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Displaying similar documents to “On Kneser solutions of the n -th order nonlinear differential inclusions”

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...

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...

Positive solutions for concave-convex elliptic problems involving p ( x ) -Laplacian

Makkia Dammak, Abir Amor Ben Ali, Said Taarabti (2022)

Mathematica Bohemica

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We study the existence and nonexistence of positive solutions of the nonlinear equation - Δ p ( x ) u = λ k ( x ) u q ± h ( x ) u r in Ω , u = 0 on Ω where Ω N , N 2 , is a regular bounded open domain in N and the p ( x ) -Laplacian Δ p ( x ) u : = div ( | u | p ( x ) - 2 u ) is introduced for a continuous function p ( x ) > 1 defined on Ω . The positive parameter λ induces the bifurcation phenomena. The study of the equation (Q) needs generalized Lebesgue and Sobolev spaces. In this paper, under suitable assumptions, we show that some variational methods still work. We use them to prove the existence of positive...

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 &gt; 0 in Ω ε ω , u = 0 on Ω ε ω , where Ω and ω are smooth bounded domains in N , N 3 , ε &gt; 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.

Traceability in { K 1 , 4 , K 1 , 4 + e } -free graphs

Wei Zheng, Ligong Wang (2019)

Czechoslovak Mathematical Journal

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A graph G is called { H 1 , H 2 , , H k } -free if G contains no induced subgraph isomorphic to any graph H i , 1 i k . We define σ k = min i = 1 k d ( v i ) : { v 1 , , v k } is an independent set of vertices in G . In this paper, we prove that (1) if G is a connected { K 1 , 4 , K 1 , 4 + e } -free graph of order n and σ 3 ( G ) n - 1 , then G is traceable, (2) if G is a 2-connected { K 1 , 4 , K 1 , 4 + e } -free graph of order n and | N ( x 1 ) N ( x 2 ) | + | N ( y 1 ) N ( y 2 ) | n - 1 for any two distinct pairs of non-adjacent vertices { x 1 , x 2 } , { y 1 , y 2 } of G , then G is traceable, i.e., G has a Hamilton path, where K 1 , 4 + e is a graph obtained by joining a pair of non-adjacent vertices in a K 1 , 4 .

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...

Nonlinear fourth order problems with asymptotically linear nonlinearities

Abir Amor Ben Ali, Makkia Dammak (2024)

Mathematica Bohemica

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We investigate some nonlinear elliptic problems of the form Δ 2 v + σ ( x ) v = h ( x , v ) in Ω , v = Δ v = 0 on Ω , ( P ) where Ω is a regular bounded domain in N , N 2 , σ ( x ) a positive function in L ( Ω ) , and the nonlinearity h ( x , t ) is indefinite. We prove the existence of solutions to the problem (P) when the function h ( x , t ) is asymptotically linear at infinity by using variational method but without the Ambrosetti-Rabinowitz condition. Also, we consider the case when the nonlinearities are superlinear and subcritical.

2-Cohomology of semi-simple simply connected group-schemes over curves defined over p -adic fields

Jean-Claude Douai (2013)

Journal de Théorie des Nombres de Bordeaux

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Let X be a proper, smooth, geometrically connected curve over a p -adic field k . Lichtenbaum proved that there exists a perfect duality: Br ( X ) × Pic ( X ) / between the Brauer and the Picard group of X , from which he deduced the existence of an injection of Br ( X ) in P X Br ( k P ) where P X and k P denotes the residual field of the point P . The aim of this paper is to prove that if G = G ˜ is an X e t - scheme of semi-simple simply connected groups (s.s.s.c groups), then we can deduce from Lichtenbaum’s results...

Further generalized versions of Ilmanen’s lemma on insertion of C 1 , ω or C loc 1 , ω functions

Václav Kryštof (2021)

Commentationes Mathematicae Universitatis Carolinae

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The author proved in 2018 that if G is an open subset of a Hilbert space, f 1 , f 2 : G continuous functions and ω a nontrivial modulus such that f 1 f 2 , f 1 is locally semiconvex with modulus ω and f 2 is locally semiconcave with modulus ω , then there exists f C loc 1 , ω ( G ) such that f 1 f f 2 . This is a generalization of Ilmanen’s lemma (which deals with linear modulus and functions on an open subset of n ). Here we extend the mentioned result from Hilbert spaces to some superreflexive spaces, in particular to L p spaces, p [ 2 , ) . We...

On k -Pell numbers which are sum of two Narayana’s cows numbers

Kouèssi Norbert Adédji, Mohamadou Bachabi, Alain Togbé (2025)

Mathematica Bohemica

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For any positive integer k 2 , let ( P n ( k ) ) n 2 - k be the k -generalized Pell sequence which starts with 0 , , 0 , 1 ( k terms) with the linear recurrence P n ( k ) = 2 P n - 1 ( k ) + P n - 2 ( k ) + + P n - k ( k ) for n 2 . Let ( N n ) n 0 be Narayana’s sequence given by N 0 = N 1 = N 2 = 1 and N n + 3 = N n + 2 + N n . The purpose of this paper is to determine all k -Pell numbers which are sums of two Narayana’s numbers. More precisely, we study the Diophantine equation P p ( k ) = N n + N m in nonnegative integers k , p , n and m .

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...

Explicit birational geometry of threefolds of general type, I

Jungkai A. Chen, Meng Chen (2010)

Annales scientifiques de l'École Normale Supérieure

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Let V be a complex nonsingular projective 3-fold of general type. We prove P 12 ( V ) : = dim H 0 ( V , 12 K V ) &gt; 0 and P m 0 ( V ) &gt; 1 for some positive integer m 0 24 . A direct consequence is the birationality of the pluricanonical map ϕ m for all m 126 . Besides, the canonical volume Vol ( V ) has a universal lower bound ν ( 3 ) 1 63 · 126 2 .

The tangent function and power residues modulo primes

Zhi-Wei Sun (2023)

Czechoslovak Mathematical Journal

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Let p be an odd prime, and let a be an integer not divisible by p . When m is a positive integer with p 1 ( mod 2 m ) and 2 is an m th power residue modulo p , we determine the value of the product k R m ( p ) ( 1 + tan ( π a k / p ) ) , where R m ( p ) = { 0 < k < p : k is an m th power residue modulo p } . In particular, if p = x 2 + 64 y 2 with x , y , then k R 4 ( p ) 1 + tan π a k p = ( - 1 ) y ( - 2 ) ( p - 1 ) / 8 .

A density version of the Carlson–Simpson theorem

Pandelis Dodos, Vassilis Kanellopoulos, Konstantinos Tyros (2014)

Journal of the European Mathematical Society

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We prove a density version of the Carlson–Simpson Theorem. Specifically we show the following. For every integer k 2 and every set A of words over k satisfying lim sup n | A [ k ] n | / k n > 0 there exist a word c over k and a sequence ( w n ) of left variable words over k such that the set c { c w 0 ( a 0 ) . . . w n ( a n ) : n and a 0 , . . . , a n [ k ] } is contained in A . While the result is infinite-dimensional its proof is based on an appropriate finite and quantitative version, also obtained in the paper.

Existence of renormalized solutions for some degenerate and non-coercive elliptic equations

Youssef Akdim, Mohammed Belayachi, Hassane Hjiaj (2023)

Mathematica Bohemica

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This paper is devoted to the study of some nonlinear degenerated elliptic equations, whose prototype is given by t 2 - div ( b ( | u | ) | u | p - 2 u ) + d ( | u | ) | u | p = f - div ( c ( x ) | u | α ) in Ω , u = 0 on Ω , t where Ω is a bounded open set of N ( N 2 ) with 1 < p < N and f L 1 ( Ω ) , under some growth conditions on the function b ( · ) and d ( · ) , where c ( · ) is assumed to be in L N ( p - 1 ) ( Ω ) . We show the existence of renormalized solutions for this non-coercive elliptic equation, also, some regularity results will be concluded.