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Gao and Chen AdvancesinDif f erenceEquations (2016) 2016:314 DOI 10.1186/s13662-016-1034-9RESEARCH Open AccessExistence of solutions for a class ofnonlinear higher-order fractional differentialequation with fractional nonlocalboundary conditionYabing Gao and Pengyu Chen ** Correspondence:chpengyu123@163.comDepartment of Mathematics,Northwest Normal University,Lanzhou, 730070, People’s Republicof ChinaAbstractIn this paper, we study the existence of solutions for a class of nonlinear higher-orderfractional dif f erential equation with fractional nonlocal boundary condition by usingthe monotone iterative technique based on the method of upper and lower solutionsand give a specif i c iterative equation about its solutions.MSC: 26A33; 34B10; 34B15Keywords: nonlinear fractional dif f erential equation; nonlocal boundary valueproblem; monotone iterative technique of upper and lower solutions1 IntroductionWe consider the existence of solutions for the following nonlinear fractional dif f erentialequation with nonlocal boundary value condition:???C D α? + u(t) = f(t,u(t),C D β ?? + u(t),C D β ?? + u(t),...,C D β n–?? +u(t)), ? < t < ?,u (j) (?) = ?, u (n–?) (?) = ρI γ? + u(?),j = ?,?,...,n–?,(?)where n – ? < α < n is a real number, n ≥ ?,C D α? + ,C D β i? + , i = ?,?,...,n – ?, i – ? < β i < i isthe standard Caputo fractional derivative, I γ? +is the standard Riemann-Liouville integral,? < γ, and ? < ρ < ?(n+γ). The nonlinear term f : [?,?]×R n → R is continuous.Theboundaryvalueproblemoffractionalequationshasemergedasanewbranchinthef i elds of dif f erential equations for their deep backgrounds. In recent years, it is popularand important because the subject of fractional calculus frequently appears in variousf i elds such as physics, chemistry, biology, economics, control theory, signal and imageprocessing, and blood f l ow phenomenon. For more details about fractional calculus andfractionaldif f erentialequations,wereferthereadertothemonographsbyMillerandRoss[?], Heikkila et al. [?], Podlubny [?], Hilfer [?], and Kilbas et al. [?], the survey by Agarwalet al. [?], and the papers [?–??]. Many scholars have studied the existence for nonlinearfractionaldif f erentialequationswithavarietyofboundaryconditions;see[??–??]andthereferences therein. However, sometimes it is better to impose integral conditions because© The Author(s) 2016. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/),whichpermitsunrestricteduse,distribution,andreproductioninanymedium,pro-vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, andindicate if changes were made.