- Open Access
Nonlocal Hadamard fractional integral conditions for nonlinear Riemann-Liouville fractional differential equations
© Tariboon et al.; licensee Springer. 2014
Received: 3 August 2014
Accepted: 24 November 2014
Published: 9 December 2014
In this paper, we introduce a new class of boundary value problems consisting of a fractional differential equation of Riemann-Liouville type, , , subject to the Hadamard fractional integral conditions , . Existence and uniqueness results are obtained by using a variety of fixed point theorems. Examples illustrating the results obtained are also presented.
MSC: 34A08, 34B15.
where , is the standard Riemann-Liouville fractional derivative of order q, is the Hadamard fractional integral of order , , , and , are real constants such that .
Several interesting and important results concerning existence and uniqueness of solutions, stability properties of solutions, analytic and numerical methods of solutions for fractional differential equations can be found in the recent literature on the topic and the search for more and more results is in progress. Fractional-order operators are nonlocal in nature and take care of the hereditary properties of many phenomena and processes. Fractional calculus has also emerged as a powerful modeling tool for many real world problems. For examples and recent development of the topic, see –. However, it has been observed that most of the work on the topic involves either Riemann-Liouville or Caputo type fractional derivatives. Besides these derivatives, the Hadamard fractional derivative is another kind of fractional derivative that was introduced by Hadamard in 1892 . This fractional derivative differs from the other ones in the sense that the kernel of the integral (in the definition of the Hadamard derivative) contains a logarithmic function of an arbitrary exponent. For background material of the Hadamard fractional derivative and integral, we refer to , –.
In the present paper we initiate the study of boundary value problems like (1.1)-(1.2), in which we combine Riemann-Liouville fractional differential equations subject to the Hadamard fractional integral boundary conditions. The key tool for this combination is Property 2.25 from , p.113. To the best of the authors’ knowledge this is the first paper dealing with the Riemann-Liouville fractional differential equation subject to Hadamard type integral boundary conditions.
Several new existence and uniqueness results are obtained by using a variety of fixed point theorems. Thus, in Theorem 3.1 we present an existence and uniqueness result via Banach’s fixed point theorem, while in Theorems 3.2 and 3.3 we give two other existence and uniqueness results via Banach’s fixed point theorem and Hölder inequality and nonlinear contractions, respectively. In the sequel existence results are obtained in Theorem 3.4, via Krasnoselskii’s fixed point theorem, in Theorem 3.5 via Leray-Schauder’s nonlinear alternative and finally in Theorem 3.7 via Leray-Schauder’s degree theory. Examples illustrating the results obtained are also presented.
In this section, we introduce some notations and definitions of fractional calculus and present preliminary results needed in our proofs later.
where , denotes the integer part of a real number q. Here Γ is the Gamma function defined by .
where , provided the integral exists.
provided the integral exists.
where, , and.
where, , and.
3 Main results
where for .
3.1 Existence and uniqueness result via Banach’s fixed point theorem
(H1):there exists a constantsuch that, for eachand.
We transform the BVP (1.1)-(1.2) into a fixed point problem, , where the operator is defined as in (3.1). Observe that the fixed points of the operator are solutions of problem (1.1)-(1.2). Applying the Banach contraction mapping principle, we shall show that has a unique fixed point.
which implies that .
which implies that . As , is a contraction. Therefore, we deduce, by the Banach contraction mapping principle, that has a fixed point which is the unique solution of the boundary value problem (1.1)-(1.2). The proof is completed. □
3.2 Existence and uniqueness result via Banach’s fixed point theorem and Hölder inequality
Suppose that: is a continuous function satisfying the following assumption:
(H2):, for, and, .
It follows that is contraction mapping. Hence Banach’s fixed point theorem implies that has a unique fixed point, which is the unique solution of the boundary value problem (1.1)-(1.2). The proof is completed. □
3.3 Existence and uniqueness result via nonlinear contractions
(Boyd and Wong) 
Let E be a Banach space and letbe a nonlinear contraction. Thenhas a unique fixed point in E.
Letbe a continuous function satisfying the assumption:
(H3):, for, , whereis continuous andthe constant defined by
Note that the function Ψ satisfies and for all .
This implies that . Therefore is a nonlinear contraction. Hence, by Lemma 3.1 the operator has a unique fixed point which is the unique solution of the boundary value problem (1.1)-(1.2). This completes the proof. □
3.4 Existence result via Krasnoselskii’s fixed point theorem
(Krasnoselskii’s fixed point theorem) 
Let M be a closed, bounded, convex, and nonempty subset of a Banach space X. Let A, B be the operators such that (a) whenever; (b) A is compact and continuous; (c) B is a contraction mapping. Then there existssuch that.
Letbe a continuous function satisfying (H1). In addition we assume that:
(H4):, , and.
This shows that . It is easy to see using (3.5) that is a contraction mapping.
Now we prove the compactness of the operator .
which is independent of x and tend to zero as . Thus, is equicontinuous. So is relatively compact on . Hence, by Arzelá-Ascoli’s theorem, is compact on . Thus all the assumptions of Lemma 3.2 are satisfied. So the conclusion of Lemma 3.2 implies that the boundary value problem (1.1)-(1.2) has at least one solution on . □
3.5 Existence result via Leray-Schauder’s nonlinear alternative
(Nonlinear alternative for single valued maps) 
either has a fixed point in , or
there is a (the boundary of U in C) and with .
where Ω is defined by (3.2).
As , the right-hand side of the above inequality tends to zero independently of . Therefore by Arzelá-Ascoli’s theorem the operator is completely continuous.
We see that the operator is continuous and completely continuous. From the choice of U, there is no such that for some . Consequently, by the nonlinear alternative of Leray-Schauder type, we deduce that has a fixed point which is a solution of the boundary value problem (1.1)-(1.2). This completes the proof. □
3.6 Existence result via Leray-Schauder’s degree theory
Letbe a continuous function. Suppose that
where Ω is defined by (3.2).
If , inequality (3.8) holds. This completes the proof. □
In this section, we present some examples to illustrate our results.
Thus . Hence, by Theorem 3.1, the boundary value problem (4.1) has a unique solution on .
Hence, by Theorem 3.2, the boundary value problem (4.2) has a unique solution on .
Hence, by Theorem 3.3, the boundary value problem (4.3) has a unique solution on .
Hence, by Theorem 3.4, the boundary value problem (4.4) has at least one solution on .
which implies that . Hence, by Theorem 3.6, the boundary value problem (4.5) has at least one solution on .
Hence, by Theorem 3.7, the boundary value problem (4.6) has at least one solution on .
Sotiris K Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
This research was funded by King Mongkut’s University of Technology North Bangkok, Thailand. Contract no. KMUTNB-GEN-58-09.
- Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.Google Scholar
- Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.Google Scholar
- Baleanu D, Diethelm K, Scalas E, Trujillo JJ: Fractional Calculus Models and Numerical Methods. World Scientific, Boston; 2012.Google Scholar
- Agarwal RP, Zhou Y, He Y: Existence of fractional neutral functional differential equations. Comput. Math. Appl. 2010, 59: 1095-1100. 10.1016/j.camwa.2009.05.010MathSciNetView ArticleGoogle Scholar
- Baleanu D, Mustafa OG, Agarwal RP:On -solutions for a class of sequential fractional differential equations. Appl. Math. Comput. 2011, 218: 2074-2081. 10.1016/j.amc.2011.07.024MathSciNetView ArticleGoogle Scholar
- Ahmad B, Nieto JJ: Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions. Bound. Value Probl. 2011., 2011: 10.1186/1687-2770-2011-36Google Scholar
- Ahmad B, Ntouyas SK, Alsaedi A: New existence results for nonlinear fractional differential equations with three-point integral boundary conditions. Adv. Differ. Equ. 2011., 2011: 10.1155/2011/107384Google Scholar
- O’Regan D, Stanek S: Fractional boundary value problems with singularities in space variables. Nonlinear Dyn. 2013, 71: 641-652. 10.1007/s11071-012-0443-xMathSciNetView ArticleGoogle Scholar
- Ahmad B, Ntouyas SK, Alsaedi A: A study of nonlinear fractional differential equations of arbitrary order with Riemann-Liouville type multistrip boundary conditions. Math. Probl. Eng. 2013., 2013:Google Scholar
- Ahmad B, Nieto JJ: Boundary value problems for a class of sequential integrodifferential equations of fractional order. J. Funct. Spaces Appl. 2013., 2013: 10.1155/2013/149659Google Scholar
- Zhang L, Ahmad B, Wang G, Agarwal RP: Nonlinear fractional integro-differential equations on unbounded domains in a Banach space. J. Comput. Appl. Math. 2013, 249: 51-56. 10.1016/j.cam.2013.02.010MathSciNetView ArticleGoogle Scholar
- Liu X, Jia M, Ge W: Multiple solutions of a p -Laplacian model involving a fractional derivative. Adv. Differ. Equ. 2013., 2013: 10.1186/1687-1847-2013-126Google Scholar
- Sudsutad W, Tariboon J: Boundary value problems for fractional differential equations with three-point fractional integral boundary conditions. Adv. Differ. Equ. 2012., 2012: 10.1186/1687-1847-2012-93Google Scholar
- Sudsutad W, Tariboon J: Existence results of fractional integro-differential equations with m -point multi-term fractional order integral boundary conditions. Bound. Value Probl. 2012., 2012: 10.1186/1687-2770-2012-94Google Scholar
- Hadamard J: Essai sur l’étude des fonctions données par leur développement de Taylor. J. Math. Pures Appl. 1892, 8: 101-186.Google Scholar
- Butzer PL, Kilbas AA, Trujillo JJ: Compositions of Hadamard-type fractional integration operators and the semigroup property. J. Math. Anal. Appl. 2002, 269: 387-400. 10.1016/S0022-247X(02)00049-5MathSciNetView ArticleGoogle Scholar
- Butzer PL, Kilbas AA, Trujillo JJ: Fractional calculus in the Mellin setting and Hadamard-type fractional integrals. J. Math. Anal. Appl. 2002, 269: 1-27. 10.1016/S0022-247X(02)00001-XMathSciNetView ArticleGoogle Scholar
- Butzer PL, Kilbas AA, Trujillo JJ: Mellin transform analysis and integration by parts for Hadamard-type fractional integrals. J. Math. Anal. Appl. 2002, 270: 1-15. 10.1016/S0022-247X(02)00066-5MathSciNetView ArticleGoogle Scholar
- Kilbas AA: Hadamard-type fractional calculus. J. Korean Math. Soc. 2001, 38: 1191-1204.MathSciNetGoogle Scholar
- Kilbas AA, Trujillo JJ: Hadamard-type integrals as G -transforms. Integral Transforms Spec. Funct. 2003, 14: 413-427. 10.1080/1065246031000074443MathSciNetView ArticleGoogle Scholar
- Jarad F, Abdeljawad T, Baleanu D: Caputo-type modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2012., 2012: 10.1186/1687-1847-2012-142Google Scholar
- Gambo YY, Jarad F, Baleanu D, Abdeljawad T: On Caputo modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2014., 2014: 10.1186/1687-1847-2014-10Google Scholar
- Boyd DW, Wong JSW: On nonlinear contractions. Proc. Am. Math. Soc. 1969, 20: 458-464. 10.1090/S0002-9939-1969-0239559-9MathSciNetView ArticleGoogle Scholar
- Krasnoselskii MA: Two remarks on the method of successive approximations. Usp. Mat. Nauk 1955, 10: 123-127.MathSciNetGoogle Scholar
- Granas A, Dugundji J: Fixed Point Theory. Springer, New York; 2003.View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd.Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.