Periodic solutions for p(t)-Lienard equations with a singular nonlinearity of attractive type

Calin Serban; Jean Mawhin; Petru Jebelean

Periodic solutions for p(t)-Lienard equations with a singular nonlinearity of attractive type

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Pith's one-line read The paper proves that an attractive singular p(t)-Lienard equation has a T-periodic solution whenever some level alpha of the restoring term g sits above max h and g eventually stays below the mean of h.

desk verdict Solid, honest extension of the constant-exponent attractive singular Lienard existence theory to the p(t)-Laplacian setting; the proof is transparent and the main caveat is a normal reliance on an imported continuation theorem. read the letter →

arxiv 2506.04927 v2 pith:DFAHG5GX submitted 2025-06-05 math.AP math.CA

Petru Jebelean , Jean Mawhin , Calin Serban This is my paper

classification math.APmath.CA MSC 34B1634C2534B1547H11

keywords p(t)-LaplacianattractivesingularityLienardequationperiodicsolutionsBrouwerdegreeloweranduppersingularnonlinearity

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper establishes a sufficient condition for T-periodic positive solutions of a Lienard equation whose diffusion term is the variable-exponent p(t)-Laplacian and whose restoring force g may blow up at zero, an attractive singularity. The condition is one-sided: some level $\alpha>0$ has $g(\alpha)$ at least the maximum of the forcing $h$, and $g(x)$ stays below the mean value of $h$ for all large $x$. Under that condition a solution exists and is trapped above $\alpha$. When $g$ is positive, blows up at zero, and decays to zero at infinity, the condition becomes a clean if-and-only-if: solutions exist exactly when the forcing has positive mean. The result matters because it extends a classical constant-exponent theory to the variable-exponent setting without a variational structure.

What carries the argument

The load-bearing mechanism is a continuation theorem (Theorem 2.3) for the variable-exponent operator $\phi_{p(t)}(x)=|x|^{p(t)-2}x$, imported from recent work on generalized variable-exponent systems; it packages existence for the homotopy problems (2.2) and (3.2) into a Brouwer-degree condition on a one-dimensional averaged equation. Two internal tools do the heavy lifting: a priori estimates that control $\|u'\|_\infty$ through an energy identity and Sobolev and Poincare-Wirtinger inequalities, and a lower/upper solution theorem (Theorem 3.3), proved directly from the continuation theorem, which converts two bracketing functions into a genuine solution. The upper solution is manufactured by Proposition 2.8, which says zero-mean solutions of an auxiliary Lienard equation are bounded independently of the additive constant, so a large shift lands them in the region where $g < \bar h$.

What would settle it

A concrete counterexample would settle it: find continuous $p$, $f$, $h$ and attractive $g$ satisfying $g(\alpha) \ge \max h$ for some $\alpha$ and $\limsup_{x\to+\infty} g(x) < \bar h$, yet with no positive $T$-periodic solution; a natural place to look is a non-monotone $g$ that dips below $h$ at intermediate arguments, with $p(t)$ genuinely nonconstant.

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Extended reading notes

Core claim

Theorem 4.1 states that if there exists $\alpha > 0$ with $g(\alpha) \ge \max_{[0,T]} h$ and $\limsup_{x\to+\infty} g(x) < \bar h$, where $\bar h$ is the mean of $h$ on $[0,T]$, then problem (1.1) has at least one solution $u$ with $u(t) \ge \alpha$ for every $t$. The proof uses the constant $\alpha$ as a lower solution and constructs an upper solution $\beta$ by adding a zero-mean periodic solution of an auxiliary Lienard equation to a large constant; bounds on the added solution are independent of the constant, which keeps $\beta$ in the region where $g$ is below $\bar h$. Corollary 4.3 draws the if-and-only-if consequence for attractive $g$: if $g$ is positive, blows up at zero, and tends to zero at infinity, then solutions exist exactly when $\bar h > 0$.

Load-bearing premise

The proof leans on a previously established continuation theorem for a general class of variable-exponent operators; if that theorem does not cover the specific scalar p(t)-Laplacian operator exactly as applied here, the existence chain collapses even though all later estimates are self-contained.

Editorial extensions

If this is right

The variable-exponent case inherits the constant-exponent existence theory for attractive singularities, closing the gap previously treated only for constant p.
For every positive continuous g that blows up at zero and decays to zero at infinity, the equation has a T-periodic solution if and only if the mean forcing is positive, with the damping f arbitrary and continuous.
The proof produces a localized solution alpha <= u <= beta, so existence comes with explicit bracketing rather than a bare fixed-point guarantee.
The repulsive-type analogue remains open, and the paper identifies it as the next natural target.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial extension: the lower/upper construction would likely survive with any auxiliary equation whose zero-mean solutions have bounds independent of the additive constant, so the limsup condition might be replaceable by a weaker eventual-boundedness assumption.
Editorial extension: a quantitative reading of Proposition 2.8 could turn the sup-norm constant K into an explicit function of the period, the minimum exponent, and the L1 norm of the zero-mean part of h, making the existence theorem constructive enough for numerical orbit searches.
Editorial extension: the if-and-only-if corollary suggests the sign of the mean forcing is the only obstruction for a whole class of attractive nonlinearities; testing non-monotone g or forcing with zero mean would probe whether the criterion is necessary beyond the positive-g case.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The short version: this is a clean extension of the known constant-exponent existence theory for attractive singular Lienard equations to the p(t)-Laplacian setting. The main theorem (4.1) is stated precisely, and Corollary 4.3 gives the expected if-and-only-if condition when g is positive, attractive at 0, and vanishes at infinity. The proof is transparent: a continuation theorem from [4], a priori estimates with a c-independent bound, then a direct lower/upper-solution argument. Nothing is fitted, and the repulsive case is explicitly left open.

What is genuinely new is the variable-exponent setting itself. [7] and [18] handle constant p; this paper completes that extension for attractive forces. The lower/upper-solution theorem (3.3) for this class is proved directly and is a useful tool in its own right.

Soft spots, in proportion: the paper leans on Theorem 2.3, imported without proof from [4]. That is a normal reliance on a published continuation theorem, and the scalar phi_{p(t)} mapping is squarely within that theorem's scope; I checked that the stated scope covers p continuous with p_->1. So it is a dependency, not a gap. The upper-solution construction in Theorem 4.1 uses the c-independent bound from Proposition 2.8; that bound is proved internally. There is a minor typo in inequality (2.7) (an exponent slip that is not used later), so it deserves a small revision, not a rejection. The proof of Corollary 4.3 uses only integration and g>0; that is fine.

The citation pattern is appropriate: the paper credits the relevant history, distinguishes the variational approach, and explicitly notes what remains open. The reader's take (ACCEPT, moderate confidence) is fair; I don't see a load-bearing flaw.

This paper is for researchers working on periodic solutions of singular ODEs or p(t)-Laplacian problems. It is not a landmark, but it is a solid, publishable contribution that closes a stated gap. I would send it to a serious referee.

Referee Report

0 major / 5 minor

Summary. This paper studies the periodic problem (1.1) for a Liénard equation driven by the p(t)-Laplacian, with a continuous restoring term g that may have an attractive singularity at 0. The main result, Theorem 4.1, states that if some positive level α satisfies g(α) ≥ max_{[0,T]} h and limsup_{x→+∞} g(x) < \bar h, then a positive T-periodic solution exists with u ≥ α. Corollary 4.3 gives the sharp necessary and sufficient condition \bar h > 0 for attractive singularities satisfying g(x) → +∞ as x → 0+ and g(x) → 0 as x → +∞. The proof combines a continuation theorem of García-Huidobro–Manásevich–Mawhin–Tanaka (Theorem 2.3), a priori estimates for auxiliary problems (Theorems 2.4 and 2.7, Proposition 2.8), and a dedicated lower/upper solutions theorem (Theorem 3.3) for the p(t)-Laplacian.

Significance. The result extends the constant-p theorem of Jebelean–Mawhin [7] to variable exponent p(t), and the iff criterion for attractive singularities is a clean and useful statement. The paper is largely self-contained: the a priori bounds in Section 2 and the lower/upper solutions argument in Section 3 are proved in detail, and the only imported ingredient is the continuation theorem from [4], which is a normal citation rather than a circular or fitted assumption. The estimates are explicit in terms of T, p_-, and the L^1 norm of the mean-zero part of h, with no parameters fitted to any data. If the result holds, it is a solid contribution to the theory of variable-exponent Liénard equations with singularities.

minor comments (5)

[Section 2, Eq. (2.7)] The displayed inequality has the wrong exponent; it should read T^{-(p_- - 1)/p_-} \|u'\|_{L^1} ≤ \|u'\|_{L^{p_-}}. The subsequent definition R_2 = T^{(p_- - 1)/p_-} R_1 uses the correct exponent, so the argument goes through, but the display should be corrected.
[Abstract and Introduction] The first displayed equation in the abstract contains a parenthesis error: it should be \left(|u'(t)|^{p(t)-2} u'(t)\right)'. Similarly, condition (R) in the Introduction has a missing closing parenthesis after min{0,h}.
[Section 3, display (3.7)] The chain of equalities in (3.7) is difficult to parse, and the inequality defining M_2 is ambiguous as typeset; adding parentheses or an explanatory sentence would help the reader see that [M_2^{p_- -1} - (2M_1/T + 1)^{p_+ -1}]/c_3 is being compared with T + 2M_1.
[Section 2, Theorem 2.3] Because the rest of the paper depends on this imported continuation theorem, the authors should add a sentence specifying how [4, Theorem 4.1] applies to the scalar mapping (t,x) ↦ |x|^{p(t)-2}x; the current phrase "immediate consequence" is terse.
[Section 3, proof of Theorem 3.3] The phrase "using the first change of variables formula" is slightly misleading; the identity in (3.7) is just the fundamental theorem of calculus applied to s(t) = |u'(t)|^{p(t)-1}. Consider rewording.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main existence theorem is derived from internal a priori estimates plus a general external continuation theorem, not from its own conclusion.

full rationale

The derivation does not fit parameters to data, does not define assumptions in terms of the conclusion, and does not rename a known result. The central result (Theorem 4.1) reduces problem (1.1) to the construction of an upper solution from Proposition 2.8; that proposition is proved inside the paper from the auxiliary equation (2.12) with a uniform bound K independent of θ, c, and u. The a priori estimates in Theorems 2.4 and 2.7 are derived by direct integration and Sobolev/Poincaré inequalities, not by assuming existence for (1.1). The lower/upper solutions theorem (Theorem 3.3) is proved in-paper from the same continuation theorem. The only external input is Theorem 2.3, imported from [4] by co-author Mawhin; this is a general continuation theorem for variable-exponent operators, published independently, whose stated assumptions do not include the target result. Its use is a normal reliance on a published theorem, not a circular reduction. Self-citations to [7] document the constant-p predecessor and are historical, not load-bearing for the p(t)-extension. Thus no circular step is exhibited.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new entities. It rests on the continuation theorem from [4] (which is not proved in the paper), standard degree theory and Sobolev inequalities, and the modeling assumption that solutions remain positive because g is singular at 0.

assumptions (5)

standard math Theorem 2.3, the continuation theorem for the scalar p(t)-Laplacian, is an immediate consequence of [4, Theorem 4.1] and is accepted without proof.
Used in Theorems 2.4, 2.7, and 3.3 to guarantee existence in the homotopy and modified problems; the present paper does not reproduce the proof.
standard math Brouwer degree properties for the finite-dimensional map L(a) in condition (H3).
Invoked in Theorems 2.4 and 3.3 to verify deg_B[L, Omega cap R, 0] != 0.
standard math Poincare-Wirtinger inequality and the Sobolev embedding W^{1,p_-} subset C([0,T]).
Used in Proposition 2.8 to obtain the c-independent bound K on ||v||_infinity.
domain assumption The mapping phi_{p(t)}(x)=|x|^{p(t)-2}x is an increasing homeomorphism on R for each t.
Used throughout, for example in Theorem 3.3 to compare alpha' and u' via the homeomorphism; this follows from p(t)>1.
domain assumption g is defined only on (0,infinity), so solutions are required to be positive; the paper constructs lower and upper solutions with positive values.
Used in Definition 3.1 and Theorem 4.1; positivity is central to the attractive singularity.

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Pith. "Pith review of Periodic solutions for p(t)-Lienard equations with a singular nonlinearity of attractive type." pith.science (2026). https://pith.science/paper/DFAHG5GX

@misc{pith2026250604927,
  author       = {Pith},
  title        = {Pith review of: Periodic solutions for p(t)-Lienard equations with a singular nonlinearity of attractive type},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DFAHG5GX}},
  note         = {Machine review of arXiv:2506.04927}
}

abstract

We are concerned with the existence of $T$-periodic solutions to an equation of type $$\left (|u'(t))|^{p(t)-2} u'(t) \right )'+f(u(t))u'(t)+g(u(t))=h(t)\quad \mbox{ in }[0,T]$$ where $p:[0,T]\to(1,\infty)$ with $p(0)=p(T)$ and $h$ are continuous on $[0,T]$, $f,g$ are also continuous on $[0,\infty)$, respectively $(0,\infty)$. The mapping $g$ may have an attractive singularity (i.e. $g(x) \to +\infty$ as $x\to 0+$). Our approach relies on a continuation theorem obtained in the recent paper M. Garc\'{i}a-Huidobro, R. Man\'{a}sevich, J. Mawhin and S. Tanaka, J. Differential Equations (2024), a priori estimates and method of lower and upper solutions.

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [4]

Garc ´ ıa-Huidobro, R

M. Garc ´ ıa-Huidobro, R. Man´ asevich, J. Mawhin and S. Tanaka, Periodic solutions for nonlinear systems of ODE’s with generalized variable exponents operators,J. Differential Equations388(2024), 34–58

work page 2024

[2] [7]

Jebelean and J

P. Jebelean and J. Mawhin, Periodic solutions of singular nonlinear perturbations of the ordinaryp- Laplacian,Adv. Nonlinear Stud.2(2002), 299–312

work page 2002

[3] [18]

Rach ˚ unkov´ a and M

I. Rach ˚ unkov´ a and M. Tvrd´ y, Periodic singular problem with quasilinear differential operator,Math. Bohem.131(3) (2006), 321–336

work page 2006

[4] [1]

Cabada, A

A. Cabada, A. Lomtatidze and M. Tvrd´ y, Periodic problem involving quasilinear differential operator and weak singularity,Adv. Nonlinear Stud.7(2007), 629–649

work page 2007

[5] [2]

De Coster and P

C. De Coster and P. Habets, The lower and upper solutions method for boundary value problems,Hand- book of Differential Equations, Elsevier/North-Holland, Amsterdam, 2004, 69–160

work page 2004

[6] [3]

De Coster and P

C. De Coster and P. Habets,Two-Point Boundary Value Problems : Lower and Upper Solutions, Math- ematics in Science and Engineering, vol. 205, Elsevier, Amsterdam, 2006

work page 2006

[7] [5]

Garc ´ ıa-Huidobro, R

M. Garc ´ ıa-Huidobro, R. Man´ asevich, J. Mawhin and S. Tanaka, Two point boundary value problems for ordinary differential systems with generalized variable exponents operators,Nonlinear Anal. Real World Appl.81(2024), 13 p

work page 2024

[8] [6]

Habets and L

P. Habets and L. Sanchez, Periodic solutions of some Li´ enard equations with singularities,Proc. Amer. Math. Soc.109(1990), 1035–1044

work page 1990