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H. R. Marasi, H. Aydi, "Existence and Uniqueness Results for Two-Term Nonlinear Fractional Differential Equations via a Fixed Point Technique", Journal of Mathematics, vol. 2021, Article ID 6670176, 7 pages, 2021. https://doi.org/10.1155/2021/6670176
Existence and Uniqueness Results for Two-Term Nonlinear Fractional Differential Equations via a Fixed Point Technique
The work addressed in this paper is to ensure the existence and uniqueness of positive solutions for initial value problems for nonlinear fractional differential equations with two terms of fractional orders. By virtue of recent fixed point theorems on mixed monotone operators, we get some new straightforward results with a wide range of applications.
In recent years, we see a great interest to consider fractional differential equations in various kinds of studies. Namely, these equations have striking applications in diverse science. Moreover, during the past decades, new areas for applications of fractional differential and integral equations with initial boundary conditions have been appeared. In the modeling of significant numbers of phenomena occurring in engineering and scientific subjects, we encountered a problem of such kinds. In multiple cases, modeling by fractional operators can also be done better than modeling by ordinary derivative operators. One can see fractional operators in applications of rheology, control, viscoelasticity, porus structures, chemical physics, electrochemistry, and significant number of other branches of science (see [1–3] for more details). For these reasons, there exist many articles about the existence and uniqueness of solutions of problems containing operators of fractional orders and using fixed point techniques [4–27]. But, the number of research studies on the study of existence and uniqueness of solutions for FDEs with two or more terms of fractional operators is limited. Fujita  studied the following specific Cauchy problem:and established some existence and uniqueness results. In , the author considered an initial value problem of the following form:where , and studied existence and uniqueness of related solutions. Based on the results of the above paper, the following exacting conditions should be satisfied on for the existence of solutions:(C1) is a continuous and differentiable function on .(C2)For , and , where is any compact subinterval of .(C3)There exists a continuous function which satisfies where . To detect a unique solution other than (C1)–(C3), the following restriction should hold.(C4)Let and be any real numbers and be the continuous function appeared in (4), then The authors in [16, 30] have studied the same problem. In , we observe that the necessary conditions to exist at least one solution for (2) and (3) are (C1), (C2), and the following condition.(C5)For a continuous function , there exist positive numbers and such that Also, the additional condition that satisfies a Lipschitz condition is needed to prove the uniqueness of the solution: where is the Lipschitz constant. In , the following general form is considered:where and and are the fractional derivative orders in the Caputo definition and . We have used an index fixed point theorem and obtained some new results about existence and also multiplicity of solutions. In this paper, we consider the same general form (8) and (9) and try to use new fixed point theorems proved for operators having mixed monotone property to establish new results with a wide range of applications for existence and uniqueness.
For convenience of readers, we provide some useful definitions and previous results which we use throughout the paper.
Definition 1. (see [1, 3]). Let . The operator defined bywhere , and , is the fractional integral of of order in Riemann–Liouville definition.
After introducing the fractional integral operator, it is time to remind the fractional differentiation definitions. There are several ways to define fractional derivative. Here, we focus on the Caputo definition.
Definition 2. (see [1, 3]). Let where . If then the fractional derivative of in Caputo sense exists on the interval , almost everywhere, and is obtained byFor the sake of convenience, here, we mention some properties of Caputo fractional operator. For the power function taking the Caputo derivative yieldsBy virtue of this formula, one can deduce . Therefore, unlike Riemann–Liouville derivative, the Caputo derivative of a constant function is zero. In the following equality:the polynomial’s degree is less than or equal to , i. e, . From the following relation, it is clear that similar to the case of integer order, the fractional integral of the Caputo fractional derivative of order requires to know the values of the function and its integer order derivatives:The results of the paper about the existence and uniqueness are expressed by the famous beta function given as follows.
Definition 3. The following integral defines the beta function:It is known thatwhere denotes the Gamma function.
The following lemma transforms problems (8) and (9) to an integral equation [29, 30], at which the nonlinear term needs to satisfy the conditions (C1) and (C2).
Lemma 1 (see ). Let and . If (C1) and (C2) hold, then is a solution of the following problem:if and only ifwhere is a continuous function and satisfies the following integral equation:wherewithHere and anywhere below, the function is the Green function.
Lemma 2. Let , and . If (C1) and (C2) hold, then is a solution of (17) and (18) if and only ifwhere satisfies the following integral equation:Now, we present the following lemma which will be useful in proving an existence and uniqueness theorem of the solution of (8) and (9).
Lemma 3. For the Green function , the following relations hold:
Proof. By making use of (14), it is easy and straightforward to prove the properties for .
Now, we provide some definitions and a fixed point theorem involving mixed monotone operators. We suggest references [19, 20] for more details.
Definition 4. Let be a Banach space and be a nonempty closed convex subset. is a cone in if(I)For any and , then (II)For any , if then where denotes the zero element of the Banach space .
If there exists a constant such that, for any with , we have , then is called a normal cone. Then, we give the definition of a partially ordered Banach space. We say that is partially ordered by if iff , for any . For arbitrary , the ordered interval is defined by . Let be an operator. If implies , then is called increasing.
Let us briefly recall that the operator is increasing in its first variable and decreasing in its second variable, if from , one can deduce . If implies that , then the operator is called positive.
In the following definition, is a cone on the Banach space .
Definition 5. (see [19, 20]). Suppose is increasing in its first variable and decreasing in its second variable, then is called a mixed monotone operator.
If for an element , , then is called a fixed point of .
Theorem 1 (see ). Suppose is a mixed monotone operator and let the following conditions hold:(i)If , then there exists such that(ii)There exist two elements such that Then, has a unique fixed point in such that . Moreover, one can construct successively the following iterates: starting from to get the fixed point of . Also, as .
3. Main Results
In this section, we prove new existence and uniqueness results for nonlinear fractional differential equations with two terms of fractional derivatives. Compared with the results of the previous studies, our results are simple and straightforward. They are expressed with the values of beta and gamma functions. For these reasons, these results can be applied for a wide range of problems. To make effective our obtained result, some examples provided at the end of section cannot be considered by techniques of [16, 30]. In order to apply the fixed point results concerning the mixed monotone operators for the study of (8) and (9), we first consider as a suitable Banach space, and let the norm on be defined by
Let us consider
It is obvious that is a normal cone in .
In the following theorem, is the function appeared in the right side of equation (8).
Theorem 2. Assume that(C6) is a continuous function which is increasing in its first variable and decreasing in its second variable .(C7)For any , let be two arbitrary elements of so that there exists such that(C8)Let be two elements of and such that Then, for equation (8) with initial conditions (9), we have a unique solution . Furthermore, by computing the iterates, For , one can find the unique solution. In other words, as n.
Proof. First, we define the operator as follows:where is the Green function (21). Taking the definition of the operator into account and Lemma 3, it is clear that a certain function is a solution of (2) and (3) iff is a fixed point of , that is, . Hence, it will be sufficient to show that satisfies the conditions of Theorem 1. By virtue of (C6), it immediately follows that is increasing in its first variable and decreasing in its second variable . On the other hand, under the hypothesis of (C7), if and are two elements of , then there exists such thatAlso, with the help of (C8), and using Lemma 3, we can choose and such thatConsequently, it is obvious that all essentials of Theorem 1 are provided by (C6)–(C8), and therefore, there is a unique positive solution so that . It is the unique solution of (8) and (9). The proof is complete.
Example 1. Let us consider the following set of nonlinear differential equations of fractional ordersubject towhere is a positive integer, is a continuous function such that , and . In connection with Theorem 2, set and . By virtue of this theorem and Lemma 3, the existence of a unique solution for problems (37) and (38) is equivalent to a unique solution for the following integral equation:where with the help of Lemma 3, one can deduce . We note that if , defined by (30), then is increasing in and decreasing in . Now, let us define and choose . Then, considering that are constant, we can show thatTherefore, using Theorem 2, we conclude that problems (37) and (38) possess a unique positive solution such that .
Example 2. In this example, we find a unique positive solution for the nonlinear fractional initial value problems of the following form:where , is a continuous function such that , and . In connection with Theorem 2, set and . By virtue of this theorem and Lemma 3, the existence of a unique solution for problems (41) and (42) is equivalent to a unique solution for the following integral equation:where with the help of Lemma 3, one deduces . We note that if are defined by (30), then is increasing in and decreasing in .
For , consider ; then, we haveNow, let us define and choose . Then, regarding that are constant functions, we getTherefore, using Theorem 2, we conclude that for problems (38)–(41), there is a unique positive solution such that for each .
In this research, we considered nonlinear fractional differential equations of arbitrary order with two terms of fractional orders. By virtue of recent fixed point theorems on normal cones for mixed monotone operators, we obtained some new results for existence and uniqueness of solutions of these equations involving initial conditions, which are more applicable compared with previous existing results. The previous results can be proved based on very strict conditions on the function . While, in our paper, the function needs to be increasing in its first variable and decreasing in its second variable . Also, establishing the conditions (C7) and (C8) is not difficult. On the other hand, our results provided a constructive approach, based on an iterative relation, to find the solution. Indeed, the interval can be generalized to . The presented examples, at the end of the paper, cannot be considered by the previous works.
No data were used to support this study.
Conflicts of Interest
The authors declare no conflicts of interest.
All authors contributed equally and significantly in writing this article.
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