Displaying similar documents to “Positive solutions to a class of elastic beam equations with semipositone nonlinearity”

A geometrically nonlinear analysis of laminated composite plates using a shear deformation theory

Giacinto Porco, Giuseppe Spadea, Raffaele Zinno (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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A shear deformation theory is developed to analyse the geometrically nonlinear behaviour of layered composite plates under transverse loads. The theory accounts for the transverse shear (as in the Reissner Mindlin plate theory) and large rotations (in the sense of the von Karman theory) suitable for simulating the behaviour of moderately thick plates. Square and rectangular plates are considered: the numerical results are obtained by a finite element computational procedure and are given...

Existence of positive solutions for a nonlinear fourth order boundary value problem

Ruyun Ma (2003)

Annales Polonici Mathematici

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We study the existence of positive solutions of the nonlinear fourth order problem u ( 4 ) ( x ) = λ a ( x ) f ( u ( x ) ) , u(0) = u’(0) = u”(1) = u”’(1) = 0, where a: [0,1] → ℝ may change sign, f(0) < 0, and λ < 0 is sufficiently small. Our approach is based on the Leray-Schauder fixed point theorem.

A geometrically nonlinear analysis of laminated composite plates using a shear deformation theory

Giacinto Porco, Giuseppe Spadea, Raffaele Zinno (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Similarity:

A shear deformation theory is developed to analyse the geometrically nonlinear behaviour of layered composite plates under transverse loads. The theory accounts for the transverse shear (as in the Reissner Mindlin plate theory) and large rotations (in the sense of the von Karman theory) suitable for simulating the behaviour of moderately thick plates. Square and rectangular plates are considered: the numerical results are obtained by a finite element computational procedure and are given...

On a singular multi-point third-order boundary value problem on the half-line

Zakia Benbaziz, Smail Djebali (2020)

Mathematica Bohemica

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We establish not only sufficient but also necessary conditions for existence of solutions to a singular multi-point third-order boundary value problem posed on the half-line. Our existence results are based on the Krasnosel’skii fixed point theorem on cone compression and expansion. Nonexistence results are proved under suitable a priori estimates. The nonlinearity f = f ( t , x , y ) which satisfies upper and lower-homogeneity conditions in the space variables x , y may be also singular at time t = 0 . Two examples...

Existence Theorems for a Fourth Order Boundary Value Problem

A. El-Haffaf (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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This paper treats the question of the existence of solutions of a fourth order boundary value problem having the following form: x ( 4 ) ( t ) + f ( t , x ( t ) , x ' ' ( t ) ) = 0 , 0 < t < 1, x(0) = x’(0) = 0, x”(1) = 0, x ( 3 ) ( 1 ) = 0 . Boundary value problems of very similar type are also considered. It is assumed that f is a function from the space C([0,1]×ℝ²,ℝ). The main tool used in the proof is the Leray-Schauder nonlinear alternative.

Blow-up of the solution to the initial-value problem in nonlinear three-dimensional hyperelasticity

J. A. Gawinecki, P. Kacprzyk (2008)

Applicationes Mathematicae

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We consider the initial value problem for the nonlinear partial differential equations describing the motion of an inhomogeneous and anisotropic hyperelastic medium. We assume that the stored energy function of the hyperelastic material is a function of the point x and the nonlinear Green-St. Venant strain tensor e j k . Moreover, we assume that the stored energy function is C with respect to x and e j k . In our description we assume that Piola-Kirchhoff’s stress tensor p j k depends on the tensor...

Positive solutions for a system of third-order differential equation with multi-point and integral conditions

Rochdi Jebari, Abderrahman Boukricha (2015)

Commentationes Mathematicae Universitatis Carolinae

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This paper concerns the following system of nonlinear third-order boundary value problem: u i ' ' ' ( t ) + f i ( t , u 1 ( t ) , , u n ( t ) , u 1 ' ( t ) , , u n ' ( t ) ) = 0 , 0 < t < 1 , i { 1 , , n } with the following multi-point and integral boundary conditions: u i ( 0 ) = 0 u i ' ( 0 ) = 0 u i ' ( 1 ) = j = 1 p β j , i u i ' ( η j , i ) + 0 1 h i ( u 1 ( s ) , , u n ( s ) ) d s where β j , i > 0 , 0 < η 1 , i < < η p , i < 1 2 , f i : [ 0 , 1 ] × n × n and h i : [ 0 , 1 ] × n are continuous functions for all i { 1 , , n } and j { 1 , , p } . Using Guo-Krasnosel’skii fixed point theorem in cone, we discuss the existence of positive solutions of this problem. We also prove nonexistence of positive solutions and we give some examples to illustrate our results.

On the solvability of a fourth-order multi-point boundary value problem

Yuqiang Feng, Xincheng Ding (2012)

Annales Polonici Mathematici

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We are concerned with the solvability of the fourth-order four-point boundary value problem ⎧ u ( 4 ) ( t ) = f ( t , u ( t ) , u ' ' ( t ) ) , t ∈ [0,1], ⎨ u(0) = u(1) = 0, ⎩ au”(ζ₁) - bu”’(ζ₁) = 0, cu”(ζ₂) + du”’(ζ₂) = 0, where 0 ≤ ζ₁ < ζ₂ ≤ 1, f ∈ C([0,1] × [0,∞) × (-∞,0],[0,∞)). By using Guo-Krasnosel’skiĭ’s fixed point theorem on cones, some criteria are established to ensure the existence, nonexistence and multiplicity of positive solutions for this problem.

Multiplicity of positive solutions for a nonlinear fourth order equation

D. R. Dunninger (2001)

Annales Polonici Mathematici

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We study the existence and multiplicity of positive solutions of the nonlinear fourth order problem ⎧ u ( 4 ) = λ f ( u ) in (0,1), ⎨ ⎩u(0) = a ≥ 0, u’(0) = a’ ≥ 0, u(1) = b ≥ 0, u(1) = -b’ ≤ 0 The methods employed are upper and lower solutions and degree theory arguments.

Positive solutions for one-dimensional singular p-Laplacian boundary value problems

Huijuan Song, Jingxue Yin, Rui Huang (2012)

Annales Polonici Mathematici

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We consider the existence of positive solutions of the equation 1 / λ ( t ) ( λ ( t ) φ p ( x ' ( t ) ) ) ' + μ f ( t , x ( t ) , x ' ( t ) ) = 0 , where φ p ( s ) = | s | p - 2 s , p > 1, subject to some singular Sturm-Liouville boundary conditions. Using the Krasnosel’skiĭ fixed point theorem for operators on cones, we prove the existence of positive solutions under some structure conditions.

On the principal eigencurve of the p-Laplacian related to the Sobolev trace embedding

Abdelouahed El Khalil, Mohammed Ouanan (2005)

Applicationes Mathematicae

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We prove that for any λ ∈ ℝ, there is an increasing sequence of eigenvalues μₙ(λ) for the nonlinear boundary value problem ⎧ Δ u = | u | p - 2 u in Ω, ⎨ ⎩ | u | p - 2 u / ν = λ ϱ ( x ) | u | p - 2 u + μ | u | p - 2 u on crtial ∂Ω and we show that the first one μ₁(λ) is simple and isolated; we also prove some results about variations of the density ϱ and the continuity with respect to the parameter λ.

Existence of solutions for a coupled system with φ -Laplacian operators and nonlinear coupled boundary conditions

Konan Charles Etienne Goli, Assohoun Adjé (2017)

Communications in Mathematics

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We study the existence of solutions of the system ( φ 1 ( u 1 ' ( t ) ) ) ' = f 1 ( t , u 1 ( t ) , u 2 ( t ) , u 1 ' ( t ) , u 2 ' ( t ) ) , a.e. t [ 0 , T ] , ( φ 2 ( u 2 ' ( t ) ) ) ' = f 2 ( t , u 1 ( t ) , u 2 ( t ) , u 1 ' ( t ) , u 2 ' ( t ) ) , a.e. t [ 0 , T ] , submitted to nonlinear coupled boundary conditions on [ 0 , T ] where φ 1 , φ 2 : ( - a , a ) , with 0 < a < + , are two increasing homeomorphisms such that φ 1 ( 0 ) = φ 2 ( 0 ) = 0 , and f i : [ 0 , T ] × 4 , i { 1 , 2 } are two L 1 -Carathéodory functions. Using some new conditions and Schauder fixed point Theorem, we obtain solvability result.

Boundedness criteria for a class of second order nonlinear differential equations with delay

Daniel O. Adams, Mathew Omonigho Omeike, Idowu A. Osinuga, Biodun S. Badmus (2023)

Mathematica Bohemica

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We consider certain class of second order nonlinear nonautonomous delay differential equations of the form a ( t ) x ' ' + b ( t ) g ( x , x ' ) + c ( t ) h ( x ( t - r ) ) m ( x ' ) = p ( t , x , x ' ) and ( a ( t ) x ' ) ' + b ( t ) g ( x , x ' ) + c ( t ) h ( x ( t - r ) ) m ( x ' ) = p ( t , x , x ' ) , where a , b , c , g , h , m and p are real valued functions which depend at most on the arguments displayed explicitly and r is a positive constant. Different forms of the integral inequality method were used to investigate the boundedness of all solutions and their derivatives. Here, we do not require construction of the Lyapunov-Krasovski functional to establish our results....

Numerical approximation of the non-linear fourth-order boundary-value problem

Svobodová, Ivona

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We consider functionals of a potential energy ψ ( u ) corresponding to 𝑎𝑛 𝑎𝑥𝑖𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦 - 𝑣𝑎𝑙𝑢𝑒 𝑝𝑟𝑜𝑏𝑙𝑒𝑚 . We are dealing with 𝑎 𝑑𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 𝑡ℎ𝑖𝑛 𝑎𝑛𝑛𝑢𝑙𝑎𝑟 𝑝𝑙𝑎𝑡𝑒 with 𝑁𝑒𝑢𝑚𝑎𝑛𝑛 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑠 . Various types of the subsoil of the plate are described by various types of the 𝑛𝑜𝑛𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑏𝑙𝑒 nonlinear term ψ ( u ) . The aim of the paper is to find a suitable computational algorithm.

On solutions of a fourth-order Lidstone boundary value problem at resonance

Mariusz Jurkiewicz (2009)

Annales Polonici Mathematici

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We consider a Lidstone boundary value problem in k at resonance. We prove the existence of a solution under the assumption that the nonlinear part is a Carathéodory map and conditions similar to those of Landesman-Lazer are satisfied.

A singular initial value problem for the equation u ( n ) ( x ) = g ( u ( x ) )

Wojciech Mydlarczyk (1998)

Annales Polonici Mathematici

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We consider the problem of the existence of positive solutions u to the problem u ( n ) ( x ) = g ( u ( x ) ) , u ( 0 ) = u ' ( 0 ) = . . . = u ( n - 1 ) ( 0 ) = 0 (g ≥ 0,x > 0, n ≥ 2). It is known that if g is nondecreasing then the Osgood condition δ 1 / s [ s / g ( s ) ] 1 / n d s < is necessary and sufficient for the existence of nontrivial solutions to the above problem. We give a similar condition for other classes of functions g.

The restriction theorem for fully nonlinear subequations

F. Reese Harvey, H. Blaine Lawson (2014)

Annales de l’institut Fourier

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Let X be a submanifold of a manifold Z . We address the question: When do viscosity subsolutions of a fully nonlinear PDE on Z , restrict to be viscosity subsolutions of the restricted subequation on X ? This is not always true, and conditions are required. We first prove a basic result which, in theory, can be applied to any subequation. Then two definitive results are obtained. The first applies to any “geometrically defined” subequation, and the second to any subequation which can be...

Multiple positive solutions of a nonlinear fourth order periodic boundary value problem

Lingbin Kong, Daqing Jiang (1998)

Annales Polonici Mathematici

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The fourth order periodic boundary value problem u ( 4 ) - m u + F ( t , u ) = 0 , 0 < t < 2π, with u ( i ) ( 0 ) = u ( i ) ( 2 π ) , i = 0,1,2,3, is studied by using the fixed point index of mappings in cones, where F is a nonnegative continuous function and 0 < m < 1. Under suitable conditions on F, it is proved that the problem has at least two positive solutions if m ∈ (0,M), where M is the smallest positive root of the equation tan mπ = -tanh mπ, which takes the value 0.7528094 with an error of ± 10 - 7 .