Existence and uniqueness of positive radial solutions for a class of quasilinear elliptic equations

The existence and uniqueness of positive radial solutions of the equations of the type [IML0001] in B_R, p>1 with Dirichlet condition are proved for λ large enough and f satisfying a condition[IML0002] is non-decreasing on [IML0003] It is also proved that all the positive solutions in C₁ ⁰(B^R) of the above equations are radially symmetric solutions for f satisfying [IML0004] and λ large enough.     

Existence and uniqueness of solutions for quasilinear elliptic systems

We obtain necessary and sufficient conditions for the existence of positive solutions for a class of sublinear Dirichlet quasilinear elliptic systems.

     

Existence,uniqueness, and stability of nonlinear evolution equations

In this paper we study existence, uniqueness, and stability of nonlinear evolution equations. We develop a new type of perturbation result for a C₀ semigroup in Banach space, where the nonlinear operators are not necessarily m-accretive or everywhere defined. Assuming that the semigroup has a smoothing property we obtain local existence, uniqueness and regularity results. We then establish a Liapunov theory which enables us to examine stability. To illustrate our theory several simple examples are presented.     

Existence and uniqueness of regular solutions of the first boundary-value problem for second-order quasilinear degenerate parabolic equations

One establishes existence and uniqueness theorems for regular generalized solutions of the first boundary-value problem for quasilinear (A, )-parabolic equations of the divergence form in the cylinder Q= X (T₁, T₂), where is a bounded domain in ⁿ n 1, with a sufficiently smooth boundary.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 127, pp. 49–67, 1983.     

Existence and uniqueness of the solutions and convergence of semi-implicit Euler methods for stochastic pantograph equations

In this paper the sufficient conditions of existence and uniqueness of the solutions for stochastic pantograph equation are given, i.e., the local Lipschitz condition and the linear growth condition. Under the Lipschitz condition and the linear growth condition it is proved that the semi-implicit Euler method is convergence with strong order .     

Existence and uniqueness of solutions to nonlinear evolution equations with locally monotone operators

In this paper we establish the existence and uniqueness of solutions for nonlinear evolution equations on a Banach space with locally monotone operators, which is a generalization of the classical result for monotone operators. In particular, we show that local monotonicity implies pseudo-monotonicity. The main results are applied to PDE of various types such as porous medium equations, reaction–diffusion equations, the generalized Burgers equation, the Navier–Stokes equation, the 3D Leray-α model and the p-Laplace equation with non-monotone perturbations.     

Existence of solutions for degenerate quasilinear elliptic equations

This paper deals with a class of degenerate quasilinear elliptic equations of the form −div(a(x,u,∇u)=g−div(f), where a(x,u,∇u) is allowed to be degenerate with the unknown u. We prove existence of bounded solutions under some hypothesis on f and g. Moreover we prove that there exists a renormalized solution in the case where g∈L¹(Ω) and f∈(L^p′(Ω))^N.     

Steepest descent evolution equations: asymptotic behavior of solutions and rate of convergence

We study the asymptotic behavior of the solutions of evolution equations of the form , where is a one-parameter family of approximations of a convex function we wish to minimize. We investigate sufficient conditions on the parametrization ensuring that the integral curves converge when towards a particular minimizer of . The speed of convergence is also investigated, and a result concerning the continuity of the limit point with respect to the parametrization is established. The results are illustrated on different approximation methods. In particular, we present a detailed application to the logarithmic barrier in linear programming.

     

Existence and uniqueness of pseudo-almost periodic solutions to some classes of partial evolution equations

Some sufficient conditions for the existence and uniqueness of pseudo-almost periodic (mild) solutions to some classes of partial evolution equations are given. We then make use of our abstract results to discuss the existence of pseudo-almost periodic solutions to some partial differential equations.     

Existence and uniqueness of solutions for delay evolution equations of second order in time

We prove some results on the existence and uniqueness of solutions for a class of evolution equations of second order in time, containing some hereditary characteristics. Our theory is developed from a variational point of view, and in a general functional setting which permits us to deal with several kinds of delay terms. In particular, we can consider terms which contain spatial partial derivatives with deviating arguments.     

Existence and uniqueness of the solutions of stochastic differential equations

The standard existence and uniqueness theorem for stochastic differential equations requires Lipschitz condition of the coefficients. In this paper, we extend these results to the case in which the coefficients are not required to be Lipschitz continuous, instead they only satisfy a ‘weak’ type of Lipschitz condition.     

Asymptotic behavior and uniqueness of blow-up solutions of quasilinear elliptic equations

In this paper, we study the asymptotic behavior of solutions of the problem Δ _p u = f (u) in Ω, u = ∞ on ∂Ω, under general conditions on the function f, where Ω _p is the p-Laplace operator. We show that the technique used by the author for the special case p = 2 works in this more general setting, and that the behavior described by various authors for the case p = 2 is easily derived from this technique for the general case.     

Existence and uniqueness of solutions of stationary transport equations

We consider the abstract kinetic equation dψ /dx = –JLψ, x ∈ [0, τ ], in a Hilbert space H. It is supposed that J = J * = J^–1, L = L * ≥ 0, ker L = 0. The following theorem is proved: if JL is similar to a self-adjoint operator, then an associated boundary problem has a unique solution. We apply this theorem to the stationary equation of Brownian motion (sgn μ)|μ |α (∂ψ /∂x) (x,μ) = (∂²ψ /∂μ²) (x,μ), 0 < x < τ, μ ∈ ℝ. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence,uniqueness, and blow‐up rate of large solutions to equations involving the Laplacian on the half line

This paper shows the existence and the uniqueness of the nonnegative viscosity solution of the singular boundary value problem for t >0, , where f is a continuous non‐decreasing function such that f (0)?0, and h is a nonnegative function satisfying the Keller–Osserman condition. Moreover, when h (u )=u ^p with p >3, we obtain the global estimates for the classic solution u (t ) and the exact blow‐up rate of it at t =0. Copyright © 2017 John Wiley & Sons, Ltd.     

Rate of convergence bound of difference schemes for quasilinear elliptical equations with solutions inW 2 1

We consider a mixed boundary-value problem for a quasilinear elliptical equation of second order in the rectangle with a generalized solution in W ₁ ² (). Exact difference scheme operators are applied to construct a first-order accurate difference scheme in the L₂ grid norm.Kiev Technological Institute of Light Industry. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 75, pp. 50–55, 1991.     

Existence and nonexistence of solutions for singular quadratic quasilinear equations

We study both existence and nonexistence of nonnegative solutions for nonlinear elliptic problems with singular lower order terms that have natural growth with respect to the gradient, whose model is

     

Existence, uniqueness and stability of Cm solutions of iterative functional equations

In this paper we discuss a relatively general kind of iterative functional equation G(x,f(x),..., fn(x))=0 (for all x∈J), where J is a connected closed subset of the real number axis R, G∈Cm(Jn+1, R), and n≥2. Using the method of approximating fixed points by small shift of maps, choosing suitable metrics on functional spaces and finding a relation between uniqueness and stability of fixed points of maps of general spaces, we prove the existence, uniqueness and stability of Cm solutions of the above equation for any integer m≥0 under relatively weak conditions, and generalize related results in reference in different aspects.