Pith. sign in

REVIEW 3 major objections 4 minor 33 references

Positive solutions of $p$-Laplacian fractional differential equations with fractional derivative boundary condition

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper proves existence and uniqueness of positive solutions to a three-point p-Laplacian fractional boundary value problem with Caputo derivative, using Krasnoselskii's fixed point theorem, the Leray-Schauder alternative, and the…

desk verdict Routine but competent fractional p-Laplacian BVP paper: the existence results are sound, but Theorem 3.5 has a range error that needs fixing before the p>2 uniqueness claim can stand. read the letter →

arxiv 1908.03966 v1 pith:BIWCGCO2 submitted 2019-08-11 math.CA

Faouzi Haddouchi This is my paper · ORCID
classification math.CA MSC 34A0826A3334B18
keywords fractionaldifferentialequationp-LaplacianpositivesolutionsCaputoderivativeGreen'sfunctionfixedpointtheoremconeexistenceanduniqueness
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

This paper studies a boundary value problem combining a fractional derivative of order between 2 and 3, the p-Laplacian operator phi_p(s)=|s|^{p-2}s, and a three-point boundary condition involving derivatives at coupled points. The main existence result (Theorem 3.1) states that when the nonlinearity f is bounded above by a phi_p-scaled constant on a whole interval and bounded below by another phi_p-scaled constant on a subinterval, with constants satisfying a ratio condition, then a positive solution exists with norm between two prescribed radii. Uniqueness is claimed under Lipschitz conditions on f, separately for the cases 12. The arguments express solutions as fixed points of an integral operator built from an explicit Green's function, then apply cone fixed point theorems and contraction estimates. A sympathetic reader would care because fractional p-Laplacian problems model non-Newtonian flow and diffusion in porous media, and this paper provides computable criteria for when such models admit positive steady states.

What carries the argument

The central object is the Green's function K(t,s)=G(t,s)+H(eta,s), with G and H explicitly defined from the fractional integral kernels, which encodes the three-point boundary condition u(1)+u'(1)=u'(eta). The solution operator A defined by Au(t)=∫$_0^{1}$ K(t,s) phi_q(∫_0^s a(tau) f(tau,u(tau)) dtau) ds maps the cone P={u>=0: min_{t in [0,rho]} u(t) >= gamma ||u||}, where gamma=(1-$eta^{{alpha-2}}$)(1-$rho^{{alpha-1}}$), into itself and is completely continuous. The Green's function estimates provide the upper bound ||Au|| <= M1 rho2 and the lower bound ||Au|| >= M2 rho1 that drive the cone-compression fixed point theorem, while Lemma 2.18's p-Laplacian difference estimates supply the contraction constants for uniqueness.

What would settle it

For a concrete check, take p>2 so that q<2, choose $\sigma$ in the interval (1/(2-q), 2/(2-q)), for example p=7/2, q=7/5, $\sigma$=1.5, and compute the $\beta$ function B($\alpha$-1, $\sigma$(q-2)+1) that appears in the contraction coefficient. For $\alpha$=5/2 and $\sigma$=1.5, the second argument is $\sigma$(q-2)+1 = 1.5*(-0.6)+1 = 0.1, which is positive, but the exponent $\sigma$(q-2) = -0.9, and the integral ∫$_0^{1}$ $s^{{-0.9}}$ ds diverges (the $\beta$ function is defined only when both arguments are positive, which requires $\sigma$(q-2)+1 > 0). Evaluating the integral numerically for $\sigma$=1.5 shows divergence, contradicting the convergence assumed in the proof.

Watch

Extended reading notes

Core claim

The central claim is Theorem 3.1: assume f and a are continuous, nonnegative, with a not identically zero on any subinterval of [0,1]. If there exist constants rho1>0, rho2>0 with rho1<rho2, M1 in (0,Lambda1], M2 in [Lambda2,infinity), and M2*rho1 < M1*rho2, such that f(t,u) <= phi_p(M1*rho2) for all u in [0,rho2] and t in [0,1], and f(t,u) >= phi_p(M2*rho1) for all u in [gamma*rho1,rho1] and t in [0,rho], then the boundary value problem (1.1)-(1.2) has at least one positive solution u in the cone P with rho1 < ||u|| < rho2. The paper also claims uniqueness in two regimes: for 1<p<2 under a bounded nonlinearity with a small Lipschitz constant (Theorem 3.4), and for p>2 under a lower-growth condition a(t)f(t,u) >= mu^$\sigma$ $t^{{sigma-1}}$ and a small Lipschitz constant (Theorem 3.5).

Load-bearing premise

The contraction proof in Theorem 3.5 relies on the inequality $\sigma$(2-q)<1 to make the integral of $s^{{sigma(q-2)}}$ converge; the assumption as stated (0<$\sigma$<2/(2-q)) allows values that make the $\beta$ function undefined, so the uniqueness claim for p>2 is not proven in that wider range.

Editorial extensions

If this is right

  • If the existence conditions of Theorem 3.1 hold, then the fractional boundary value problem has at least one positive solution whose maximum lies strictly between rho1 and rho2, providing explicit norm control useful for applications.
  • The constants Lambda1 and Lambda2 are computable from the weight function a and the kernel K, so the theorem yields a concrete test: check f against phi_p-scaled values and the ratio M2*rho1 < M1*rho2.
  • The Leray-Schauder alternative (Theorem 3.3) gives existence when the nonlinearity's maximum on a ball is small enough relative to an integral of the kernel, a condition that is often easier to verify than two-sided bounds.
  • For 1<p<2, Theorem 3.4 gives uniqueness under a boundedness condition f(t,u)<=k(t) and a Lipschitz constant below an explicitly computed threshold, ensuring the solution operator is a contraction.
  • For p>2, Theorem 3.5 claims uniqueness when the product a(t)f(t,u) grows at least like a power t^{sigma-1} and the Lipschitz constant is small, with the contraction constant expressed through beta functions.

Reading between the lines

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

  • The stated range 0<sigma<2/(2-q) in Theorem 3.5 is wider than the range for which the key integral ∫_0^1 s^{sigma(q-2)} ds converges; the proof requires sigma(2-q)<1, so for q<2 the uniqueness claim is only established when sigma < 1/(2-q). In the extra interval (1/(2-q), 2/(2-q)), the beta function B(alpha-1, sigma(q-2)+1) is not defined, leaving the contraction argument incomplete as stated.
  • The cone compression method likely extends to other multi-point boundary conditions that yield a Green's function satisfying the same two-sided bound (1-eta^{alpha-2})(1-t^{alpha-1})Phi(s) <= K(t,s) <= Phi(s), so the existence criteria could be adapted to more general nonlocal conditions.
  • The ratio condition M2*rho1 < M1*rho2 can be interpreted as a slope condition forcing the nonlinearity to cross a phi_p-scaled rectangle; this suggests a connection to shooting methods or to the idea of 'height' of nonlinearities used in other fixed-point approaches.
  • A numerical test of Theorem 3.1 is feasible: choose an explicit a and f that satisfy the bounds, then discretize the fractional boundary value problem and solve it to verify that a positive solution with norm in (rho1,rho2) actually exists, which would corroborate the abstract theorem.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The manuscript studies the three-point boundary value problem (φ_p(D^α u))' + a(t)f(t,u)=0 with Caputo derivative boundary conditions, where 2<α≤3 and φ_p is the p-Laplacian. The author constructs an explicit Green's function K(t,s), a cone P of functions satisfying min over [0,ρ] ≥ γ||u||, and a completely continuous fixed-point operator A. Existence of positive solutions is obtained by Krasnosel'skii cone compression (Theorems 3.1 and 3.2) and by the Leray-Schauder alternative (Theorem 3.3). Uniqueness is claimed for 1<p<2 in Theorem 3.4 and for p>2 in Theorem 3.5. Three examples are given to illustrate the results. The existence arguments and Green's function estimates are largely coherent, but the p>2 uniqueness theorem has an integrability-range error, and two of the examples do not satisfy hypothesis (H1).

Significance. If the technical issues are repaired, the paper would be a useful contribution to the literature on fractional p-Laplacian boundary value problems. The explicit Green's function estimates in Lemmas 2.14-2.16 are clean and lead to a transparent cone-compression argument; Theorems 3.1 and 3.3 are coherent and give checkable sufficient conditions. However, the advertised uniqueness for p>2 is not proven as stated because the parameter range in Theorem 3.5 is too permissive, and the examples that violate (H1) undermine the numerical illustrations. These are localized, fixable problems rather than a failure of the whole framework.

major comments (3)
  1. [Theorem 3.5, equation (3.6) and proof] The stated range 0<σ<2/(2−q) is too permissive. Since p>2 gives q<2, the exponent q−2 is negative; the proof uses |φ_q(x)−φ_q(y)| ≤ (q−1)(μs^σ)^{q−2}|x−y| and then integrates s^{σ(q−2)} against the Green's function. The beta integrals B(α−1,σ(q−2)+1) and B(α,σ(q−2)+1) are finite only when σ(q−2)+1>0, i.e. σ<1/(2−q). For σ in [1/(2−q),2/(2−q)) the integrand is nonintegrable and the displayed constant L is infinite, so the contraction argument does not prove uniqueness for p>2. Since continuity of a f also forces σ≥1 in any realized example, the extra interval is nonempty; the hypothesis should be restricted to σ<1/(2−q).
  2. [Section 4, Examples 4.1 and 4.2] Both examples fail hypothesis (H1), because f(t,0)=0 for all t: f(t,u)=1/2 t ln(u+1) in Example 4.1 and f(t,u)=e^{-t} sin^2 u in Example 4.2. Consequently A(0)=0, so u=0 is a fixed point. Theorem 3.3 and Theorem 3.4 require (H1) and conclude a positive solution (or a unique solution in the setting of Theorem 3.4); as written, the examples do not establish a positive solution. For Example 4.2, the contraction proof actually forces the unique fixed point to be zero, so the advertised positive uniqueness is not illustrated. The examples should be modified so that f(t,0)>0 on a set of positive measure, or the theorems should be re-stated for the case f(t,0)=0 with a separate nontriviality argument.
  3. [Theorems 3.4 and 3.5, proofs] The contraction arguments are written for arbitrary u,v∈B, but the operator A is only defined on the cone P because f and a f are only defined for u≥0. Banach's theorem is then applied on the wrong space unless f is extended to negative values and the Lipschitz conditions are verified for the extension. This is easily repaired by applying the contraction mapping theorem to the closed subset P (which is sufficient for uniqueness of positive solutions), but the current wording overstates the conclusion. The proofs should state explicitly which complete metric space is used.
minor comments (4)
  1. [Section 2, notation] The conjugate exponent q, defined by 1/p+1/q=1, is used throughout Section 2 but is not explicitly defined in the main text; define it next to problem (1.1).
  2. [Throughout] The symbol B is used both for the Banach space C([0,1],R) and for the Euler beta function, which is confusing in Sections 2 and 3.
  3. [Throughout] There are numerous typographical errors ('equa tions', 'differential', 'then u⁄∈∂U') that should be corrected before final submission.
  4. [Theorem 3.2] Theorem 3.2 is stated without proof; since it is a close analogue of Theorem 3.1, it would be helpful to say explicitly that it follows by reversing the roles of ρ1 and ρ2.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: all results are derived from explicit hypotheses (H1)-(H2) and standard external fixed point/contraction theorems.

full rationale

The paper's derivation chain is self-contained in the sense relevant to circularity. The existence results (Theorems 3.1, 3.2, 3.3) are standard Krasnoselskii and Leray-Schauder cone arguments. The constants Lambda1 and Lambda2 are explicit functionals of the data a and the Green's kernel Phi; the assumptions on f (e.g., f(t,u) <= phi_p(M1 rho2) on [0,rho2] and f(t,u) >= phi_p(M2 rho1) on [gamma rho1, rho1]) are hypotheses imposed on the nonlinearity, not consequences extracted from the conclusion. The fixed point theorems cited ([13], [4], Banach's theorem) are external and standard, and no step defines an unknown in terms of the target solution. The uniqueness theorems are direct contraction estimates: Theorem 3.4 uses the external Lemma 2.18(ii) from reference [9] and integrates the kernel against explicit beta functions; Theorem 3.5 similarly bounds the difference of phi_q via Lemma 2.18(i) and integrates K(t,s)s^{sigma(q-2)}. No parameter is fitted to a subset of data and renamed a prediction, no self-citation carries the argument, and no known result is merely renamed. The only issue I found is a non-circular correctness concern in Theorem 3.5: the statement allows 0 < sigma < 2/(2-q), but the beta integral B(alpha-1, sigma(q-2)+1) used in the contraction constant is finite only when sigma(q-2)+1 > 0, i.e. sigma < 1/(2-q); for p > 2 this leaves a nonempty interval where the displayed contraction constant is not finite. This is a proof gap, not a circularity, and therefore does not change the circularity score.

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

The paper introduces no new physical or mathematical entities and fits no parameters to data. Its conclusions rest on standard fixed point theorems and fractional calculus identities, plus the explicit regularity and nonzero hypotheses on f and a. The Green's function, cone and constants ρ, γ, Λ1, Λ2 are all constructed within the paper.

assumptions (5)
  • standard math Krasnosel'skii fixed point theorem on a cone
    Used in Theorems 3.1 and 3.2 to produce a fixed point of the operator A.
  • standard math Leray-Schauder nonlinear alternative
    Used in Theorem 3.3 to show existence.
  • standard math Banach fixed point theorem
    Used in Theorems 3.4 and 3.5 for uniqueness via contraction.
  • standard math Caputo derivative and fractional integral identities
    From Kilbas et al. and Podlubny; underpins Lemma 2.13 solution representation and Lemma 2.7.
  • domain assumption Hypotheses (H1) and (H2) on f and a
    The theorems assume f is continuous, nonnegative, f(t,0) not identically zero, and a is continuous, nonnegative and nonzero on every subinterval.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Positive solutions of $p$-Laplacian fractional differential equations with fractional derivative boundary condition." pith.science (2026). https://pith.science/paper/BIWCGCO2

@misc{pith2026190803966,
  author       = {Pith},
  title        = {Pith review of: Positive solutions of $p$-Laplacian fractional differential equations with fractional derivative boundary condition},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BIWCGCO2}},
  note         = {Machine review of arXiv:1908.03966}
}
abstract

In this paper, we show some results about the existence and the uniqueness of the positive solution for a $p$-Laplacian fractional differential equations with fractional derivative boundary condition. Our results are based on Krasnosel'skii's fixed point theorem, the nonlinear alternative of Leray-Schauder type and contraction mapping principle. Three examples are given to illustrate the applicability of our main results.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

  1. [1]

    L. E. Bobisud, Steady-state turbulent flow with reaction , Rocky Mountain J. Math., (3)21 (1991),993– 1007

    work page 1991

  2. [2]

    T. Chen, W. Liu, Anti-periodic boundary value problem for fractional differ ential equation with p- Laplacian operator. Appl. Math. Lett., (11)25 (2012), 1671–1675

    work page 2012

  3. [3]

    Cheng, W

    L. Cheng, W. Liu, Q. Ye, Boundary value problem for a coupled system of fractional di fferential equa- tions with p-Laplacian operator at resonance , Electron. J. Differential Equations, 60(2014), 12 pages

    work page 2014

  4. [4]

    Granas, J

    A. Granas, J. Dugundji, Fixed point theory, Springer Monographs in Mathematics , Springer-Verlag, New York, (2003)

    work page 2003

  5. [5]

    W. Han, L. Suli and L. Huilai, Positive solutions to p-Laplacian fractional differential equations with infinite-point boundary value conditions , Adv. Difference Equ. 2018, Paper No. 425, 15 pp. https://doi.org/10.1186/s13662-018-1886-2

    work page doi:10.1186/s13662-018-1886-2 2018

  6. [6]

    Hongling, H

    L. Hongling, H. Zhenlai, S. Shurong, Multiplicity of positive solutions for Sturm-Liouville bo undary value problems of fractional differential equations with p-Laplacian, Bound. Value Probl. 2014, 2014:26, 17 pp

    work page 2014

  7. [7]

    L. Hu, S. Zhang, Existence results for a coupled system of fractional differe ntial equations with p- Laplacian operator and infinite-point boundary conditions , Bound. Value Probl., 2017, Paper No. 88, 16 pp

    work page 2017

  8. [8]

    Z. Hu, W. Liu, J. Liu, Existence of solutions of fractional differential equation with p-Laplacian operator at resonance, Abstr. Appl. Anal. 2014, Art. ID 809637, 7 pp

    work page 2014

Show all 33 references
  1. [9]

    H. Khan, W. Chen,H. Sun, Analysis of positive solution and Hyers-Ulam stability for a class of singular fractional differential equations with p-Laplacian in Banach space , Math. Methods Appl. Sci. (9)41 (2018),3430–3440

    2018

  2. [10]

    H. Khan, Y. Li, H. Sun, A. Khan, Existence of solution and Hyers-Ulam stability for a couple d system of fractional differential equations with p-Laplacian operator, J. Nonlinear Sci. Appl. (10)10 (2017), 5219–5229

    2017

  3. [11]

    A. Khan, Y. Li, K. Shah, T. S. Khan, On coupled p-Laplacian fractional differential equations with nonlinear boundary conditions , Complexity 2017, Art. ID 8197610, 9 pp

    2017

  4. [12]

    A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equatio ns, Elsevier, Amsterdam, The Netherlands, 2006

    2006

  5. [13]

    M. A. Krasnosel’skii, Positive solutions of operator equations , P. Noordhoff, Groningen, The Nether- lands, 1964

    1964

  6. [14]

    Z. Li, W. Fanglei, and R. Yuanfang, Existence of Nontrivial Solutions for Fractional Different ial Equa- tions with p-Laplacian, Journal of Function Spaces, J. Funct. Spaces 2019, Art. ID 3 486410, 12 pp. https://doi.org/10.1155/2019/3486410. 16 F. HADDOUCHI

    2019 doi

  7. [15]

    Z. Liu, L. Lu, A class of BVPs for nonlinear fractional differential equati ons with p-Laplacian operator, Electron. J. Qual. Theory Differ. Equ., 70(2012), 16 pp

    2012

  8. [16]

    Luca, Positive solutions for a system of fractional differential e quations with p-Laplacian operator and multi-point boundary conditions , Nonlinear Anal

    R. Luca, Positive solutions for a system of fractional differential e quations with p-Laplacian operator and multi-point boundary conditions , Nonlinear Anal. Model. Control., (5)23 (2018), 771–801

    2018

  9. [17]

    Nazim I, U

    M. Nazim I, U. Sinem, Existence of solutions of fractional boundary value proble ms with p-Laplacian operator, Bound. Value Probl. 2015, 2015:99, 16 pp

    2015

  10. [18]

    Perera, M

    K. Perera, M. Squassina, Y. Yang, A note on the Dancer-Fuck spectra of the fractional p-Laplac ian and Laplacian operators , Adv. Nonlinear Anal. (1)4 (2015), 13–23

    2015

  11. [19]

    Perera, M

    K. Perera, M. Squassina, Y. Yang, Bifurcation and multiplicity results for critical fractio nal p-Laplacian problems, Math. Nachr., (2-3)289 (2016), 332–342

    2016

  12. [20]

    Podlubny, Fractional Differential Equations , Academic Press, New York, NY, USA, 1999

    I. Podlubny, Fractional Differential Equations , Academic Press, New York, NY, USA, 1999

    1999

  13. [21]

    K. R. Prasad, B. M. B. Krushna, Multiple positive solutions for a coupled system of p-Laplacian fractional order two-point boundary value problems , Int. J. Differ. Equ., 2014 (2014), 10 pages

    2014

  14. [22]

    Pucci, M

    P. Pucci, M. Xiang, B. Zhang, Existence andmultiplicity of entire solutions for fractio nal p-Kirchhoff equations, Adv. Nonlinear Anal., (1)5 (2016), 27–55

    2016

  15. [23]

    T. Shen, W. Liu, X. Shen, Existence and uniqueness of solutions for several BVPs of fr actional differ- ential equationswith p-Laplacian operator, Mediterr. J. Math., (6)13 (2016), 4623-4637

    2016

  16. [24]

    J. Tan, M. Li, Solutions of fractional differential equations with p-Laplacian operator in Banach spaces , Bound. Value Probl. 2018, Paper No. 15, 13 pp

    2018

  17. [25]

    W ang, H

    J. W ang, H. Xiang, Upper and lower solutions method for a class of singular frac tional boundary value problems with p-Laplacian Operator, Abstr. Appl. Anal. (2010) (Art. ID 971824)

    2010

  18. [26]

    Xiaosong, W

    T. Xiaosong, W. Xinchang, W. Zhiwei, O. Peichang, The existence of solutions for mixed fractional resonant boundary value problem with p(t)-Laplacian operator, J. Appl. Math. Comput., 2019, pp 1–14. https://doi.org/10.1007/s12190-019-01264-z

    2019 doi

  19. [27]

    Xiping, J

    L. Xiping, J. Mei, G. W eigao, The method of lower and upper solutions for mixed fractional four-point boundary value problem with p-Laplacian operator, Appl. Math. Lett. 65 (2017), 56–62

    2017

  20. [28]

    C. Yang, J. Yan, Positive solutions for third-order Sturm-Liouville bound ary value problems with p- Laplacian, Comput. Math. Appl., (6)59 (2010), 2059–2066

    2010

  21. [29]

    S. Ying, L. Qing and L. Xi-Lan, Existence criteria for positive solutions of p-Laplacian fractional differential equations with derivative terms , Adv. Difference Equ., 2013, 2013:119, 32 pp

    2013

  22. [30]

    Yuansheng, S

    T. Yuansheng, S. Sujing, and B. Zhanbing, Positive Solutions of Fractional Differential Equations wi th p-Laplacian, J. Funct. Spaces 2017, Art. ID 3187492, 9 pp. https://doi.o rg/10.1155/2017/3187492

    2017 doi

  23. [31]

    Yunhong, Existence of positive solutions for fractional differentia l equation involving integral bound- ary conditions with p-Laplacian operator, Adv

    L. Yunhong, Existence of positive solutions for fractional differentia l equation involving integral bound- ary conditions with p-Laplacian operator, Adv. Difference Equ. 2017, Paper No. 135, 11 pp

    2017

  24. [32]

    Yupin, L

    W. Yupin, L. Shutang, H. Zhenlai, Eigenvalue problems for fractional differential equations with mixed derivatives and generalized p-Laplacian, Nonlinear Anal. Model. Control., (6)23 (2018), 830–850

    2018

  25. [33]

    Zhenlai, L

    H. Zhenlai, L. Hongling, Z. Chao, Positive solutions for eigenvalue problems of fractional d ifferential equation with generalized p-Laplacian, Appl. Math. Comput. 257 (2015), 526–536. F aculty of Physics, University of Sciences and Technology of Oran-MB, El Mnaouar, BP 1505, 3...

    2015

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.