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Article

# Multiple Periodic Solutions for Odd Perturbations of the Discrete Relativistic Operator

by
Petru Jebelean
* and
Călin Şerban
Department of Mathematics, West University of Timişoara, 4, Blvd. V. Pârvan, 300223 Timişoara, Romania
*
Author to whom correspondence should be addressed.
Mathematics 2022, 10(9), 1595; https://doi.org/10.3390/math10091595
Submission received: 2 April 2022 / Revised: 2 May 2022 / Accepted: 7 May 2022 / Published: 8 May 2022
(This article belongs to the Special Issue Nonlinear Functional Analysis and Its Applications 2021)

## Abstract

:
We obtain the existence of multiple pairs of periodic solutions for difference equations of type where $g : R → R$ is a continuous odd function with anticoercive primitive, and $λ > 0$ is a real parameter. The approach is variational and relies on the critical point theory for convex, lower semicontinuous perturbations of $C 1$-functionals.
MSC:
39A23; 39A27; 47J20

## 1. Introduction

In this note, we are concerned with the multiplicity of solutions for difference equations with relativistic operator of type
$− Δ ϕ ( Δ u ( n − 1 ) ) = λ g ( u ( n ) ) , u ( n ) = u ( n + T ) ( n ∈ Z ) ,$
where $Δ u ( n ) = u ( n + 1 ) − u ( n )$ is the usual forward difference operator, $λ > 0$ is a real parameter, $g : R → R$ is a continuous odd function, and
$ϕ ( y ) = y 1 − y 2 ( y ∈ ( − 1 , 1 ) ) .$
In recent years, special attention has been paid to the existence and multiplicity of T-periodic solutions for problems with a discrete relativistic operator. Thus, for instance, in [1,2], variational arguments were employed to prove the solvability of systems of difference equations having the form
$Δ ϕ N ( Δ u ( n − 1 ) ) = ∇ u V ( n , u ( n ) ) + h ( n ) ( n ∈ Z ) ,$
under various hypotheses upon V and h (coerciveness, growth restriction, convexity or periodicity conditions); here, $ϕ N$ is the N-dimensional variant of $ϕ$, i.e.,
$ϕ N ( y ) = y 1 − | y | 2 ( y ∈ R N , | y | < 1 ) .$
The existence of at least $N + 1$ geometrically distinct T-periodic solutions of (2) was proved in [3], under the assumptions that h is T-periodic, $∑ j = 1 T h ( j ) = 0$, and the mapping $V ( n , x )$ is T-periodic in n and $ω i$-periodic $( ω i > 0 )$ with respect to each $x i$$( i = 1 , … , N )$. For the proof, using an idea from the differential case [4], the singular problem (2) was reduced to an equivalent non-singular one to which classical Ljusternik–Schnirelmann category methods can be applied. In addition, under some similar assumptions on V and h, were obtained in [5] using Morse theory, conditions under which system (2) has at least $2 N$ geometrically distinct T-periodic solutions.
The motivation of the present study mainly comes from paper [6], where for problems involving Fisher-Kolmogorov nonlinearities of type
$− Δ ϕ ( Δ u ( n − 1 ) ) = λ u ( n ) ( 1 − | u ( n ) | q ) , u ( n ) = u ( n + T ) ( n ∈ Z ) ,$
with $q > 0$ fixed and $λ > 0$ a real parameter, it was proved that if $λ > 8 m T$ for some $m ∈ N$ with $2 ≤ m ≤ T$, then problem (3) has at least m distinct pairs of nontrivial solutions. We also refer the interested reader to [6] for a discussion concerning the origin and steps in the study of this type of nonlinearity. In this respect, we shall see in Example 1 below that a sharper result holds true, namely,
(i)
If $λ > 8 sin 2 m π T$ with$0 ≤ m ≤ ( T − 1 ) / 2 i f T i s o d d ( T − 2 ) / 2 i f T i s e v e n ,$then problem (3) has at least $2 m + 1$ distinct pairs of nontrivial solutions.
(ii)
If T is even and $λ > 8$, then (3) has at least T distinct pairs of nontrivial solutions.
Moreover, we prove in Theorem 2 that the above statements (i) and (ii) still remain valid for a larger class of periodic problems.
As in [6], our approach to problem (1) is variational and combines a Clark-type abstract result for convex, lower semicontinuous perturbations of $C 1$-functionals, based on Krasnoselskii’s genus. However, our technique here brings the novelty that it exploits the interference of the geometry of the energy functional with fine spectral properties of the operator $− Δ 2$; recall that
$Δ 2 u ( n − 1 ) : = Δ ( Δ u ( n − 1 ) ) = u ( n + 1 ) − 2 u ( n ) + u ( n − 1 ) .$
It is worth noting that in paper [7] analogous multiplicity results are obtained in the differential case for potential systems involving parametric odd perturbations of the relativistic operator. In addition, we mention the recent paper [8], where the authors obtain the existence and multiplicity of sign-changing solutions for a slightly modified parametric problem of type (1) using bifurcation techniques.
We conclude this introductory part by briefly recalling some topics in the frame of Szulkin’s critical point theory [9], which is needed in the sequel. Let $( Y , ∥ · ∥ )$ be a real Banach space and $I : Y → ( − ∞ , + ∞ ]$ be a functional having the following structure:
$I = F + ψ ,$
where $F ∈ C 1 ( Y , R )$ and $ψ : Y → ( − ∞ , + ∞ ]$ is proper, convex and lower semicontinuous. A point $u ∈ D ( ψ )$ is said to be a critical point of $I$ if it satisfies the inequality
$〈 F ′ ( u ) , v − u 〉 + ψ ( v ) − ψ ( u ) ≥ 0 ∀ v ∈ D ( ψ ) .$
A sequence ${ u n } ⊂ D ( ψ )$ is called a (PS)-sequence if $I ( u n ) → c ∈ R$ and
$〈 F ′ ( u n ) , v − u n 〉 + ψ ( v ) − ψ ( u n ) ≥ − ε n ∥ v − u n ∥ ∀ v ∈ D ( ψ ) ,$
where $ε n → 0$. The functional $I$ is said to satisfy the (PS) condition if any (PS)-sequence has a convergent subsequence in Y.
Let $Σ$ be the collection of all symmetric subsets of $Y \ { 0 }$ which are closed in Y. The genus of a nonempty set $A ∈ Σ$ is defined as being the smallest integer k with the property that there exists an odd continuous mapping $h : A → R k \ { 0 }$; in this case, we write $γ ( A ) = k$. If such an integer does not exist, then $γ ( A ) : = + ∞$. Notice that if $A ∈ Σ$ is homeomorphic to $S k − 1$ ($k − 1$ dimension unit sphere in the Euclidean space $R k$) by an odd homeomorphism, then $γ ( A ) = k$ ([10], Corollary 5.5). For other properties and more details on the notion of genus, we refer the reader to [10,11]. The following theorem is an immediate consequence of ([9], Theorem 4.3).
Theorem 1.
Let $I$ be of type (4) with $F$ and ψ even. In addition, suppose that $I$ is bounded from below, satisfies the (PS) condition and $I ( 0 ) = 0$. If there exists a nonempty compact symmetric subset $A ⊂ Y \ { 0 }$ with $γ ( A ) ≥ k$, such that
$sup v ∈ A I ( v ) < 0 ,$
then the functional $I$ has at least k distinct pairs of nontrivial critical points.

## 2. Variational Approach and Preliminaries

To introduce the variational formulation for problem (1), let $H T$ be the space of all T-periodic $Z$-sequences in $R$, i.e., of mappings $u : Z → R$, such that $u ( n ) = u ( n + T )$ for all $n ∈ Z$. On $H T$, we consider the following inner product and corresponding norm:
$( u | v ) : = ∑ j = 1 T u ( j ) v ( j ) , ∥ u ∥ = ∑ j = 1 T | u ( j ) | 2 1 / 2 ,$
which makes it a Hilbert space. In addition, for each $u ∈ H T$, we set
$u ¯ : = 1 T ∑ j = 1 T u ( j ) , u ˜ : = u − u ¯ .$
It is not difficult to check that
$| u ˜ ( i ) | ≤ T 1 2 ∑ j = 1 T | Δ u ( j ) | 2 1 / 2 ( i ∈ { 1 , … , T } ) .$
Now, let the closed convex subset K of $H T$ be defined by
$K : = { u ∈ H T : | Δ u | ∞ ≤ 1 } ,$
where $| Δ u | ∞ : = max i = 1 , … , T | Δ u ( i ) | .$ Then, from (5), one has
for all $u ∈ K$. We introduce the even functions
where $Φ ( y ) = 1 − 1 − y 2 ( y ∈ [ − 1 , 1 ] )$ and
$G λ ( u ) = − λ ∑ j = 1 T G ( u ( j ) ) ( u ∈ H T ) ,$
with G the primitive
$G ( x ) = ∫ 0 x g ( τ ) d τ ( x ∈ R ) .$
It is not difficult to see that $Ψ$ is convex and lower semicontinuos, while $G λ$ is of class $C 1$, its derivative being given by
$〈 G λ ′ ( u ) , v 〉 = − λ ∑ j = 1 T g ( u ( j ) ) v ( j ) ( u , v ∈ H T ) .$
Then, the functional $I λ : H T → ( − ∞ , + ∞ ]$ associated to (1) is
$I λ = Ψ + G λ$
and it is clear that it has the structure required by Szulkin’s critical point theory. A solution of problem (1) is an element $u ∈ H T$ such that $| Δ u ( n ) | < 1$, for all $n ∈ Z$, which satisfies the equation in (1). The following result reduces the search of solutions of problem (1) to finding critical points of $I λ$.
Proposition 1.
Any critical point of $I λ$ is a solution of problem (1).
Proof.
Let $e ∈ H T$. By virtue of Lemmas 5 and 6 in [1], the problem
$Δ ϕ ( Δ u ( n − 1 ) ) = u ¯ + e ( n ) , u ( n ) = u ( n + T ) ( n ∈ Z )$
has a unique solution $u e$, which is also the unique solution of the variational inequality
$∑ j = 1 T Φ [ Δ v ( j ) ] − Φ [ Δ u ( j ) ] + u ¯ ( v ¯ − u ¯ ) + e ( j ) ( v ( j ) − u ( j ) ) ≥ 0 , ∀ v ∈ K$
([6], Proposition 3.1). Next, let $w ∈ K$ be a critical point of $I λ$. Then, for any $v ∈ K ,$ one has
$∑ j = 1 T Φ [ Δ v ( j ) ] − Φ [ Δ w ( j ) ] − λ g ( w ( j ) ) ( v ( j ) − w ( j ) ) ≥ 0 ,$
which can be written as
$∑ j = 1 T { Φ [ Δ v ( j ) ] − Φ [ Δ w ( j ) ] + w ¯ ( v ( j ) − w ( j ) ) } − ∑ j = 1 T λ g ( w ( j ) ) + w ¯ ( v ( j ) − w ( j ) ) ≥ 0 .$
Hence, w is a solution of the variational inequality
$∑ j = 1 T { Φ [ Δ v ( j ) ] − Φ [ Δ w ( j ) ] + w ¯ ( v ¯ − w ¯ ) + e w ( j ) ( v ( j ) − w ( j ) ) } ≥ 0 , ∀ v ∈ K ,$
with $e w ∈ H T$ being given by $e w ( n ) = − λ g ( w ( n ) ) − w ¯$$( n ∈ Z )$.
Therefore, by (8) and the uniqueness of the solution of (7), we obtain that, in fact, w solves problem (1). □
Proposition 2.
If G is anticoercive, i.e.,
$lim | x | → + ∞ G ( x ) = − ∞ ,$
then $I λ$ is bounded from below and satisfies the (PS) condition.
Proof.
From (9) we have that $− G ,$ hence $G λ$, are bounded from below on $R$, respectively on $H T$. This, together with the fact that $Ψ$ is bounded from below, ensure that the same is true for $I λ$.
To see that $I λ$ satisfies the (PS) condition, let ${ u n } ⊂ K$ be a (PS)-sequence. Assuming by contradiction that ${ | u ¯ n | }$ is not bounded, we may suppose, going, if necessary, to a subsequence, that $| u ¯ n | → + ∞$. Then, by virtue of (6) and (9), we deduce that $I λ ( u n ) → − ∞$, contradicting the fact that ${ I λ ( u n ) }$ is convergent. Consequently, ${ | u ¯ n | }$ is bounded. This, together with shows that ${ u n }$ is bounded in the finite-dimensional space $H T$; hence, it contains a convergent subsequence. □
Remark 1.
Notice that until here in this section, no parity assumptions on the continuous function $g : R → R$ must be required.
We end this section by reviewing some spectral properties of the operator $− Δ 2$, which is needed in the sequel. A real number $λ ∈ R$ is said to be an eigenvalue of $− Δ 2$ on $H T$, if there is some $u ∈ H T \ { 0 H T }$ such that
$− Δ 2 u ( n − 1 ) = λ u ( n ) , ( n ∈ Z )$
and in this case, u is called eigensequence corresponding to the eigenvalue $λ$. On account of the periodicity of u, relation (10) is equivalent to the system
$− u ( 2 ) + 2 u ( 1 ) − u ( T ) = λ u ( 1 ) − u ( 3 ) + 2 u ( 2 ) − u ( 1 ) = λ u ( 2 ) ⋮ − u ( T ) + 2 u ( T − 1 ) − u ( T − 2 ) = λ u ( T − 1 ) − u ( 1 ) + 2 u ( T ) − u ( T − 1 ) = λ u ( T ) .$
If we consider the particular circulant matrix
$M T : = 2 − 1 0 ⋯ 0 0 − 1 − 1 2 − 1 ⋯ 0 0 0 ⋮ 0 0 0 ⋯ − 1 2 − 1 − 1 0 0 ⋯ 0 − 1 2$
then, having in view (11), the eigenvalues of $− Δ 2$ are precisely the characteristic roots of $M T .$ In addition, if $y = ( y 1 , … , y T ) ∈ R T \ { 0 R T }$ is an eigenvector corresponding to a characteristic root $λ$, then its extension $u y ∈ H T$, defined by $u y ( i ) = y i$ for $i = 1 , T ¯$, is an eigensequence corresponding to the eigenvalue $λ .$ This means that an orthonormal basis of eigensequences $u 1 , … , u T$ can be constructed from an orthonormal basis of eigenvectors $x 1 , … , x T$ of $M T$ by extending $x i$ in $H T$ ($i = 1 , T ¯$) as above.
From ([12], p. 38), we know that the characteristic roots of $M T$, hence the eigenvalues of $− Δ 2$, are $4 sin 2 i π / T$ ($i = 0 , T − 1 ¯$). We can label them according to the parity of T as follows:
$T odd ̲ :$
$λ 0 = 0 , λ 2 k − 1 = λ 2 k = 4 sin 2 k π T , k = 1 , … , T − 1 2 ;$
$T even ̲ :$
$λ 0 = 0 , λ 2 k − 1 = λ 2 k = 4 sin 2 k π T , k = 1 , … , T − 2 2 , λ T − 1 = 4 .$
In both cases, we consider an orthonormal basis $e 0 , … , e T − 1$ in $H T$, such that $e i$ is an eigensequence corresponding to $λ i$ ($i = 0 , T − 1 ¯$). Observe that, by multiplying equality (10) by arbitrary $v ∈ H T$ and using summation by parts formula, one obtains that if $u ∈ H T$ and $λ ∈ R$ satisfy (10), then
$∑ j = 1 T Δ u ( j ) Δ v ( j ) = λ ( u | v ) .$
This yields
$∑ j = 1 T Δ e i ( j ) Δ e k ( j ) = λ k δ i k ( i , k ∈ { 0 , … , T − 1 } ) ,$
where $δ i k$ stands for the Kronecker delta function.

## 3. Main Result

Our main result is given in the following.
Theorem 2.
Assume that $g : R → R$ is a continuous odd function and that G satisfies (9) together with
$lim inf x → 0 2 G ( x ) x 2 ≥ 1 .$
Then, the following hold true:
(i)
If
then problem (1) has at least $2 m + 1$ distinct pairs of nontrivial solutions.
(ii)
If T is even and
$λ > 8 ( = 2 λ T − 1 ) ,$
then (1) has at least T distinct pairs of nontrivial solutions.
Proof.
We show $( i )$ in the odd case because the even case follows by exactly the same arguments, and under assumption (15), a quite similar strategy works by simply replacing “$2 m$” with “$T − 1$”.
Thus, let $0 ≤ m ≤ ( T − 1 ) / 2 .$ On account of Theorem 1 and Propositions 1 and 2, we have to prove that there exists a nonempty compact symmetric subset $A m ⊂ H T \ { 0 }$ with $γ ( A m ) ≥ 2 m + 1$, such that
$sup v ∈ A m I λ ( v ) < 0 .$
Since $λ > 2 λ 2 m$, we can choose $ε ∈ ( 0 , 1 )$, so that $λ > 2 λ 2 m / ( 1 − ε )$. Then, by virtue of (13), there exists $δ > 0$ such that
$2 G ( x ) ≥ ( 1 − ε ) x 2 as | x | ≤ δ .$
Next, we introduce the set
$A m : = ∑ k = 0 2 m α k e k : α 0 2 + ⋯ + α 2 m 2 = ρ 2 ,$
where $ρ$ is a positive number, which is chosen $≤ min 1 2 2 m + 1 , δ$.
Then, it is not difficult to see that the odd mapping $H : A m → S 2 m$ defined by
$H ∑ k = 0 2 m α k e k = α 0 ρ , α 1 ρ … , α 2 m ρ$
is a homeomorphism between $A m$ and $S 2 m$; therefore, $γ ( A m ) = 2 m + 1$.
We have that $A m ⊂ K .$ Indeed, let $v = ∑ k = 0 2 m α k e k ∈ A m$. Then, for all $j ∈ { 1 , … , T }$, we obtain
$| Δ v ( j ) | ≤ ∑ k = 0 2 m α k e k ( j + 1 ) + ∑ k = 0 2 m α k e k ( j ) ≤ 2 ∑ k = 0 2 m | α k | ≤ 2 2 m + 1 ∑ k = 0 2 m α k 2 1 / 2 = 2 ρ 2 m + 1$
and since $ρ ≤ 1 / ( 2 2 m + 1 )$, one has $| Δ v | ∞ ≤ 1$, which shows that $v ∈ K$. On the other hand, using (12), we obtain
$∑ j = 1 T | Δ v ( j ) | 2 = ∑ j = 1 T Δ ∑ k = 0 2 m α k e k ( j ) 2 = ∑ j = 1 T ∑ k = 0 2 m α k Δ e k ( j ) 2 = ∑ j = 1 T ∑ k = 0 2 m α k 2 ( Δ e k ( j ) ) 2 + ∑ i , k = 0 i ≠ k 2 m α i α k Δ e k ( j ) Δ e i ( j ) = ∑ k = 0 2 m α k 2 ∑ j = 1 T ( Δ e k ( j ) ) 2 + ∑ i , k = 0 i ≠ k 2 m α i α k ∑ j = 1 T Δ e k ( j ) Δ e i ( j ) = ∑ k = 0 2 m λ k α k 2 ≤ λ 2 m ∑ k = 0 2 m α k 2 = λ 2 m ρ 2 .$
In addition, it is clear that
$∑ j = 1 T v ( j ) 2 = ∥ v ∥ 2 = ( v | v ) = ∑ k = 0 2 m α k 2 = ρ 2 .$
Then, from (17), (19), (20) and $| v ( j ) | ≤ ρ ≤ δ$ ($j ∈ { 1 , … , T }$), it follows that
$I λ ( v ) = Ψ ( v ) + G λ ( v ) ≤ ∑ j = 1 T | Δ v ( j ) | 2 − λ 2 ( 1 − ε ) ∑ j = 1 T | v ( j ) | 2 ≤ ρ 2 λ 2 m − λ 2 ( 1 − ε ) ρ 2 = ρ 2 2 λ 2 m − λ ( 1 − ε ) 2 < 0 .$
Therefore, (16) holds true and the proof of $( i )$ is complete. □
Example 1.
If (14) holds true, then problem (3) has at least $2 m + 1$ distinct pairs of nontrivial solutions. In addition, if T is even, under assumption (15), problem (3) has at least T distinct pairs of nontrivial solutions. Notice that besides the trivial solution, problem (3) always has the pair of constant solutions $u ≡ ± 1$, and these are the only constant nontrivial solutions of (3). Therefore, problem (3) has at least $2 m$ (resp. $T − 1$) distinct pairs of nonconstant solutions if hypothesis (14) is satisfied (resp. (15) holds true).
Consider the eigenvalue type problem
$− Δ ϕ ( Δ u ( n − 1 ) ) = λ u ( n ) + h ( u ( n ) ) , u ( n ) = u ( n + T ) ( n ∈ Z )$
and set $H ( x ) = ∫ 0 x h ( τ ) d τ$$( x ∈ R ) .$
Corollary 1.
If the continuous function $h : R → R$ is odd and
$lim inf x → 0 H ( x ) x 2 ≥ 0 , lim x → + ∞ H ( x ) x 2 = − ∞ ,$
then the conclusions $( i )$ and $( i i )$ of Theorem 2 remain valid with (21) instead of (1).
Proof.
Theorem 2 applies to the problem
$− Δ ϕ ( Δ u ( n − 1 ) ) = λ u ( n ) + h ( u ( n ) ) λ , u ( n ) = u ( n + T ) ( n ∈ Z ) .$
Theorem 2 can be employed to derive the multiplicity of nontrivial solutions of autonomous non-parametric problems having the form
$− Δ ϕ ( Δ u ( n − 1 ) ) = f ( u ( n ) ) , u ( n ) = u ( n + T ) ( n ∈ Z ) .$
Setting $F ( x ) = ∫ 0 x f ( τ ) d τ$ $( x ∈ R )$, we have the following.
Corollary 2.
Assume that $f : R → R$ is a continuous odd function and that
$lim x → + ∞ F ( x ) = − ∞ .$
Then, the following hold true:
(i)
If
then problem (22) has at least $2 m + 1$ distinct pairs of nontrivial solutions.
(ii)
If T is even and
$lim inf x → 0 F ( x ) x 2 > 4 ,$
then (22) has at least T distinct pairs of nontrivial solutions.
Proof.
From (24), there exists $λ ¯ > 0$ such that
$lim inf x → 0 2 F ( x ) x 2 ≥ λ ¯ > 8 sin 2 m π T$
and the result follows from Theorem 2 with $g ( x ) = f ( x ) / λ ¯$; a similar argument works when (25) is fulfilled. □
Example 2.
Let $f a : R → R$ be given by
$f a ( x ) = 2 x sin | x | − 1 2 − x | x | − 1 2 cos | x | − 1 2 2 + 2 a x − 4 x 3 ( x ∈ R ) .$
Then,
$F a ( x ) = x 2 sin | x | − 1 2 + a − x 2 ( x ∈ R )$
and by Corollary 2, we obtain that, if
then the equation
$− Δ ϕ ( Δ u ( n − 1 ) ) = f a ( u ( n ) ) ( n ∈ Z )$
has at least$2 m + 1$distinct pairs of nontrivial T-periodic solutions, while if T is even and$a > 5$, then (26) has at least T distinct pairs of nontrivial T-periodic solutions.
Remark 2.
A multiplicity result for odd perturbations of the discrete p-Laplacian operator is obtained in [13] using a Clark-type result in the frame of the classical critical point theory.

## Author Contributions

Writing—original draft, P.J. and C.Ş.; Writing—review & editing, P.J. and C.Ş. All authors contributed equally to the manuscript. All authors have read and agreed to the published version of the manuscript.

## Funding

This research received no external funding.

Not applicable.

Not applicable.

Not applicable.

## Acknowledgments

The authors are grateful to the anonymous referees for the very useful comments and suggestions, which helped them to improve the presentation of the paper.

## Conflicts of Interest

The authors declare no conflict of interest.

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Jebelean, P.; Şerban, C. Multiple Periodic Solutions for Odd Perturbations of the Discrete Relativistic Operator. Mathematics 2022, 10, 1595. https://doi.org/10.3390/math10091595

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Jebelean P, Şerban C. Multiple Periodic Solutions for Odd Perturbations of the Discrete Relativistic Operator. Mathematics. 2022; 10(9):1595. https://doi.org/10.3390/math10091595

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Jebelean, Petru, and Călin Şerban. 2022. "Multiple Periodic Solutions for Odd Perturbations of the Discrete Relativistic Operator" Mathematics 10, no. 9: 1595. https://doi.org/10.3390/math10091595

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