Displaying similar documents to “Oscillation of deviating differential equations”

Nonrectifiable oscillatory solutions of second order linear differential equations

Takanao Kanemitsu, Satoshi Tanaka (2017)

Archivum Mathematicum

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The second order linear differential equation ( p ( x ) y ' ) ' + q ( x ) y = 0 , x ( 0 , x 0 ] is considered, where p , q C 1 ( 0 , x 0 ] , p ( x ) > 0 , q ( x ) > 0 for x ( 0 , x 0 ] . Sufficient conditions are established for every nontrivial solutions to be nonrectifiable oscillatory near x = 0 without the Hartman–Wintner condition.

Oscillation criteria for fourth order half-linear differential equations

Jaroslav Jaroš, Kusano Takaŝi, Tomoyuki Tanigawa (2020)

Archivum Mathematicum

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Criteria for oscillatory behavior of solutions of fourth order half-linear differential equations of the form ( | y ' ' | α sgn y ' ' ) ' ' + q ( t ) | y | α sgn y = 0 , t a > 0 , A where α > 0 is a constant and q ( t ) is positive continuous function on [ a , ) , are given in terms of an increasing continuously differentiable function ω ( t ) from [ a , ) to ( 0 , ) which satisfies a 1 / ( t ω ( t ) ) d t < .

Oscillation criteria for two dimensional linear neutral delay difference systems

Arun Kumar Tripathy (2023)

Mathematica Bohemica

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In this work, necessary and sufficient conditions for the oscillation of solutions of 2-dimensional linear neutral delay difference systems of the form Δ x ( n ) + p ( n ) x ( n - m ) y ( n ) + p ( n ) y ( n - m ) = a ( n ) b ( n ) c ( n ) d ( n ) x ( n - α ) y ( n - β ) are established, where m > 0 , α 0 , β 0 are integers and a ( n ) , b ( n ) , c ( n ) , d ( n ) , p ( n ) are sequences of real numbers.

On asymptotic behavior of solutions to Emden-Fowler type higher-order differential equations

Irina Astashova (2015)

Mathematica Bohemica

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For the equation y ( n ) + | y | k sgn y = 0 , k > 1 , n = 3 , 4 , existence of oscillatory solutions y = ( x * - x ) - α h ( log ( x * - x ) ) , α = n k - 1 , x < x * , is proved, where x * is an arbitrary point and h is a periodic non-constant function on . The result on existence of such solutions with a positive periodic non-constant function h on is formulated for the equation y ( n ) = | y | k sgn y , k > 1 , n = 12 , 13 , 14 .

Solutions of an advance-delay differential equation and their asymptotic behaviour

Gabriela Vážanová (2023)

Archivum Mathematicum

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The paper considers a scalar differential equation of an advance-delay type y ˙ ( t ) = - a 0 + a 1 t y ( t - τ ) + b 0 + b 1 t y ( t + σ ) , where constants a 0 , b 0 , τ and σ are positive, and a 1 and b 1 are arbitrary. The behavior of its solutions for t is analyzed provided that the transcendental equation λ = - a 0 e - λ τ + b 0 e λ σ has a positive real root. An exponential-type function approximating the solution is searched for to be used in proving the existence of a semi-global solution. Moreover, the lower and upper estimates are given for such a solution.

Existence and multiplicity of solutions for a p ( x ) -Kirchhoff type problem via variational techniques

A. Mokhtari, Toufik Moussaoui, D. O’Regan (2015)

Archivum Mathematicum

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This paper discusses the existence and multiplicity of solutions for a class of p ( x ) -Kirchhoff type problems with Dirichlet boundary data of the following form - a + b Ω 1 p ( x ) | u | p ( x ) d x div ( | u | p ( x ) - 2 u ) = f ( x , u ) , i n Ω u = 0 o n Ω , where Ω is a smooth open subset of N and p C ( Ω ¯ ) with N < p - = inf x Ω p ( x ) p + = sup x Ω p ( x ) < + , a , b are positive constants and f : Ω ¯ × is a continuous function. The proof is based on critical point theory and variable exponent Sobolev space theory.

A note on the existence of solutions with prescribed asymptotic behavior for half-linear ordinary differential equations

Manabu Naito (2024)

Mathematica Bohemica

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The half-linear differential equation ( | u ' | α sgn u ' ) ' = α ( λ α + 1 + b ( t ) ) | u | α sgn u , t t 0 , is considered, where α and λ are positive constants and b ( t ) is a real-valued continuous function on [ t 0 , ) . It is proved that, under a mild integral smallness condition of b ( t ) which is weaker than the absolutely integrable condition of b ( t ) , the above equation has a nonoscillatory solution u 0 ( t ) such that u 0 ( t ) e - λ t and u 0 ' ( t ) - λ e - λ t ( t ), and a nonoscillatory solution u 1 ( t ) such that u 1 ( t ) e λ t and u 1 ' ( t ) λ e λ t ( t ).

Correct solvability of a general differential equation of the first order in the space L p ( )

Nina A. Chernyavskaya, Leonid A. Shuster (2015)

Archivum Mathematicum

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We consider the equation - r ( x ) y ' ( x ) + q ( x ) y ( x ) = f ( x ) , x where f L p ( ) , p [ 1 , ] ( L ( ) : = C ( ) ) and 0 < r C ( ) , 0 q L 1 ( ) . We obtain minimal requirements to the functions r and q , in addition to (), under which equation () is correctly solvable in L p ( ) , p [ 1 , ] .

Necessary and sufficient conditions for oscillation of second-order differential equations with nonpositive neutral coefficients

Arun K. Tripathy, Shyam S. Santra (2021)

Mathematica Bohemica

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In this work, we present necessary and sufficient conditions for oscillation of all solutions of a second-order functional differential equation of type ( r ( t ) ( z ' ( t ) ) γ ) ' + i = 1 m q i ( t ) x α i ( σ i ( t ) ) = 0 , t t 0 , where z ( t ) = x ( t ) + p ( t ) x ( τ ( t ) ) . Under the assumption ( r ( η ) ) - 1 / γ d η = , we consider two cases when γ > α i and γ < α i . Our main tool is Lebesgue’s dominated convergence theorem. Finally, we provide examples illustrating our results and state an open problem.