1. Introduction
The stability theory of functional equations has advanced significantly in the last thirty years. This topic has benefited greatly from the contributions of others (see [
1,
2,
3,
4,
5,
6,
7]). Our findings are related to recent works [
3,
8] (where integral and differential equations are considered), as well as papers of [
9,
10] (where the Ulam–Hyers stability for operatorial equations and inclusions is examined). For more details on Ulam–Hyers stability (see [
11,
12,
13,
14,
15,
16,
17,
18,
19,
20,
21,
22,
23]).
In 2013, Rabha [
24] discussed different types of the generalized Ulam–Hyers stability for a univalent solution and studied the existence and uniqueness of a solution.
Since then, there have been many contributions in the form of generalization, refinement, and modification on this subject. In particular, in [
25], the authors studied Ulam-type stabilities for Volterra delay integrodifferential equations on a finite integral.
This paper aims to discuss different types of Ulam stability of the form
where
is the fractional derivative of
of order
and
2. Preliminaries
In this section, we outline a list of important notations, definitions, and lemmas that will be used in our main results.
Definition 1. Following this [26,27], we realize the Riemann–Liouville derivatives and integral of fractional order α as follows Definition 2. Equation (1) is Hyers–Ulam stable if s.t. and , with of Equation (1) s.t. Definition 3. Equation (1) is a generalized Hyers–Ulam stable if s.t. for of (3), of Equation (1), with Definition 4. Equation (1) is Ulam–Hyers–Rassias stable if s.t. and , with of Equation (1) s.t.where Remark 1. A function is a solution of (3) if (depends on ) s.t. - (a)
- (b)
Similarly, similar arguments apply to inequality (
4).
Remark 2. If satisfies (3), then is a solution to the following inequality Indeed, if satisfies (3), then by Remark 1, we get Similar estimates can also be obtained for the inequality (4). The following inequality is the key to obtaining our main results.
Lemma 1 (A generalized Gronwall lemma [
28]).
Let is a locally integrable function on , and is a nondecreasing continuous function on (constant), and suppose that is locally integrable on with Definition 5 ([
28]).
Assume that is a metric space. An operator is a Picard operator if s.t.- (a)
where is the fixed point set of
- (b)
converges to
Lemma 2 ([
28]).
Assume that is an ordered metric space and is an increasing Picard operator Then for while Definition 6 (Contraction principle). Every contraction in a complete metric space admits a unique fixed point.
3. Ulam Stabilities for Nonlinear Fractional Delay Differential
Equations
In this part, we are going to provide our results of Hyers–Ulam’s stability for Equation (
1).
Theorem 1. Equation (1) has a unique solution in and Ulam–Hyers–Rassias stable w.s.t. the function φ if - (A1)
and are continuous with the Lipschitz condition: and
- (A2)
- (A3)
is a positive continuous nondecreasing function and s.t.
Proof . (i) We first note that in view of (A1), Equation (
1) is equivalent to the following integral equations:
where
and
Consider the Banach space
with Chebyshev norm
and define the operator
by
Using the contraction principle, we show that
has a fixed point. In fact, it is clear that
Next, for any
, we get
As
the operator
is a contraction on the complete space
Hence,
has a fixed point
which is a solution of Equation (
1).
(ii) Assume that
is a solution to the inequality (
4). Denote by
the unique solution of the problem:
Then assumption (A1) allows to write the following integral equation (equivalent to the above problem):
If
satisfies the inequality (
4), then using assumption (A3) and Remarks 1 and 2, we obtain
Note that
for
Next, by the assumption (A1), Equation (
5) and the estimate in (
6), for any
we can write
According to above inequality, we consider the operator
defined by
Next, we prove that
G is a Picard operator. To this end, observe first that for any
we have
Now, using
we have
As
G is a contraction on
using Banach contraction principle,
G is a Picard operator and
Then, for
, we have
As
is increasing, then
for
and hence
Next, applying the inequality given in Lemma 1, we obtain
As
is positive and nondecreasing, then
So, clearly, if we put
then
Taking
then we get
For
the inequality (
7) leads to
So, we have proved that
is an increasing Picard operator for
and
Thus, applying the abstract Gromwell lemma (Lemma 2), we get
implying that
Hence, Equation (
1) is Ulam–Hyers–Rassias stable with respect to
□
Corollary 1. Assume that F and g in (1) satisfy the hypothesis (A1), (A2) and (A3). Then the problem (1) has a unique solution and Ulam–Hyers stable. Proof. By taking
in Theorem 1, we obtain
and the result follows. □
Remark 3. It is easy to show that (1) has generalized Ulam–Hyers stability by taking in Corollary 1. 4. Applications
In this section, we consider some important particular cases of the problem (
1).
Example 1. Let and Then we get the following special case of the problem (1): Now, consider the following inequality:where , ϵ and φ are as specified in Section 3. Using Theorem 1, we obtain the following result.
Theorem 2. If and satisfying (A1), (A2), and (A3), then the problem (8) has a unique solution and Ulam–Hyers–Rassias stable. Example 2. Let Then we get the specific case of the problem(1):which is an initial value problem for a nonlinear Volterra fractional delay integrodifferential equation. Consider the inequality:where ϵ and φ are as defined in Section 3. Applying Theorem 1, we arrive at the following result for the problem (
9).
Theorem 3. if and satisfying (A1), (A2), and (A3), then the problem (8) has a unique solution and Ulam–Hyers–Rassias stable. Other Ulam type stability results for Equation (
9) can be obtained by Using the corresponding results from
Section 3.
5. Examples
In this part, we offer two examples to demonstrate the key findings.
Example 3. Consider the non-linear fractional delay differential equations As and f is continuous Thus the Lipschitz constant is . Moreover, we have Thus, according to Corollary 1, (10) is Ulam–Hyers stable. Example 4. Consider the non-linear fractional delay differential equations As , f is continuous and . Now Thus, according to Corollary 1, (11) is Ulam–Hyers stable. 6. Conclusions
In this manuscript, we discussed different types of Ulam stability for the first-order nonlinear fractional delay differential equation in the problem (
1), using a generalized Gronwall’s inequality and Picard operator theory, we discussed some applications to illustrate the stability results obtained in the case of a finite interval. Our obtained results generalize those of Otrocol [
29] in the case take
7. Future Direction
It could be interesting to study different future types of Ulam stability for first-order nonlinear fractional Volterra integral equations. It is also interesting to discuss different future types of Ulam stability for the case of impulsive Volterra delay integrodifferential equations. Moreover we expect to get in these cases richer results with more attributes.
Author Contributions
All authors contributed equally. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
Not applicable.
Acknowledgments
Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R45), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
Conflicts of Interest
The authors declare no conflict of interest.
References
- Guo, D.; Lakshmikantham, V.; Liu, X. Nonlinear Integral Equations in Abstract Spaces; Kuwer: Dordrecht, The Netherlands, 1996; p. 373. [Google Scholar]
- Hyers, D.H.; Isac, G.; Rassias, T.M. Stability of Functional Equations in Several Variables; Progr. Nonlinear Differential Equations Appl., 34; Springer: Boston, MA, USA; Birkhäuser: Basel, Switzerland, 1998. [Google Scholar]
- Jung, S.M. A fixed point approach to the stability of a Volterra integral equation. Fixed Point Theory Appl. 2007, 2007, 57064. [Google Scholar] [CrossRef] [Green Version]
- Kolmanovskií, V.; Myshkis, A. Applied Theory of Functional-Differential Equations, Math. Appl. (Soviet Ser.); Kluwer: Dordrecht, The Netherlands, 1992; p. 85. [Google Scholar]
- Radu, V. The fixed point alternative and the stability of functional equations. Fixed Point Theory 2003, 4, 91–96. [Google Scholar]
- Rassias, T.M. On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72, 297–300. [Google Scholar] [CrossRef]
- Ulam, S.M. A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics; Interscience: New York, NY, USA; London, UK, 1960; p. 8. [Google Scholar]
- Castro, L.P.; Ramos, A. Hyers–Ulam–Rassias stability for a class of nonlinear Volterra integral equations. Banach J. Math. Anal. 2009, 3, 36–43. [Google Scholar] [CrossRef]
- Bota-Boriceanu, M.F.; Petrusel, A. Ulam-Hyers stability for operatorial equations. An. Stiintifice Ale Univ. 2011, 57, 65–74. [Google Scholar] [CrossRef]
- Petru, T.P.; Petruşel, A.; Yao, J.C. Ulam–Hyers stability for operatorial equations and inclusions via nonself operators. Taiwan. J. Math. 2011, 15, 2195–2212. [Google Scholar] [CrossRef]
- Brzdęk, J.; Eghbali, N. On approximate solutions of some delayed fractional differential equations. Appl. Math. Lett. 2016, 54, 31–35. [Google Scholar] [CrossRef]
- Huang, J.; Li, Y. Hyers-Ulam stability of linear functional differential equations. J. Math. Anal. Appl. 2015, 426, 1192–1200. [Google Scholar] [CrossRef]
- Jan, M.N.; Zaman, G.; Ahmad, I.; Ali, N.; Nisar, K.S.; Abdel-Aty, A.H.; Zakarya, M. Existence Theory to a Class of Fractional Order Hybrid Differential Equations. Fractals 2022, 30, 2240022. [Google Scholar] [CrossRef]
- Li, Y.; Shen, Y. Hyers Ulam stability of linear differential equations of second order. Appl. Math. 2010, 23, 306–309. [Google Scholar] [CrossRef] [Green Version]
- Li, T.; Zada, A.; Faisal, S. Hyers Ulam stability of nth order linear differential equations. J. Nonlinear Sci. 2016, 9, 2070–2075. [Google Scholar] [CrossRef]
- Miura, T.; Miyajima, S.; Takahasi, S.E. A characterization of Hyers-Ulam stability of first order linear differential operators. J. Math. Anal. 2003, 286, 136–146. [Google Scholar] [CrossRef] [Green Version]
- Rezapour, S.; Abbas, M.I.; Etemad, S.; Minh, D.N. On a multi-point p p-Laplacian fractional differential equation with generalized fractional derivatives. Math. Methods Appl. Sci. 2022; in press. [Google Scholar]
- Rezapour, S.; Souid, M.S.; Bouazza, Z.; Hussain, A.; Etemad, S. On the fractional variable order thermostat model: Existence theory on cones via piece-wise constant functions. J. Funct. Spaces, 2022; in press. [Google Scholar]
- Saker, S.; Kenawy, M.; AlNemer, G.; Zakarya, M. Some fractional dynamic inequalities of Hardy’s type via conformable calculus. Mathematics 2020, 8, 434. [Google Scholar] [CrossRef] [Green Version]
- Tang, S.; Zada, A.; Faisal, S.; El-Sheikh, M.M.A.; Li, T. Stability of higher-order nonlinear impulsive differential equations. J. Nonlinear Sci. Appl. 2016, 9, 4713–4721. [Google Scholar] [CrossRef] [Green Version]
- Turab, A.; Mitrović, Z.D.; Savić, A. Existence of solutions for a class of nonlinear boundary value problems on the hexasilinane graph. Adv. Differ. Equ. 2021, 2021, 494. [Google Scholar] [CrossRef]
- Zada, A.; Ali, W.; Farina, S. Hyers Ulam stability of nonlinear differential equations with fractional integrable impulses. Math. Methods 2017, 40, 5502–5514. [Google Scholar] [CrossRef]
- Zada, A.; Faisal, S.; Li, Y. Hyers Ulam Rassias stability of non linear delay differential equations. J. Nonlinear Sci. 2017, 504, 510. [Google Scholar] [CrossRef] [Green Version]
- Ibrahim, R.W. Ulam Stability of Boundary Value Problem. J. Math. 2013, 37, 287–297. [Google Scholar]
- Kucche, K.D.; Shikhare, P.U. Ulam Stabilities for Nonlinear Voltera Delay Integro-differential Equations. J. Contemp. Math. Anal. 2019, 54, 276–287. [Google Scholar] [CrossRef]
- Podlubny, I. Fractional Differential Equations; Academic Press: San Diego, CA, USA, 1999. [Google Scholar]
- Srivastava, H.M.; Owa, S. Univalent Functions, Fractional Calculus, and Their Applications; Halsted Press: Sydney, Australia; John Wiley and Sons: New York, NY, USA; Chichester, UK; Brisban, Australia; Toronto, ON, Canada, 1989. [Google Scholar]
- Ye, H.; Gao, J.; Ding, Y. A generalized Gronwall inequality and its application to a fractional differential equation. J. Math. Anal. Appl. 2007, 328, 1075–1081. [Google Scholar] [CrossRef] [Green Version]
- Otrocol, D.; Ilea, V. Ulam stability for a delay differential equation. Cent. Eur. J. Math. 2013, 11, 1296–1303. [Google Scholar] [CrossRef] [Green Version]
| Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |
© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).