Abstract
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.
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1 Introduction
In this paper, we concentrate on the study of existence and uniqueness of solutions for the nonlinear Riemann-Liouville fractional differential equation with nonlocal Hadamard fractional integral boundary conditions of the form
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 [1]–[14]. 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 [15]. 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 [2], [16]–[22].
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 [2], 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.
2 Preliminaries
In this section, we introduce some notations and definitions of fractional calculus and present preliminary results needed in our proofs later.
Definition 2.1
The Riemann-Liouville fractional derivative of order of a continuous function is defined by
where , denotes the integer part of a real number q. Here Γ is the Gamma function defined by .
Definition 2.2
The Riemann-Liouville fractional integral of order of a continuous function is defined by
Definition 2.3
The Hadamard derivative of fractional order q for a function is defined as
where , provided the integral exists.
Definition 2.4
The Hadamard fractional integral of order of a function , for all , is defined as
provided the integral exists.
Lemma 2.1
([2], p.113])
Letand. Then the following formulas:
hold.
Lemma 2.2
Letand. Then the fractional differential equation
has a unique solution
where, , and.
Lemma 2.3
Let. Then forwe have
where, , and.
Lemma 2.4
Let, , , , , , and. Then the nonlocal Hadamard fractional integral problem for the nonlinear Riemann-Liouville fractional differential equation
subject to the boundary conditions
has a unique solution given by
where
Proof
Using Lemmas 2.2-2.3, (2.1) can be expressed as an equivalent integral equation
for . The first condition of (2.2) implies that . Taking the Hadamard fractional integral of order for (2.5) and using the property of the Hadamard fractional integral , we get
The second condition of (2.2) implies that
Thus,
Substituting the values of and in (2.5), we obtain the solution (2.3). □
3 Main results
Throughout this paper, for convenience, we use the following expressions:
for and
where for .
Let denote the Banach space of all continuous functions from to ℝ endowed with the norm defined by . As in Lemma 2.4, we define an operator by
It should be noticed that problem (1.1)-(1.2) has a solution if and only if the operator has fixed points.
In the following, for the sake of convenience, we set a constant
In the following subsections we prove existence, as well as existence and uniqueness results, for the boundary value problem (1.1)-(1.2) by using a variety of fixed point theorems.
3.1 Existence and uniqueness result via Banach’s fixed point theorem
Theorem 3.1
Assume that:
(H1):there exists a constantsuch that, for eachand.
If
where Ω is defined by (3.2), then the boundary value problem (1.1)-(1.2) has a unique solution on.
Proof
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.
We let , and choose
Now, we show that , where . For any , we have
which implies that .
Next, we let . Then for , we have
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
Theorem 3.2
Suppose that: is a continuous function satisfying the following assumption:
(H2):, for, and, .
Denote. If
then the boundary value problem (1.1)-(1.2) has a unique solution.
Proof
For and for each , by Hölder’s inequality, we have
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
Definition 3.1
Let E be a Banach space and let be a mapping. is said to be a nonlinear contraction if there exists a continuous nondecreasing function such that and for all with the property:
Lemma 3.1
(Boyd and Wong) [23]
Let E be a Banach space and letbe a nonlinear contraction. Thenhas a unique fixed point in E.
Theorem 3.3
Letbe a continuous function satisfying the assumption:
(H3):, for, , whereis continuous andthe constant defined by
Then the boundary value problem (1.1)-(1.2) has a unique solution.
Proof
We define the operator as in (3.1) and the continuous nondecreasing function by
Note that the function Ψ satisfies and for all .
For any and for each , we have
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
Lemma 3.2
(Krasnoselskii’s fixed point theorem) [24]
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.
Theorem 3.4
Letbe a continuous function satisfying (H1). In addition we assume that:
(H4):, , and.
Then the boundary value problem (1.1)-(1.2) has at least one solution onprovided
Proof
Setting and choosing
(where Ω is defined by (3.2)), we consider . We define the operators and on by
For any , we have
This shows that . It is easy to see using (3.5) that is a contraction mapping.
Continuity of f implies that the operator is continuous. Also, is uniformly bounded on as
Now we prove the compactness of the operator .
We define , and consequently we have
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
Theorem 3.5
(Nonlinear alternative for single valued maps) [25]
Let E be a Banach space, C a closed, convex subset of E, U an open subset of C and. Suppose thatis a continuous, compact (that is, is a relatively compact subset of C) map. Then:
-
(i)
either has a fixed point in , or
-
(ii)
there is a (the boundary of U in C) and with .
Theorem 3.6
Assume that:
(H5):there exist a continuous nondecreasing functionand a functionsuch that
(H6):there exists a constantsuch that
where Ω is defined by (3.2).
Then the boundary value problem (1.1)-(1.2) has at least one solution on.
Proof
Let the operator be defined by (3.1). Firstly, we shall show that maps bounded sets (balls) into bounded sets in. For a number , let be a bounded ball in . Then for we have
and consequently,
Next we will show that maps bounded sets into equicontinuous sets of. Let with and . Then we have
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.
Let x be a solution. Then, for , and following similar computations to the first step, we have
which leads to
In view of (H6), there exists M such that . Let us set
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
Theorem 3.7
Letbe a continuous function. Suppose that
(H5):there exist constantsandsuch that
where Ω is defined by (3.2).
Then the boundary value problem (1.1)-(1.2) has at least one solution on.
Proof
We define an operator as in (3.1). In view of the fixed point problem
We shall prove the existence of at least one solution satisfying (3.7). Set a ball , as
where a constant radius . Hence, we shall show that satisfies the condition
We set
As shown in Theorem 3.6 we see that the operator is continuous, uniformly bounded, and equicontinuous. Then, by Arzelá-Ascoli’s theorem, a continuous map defined by is completely continuous. If (3.8) holds, then the following Leray-Schauder degrees are well defined, and by the homotopy invariance of topological degree it follows that
where I denotes the unit operator. By the nonzero property of the Leray-Schauder degree, for at least one . Let us assume that for some and for all so that
which, on taking the norm and solving for , yields
If , inequality (3.8) holds. This completes the proof. □
4 Examples
In this section, we present some examples to illustrate our results.
Example 4.1
Consider the following nonlocal Hadamard fractional integral conditions for a nonlinear Riemann-Liouville fractional differential equation:
Here , , , , , , , , , , , , and . Since , (H1) is satisfied with . By using a Maple program, we can find that
Thus . Hence, by Theorem 3.1, the boundary value problem (4.1) has a unique solution on .
Example 4.2
Consider the following nonlocal Hadamard fractional integral conditions for a nonlinear Riemann-Liouville fractional differential equation:
Here , , , , , , , , , , , , , , and . Since , then (H2) is satisfied with and . By using a Maple program, we can show that
Hence, by Theorem 3.2, the boundary value problem (4.2) has a unique solution on .
Example 4.3
Consider the following nonlocal Hadamard fractional integral conditions for a nonlinear Riemann-Liouville fractional differential equation:
Here , , , , , , , , , , , , and . We choose and
Clearly,
Hence, by Theorem 3.3, the boundary value problem (4.3) has a unique solution on .
Example 4.4
Consider the following nonlocal Hadamard fractional integral conditions for a nonlinear Riemann-Liouville fractional differential equation:
Here , , , , , , , , , , , , , , , , , , and . Since , (H1) is satisfied with . By a Maple program, we show that
Clearly,
Hence, by Theorem 3.4, the boundary value problem (4.4) has at least one solution on .
Example 4.5
Consider the following nonlocal Hadamard fractional integral conditions for a nonlinear Riemann-Liouville fractional differential equation:
Here , , , , , , , , , , , , and . It is easy to verify that
Clearly,
Choosing and , we can show that
which implies that . Hence, by Theorem 3.6, the boundary value problem (4.5) has at least one solution on .
Example 4.6
Consider the following nonlocal Hadamard fractional integral conditions for a nonlinear Riemann-Liouville fractional differential equation:
Here , , , , , , , , , and . We can show that
Since
(H5) is satisfied with and such that
Hence, by Theorem 3.7, the boundary value problem (4.6) has at least one solution on .
Authors’ information
Sotiris K Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
References
Podlubny I: Fractional Differential Equations. Academic Press, San Diego; 1999.
Kilbas AA, Srivastava HM, Trujillo JJ: Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.
Baleanu D, Diethelm K, Scalas E, Trujillo JJ: Fractional Calculus Models and Numerical Methods. World Scientific, Boston; 2012.
Agarwal RP, Zhou Y, He Y: Existence of fractional neutral functional differential equations. Comput. Math. Appl. 2010, 59: 1095-1100. 10.1016/j.camwa.2009.05.010
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.2011.07.024
Ahmad B, Nieto JJ: Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions. Bound. Value Probl. 2011., 2011: 10.1186/1687-2770-2011-36
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: 10.1155/2011/107384
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-x
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:
Ahmad B, Nieto JJ: Boundary value problems for a class of sequential integrodifferential equations of fractional order. J. Funct. Spaces Appl. 2013., 2013: 10.1155/2013/149659
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.2013.02.010
Liu X, Jia M, Ge W: Multiple solutions of a p -Laplacian model involving a fractional derivative. Adv. Differ. Equ. 2013., 2013: 10.1186/1687-1847-2013-126
Sudsutad W, Tariboon J: Boundary value problems for fractional differential equations with three-point fractional integral boundary conditions. Adv. Differ. Equ. 2012., 2012: 10.1186/1687-1847-2012-93
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: 10.1186/1687-2770-2012-94
Hadamard J: Essai sur l’étude des fonctions données par leur développement de Taylor. J. Math. Pures Appl. 1892, 8: 101-186.
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)00049-5
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-X
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)00066-5
Kilbas AA: Hadamard-type fractional calculus. J. Korean Math. Soc. 2001, 38: 1191-1204.
Kilbas AA, Trujillo JJ: Hadamard-type integrals as G -transforms. Integral Transforms Spec. Funct. 2003, 14: 413-427. 10.1080/1065246031000074443
Jarad F, Abdeljawad T, Baleanu D: Caputo-type modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2012., 2012: 10.1186/1687-1847-2012-142
Gambo YY, Jarad F, Baleanu D, Abdeljawad T: On Caputo modification of the Hadamard fractional derivatives. Adv. Differ. Equ. 2014., 2014: 10.1186/1687-1847-2014-10
Boyd DW, Wong JSW: On nonlinear contractions. Proc. Am. Math. Soc. 1969, 20: 458-464. 10.1090/S0002-9939-1969-0239559-9
Krasnoselskii MA: Two remarks on the method of successive approximations. Usp. Mat. Nauk 1955, 10: 123-127.
Granas A, Dugundji J: Fixed Point Theory. Springer, New York; 2003.
Acknowledgements
This research was funded by King Mongkut’s University of Technology North Bangkok, Thailand. Contract no. KMUTNB-GEN-58-09.
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Tariboon, J., Ntouyas, S.K. & Sudsutad, W. Nonlocal Hadamard fractional integral conditions for nonlinear Riemann-Liouville fractional differential equations. Bound Value Probl 2014, 253 (2014). https://doi.org/10.1186/s13661-014-0253-9
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DOI: https://doi.org/10.1186/s13661-014-0253-9