Existence of positive solutions for the system of higher order two-point boundary value problems

In this paper, we establish the existence of at least one and two positive solutions for the system of higher order boundary value problems by using the Krasnosel’skii fixed point theorem.     

Fourth-Order p-Laplacian Nonlinear Systems via the Vector Version of Krasnosel’skiĭ’s Fixed Point Theorem

In this paper, existence results for a fourth-order nonlinear system are obtained. Both classical and vector versions of the Krasnosel’skiĭ’s fixed point theorem are used and a comparison of the obtained results to those from the literature is provided.     

A vector version of Krasnosel’skiĭ’s fixed point theorem in cones and positive periodic solutions of nonlinear systems

A new version of Krasnosel’skiĭ’s fixed point theorem in cones is established for systems of operator equations, where the compression-expansion conditions are expressed on components. In applications, this allows the nonlinear term of a system to have different behaviors both in components and in variables.     

Affine properties and injectivity of quasi-isometries

We approximateε-quasi-isometries between finite-dimensional Banach spaces by linear near-isometries. In this way we improve and extend a theorem of John. We also improve results of Gevirtz on injectivity criteria for quasi-isometries. Our approach is to show thatε-quasi-isometries almost satisfy the Jensen functional equation and to use then known facts about linear approximation of approximate solutions of Jensen’s equation.     

Approximate Inertial Manifolds of Strongly Damped Nonlinear Wave Equations

In this paper we consider a class of strongly damped nonlinear wave equations. By the transformation of unknown functions and decomposition of operators, we construct a family of approximate inertial manifolds, and obtain the estimate of orders of approximation of such manifolds to solution orbits.     

A new method for solving the nonlinear second-order boundary value differential equations

In this paper we use measure theory to solve a wide range of second-order boundary value ordinary differential equations. First, we transform the problem to a first order system of ordinary differential equations (ODE’s) and then define an optimization problem related to it. The new problem is modified into one consisting of the minimization of a linear functional over a set of Radon measures; the optimal measure is then approximated by a finite combination of atomic measures and the problem converted approximatly to a finite-dimensional linear programming problem. The solution to this problem is used to construct the approximate solution of the original problem. Finally we get the error functionalE (we define in this paper) for the approximate solution of the ODE’s problems.     

Superconvergence in the generalized finite element method

In this paper, we address the problem of the existence of superconvergence points of approximate solutions, obtained from the Generalized Finite Element Method (GFEM), of a Neumann elliptic boundary value problem. GFEM is a Galerkin method that uses non-polynomial shape functions, and was developed in (Babuška et al. in SIAM J Numer Anal 31, 945–981, 1994; Babuška et al. in Int J Numer Meth Eng 40, 727–758, 1997; Melenk and Babuška in Comput Methods Appl Mech Eng 139, 289–314, 1996). In particular, we show that the superconvergence points for the gradient of the approximate solution are the zeros of a system of non-linear equations; this system does not depend on the solution of the boundary value problem. For approximate solutions with second derivatives, we have also characterized the superconvergence points of the second derivatives of the approximate solution as the roots of a system of non-linear equations. We note that smooth generalized finite element approximation is easy to construct. I. Babuška’s research was partially supported by NSF Grant # DMS-0341982 and ONR Grant # N00014-99-1-0724. U. Banerjee’s research was partially supported by NSF Grant # DMS-0341899. J. E. Osborn’s research was supported by NSF Grant # DMS-0341982.     

Schlichting’s theorem for approximate subgroups

We prove Schlichting’s theorem for approximate subgroups: if $${\mathcal {X}}$$ is a uniform family of commensurable approximate subgroups in some ambient group, then there exists an invariant approximate subgroup commensurable with $${\mathcal {X}}$$.     

On Ha＇s Version of Set-valued Ekeland＇s Variational Principle

By using the concept of cone extensions and Dancs-Hegedus-Medvegyev theorem, Ha [Some variants of the Ekeland variational principle for a set-valued map. J. Optim. Theory Appl., 124, 187–206 (2005)] established a new version of Ekeland’s variational principle for set-valued maps, which is expressed by the existence of strict approximate minimizer for a set-valued optimization problem. In this paper, we give an improvement of Ha’s version of set-valued Ekeland’s variational principle. Our proof is direct and it need not use Dancs-Hegedus-Medvegyev theorem. From the improved Ha’s version, we deduce a Caristi-Kirk’s fixed point theorem and a Takahashi’s nonconvex minimization theorem for set-valued maps. Moreover, we prove that the above three theorems are equivalent to each other.     

On minimization of one integral functional by the Ritz method

Using the variational method, we investigate a nonlinear problem with a Bernoulli condition in the form of an inequality on a free boundary. We prove a solvability theorem and establish the convergence of an approximate solution obtained by the Ritz method to the exact solution in certain metrics. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 10, pp. 1385–1394, October, 2006.     

Hyperbolicity criterion for periodic solutions of functional-differential equations with several delays

In this paper, a hyperbolicity criterion for periodic solutions of nonlinear functional-differential equations is constructed in terms of zeros of the characteristic function. In the earlier papers in this area, necessary and sufficient conditions were different from each other. Moreover, it was assumed that if the period of the investigated solution is irrational, then that solution admits a rational approximation. In this paper, we obtain necessary and sufficient conditions of the hyperbolicity. It is proved (and the proof is constructive) that a rational approximation exists for any irrational period. All the results are obtained for the case of several rational delays. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 21, Proceedings of the Seminar on Differential and Functional Differential Equations Supervised by A. Skubachevskii (Peoples’ Friendship University of Russia), 2007.     

On the application of Ateb-functions to the construction of a solution of a nonlinear Klein-Gordon equation

For a nonlinear Klein-Gordon equation, we construct the first approximation of an asymptotic solution by using Ateb-functions. The resonance and nonresonance cases are considered. “L’vivs’ka Politeknika” University, Lviv. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 6, pp. 872–877, June, 1997.     

A hybrid method for computing Lyapunov exponents

In this paper we propose a numerical method for computing all Lyapunov coefficients of a discrete time dynamical system by spatial integration. The method extends an approach of Aston and Dellnitz (Comput Methods Appl Mech Eng 170:223–237, 1999) who use a box approximation of an underlying ergodic measure and compute the first Lyapunov exponent from a spatial average of the norms of the Jacobian for the iterated map. In the hybrid method proposed here, we combine this approach with classical QR-oriented methods by integrating suitable R-factors with respect to the invariant measure. In this way we obtain approximate values for all Lyapunov exponents. Assuming somewhat stronger conditions than those of Oseledec’ multiplicative theorem, these values satisfy an error expansion that allows to accelerate convergence through extrapolation. W.-J. Beyn and A. Lust was supported by CRC 701 ‘Spectral Analysis and Topological Methods in Mathematics’. The paper is mainly based on the PhD thesis [27] of A. Lust.     

Star-shaped sets in normed spaces

We prove a generalization of the Krasnosel’ski theorem on star-shaped sets. Usingd-segments inn-dimensional Minkowski spaces instead of usual segments, the notions “d-visibility” and “d-star-shapedness” are introduced. Our main aim is to give necessary and sufficient conditions ford-star-shapedness in finite-dimensional normed spaces.     

Frequency methods in the theory of bounded solutions of nonlinear <Emphasis Type="Italic">n</Emphasis>th-order differential equations (existence,almost periodicity,and stability)

We obtain new conditions for the existence of bounded solutions of higher-order nonlinear differential equations. In addition to the classical contraction mapping principle, A.N. Tikhonov’s fixed-point principle is used in the proof of existence theorems. Assertions dealing with the stability of a bounded solution are derived directly from the corresponding results obtained by M.A. Krasnosel’skii and A.V. Pokrovskii.     

Constructive notions of equicontinuity

In the informal setting of Bishop-style constructive reverse mathematics we discuss the connection between the antithesis of Specker’s theorem, Ishihara’s principle BD-N, and various types of equicontinuity. In particular, we prove that the implication from pointwise equicontinuity to uniform sequential equicontinuity is equivalent to the antithesis of Specker’s theorem; and that, for a family of functions on a separable metric space, the implication from uniform sequential equicontinuity to uniform equicontinuity is equivalent to BD-N.      

The Hukuhara–Kneser property for a nonlinear integral equation

Applying a structure theorem of Krasnosel’skii and Perov, we show that the solution set of a nonlinear integral equation satisfies the classical Hukuhara–Kneser property.     

Stability of Functional Equations in Several Variables

We prove a generalization of Hyers＇ theorem on the stability of approximately additive mapping and a generalization of Badora＇s theorem on an approximate ring homomorphism. We also obtain a more general stability theorem, which gives the stability theorems on Jordan and Lie homomorphisms. The proofs of the theorems given in this paper follow essentially the D. H. Hyers-Th. M. Rassias approach to the stability of functional equations connected with S. M. Ulam＇s problem.     

Approximations of almost periodic functions by entire ones

In this paper we propose a new proof of the well-known theorem by S. N. Bernstein, according to which among entire functions which give on (−∞,∞) the best uniform approximation of order σ of periodic functions there exists a trigonometric polynomial whose order does not exceed σ. We also prove an analog of this Bernstein theorem and an analog of the Jackson theorem for uniform almost periodic functions with an arbitrary spectrum.     

An application of the algorithm of the cyclic coordinate descent in multidimensional optimization problems with constrained speed

In this paper we present an application of the algorithm of the cyclic coordinate descent in multidimensional variational problems with constrained speed, the physical motivation of the problem being the optimization of hydrothermal systems. The proof of the convergence of the succession generated by the algorithm was based on the use of an appropriate adaptation of Zangwill’s global theorem of convergence. We have also included an algorithm for the formal construction of the descending succession (the solution of an optimum control problem), the approximation of which we carried out using an adaptation of the Euler method in conjunction with a procedure inspired by the shooting method.