• Title/Summary/Keyword: bounded and periodic solutions

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BOUNDED WEAK SOLUTION FOR THE HAMILTONIAN SYSTEM

  • Choi, Q-Heung;Jung, Tacksun
    • Korean Journal of Mathematics
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    • v.21 no.1
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    • pp.81-90
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    • 2013
  • We investigate the bounded weak solutions for the Hamiltonian system with bounded nonlinearity decaying at the origin and periodic condition. We get a theorem which shows the existence of the bounded weak periodic solution for this system. We obtain this result by using variational method, critical point theory for indefinite functional.

SIGN CHANGING PERIODIC SOLUTIONS OF A NONLINEAR WAVE EQUATION

  • Jung, Tacksun;Choi, Q-Heung
    • Korean Journal of Mathematics
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    • v.16 no.2
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    • pp.243-257
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    • 2008
  • We seek the sign changing periodic solutions of the nonlinear wave equation $u_{tt}-u_{xx}=a(x,t)g(u)$ under Dirichlet boundary and periodic conditions. We show that the problem has at least one solution or two solutions whether $\frac{1}{2}g(u)u-G(u)$ is bounded or not.

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TIME PERIODIC SOLUTION FOR THE COMPRESSIBLE MAGNETO-MICROPOLAR FLUIDS WITH EXTERNAL FORCES IN ℝ3

  • Qingfang Shi;Xinli Zhang
    • Journal of the Korean Mathematical Society
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    • v.60 no.3
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    • pp.587-618
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    • 2023
  • In this paper, we consider the existence of time periodic solutions for the compressible magneto-micropolar fluids in the whole space ℝ3. In particular, we first solve the problem in a sequence of bounded domains by the topological degree theory. Then we obtain the existence of time periodic solutions in ℝ3 by a limiting process.

ON GENERALIZED FLOQUET SYSTEMS II

  • EI-Owaidy, H.;Zagrout, A.A.
    • Kyungpook Mathematical Journal
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    • v.27 no.1
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    • pp.35-41
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    • 1987
  • Consider the system (i) x'=Ax. Let ${\Phi}$ be its fundamental matrix solution. If there is w>0 such that A(t+w)-A(t) commutes with ${\Phi}$ for all t, then we call this system a "generalized" Floquet system or a "G. F. system". We show that $A(t+w)-A(t)=B_1$=constant if and only if $A(t)=C+B_1t/w+Q(t)$, Q is periodic of period w>0. For this A(t) We prove that if all eigenvalues of $B_1$ have negative real parts, then the origin is asumptotically stable. We find a growth condition for a continuous D(t) which guarantees that all solutions of z'=[A(t)+D(t)]z are bounded if all solutions of the G. F. system (i) are bounded. Combining the foregoing results yield a class of perturbed G. F. operators all of whose solutions are bounded.

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ON STABILITY AND BIFURCATION OF PERIODIC SOLUTIONS OF DELAY DIFFERENTIAL EQUATIONS

  • EL-SHEIKH M. M. A.;EL-MAHROUF S. A. A.
    • Journal of applied mathematics & informatics
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    • v.19 no.1_2
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    • pp.281-295
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    • 2005
  • The purpose of this paper is to study a class of delay differential equations with two delays. First, we consider the existence of periodic solutions for some delay differential equations. Second, we investigate the local stability of the zero solution of the equation by analyzing the corresponding characteristic equation of the linearized equation. The exponential stability of a perturbed delay differential system with a bounded lag is studied. Finally, by choosing one of the delays as a bifurcation parameter, we show that the equation exhibits Hopf and saddle-node bifurcations.

ON THE RECURSIVE SEQUENCE X_{n+1} = $\alpha$ - (X_n/X_n-1)

  • YAN XING XUE;LI WAN TONG;ZHAO ZHU
    • Journal of applied mathematics & informatics
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    • v.17 no.1_2_3
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    • pp.269-282
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    • 2005
  • We study the global asymptotic stability, global attractivity, boundedness character, and periodic nature of all positive solutions and all negative solutions of the difference equation $$x_{n+1}\;=\;{\alpha}-{\frac{x_{n-1}}{x_{n}},\;n=0,1,\;{\cdots}$$, where ${\alpha}\;\in\; R$ is a real number, and the initial conditions $x_{-1},\;x_0$ are arbitrary real numbers.

CIRCULAR SPECTRUM AND ASYMPTOTIC PERIODIC SOLUTIONS TO A CLASS OF NON-DENSELY DEFINED EVOLUTION EQUATIONS

  • Le Anh Minh;Nguyen Ngoc Vien
    • Communications of the Korean Mathematical Society
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    • v.38 no.4
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    • pp.1153-1162
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    • 2023
  • In this paper, for the bounded solution of the non-densely defined non-autonomous evolution equation, we present the condition for asymptotic periodicity by using the circular spectral theory of functions on the half line and the extrapolation theory of non-densely defined evolution equation.

DYNAMICS OF A ONE-PREY AND TWO-PREDATOR SYSTEM WITH TWO HOLLING TYPE FUNCTIONAL RESPONSES AND IMPULSIVE CONTROLS

  • Baek, Hunki
    • Journal of the Korean Society for Industrial and Applied Mathematics
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    • v.16 no.3
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    • pp.151-167
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    • 2012
  • In this paper, we investigate the dynamic behaviors of a one-prey and two-predator system with Holling-type II functional response and defensive ability by introducing a proportion that is periodic impulsive harvesting for all species and a constant periodic releasing, or immigrating, for predators at different fixed time. We establish conditions for the local stability and global asymptotic stability of prey-free periodic solutions by using Floquet theory for the impulsive equation, small amplitude perturbation skills. Also, we prove that the system is uniformly bounded and is permanent under some conditions via comparison techniques. By displaying bifurcation diagrams, we show that the system has complex dynamical aspects.

STABILIZATION OF 2D g-NAVIER-STOKES EQUATIONS

  • Nguyen, Viet Tuan
    • Communications of the Korean Mathematical Society
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    • v.34 no.3
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    • pp.819-839
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    • 2019
  • We study the stabilization of 2D g-Navier-Stokes equations in bounded domains with no-slip boundary conditions. First, we stabilize an unstable stationary solution by using finite-dimensional feedback controls, where the designed feedback control scheme is based on the finite number of determining parameters such as determining Fourier modes or volume elements. Second, we stabilize the long-time behavior of solutions to 2D g-Navier-Stokes equations under action of fast oscillating-in-time external forces by showing that in this case there exists a unique time-periodic solution and every solution tends to this periodic solution as time goes to infinity.