sayed, A. M. A.: Nonlinear functional differential equations of arbitrary orders, Nonlinear anal. 33, 181-186 (1998) · Zbl 0934.34055 · doi:10.1016/S0362-546X(97)00525-7
[6]Feng, J.; Yong, Z.: Existence of solutions for a class of fractional boundary value problems via critical point theory, Comput. math. Appl. 62, 1181-1199 (2011)
[7]Kilbas, A. A.; Trujillo, J. J.: Differential equations of fractional order: methods, results and problems I, Appl. anal. 78, 153-192 (2001) · Zbl 1031.34002 · doi:10.1080/00036810108840931
[8]Kilbas, A. A.; Trujillo, J. J.: Differential equations of fractional order: methods, results and problems II, Appl. anal. 81, 435-493 (2002) · Zbl 1033.34007 · doi:10.1080/0003681021000022032
[9]Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, North-holland mathematics studies 204 (2006)
[10]Lakshmikantham, V.; Vatsala, A. S.: Basic theory of fractional differential equations, Nonlinear anal. TMA 69, No. 8, 2677-2682 (2008) · Zbl 1161.34001 · doi:10.1016/j.na.2007.08.042
[11]Leggett, R. W.; Williams, L. R.: Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana univ. Math. J. 28, 673-688 (1979) · Zbl 0421.47033 · doi:10.1512/iumj.1979.28.28046
[12]Mainardi, F.; Luchko, Y.; Pagnini, G.: The fundamental solution of the space-time fractional diffusion equation, Fract. calc. Appl. anal. 4, 153-192 (2001) · Zbl 1054.35156
[13]Miller, K. S.; Ross, B.: An introduction to the fractional calculus and fractional differential equations, (1993)
[14]Momani, S.; Odibat, Z.: A novel method for nonlinear fractional partial differential equations: combination of DTM and generalized Taylor’s formula, J. comput. Appl. math. 220, No. 1 – 2, 85-95 (2008) · Zbl 1148.65099 · doi:10.1016/j.cam.2007.07.033
[15]Odibat, Z.; Momani, S.: An algorithm for the numerical solution of differential equations of fractional order, J. appl. Math. inform. 26, No. 1-2, 15-27 (2008)
[16]Odibat, Z.; Momani, S.: Numerical methods for nonlinear partial differential equations of fractional order, Appl. math. Modelling 32, No. 12, 28-39 (2008) · Zbl 1133.65116 · doi:10.1016/j.apm.2006.10.025
[17]Odibat, Z.; Momani, S.: Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order, Chaos solitons fractals 36, No. 1, 167-174 (2008) · Zbl 1152.34311 · doi:10.1016/j.chaos.2006.06.041
[18]Podlubny, I.: Fractional differential equations, (1999)
[19]Podlubny, I.: The Laplace transform method for linear differential equations of fractional order, (1994)
[20]Samko, G.; Kilbas, A.; Marichev, O.: Fractional integrals and derivatives: theory and applications, (1993) · Zbl 0818.26003
[21]Schneider, W.; Wyss, W.: Fractional diffusion and wave equations, J. math. Phys. 30, 134-144 (1989) · Zbl 0692.45004 · doi:10.1063/1.528578