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Mathematical Problems in Engineering
VolumeΒ 2011, Article IDΒ 128479, 15 pages
http://dx.doi.org/10.1155/2011/128479
Research Article

The Stability of Some Differential Equations

1Department of Mathematics, Chalous Branch, Islamic Azad University (IAU), Chalous 46615, Iran
2Department of Mathematics, National Technical University of Athens, Zografou Campus, 157 80 Athens, Greece
3Department of Mathematics, Science and Research Branch, Islamic Azad University (IAU), Tehran 14778, Iran
4Department of Mathematics and Computer Science, Amirkabir University of Technology, 424 Hafez Ave., Tehran, Iran

Received 17 October 2011; Accepted 17 November 2011

Academic Editor: Alexander P.Β Seyranian

Copyright Β© 2011 M. M. Pourpasha et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We generalize the results obtained by Jun and Min (2009) and use fixed point method to obtain the stability of the functional equation 𝑓(π‘₯+𝜎(𝑦))=𝐹[𝑓(π‘₯),𝑓(𝑦)], for a class of functions of a vector space into a Banach space where 𝜎 is an involution. Then we obtain the stability of the differential equations of the form π‘¦ξ…ž=𝐹[π‘ž(π‘₯),𝑃(π‘₯)𝑦(π‘₯)].

1. Introduction and Preliminary

The stability problem of functional equations originated from a question of Ulam [1] in 1940, concerning the stability of group homomorphisms.

The stability concept that was introduced by Rassias’ theorem [2] in 1978 provided a large influence to a number of mathematicians to develop the notion of what is known today by the term Hyers-Ulam-Rassias stability of the linear mapping. Since then, the stability of several functional equations has been extensively investigated by several mathematicians, see [3–5]. They have many applications in Information Theory, Physics, Economic Theory, and Social and Behavior Sciences.

In 1996, Isac and Rassias [6] were the first to use the fixed point methods to investigate the Hyers-Ulam-Rassias stability.

Let 𝑋 be a set. A function π‘‘βˆΆπ‘‹Γ—π‘‹β†’[0,∞] is called a generalized metric on 𝑋 if and only if 𝑑 satisfies

(1) 𝑑(π‘₯,𝑦)=0, if and only if π‘₯=𝑦,

(2) 𝑑(π‘₯,𝑦)=𝑑(𝑦,π‘₯), for all π‘₯,π‘¦βˆˆπ‘‹,

(3) 𝑑(π‘₯,𝑧)≀𝑑(π‘₯,𝑦)+𝑑(𝑦,𝑧), for all π‘₯,𝑦,π‘§βˆˆπ‘‹.

Note that the only substantial difference of the generalized metric from the metric is that the range of generalized metric includes the infinity.

We now introduce one of fundamental results of fixed point theory. For the proof, refer to [7]. For an extensive theory of fixed point theorems and other nonlinear methods, the reader is referred to the book of Hyers et al. [8].

Theorem 1.1. Let (𝑋,𝑑) be a generalized complete metric space. Assume that π½βˆΆπ‘‹β†’π‘‹ is a strictly contractive operator with the Lipschitz constant 0<𝐿<1. If there exists a nonnegative integer π‘˜ such that 𝑑(π½π‘˜+1𝑓,π½π‘˜π‘“)<∞ for some π‘“βˆˆπ‘‹, then the followings are true: (a) the sequence {𝐽𝑛𝑓} converges to a fixed point π‘“βˆ— of 𝐽,(b)π‘“βˆ— is the unique fixed point of 𝐽 inπ‘‹βˆ—=ξ€½ξ€·π½π‘”βˆˆπ‘‹βˆΆπ‘‘π‘˜ξ€Έξ€Ύπ‘“,𝑔<∞,(1.1)(c) if π‘”βˆˆπ‘‹βˆ—, then𝑑𝑔,π‘“βˆ—ξ€Έβ‰€11βˆ’πΏπ‘‘(𝐽𝑔,𝑔).(1.2)

2. Stability of the Generalized Functional Equations

The stability problem for a general equation of the form[]𝑓(𝐺(π‘₯,𝑦))=𝐻𝑓(π‘₯),𝑓(𝑦)(2.1) was investigated by Cholewa [9] in 1984. Indeed, Cholewa proved the superstability of the above equation under some additional assumptions on the functions and spaces involved.

Recently, Jung and Min [10] applied the fixed point method to the investigate the stability of functional equation[]𝑓(π‘₯+𝑦)=𝐹𝑓(π‘₯),𝑓(𝑦).(2.2)

In this section, we generalized the Jun and Min’s results and use fixed point approach to obtain the stability of the functional equation[]𝑓(π‘₯+𝜎(𝑦))=𝐹𝑓(π‘₯),𝑓(𝑦)(2.3) for a class of functions of a vector space into a Banach space where 𝜎 is an involution.

Theorem 2.1. Let 𝑋 and (π‘Œ,β€–β‹…β€–) be a vector space over 𝐾 and a Banach space over 𝐾, respectively. Let (𝑋×𝑋,β€–β‹…β€–2) be a Banach space over 𝐾. Assume that πΉβˆΆπ‘‹Γ—π‘‹β†’π‘Œ is a bounded linear transformation, whose norm is denoted by ‖𝐹‖, satisfying 𝐹(𝐹(𝑒,𝑒),𝐹(𝑣,𝑣))=𝐹(𝐹(𝑒,𝑣),𝐹(𝑒,𝑣))(2.4) for all 𝑒,π‘£βˆΆπ‘‹β†’π‘‹ and there exists a real number πœ…>0 with β€–(𝑒(π‘₯),𝑒(𝜎(π‘₯)))βˆ’(𝑣(π‘₯),𝑣(𝜎(π‘₯)))β€–2β‰€πœ…β€–π‘’(π‘₯)βˆ’π‘£(π‘₯)β€–(2.5) for all 𝑒,π‘£βˆΆπ‘‹β†’π‘‹. Moreover, assume that πœ‘βˆΆπ‘‹Γ—π‘‹β†’[0,∞) is a given function satisfying πœ‘(π‘₯,𝜎(𝑦))β‰€πœ‘(2π‘₯,2𝑦)(2.6) for all π‘₯,π‘¦βˆˆπ‘‹. If πœ…β€–πΉβ€–<1 and a function π‘“βˆΆπ‘‹β†’π‘Œ satisfies the inequality []‖𝑓(π‘₯+𝜎(𝑦))βˆ’πΉπ‘“(π‘₯),𝑓(𝑦)β€–β‰€πœ‘(π‘₯,𝑦)(2.7) for any π‘₯,π‘¦βˆˆπ‘‹, then there exists a unique solution π‘“βˆ—βˆΆπ‘‹β†’π‘Œ of (2.3) such that ‖𝑓(π‘₯)βˆ’π‘“βˆ—1(π‘₯)‖≀1βˆ’πœ…β€–πΉβ€–πœ‘(π‘₯,π‘₯).(2.8)

Proof. First, we denote by 𝑋 the set of all functions β„ŽβˆΆπ‘‹β†’π‘Œ and by d the generalized metric on 𝑋 defined as 𝑑[(𝑔,β„Ž)=inf𝐢∈0,∞)βˆΆβ€–π‘”(π‘₯)βˆ’β„Ž(π‘₯)β€–β‰€πΆπ‘€ξ…ž(π‘₯,𝑦),βˆ€π‘₯∈𝐸1ξ€Ύ.(2.9) By a similar method used at the proof of [4, Theorem 3.1], we can show that (𝑋,𝑑) is a generalized complete metric space. Now, let us define an operator π½βˆΆπ‘‹β†’π‘‹ by ξ‚ƒβ„Žξ‚€π‘₯(π½β„Ž)(π‘₯)=𝐹2ξ‚ξ‚€πœŽξ‚€π‘₯,β„Ž2(2.10) for every π‘₯βˆˆπ‘‹. We assert that 𝐽 is strictly contractive on 𝑋. Given 𝑔,β„Žβˆˆπ‘‹, let 𝐢∈[0,∞] be an arbitrary constant with 𝑑(𝑔,β„Ž)≀𝐢, that is, ‖𝑔(π‘₯)βˆ’β„Ž(π‘₯)β€–β‰€πΆπœ‘(π‘₯,𝑦)(2.11) for each π‘₯βˆˆπ‘‹. By (2.5), (2.6), (2.10), and (2.11), we have ‖‖‖‖𝐹𝑔π‘₯𝐽𝑔(π‘₯)βˆ’π½β„Ž(π‘₯)‖≀2ξ‚ξ‚€πœŽξ‚€π‘₯,𝑔2ξ‚ƒβ„Žξ‚€π‘₯ξ‚ξ‚ξ‚„βˆ’πΉ2ξ‚ξ‚€πœŽξ‚€π‘₯,β„Ž2‖‖‖‖‖‖𝑔π‘₯≀‖𝐹‖2ξ‚ξ‚€πœŽξ‚€π‘₯,𝑔2βˆ’ξ‚ƒβ„Žξ‚€π‘₯2ξ‚ξ‚€πœŽξ‚€π‘₯,β„Ž2‖‖‖‖‖‖𝑔π‘₯ξ‚ξ‚ξ‚„β‰€β€–πΉβ€–πœ…2π‘₯βˆ’β„Ž2‖‖‖π‘₯β‰€β€–πΉβ€–πœ…πΆπœ‘2,π‘₯2ξ‚β‰€β€–πΉβ€–πœ…πΆπœ‘(π‘₯,𝑦)(2.12) for every π‘₯βˆˆπ‘‹. Then, from (2.9) we have 𝑑(𝐽𝑔,π½β„Ž)β‰€πœ…β€–πΉβ€–π‘‘(𝑔,β„Ž) for any 𝑔,β„Žβˆˆπ‘‹, where πœ…β€–πΉβ€– is the Lipschitz constant with 0<πœ…β€–πΉβ€–<1. Thus, 𝐽 is strictly contractive.
Now, we claim that 𝑑(𝐽𝑓,𝑓)β‰€βˆž. Replacing π‘₯/2 by π‘₯ and 𝜎(π‘₯/2) by 𝑦 in (2.7), then it follows from (2.6) and (2.10) that ‖‖‖𝑓π‘₯2ξ‚€πœŽξ‚€π‘₯+𝜎2𝑓π‘₯ξ‚ξ‚ξ‚βˆ’πΉ2ξ‚ξ‚€πœŽξ‚€π‘₯,𝑓2β€–β€–β€–ξ‚€π‘₯ξ‚ξ‚ξ‚„β‰€πœ‘2ξ‚€π‘₯,𝜎2ξ‚€π‘₯‖𝑓(π‘₯)βˆ’(𝐽𝑓)(π‘₯)β€–β‰€πœ‘2ξ‚€π‘₯,𝜎2ξ‚ξ‚β‰€πœ‘(π‘₯,π‘₯)(2.13) for every π‘₯βˆˆπ‘‹. Then, 𝑑(𝐽𝑓,𝑓)≀1β‰€βˆž.(2.14) Now, it follows from Theorem 1.1(a) that there exists a function π‘“βˆ—βˆΆπΈ1→𝐸2 which is a fixed point of 𝐽, such that limπ‘›β†’βˆžπ‘‘ξ€·π½π‘›π‘“,π‘“βˆ—ξ€Έ=0.(2.15) From Theorem 1.1(c), we get 𝑑𝐽𝑛𝑓,π‘“βˆ—ξ€Έβ‰€111βˆ’πœ…β€–πΉβ€–π‘‘(𝐽𝑓,𝑓)≀1βˆ’πœ…β€–πΉβ€–ξ…ž(2.16) which implies the validity of (2.8). According to Theorem 1.1(b), π‘“βˆ— is the unique fixed point of 𝐽 with 𝑑(𝑓,π‘“βˆ—)<∞.
We now assert that β€–β€–(𝐽𝑛𝑓)(π‘₯+𝜎(𝑦))βˆ’πΉ(𝐽𝑛𝑓)(π‘₯),(𝐽𝑛𝑓‖‖≀))(𝑦)(πœ…β€–πΉβ€–π‘›πœ‘(π‘₯,π‘₯)(2.17) for all π‘›βˆˆπ‘ and π‘₯,π‘¦βˆˆπ‘‹. Indeed, it follows from (2.4), (2.5), (2.6), (2.7), and (2.10) that β€–[]β€–=‖‖‖𝐹𝑓(𝐽𝑓)(π‘₯+𝜎(𝑦))βˆ’πΉ(𝐽𝑓)(π‘₯),(𝐽𝑓)(𝑦)π‘₯+𝜎(𝑦)2ξ‚Άξ‚΅πœŽξ‚΅,𝑓π‘₯+𝜎(𝑦)2𝐹𝑓π‘₯ξ‚Άξ‚Άξ‚Ήβˆ’πΉ2ξ‚ξ‚€πœŽξ‚€π‘₯,𝑓2𝑓𝑦,𝐹2ξ‚ξ‚€πœŽξ‚€π‘¦,𝑓2‖‖‖‖‖‖𝑓≀‖𝐹‖π‘₯+𝜎(𝑦)2ξ‚Άξ‚΅πœŽξ‚΅,𝑓π‘₯+𝜎(𝑦)2βˆ’ξ‚ƒπΉξ‚ƒπ‘“ξ‚€π‘₯ξ‚Άξ‚Άξ‚Ή2ξ‚ξ‚€πœŽξ‚€π‘₯,𝑓2𝑓𝑦,𝐹2ξ‚ξ‚€πœŽξ‚€π‘¦,𝑓2β€–β€–β€–β€–β€–β€–π‘“ξ‚΅ξ‚ξ‚ξ‚„ξ‚„β‰€β€–πΉβ€–πœ…π‘₯+𝜎(𝑦)2ξ‚Άβˆ’ξ‚ƒπΉξ‚ƒπ‘“ξ‚€π‘₯2𝑦,𝑓2β€–β€–β€–ξ‚€π‘₯ξ‚ξ‚„ξ‚„β‰€β€–πΉβ€–πœ…πœ‘2,𝑦2ξ‚β‰€β€–πΉβ€–πœ…πœ‘(π‘₯,π‘₯)(2.18) for any π‘₯,π‘¦βˆˆπ‘‹. Then, it follows from (2.4), (2.5), (2.6), (2.10), and (2.17) that ‖‖𝐽𝑛+1𝑓𝐽(π‘₯+𝜎(𝑦))βˆ’πΉξ€Ίξ€·π‘›+1𝑓𝐽(π‘₯),𝑛+1𝑓‖‖=‖‖‖𝐹𝐽(𝑦)𝑛𝑓π‘₯+𝜎(𝑦)2ξ‚Ά,π½π‘›π‘“ξ‚΅πœŽξ‚΅π‘₯+𝜎(𝑦)2ξ‚ƒπΉξ‚ƒπ½ξ‚Άξ‚Άξ‚Ήβˆ’πΉπ‘›π‘“ξ‚€π‘₯2,π½π‘›π‘“ξ‚€πœŽξ‚€π‘₯2𝐽,𝐹𝑛𝑓𝑦2,π½π‘›π‘“ξ‚€πœŽξ‚€π‘¦2β€–β€–β€–β€–β€–β€–π½ξ‚ξ‚ξ‚„ξ‚„β‰€β€–πΉβ€–πœ…π‘›π‘“ξ‚΅π‘₯+𝜎(𝑦)2ξ‚Άβˆ’ξ‚ƒπΉξ‚ƒπ½π‘›π‘“ξ‚€π‘₯2,𝐽𝑛𝑓𝑦2‖‖‖≀(β€–πΉβ€–πœ…)𝑛+1πœ‘ξ‚€π‘₯2,𝑦2≀(β€–πΉβ€–πœ…)𝑛+1πœ‘(π‘₯,π‘₯)(2.19) for all π‘›βˆˆπ‘, which proves the validity of (2.17).
Finally, we prove that π‘“βˆ—(π‘₯+𝜎(𝑦))=𝐹[π‘“βˆ—(π‘₯),π‘“βˆ—(𝑦)] for any π‘₯,π‘¦βˆˆπ‘‹. Since 𝐹 is continuous as a bounded linear transformation, it follows from (2.15) and (2.17) that β€–β€–π‘“βˆ—ξ€Ίπ‘“(π‘₯+𝜎(𝑦))βˆ’πΉβˆ—(π‘₯),π‘“βˆ—ξ€»β€–β€–(𝑦)=limπ‘›β†’βˆžβ€–β€–β€–π½π‘›π‘“ξ‚΅π‘₯+𝜎(𝑦)2ξ‚Άβˆ’ξ‚ƒπΉξ‚ƒπ½π‘›π‘“ξ‚€π‘₯2,𝐽𝑛𝑓𝑦2‖‖‖≀limπ‘›β†’βˆž(β€–πΉβ€–πœ…)π‘›πœ‘(π‘₯,π‘₯)=0(2.20) for all π‘₯,π‘¦βˆˆπ‘‹, which implies that π‘“βˆ— is a solution of  (2.7).

Corollary 2.2. Let 𝑋 and (π‘Œ,β€–β‹…β€–) be a vector space over 𝐾 and a Banach space over 𝐾, respectively, and let (π‘ŒΓ—π‘Œ,β€–β‹…β€–2) be a Banach space over 𝐾. Assume that πΉβˆΆπ‘ŒΓ—π‘Œβ†’π‘Œ is a bounded linear transformation, whose norm is denoted by 𝐹, satisfying condition (2.4) and that there exists a real number πœ…>0 satisfying condition (2.5). If πœ…β€–πΉβ€–<1 and a function π‘“βˆΆπ‘‹β†’π‘Œ satisfies the inequality [𝑓]‖𝑓(π‘₯+𝑦)βˆ’πΉ(π‘₯),𝑓(𝑦)β€–β‰€πœ–β€–π‘₯‖𝑝+‖𝑦‖𝑝(2.21) for all π‘₯,π‘¦βˆˆπ‘‹ and for some nonnegative real constants πœƒ and 𝑝, then there exists a unique solution π‘“βˆ—βˆΆπ‘‹β†’π‘Œ of 1.2 such that ‖𝑓(π‘₯)βˆ’π‘“βˆ—(π‘₯)‖≀2πœƒ1βˆ’πœ…β€–πΉβ€–β€–π‘₯‖𝑝(2.22) for all π‘₯βˆˆπ‘‹.

Example 2.3. Assume that 𝑋=π‘Œ=β„‚, and consider the Banach spaces (β„‚,|β‹…|) and (β„‚Γ—β„‚,|β‹…|2), where we define |(𝑒(𝑑),𝑣(𝑑))|2=√|𝑒|2+|𝑣|2 for all 𝑒,π‘£βˆΆβ„‚β†’β„‚. Let 𝐴 and 𝐡 be fixed complex numbers with √|𝐴|+|𝐡|<1/2, and let πΉβˆΆβ„‚Γ—β„‚β†’β„‚ be a linear transformation defined by 𝐹(𝑒(𝑑),𝑣(𝑑))=𝐴𝑒(𝑑)+𝐡𝑣(𝑑).(2.23) Then it is easy to show that 𝐹 satisfies condition (2.13).
If 𝑒 and 𝑣 are complex numbers satisfying |(𝑒(𝑑),𝑣(𝑑))|2≀1 for all 𝑑, then ||||≀||𝐴||||𝐡||||𝐴||+||𝐡||𝐹(𝑒,𝑣)|𝑒|+|𝑣|≀.(2.24) Thus, we get ξ€½||𝐹||||||‖𝐹‖≀sup(𝑒,𝑣)βˆΆπ‘’,π‘£βˆˆβ„‚with(𝑒,𝑣)2≀||𝐴||+||𝐡||≀1,(2.25) which implies the boundedness of the linear transformation 𝐹.
On the other hand, we obtain β€–(𝑒(π‘₯),𝑒(𝜎(π‘₯)))βˆ’(𝑣(π‘₯),𝑣(𝜎(π‘₯)))β€–2β‰€βˆš2‖𝑒(π‘₯)βˆ’π‘£(π‘₯)β€–(2.26) for any 𝑒,π‘£βˆˆβ„‚, then we have βˆšβ€–πΉβ€–πœ…β‰€2ξ€·||𝐴||+||𝐡||≀1.(2.27) If the function π‘“βˆΆβ„‚β†’β„‚ satisfies the inequality β€–β€–π‘“βˆ—ξ€Ίπ‘“(π‘₯+𝜎(𝑦))βˆ’πΉβˆ—(π‘₯),π‘“βˆ—ξ€»β€–β€–(𝑦)(2.28) for all π‘₯,π‘¦βˆˆβ„‚ and for some πœ€>0, then Corollary 2.2 (with πœƒ=πœ€/2 and 𝑝=0) implies that there exists a unique function π‘“βˆ—βˆΆβ„‚β†’β„‚ such that β€–β€–π‘“βˆ—ξ€Ίπ‘“(π‘₯+𝜎(𝑦))=πΉβˆ—(π‘₯),π‘“βˆ—ξ€»β€–β€–(𝑦)(2.29) for all π‘₯,π‘¦βˆˆβ„‚ and ||π‘“βˆ—||β‰€πœ€(π‘₯)βˆ’π‘“(π‘₯)√1βˆ’2ξ€·||𝐴||+||𝐡||ξ€Έ(2.30) for any π‘₯βˆˆβ„‚.

3. Stability of the Generalized Differential Equations

Let π‘Œ be a normed space, and let 𝐼 be an open interval. Assume that for any function π‘¦βˆΆπΌβ†’π‘Œ satisfying the differential inequalityβ€–β€–π‘Žπ‘›(π‘₯)𝑦(𝑛)(π‘₯)+π‘Žπ‘›βˆ’1(π‘₯)𝑦(π‘›βˆ’1)(π‘₯)+β‹―+π‘Ž1(π‘₯)π‘¦ξ…ž(π‘₯)+π‘Ž0β€–β€–(π‘₯)𝑦(π‘₯)+β„Ž(π‘₯)β‰€πœ€(3.1) for all π‘₯∈𝐼 and for some πœ€β‰₯0, there exists a solution 𝑦0βˆΆπΌβ†’π‘Œ of the differential equationπ‘Žπ‘›(π‘₯)𝑦(𝑛)(π‘₯)+π‘Žπ‘›βˆ’1(π‘₯)𝑦(π‘›βˆ’1)(π‘₯)+β‹―+π‘Ž1(π‘₯)π‘¦ξ…ž(π‘₯)+π‘Ž0(π‘₯)𝑦(π‘₯)+β„Ž(π‘₯)=0(3.2) such that ‖𝑦(π‘₯)βˆ’π‘¦0(π‘₯)‖≀𝐾(πœ€) for any π‘₯∈𝐼, where 𝐾(πœ€) is an expression of πœ€ only. Then, we say that the above differential equation has the Hyers-Ulam stability.

If the above statement is also true when we replace πœ€ and 𝐾(πœ€) by πœ‘(π‘₯) and Ξ¦(π‘₯), where πœ‘,Ξ¦βˆΆπΌβ†’[0,∞) are functions not depending on 𝑦 and 𝑦0 explicitly, then we say that the corresponding differential equation has the Hyers-Ulam-Rassias stability (or the generalized Hyers-Ulam stability).

We may apply these terminologies for other differential equations. For more detailed definitions of the Hyers-Ulam stability and the Hyers-Ulam-Rassias stability, refer to [11, 12].

In 1998, Alsina and Ger investigated the Hyers-Ulam stability of differential equations. They proved in [13] that if a differentiable function π‘¦βˆΆπΌβ†’β„ satisfies the differential inequality |π‘¦ξ…ž(𝑑)βˆ’π‘¦(𝑑)|β‰€πœ€, where 𝐼 is an open subinterval of ℝ, then there exists a differentiable function 𝑦0βˆΆπΌβ†’β„ satisfying π‘¦ξ…ž0(𝑑)=𝑦0(𝑑) and |𝑦0(𝑑)βˆ’π‘¦(𝑑)|≀3πœ€ for any π‘‘βˆˆπΌ.

Alsina and Ger’s results have been generalized by Takahasi et al. [14]. They proved that the Hyers-Ulam stability holds for the Banach space-valued differential equation π‘¦ξ…ž(π‘₯)=πœ†π‘¦(π‘₯) (see also [15]).

Recently, Takahasi et al. also proved the Hyers-Ulam stability of linear differential equations of first order, π‘¦ξ…ž(π‘₯)+𝑔(π‘₯)𝑦(π‘₯)=0, where 𝑔(π‘₯) is a continuous function, and they also proved the Hyers-Ulam stability of linear differential equations of other types (see [16–18]).

In this section, for a bounded and continuous function 𝐹(π‘₯,𝑦), we will adopt the idea of CΜ†adariu and Radu [19, 20] and prove the Hyers-Ulam-Rassias stability as well as the Hyers-Ulam stability of the differential equations of the formπ‘¦ξ…ž(π‘₯)=𝐹(π‘ž(π‘₯),𝑝(π‘₯)𝑦(π‘₯)).(3.3)

Theorem 3.1. For given real numbers π‘Ž and 𝑏 with π‘Ž<𝑏, let 𝐼=[π‘Ž,𝑏] be a closed interval and choose π‘βˆˆπΌ. Let 𝐾 and 𝐿 be positive constants with 0<𝐾𝐿<1. Assume that πΉβˆΆπΌΓ—β„β†’β„ is a continuous function which satisfies a Lipschitz condition ||||||||𝐹(π‘₯,𝑦)βˆ’πΉ(π‘₯,𝑧)β‰€πΏπ‘¦βˆ’π‘§(3.4) for any π‘₯∈𝐼 and 𝑦,π‘§βˆˆβ„. If a continuously differentiable function π‘¦βˆΆπΌβ†’β„ satisfies ||π‘¦ξ…ž||(π‘₯)βˆ’πΉ(π‘ž(π‘₯),𝑝(π‘₯)𝑦(π‘₯))β‰€πœ‘(π‘₯)(3.5) for all π‘₯∈𝐼, where 𝑝(π‘₯),π‘ž(π‘₯) are continuous functions in which |𝑝(π‘₯)|≀𝑐 and πœ‘βˆΆπΌβ†’(0,∞) is a continuous function with ||||ξ€œπ‘₯π‘πœ‘||||(𝑑)π‘‘π‘‘β‰€πΎπœ‘(π‘₯)(3.6) for each π‘₯∈𝐼, then there exists a unique continuous function 𝑦0βˆΆπΌβ†’β„ such that 𝑦0ξ€œ(π‘₯)=𝑦(𝑐)+π‘₯𝑐𝐹(π‘ž(π‘₯),𝑝(π‘₯)𝑦(π‘₯))𝑑𝑑(3.7) (consequently, 𝑦0 is a solution to (2.15)) and ||𝑦(π‘₯)βˆ’π‘¦0||≀𝐾(π‘₯)πœ‘1βˆ’πΎπΏ(π‘₯)(3.8) for all π‘₯∈𝐼.

Proof. Let us define a set 𝑋 of all continuous functions π‘“βˆΆπΌβ†’β„ by 𝑋={π‘“βˆΆπΌβ†’β„βˆ£π‘“iscontinuous}(3.9) and introduce a generalized metric on 𝑋 as follows: 𝑑[||𝑓||ξ€Ύ(𝑓,𝑔)=inf𝐢∈0,∞)∢(π‘₯)βˆ’π‘”(π‘₯)β‰€πΆπœ‘(π‘₯),βˆ€π‘₯∈𝐼.(3.10) By a similar method used at the proof of [4, Theorem 3.1], we assert that (𝑋,𝑑) is complete. Let {β„Žπ‘›} be a Cauchy sequence in (𝑋,𝑑).
Then, for any πœ€>0, there exists an integer π‘πœ€>0 such that 𝑑(β„Žπ‘š,β„Žπ‘›)β‰€πœ€ for all π‘š,π‘›βˆˆπ‘πœ€. It further follows from (3.10) that βˆ€πœ€>0βˆƒπ‘πœ€βˆˆβ„•βˆ€π‘š,π‘›βˆˆπ‘πœ€||β„Žβˆ€π‘₯βˆˆπΌβˆΆπ‘š(π‘₯)βˆ’β„Žπ‘›||(π‘₯)β‰€πœ€πœ‘(π‘₯).(3.11)Equation (3.11) implies that {β„Žπ‘›(π‘₯)} is a Cauchy sequence in ℝ. Since ℝ is complete, {β„Žπ‘›(π‘₯)} converges for each π‘₯∈𝐼. Thus, we can define a function β„ŽβˆΆπΌβ†’β„ by β„Ž(π‘₯)=limπ‘›β†’βˆžβ„Žπ‘›(π‘₯).(3.12) Let π‘š increase to infinity, then by (3.11) we have βˆ€πœ€>0βˆƒπ‘πœ€βˆˆβ„•βˆ€π‘›βˆˆπ‘πœ€||βˆ€π‘₯βˆˆπΌβˆΆβ„Ž(π‘₯)βˆ’β„Žπ‘›||(π‘₯)β‰€πœ€πœ‘(π‘₯).(3.13)
Since πœ‘ is bounded on 𝐼, {β„Žπ‘›} converges uniformly to β„Ž. Hence, β„Ž is continuous and β„Žβˆˆπ‘‹.
Further, considering (3.10) and (3.13), then βˆ€πœ€>0βˆƒπ‘πœ€βˆˆβ„•βˆ€π‘›βˆˆπ‘πœ€ξ€·βˆΆπ‘‘β„Ž,β„Žπ‘›ξ€Έβ‰€πœ€.(3.14) Then, the Cauchy sequence {β„Žπ‘›} converges to β„Ž in (𝑋,𝑑). Hence, (𝑋,𝑑) is complete.
Now, define the operator π½βˆΆπ‘‹β†’π‘‹ by ξ€œ(𝐽𝑓)(π‘₯)=𝑦(𝑐)+π‘₯𝑐𝐹(π‘ž(π‘₯),𝑝(π‘₯)𝑓(π‘₯))𝑑𝑑π‘₯∈𝐼(3.15) for all π‘“βˆˆπ‘‹. (Indeed, according to the Fundamental Theorem of Calculus, 𝐽𝑓 is continuously differentiable on 𝐼, since 𝐹 and 𝑓 are continuous functions. Hence, we may conclude that π½π‘“βˆˆπ‘‹.) We prove that 𝐽 is strictly contractive on 𝑋. For any 𝑓,π‘”βˆˆπ‘‹, let πΆπ‘“π‘”βˆˆ[0,∞] be an arbitrary constant with 𝑑(𝑓,𝑔)≀𝐢𝑓𝑔, then, by (2.15), we have ||||𝑓(π‘₯)βˆ’π‘”(π‘₯)β‰€πΆπ‘“π‘”πœ‘(π‘₯)(3.16) for any π‘₯∈𝐼. It then follows from (3.4), (3.6), (3.10), (3.15), and (3.16) that ||||≀||||ξ€œ(𝐽𝑓)(π‘₯)βˆ’(𝐽𝑔)(π‘₯)π‘₯𝑐||||≀||||ξ€œ{𝐹(π‘ž(π‘₯),𝑝(π‘₯)𝑓(π‘₯))βˆ’πΉ(π‘ž(π‘₯),𝑝(π‘₯)𝑔(π‘₯))}𝑑𝑑π‘₯𝑐||||||||||||ξ€œπΉ(π‘ž(π‘₯),𝑝(π‘₯)𝑓(π‘₯))βˆ’πΉ(π‘ž(π‘₯),𝑝(π‘₯)𝑔(π‘₯))𝑑𝑑≀𝐿π‘₯𝑐||𝑓||||||(𝑑)βˆ’π‘”(𝑑)𝑑𝑑≀𝐿𝐢𝑓𝑔||||ξ€œπ‘₯𝑐||||πœ‘(𝑑)π‘‘π‘‘β‰€πΎπΏπΆπ‘“π‘”πœ‘(π‘₯)(3.17) for all π‘₯∈𝐼. Then, 𝑑(𝐽𝑓,𝐽𝑔)≀𝐾𝐿𝐢𝑓𝑔. Hence, we can conclude that 𝑑(𝐽𝑓,𝐽𝑔)≀𝐾𝐿𝑑(𝑓,𝑔) for any 𝑓,π‘”βˆˆπ‘‹ (note that 0<𝐾𝐿<1). It follows from (3.9) and (3.15) that for an arbitrary 𝑔0βˆˆπ‘‹, there exists a constant 0<𝐢<1 with ||𝐽𝑔0ξ€Έ(π‘₯)βˆ’π‘”0||=||||ξ€œ(π‘₯)𝑦(𝑐)+π‘₯𝑐𝐹𝑝(𝑑),𝑔0ξ€Έ(𝑑)π‘‘π‘‘βˆ’π‘”0||||(π‘₯)β‰€πΆπœ‘(π‘₯)(3.18) for all π‘₯∈𝐼, since 𝐹(π‘₯,𝑔0(π‘₯)) and 𝑔0(π‘₯) are bounded on 𝐼 and minπ‘₯βˆˆπΌπœ‘(π‘₯)>0. Thus, (3.10) implies that 𝑑𝐽𝑔0,𝑔0ξ€Έ<∞.(3.19) Therefore, according to Theorem 1.1(a), there exists a continuous function 𝑦0βˆΆπΌβ†’β„ such that 𝐽𝑛𝑔0→𝑦0 in (𝑋,𝑑) and 𝐽𝑦0=𝑦0, that is, 𝑦0 satisfies (3.7) for every π‘₯∈𝐼. For any π‘”βˆˆπ‘‹, since 𝑔 and 𝑔0 are bounded on 𝐼 and minπ‘₯βˆˆπΌπœ‘(π‘₯)>0, there exists a constant 0<𝐢𝑔<1 such that ||𝑔(π‘₯)βˆ’π‘”0||(π‘₯)β‰€πΆπ‘”πœ‘(π‘₯)(3.20) for any π‘₯∈𝐼. Hence, we have 𝑑(𝑔0,𝑔)<∞ for all π‘”βˆˆπ‘‹, that is, {π‘”βˆˆπ‘‹βˆ£π‘‘(𝑔0,𝑔)<∞}=𝑋. Hence, in view of Theorem 1.1(b), we conclude that 𝑦0 is the unique continuous function with the property (3.7).
On the other hand, it follows from (3.5) that βˆ’πœ‘(π‘₯)β‰€π‘¦ξ…ž(π‘₯)βˆ’πΉ(π‘ž(π‘₯),𝑝(π‘₯)𝑦(π‘₯))β‰€πœ‘(π‘₯)(3.21) for all π‘₯∈𝐼. If we integrate each term in the above inequality from 𝑐 to π‘₯, then we obtain ||||π‘¦ξ€œ(π‘₯)βˆ’π‘¦(𝑐)βˆ’π‘₯𝑐𝐹||||≀||||ξ€œ(π‘ž(π‘₯),𝑝(π‘₯)𝑦(π‘₯))𝑑𝑑π‘₯π‘πœ‘||||(𝑑)𝑑𝑑(3.22) for any π‘₯∈𝐼. Thus, by (3.6) and (3.15), we get ||||𝑦(π‘₯)βˆ’π½π‘¦(π‘₯)β‰€πΎπœ‘(π‘₯)(3.23) for each π‘₯∈𝐼, which implies that 𝑑(𝐽𝑦,𝑦)≀𝐾.(3.24) Finally, Theorem 1.1(c) and (3.24) implys that 𝑑𝐽𝑦,𝑦0≀1𝐾1βˆ’πΎπΏπ‘‘(𝐽𝑦,𝑦)≀1βˆ’πΎπΏ,(3.25) which means that inequality (3.24) holds true for all π‘₯∈𝐼.

Now, we prove the last theorem for unbounded intervals. Also we show that Theorem 3.1 is also true if 𝐼 is replaced by an unbounded interval such as (βˆ’βˆž,𝑏], ℝ, or [π‘Ž,∞).

Theorem 3.2. For given real numbers π‘Ž and 𝑏, let 𝐼 denote either (βˆ’βˆž,𝑏], ℝ, or [π‘Ž,∞). Set either 𝑐=π‘Ž for 𝐼=[π‘Ž,∞) or 𝑐=𝑏 for 𝐼=(βˆ’βˆž,𝑏] or 𝑐 is a fixed real number if 𝐼=ℝ. Let 𝐾 and 𝐿 be positive constants with 0<𝐾𝐿<1. Assume that πΉβˆΆπΌΓ—β„β†’β„ is a continuous function which satisfies Lipschitz condition (3.4) for any π‘₯∈𝐼 and 𝑦,π‘§βˆˆβ„. If a continuously differentiable function π‘¦βˆΆπΌβ†’β„ satisfies ||π‘¦ξ…ž||(π‘₯)βˆ’πΉ(π‘ž(π‘₯),𝑝(π‘₯)𝑦(π‘₯))β‰€πœ‘(π‘₯)(3.26) for all π‘₯∈𝐼, where 𝑝(π‘₯) is a continuous function and πœ‘βˆΆπΌβ†’(0,∞) is a continuous function satisfying condition (3.6) for each π‘₯∈𝐼, then there exists a unique continuous function 𝑦0βˆΆπΌβ†’β„ such that ||𝑦(π‘₯)βˆ’π‘¦0||≀𝐾(π‘₯)πœ‘1βˆ’πΎπΏ(π‘₯)(3.27) for all π‘₯∈𝐼.

Proof. We prove for 𝐼=ℝ. The other cases can be proved similarly. For any π‘›βˆˆβ„•, we define 𝐼𝑛=[π‘βˆ’π‘›,𝑐+𝑛]. (We set 𝐼𝑛=[π‘βˆ’π‘›,𝑏] for 𝐼=(βˆ’βˆž,𝑏] and 𝐼𝑛=[π‘Ž,π‘Ž+𝑛] for 𝐼=[π‘Ž,∞)). By Theorem 3.1, there exists a unique continuous function π‘¦π‘›βˆΆπΌπ‘›β†’β„ such that π‘¦π‘›ξ€œ(π‘₯)=𝑦(𝑐)+π‘₯π‘πΉξ€·π‘ž(π‘₯),𝑝(π‘₯)𝑦𝑛||𝑦(𝑑)𝑑𝑑(3.28)𝑛||≀𝐾(π‘₯)βˆ’π‘¦(π‘₯)1βˆ’πΎπΏπœ‘(π‘₯)(3.29) for all π‘₯βˆˆπΌπ‘›. The uniqueness of 𝑦𝑛 implies that, if π‘₯βˆˆπΌπ‘›, then 𝑦𝑛(π‘₯)=𝑦𝑛+1(π‘₯)=𝑦𝑛+2(π‘₯)=β‹―.(3.30) For any π‘₯βˆˆβ„, define 𝑛(π‘₯)βˆˆβ„• as 𝑛(π‘₯)=minπ‘›βˆˆπ‘βˆ£π‘₯βˆˆπΌπ‘›ξ€Ύ.(3.31) Moreover, define a function 𝑦0βˆΆβ„β†’β„ by 𝑦0(π‘₯)=𝑦𝑛(π‘₯)(π‘₯),(3.32) and we assert that 𝑦0 is continuous. For an arbitrary π‘₯1βˆˆβ„, we choose the integer 𝑛1=𝑛(π‘₯1). Then, π‘₯1 belongs to the interior of 𝐼𝑛1+1 and there exists an πœ€>0 such that 𝑦0(π‘₯)=𝑦𝑛1+1(π‘₯) for all π‘₯ with π‘₯1βˆ’πœ€<π‘₯<π‘₯1+πœ€. Since 𝑦𝑛1+1 is continuous at π‘₯1, so is 𝑦0. That is, 𝑦0 is continuous at π‘₯1 for any π‘₯1βˆˆβ„.
We will now show that 𝑦0 satisfies (3.8) for all π‘₯βˆˆβ„. For an arbitrary π‘₯βˆˆβ„, we choose the integer 𝑛(π‘₯). Then, it holds that π‘₯βˆˆπΌπ‘›(π‘₯) and it follows from (3.28) and (3.32) that 𝑦0(π‘₯)=𝑦𝑛(π‘₯)ξ€œ(π‘₯)=𝑦(𝑐)+π‘₯π‘πΉξ€·π‘ž(π‘₯),𝑝(π‘₯)𝑦𝑛(π‘₯)ξ€Έξ€œ(𝑑)𝑑𝑑=𝑦(𝑐)+π‘₯π‘πΉξ€·π‘ž(π‘₯),𝑝(π‘₯)𝑦0ξ€Έ(𝑑)𝑑𝑑(3.33) since 𝑛(𝑑)≀𝑛(π‘₯) for any π‘‘βˆˆπΌπ‘›(π‘₯). Then, from (3.30) and (3.32) we have 𝑦𝑛(π‘₯)(𝑑)=𝑦𝑛(𝑑)(𝑑)=𝑦0(𝑑).(3.34) Since π‘₯βˆˆπΌπ‘›(π‘₯) for every π‘₯βˆˆβ„, by (3.29) and (3.32), we have ||𝑦0||=||𝑦(π‘₯)βˆ’π‘¦(π‘₯)𝑛(π‘₯)||≀𝐾(π‘₯)βˆ’π‘¦(π‘₯)πœ‘1βˆ’πΎπΏ(π‘₯)(3.35) for any π‘₯βˆˆβ„.
Finally, we show that 𝑦0 is unique. Let 𝑧0βˆΆβ„β†’β„ be another continuous function which satisfies (3.8), with 𝑧0 in place of 𝑦0, for all π‘₯βˆˆβ„. Suppose π‘₯ is an arbitrary real number. Since the restrictions 𝑦0βˆ£πΌπ‘›(π‘₯)(=𝑦𝑛(π‘₯)) and 𝑧0βˆ£πΌπ‘›(π‘₯) both satisfy (3.7) and (3.8) for all π‘₯βˆˆπΌπ‘›(π‘₯), the uniqueness of 𝑦𝑛(π‘₯)=𝑦0βˆ£πΌπ‘›(π‘₯) implies that 𝑦0(π‘₯)=𝑦0βˆ£πΌπ‘›(π‘₯)(π‘₯)=𝑧0βˆ£πΌπ‘›(π‘₯)(π‘₯)=𝑧0(π‘₯)(3.36) as required.

Corollary 3.3. Given π‘βˆˆβ„ and π‘Ÿ>0, let 𝐼 denote a closed ball of radius π‘Ÿ and centered at 𝑐, that is, 𝐼={π‘₯βˆˆβ„βˆ£π‘βˆ’π‘Ÿβ‰€π‘₯≀𝑐+π‘Ÿ}, and let πΉβˆΆπΌΓ—β„β†’β„ be a continuous function which satisfies a Lipschitz condition (3.4) for all π‘₯∈𝐼 and 𝑦,π‘§βˆˆπ‘…, where 𝐿 is a constant with 0<πΏπ‘Ÿ<1. If a continuously differentiable function π‘¦βˆΆπΌβ†’β„ satisfies the differential inequality ||π‘¦ξ…ž||(π‘₯)βˆ’πΉ(π‘₯,𝑦(π‘₯))β‰€πœ€(3.37) for all π‘₯∈𝐼 and for some πœ€β‰₯0, then there exists a unique continuous function 𝑦0βˆΆπΌβ†’β„ satisfying (3.7) and ||𝑦(π‘₯)βˆ’π‘¦0||β‰€π‘Ÿ(π‘₯)1βˆ’π‘ŸπΏπœ€(3.38) for any π‘₯∈𝐼.

Example 3.4. We choose positive constants 𝐾 and 𝐿 with 𝐾𝐿<1. For a positive number πœ€<2𝐾, let 𝐼=[0,2πΎβˆ’πœ€] be a closed interval. Given a polynomial 𝑝(π‘₯), we assume that a continuously differentiable function π‘¦βˆΆπΌβ†’β„ satisfies ||π‘¦ξ…ž||(π‘₯)βˆ’πΏπ‘¦(π‘₯)βˆ’π‘(π‘₯)≀π‘₯+πœ€(3.39) for all π‘₯∈𝐼. If we set 𝐹(π‘₯,𝑦)=𝐿𝑦+𝑝(π‘₯) and πœ‘(π‘₯)=π‘₯+πœ€, then the above inequality has the identical form. Moreover, we obtain ||||ξ€œπ‘₯π‘πœ‘||||=π‘₯(𝑑)𝑑𝑑22+πœ€π‘₯β‰€πΎπœ‘(π‘₯)(3.40) for each π‘₯∈𝐼, since πΎπœ‘(π‘₯)βˆ’π‘₯2/2βˆ’πœ€π‘₯β‰₯0 for all π‘₯∈𝐼. By Theorem 3.1, there exists a unique continuous function 𝑦0βˆΆπΌβ†’β„ such that 𝑦0ξ€œ(π‘₯)=𝑦(0)+π‘₯0||𝑦{𝐿𝑦(𝑑)βˆ’π‘(𝑑)}𝑑𝑑0||≀𝐾(π‘₯)βˆ’π‘¦(π‘₯)1βˆ’πΎπΏ(π‘₯+πœ€)(3.41) for any π‘₯∈𝐼.

Example 3.5. Let π‘Ž be a constant greater than 1 and choose a constant 𝐿 with 0<𝐿<lnπ‘Ž. Given an interval 𝐼=[0,1) and a polynomial 𝑝(π‘₯), suppose π‘¦βˆΆπΌβ†’β„ is a continuously differentiable function satisfying ||π‘¦ξ…ž||(π‘₯)βˆ’πΏπ‘¦(π‘₯)βˆ’π‘(π‘₯)β‰€π‘Žπ‘₯(3.42) for all π‘₯∈𝐼. If we set πœ‘(π‘₯)=π‘Žπ‘₯, then we have ||||ξ€œπ‘₯π‘πœ‘||||≀1(𝑑)π‘‘π‘‘πœ‘lnπ‘Ž(π‘₯)(3.43) for any π‘₯∈𝐼. By Theorem 3.2, there exists a unique continuous function 𝑦0βˆΆπΌβ†’β„ with 𝑦0ξ€œ(π‘₯)=𝑦(0)+π‘₯0||𝑦{𝐿𝑦(𝑑)βˆ’π‘(𝑑)}𝑑𝑑0||≀1(π‘₯)βˆ’π‘¦(π‘₯)π‘Žlnπ‘Žβˆ’πΏπ‘₯(3.44) for any π‘₯∈𝐼.

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