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The famous J.C.P. Miller formula provides a recurrence algorithm for the composition B a ∘ f , where B a is the formal binomial series and f is a formal power series, however it requires that f has to be a nonunit. In this paper we provide the general J.C.P. Miller formula which eliminates the requirement of nonunitness of f and, instead, we establish a necessary and sufficient condition for the existence of the composition B a ∘ f. We also provide the general J.C.P. Miller recurrence algorithm for computing the coefficients of that composition, if B a ∘ f is well defined, obviously. Our generalizations cover both the case in which f is a one–variable formal power series and the case in which f is a multivariable formal power series. In the central part of this article we state, using some combinatorial techniques, the explicit form of the general J.C.P. Miller formula for one-variable case. As applications of these results we provide an explicit formula for the inverses of polynomials and formal power series for which the inverses exist, obviously. We also use our results to investigation of approximate solution to a differential equation which cannot be solved in an explicit way. [ABSTRACT FROM AUTHOR]

In this paper we examine two basic topological properties of partial metric spaces, namely compactness and completeness. Our main result claims that in these spaces compactness is equivalent to sequential compactness. We also show that Hausdorff compact partial metric spaces are metrizable. In the second part of this article we discuss the significance of bottom sets of partial metric spaces in fixed point theorems for mappings acting in these spaces. [ABSTRACT FROM AUTHOR]

The aim of this paper is to generalize the Wigner Theorem to real normed spaces. A normed space is said to have the Wigner Property if the Wigner Theorem holds for it. We prove that every two-dimensional real normed space has the Wigner Property. We also study the Wigner Property of real normed spaces of dimension at least three. It is also shown that strictly convex real normed spaces possess the Wigner Property. [ABSTRACT FROM AUTHOR]

In this paper, we give some necessary and sufficient conditions under which the topology generated by a partial metric is equivalent to the topology generated by a suitably defined metric. Next, we study some new extensions of the Generalized Banach Contraction Principle to partial metric spaces. Moreover, we draw a particular attention to the space of all sequences showing, in particular, that some well-known fixed point theorems for ultrametric spaces, can be used for operators acting in that space. We illustrate our considerations by suitable examples and counterexamples. [ABSTRACT FROM AUTHOR]

The aim of this note is to give a simple topological proof of the well-known result concerning continuity of roots of polynomials. We also consider a more general case with polynomials of a higher degree approaching a given polynomial. We then examine the continuous dependence of solutions of linear differential equations with constant coefficients. [ABSTRACT FROM AUTHOR]

We analyze topological properties of metric spaces obtained by using Száz’s construction, which we used to call generalized “linking construction.” In particular, we provide necessary and sufficient conditions for completeness of metric spaces obtained in this way. Moreover, we examine the relation between Száz’s construction and the “linking construction.” A particular attention is drawn to the “floor” metric, the analysis of which provides some interesting observations. [ABSTRACT FROM AUTHOR]

In this article we consider the topology on the set of formal Laurent series induced by the ultrametric defined via the order. In particular, we establish that the product of formal Laurent series, considered in [GAN, X. X.--BUGAJEWSKI, D.:On formal Laurent series, Bull. Braz. Math. Soc. 42 (2011), 415-437], is not continuous. We also show some applications of fixed point theorems to some nonlinear mappings defined on the space of formal power series or on the space of formal Laurent series. Finally, in the second part of the paper, we propose another approach to study of dot product and multiplication of formal Laurent series, in particular establishing integral representation of dot product and convolution representation of multiplication. [ABSTRACT FROM AUTHOR]

In this paper we introduce new conditions on a kernel of a linear Fredholm integral operator which turn out to be sufficient and necessary for that operator to map the space of functions of bounded variation in the sense of Jordan into itself. Furthermore, we apply those conditions to deal with the problem of the existence of solutions to nonlinear Hammerstein equations in that space. To achieve our goals we will use a topological degree approach as well as a fixed point approach. [ABSTRACT FROM AUTHOR]

The aim of this article is to prove a few Leggett–Williams type theorems, in particular for a more general class of mappings than compact ones. We examine also invariant directions which satisfy a certain additional condition formulated in terms of a given seminorm. As applications of these results we prove a few existence results concerning positive solutions to Hammerstein integral equations or to a two-point boundary value problem. [ABSTRACT FROM AUTHOR]

Abstract: In the first part of the paper we examine mappings of higher order from a general point of view, that is, in normed spaces of bounded real-valued functions defined on . Particular attention is paid to the relation of such mappings with the so-called autonomous superposition operators. Next we investigate mappings of higher order in Banach spaces of almost periodic functions and their perturbations. We also give necessary and sufficient conditions guaranteeing that a nonautonomous superposition operator acts in the space of almost periodic functions in the sense of Levitan and is uniformly continuous. In the Banach space of bounded almost periodic functions in the sense of Levitan we discuss mappings of higher order and a convolution operator. Some applications to nonlinear differential and integral equations are given. [Copyright &y& Elsevier]

Several kinds of formal Laurent series have been introduced with some restrictions so far. This paper systematically sets up a natural definition and structure of formal Laurent series without those restrictions, including introducing a multiplication between formal Laurent series. This paper also provides some results on the algebraic structure of the space of formal Laurent series, denoted by $$\mathbb{L}$$. By means of the results of the generalized composition of formal power series, we define a composition of a Laurent series with a formal power series and provide a necessary and sufficient condition for the existence of such compositions. The calculus about formal Laurent series is also introduced. [ABSTRACT FROM AUTHOR]

Abstract: The aim of this paper is to prove a collection of fixed-point theorems for mappings which can be roughly called generalized contractions or their perturbations. In particular, we are going to consider operators (single-valued or multi-valued) in Banach spaces with a quasimodulus, in hyperconvex subsets of normed spaces, or finally in non-Archimedean spaces. A particular attention will be paid to Krasnoselskii-type fixed-point theorems as well as to a Schaefer-type fixed-point theorem. Some applications to nonlinear functional-integral equations will be given. Our results extend and complement some commonly known theorems. [Copyright &y& Elsevier]

Abstract: The main purpose of this paper is to prove a collection of new fixed point theorems for so-called weakly F-contractive mappings. By analogy, we introduce also a class of strongly F-expansive mappings and we prove fixed point theorems for such mappings. We provide a few examples, which illustrate these results and, as an application, we prove an existence and uniqueness theorem for the generalized Fredholm integral equation of the second kind. Finally, in Appendix A, we apply the Mönch fixed point theorem to prove two results on the existence of approximate fixed points of some continuous mappings. [Copyright &y& Elsevier]

We discuss conditions which do ensure the almost automorphy of the convolution function f*h of f with h where f is almost automorphic and h is a (Lebesgue) measurable function satisfying some additional assumptions. Next we make extensive use of the obtained result to examine almost automorphic solutions to some differential and functional equations as well as a Volterra-like equation. [ABSTRACT FROM AUTHOR]

Abstract: The main purpose of this paper is to deal with almost automorphic and asymptotically almost automorphic solutions of the initial value problem as well as the nonlinear Volterra integral equation in Banach spaces. We obtain a collection of existence results of such solutions to these equations. We investigate also a topological structure of such solution sets. Moreover, we prove Aronszajn-type theorems for solutions of the initial value problem as well as the nonlinear Volterra integral equation, defined on the whole real line. [Copyright &y& Elsevier]

Abstract: In this paper we deal with almost periodic functions with values in a Fréchet space. We apply obtained results to prove the existence of solutions of the initial value problem as well as the Volterra integral equation in this class of functions. We also introduce and investigate asymptotically almost periodic functions with values in a Fréchet space. [Copyright &y& Elsevier]

In this paper we investigate solutions of nonlinear Hammerstein and Volterra–Hammerstein integral equations in the space of functions of bounded φ-variation in the sense of Young. We prove the existence and in some cases the existence and uniqueness of local and global solutions in this class. Real-valued as well as vector-valued functions are under our consideration. The method of our proofs is based on an application of the Banach contraction principle as well as the Leray–Schauder alternative for contractions. [Copyright &y& Elsevier]