Functional equations and group substitutions

Motivated by some investigations of Babbage and a method of solving certain functional equations arising in competition problems, we investigate a class of functional equations and prove a local existence and uniqueness theorem for them. The main tools of the proof are the Inverse Function Theorem and the Global Existence and Uniqueness Theorem.     

Quasilinear asymptotically periodic Schrödinger equations with subcritical growth

In this article the existence of one solution for a class of asymptotically periodic equations in the euclidean space is established. The basic tools employed here are the Mountain Pass Theorem and the Concentration-Compactness Principle. By using a change of variable, the quasilinear equation is reduced to a semilinear equation, whose respective associated functional is well defined in H¹(R^N) and satisfies the geometric hypotheses of the Mountain Pass Theorem.     

A strong open mapping theorem for surjections from cones onto Banach spaces

We show that a continuous additive positively homogeneous map from a closed not necessarily proper cone in a Banach space onto a Banach space is an open map precisely when it is surjective. This generalization of the usual Open Mapping Theorem for Banach spaces is then combined with Michael's Selection Theorem to yield the existence of a continuous bounded positively homogeneous right inverse of such a surjective map; a strong version of the usual Open Mapping Theorem is then a special case. As another consequence, an improved version of the analogue of Andô's Theorem for an ordered Banach space is obtained for a Banach space that is, more generally than in Andô's Theorem, a sum of possibly uncountably many closed not necessarily proper cones. Applications are given for a (pre)-ordered Banach space and for various spaces of continuous functions taking values in such a Banach space or, more generally, taking values in an arbitrary Banach space that is a finite sum of closed not necessarily proper cones.     

General methods for converting impulsive fractional differential equations to integral equations and applications

In this paper, we propose the concepts of Caputo fractional derivatives and Caputo type Hadamard fractional derivatives for piecewise continuous functions. We obtain general solutions of four classes of impulsive fractional differential equations (Theorem 3.1–Theorem 3.4) respectively. These results are applied to converting boundary value problems for impulsive fractional differential equations to integral equations. Some comments are made on recently published papers (see Section 4).     

On the singular limit for slightly compressible fluids

In this paper we study the motion of slightly compressible inviscid fluids. We prove that the solution of the corresponding system of nonlinear partial differential equations converges (uniformly) in the strong norm (that of the data space) to the solution of the incompressible equations, as the Mach number goes to zero (see Theorem 1.2). Actually, our proof applies to a large class of singular limit problems as shown in the Theorem 2.2.     

Asymptotic Estimation for Some Nonlinear Delay Differential Equations

This paper deals with the asymptotic investigation of a nonlinear differential equation with variable time delay. We impose some growth restrictions on the nonlinear delayed term and introduce certain auxiliary differential and functional nondifferential equations. Then some comparison results are given and applied to particular cases of the studied nonlinear equation. Received: January 9, 2006. Revised: October 29, 2007.     

On Continuous and Smooth Solutions of the Schröder Equation in Normed Spaces

The problem of the existence of continuous and smooth solutions of the Schr?der equation defined on normed spaces is examined. This research was supported by the Silesian University Mathematics Department (Linear functional equations program).     

Characteristic functional equations of polynomials and the morera-carleman theorem

Several characteristic functional equations satisfied by classes of polynomials of bounded degree are examined in connection with certain generalizations of the Morera-Carleman Theorem. Certain functional equations which have nonanalytic polynomial solutions are also considered.     

Almost sure explosion of solutions to stochastic differential equations

This paper is concerned with the problem of explosive solutions for a class of stochastic differential equations. Our main results are presented as two theorems. Theorem 1 is concerned with the existence of explosive solutions with positive probability under certain sufficient conditions. With some additional mild conditions, it is shown in Theorem 2 that the explosion will occur almost surely. The methods of auxiliary functions and cycles are used in the proofs. Several remarks about their applications are given.     

Corrigendum and addendum to “The strong approximation property and the weak bounded approximation property” [J. Funct. Anal. 266 (2014) 5439–5447]

We correct a mistake in the proof of [2, Theorem 1.1] and a proof of the main result in [2] is presented without depending on [2, Theorem 1.1].     

On an application of hypoellipticity to solutions of functional equations

Summary We study the regularity of solutions of functional equations of a generalized mean value type. In this paper we give sufficient conditions for the regularity by using hypoellipticity which is a concept of the theory of partial differential equations. We also give an affirmative answer to a conjecture of H. wiatak. A part of the results was announced in the comprehensive paper [8] on our joint works. To prove the regularity of solutions of functional equations is one of the central problems in the theory of functional equations (see [1]).     

A remark on entire solutions of quasilinear elliptic equations

By applying a main comparison theorem of Pucci and Serrin (2007) [2] we cover, for general equations of p-Laplace type, the open cases of Theorems B, D, E of Farina and Serrin (submitted for publication) [1] as described in Problems 2 and 3 of Section 12 of Farina and Serrin (submitted for publication) [1]. Moreover, we provide significant improvements of Theorem C and Theorem 5 of Farina and Serrin (submitted for publication) [1], the latter in the context of mean curvature type operators, see Theorem 1.3 and Theorems 5.2-5.4 below.Finally, Theorem 1.1 provides a new Liouville theorem outside the context of work in Farina and Serrin (submitted for publication) [1].     

YoshiZawa型周期解定理和Massera型周期解定理研究进展简介

微分方程解的有界性和周期解的存在性是檄分方程理论研究中的两个重要课题，二者之间有着紧密的联系．在解的有界性与周期解的存在性的研究中，Yoshizawa周期解定理和Massera周期解定理是非常重要的结果，具有重要的理论意义和应用价值．本文以Yoshizawa型周期解定理和Massera型周期解定理的研究为主，简要介绍泛函微分方程周期解理论研究方面的一些新进展。     

Applications of Schauder's Fixed-point theorem with respect to iterated functional equations

In this paper, we establish the existence and uniqueness of continuous solutions for two classes of iterated functional equations by using Schauder's Fixed-Point Theorem.     

On operatorsT such thatf(T) is hypercyclic

We give conditions such that an operator given by the Dunford-Taylor functional calculus is supercyclic or hypercyclic. Indeed, we improve [15, Theorem 1].tbermudeull.esvivienmath.msstate.eduPartially supported by DGICYT Grant PB97-1489 (Spain)     

Periodic solutions of non-linear discrete Volterra equations with finite memory

In this paper we discuss the existence of periodic solutions of discrete (and discretized) non-linear Volterra equations with finite memory. The literature contains a number of results on periodic solutions of non-linear Volterra integral equations with finite memory, of a type that arises in biomathematics. The “summation” equations studied here can arise as discrete models in their own right but are (as we demonstrate) of a type that arise from the discretization of such integral equations. Our main results are in two parts: (i) results for discrete equations and (ii) consequences for quadrature methods applied to integral equations. The first set of results are obtained using a variety of fixed-point theorems. The second set of results address the preservation of properties of integral equations on discretizing them. The effect of weak singularities is addressed in a final section. The detail that is presented, which is supplemented using appendices, reflects the differing prerequisites of functional analysis and numerical analysis that contribute to the outcomes.     

A desarguesian theorem for algebraic combinatorial geometries

The points of an algebraic combinatorial geometry are equivalence classes of transcendentals over a fieldk; two transcendentals represent the same point when they are algebraically dependent overk. The points of an algebraically closed field of transcendence degree two (three) overk are the lines (resp. planes) of the geometry. We give a necessary and sufficient condition for two coplanar lines to meet in a point (Theorem 1) and prove the converse of Desargues’ theorem for these geometries (Theorem 2). A corollary: the “non-Desargues” matroid is non-algebraic. The proofs depend on five properties (or postulates). The fifth of these is a deep property first proved by Ingleton and Main [3] in their paper showing that the Vámos matroid is non-algebraic.     

泛函微分方程周期边值问题的正解

讨论一阶泛函微分方程的周期边值问题,利用锥上的不动点定理给出了正解的存在性和多重性的简捷的判别条件.     

Existence theorems of periodic solutions for fourth-order nonlinear functional difference equations

By using of the critical point method, the existence of periodic solutions for fourth-order nonlinear functional difference equations is obtained. The main approaches used in our paper are variational techniques and the Saddle Point Theorem. The problem is to solve the existence of periodic solutions of fourth-order nonlinear functional difference equations. Results obtained generalize and complement the existing one.