Generalized linear differential equation using Hyers-Ulam stability approach

1.   Introduction

The challenge of stability with respect to the functional equation stemmed from an issue of Ulam ^[1] concerning the strength of gathering homomorphisms. Suppose $G_1$ is a group, and $G_2$ is a measurement group with metric $d(., .)$. Given $\in > 0$, does a $\delta > 0$ exist to such an extent that if a mapping $H:G_1 \rightarrow G_2$ fulfills the imbalance $d(H({\varkappa} \nu), H({\varkappa})H(\nu)) < \delta$ with respect to ${\varkappa}, \nu \in G_1$, and as such, there exists a homomorphism $h:G_1 \rightarrow G_2$ with $d(H({\varkappa}), h({\varkappa})) < {\epsilon}$ with respect to ${\varkappa} \in G_1$? If the mapping is almost a homomorphism, and as such, there exists a true homomorphism of ${s}$, what would be the error that could reasonably be expected?

The problem from the instance of roughly additive mappings was formulated by Hyers ^[2] with $G_1$ and $G_2$ as the Banach spaces. Then, Rassias (see ^[3]) summed up the effects of the study of Hyers. Since then, the dependability issues of practical conditions have been widely examined by researchers, e.g. see ^{[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]}).

Supposedly, the study of Ozawa ^[8] was among the first attempts on managing the $H - U$ stability of differential equations. In ^[5], the $H -U$ stability of differential condition ${\psi}'({\varkappa}) = {\psi}({\varkappa})$ was analyzed. Then, the studies of ^[18,19] have been further extended to the Banach space differential condition ${\psi}'({\varkappa}) = \lambda {\psi}({\varkappa})$. Utilizing a direct strategy, cycle technique, find point technique, and open mapping theorem, the $H-U$ stability of certain classes of useful fractional differential equations have been explored, e.g. see ^{[1,12,13,18,19,20]}).

In this paper, we investigate the Hyers-Ulam stability of linear differential equation of the fourth order. Specifically, ${\psi}$ is an interact arrangement of the following differential equation:

where ${\psi} \in c^4 [{\ell}, {\mu}], {\Psi} \in [{\ell}, {\mu}]$. We demonstrate that ${\psi}^{iv} ({\varkappa}) + {\xi}_1 {\psi}{'''} ({\varkappa})+ {\xi}_2 {\psi}{''} ({\varkappa}) + {\xi}_3 {\psi}' ({\varkappa}) + {\xi}_4 {\psi}({\varkappa}) = {\Psi}({\varkappa})$ has the Hyers-Ulam stability. A numerical example is provided to illustrate the proposed method.

Moreover, the effects of $H-U$ stability for the first order differential conditions were studied in ^[14,16,19]. These studies focused on the non-homogeneous straight differential equation of the first order, i.e.,

Jung ^[14] demonstrated the $H -U$ stability with respect to the differential condition of the following form:

and further applied the outcome to examine the $H-U$ stability of the following differential equation

Then, Wang, Zhon and Sun ^[20] examined the ${H-U}$ stability of the the first order linear differential condition, i.e.,

In this study, we study the following implication of $H-U$ stability.

Definition 1.1. We denote that Eq 1.2 has the $H-U$ Stability if there exists a steady ${\lambda} > 0$ with the accompanying property of: for every ${\epsilon} > 0, {\psi} \in c^2 [{\ell}, {\mu}]$, if

as such, there exists some ${u} \in c^2 [{\ell}, {\mu}]$ that fulfill:

such that $|{\psi}({\varkappa}) -u({\varkappa}) | < {\lambda} \epsilon$. We denote such ${\lambda}$ a $H-U$ stability constant for Eq 1.2.

Definition 1.2. We denote that the extension of Eq 1.2 has the $H-U$ stability, if there exists a steady ${\lambda} > 0$ with the accompanying property of: for every ${\epsilon} > 0, {\psi} \in c^3 [{\ell}, {\mu}]$, if

As such, there exist some ${u} \in c^3 [{\ell}, {\mu}]$ that fulfill

such that $|{\psi}({\varkappa}) -u({\varkappa}) | < {\lambda} \epsilon$. We denote such ${\lambda}$ a $H-U$ stability constant for Equation 1.6.

Definition 1.3. We denote that the extension of Eq 1.6 has the $H-U$ stability, if there exists a steady ${\lambda} > 0$ with the accompanying property of: for every ${\epsilon} > 0, {\psi} \in c^4 [{\ell}, {\mu}]$, if

as such, there exist some ${u} \in c^4 [{\ell}, {\mu}]$ that fulfill

such that $|{{\psi}}({\varkappa}) -u({\varkappa}) | < {\lambda} \epsilon$. We denote such ${\lambda}$ a $H-U$ stability constant for Eq 1.8.

2.   Main results

Now, the key results of this study are given in the following hypothesis.

Lemma 2.1. The differential equation $j$ ${\psi}^{iv} ({\varkappa}) + {\xi}_1 {\psi}{'''} ({\varkappa})+ {\xi}_2 {\psi}{''} ({\varkappa}) + {\xi}_3 {\psi}' ({\varkappa}) + {\xi}_4 {\psi}({\varkappa}) = {\Psi}({\varkappa})$ has the Hyers -Ulam stability, where ${\psi} \in c^4 [{\ell}, {\mu}]$ and ${\Psi} \in [{\ell}, {\mu}]$.

Proof. Assume that $u_1, u_2, u_3$, and $u_4$ are the roots of $\nu^4 + {\xi}_1 \nu^3 + p_2 \nu^2 + p_3 \nu + p_4 = 0$ with $q_1 = \mathbb{R} u_1, q_2 = \mathbb{R} u_2, q_3 = \mathbb{R} u_4$, and $q_4 = \mathbb{R} u_3$. Here $\mathbb{R}$ means the real parts.

Suppose $\epsilon > 0$ and ${\psi} \in c^4 [{\ell}, {\mu}]$

and let

we obtain

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. As such,

with respect to ${\varkappa} \in [{\ell}, {\mu}]$, it yields that

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. Utilizing the above condition, we obtain

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. Similarly, $g_1$ fulfills

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. Multiplying the above condition by $e^{-u_1 ({\varkappa} - {\ell})}$ yields

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. Without loss of generality, we accept that $u_1 > 1$; therefore

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. Integrating 2.7 from ${\varkappa}$ to ${\mu}$, we obtain

with respect to ${\varkappa} \in [{\ell}, {\mu}]$; therefore

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. The above condition yields

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. Multiplying 2.10 by $e^{u_1 ({\varkappa} - {\ell})}$ on both sides, we obtain

therefore

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. Let

then $\zeta({\varkappa})$ fulfills $\zeta'({\varkappa}) = u_1 \zeta ({\varkappa}) + {\Psi} ({\varkappa})$ with respect to ${\varkappa} \in [{\ell}, {\mu}]$. It satisfies the inequality of

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. If ${\xi} \neq 0$, then

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. If ${\xi} = 0$, then

with respect to ${\varkappa} {\epsilon} [{\ell}, {\mu}]$. Therefore

Theorem 2.2. The differential equation ${\psi}^{iv} ({\varkappa}) + {\xi}_1 {\psi}{'''} ({\varkappa})+ {\xi}_2 {\psi}{''} ({\varkappa}) + {\xi}_3 {\psi}' ({\varkappa}) + {\xi}_4 {\psi}({\varkappa}) = {\Psi}({\varkappa})$ has the $H - U$ stability, where ${\psi} \in c^4 [{\ell}, {\mu}]$ and ${\Psi} \in [{\ell}, {\mu}]$. Therefore

with respect to ${\varkappa} \in [{\ell}, {\mu}]$.

Proof. Similar to the proof of Lemma 2.1. Let $H({\varkappa}) = {\psi}'({\varkappa})-u_2 {\psi} ({\varkappa})$ by $H'({\varkappa}) = {\psi}{''}({\varkappa})-u_1 {\psi}' ({\varkappa})$ and let ${\epsilon} > 0; {\psi} \in c^4 [{\ell}, {\mu}]$.

In addition,

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. Therefore

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. Equivalently $H$ fulfills

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. Multiplying 2.19 by $e^{-u_4 ({\varkappa} - {\ell})}$ on both sides yields

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. Without loss of generality, we accept that $u_4 > 1$; therefore

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. Integrating 2.21 from ${\varkappa}$ to ${\mu}$, we obtain

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. Based on 2.22, we obtain

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. Multiplying 2.23 by the function $e^{-u_4 ({\varkappa} - {\ell})}$, we obtain

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. Based on 2.24, we obtain

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. Let ${\lambda} ({\varkappa}) = H({\mu})e^{-u_4 ({\varkappa} - {\mu})} - e^{u_4 {\varkappa}} \int_{{\varkappa}}^{{\mu}} \zeta ({s}) e^{-u_4 {s}} d {s}$ with respect to ${\varkappa} \in [{\ell}, {\mu}]$. Then

Therefore

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. If ${\psi} \neq 0$, then

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. If ${\psi} = 0$, then

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. Based on 2.16, we obtain

with respect to ${\varkappa} \in [{\ell}, {\mu}]$.

Theorem 2.3. The DE ${\psi}^{iv} ({\varkappa}) + {\xi}_1 {\psi}{'''} ({\varkappa})+ {\xi}_2 {\psi}{''} ({\varkappa}) + {\xi}_3 {\psi}' ({\varkappa}) + {\xi}_4 {\psi}({\varkappa}) = {\Psi}({\varkappa})$ has the Hyers Ulam stability, where ${\psi} \in c^4 [{\ell}, {\mu}]$ and with respect to ${\varkappa} \in [{\ell}, {\mu}]$, $|u({\varkappa}) - {\psi} ({\varkappa})| \leq T$

where

with respect to ${\varkappa} \in [{\ell}, {\mu}]$.

Proof. Based on Theorem 2.2, let us choose

by

Then

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. As such, we have

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. Equivalently, ${\psi}$ fulfills

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. Multiplying the condition by the function $e^{-u_3 ({\varkappa} - {\ell})}$

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. Without loss of generality, we accept that $u_3 > 1$. Then

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. Integrating 2.33 from ${\varkappa}$ to ${\mu}$, we obtain

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. Based on 2.34, we obtain

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. Again, multiplying the condition by function $e^{-u_3 ({\varkappa} - {\ell})}$ yields

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. Based on 2.36, we have

for all ${\varkappa} \in [{\ell}, {\mu}]$. Let $u_2 ({\varkappa}) = {\psi} ({\mu}) e^{-u_3 ({\varkappa} - {\mu})} - e^{u_3 {{\varkappa}}} \int_{{\varkappa}}^{{\mu}} {\lambda} ({s}) e^{-u_3 ({s} - {\ell})} d {s}$, then $u'_2 ({\varkappa}) - u_3 u_2 ({\varkappa}) - {\lambda} ({\varkappa}) = 0$ by $u'_2 ({\varkappa}) = u_3 u_2 ({\varkappa}) + {\lambda} ({\varkappa})$, for all ${\varkappa} \in [{\ell}, {\mu}]$. Therefore

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. If $\sigma \neq 0$, then

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. If $\sigma = 0$, then

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. Based on 2.40, we obtain

$|u({\varkappa}) - {\psi} ({\varkappa})| \leq T$, where

with respect to ${\varkappa} \in [{\ell}, {\mu}]$.

Theorem 2.4. The differential equation ${\psi}^{iv} ({\varkappa}) + {\xi}_1 {\psi}{'''} ({\varkappa})+ {\xi}_2 {\psi}{''} ({\varkappa}) + {\xi}_3 {\psi}' ({\varkappa}) + {\xi}_4 {\psi}({\varkappa}) = {\Psi}({\varkappa})$ has the Hyers Ulam stability, where ${\psi} \in c^4 [{\ell}, {\mu}]$ and with respect to ${\varkappa} \in [{\ell}, {\mu}]$, as such

$|{\Psi}({\varkappa}) - \zeta ({\varkappa})| \leq \Theta$, where

with respect to ${\varkappa} \in [{\ell}, {\mu}]$.

Proof. Similar to the proof of theorem 2.3, let $\in > 0$ and ${\psi} \in c^4 [{\ell}, {\mu}]$.

Consider

we obtain

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. As such

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. If follows from 2.42 that

So

for all ${\varkappa} \in [{\ell}, {\mu}]$. Equivalently, $\zeta$ fulfills

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. Multiplying the formula by the function $e^{-u_2 ({\varkappa} - {\ell})}$ satisfies

with respect to ${\varkappa} {\epsilon} [{\ell}, {\mu}]$. Without loss of generality, we accept that $u_2 > 1$. As such

for all ${\varkappa} \in [{\ell}, {\mu}]$. Integrating 2.45 from ${\varkappa}$ to ${\mu}$. As such

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. It follows from 2.48 that

for all ${\varkappa} \in [{\ell}, {\mu}]$. Multiplying the formula by the function $e^{-u_2 ({\varkappa} - {\ell})}$, we obtain

with respect to ${\varkappa} \in [{\ell}, {\mu}]$.

Let ${\Psi} ({\varkappa}) = \zeta ({\mu}) e^{-u_2 (2H} - e^{u_2 {{\varkappa}}} \int_{{\varkappa}}^{{\mu}} H({s}) e^{-u_2 ({s} - {\ell})} d {s}$. As such, ${\Psi}({\varkappa})$ satisfies ${\Psi}' ({\varkappa}) - u_2 {\Psi} ({\varkappa}) - H({\varkappa}) = 0$ by

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. As such,

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. If ${\nu} \neq 0$, then

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. If ${\nu} = 0$, then

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. If follows from 2.41 that

with respect to ${\varkappa} \in [{\ell}, {\mu}]$.

3.   Illustrative examples

Two examples to illustrate the results in this study are provided, as follows.

Example 3.1. Consider a differential equation of the form $\sigma^{iv} ({\varkappa})+ 2\sigma{'''} ({\varkappa}) + \sigma{''} ({\varkappa}) = {\Psi} ({\varkappa}); {\varkappa} \in [2, 3]$.

Suppose ${\epsilon} > 0$, as such

with respect to ${\varkappa} \in [2, 3]$. Suppose $\lambda = 1$, then

with respect to ${\varkappa} \in [2, 3]$. The conditions 2.16, 2.18 and 2.30 of Theorem 2.4 are satisfied. Therefore, there is a function ${\varkappa} \in c^4[2, 3]$, which is a mild solution of $u^{iv} ({\varkappa})+ 2u{'''}({\varkappa}) + u{''} ({\varkappa}) = {\Psi} ({\varkappa})$ that is satisfied by 2.54.

Example 3.2. Consider a differential equation of the form $\sigma^{iv} ({\varkappa})+ \sigma{'''} ({\varkappa}) + \sigma{''} ({\varkappa}) = {\Psi} ({\varkappa}); {\varkappa} \in [3, 2]$.

Suppose ${\epsilon} > 0$, and ${\psi}\in [3, 2]$, such that

with respect to ${\varkappa}\in [3, 2]$. We take

with respect to ${\varkappa}\in [3, 2]$. Then,

with respect to ${\varkappa} \in [3, 2]$. As such,

with respect to ${\varkappa} \in [{\ell}, {\mu}]$. The conditions 2.16, 2.18 and 2.30 of Theorem 2.4 are satisfied. Therefore, there is a function ${\varkappa} \in c^4[3, 2]$, which is a mild solution of $u^{iv} ({\varkappa})+ u{'''}({\varkappa}) + u{''} ({\varkappa}) = {\Psi} ({\varkappa})$ that is satisfied by 2.54.

4.   Conclusions

We have investigated the Hyers-Ulam stability with respect to the linear differential condition of fourth order in this study. The effectiveness of the proposed method has been illustrated in the examples.

5.   Acknowledgements

This research is made possible through financial support from the Phuket Rajabhat University, Thailand, and the Thailand Research Fund (RSA6280004). The authors are grateful to the Phuket Rajabhat University, Thailand, and the Thailand Research Fund (RSA6280004) for supporting this research. The authors are grateful to the referees for their valuable comments and suggestions.

Conflict of interest

The authors declares no conflict of interest.