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Axioms
, Volume 12 (5) – Apr 23, 2023

/lp/multidisciplinary-digital-publishing-institute/lyapunov-functionals-in-integral-equations-VpS0mO0Bpw

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- 10.3390/axioms12050410
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axioms Article 1, 2 Youssef N. Raffoul * and Joseph Raffoul Department of Mathematics, University of Dayton, Dayton, OH 45469-2316, USA Electrical and Computer Engineering Department, University of Dayton, Dayton, OH 45469-2316, USA; raffoulj1@udayton.edu * Correspondence: yraffoul1@udayton.edu Abstract: Lyapunov functions/functionals have found their footing in Volterra integro-differential equations. This is not the case for integral equations, and it is therefore further explored in this paper. In this manuscript, we utilize Lyapunov functionals combined with Laplace transform to qualitatively analyze the solutions of the integral equation In addition, we extend our method to nonlinear integral equations, integral equations with inﬁnite delay, and integral equations with several kernels. We mention that Laplace transform has been used to solve integral equations of convolution types but has never been applied directly to integral equations that are not of the convolution type. In addition, our method allows us to ﬁnd the upper estimates, and our necessary conditions are easy to verify. Keywords: integral equation; nonlinear; boundedness; uniform; stability; Laplace transform; Lyapunov functionals; inﬁnite delay MSC: 34D20; 39A10; 39A12; 40A05; 45J05 1. Introduction Lyapunov functions/functionals have a long history of successful use in ordinary differential equations, functional differential equations, and Volterra integro-differential equations. The literature is vast, and we refer the reader to the most prominent results given in [1–8]. Most scientiﬁc ﬁelds are directly or indirectly involved with differential or in- Citation: Raffou, Y.N.; Raffoul, J. tegral equations. Additionally, a lot of the issues call for quite precise qualitative outcomes. Lyapunov Functionals in Integral In particular, it is imperative to consider the following issues when dealing with certain Equations. Axioms 2023, 12, 410. problems, for example, in the case that a convenient approximation cannot be used in place https://doi.org/10.3390/ of the function. Moreover, it is of great beneﬁt to understand how each solution behaves as axioms12050410 well as understand how solutions behave over a very long period of time. It is challenging Academic Editors: Simeon Reich and to achieve all three requirements, even with the most sophisticated computational tech- Patricia J. Y. Wong niques. However, A. M. Lyapunov, a Russian mathematician, developed a straightforward approach that satisﬁed those requirements for ordinary differential equations more than a Received: 9 March 2023 Revised: 13 April 2023 century ago. His approach is now known as the “Lyapunov direct method”. Accepted: 17 April 2023 Many researchers differentiate an integral equation before using Lyapunov’s direct Published: 23 April 2023 approach on it. Miller, in [7], considered a system of integral equations transferred to a system of integro-differential equations and used the notion of the Lyapunov direct method to analyze the solutions. The given functions are not differentiable, which makes this procedure complex and challenging. Furthermore, it is well known that differentiation Copyright: © 2023 by the authors. causes roughness, whereas integration produces smoothness; as a result, differentiation Licensee MDPI, Basel, Switzerland. might produce results that might not be applicable to or even hold for the original problem. This article is an open access article T. A. Burton in [1] compiled a collection of recent results and papers on integral equations. distributed under the terms and His work contains clever ways of constructing Lyapunov functions/functionals for integral conditions of the Creative Commons equations. Burton utilizes Lyapunov functionals along with the resolvent to arrive at Attribution (CC BY) license (https:// boundedness and stability results. In [9], the authors extended some of the arguments of [1] creativecommons.org/licenses/by/ to Caputo integral equations and arrived at boundedness and stability results. Researchers 4.0/). Axioms 2023, 12, 410. https://doi.org/10.3390/axioms12050410 https://www.mdpi.com/journal/axioms Axioms 2023, 12, 410 2 of 13 and scientists periodically use Laplace transform to solve an integral equation of the convolution type. No one up till now has been able to use Laplace transform on integral equations that are not of convolution type. That is why we believe that the results of this this paper are signiﬁcant and innovative. As we have mentioned, the Lyapunov method is well established in the study of integro-differential equations. For example, in Ref. [10], the authors considered the nonlin- ear integro-differential equation y (t) = A(t)y + f (y) + C(t, s)h(y(s))ds + p(t), y(0) = y , where A, f (y), p, and h(y) are scalar functions that are continuous, use Lyapunov func- tionals combined with Laplace transform, and provide qualitative results concerning the equation’s solution. Our approach is a novel method of analyzing solutions to integral equations. This, by itself, should spark an outburst of new research in integral equations and related topics. This paper is organized into the following sections. In Section 2, we consider linear equations and utilize Lyapunov functionals combined with Laplace transform and obtain boundedness and existence results concerning solutions. In Section 3, we extend the results of Section 2 to nonlinear integral equations. Section 4 is devoted to integral equations with inﬁnite delay and integral equations with several kernels. Examples will be fully worked out in the relevant sections. The following is the deﬁnition of Laplace transform. We say the function x(t) is of an exponential order for t 0, if there are constants m 0 and c such that ct jxj me f or all t 0. Let x(t) be a piecewise continuous function that is deﬁned for t 0 and of exponential order. Then, the Laplace transform L(x)(s) of x(t) is deﬁned by the integral +¥ st L(x)(s) = e x(t)dt, where s is a real number chosen so that the improper integral exists. Below, we brieﬂy introduce the notion of a Lyapunov function/functional. The deﬁnitions below are of general types, and hence they can be adjusted to suit different types of differential equations or integral equations. Let D be an open subset of R containing x = 0. Deﬁne V : [0,¥) D ! [0,¥), where V is any differentiable scalar function. If z : [0,¥) ! R is any differentiable function, then V(t) := V(t, z(t)) is a scalar function of t, and using the chain rule we can compute its derivative, ¶V ¶V ¶V 0 0 0 V (t) = z (t) + . . . + z (t) + . ¶x ¶x ¶t n n For emphasis, let D be an open subset of R containing x = 0, and f : [0,¥) D ! R with f (t, 0) = 0. Assume the existence of the unknown solution x : [0,¥) ! R of the system x = f (t, x), (1) where 0 1 f (t, x) B C f (t, x) B C f (t, x) = . B C @ A f (t, x) n Axioms 2023, 12, 410 3 of 13 Thus, x and f are n vectors. Then, it follows from the above argument that ¶V(t, x(t)) V (t, x(t)) = f (t, x(t)) + . . . ¶x ¶V(t, x(t)) ¶V + f (t, x(t)) + . (2) ¶x ¶t Thus, expression (2) deﬁnes the derivative of the function V(t, x) along the unknown solutions of (1). Let D be the subset deﬁned above. Deﬁnition 1. A continuous autonomous function V : D ! [0,¥) is positive deﬁnite if V(0) = 0 and V(x) > 0 if x 6= 0. V is said to be negative deﬁnite if V is positive deﬁnite. It is customary to deﬁne a Lyapunov function by the next deﬁnition. This is the case when the function f in (1) does not explicitly depend on time t, or system (1) is autonomous. Deﬁnition 2. Let V : D ! [0,¥), have continuous ﬁrst partial derivatives. If V is positive deﬁnite and ¶V(x(t)) ¶V(x(t)) V (x(t)) = f (x(t)) + . . . + f (x(t)) 0, ¶x ¶x 1 n for x 2 D and x 6= 0, then V is called a Lyapunov function for system (1). If the inequality is strict, that is, V (x(t)) < 0, then V is said to be a strict Lyapunov function. For the sake of this paper, we adopt the following deﬁnition of a Lyapunov function. Deﬁnition 3. Let M and t be positive constants. Let V be deﬁned as in Deﬁnition 2. If V (x(t)) tjxj + M, for x 2 D, and x 6= 0, then V is called a Lyapunov function for system (1). The literature on the use of Lyapunov functions/functionals in differential, functional differential equations are vast, and we refer the reader to [1–5,11,12]. For the rest of the paper, we use the notation V(t) := V(x(t)), where x is the unknown solution of (1). 2. Linear Integral Equations We begin by considering the linear and scalar integral equation y(t) = a(t) C(t, s)y(s)ds, (3) where y : R ! R, a : [0,¥) ! R is continuous and C : [0,¥) [0,¥) ! R is continuous for 0 s t < ¥. If C and a are differentiable, we can differentiate (3) to obtain a Volterra integro-differential equation, which we can then analyze using the method of [10]. However, because differentiability is such a signiﬁcant criterion, we might not always have that luxury. We want to be clear that the approach we use in this work is completely Axioms 2023, 12, 410 4 of 13 distinct from any approach offered in the book [1]. However, for more reading on the subject of Volterra integro-differential equations, we refer to [6–8,13,14]. We begin with the following lemma. Lemma 1. Suppose there is a differentiable function y : [0,¥) ! (0,¥) such that y (t s) jC(t, s)j, 0 s t < ¥, (4) and 0 < y(0) < 1. If y(t) is any solution of (3) and if the Lyapunov function V is deﬁned by V(t) = y(t s)jy(s)jds, (5) then there exists a constant a 2 (0, 1), such that V (t) ajy(t)j + M, (6) where a = 1 y(0), and M = maxja(t)j. t0 Proof. Let V be deﬁned by (5) and y(t) be a solution of (3). Then, differentiating V with respect to t gives 0 0 V (t) = y(0)jy(t)j + y (t s)jy(s)jds y(0)jy(t)j jC(t, s)jjy(s)jds. (7) Now, from (3) we have that jy(t)jja(t)j jC(t, s)jjy(s)jds. Substituting into (7), we arrive at V (t) y(0)jy(t)j +ja(t)jjy(t)j ajy(t)j + M. Proposition 1. If f : [0,¥) ! [0,¥) is uniformly continuous, and f 2 L ([0,¥)), then lim f (x) = 0. x!¥ Proof. Suppose the contrary, that is, f does not converge to zero. Then, there is an e > 0 such that we can deﬁne an increasing sequence x so that x ! ¥, so we have f (x ) > e. n n n Since f is uniformly continuous, d > 0 exists such that jx yj < d, implies that j f (x) f (y)j < . By referring to the subsequence, we may suppose that x > x + 1, for each n+1 n n = 1, 2, . . . Since the intervals (x , x + d), n = k, k + 1, . . . are disjointed, we have that n n Z Z h i x +d x +d n n e e f (y)dy f (x ) dy > d. 2 2 x x n n Axioms 2023, 12, 410 5 of 13 Summing these intervals, we see that f (x)dx = ¥, which is a contradiction. This completes the proof. Lemma 2. Let b : [0,¥) ! [0,¥) be uniformly continuous such that fy(t s) + agb(s)ds = 1, t 0. (8) Let y be deﬁned in Lemma 1 and if y(t)dt < ¥, (9) then b(t) 2 L [0,¥), (10) and b(t) ! 0 as t ! ¥. Proof. Since y > 0 and due to (9), we have from (8) that ab(s)ds 1, or b(s)ds . Taking the limit at t ! ¥, we obtain lim b(s)ds , t!¥ a This proves b(t) 2 L [0,¥), since the term on the right-hand side is independent of t. Since b 0 for all t 0, uniformly continuous, and b(t) 2 L [0,¥), it follows from Proposition 1 that b(t) ! 0 as t ! ¥. This completes the proof. Remark 1. The results of Lemma 2 imply that a positive constant F exists such that b(t) F. Theorem 1. Assume the hypotheses of Lemmata 1 and 2 hold. In addition, we assume that b and y are of exponential orders. If y(t) is any solution of (3), then jy(t)j . Proof. Let denote the convolution between two functions. By taking the Laplace trans- form in (8), we arrive at Z Z t t L( y(t s)b(s)ds) + L( ab(s)ds) = L(1). 0 0 Or, L(y b) + aL(1 b) = . In particular, L(y)L(b) + aL(1)L(b) = . s Axioms 2023, 12, 410 6 of 13 Solving for L(b) gives L(b) = . (11) (L(y) + a )s Due to (6), there is a non-negative function h : [0,¥) ! [0,¥) that is of exponential order such that V (t) := ajyj + M h(t). Taking the Laplace transform and using V(0) = 0, we arrive at sL(V) = aL(jyj) + L(h). This yields M 1 L(V) = aL(jyj) + L(h) . s s Taking the Laplace transform in (5), we obtain L(V) = L(y)L(jyj). Comparing the last two expressions and solving for L(jyj), we obtain L(h) L(jyj) = a + sL(y) L(h) [ + L(y)]s = [ L(h)]L(b) = L(b) L(h)L(b) Z Z t t = L( Mb(s)ds) L( h(t s)b(s)ds). (12) 0 0 Taking the inverse Laplace transform in (12), we obtain Z Z t t jyj = M b(s)ds h(t s)b(s)ds 0 0 or jy(t)j M b(s)ds . This completes the proof. We display the following simple example. Note that the ﬁgures accompanying the several examples are numerically approximated. The approximate solutions are obtained using the iterative method, y (t) = a(t) + C(x, s)y (s)ds, n = 0, 1, ..., (13) n+1 where y (x) = a(t). The sequence converges to the approximate solution as the number of iterations approaches ¥. Example 1. Consider the integral equation y(t) = cos(t) y(s)ds. (14) (t + 2 s) 0 Axioms 2023, 12, 410 7 of 13 1 1 Then, we have a(t) = cos(t) and C(t, s) = . Set y(t) = . Then, it 3 2 (t + 2 s) (t + 2) follows that dt < ¥. (t + 2) In addition, y (t) = and hence (t+2) 2 1 = y (t s) < = jC(t, s)j. 3 3 (t + 2 s) (t + 2 s) Moreover, 1 3 a = 1 y(0) = 1 = 2 (0, 1). 4 4 Thus, by Theorem 1 any solution y(t) of (14) satisﬁes M 4 jy(t)j = , a 3 since M = max j cos(t)j = 1. t0 We refer to Figure 1. for the upper bound on the solution. Figure 1. Using MATLAB, the graph shows that the upper bound of this approximation at t = 0 is 1. 3. Nonlinear Integral Equations Now, we extend the results of Section 2 to the nonlinear and scalar integral equations of the form y(t) = a(t) C(t, s)h(y(s))ds, (15) where the continuity of a and C are the same as in Section 2 and the function h is continuous in y and satisﬁes the growth condition jh(y)j ljyj, (16) for positive constant l. The transition from the linear case to nonlinear case is not difﬁcult, but nevertheless some of the details must be provided. The next lemma is parallel to Lemma 1. Axioms 2023, 12, 410 8 of 13 Lemma 3. Assume (16), and suppose there is a differentiable function y : [0,¥) ! (0,¥) such that y (t s) jC(t, s)j, 0 s t < ¥, (17) and 0 < y(0) < 1. If y(t) is any solution of (15) and if the Lyapunov function V is deﬁned by V(t) = l y(t s)jy(s)jds, (18) then a constant a 2 (0, 1), exists such that V (t) ajy(t)j + M, (19) where such that a = 1 ly(0), and M = maxja(t)j. t0 Proof. Let V be deﬁned by (18) and y(t) be a solution of (15). Then, differentiating V with respect to t gives 0 0 V (t) ly(0)jy(t)j + l y (t s)jy(s)jds ly(0)jy(t)j l jC(t, s)jjy(s)jds. (20) Now, from (15) we have that jy(t)jja(t)j l jC(t, s)jjy(s)jds. Substituting into (20), we arrive at V (t) ly(0)jy(t)j +ja(t)jjy(t)j ajy(t)j + M. Similarly, the next lemma is parallel to Lemma 2. Its proof is identical to Lemma 1, and it will be omitted. Lemma 4. Assume (9), and let b(t) 0 be a scalar function that is uniformly continuous on [0,¥) and be deﬁned by fly(t s) + agb(s)ds = 1. (21) Then, b(t) 2 L [0,¥), and b(t) ! 0 as t ! ¥. We state our results in the next theorem, which is parallel to Theorem 1. Theorem 2. Assume the hypotheses of Lemmata 3 and 4 hold. If y(t) is any solution of (15), then jy(t)j . Proof. By taking the Laplace transform in (21), we arrive at Z Z t t lL( y(t s)b(s)ds) + L( ab(s)ds) = L(1). 0 0 Axioms 2023, 12, 410 9 of 13 In particular, lL(y)L(b) + aL(1)L(b) = . Solving for L(b) gives L(b) = . (lL(y) + a )s Due to (19), there is a non-negative function h : [0,¥) ! [0,¥) of an exponential order such that V (t) := ajyj + M h(t). By taking the Laplace transform and by considering V(0) = 0, we have that M 1 L(V) = aL(jyj) + L(h) . s s Taking the Laplace transform in (18), we obtain L(V) = lL(y)L(jyj). Comparing the last two expressions and solving for L(jyj), we obtain L(h) L(jyj) = [ + lL(y)]s = [ L(h)]L(b) Z Z t t = L( Mb(s)ds) L( h(t s)b(s)ds). (22) 0 0 Taking the inverse Laplace transform in (22), we obtain Z Z t t jy(t)j = M b(s)ds h(t s)b(s)ds 0 0 or jy(t)j M b(s)ds , where a 2 (0, 1), such that y(0) 2 (0, 1) with a = 1 ly(0), and M = maxja(t)j. t0 This completes the proof. Now, we offer an example. Example 2. Consider the nonlinear integral equation (t+3s) y(t) = cos(t) e sin(y(s))ds. (23) (t+3s) Then, a(t) = cos(t), h(y) = sin(y), and C(t, s) = e . Then, we have, l = 1. Let (t+3) y(t) = e . Then, it follows that (t+3) e dt < ¥. 0 Axioms 2023, 12, 410 10 of 13 0 (t+3) 0 (t+3s) In addition, j (t) = e , which implies that y (t s) = e = jC(t, s)j, for 0 s t < ¥. Thus, condition (17) is satisﬁed. Moreover, a = 1 ly(0) = 1 e 2 (0, 1). Thus, by Theorem 2 any solution y(t) of (23) satisﬁes M 1 jy(t)j = , a 1 e since M = max j cos(t)j = 1. t0 We refer to Figure 2. for the upper bound on the solution. Figure 2. Using MATLAB, the graph shows the upper bound of this approximation at t = 0 is 1. 4. Inﬁnite Delay and Several Kernels In this section, we extend the method to integral equations with inﬁnite delay if the history of the solution is known and is a continuous function. Additionally, we generalize the concept to integral equations with several kernels. We begin by considering scalar integral equations with inﬁnite delay of the form y(t) = b(t) C(t, s)g(y(s))ds, (24) where b, C, and g are continuous. We assume the solution exists under some conditions. To specify a solution of (24), we require a continuous initial function j : (¥, 0] ! R, with j(0) = a(0), where a(t) := b(t) C(t, s)g(j(s))ds is continuous so that y(t) = a(t) C(t, s)g(y(s))ds, t 0 (25) 0 Axioms 2023, 12, 410 11 of 13 is basically of the form of (15). With this set up, a function y(t) is said to be a solution of (24), if y(t) = j(t), for t 0, and y(t) satisﬁes (24) for t 0. Finally, Theorem 2 is exactly what would one needs to obtain boundedness results. We end this paper with the extension to integral equations with N number of ker- nels and N number of nonlinear functions in y. Thus, we consider the scalar nonlinear integral equation y(t) = a(t) C (t, s)h (y(s))ds, t 0 (26) å i i i=1 where all functions are scalars and continuous on their respective domains. The functions h , i = 1 . . . N are continuous and satisfy the growth condition jh (y)j l jyj, i = 1 . . . N (27) i i for positive constants l . Under this set up, the conditions of Lemmata 3 and 4 can be easily modiﬁed as seen next. Suppose there are differentiable functions y : [0,¥) ! (0,¥) for i = 1, . . . N, such that N N y (t s) l C (t, s) , 0 s t < ¥. (28) å å i i i=1 i=1 Moreover, if we assume the existence of a scalar function b(t) that is uniformly continuous on [0,¥); then, we may redeﬁne (21) as follows: f l y (t s) + agb(s)ds = 1. (29) å i i i=1 If y(t) is any solution of (26), then (18) is modiﬁed and given by V(t) = l y (t s)jy(s)jds, (30) å i i i=1 Considering the above modiﬁcations, one can easily conclude the following theorem. Theorem 3. Assume conditions (27), (28), and y (t)dt < ¥. å i i=1 If y(t) is any solution of (26), then jy(t)j , where the constant a 2 (0, 1), such that N N 0 < l y (0) < 1 with a = 1 l y (0), and M = maxja(t)j. å i i å i i t0 i=1 i=1 Now, we offer an example. Example 3. Consider the nonlinear integral equation t (t+3s) y(t) = te sin(y(s)) + e y(s) ds. (31) (t + 2 s) 0 Axioms 2023, 12, 410 12 of 13 1 (t+3s) Consequently, a(t) = te , h(y) = sin(y), and C (t, s) = , C (t, s) = e . Then, 1 2 (t+2s) (t+3) we have, l = l = 1 and M = e . Let y (t) = , y (t) = e . Then, it follows that 1 2 1 2 (t + 2) y (t)dt < ¥, i = 1, 2. In addition, (28) is satisﬁed for 0 s t < ¥. Moreover, 1 3 3 3 a = 1 l y (0) l y (0)) = 1 e = e 2 (0, 1). 1 1 2 2 4 4 Thus, by Theorem 3 any solution y(t) of (31) satisﬁes M 1/4 jy(t)j = . a 3/4 e We refer to Figure 3. for the upper bound on the solution. Figure 3. Using MATLAB, the graph shows the upper bound of this approximation is 0.3345. Author Contributions: Every author contributed equally to the development of this paper. All au- thors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Acknowledgments: The authors would like to thank the anonymous referees for their suggestions and careful reading of our paper. Conﬂicts of Interest: The authors declare no conﬂict of interest. References 1. Burton, T.A. Liapunov Functionals for Integral Equations; Trafford Publishing: Bloomington, IN, USA, 2008. 2. Brauer, F.; Nohel, J.A. Qualitative Theory of Ordinary Differential Equations; Dover: New York, NY, USA, 1969. 3. Coddington, E.A.; Levinson, N. Theory of Ordinary Differential Equations; McGraw-Hill Book Company: New York, NY, USA; London, UK, 1955. Axioms 2023, 12, 410 13 of 13 4. Cushing, J.M. Integro-Differential Equations and Delay Models in Population Dynamics; Lecture Notes in Biomathematics; Springer: Berlin, Germany; New York, NY, USA, 1977; Volume 20. 5. Driver, R.D. Introduction to Ordinary Differential Equations; Harper & Row, Publishers: New York, NY, USA, 1978. 6. Messina, E.; Raffoul, Y.N.; Vecchio, A. Qualitative Analysis of Dynamic Equations on Time Scales Using Lyapunov Functions. Differ. Equ. Appl. 2022, 14, 137–143. [CrossRef] 7. Miller, R.K. Nonlinear Volterra Integral Equations; Benjamin: New York, NY, USA, 1971. 8. Raffoul, Y. Advanced Differential Equations; Academic Press: Cambridge, MA, USA, 2023. 9. Wang, M.; Saleem, N.; Bashir, S.; Zhou, M. Fixed point of modiﬁed F-Contraction with an application. Axioms 2022, 11, 413. [CrossRef] 10. Alhamadi, F.; Raffoul, Y.N.; Alharbi, S. Boundedness and Stability of Solutions of Nonlinear Volterra Integro-Differential Equations. Adv. Dyn. Syst. Appl. 2018, 13, 19–31. 11. Burton, T.A. Stability and Periodic Solutions of Ordinary and Functional Differential Equations; Academic Press: New York, NY, USA, 1985. 12. Burton, T.A. Stability by Fixed Point Theory for Functional Differential Equations; Dover: New York, NY, USA, 2006. 13. Islam, M.; Raffoul, Y. Bounded Solutions of Almost Linear Volterra Equation. Adv. Dyn. Syst. Appl. 2012, 7, 2. 14. Tunc, C. New stability and boundedness results to Volterra integro-differential equations with delay. J. Egypt. Math. Soc. 2016, 24, 210–213. [CrossRef] Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Axioms – Multidisciplinary Digital Publishing Institute

**Published: ** Apr 23, 2023

**Keywords: **integral equation; nonlinear; boundedness; uniform; stability; Laplace transform; Lyapunov functionals; infinite delay

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