Displaying similar documents to “On the solvability of a fourth-order multi-point boundary value problem”

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.

Positive solutions of a fourth-order differential equation with integral boundary conditions

Seshadev Padhi, John R. Graef (2023)

Mathematica Bohemica

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We study the existence of positive solutions to the fourth-order two-point boundary value problem u ' ' ' ' ( t ) + f ( t , u ( t ) ) = 0 , 0 < t < 1 , u ' ( 0 ) = u ' ( 1 ) = u ' ' ( 0 ) = 0 , u ( 0 ) = α [ u ] , where α [ u ] = 0 1 u ( t ) d A ( t ) is a Riemann-Stieltjes integral with A 0 being a nondecreasing function of bounded variation and f 𝒞 ( [ 0 , 1 ] × + , + ) . The sufficient conditions obtained are new and easy to apply. Their approach is based on Krasnoselskii’s fixed point theorem and the Avery-Peterson fixed point theorem.

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.

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.

Region of variability for spiral-like functions with respect to a boundary point

S. Ponnusamy, A. Vasudevarao, M. Vuorinen (2009)

Colloquium Mathematicae

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For μ ∈ ℂ such that Re μ > 0 let μ denote the class of all non-vanishing analytic functions f in the unit disk with f(0) = 1 and R e ( 2 π / μ z f ' ( z ) / f ( z ) + ( 1 + z ) / ( 1 - z ) ) > 0 in . For any fixed z₀ in the unit disk, a ∈ ℂ with |a| ≤ 1 and λ ∈ ̅, we shall determine the region of variability V(z₀,λ) for log f(z₀) when f ranges over the class μ ( λ ) = f μ : f ' ( 0 ) = ( μ / π ) ( λ - 1 ) a n d f ' ' ( 0 ) = ( μ / π ) ( a ( 1 - | λ | ² ) + ( μ / π ) ( λ - 1 ) ² - ( 1 - λ ² ) ) . In the final section we graphically illustrate the region of variability for several sets of parameters.

Existence results and iterative method for fully third order nonlinear integral boundary value problems

Quang A Dang, Quang Long Dang (2021)

Applications of Mathematics

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We consider the boundary value problem u ' ' ' ( t ) = f ( t , u ( t ) , u ' ( t ) , u ' ' ( t ) ) , 0 < t < 1 , u ( 0 ) = u ' ( 0 ) = 0 , u ( 1 ) = 0 1 g ( s ) u ( s ) d s , where f : [ 0 , 1 ] × 3 + , g : [ 0 , 1 ] + are continuous functions. The case when f = f ( u ( t ) ) was studied in 2018 by Guendouz et al. Using the fixed-point theory on cones they established the existence of positive solutions. Here, by the method developed by ourselves very recently, we establish the existence, uniqueness and positivity of the solution under easily verified conditions and propose an iterative method for finding the solution. Some examples demonstrate the validity of the...

Boundedness results of solutions to the equation x ′′′ + a x ′′ + g ( x ) x + h ( x ) = p ( t ) without the hypothesis h ( x ) sgn x 0 for | x | > R .

Ján Andres (1986)

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

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Per l'equazione differenziale ordinaria non lineare del 3° ordine indicata nel titolo, studiata da numerosi autori sotto l'ipotesi h ( x ) sgn x 0 f o r | x | > R , si dimostra l'esistenza di almeno una soluzione limitata sopprimendo l'ipotesi suddetta.

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.

E 1 -degeneration and d ' d ' ' -lemma

Tai-Wei Chen, Chung-I Ho, Jyh-Haur Teh (2016)

Commentationes Mathematicae Universitatis Carolinae

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For a double complex ( A , d ' , d ' ' ) , we show that if it satisfies the d ' d ' ' -lemma and the spectral sequence { E r p , q } induced by A does not degenerate at E 0 , then it degenerates at E 1 . We apply this result to prove the degeneration at E 1 of a Hodge-de Rham spectral sequence on compact bi-generalized Hermitian manifolds that satisfy a version of d ' d ' ' -lemma.

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....

Couples of lower and upper slopes and resonant second order ordinary differential equations with nonlocal boundary conditions

Jean Mawhin, Katarzyna Szymańska-Dębowska (2016)

Mathematica Bohemica

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A couple ( σ , τ ) of lower and upper slopes for the resonant second order boundary value problem x ' ' = f ( t , x , x ' ) , x ( 0 ) = 0 , x ' ( 1 ) = 0 1 x ' ( s ) d g ( s ) , with g increasing on [ 0 , 1 ] such that 0 1 d g = 1 , is a couple of functions σ , τ C 1 ( [ 0 , 1 ] ) such that σ ( t ) τ ( t ) for all t [ 0 , 1 ] , σ ' ( t ) f ( t , x , σ ( t ) ) , σ ( 1 ) 0 1 σ ( s ) d g ( s ) , τ ' ( t ) f ( t , x , τ ( t ) ) , τ ( 1 ) 0 1 τ ( s ) d g ( s ) , in the stripe 0 t σ ( s ) d s x 0 t τ ( s ) d s and t [ 0 , 1 ] . It is proved that the existence of such a couple ( σ , τ ) implies the existence and localization of a solution to the boundary value problem. Multiplicity results are also obtained.

A compactness result in thin-film micromagnetics and the optimality of the Néel wall

Radu Ignat, Felix Otto (2008)

Journal of the European Mathematical Society

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In this paper, we study a model for the magnetization in thin ferromagnetic films. It comes as a variational problem for S 1 -valued maps m ' (the magnetization) of two variables x ' : E ε ( m ' ) = ε | ' · m ' | 2 d x ' + 1 2 | ' | - 1 / 2 ' · m ' 2 d x ' . We are interested in the behavior of minimizers as ε 0 . They are expected to be S 1 -valued maps m ' of vanishing distributional divergence ' · m ' = 0 , so that appropriate boundary conditions enforce line discontinuities. For finite ε > 0 , these line discontinuities are approximated by smooth transition layers, the so-called Néel...

On a divisibility problem

Shichun Yang, Florian Luca, Alain Togbé (2019)

Mathematica Bohemica

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Let p 1 , p 2 , be the sequence of all primes in ascending order. Using explicit estimates from the prime number theory, we show that if k 5 , then ( p k + 1 - 1 ) ! ( 1 2 ( p k + 1 - 1 ) ) ! p k ! , which improves a previous result of the second author.

On hyperbolic partial differential equations in Banach spaces

Bogdan Rzepecki (1986)

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

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Viene dimostrata l'esistenza di soluzioni del problema di Darboux per l'equazione iperbolica z x y ′′ = f ( x , y , z , Z x , z y ) sul planiquarto x 0 , y 0 . Qui, f è una funzione continua, con valori in uno spazio Banach che soddisfano alcune condizioni di regolarità espresse in termini della misura di non-compattezza α .

Existence and multiplicity of solutions for a class of damped vibration problems with impulsive effects

Jianwen Zhou, Yongkun Li (2011)

Annales Polonici Mathematici

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Some sufficient conditions on the existence and multiplicity of solutions for the damped vibration problems with impulsive effects ⎧ u”(t) + g(t)u’(t) + f(t,u(t)) = 0, a.e. t ∈ [0,T ⎨ u(0) = u(T) = 0 ⎩ Δ u ' ( t j ) = u ' ( t j - u ' ( t ¯ j ) = I j ( u ( t j ) ) , j = 1,...,p, are established, where t = 0 < t < < t p < t p + 1 = T , g ∈ L¹(0,T;ℝ), f: [0,T] × ℝ → ℝ is continuous, and I j : , j = 1,...,p, are continuous. The solutions are sought by means of the Lax-Milgram theorem and some critical point theorems. Finally, two examples are presented to illustrate the effectiveness...