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 and uniqueness of strictly nondecreasing and positive solution for a fractional three-point boundary value problem. (English) Zbl 1235.34024

Summary: We consider the following nonlinear fractional three-point boundary value problem

D 0+ α u(t)+f(t,u(t))=0,0<t<1,3<α4,u(0)=u ' (0)=u '' (0)=0,u '' (1)=βu '' (η),

where D 0+ α is the standard Riemann-Liouville fractional derivative. By using a fixed point theorem in partially ordered sets, we obtain sufficient conditions for the existence and uniqueness of positive and nondecreasing solution to the above boundary value problem.

MSC:
34A08Fractional differential equations
34B10Nonlocal and multipoint boundary value problems for ODE
34B18Positive solutions of nonlinear boundary value problems for ODE
47N20Applications of operator theory to differential and integral equations
References:
[1]Benchohra, M.; Henderson, J.; Ntouyas, S. K.; Ouahab, A.: Existence results for fractional order functional differential equations with infinite delay, J. math. Anal. appl. 338, 1340-1350 (2008) · Zbl 1209.34096 · doi:10.1016/j.jmaa.2007.06.021
[2]Lakshmikantham, V.: Theory of fractional functional differential equations, Nonlinear anal. 69, 3337-3343 (2008) · Zbl 1162.34344 · doi:10.1016/j.na.2007.09.025
[3]Zhou, Y.: Existence and uniqueness of fractional functional differential equations with unbounded delay, Int. J. Dyn. syst. Differ. equ. 1, 239-244 (2008) · Zbl 1175.34081 · doi:10.1504/IJDSDE.2008.022988
[4]Zhou, Y.; Jiao, F.: Nonlocal Cauchy problem for fractional evolution equations, Nonlinear anal. RWA 11, 4465-4475 (2010)
[5]Zhou, Y.; Jiao, F.; Li, J.: Existence and uniqueness for fractional neutral differential equations with infinite delay, Nonlinear anal. TMA 71, 3249-3256 (2009) · Zbl 1177.34084 · doi:10.1016/j.na.2009.01.202
[6]Zhou, Y.; Jiao, F.; Li, J.: Existence and uniqueness for p-type fractional neutral differential equations, Nonlinear anal. TMA 71, 2724-2733 (2009) · Zbl 1175.34082 · doi:10.1016/j.na.2009.01.105
[7]Samko, S. G.; Kilbas, A. A.; Marichev, O. I.: Fractional integrals and derivatives. Theory and applications, (1993) · Zbl 0818.26003
[8]Podlubny, I.: Fractional differential equations, Mathematics in sciences and engineering 198 (1999) · Zbl 0924.34008
[9]Kilbas, A. A.; Srivastava, H. M.; Trujillo, J. J.: Theory and applications of fractional differential equations, North-holland mathematics studies 204 (2006)
[10]Diethelm, K.: The analysis of fractional differential equations, (2010)
[11]Lakshmikantham, V.; Vatsala, A. S.: Basic theory of fractional differential equations, Nonlinear anal. 69, 2677-2682 (2008) · Zbl 1161.34001 · doi:10.1016/j.na.2007.08.042
[12]Lakshmikantham, V.; Vatsala, A. S.: General uniqueness and monotone iterative technique for fractional differential equations, Appl. math. Lett. 21, 828-834 (2008) · Zbl 1161.34031 · doi:10.1016/j.aml.2007.09.006
[13]Bai, Z.; Lü, H.: Positive solutions for boundary value problem of nonlinear fractional differential equation, J. math. Anal. appl. 311, 495-505 (2005) · Zbl 1079.34048 · doi:10.1016/j.jmaa.2005.02.052
[14]El-Sayed, A. M. A.; El-Mesiry, A. E. M.; El-Saka, H. A. A.: On the fractional-order logistic equation, Appl. math. Lett. 20, 817-823 (2007) · Zbl 1140.34302 · doi:10.1016/j.aml.2006.08.013
[15]El-Shahed, M.: Positive solutions for boundary value problem of nonlinear fractional differential equation, Abstr. appl. Anal. 2007, pp. 8 (2007) · Zbl 1149.26012 · doi:10.1155/2007/10368
[16]Bai, C.: Positive solutions for nonlinear fractional differential equations with coefficient that changes sign, Nonlinear anal. 64, 677-685 (2006) · Zbl 1152.34304 · doi:10.1016/j.na.2005.04.047
[17]Bai, C.: Triple positive solutions for a boundary value problem of nonlinear fractional differential equation, Electron. J. Qualitative theory diff. Equ., No. 24, 1-10 (2008) · Zbl 1183.34005 · doi:emis:journals/EJQTDE/2008/200824.html
[18]Zhang, S.: Existence of solution for a boundary value problem of fractional order, Acta math. Sci. 26, 220-228 (2006) · Zbl 1106.34010 · doi:10.1016/S0252-9602(06)60044-1
[19]Liang, S.; Zhang, J. H.: Positive solutions for boundary value problems of nonlinear fractional differential equation, Nonlinear anal. 71, 5545-5550 (2009) · Zbl 1185.26011 · doi:10.1016/j.na.2009.04.045
[20]Li, C. F.; Luo, X. N.; Zhou, Y.: Existence of positive solutions of the boundary value problem for nonlinear fractional differential equations, Comput. math. Appl. 59, 1363-1375 (2010) · Zbl 1189.34014 · doi:10.1016/j.camwa.2009.06.029
[21]Mena, J. Caballero; Harjani, J.; Sadarangani, K.: Existence and uniqueness of positive and nondecreasing solutions for a class of singular fractional boundary value problems, Bound. value probl. 2009, pp. 10 (2009) · Zbl 1182.34005 · doi:10.1155/2009/421310
[22]Harjani, J.; Sadarangani, K.: Fixed point theorems for weakly contractive mappings in partially ordered sets, Nonlinear anal. 71, 3403-3410 (2009) · Zbl 1221.54058 · doi:10.1016/j.na.2009.01.240
[23]Nieto, J. J.; Rodríguez-López, R.: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations, Order 22, 223-239 (2005) · Zbl 1095.47013 · doi:10.1007/s11083-005-9018-5
[24]Nieto, J. J.; Rodríguez-López, R.: Fixed point theorems in ordered abstract spaces, Proc. amer. Math. soc. 135, No. 8, 2505-2517 (2007) · Zbl 1126.47045 · doi:10.1090/S0002-9939-07-08729-1
[25]O’regan, D.; Petrusel, A.: Fixed point theorems for generalized contractions in ordered metric spaces, J. math. Anal. appl. 341, 1241-1252 (2008) · Zbl 1142.47033 · doi:10.1016/j.jmaa.2007.11.026