Abstract
In this paper, by using a fixed point result on ordered metric spaces, we prove the existence and uniqueness of a solution of the nonlinear fractional differential equation (, ) via the periodic boundary condition , where and is a continuous increasing function and denotes the Caputo fractional derivative of order α. Also, we solve it by using the anti-periodic boundary conditions with and with and separately.
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1 Introduction
Fractional calculus started to be used intensively as an important tool in several branches of science and engineering (see, for example, [1–20] and the references therein). This type of calculus has an important impact in describing the dynamics of complex phenomena [1–4, 17]. During the last few years, some new experimental confirmations have appeared in the literature in addition to the ones already established in chemistry, engineering, biology, physics, etc. As a result, the fractional differential equations were investigated intensively during the last few years. A special attention was devoted to the solvability of linear initial fractional differential equations on terms of special functions. On the other hand, the fixed point theory has wide applications in several fields (see Ref. [21] and the references therein) and it is continuously developing. Having these above mentioned things in mind, in this manuscript we have the main aim to prove the existence and uniqueness of a nonlinear fractional differential equation of Caputo type with the help of results obtained within ordered metric spaces. The paper is organized as follows. After the introductory part, in the second section, we present some of the basic tools needed in the rest of the manuscript. The third section is devoted to the main result as well as to the illustrative examples. Finally, the manuscript ends with our conclusions.
2 Basic tools
Recall that for a continuous function , the Caputo derivative of fractional order α is defined as follows.
Definition 2.1 The Caputo fractional derivative of order α for a continuous function f is defined by
Definition 2.2 The Riemann-Liouville fractional integral of order α is defined by
whenever the integral exists (see, for example, [2, 3] and [18]).
Theorem 2.1 Let be a complete ordered metric space, and let f be an increasing self-map on M. Suppose that there exist and such that and for all with . Assume that either f is continuous or M has the property that if is an increasing sequence and for some , then for all n. Also, suppose that for each , there exists such that z is comparable with x and y. Then f has a unique fixed point.
3 Main result
We study the existence and uniqueness of a solution for fractional differential equations with two types of boundary conditions.
3.1 Existence results for a nonlinear fractional differential equation in ordered metric spaces
Consider the nonlinear fractional differential equation:
via the boundary condition , where and is a continuous function. A function β is called a lower solution of the fractional boundary value problem whenever and β satisfies and . Let Ω denote the class of the functions satisfying the condition implies . Also, let Φ denote the class of increasing functions such that for all and . By using the following result of [21], we study the existence of solution for the above nonlinear fractional differential equation. One can compare it with the relation (6) in [13]. This nonlinear fractional differential equation has been solved for and different boundary value conditions [22]. Also, our technique in this section was used in [23] for solving another nonlinear fractional differential equation.
Lemma 3.1 Suppose that , and . Then the problem with the boundary value condition is equivalent to the fractional integral equation
where
Proof From , and boundary conditions, one can see that . By the definition of the fractional integral, we get
Thus,
This completes the proof. □
Now, we are ready to state and prove our main result.
Theorem 3.2 Consider the nonlinear fractional differential equation (, ) via the boundary condition , where , , is an increasing continuous function and for all with . If there exists a lower solution for the problem, then the problem has a unique solution.
Proof Note that by using Lemma 3.1, the problem is equivalent to the integral equation , where
Define by . Note that is a fixed point of F if and only if u is a solution of the problem. Let . Define the order ≤ on M by if and only if for all . Then is a complete ordered metric space, where . It is easy to check that F is an increasing self-map. Thus,
for all . Since the function ϕ is increasing and , we get
and so
for all . Now, let β be a lower solution for the problem. We show that . Since β is a lower solution for the problem, we get and , and so
for all . Since , we have . Thus, we obtain for all . Thus, . Note that the space has the property that if is an increasing sequence and for some , then for all n. Therefore by using Theorem 2.1, F has a unique fixed point. □
Example 3.1
Consider the nonlinear fractional differential equation
via the periodic boundary condition , where and the function is defined by . Note that
for all with . Then, by using Theorem 3.2, the existence of a lower solution for the problem provides the existence of a unique solution.
Example 3.2 Let be given. Consider the nonlinear fractional differential equation
via the boundary condition , where and the function is defined by . Note that
for all with . Then, by using Theorem 3.2, the existence of a lower solution for the problem provides the existence of a unique solution.
3.2 Existence result for a nonlinear fractional differential equation with the anti-periodic boundary condition
In this section, we solve the nonlinear fractional differential equation
via the boundary condition with , where and is an increasing continuous function and .
Lemma 3.3 Suppose that , and . Then the problem with the boundary value condition with is equivalent to the fractional integral equation , where
Proof It is easy to see that . Hence,
and so . Thus,
This completes the proof. □
Now, we present the next result.
Theorem 3.4 Consider the nonlinear fractional differential equation via the boundary condition with , where , , is an increasing continuous function and and for all with . If there exists a lower solution for the problem, then the problem has a unique solution.
Proof Note that by using Lemma 3.3, the problem is equivalent to the integral equation , where
Define by . Note that is a fixed point of F if and only if u is a solution of the problem. Let . Define the order ≤ on M by if and only if for all . Then is a complete ordered metric space, where . It is easy to check that F is an increasing self-map. Thus,
for all . Since the function ϕ is increasing and , we get , and so
for all . Now, let β be a lower solution for the problem. We show that . Since β is a lower solution for the problem, we get , and . Hence,
and so for all . Since , and
Thus,
and so
Hence, , that is, . Note that the space has the property that if is an increasing sequence and for some , then for all n. Therefore, by using Theorem 2.1, F has a unique fixed point. □
3.3 Existence result for a nonlinear fractional differential equation with another boundary condition
In this section, we solve the nonlinear fractional differential equation
via the boundary condition with and , where and is an increasing continuous function and .
Lemma 3.5 Suppose that , and . Then the problem with the boundary value condition with and is equivalent to the fractional integral equation , where
Proof It is easy to see that , and so
Hence, . Since , , and so
Thus,
This completes the proof. □
Now, we give the following result.
Theorem 3.6 Consider the nonlinear fractional differential equation via the boundary condition with and , where , , is an increasing continuous function and and for all with . If there exists a lower solution for the problem, then the problem has a unique solution.
Proof Note that by using Lemma 3.5, the problem is equivalent to the integral equation , where
Define by . Note that is a fixed point of F if and only if u is a solution of the problem. Let . Define the order ≤ on M by if and only if for all . Then is a complete ordered metric space, where . It is easy to check that F is an increasing self-map. Thus,
for all . Since the function ϕ is increasing and , we get , and so
for all . Now, let β be a lower solution for the problem. We show that . Since β is a lower solution for the problem, we get , and . Hence, , and so
for all . Since , . Thus,
and so . Hence, . Note that the space has the property that if is an increasing sequence and for some , then for all n. Therefore by using Theorem 2.1, F has a unique fixed point. □
Conclusions
In this manuscript, by making use of fixed point techniques on ordered metric spaces, the existence and uniqueness of the solution of a nonlinear fractional differential equation with periodic and anti-periodic boundary conditions were investigated. To sustain our results, two illustrative examples were analyzed in detail.
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Acknowledgements
This work is partially supported by the Scientific and Technical Research Council of Turkey. Research of the second and third authors was supported by Azarbaidjan Shahid Madani University.
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Baleanu, D., Mohammadi, H. & Rezapour, S. On a nonlinear fractional differential equation on partially ordered metric spaces. Adv Differ Equ 2013, 83 (2013). https://doi.org/10.1186/1687-1847-2013-83
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DOI: https://doi.org/10.1186/1687-1847-2013-83