Existence and multiplicity of positive solutions to a fourth-order multi-point boundary value problem

Cheikh Guendouz; Faouzi Haddouchi; Slimane Benaicha

Existence and multiplicity of positive solutions to a fourth-order multi-point boundary value problem

T0 review · 0 major / 7 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that a fourth-order multi-point integral boundary value problem has one or two positive solutions depending on growth ratios of the nonlinear term.

desk verdict A correct, entirely standard cone fixed-point paper whose only novelty is the specific mixed integral-plus-multi-point boundary condition; no red flags, modest contribution. read the letter →

arxiv 1908.08598 v1 pith:CBQQXIC7 submitted 2019-08-22 math.CA

Faouzi Haddouchi , Cheikh Guendouz , Slimane Benaicha This is my paper · ORCID

classification math.CA MSC 34B1534B18

keywords positivesolutionsfourth-orderboundaryvalueproblemmulti-pointconditionsintegralconditionKrasnoselskiifixedpointtheoremconeGreen'sfunctionmultiplicity

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 existence and multiplicity of positive solutions for the fourth-order equation $u''''(t)+f(t,u(t))=0$ under the boundary conditions $u'(0)=u'(1)=u''(0)=0$ and $u(0)=\alpha\int_0^1 u(s)\,ds+\sum_{i=1}^n \beta_i u(\eta_i)$, with nonnegative coefficients summing to less than one. The main claim is that the asymptotic ratios of $f$ at zero and infinity decide the count: if the minimum of $f(t,u)/u$ blows up at zero while the maximum tends to zero at infinity, or vice versa, there is at least one positive solution; with one additional local bound there are at least two. The proof treats solutions as fixed points of an integral operator whose kernel is an explicit Green's function, and uses a cone of functions whose minimum on a fixed interior subinterval is comparable to their sup-norm. A sympathetic reader would care because the result covers nonlinearities that are only continuous, not monotone or differentiable, and because the boundary condition mixes a derivative condition, an integral term, and many interior sampling points.

What carries the argument

The central object is the Green's function $H(t,s)$ of Lemma 2.5, built from $G(t,s)=\frac16 t^3(1-s)^2-\frac16(t-s)^3$ for $s\le t$ and $G(t,s)=\frac16 t^3(1-s)^2$ for $t\le s$, together with the integral and multi-point boundary terms. Its two-sided bound $\theta^3 e(s)\le G(t,s)\le e(s)$ with $e(s)=\frac16 s(1-s)^2$ on $[\theta,1-\theta]$ is what converts pointwise growth estimates on $f$ into sup-norm inequalities, via the cone $K$. Krasnoselskii's fixed point theorem on cones, quoted as Theorem 2.4, is the engine that turns expansion and compression of $T$ on sphere boundaries into the existence of one or two fixed points.

What would settle it

For fixed coefficients, compute the unique solution of $u''''+y=0$ with the given boundary conditions for a peaked continuous nonnegative $y$, evaluate $\min_{t\in[\theta,1-\theta]}u(t)/\|u\|$, and check whether it is ever below $\theta^3(1-2\theta)$; one such counterexample would disprove Lemma 2.7, which is used in all four theorems.

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Extended reading notes

Core claim

On the problem (1.11)-(1.12), the paper's claim is: if the minimum of $f(t,u)/u$ over $t$ tends to $+\infty$ as $u\to 0^+$ while the maximum tends to $0$ as $u\to +\infty$, or if the maximum tends to $0$ at $0$ while the minimum tends to $+\infty$ at infinity, then there is at least one positive solution; and if both endpoint trends are the same, supplemented by a local bound at an intermediate level, then there are at least two positive solutions whose sup-norms lie on opposite sides of that intermediate level. The proof shows that the integral operator $Tu(t)=\int_0^1 H(t,s)f(s,u(s))\,ds$ has fixed points in the cone $K=\{u\ge 0:\min_{t\in[\theta,1-\theta]}u(t)\ge \theta^3(1-2\theta)\|u\|\}$; because the Green's function $H$ is nonnegative, fixed points are exactly nonnegative solutions, and the cone inequality makes them strictly positive on $(0,1)$.

Load-bearing premise

The argument collapses without the imported pointwise bound $\theta^3 e(s)\le G(t,s)\le e(s)$ for $t\in[\theta,1-\theta]$, which is used to prove the cone inequality $\min_{[\theta,1-\theta]}u\ge \theta^3(1-2\theta)\|u\|$; every fixed-point step in Theorems 3.1, 4.1, and 4.2 depends on that cone.

Editorial extensions

If this is right

If $f$ grows faster than linearly at zero and slower than linearly at infinity, or the reverse, the boundary value problem has at least one positive solution; no monotonicity or differentiability is required.
The two multiplicity theorems produce explicitly separated solutions, $0<\|u_1\|<\rho<\|u_2\|$, so the two solutions are distinguishable by size.
Every solution found obeys the interior concentration estimate $\min_{t\in[\theta,1-\theta]}u(t)\ge \theta^3(1-2\theta)\|u\|$, so positivity is uniform on a whole subinterval, not merely pointwise.
The examples show the hypotheses are checkable by elementary inequalities, such as $f(t,u)=t+|\cos u|$ for the single-solution case and $f(t,u)=(1+t)e^u$ for the two-solution case.

Reading between the lines

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

The same cone-and-Green's-function template should extend to other fourth-order nonlocal boundary conditions as long as the associated Green's function admits a two-sided estimate of the form $\theta^\gamma e(s)\le G(t,s)\le e(s)$; the exponent $\gamma$ would simply replace $3$ in the cone factor.
Because the proof uses only the asymptotic ratios $f_0,f^0,f_\infty,f^\infty$, nonlinearities with oscillations, such as the cosine term in Example 5.1, are covered; a natural stress test is to amplify those oscillations and see numerically whether the two solution norms remain separated.
The constants $\Lambda_1=6k$ and $\Lambda_2=\Psi^{-1}$ are explicit in $\alpha,\beta_i,\eta_i,\theta$, so one could optimize over $\theta$ to widen the admissible parameter ranges in the multiplicity examples, a check the paper leaves implicit.

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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 paper is a perfectly competent, textbook application of Krasnoselskii's cone fixed point theorem to a fourth-order boundary value problem. The only real novelty is the boundary condition (1.12), a convex combination of an integral term and a finite sum of interior point evaluations. That specific condition does not appear in the references I know. The fixed point machinery is standard: build the Green's function, prove a positivity and a Harnack-type inequality, define a cone, verify expansion/compression, apply the theorem.

The proofs check out. I went through the Green's function derivation in Lemma 2.5 and the cone inequality in Lemma 2.7; both are correct. The paper imports Lemma 2.6 from the authors' own earlier paper [2] for the lower bound θ^3 e(s) ≤ G(t,s) on [θ,1−θ], and that is the load-bearing estimate. The stress-test note confirms it is true, and I agree. Still, a referee should ask for a proof or a precise reference statement, because every fixed point argument in the paper rests on it.

The norm estimates in Theorems 3.1, 4.1 and 4.2 are consistent. The multiplicity results really do give two solutions with the stated norm ordering, and the examples, while not elegant, do illustrate the hypotheses. The last example leans on a Mathematica check of a polynomial inequality, which is acceptable but slightly ugly.

The soft spots are minor. The paper is not self-contained at the one point that matters, the imported lemma. The introduction lumps many references without much discrimination. There are also the usual typos in the arXiv text. None of that undermines the math.

Who is this for? Someone working on nonlocal fourth-order boundary value problems who needs a concrete new example of a boundary condition combining integral and multi-point terms. It will not change the shape of the field, but it is a solid, citable result for the specialized literature. I would send it to peer review at an appropriate journal. I wouldn't bring it to a general reading group, and I wouldn't cite it in my own work, but that is a reflection of my interests, not the paper's quality.

Referee Report

0 major / 7 minor

Summary. The paper studies the fourth-order boundary value problem u'''' + f(t,u)=0 on (0,1) with the nonlocal boundary conditions u'(0)=u'(1)=u''(0)=0 and u(0)=α∫_0^1 u(s)ds + Σ_{i=1}^n β_i u(η_i), under the assumptions f∈C([0,1]×[0,∞),[0,∞)), α,β_i≥0, 0<η_1<...<η_n<1, and α+Σβ_i<1. The authors construct the Green's function H for the linear problem (Lemma 2.5), establish a cone-invariance estimate min_{[θ,1-θ]} u ≥ θ^3(1-2θ)‖u‖ for nonnegative forcing (Lemma 2.7), and then apply Krasnoselskii's cone expansion/compression theorem. Theorem 3.1 gives one positive solution under either (H1) f_0=∞ and f^∞=0 or (H2) f^0=0 and f_∞=∞. Theorems 4.1 and 4.2 give two positive solutions under local upper/lower bounds on f, with the claimed norm separation 0<‖u1‖<ρ<‖u2‖. Four examples illustrate the results.

Significance. If the results hold, this is a technically sound extension of known fourth-order two-point and multi-point boundary value results to a combined integral/multi-point nonlocal condition. The Green's function is explicit, the cone estimate is worked out in detail, and the fixed-point arguments are standard but carefully checked. The paper does not ship machine-checked proofs, but the estimates are explicit and can be verified by hand, and the examples are nontrivial. The contribution is incremental rather than groundbreaking, but it is a solid and useful addition to the literature on positive solutions of nonlocal fourth-order BVPs, provided the small gaps and presentation issues listed below are addressed.

minor comments (7)

[Section 3, Theorem 3.1] The notation for the limits is confusing: (H1) should read f_0=∞ and f^∞=0, and (H2) should read f^0=0 and f_∞=∞, with clearly distinguished superscript and subscript symbols. As printed, the expression 'f∞=0' in (H1) could be misread as the lower limit at infinity, which is not the hypothesis used in the proof.
[Section 4, after (4.5) and (4.10)] The strict norm inequalities 0<‖u1‖<ρ1<‖u2‖ (resp. <ρ2<) do not follow directly from the inequalities as written, because the Krasnoselskii theorem yields fixed points in closed annuli and the boundary estimates are stated as non-strict. Please justify the strictness explicitly, for instance by observing that the estimate on ∂Ω_{ρ1} is actually strict since ∫_0^1 e(s)ds=1/72<1/6, that (H6) prevents a fixed point on ∂Ω_{ρ2}, or by choosing M1<Λ1 and M2>Λ2 when the hypotheses allow.
[Section 2, Lemma 2.6] The lower bound θ^3 e(s) ≤ G(t,s) for t∈[θ,1-θ] is imported from [2, Lemma 2.3] without proof. Since this estimate is load-bearing for the cone inequality in Lemma 2.7 and hence for all the main theorems, a short proof or a precise restatement of the relevant part of [2] would make the paper self-contained.
[Section 4, proof of Theorem 4.1] The symbol ρ* is used for both the inner radius (ρ*∈(0,ρ1)) and the outer radius (ρ*≥ρ̄*/θ^3(1-2θ)). Using distinct symbols, such as r and R, would remove the ambiguity and make the nesting Ω_r⊂Ω_{ρ1}⊂Ω_R clear.
[Section 2, Definition 2.3 and Theorem 3.1] It is not explicitly shown that the fixed points in K are positive on the whole interval (0,1), rather than merely nonnegative with a positive lower bound on [θ,1-θ]. Since H(t,s)>0 for t∈(0,1), s∈[0,1] a.e., and the fixed point is not identically zero by the annulus construction, this follows, but a sentence would be helpful.
[Section 5, Example 5.4] In the displayed computation of Ψ, the integral ∫_{1/2}^{θ} appears where ∫_θ^{1/2} is clearly intended; please correct. Also, the verification that 10^9 θ^{12}(1-2θ)^5(103+206θ-212θ^2+8θ^3) ≥ 1 for θ∈[17/125,12/25] is delegated to Mathematica; an analytic justification or a clear plot would be preferable, though the claim is plausible.
[Throughout] There are several typographical and grammatical errors that should be corrected, such as 'Both one positive solutions' in the proof of Theorem 4.1 and the broken words in the title and abstract ('MUL TIPLICITY', 'pos itive'). These do not affect the mathematics but should be cleaned up before publication.

Circularity Check

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No significant circularity: the derivation is self-contained and the imported Green's-function estimate is elementary and independently checkable.

full rationale

The paper's central claims (Theorems 3.1, 4.1, 4.2) are proved by a standard cone fixed-point argument: the Green's function H(t,s) is constructed explicitly from the linear boundary value problem in Lemma 2.5, the cone invariance and the norm-type compression/expansion inequalities follow from direct estimates of this explicit kernel, and the existence conclusions are then forced by the Krasnoselskii fixed-point theorem. Nothing is fitted to data, and no 'prediction' is defined in terms of the target conclusion. The only imported component is Lemma 2.6(iii), the lower bound θ³e(s) ≤ G(t,s) for t∈[θ,1−θ], which is cited from the authors' earlier paper [2] rather than reproved. This is a self-citation, but it is not circular in any load-bearing sense: the estimate is an elementary, parameter-free inequality for the explicit polynomial kernel G(t,s) that any reader can verify independently, and the paper's own proof of Lemma 2.7 shows exactly how the estimate is used. The hypotheses (H1)-(H6) are assumptions on the nonlinear term f, not on the solution, and the constants Φ, Ψ, Λ₁, Λ₂ are explicit expressions in the problem data, not calibration parameters chosen to match the conclusions. Thus the central derivations do not reduce to their inputs by construction or by an unverified self-citation chain.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to data. The only background imports are standard fixed point and compactness theorems and an elementary Green's function estimate. No new entities are postulated.

assumptions (4)

standard math Krasnoselskii fixed point theorem on cones (Theorem 2.4)
Used to obtain fixed points of the cone operator T in Theorems 3.1, 4.1, and 4.2.
standard math Arzela-Ascoli theorem
Used in Lemma 2.9 to prove that T is completely continuous.
standard math Green's function estimate θ^3 e(s) ≤ G(t,s) ≤ e(s) on [θ,1-θ] (Lemma 2.6, from [2])
Imported from the authors' earlier paper; it is a parameter-free calculus estimate and is load-bearing for the cone lower bound.
domain assumption Continuity and nonnegativity of f, and α, β_i ≥ 0 with α+Σβ_i < 1 ((C1)-(C3))
These assumptions define the problem class and ensure the Green's function operator preserves positivity and the cone.

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Cite this review

Pith. "Pith review of Existence and multiplicity of positive solutions to a fourth-order multi-point boundary value problem." pith.science (2026). https://pith.science/paper/CBQQXIC7

@misc{pith2026190808598,
  author       = {Pith},
  title        = {Pith review of: Existence and multiplicity of positive solutions to a fourth-order multi-point boundary value problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CBQQXIC7}},
  note         = {Machine review of arXiv:1908.08598}
}

read the original abstract

In this paper, we study the existence and multiplicity of positive solutions for a nonlinear fourth-order with multi-point boundary conditions involving an integral boundary condition. The main tool is Krasnosel'skii fixed point theorem on cones.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

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Pith tools

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

[1] [2]

Benaicha, F

S. Benaicha, F. Haddouchi, Positive solutions of a nonlinear fourth-order integral bo undary value prob- lem, An. Univ. Vest Timi¸ s. Ser. Mat.-Inform., 54(1) (2016), 73-86

work page 2016

[2] [1]

D. R. Anderson, R. I. Avery, A fourth-order four-point right focal boundary value probl em, Rocky Mountain J. Math., 36(2) (2006), 367–380

work page 2006

[3] [3]

C. Bai, J. Fang, Existence of multiple positive solutions for nonlinear m-p oint boundary value problems , J. Math. Anal. Appl., 281(1) (2003), 76–85

work page 2003

[4] [4]

J. R. Graef, C. Qian, B. Yang, A three point boundary value problem for nonlinear fourth or der diﬀer- ential equations , J. Math. Anal. Appl., 287(1) (2003), 217–233

work page 2003

[5] [5]

X. Han, H. Gao, J. Xu, Existence of positive solutions for nonlocal fourth-order boundary value problem with variable parameter , Fixed Point Theory Appl., 2011 (2011), Art. ID 604046, 11 pages

work page 2011

[6] [6]

P. Kang, Z. W ei, J. Xu, Positive solutions to fourth-order singular boundary valu e problems with integral boundary conditions in abstract spaces , Appl. Math. Comput., 206(1) (2008), 245–256

work page 2008

[7] [7]

M. A. Krasnosel’skii, Positive solutions of operator equations , P. Noordhoﬀ, Groningen, The Nether- lands, 1964. POSITIVE SOLUTIONS FOR A FOURTH-ORDER BOUNDARY V ALUE PROBL EM 13

work page 1964

[8] [8]

Li, Existence of positive solutions for the cantilever beam equ ations with fully nonlinear terms , Nonlinear Anal

Y. Li, Existence of positive solutions for the cantilever beam equ ations with fully nonlinear terms , Nonlinear Anal. Real W orld Appl., 27 (2016), 221–237

work page 2016