# 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)
Unbounded solutions for a fractional boundary value problems on the infinite interval. (English) Zbl 1193.34008

Summary: We consider the fractional boundary value problem

$\left\{\begin{array}{c}{D}_{0+}^{a}u\left(t\right)+f\left(t,u\left(t\right)\right)=0,\phantom{\rule{1.em}{0ex}}t\in \left(0,\infty \right),\alpha \in \left(1,2\right),\hfill \\ u\left(0\right)=0,\phantom{\rule{1.em}{0ex}}\underset{t\to \infty }{lim}{D}_{0+}^{a-1}u\left(t\right)=\beta u\left(\xi \right),\hfill \end{array}\right\$

where $D$ is the standard Riemann-Liouville fractional derivative. By means of fixed point theorems, sufficient conditions are obtained that guarantee the existence of solutions to the above boundary value problem.

##### MSC:
 34A08 Fractional differential equations 34B18 Positive solutions of nonlinear boundary value problems for ODE 34B40 Boundary value problems for ODE on infinite intervals 47N20 Applications of operator theory to differential and integral equations
##### References:
 [1] Ahmeda, E., Elgazzar, A.S.: On fractional order differential equations model for nonlocal epidemics. Physica A 379, 607–614 (2007) · doi:10.1016/j.physa.2007.01.010 [2] He, J.H.: Some applications of nonlinear fractional differential equations and their approximations. Bull. Sci. Technol. 15, 86–90 (1999) [3] He, J.H.: Nonlinear oscillation with fractional derivative and its applications. In: International Conference on Vibrating Engineering, Dalian, China, pp. 288–291 (1998) [4] He, J.H.: Approximate analytical solution for seepage flow with fractional derivatives in porous media. Comput. Methods Appl. Mech. Eng. 167, 57–68 (1998) · Zbl 0942.76077 · doi:10.1016/S0045-7825(98)00108-X [5] Mahmoud, M.E.: Some probability densities and fundamental solutions of fractional evolution equations. Chaos Solitons Fractals 14, 433–440 (2001) [6] Xu, H.: Analytical approximations for a population growth model with fractional order. Commun. Nonlinear Sci. Numer. Simul. (2008). doi: 10.1016/j.cnsns.2008.07.006 [7] Nakhushev, A.M.: The Sturm-Liouville problem for a second order ordinary differential equation with fractional derivatives in the lower terms. Dokl. Akad. Nauk SSSR 234, 308–311 (1977) [8] Aleroev, T.S.: The Sturm-Liouville problem for a second order ordinary differential equation with fractional derivatives in the lower terms. Differ. Uravn. 18, 341–342 (1982) [9] Bai, Z.B., Lü, H.S.: 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 [10] Zhao, X., Ge, W.: Existence of at least three positive solutions for multi-point boundary value problem on infinite intervals with p-Laplacian operator. J. Appl. Math. Comput. 28, 391–403 (2008) · Zbl 1155.34307 · doi:10.1007/s12190-008-0113-9 [11] Guo, D.: A class of second-order impulsive integro-differential equations on unbounded domain in a Banach space. Appl. Math. Comput. 125, 59–77 (2002) · Zbl 1030.45011 · doi:10.1016/S0096-3003(00)00115-6 [12] Yan, B., Liu, Y.: Unbounded solutions of the singular boundary value problems for second order differential equations on the half-line. Appl. Math. Comput. 147, 629–644 (2004) · Zbl 1045.34009 · doi:10.1016/S0096-3003(02)00801-9 [13] Agarwal, R.P., Meehan, M., O’Regan, D.: Fixed Point Theory and Applications. Cambridge University Press, Cambridge (2001) [14] Liu, Y.S.: Boundary value problem for second order differential equations on unbounded domain. Acta Anal. Funct. Appl. 4, 211–216 (2002)