A novel dual approach to nonlinear semigroups of Lipschitz operators

Lipschitzian semigroup refers to a one-parameter semigroup of Lipschitz operators that is strongly continuous in the parameter. It contains -semigroup, nonlinear semigroup of contractions and uniformly -Lipschitzian semigroup as special cases. In this paper, through developing a series of Lipschitz dual notions, we establish an analysis approach to Lipschitzian semigroup. It is mainly proved that a (nonlinear) Lipschitzian semigroup can be isometrically embedded into a certain -semigroup. As application results, two representation formulas of Lipschitzian semigroup are established, and many asymptotic properties of -semigroup are generalized to Lipschitzian semigroup.

     

On generation of nonlinear operator semigroups and nonlinear evolution operators

Generation of nonlinear operator semigroups and nonlinear evolution operators are proved by a different method, based on the theory of difference equations.     

Balanced allocation on graphs: A random walk approach

We propose algorithms for allocating n sequential balls into n bins that are interconnected as a d‐regular n‐vertex graph G, where d ≥ 3 can be any integer. In general, the algorithms proceeds in n succeeding rounds. Let ? > 0 be an integer, which is given as an input to the algorithms. In each round, ball 1 ≤ t ≤ n picks a node of G uniformly at random and performs a nonbacktracking random walk of length ? from the chosen node and simultaneously collects the load information of a subset of the visited nodes. It then allocates itself to one of them with the minimum load (ties are broken uniformly at random). For graphs with sufficiently large girths, we obtain upper and lower bounds for the maximum number of balls at any bin after allocating all n balls in terms of ?, with high probability.     

A Trotter-Lie formula for compactly generated semigroups of nonlinear operators

A Trotter-Lie type formula for semigroups of nonlinear operators on a Banach space X into itself, which are generated by compact operators, is proved.     

Fredholm operators,evolution semigroups,and periodic solutions of nonlinear periodic systems

Let X be a Banach space and L the generator of the evolution semigroup associated with the τ  -periodic evolutionary process _{{U(t,s)}t≥s}${\{U(t,s)\}}_{t\ge s}$ on the space _Pτ(X)$P_{\tau }(X)$ of all τ-periodic continuous X  -valued functions. We give criteria for the existence of periodic solutions to nonlinear systems of the form Lp=−?F(p,?)$Lp=-?F(p,?)$ under the condition that 1 is a normal eigenvalue of the monodromy operator U(τ,0)$U(\tau ,0)$. The proof is based on a new decomposition of the space _Pτ(X)$P_{\tau }(X)$ by constructing a right inverse of L.     

A finite difference scheme for nonlinear ultra-parabolic equations

In this paper, our aim is to study a numerical method for an ultraparabolic equation with nonlinear source function. Mathematically, the bibliography on initial–boundary value problems for ultraparabolic equations is not extensive although the problems have many applications related to option pricing, multi-parameter Brownian motion, population dynamics and so forth. In this work, we present the approximate solution by virtue of finite difference scheme and Fourier series. For the nonlinear case, we use an iterative scheme by linear approximation to get the approximate solution and obtain error estimates. A numerical example is given to justify the theoretical analysis.     

A direct approach to the weighted admissibility of observation operators for bounded analytic semigroups

In this note we adopt the approach in Bonnit et al. (Czechoslov. Math. J. 60(2):527–539, 2010) to give a direct proof of some recent results in Haak and Le Merdy (Houst. J. Math., 2005) and Haak and Kunstmann (SIAM J. Control Optim. 45:2094–2118, 2007) which characterizes the L ^p -admissibility of type α depending on p of unbounded observation operators for bounded analytic semigroups.     

A conservative nonlinear difference scheme for the viscous Cahn-Hilliard equation

Numerical solutions for the viscous Cahn-Hilliard equation are considered using the crank-Nicolson type finite difference method which conserves the mass. The corresponding stability and error analysis of the scheme are shown. The decay speeds of the solution inH ¹-norm are shown. We also compare the evolution of the viscous Cahn-Hilliard equation with that of the Cahn-Hilliard equation numerically and computationally, which has been given as an open question in Novick-Cohen[13].     

A difference scheme for the biharmonic equation with nonlinear boundary condition

In a rectangular domain we construct a grid scheme by applying the operators of exact difference schemes. We study an estimate of the rate of convergence of the grid scheme in the grid norm L₂(). It is shown that in the case when the solution of the differential problem belongs to the space W ₂ ^k (), k (3/2,2] the order of precision of the proposed scheme is O(h^k–3/2), and in the linear case it is O(h^k).Translated fromVychislitel'naya i Prikladnaya Matematika, Issue 71, 1990, pp. 3–14.     

A finite difference scheme for the nonlinear time-fractional partial integro-differential equation

In this paper, a finite difference scheme is proposed for solving the nonlinear time-fractional integro-differential equation. This model involves two nonlocal terms in time, ie, a Caputo time-fractional derivative and an integral term with memory. The existence of numerical solutions is shown by the Leray-Schauder theorem. And we obtain the discrete L₂ stability and convergence with second order in time and space by the discrete energy method. Then the uniqueness of numerical solutions is derived. Moreover, an iterative algorithm is designed for solving the derived nonlinear system. Numerical examples are presented to validate the theoretical findings and the efficiency of the proposed algorithm.     

Analytic semigroups,degenerate elliptic operators and applications to nonlinear cauchy problems

In this paper we prove L^p () and -norm estimates for the solution of the elliptic equation:

A note on the derivation of filter regularization operators for nonlinear evolution equations

Despite the strong focus of regularization on ill-posed problems, the general construction of such methods has not been fully explored. Moreover, many previous studies cannot be clearly adapted to handle more complex scenarios, albeit the greatly increasing concerns on the improvement of wider classes. In this note, we rigorously study a general theory for filter regularized operators in a Hilbert space for nonlinear evolution equations which have occurred naturally in different areas of science. The starting point lies in problems that are in principle ill-posed with respect to the initial/final data – these basically include the Cauchy problem for nonlinear elliptic equations and the backward-in-time nonlinear parabolic equations. We derive general filters that can be used to stabilize those problems. Essentially, we establish the corresponding well-posed problem whose solution converges to the solution of the ill-posed problem. The approximation can be confirmed by the error estimates in the Hilbert space. This work improves very much many papers in the same field of research.     

On the perturbation algorithm for the semidiscrete scheme for the evolution equation and estimation of the approximate solution error using semigroups

In a Banach space, for the approximate solution of the Cauchy problem for the evolution equation with an operator generating an analytic semigroup, a purely implicit three-level semidiscrete scheme that can be reduced to two-level schemes is considered. Using these schemes, an approximate solution to the original problem is constructed. Explicit bounds on the approximate solution error are proved using properties of semigroups under minimal assumptions about the smoothness of the data of the problem. An intermediate step in this proof is the derivation of an explicit estimate for the semidiscrete Crank–Nicolson scheme. To demonstrate the generality of the perturbation algorithm as applied to difference schemes, a four-level scheme that is also reduced to two-level schemes is considered.     

A series solution for nonlinear differential equations using delta operators

In this paper, we shall generalize our previous results [1] to the case of series expansion in powers of several polynomials. For this, we shall extend the ideas of delta operators and their basic polynomial sequences, introduced in conjunction with the algebra (over a field of characteristic zero) of all polynomials in one variable [2] to the algebra (over a field of characteristic zero) of all polynomials in n indeterminates. We apply this technique to derive the formal power series expansion of the input-output map describing a nonlinear system with polynomial inputs.     

On the difference scheme for a nonlinear diffusion-reaction-type problem

Published in Lietuvos Matematikos Rinkinys, Vol. 33, No. 1, pp. 16–29, January–March, 1993.     

A product formula for semigroups of Lipschitz operators associated with abstract quasilinear evolution equations

The notion of semigroups of Lipschitz operators associated with abstract quasilinear evolution equations is introduced and a product formula for such semigroups is established. The product formula obtained in the paper is applied to the solvability of the Cauchy problem for a first order quasilinear system through a finite difference scheme of the Lax‐Friedrichs type.     

广义非线性Schroedinger方程的一个新的守恒差分格式

对广义非线性Schroedinger方程提出了一种新的差分格式。揭示了该差分格式满足两个守恒律，并证明该格式的收敛性和稳定性．数值实验结果表明，新的差分格式优于Crank-Nicolson格式以及Zhang Fei等人提出的格式。