2015
DOI: 10.1007/s40840-015-0126-0
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Fractional Boundary Value Problems with Integral and Anti-periodic Boundary Conditions

Abstract: In this paper, we consider a class of boundary value problems of fractional differential equations with integral and anti-periodic boundary conditions, which is a new type of mixed boundary condition. Using the contraction mapping principle, Krasnosel'skii fixed point theorem, and Leray-Schauder degree theory, we obtain some results of existence and uniqueness. Finally, several examples are provided for illustrating the applications of our theoretical analysis.

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Cited by 22 publications
(17 citation statements)
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“…Then, using some fixed point theorems such as the Banach contraction mapping principle and Schaefer's fixed point theorem, we prove the existence and uniqueness of solutions. Since the problem considered in [24] is a special case of our problem, the results of this work can be regarded as generalizations of those results.…”
Section: Resultsmentioning
confidence: 81%
See 1 more Smart Citation
“…Then, using some fixed point theorems such as the Banach contraction mapping principle and Schaefer's fixed point theorem, we prove the existence and uniqueness of solutions. Since the problem considered in [24] is a special case of our problem, the results of this work can be regarded as generalizations of those results.…”
Section: Resultsmentioning
confidence: 81%
“…Their equation in problem (2) is a generalization of the classical relaxation equation and governs some fractional relaxation processes. Analyzing the higher-order fractional differential equations like that in problem (1), some new research papers considered not only antiperiodic boundary conditions but also mixed-type boundary conditions which are composed of both integral and antiperiodic boundary conditions (see [21][22][23][24][25]). Xu [24] obtained new existence and uniqueness results for the following single-term fractional differential equations with integral and antiperiodic boundary conditions by means of the Krasnosel'skii fixed point theorem, contraction mapping principle, and Leray-Schauder degree theory:…”
Section: Introductionmentioning
confidence: 99%
“…Despite the fact that inside the start, fractional calculus had an advancement as a simply purely mathematical idea, in current quite a while its utilization had moreover unfurl into numerous fields such as physics, mechanics, chemistry, biology, engineering, bioengineering and electrochemistry, e.g., [1,2,3,4,5,6,7,8,9] and the references therein. So in the literature, several studies handled comparable topics to various operators, as an instance, Riemann-Liouville [10,11], Caputo [12,13], Erdelyi-Kober [14,15], generalized Caputo [16,17], Hilfer [2], generalized Hilfer [18], Hadamard [19,20], generalized Hadamard [21], Katugampola [22,23], generalized Katugampola [24], Caputo-Fabrizio [25], Atangana-Baleanu [26], etc.…”
Section: Introductionmentioning
confidence: 99%
“…In addition, with the application of anti-periodic boundary value problems in various mathematical models and physical processes has been widely applied, the integral boundaries are also widely used in heat conduction, chemical engineering, groundwater flow, thermoelasticity, plasma physics and other fields. As a result, more and more studies have been made on this kind of problems [10] [11] [12] (anti-periodic boundary value problems, anti-periodic boundary value problems with integral boundaries). However, the indefinite sign of solutions of nonlinear differential equations determines that some problems (anti-periodic boundary value problems and their generalizations) cannot be studied directly by the method of upper and lower solutions for monotone iteration.…”
Section: Introductionmentioning
confidence: 99%