Periodic viable trajectories of differential inclusions

In this paper, the periodic viable trajectories of differential inclusions are discussed. Firstly, a simplified property of differential inclusions is given. Then, an existence theorem of periodic viable trajectories of differential inclusions in a finite dimensional space is proved. With the above results and Galerkin’s approximation, an existence theorem of periodic viable trajectories of partial differential inclusions in a Hilbert space is proved.     

Differential inclusions with Hukuhara derivative

We introduce two types of differential inclusions with Hukuhara derivative and consider properties of these inclusions. For differential inclusions of the second type, we give different definitions of a solution, prove theorems on the existence of an ordinary solution, and establish the compactness of the set of these solutions. __________ Translated from Neliniini Kolyvannya, Vol. 10, No. 2, pp. 229–246, April–June, 2007.     

A second-order multivalued boundary-value problem and impulsive neutral functional differential inclusions in Banach spaces

In this paper, by using the new fixed-point theorem of O'Regan and Precup and a noncompact measure, we study the existence of solutions of a second-order multivalued boundary-value problem in Banach spaces and the existence of a mild solution for impulsive neutral functional differential inclusions in Banach spaces. The compactness conditions and the upper-semicontinuity conditions for multivalued integral operators are weakened in this paper. Published in Neliniini Kolyvannya, Vol. 11, No. 2, pp. 191–207, April–June, 2008.     

On a class of generalized nonlinear implicit quasivariational inclusions

ＩｎｔｒｏｄｕｃｔｉｏｎＶａｒｉａｔｉｏｎａｌｉｎｅｑｕａｌｉｔｙｔｈｅｏｒｙａｎｄｃｏｍｐｌｅｍｅｎｔａｒｉｔｙｐｒｏｂｌｅｍｔｈｅｏｒｙｈａｖｅｂｅｃｏｍｅｖｅｒｙｅｆｆｅｃｔｉｖｅａｎｄｐｏｗｅｒｆｕｌｔｏｏｌｓｆｏｒｓｔｕｄｙｉｎｇａｗｉｄｅｒａｎｇｅｏｆｐｒｏｂｌｅｍｓａｒｉｓｉｎｇｉｎｍｅｃｈａｎｉｃｓ,ｍａｔｈｅｍａｔｉｃａｌｐｒｏｇｒａｍｍｉｎｇ,ｏｐｔｉｍｉｚａｔｉｏｎａｎｄｃｏｎｔｒｏｌ,ｅｑｕｉｌｉｂｒｉｕｍｔｈｅｏｒｙｏｆｅｃｏｎｏｍｉｃｓ,ｍａｎａｇｅｍｅｎｔｓｃｉｅｎ…     

Singular-perturbation method for nonlinear second-order evolution inclusions with Volterra operators

We consider a class of second-order operator differential inclusions with Volterra-type operators. Using the singular-perturbation method, we study the problem of the existence of a solution of the Cauchy problem for these inclusions. Important a priori estimates are obtained for solutions and their derivatives. We give an example that illustrates the proposed approach to the analysis of the problem under consideration. Translated from Neliniini Kolyvannya, Vol. 12, No. 1, pp. 27–43, January–March, 2009.     

Numerical approximation of a class of discontinuous systems of fractional order

In this paper we investigate the possibility to formulate an implicit multistep numerical method for fractional differential equations, as a discrete dynamical system to model a class of discontinuous dynamical systems of fractional order. For this purpose, the problem is continuously transformed into a set-valued problem, to which the approximate selection theorem for a class of differential inclusions applies. Next, following the way presented in the book of Stewart and Humphries (Dynamical Systems and Numerical Analysis, Cambridge University Press, Cambridge, 1996) for the case of continuous differential equations, we prove that a variant of Adams?CBashforth?CMoulton method for fractional differential equations can be considered as defining a discrete dynamical system, approximating the underlying discontinuous fractional system. For this purpose, the existence and uniqueness of solutions are investigated. One example is presented.     

Periodic Solutions of Linear Impulsive Differential Inclusions

For linear differential inclusions with pulse action at fixed times, we establish conditions for the existence of periodic ordinary solutions and R-solutions.__________Translated from Neliniini Kolyvannya, Vol. 7, No. 4, pp. 495–515, October–December, 2004.     

Nonconvex-valued impulsive functional differential inclusions with variable times

We use the Schaefer fixed-point theorem, combined with the Bressan-Colombo selection theorem for lower-semicontinuous multivalued operators with nonempty closed and decomposable values, for the investigation of the existence of solutions for first-and second-order impulsive functional differential inclusions with variable times. Published in Neliniini Kolyvannya, Vol. 10, No. 4, pp. 443–463, October–December, 2007.     

Existence of solutions for implicit fuzzy differential inclusions

A class of implicit fuzzy differential inclusions(IFDIs) are introduced and studied.Some existence theorems under different conditions are proved with the selection theorems for the open situation and the closed situation,respectively.A viable solution for a closed IFDI is proved to exist under the tangential condition.As an application,an implicit fuzzy differential equation,which comes from the drilling dynamics in petroleum engineering,is analyzed numerically.The obtained results can improve and extend some known results for fuzzy differential inclusions(FDIs) and fuzzy differential equations(FDEs),which might be helpful in the analysis of fuzzy dynamic systems.     

Existence of solutions for parabolic type evolution differential inclusions and the property of the solution set

IntroductionInrecenttenyears,becauseoftheneedofthecontroltheoryandeconomics,thetheoryofdifferentialinclusionshasbeendevelopingfast,andsome~tresultSindifferenhalinclusionshavebeenobtained.Asystematicstudyondifferenhalinclusionscanbefoundin[l].Inordertosuitcontroltheory,thefollowingsendlineardifferentialinclusionswereconsideredin[2~4]:x'(t) Ax(t)6F(t,x(t)),a.e.teI,x(0)=xo'(l)xthereAgeneratesalinearsendgroup(compactorequicolltinuouS),XisasepotleBanachspaCe,I=[0,TiisanintervalofR,F:IxX~Zx\0i…     

MULTI-VALUED QUASI VARLATIONAL INCLUSIONS IN BANACH SPACES

The purpose is to introduce and study a class of more general multivalued quasi variational inclusions in Banach spaces.By using the resolvent operator technique some existence theorem of solutions and iterative approximation for solving this kind of multivalued quasi variational inclusions are established.The results generalize,improve and unify a number of Noor's and others' recent results.     

Fixed-Point Theory for Weakly Sequentially Upper-Semicontinuous Maps with Applications to Differential Inclusions

We present new fixed-point theorems for weakly sequentially upper-semicontinuous maps. These results are then used to establish existence principles for second-order differential equations and inclusions.     

Self-similar singular solution of fast diffusion equation with gradient absorption terms

The self-similar singular solution of the fast diffusion equation with nonlinear gradient absorption terms are studied. By a self-similar transformation, the self-similar solutions satisfy a boundary value problem of nonlinear ordinary differential equation (ODE). Using the shooting arguments, the existence and uniqueness of the solution to the initial data problem of the nonlinear ODE are investigated, and the solutions are classified by the region of the initial data. The necessary and sufficient condition for the existence and uniqueness of self-similar very singular solutions is obtained by investigation of the classification of the solutions. In case of existence, the self-similar singular solution is very singular solution.     

On the periodic solutions of differential inclusions and applications

The periodic problem of evolution inclusion is studied and its results are used toestablish existence theorems of periodic solutions of a class of semi-linear differential inclusion.Also existence theorem of the extreme solutions and the strong relaxation theorem are given forthis class of semi-linear differential inclusion. An application to some feedback control systems isdiscussed.     

OGY method for a class of discontinuous dynamical systems

In this paper, we prove that the OGY method to control unstable periodic orbits (UPOs) of continuous-time systems can be applied to a class of systems discontinuous with respect the state variable, by using a generalized derivative. Because the discontinuous problem may have not classical solutions, the initial value problem is transformed into a set-valued problem via Filippov regularization. The existence of the ingredients necessary to apply OGY method (UPO, Poincaré map and stable and unstable directions) is proved and the numerically implementation is explained. Another possible way analyzed in this paper is the continuous approximation of the underlying initial value problem, via Cellina??s theorem for differential inclusions. Thus, the problem is approximated by a continuous initial value problem, and the OGY method can be applied as usual.     

Stationary Solutions for Stochastic Differential Equations Driven by Lévy Processes

In the paper, stationary solutions of stochastic differential equations driven by Lévy processes are considered. And the existence of these stationary solutions follows from the theory of random dynamical systems and their attractors. Moreover, under a one-sided Lipschitz continuity condition and a temperedness condition, Itô and Marcus stochastic differential equations driven by Lévy processes are proved to have stationary solutions. Besides, continuous dependence of stationary solutions on drift coefficients of these equations is presented.     

TWO POINTS BOUNDARY VALUE PROBLEMS IN BANACH SPACES

（王志华）（张凤祥）ＴＷＯＰＯＩＮＴＳＢＯＵＮＤＡＲＹＶＡＬＵＥＰＲＯＢＬＥＭＳＩＮＢＡＮＡＣＨＳＰＡＣＥＳ￥ＷａｎｇＺｈｉｈｕａ；ＺｈａｎｇＦｅｎｇｘｉａｎｇ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ．ＬａｎｚｈｏｕＵｎｉｖｅｒｓｉｔｙ．Ｌａ...     

MULTI-VALUED QUASI VARIATIONAL INCLUSIONS IN BANACH SPACES

The purpose is to introduce and study a class of more general multivalued quasi variational inclusions in Banach spaces. By using the resolvent operator technique some existence theorem of solutions and iterative approximation for solving this kind of multivalued quasi variational inclusions are established. The results generalize, improve and unify a number of Noor‘s and others‘ recent results.     

Existence and Approximation of Solutions to Variational Inclusions with Accretive Mappings in Banach Spaces

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｒｏｕｇｈｏｕｔｔｈｉｓｐａｐｅｒｗｅｓｕｐｐｏｓｅｔｈａｔＸｉｓａｒｅａｌＢａｎａｃｈｓｐａｃｅ,Ｘ ｉｓｉｔｓｄｕａｌｓｐａｃｅ ,〈· ,·〉ｉｓｔｈｅｐａｉｒｉｎｇｏｆＸａｎｄＸ .Ｄ(Ｔ)ａｎｄＲ(Ｔ)ｄｅｎｏｔｅｔｈｅｄｏｍａｉｎａｎｄｔｈｅｒａｎｇｅｏｆＴｒｅｓｐｅｃｔｉｖｅｌｙ .ＬｅｔＴ ,Ａ :Ｘ →Ｘ ,ｇ :Ｘ→Ｘ ｂｅｔｈｒｅｅｍａｐｐｉｎｇｓａｎｄφ :Ｘ →Ｒ∪ ∞ ａｐｒｏｐｅｒｃｏｎｖｅｘｌｏｗｅｒｓｅｍｉ＿ｃｏｎｔｉｎｕｏｕｓｆｕｎｃｔｉｏｎ .Ｉｎ 1 999,ｔｈ…     

On a nonlinear hyperbolic variational equation: I. Global existence of weak solutions

We study the nonlinear hyperbolic partial differential equation, (u _t+uu_x)_x=1/2u _x ² . This partial differential equation is the canonical asymptotic equation for weakly nonlinear solutions of a class of hyperbolic equations derived from variational principles. In particular, it describes waves in a massive director field of a nematic liquid crystal.Global smooth solutions of the partial differential equation do not exist, since their derivatives blow up in finite time, while weak solutions are not unique. We therefore define two distinct classes of admissible weak solutions, which we call dissipative and conservative solutions. We prove the global existence of each type of admissible weak solution, provided that the derivative of the initial data has bounded variation and compact support. These solutions remain continuous, despite the fact that their derivatives blow up.There are no a priori estimates on the second derivatives in any L ^p space, so the existence of weak solutions cannot be deduced by using Sobolev-type arguments. Instead, we prove existence by establishing detailed estimates on the blowup singularity for explicit approximate solutions of the partial differential equation.We also describe the qualitative properties of the partial differential equation, including a comparison with the Burgers equation for inviscid fluids and a number of illustrative examples of explicit solutions. We show that conservative weak solutions are obtained as a limit of solutions obtained by the regularized method of characteristics, and we prove that the large-time asymptotic behavior of dissipative solutions is a special piecewise linear solution which we call a kink-wave.