Displaying similar documents to “Stieltjes differential problems with general boundary value conditions. Existence and bounds of solutions”

Sharp upper bounds for a singular perturbation problem related to micromagnetics

Arkady Poliakovsky (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We construct an upper bound for the following family of functionals { E ε } ε > 0 , which arises in the study of micromagnetics: E ε ( u ) = Ω ε | u | 2 + 1 ε 2 | H u | 2 . Here Ω is a bounded domain in 2 , u H 1 ( Ω , S 1 ) (corresponding to the magnetization) and H u , the demagnetizing field created by u , is given by div ( u ˜ + H u ) = 0 in 2 , curl H u = 0 in 2 , where u ˜ is the extension of u by 0 in 2 Ω . Our upper bound coincides with the lower bound obtained by Rivière and Serfaty.

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.

The periodic Ambrosetti-Prodi problem for nonlinear perturbations of the p-Laplacian

Jean Mawhin (2006)

Journal of the European Mathematical Society

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We prove an Ambrosetti–Prodi type result for the periodic solutions of the equation ( | u ' | p 2 u ' ) ) ' + f ( u ) u ' + g ( x , u ) = t , when f is arbitrary and g ( x , u ) + or g ( x , u ) when | u | . The proof uses upper and lower solutions and the Leray–Schauder degree.

On boundary value problems for systems of nonlinear generalized ordinary differential equations

Malkhaz Ashordia (2017)

Czechoslovak Mathematical Journal

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A general theorem (principle of a priori boundedness) on solvability of the boundary value problem d x = d A ( t ) · f ( t , x ) , h ( x ) = 0 is established, where f : [ a , b ] × n n is a vector-function belonging to the Carathéodory class corresponding to the matrix-function A : [ a , b ] n × n with bounded total variation components, and h : BV s ( [ a , b ] , n ) n is a continuous operator. Basing on the mentioned principle of a priori boundedness, effective criteria are obtained for the solvability of the system under the condition x ( t 1 ( x ) ) = ( x ) · x ( t 2 ( x ) ) + c 0 , where t i : BV s ( [ a , b ] , n ) [ a , b ] ( i = 1 , 2 ) and : BV s ( [ a , b ] , n ) n are continuous...

Lower semicontinuous envelopes in W 1 , 1 × L p

Ana Margarida Ribeiro, Elvira Zappale (2014)

Banach Center Publications

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The lower semicontinuity of functionals of the type Ω f ( x , u , v , u ) d x with respect to the ( W 1 , 1 × L p ) -weak* topology is studied. Moreover, in absence of lower semicontinuity, an integral representation in W 1 , 1 × L p for the lower semicontinuous envelope is also provided.

Persistence of Coron’s solution in nearly critical problems

Monica Musso, Angela Pistoia (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We consider the problem - Δ u = u N + 2 N - 2 + λ in Ω ε ω , u > 0 in Ω ε ω , u = 0 on Ω ε ω , where Ω and ω are smooth bounded domains in N , N 3 , ε > 0 and λ . We prove that if the size of the hole ε goes to zero and if, simultaneously, the parameter λ goes to zero at the appropriate rate, then the problem has a solution which blows up at the origin.

On Kneser solutions of the n -th order nonlinear differential inclusions

Martina Pavlačková (2019)

Czechoslovak Mathematical Journal

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The paper deals with the existence of a Kneser solution of the n -th order nonlinear differential inclusion x ( n ) ( t ) - A 1 ( t , x ( t ) , ... , x ( n - 1 ) ( t ) ) x ( n - 1 ) ( t ) - ... - A n ( t , x ( t ) , ... , x ( n - 1 ) ( t ) ) x ( t ) for a.a. t [ a , ) , where a ( 0 , ) , and A i : [ a , ) × n , i = 1 , ... , n , are upper-Carathéodory mappings. The derived result is finally illustrated by the third order Kneser problem.

Existence of positive solutions for second order m-point boundary value problems

Ruyun Ma (2002)

Annales Polonici Mathematici

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Let α,β,γ,δ ≥ 0 and ϱ:= γβ + αγ + αδ > 0. Let ψ(t) = β + αt, ϕ(t) = γ + δ - γt, t ∈ [0,1]. We study the existence of positive solutions for the m-point boundary value problem ⎧u” + h(t)f(u) = 0, 0 < t < 1, ⎨ α u ( 0 ) - β u ' ( 0 ) = i = 1 m - 2 a i u ( ξ i ) γ u ( 1 ) + δ u ' ( 1 ) = i = 1 m - 2 b i u ( ξ i ) , where ξ i ( 0 , 1 ) , a i , b i ( 0 , ) (for i ∈ 1,…,m-2) are given constants satisfying ϱ - i = 1 m - 2 a i ϕ ( ξ i ) > 0 , ϱ - i = 1 m - 2 b i ψ ( ξ i ) > 0 and Δ : = - i = 1 m - 2 a i ψ ( ξ i ) ϱ - i = 1 m - 2 a i ϕ ( ξ i ) ϱ - i = 1 m - 2 b i ψ ( ξ i ) - i = 1 m - 2 b i ϕ ( ξ i ) < 0 . We show the existence of positive solutions if f is either superlinear or sublinear by a simple application of a fixed point theorem in cones. Our result extends a result established by Erbe and...

Sobolev versus Hölder local minimizers and existence of multiple solutions for a singular quasilinear equation

Jacques Giacomoni, Ian Schindler, Peter Takáč (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We investigate the following quasilinear and singular problem, t o 2 . 7 c m - Δ p u = λ u δ + u q in Ω ; u | Ω = 0 , u &gt; 0 in Ω , t o 2 . 7 c m (P) where Ω is an open bounded domain with smooth boundary, 1 &lt; p &lt; , p - 1 &lt; q p * - 1 , λ &gt; 0 , and 0 &lt; δ &lt; 1 . As usual, p * = N p N - p if 1 &lt; p &lt; N , p * ( p , ) is arbitrarily large if p = N , and p * = if p &gt; N . We employ variational methods in order to show the existence of at least two distinct (positive) solutions of problem (P) in W 0 1 , p ( Ω ) . While following an approach due to Ambrosetti-Brezis-Cerami, we need to prove two new results of separate interest: a strong comparison principle...

On a sequence formed by iterating a divisor operator

Bellaouar Djamel, Boudaoud Abdelmadjid, Özen Özer (2019)

Czechoslovak Mathematical Journal

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Let be the set of positive integers and let s . We denote by d s the arithmetic function given by d s ( n ) = ( d ( n ) ) s , where d ( n ) is the number of positive divisors of n . Moreover, for every , m we denote by δ s , , m ( n ) the sequence d s ( d s ( ... d s ( d s ( n ) + ) + ... ) + ) m -times = d s ( n ) for m = 1 , d s ( d s ( n ) + ) for m = 2 , d s ( d s ( d s ( n ) + ) + ) for m = 3 , We present classical and nonclassical notes on the sequence ( δ s , , m ( n ) ) m 1 , where , n , s are understood as parameters.

On behavior of solutions to a chemotaxis system with a nonlinear sensitivity function

Senba, Takasi, Fujie, Kentarou

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In this paper, we consider solutions to the following chemotaxis system with general sensitivity τ u t = Δ u - · ( u χ ( v ) ) in Ω × ( 0 , ) , η v t = Δ v - v + u in Ω × ( 0 , ) , u ν = u ν = 0 on Ω × ( 0 , ) . Here, τ and η are positive constants, χ is a smooth function on ( 0 , ) satisfying χ ' ( · ) > 0 and Ω is a bounded domain of 𝐑 n ( n 2 ). It is well known that the chemotaxis system with direct sensitivity ( χ ( v ) = χ 0 v , χ 0 > 0 ) has blowup solutions in the case where n 2 . On the other hand, in the case where χ ( v ) = χ 0 log v with 0 < χ 0 1 , any solution to the system exists globally in time and is bounded. We present a sufficient condition for the boundedness...

Existence and multiplicity of solutions for a fractional p -Laplacian problem of Kirchhoff type via Krasnoselskii’s genus

Ghania Benhamida, Toufik Moussaoui (2018)

Mathematica Bohemica

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We use the genus theory to prove the existence and multiplicity of solutions for the fractional p -Kirchhoff problem - M Q | u ( x ) - u ( y ) | p | x - y | N + p s d x d y p - 1 ( - Δ ) p s u = λ h ( x , u ) in Ω , u = 0 on N Ω , where Ω is an open bounded smooth domain of N , p > 1 , N > p s with s ( 0 , 1 ) fixed, Q = 2 N ( C Ω × C Ω ) , λ > 0 is a numerical parameter, M and h are continuous functions.

On the balanced domination of graphs

Baogen Xu, Wanting Sun, Shuchao Li, Chunhua Li (2021)

Czechoslovak Mathematical Journal

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Let G = ( V G , E G ) be a graph and let N G [ v ] denote the closed neighbourhood of a vertex v in G . A function f : V G { - 1 , 0 , 1 } is said to be a balanced dominating function (BDF) of G if u N G [ v ] f ( u ) = 0 holds for each vertex v V G . The balanced domination number of G , denoted by γ b ( G ) , is defined as γ b ( G ) = max v V G f ( v ) : f is a BDF of G . A graph G is called d -balanced if γ b ( G ) = 0 . The novel concept of balanced domination for graphs is introduced. Some upper bounds on the balanced domination number are established, in which one is the best possible bound and the rest are sharp, all the...

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.

Subclasses of typically real functions determined by some modular inequalities

Leopold Koczan, Katarzyna Trąbka-Więcław (2010)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

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Let T be the family of all typically real functions, i.e. functions that are analytic in the unit disk Δ : = { z : | z | < 1 } , normalized by f ( 0 ) = f ' ( 0 ) - 1 = 0 and such that Im z Im f ( z ) 0 for z Δ . Moreover, let us denote: T ( 2 ) : = { f T : f ( z ) = - f ( - z ) for z Δ } and T M , g : = { f T : f M g in Δ } , where M > 1 , g T S and S consists of all analytic functions, normalized and univalent in Δ .We investigate  classes in which the subordination is replaced with the majorization and the function g is typically real but does not necessarily univalent, i.e. classes { f T : f M g in Δ } , where M > 1 , g T , which we denote...

Linear FDEs in the frame of generalized ODEs: variation-of-constants formula

Rodolfo Collegari, Márcia Federson, Miguel Frasson (2018)

Czechoslovak Mathematical Journal

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We present a variation-of-constants formula for functional differential equations of the form y ˙ = ( t ) y t + f ( y t , t ) , y t 0 = ϕ , where is a bounded linear operator and ϕ is a regulated function. Unlike the result by G. Shanholt (1972), where the functions involved are continuous, the novelty here is that the application t f ( y t , t ) is Kurzweil integrable with t in an interval of , for each regulated function y . This means that t f ( y t , t ) may admit not only many discontinuities, but it can also be highly oscillating and yet, we are...

Nontrivial solutions to boundary value problems for semilinear Δ γ -differential equations

Duong Trong Luyen (2021)

Applications of Mathematics

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In this article, we study the existence of nontrivial weak solutions for the following boundary value problem: - Δ γ u = f ( x , u ) in Ω , u = 0 on Ω , where Ω is a bounded domain with smooth boundary in N , Ω { x j = 0 } for some j , Δ γ is a subelliptic linear operator of the type Δ γ : = j = 1 N x j ( γ j 2 x j ) , x j : = x j , N 2 , where γ ( x ) = ( γ 1 ( x ) , γ 2 ( x ) , , γ N ( x ) ) satisfies certain homogeneity conditions and degenerates at the coordinate hyperplanes and the nonlinearity f ( x , ξ ) is of subcritical growth and does not satisfy the Ambrosetti-Rabinowitz (AR) condition.