On the solvability of some nonlinear Elliptic problems

The solvability of second-order nonlinear elliptic equations in weighted Sobolev spaces is analyzed. An additional condition ensuring the solvability of such equations is that the average of the desired solution over some circle of fixed radius is zero. Examples are equations containing a weighted p-Laplacian and the Euler equations.     

On the solution of physically nonlinear quasistatic problems in viscoelasticity

The authors propose a method of solving a Volterra system of nonlinear integral equations. The method is based on Cauchy's expression for multiplication of a power series. As an illustration of the method, the authors consider the flexure of rectangular plates of physically nonlinear viscoelastic materials.Institute of Cybernetics and Computer Center, Academy of Sciences of the Uzbek SSR, Tashkent. Translated from Mekhanika Polimerov, Vol. 9, No. 3, pp. 558–561, May–June, 1973.     

On the solvability of boundary value problems for nonlinear differential systems

We obtain in a sense optimal tests for the solvability of the nonlinear boundary value problem

$$\frac{{dx}}{{dt}} = f(t,x),x(a) = h(x,(b)),$$

where the function f: [a, b] × ? ⁿ → ? ⁿ belongs to the Carathéodory class and the function h: ? ⁿ → ? ⁿ is continuous.

     

On solvability of boundary value problems for higher order nonlinear hyperbolic equations

In the rectangle Ω=[0,a]×[0,b] for the nonlinear hyperbolic equation

the boundary value problems of the type

are considered, where and are linear bounded functionals.Sufficient conditions of solvability and unique solvability of the general problem and its particular cases (Nicoletti type, Dirichlet, Lidstone and Periodic problems) are established.     

The problem of the solvability of nonlinear two-point boundary-value problems

We establish sufficient conditions for the solvability of boundary-value problems of the form $$\begin{gathered} u'' = f(t,u,u'); \hfill \\ \begin{array}{*{20}c} {(u(0),} & {u'(0)) \in S_0 ,} & {(u(1),} & {u'(1)) \in S_1 .} \\ \end{array} \hfill \\ \end{gathered} $$      

On the solvability of a nonlinear pseudoparabolic problem

This paper is concerned with the study of an initial boundary value problem for a nonlinear second order pseudoparabolic equation arising in the unidirectional flow of a thermodynamic compatible third grade fluid. We establish some a priori bounds for the solution and prove its existence.     

Conditions for one-valued solvability of nonlinear stationary heat-conduction problems

We establish conditions for existence and uniqueness of nonnegative solutions of nonlinear stationary heat-conduction problems, the Dirichlet, problem and the Neumann one, with regard for the dependence of the heat-conduction coefficient and inner heat sources on temperature. Lvov University, Lvov. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51., No. 4, pp. 562–567, April, 1999.     

On a method of solving physically nonlinear problems in the bending of thick rectangular plates

We study the three-dimensional boundary-value problem of the physically nonlinear theory of elasticity for bending of a homogeneous isotropic thick plate by transverse forces. We assume that the intensity of the load is such that the connection between the stresses and small strains within the elastic limit can be described by nonlinear relations in the form of H. Kauderer. We develop an approximate analytic method that makes it possible to reduce the original nonlinear boundary-value problem to a recursive sequence of corresponding linear boundary-value problems. Translated fromMatematichni Metodi i Fiziko-mekhanichni Polya, No. 40, 1997, pp. 30–35.     

On the solvability of equations of a nonlinear viscous-ideal fluid

A model of the one-dimensional motion of a viscous compressible fluid is considered. A class of nonlinear stressed-state equations for which the initial boundary-value problem has global (in time) solutions in the class of generalized solutions satisfying the energy identity is described. In particular, media exhibiting viscous properties only for large strain rates are studied. Translated fromMatematicheskie Zametki, Vol. 68, No. 1, pp. 13–23, July, 2000.     

On the solvability of nonlinear operator equations in normed spaces

Sunto Si studia una classe di applicazioni continue negli spazi normati e si dimostrano, per tali applicazioni, proprietà analoghe a quelle della teoria del grado di Leray- Schauder. Si danno esempi di problemi ai limiti per equazioni differenziali ordinarie che possono essere agevolmente trattati usando la teoria sviluppata nel presente lavoro.     

On the solvability of nonlinear differential inequalities with singular coefficients

On the basis of the nonlinear capacity method, we obtain exact solvability conditions for some classes of stationary and evolution nonlinear differential inequalities with singular coefficients.     

Global solvability of initial boundary-value problems for nonlinear analogs of the Boussinesq equation

The solvability of the natural (first, second, and mixed) initial boundary-value problems for nonlinear analogs of the Boussinesq equation is studied. Uniqueness theorems for regular solutions and global solvability theorems are proved.     

On polynomial solvability of two multiprocessor scheduling problems

We discuss a new approach for solving multiprocessor scheduling problems by using and improving results for guillotine pallet loading problem. We introduce a new class of schedules by analogy with the guillotine restriction for cutting stock problem and show that many well-known algorithms from classical scheduling theory construct schedules only from this class. We also consider two multiprocessor scheduling problems and prove that they can be solved in polynomial time.     

On the solvability of nonlocal nonstationary problems with double degeneration

We study the solvability in Sobolev spaces of the first boundary value problem for a nonlinear evolution equation degenerating both on the solution and on the solution gradient. We consider the case in which the spatial operator can depend on a nonlocal characteristic of a solution, for example, on an integral characteristic. The theorem is proved with the use of the time discretization method. To study the solvability of the spatial problems arising in the course of the proof, we use the Galerkin method.