Stability of Jensen functional equation in intuitionistic fuzzy normed space

In this paper, we determine some stability results concerning the Jensen functional equation 2f((x+y)/2)=f(x)+f(y) in intuitionistic fuzzy normed spaces (IFNS). We define the intuitionistic fuzzy continuity of the Jensen mappings and prove that the existence of a solution for any approximately Jensen mapping implies the completeness of IFNS.     

Piece adding technique for convex maximization problems

In this article we provide an algorithm, where to escape from a local maximum y of convex function f over D, we (locally) solve piecewise convex maximization max{min{f (x) − f (y), p _y (x)} | x ∈ D} with an additional convex function p _y (·). The last problem can be seen as a strictly convex improvement of the standard cutting plane technique for convex maximization. We report some computational results, that show the algorithm efficiency.     

Some Remarks about the FM-partners of <Emphasis Type="Italic">K</Emphasis>3 Surfaces with Picard Numbers 1 and 2

In this paper we describe some results about K3 surfaces with Picard number 1 and 2. In particular, we give a new simple proof of a theorem due to Oguiso which shows that, given an integer N, there is a K3 surface with Picard number 2 and at least N non-isomorphic FM-partners. We describe also the Mukai vectors of the moduli spaces associated to the FM-partners of K3 surfaces with Picard number 1.     

A generalization of the Looman-Menchoff theorem

In this paper we give a generalization of the classical Looman-Menchoff theorem:If f is a complex-valued continuous function of a complex variable in a domain G, f has partial derivatives f _x and f _y everywhere in G and the Cauchy Riemann equations f _x +if _y = 0are satisfied almost everywhere, then f is holomorphic in G. From our generalization of this theorem, we deduce a theroem of Sindalovskii [9] as a corollary and also answer some of the questions raised in [9]. We note in this context that, as far as we know, Sindalovskii’s result is the best published to date in this area.     

Positive solutions for quasilinear second order differential equation

It is well known that the Krasnoselskii's fixed point theorem is very very important. It was extensively used for studying the boundary value problems. In this article, the Krasnoselskii's fixed point theorem is extended. The new fixed point theorem is obtained. The second order quasilinear differential equation (Φ (y′))′+a(t)f(t,y,y′)=0,, 0<t<1 subject to mixed boundary condition is studied, where f is a nonnegative continuous function, Φ (v)= |v|^p-2 v, p>1. We show the existence of at least one positive solution by using the new fixed point theorem in cone.     

Über Picardmengen ganzer Funktionen

Following Lehto we call a set E in the complex plane a Picard set for integral functions, if every non-rational integral function omits at most one finite value in the complement (with respect to the plane) of E. The existence of non-trivial Picard sets was proved by Lehto [3]. The aim of this paper is to give a new criterion for denumberable point sets E to be Picard sets for integral functions. In some way the criterion given by theorem 1 is an extension of the result on Picard sets for integral functions given by Toppila [5] and improves the criterion given by the author in [6].     

Hyperstability of the Cauchy equation on restricted domains

We show that a very classical result, proved by T. Aoki, Z. Gajda and Th. M. Rassias and concerning the Hyers–Ulam stability of the Cauchy equation f(x+y)=f(x)+f(y), can be significantly improved. We also provide some immediate applications of it (among others for the cocycle equation, which is useful in characterizations of information measures). In particular, we give a solution to a problem that was formulated more than 20 years ago and concerned optimality of some estimations. The proof of that result is based on a fixed point theorem.     

New fifth‐order two‐derivative Runge‐Kutta methods with constant and frequency‐dependent coefficients

Two‐derivative Runge‐Kutta methods are Runge‐Kutta methods for problems of the form y′ = f(y) that include the second derivative y′′ = g(y) = f ′(y)f(y) and were developed in the work of Chan and Tsai. In this work, we consider explicit methods and construct a family of fifth‐order methods with three stages of the general case that use several evaluations of f and g per step. For problems with oscillatory solution and in the case that a good estimate of the dominant frequency is known, methods with frequency‐dependent coefficients are used; there are several procedures for constructing such methods. We give the general framework for the construction of methods with variable coefficients following the approach of Simos. We modify the above family to derive methods with frequency‐dependent coefficients following this approach as well as the approach given by Vanden Berghe. We provide numerical results to demonstrate the efficiency of the new methods using three test problems.     

Finiteness theorems for the Picard objects of an algebraic stack

We prove some finiteness theorems for the Picard functor of an algebraic stack, in the spirit of SGA 6, exp. XII and XIII. In particular, we give a stacky version of Raynaud?s relative representability theorem, we give sufficient conditions for the existence of the torsion component of the Picard functor, and for the finite generation of the Néron–Severi groups or of the Picard group itself. We give some examples and applications. In Appendix A, we prove the semicontinuity theorem for a (non-necessarily tame) algebraic stack.     

Nondifferentiable mathematical programming and convex-concave functions

For a convex-concave functionL(x, y), we define the functionf(x) which is obtained by maximizingL with respect toy over a specified set. The minimization problem with objective functionf is considered. We derive necessary conditions of optimality for this problem. Based upon these necessary conditions, we define its dual problem. Furthermore, a duality theorem and a converse duality theorem are obtained. It is made clear that these results are extensions of those derived in studies on a class of nondifferentiable mathematical programming problems.This work was supported by the Japan Society for the Promotion of Sciences.     

Stability of the Quadratic Equation of Pexider Type

We will investigate the stability problem of the quadratic equation (1) and extend the results of Borelli and Forti, Czerwik, and Rassias. By applying this result and an improved theorem of the author, we will also prove the stability of the quadratic functional equation of Pexider type,f ₁ (x +y) + f₂(x -y) =f ₃(x) +f ₄(y), for a large class of functions.     

Asymptotic fixed point theory and the beer barrel theorem

In Sections 2 and 3 of this paper we refine and generalize theorems of Nussbaum (see [42]) concerning the approximate fixed point index and the fixed point index class. In Section 4 we indicate how these results imply a wide variety of asymptotic fixed point theorems. In Section 5 we prove a generalization of the mod p theorem: if p is a prime number, f belongs to the fixed point index class and f satisfies certain natural hypothesis, then the fixed point index of f ^p is congruent mod p to the fixed point index of f. In Section 6 we give a counterexample to part of an asymptotic fixed point theorem of A. Tromba [55]. Sections 2, 3, and 4 comprise both new and expository material. Sections 5 and 6 comprise new results. This paper is dedicated to Felix Browder on the occasion of his eightieth birthday and in recognition of his many contributions to nonlinear analysis     

A fixed point approach to stability of a quadratic equation

Using the fixed point alternative theorem we establish the orthogonal stability of the quadratic functional equation of Pexider type f (x+y)+g(x−y) = h(x)+k(y), where f, g, h, k are mappings from a symmetric orthogonality space to a Banach space, by orthogonal additive mappings under a necessary and sufficient condition on f.     

Mean-field backward stochastic differential equations driven by G-Brownian motion and related partial differential equations

In this paper, we study mean-field backward stochastic differential equations driven by G-Brownian motion (G-BSDEs). We first obtain the existence and uniqueness theorem of these equations. In fact, we can obtain local solutions by constructing Picard contraction mapping for Y term on small interval, and the global solution can be obtained through backward iteration of local solutions. Then, a comparison theorem for this type of mean-field G-BSDE is derived. Furthermore, we establish the connection of this mean-field G-BSDE and a nonlocal partial differential equation. Finally, we give an application of mean-field G-BSDE in stochastic differential utility model.     

An upper and lower solution approach for singular boundary value problems with sign changing non‐linearities

We obtain via Schauder's fixed point theorem new results for singular second‐order boundary value problems where our non‐linear term f(t,y,z) is allowed to change sign. In particular, our problem may be singular at y=0, t=0 and/or t=1. Copyright © 2002 John Wiley & Sons, Ltd.     

A tauberian theorem for distributional point values

We give a tauberian theorem for boundary values of analytic functions. We prove that if is the distributional limit of the analytic function F defined in a region of the form (a, b) × (0, R), if F (x ₀ + iy)→ γ as y → 0⁺, and if f is distributionally bounded at x = x ₀, then f (x ₀) = γ distributionally. As a consequence of our tauberian theorem, we obtain a new proof of a tauberian theorem of Hardy and Littlewood. Received: 10 December 2007     

Algebraic form of a theorem of Hahn—Banach type for lattice manifolds

Let E be a vector lattice. A linear functionalf on E is called a lattice homomorphism iff(sup (x, y)) = max (f(x),f(y)) for all x, y E. For lattice homomorphisms a theorem of Hahn—Banach type is valid. In this note we prove an algebraic analog of this theorem.Translated from Matematicheskie Zametki, Vol. 16, No. 4, pp. 595–600, October, 1974.In conclusion the author expresses his thanks to D. A. Raikov for his statement of the problem and his interest in my work.     

A Method for Global Approximation of the Initial Value Problem

For the numerical solution of the initial value problem a parallel, global integration method is derived and studied. It is a collocation method. If f(x,y)f(x) the method coincides with the Filippi's modified Clenshaw–Curtis quadrature [11]. Two numerical algorithms are considered and implemented, one of which is the application of the new method to Picard iterations, so it is a waveform relaxation technique [3]. Numerical experiments are favourably compared with the ones given by the known GAM [2], GBS [14] and Sarafyan [18] methods.     

Two types of minimax theorems for vector-valued functions

It seems that minimax theorems for vector-valued functions found in recent papers have something in common. Taking note of this, we improve several results in the author's recent works and state two types of minimax theorems for vector-valued functions. One theorem refers to functions with some special convexity properties; the other theorem refers to separated functions of the typef(x, y)=u(x)+v(y). The proofs are based on the existence of weak cone saddle points off and on a condition about a pointed convex cone which induces a partial ordering in the image space off. We need the condition (C{0})+clC–C, which implies the Sterna-Karwat condition for a convex coneC of a Hausdorff topological vector space.The author thanks the referees for their valuable suggestions on the original draft. Also, he is grateful to S. Yoshiara for his useful suggestions on the English presentation.     

On stability of a cubic functional equation in intuitionistic fuzzy normed spaces

In this paper, we determine some stability results concerning the cubic functional equation

f(2x+y)+f(2x-y)=2f(x+y)+2f(x-y)+12f(x)