Abstract
This article studies the existence theory of an innovation type of generalized proportional fractional differential equations with the assistance of the technique of Kuratowski measure on noncompactness combined with the fixed-point theorem of Mönch. Also, we use Lebesgue’s dominated convergence theorem and Arzelá–Ascoli fixed point theorem on existence and uniqueness results. An application is also presented by employing two illustrative iks, which enrich our outcomes.
1. Introduction
The idea of a measure of noncompactness was presented by Kuratowski in 1958 for the first time. In fact, for any bounded subset of a metric space , the denoted is defined to be the infimum of numbers such that can be covered by a finite number of sets of diameter less than .
The first who used the index to the nonlinear analysis was Darbo [1]. The technique is a very useful tool for the existence solution of integral equations, which is well presented by Banaś et al. [2]. In 2008, Benchohra et al. considered the for a nonlinear fractional differential equationwhere is the Caputo fractional derivative of order and is a given function [3]. Some authors have been utilized comparative strategies through K technique to diverse sorts of s [4,5]. In 2016, Kiataramkul et al studied the generalized Langevin and Sturm–Liouville fractional differential equations (L-SLs) of Hadamard type, with antiperiodic boundary conditions ( s) of the formwhere denotes the Caputo-type Hadamard fractional derivative of order , is continuous, with , and [6].
In this manuscript, we consider an innovation class of s by combining Langevin and Sturm–Liouville fractional differential equations (L-SL s). This work discusses the existence theory of the generalized proportional fractional differential equations with the help of the technique of K combined with the Mönch fixed-point theorem. To be more precise, we initiate the study of the existence, uniqueness, and different types of Ulam stability for the following generalized proportional Langevin and Sturm–Liouville fractional differential problems ( L-SL s) with variable coefficients (s) and s
So that denotes the generalized proportional Caputo fractional derivatives of order , , , is continuous, with , and . Our proposed method is essentially based on the result given by Banaś et al. [2]. Therefore, it is emphasized that the K technique is implemented for the first time on the L-SL s.
This article is arranged as follows: in Section 2, we provide the definitions and preliminary lemmas that we will use to verify our main theorems. In Section 3, we establish the existence and uniqueness of solutions to problem (3) by the K -method. In Section 4, we discuss some types of fractional . We give an application by presenting two examples to illustrate numerical findings in Section 5. Finally, we conclude our exposition.
2. Preliminary Notions
We consider the space with the norm
And denote by the class of all bounded mappings in . Regard as a Banach space of all measurable Bochner integrable mappings like as which is endowed with the norm
Now we look at the specifications of the concept of K .
Definition 1. (see [2, 7]). The mapping denoted by for , is named as the , ifwhere
Proposition 1 (see [2, 7]). The fulfills the following:(1) whenever , ;(2) iff is relatively compact;(3), where and represent the closure and the convex hull of , respectively;(4);(5).Let stands for the set of functions , , set , and
Theorem 1 (see [8–10]). The continuous self function on convex bounded admits a fixed-point whenever
Lemma 1 (see [10]). Let be convex bounded and closed, be Carathéodory, and so that and every bounded set ,From equicontinuous of , we haveHere .
Definition 2. (see [11]) The integral of of order isFor , where is Riemann–Liouville fractional integral and the Caputo type derivative of order iswhere .
Lemma 2 (see [11]). For the map , we have
Proposition 2 (see [11]). Let be such that and . Then, for any , we have(i);(ii);(iii).
3. Main Results
We investigate the given L-SL s (3) and present the solution’s characterization in relation to it.
Definition 3. By a solution of the L-SL s with s and s (3), we mean a measurable function such that ,And sare fulfilled on .
Lemma 3. Let and with . Then, the solution of the linear L-SL s with s and s and is given bywhere
Proof. Taking the - integral to q of (3.1), we getFrom the s of system (3.1), we obtainTaking the - integral to (3.4), we obtainUsing the condition of (18), we haveSo, we derive (19) by substituting (24), and the proof is ended.
Clearly, by using Lemma 2.7, the system (18) is immediately established whenever we apply the --derivative and --derivative to both sides of (19). On the basis of Lemma 3, the solutions of (3) are corresponding to -integral equation in the following format:
3.1. Existence Result via -Method
We consider the following hypotheses: (H1) fulfills the Carathéodory criterion; (H2) s.t. (H3) For any and each bounded measurable set , where is the and .
Then, let us set
Theorem 2. Let us (H1)-(H3) holds. IfwhereThen the system of GPFDEs (3) possesses a solution on .
Proof. We define the self-operator on by. Evidently, the fixed-point is a solution of the L-SL s (3). We defineWe shall follow the proof in three steps.
Step 1. Let be a sequence with in . Then , one may writeBy continuity and from (H1), the function tends uniformly to . In accordance with Lebesgue’s dominated convergence theorem, tends uniformly to , that is . Hence is sequentially continuous.
Step 2. Let is take , by (H2), and assume that , we obtain. By using the propertyFor , we haveHence, we getIndeed .
Step 3. By Step 2, the is bounded. Let , and , soThenBy using the propertyFor , we getAs , the right-hand side of inequality (42) tends to zero. This means that is equicontinuous. Now, let withSince is bounded and equicontinuous, the function is continuous on . By assumption (H2) and using the propertyFor , we get.Indeed,Equation (29) implies that or , . Hence is relatively compact in and so Arzelá–Ascoli theorem implies that is relatively compact in . Now, from Theorem 3, we conclude that the problem (3) has a solution which is a fixed point of operator .
3.2. Uniqueness Criterion
Theorem 3. Let the following assumptions hold: (H4) is a continuous function. (H5) there exists a nonnegative constant , such thatThen, problem (3) has a unique solution on , provided thatwhere and are given by (30) and (31), respectively.
Proof. In the first step, we show that , where the operator is defined by Equation (32), , and we choose a real number , such thatFor any , using (H5), we occurBy using the propertyFor and (H5) implies thatWhich implies after taking the norm on . Thus, maps into itself. For , we obtain by using the notations in (30) and (31) thatBy using the propertyFor , we haveConsequently, we getTherefore, is a contraction. Hence, the operator has a unique fixed point, which corresponds to a unique solution of the (3) on .
4. UH and UHR Stability of System
In the recent section, we are interested to study UH and UHR stability of the system (3). The system (3) is UH stable if such that, for each and for each satisfying
There exists a unique solution of (3) with
The system (3) is generalized UH stable (GUH) if there exists and such that for each and for each satisfying (57), there exists a unique solution of (3) with .
Remark 1. A function is a solution of inequality (57) if (which depends on solution ) such that, i) ; ii)
Theorem 4. Suppose that the condition and inequality (48) are fulfilled. Then, the solution of (3) is UH and GUH stable.
Proof. Let and let be a function that satisfies the inequality (57) and let the unique solution of the problemLemma 3 implies thatHence by Remark 1, we haveAgain Lemma 3 implies thatOn the other hand, we have, for each ,Hence using part 1 of Remark 1 and (H5), we can getIn consequence, it follows thatIf we let , then, the UH stability condition is satisfied. More generally, for the generalized UH stability condition is also satisfied. This completes the proof.
5. Application and Numerical Examples
The notations and terminologies in this section are adopted from [11–13]. In control theory, a proportional derivative controller for controller output at time has the following equation:where , , and are the proportional gain, the derivative gain, and the input deviation (or the error between the state variable and the process variable), respectively. Recent investigations have shown that has direct incorporation in the control of complex networks models (see [14] for more details).
Let us consider the continuous functions such that,
For and , for , , respectively. Then, Anderson et al. [21] defined the proportional derivative of order byHere , provided that the right-hand side exists and is a type of proportional gain is a type of derivative gain , is the error, and is the controller output (for more details refer to [15]). We next restrict ourselves to the case that
Therefore,
Clearly,
Thus, equation (72) is considered to be more general than the conformable derivative, which evidently does not tend to the original functions as tends to 0.
Now, we present some examples to illustrate our results.
Example 1. Based on (3), we consider an L-SL s asFor . ClearlyBy takingHypothesis (H1) is held and we havewhereBy using Equations (20), (30), and (31), we gothere , , and , andWe can see the results of , , and for in Table 1. These results are plotted in Figure 1. By using the MATLAB program (Algorithm 1) according to (29), we findThus from Theorem 2, the system of L-SL s (74) possesses a solution on .
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Example 2. By taking , , and , let us have the following system of L-SL s:For . We chooseAndSo, for and , we havewhere . By using Equations (20), (30), and (31), we gotHere , andTable 2 shows these results. Also, we can see from the graphical representation of , , and in Figures 2 and 3 whenever . By using the MATLAB program (Algorithm 2) according to (48), we findThus, from Theorem 3 we conclude that the system of L-SL s (83) has a unique solution on .
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6. Conclusion
With respect to the oddity of the display composition, no commitments exist, as far as we know, concerning the existence theory of the s (3) with the assistance of the strategy of K combined with the Mönch fixed-point theorem. As a result, the objective of this paper is to enhance this scholastic zone by means of modern procedures based on an uncommon idea of Kuratowski measures. Subsequently, the and stability were established for the proposed nonlinear L-SL s (3). We initiate the study of the existence, uniqueness, and different types of for the L-SL s (3) with s and s. Moreover, two examples were presented as an illustration of the obtained theory. With respect to another research venture, we are attending to proceed the investigation of such combined structures of physical and mathematical models by utilizing nonsingular fractional operators which provide more exact numerical results.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest with this study.
Authors’ Contributions
Abdellatif Boutiara contributed to actualization, methodology, formal analysis, validation, investigation, and initial draft. Mohammed K. A. Kaabar contributed to actualization, methodology, formal analysis, validation, investigation, initial draft, supervision of the original draft, and editing. Zailan Siri contributed to actualization, validation, methodology, formal analysis, investigation, and initial draft. Mohammad Esmael Samei contributed to actualization, methodology, formal analysis, validation, investigation, software, simulation, and the initial draft. Xiao-Guang Yue contributed to actualization, validation, methodology, formal analysis, investigation, and initial draft. All authors read and approved the final version.
Acknowledgments
The fourth author was supported by Bu-Ali Sina University.