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Uncountably many solutions of a system of third order nonlinear differential equations

Min Liu (2011)

Commentationes Mathematicae Universitatis Carolinae

In this paper, we aim to study the global solvability of the following system of third order nonlinear neutral delay differential equations d d t r i ( t ) d d t λ i ( t ) d d t x i ( t ) - f i ( t , x 1 ( t - σ i 1 ) , x 2 ( t - σ i 2 ) , x 3 ( t - σ i 3 ) ) + d d t r i ( t ) d d t g i ( t , x 1 ( p i 1 ( t ) ) , x 2 ( p i 2 ( t ) ) , x 3 ( p i 3 ( t ) ) ) + d d t h i ( t , x 1 ( q i 1 ( t ) ) , x 2 ( q i 2 ( t ) ) , x 3 ( q i 3 ( t ) ) ) = l i ( t , x 1 ( η i 1 ( t ) ) , x 2 ( η i 2 ( t ) ) , x 3 ( η i 3 ( t ) ) ) , t t 0 , i { 1 , 2 , 3 } in the following bounded closed and convex set Ω ( a , b ) = x ( t ) = ( x 1 ( t ) , x 2 ( t ) , x 3 ( t ) ) C ( [ t 0 , + ) , 3 ) : a ( t ) x i ( t ) b ( t ) , t t 0 , i { 1 , 2 , 3 } , where σ i j > 0 , r i , λ i , a , b C ( [ t 0 , + ) , + ) , f i , g i , h i , l i C ( [ t 0 , + ) × 3 , ) , p i j , q i j , η i j C ( [ t 0 , + ) , ) for i , j { 1 , 2 , 3 } . By applying the Krasnoselskii fixed point theorem, the Schauder fixed point theorem, the Sadovskii fixed point theorem and the Banach contraction principle, four existence results of uncountably many bounded positive solutions of the system are established.

Uniform attractors in sup-norm for semi linear parabolic problem and application to the robust stability theory

Oleksiy Kapustyan, Olena Kapustian, Oleksandr Stanzytskyi, Ihor Korol (2023)

Archivum Mathematicum

In this paper we establish the existence of the uniform attractor for a semi linear parabolic problem with bounded non autonomous disturbances in the phase space of continuous functions. We applied obtained results to prove the asymptotic gain property with respect to the global attractor of the undisturbed system.

Unique solvability of a linear problem with perturbed periodic boundary values

Bahman Mehri, Mohammad H. Nojumi (1999)

Czechoslovak Mathematical Journal

We investigate the problem with perturbed periodic boundary values y ' ' ' ( x ) + a 2 ( x ) y ' ' ( x ) + a 1 ( x ) y ' ( x ) + a 0 ( x ) y ( x ) = f ( x ) , y ( i ) ( T ) = c y ( i ) ( 0 ) , i = 0 , 1 , 2 ; 0 < c < 1 with a 2 , a 1 , a 0 C [ 0 , T ] for some arbitrary positive real number T , by transforming the problem into an integral equation with the aid of a piecewise polynomial and utilizing the Fredholm alternative theorem to obtain a condition on the uniform norms of the coefficients a 2 , a 1 and a 0 which guarantees unique solvability of the problem. Besides having theoretical value, this problem has also important applications since decay is a phenomenon that all...

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