Cell decomposition and definable functions for weak p‐adic structures

We develop a notion of cell decomposition suitable for studying weak p‐adic structures (reducts of p‐adic fields where addition and multiplication are not (everywhere) definable). As an example, we consider a structure with restricted addition.     

Peters type polynomials and numbers and their generating functions: Approach with p‐adic integral method

The aim of this paper is to introduce and investigate some of the primary generalizations and unifications of the Peters polynomials and numbers by means of convenient generating functions and p‐adic integrals method. Various fundamental properties of these polynomials and numbers involving some explicit series and integral representations in terms of the generalized Stirling numbers, generalized harmonic sums, and some well‐known special numbers and polynomials are presented. By using p‐adic integrals, we construct generating functions for Peters type polynomials and numbers (Apostol‐type Peters numbers and polynomials). By using these functions with their partial derivative eqautions and functional equations, we derive many properties, relations, explicit formulas, and identities including the Apostol‐Bernoulli polynomials, the Apostol‐Euler polynomials, the Boole polynomials, the Bernoulli polynomials, and numbers of the second kind, generalized harmonic sums. A brief revealing and historical information for the Peters type polynomials are given. Some of the formulas given in this article are given critiques and comments between previously well‐known formulas. Finally, two open problems for interpolation functions for Apostol‐type Peters numbers and polynomials are revealed.     

On certain character sums over p ‐adic rings and their L ‐functions

In this paper we present a new method for evaluating exponential sums associated to a restricted power series in one variable modulo p^l , a power of a prime. We show that for sufficiently large l, these sums can be expressed in terms of Gauss sums. Moreover, we study the associated L ‐functions; we show that they are rational, then we determine their degrees and the weights as Weil numbers of their reciprocal roots and poles. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Carleson measures and the BMO space on the p ‐adic vector space

For a prime number p, let Q _p be the p ‐adic field and let Q _p ^d denote a vector space over Q _p which consists of all d ‐tuples of Q _p . Then we study the p ‐adic version of the Calderón–Zygmund decomposition, Carleson measures on the vector space Q _p ^{d +1} and the space BMO ( Q _p ^d ) of functions of bounded mean oscillation on Q _p ^d . In particular, it turns out that the operator norms of various oncoming operators are independent of the dimension d and the prime number p, which is one of the big differences from that of the Euclidean case. Interestingly, the independence of the dimension d and p makes it possible to develop Harmonic Analysis on the infinite dimensional p ‐adic vector space as the importance had already been pointed out in the Euclidean case (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the characterization of p ‐adic Colombeau–Egorov generalized functions by their point values

We show that contrary to recent papers by S. Albeverio, A. Yu. Khrennikov and V. Shelkovich, point values do not determine elements of the so‐called p ‐adic Colombeau–Egorov algebra ??(?ⁿ _p ) uniquely. We further show in a more general way that for an Egorov algebra ??(M, R) of generalized functions on a locally compact ultrametric space (M, d) taking values in a nontrivial ring, a point value characterization holds if and only if (M, d) is discrete. Finally, following an idea due to M. Kunzinger and M. Oberguggenberger, a generalized point value characterization of ??(M, R) is given. Elements of ??(?ⁿ _p ) are constructed which differ from the p ‐adic δ ‐distribution considered as an element of ??(?ⁿ _p ), yet coincide on point values with the latter. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Comparison between overconvergent de Rham‐Witt and crystalline cohomology for projective and smooth varieties

We prove that for a projective smooth scheme X the hypercohomology of the overconvergent de Rham‐Witt complex is canonically isomorphic to crystalline cohomology.     

Limit behaviour of solutions to equivalued surface boundary value problem for p‐Laplacian equations

In this paper, we discuss the limit behaviour of solutions to boundary value problem with equivalued surface for p‐Laplacian equations when the equivalued surface boundary shrinks to a point in certain way. Copyright © 2000 John Wiley & Sons, Ltd.     

Limit behaviour of solutions to equivalued surface boundary value problem for p‐Laplacian equations: II

In this paper (which is a continuation of Part‐I), we discuss the limit behaviour of solutions to boundary value problem with equivalued surface for p‐Laplacian equations in the case of 1<p?2?1/N when the equivalued surface boundary shrinks to a point in certain way. Copyright © 2001 John Wiley & Sons, Ltd.     

Critical extinction and blow‐up exponents for fast diffusive p‐Laplacian with sources

We discuss and determine the critical extinction and blow‐up exponents for the homogeneous Dirichlet boundary value problem of the fast diffusive p‐Laplacian with sources. Copyright © 2007 John Wiley & Sons, Ltd.     

On quasi‐varieties of multiple valued logic models

We extend the concept of quasi‐variety of first‐order models from classical logic to multiple valued logic (MVL) and study the relationship between quasi‐varieties and existence of initial models in MVL. We define a concept of ‘Horn sentence’ in MVL and based upon our study of quasi‐varieties of MVL models we derive the existence of initial models for MVL ‘Horn theories’. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Existence of ground state sign‐changing solutions for p‐Laplacian equations of Kirchhoff type

In this paper, we prove the existence of ground state sign‐changing solutions for the following class of elliptic equation: where , and K(x) are positive continuous functions. Firstly, we obtain one ground state sign‐changing solution u_b by using some new analytical skills and non‐Nehari manifold method. Furthermore, the energy of u_b is strictly larger than twice that of the ground state solutions of Nehari type. We also establish the convergence property of u_b as the parameter b↘0. Copyright © 2017 John Wiley & Sons, Ltd.     

AMLI preconditioner for the p‐version of the FEM

From the literature it is known that the conjugate gradient method with domain decomposition preconditioners is one of the most efficient methods for solving systems of linear algebraic equations resulting from p‐version finite element discretizations of elliptic boundary value problems. One ingredient of such a preconditioner is a preconditioner related to the Dirichlet problems. In the case of Poisson's equation, we present a preconditioner for the Dirichlet problems which can be interpreted as the stiffness matrix K_h,k resulting from the h‐version finite element discretization of a special degenerated problem. We construct an AMLI preconditioner C_h,k for the matrix K_h,k and show that the condition number of C K_h,k is independent of the discretization parameter. This proof is based on the strengthened Cauchy inequality. The theoretical result is confirmed by numerical examples. Copyright © 2003 John Wiley & Sons, Ltd.     

Existence and multiplicity of nontrivial solutions for nonlinear fractional differential systems with p‐Laplacian via critical point theory

In this paper, the existence and multiplicity of nontrivial solutions are obtained for nonlinear fractional differential systems with p‐Laplacian by combining the properties of fractional calculus with critical point theory. Firstly, we present a result that a class of p‐Laplacian fractional differential systems exists infinitely many solutions under the famous Ambrosetti‐Rabinowitz condition. Then, a criterion is given to guarantee that the fractional systems exist at least 1 nontrivial solution without satisfying Ambrosetti‐Rabinowitz condition. Our results generalize some existing results in the literature.     

On models of exponentiation. Identities in the HSI‐algebra of posets

We prove that Wilkie's identity holds in those natural HSI‐algebras where each element has finite decomposition into components. Further, we construct a bunch of HSI‐algebras that satisfy all the identities of the set of positive integers ?. Then, based on the constructed algebras, we prove that the identities of ? hold in the HSI‐algebra of finite posets when the value of each variable is a poset having an isolated point. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Decay estimates for the wave equation of p‐Laplacian type with dissipation of m‐Laplacian type

In this paper, we study decay properties of solutions to the wave equation of p‐Laplacian type with a weak dissipation of m‐Laplacian type. Copyright © 2006 John Wiley & Sons, Ltd.     

Positive solutions for multi‐point boundary value problems of fractional differential equations with p‐Laplacian

This paper investigates the existence of solutions for multi‐point boundary value problems of higher‐order nonlinear Caputo fractional differential equations with p‐Laplacian. Using the five functionals fixed‐point theorem, the existence of multiple positive solutions is proved. An example is also given to illustrate the effectiveness of ourmain result. Copyright © 2015 John Wiley & Sons, Ltd.     

Limit behaviour of the solution of an evolution boundary‐value problem involving the p‐Laplacian operator for the case of an equivalued condition on a shrinking surface

In this paper, we discuss the limit behaviour of the solution of an evolution boundary‐value problem involving the p‐Laplacian operator for the case of an equivalued condition on a shrinking surface. Copyright © 2004 John Wiley & Sons, Ltd.     

Multiple solutions for a noncooperative Kirchhoff‐type system involving the fractional p‐Laplacian and critical exponents

In this paper, we use the Limit Index Theory due to Li 19 and the fractional version of concentration compactness principle to study the multiplicity of solutions for a class of noncooperative fractional p‐Laplacian elliptic system with homogeneous Dirichlet boundary conditions involving the critical exponents.     

Positive radial solutions of p‐Laplacian equation with sign changing nonlinear sources

This paper is concerned with the p‐Laplacian equation with singular sources which is allowed to change sign in a ball. The singularity may occur at the zero points of solutions and on the boundary. Using the upper and lower solutions method, we establish the existence of positive radial solutions for the problem considered. Copyright © 2006 John Wiley & Sons, Ltd.     

Maximal ℓp‐regularity of multiterm fractional equations with delay

We provide a characterization for the existence and uniqueness of solutions in the space of vector‐valued sequences ${?}_p(?,X)$ for the multiterm fractional delayed model in the form $${\mathrm{\Delta }}^{\alpha }u(n)+\lambda {\mathrm{\Delta }}^{\beta }u(n)=Au(n)+u(n?\tau )+f(n),n\in ?,\alpha ,\beta \in {?}_{+},\tau \in ?,\lambda \in ?,$$ where X is a Banach space, A is a closed linear operator with domain D(A) defined on X, $f\in {?}_p(?,X)$ and Δ^Γ denotes the Grünwald–Letkinov fractional derivative of order Γ > 0. We also give some conditions to ensure the existence of solutions when adding nonlinearities. Finally, we illustrate our results with an example given by a general abstract nonlinear model that includes the fractional Fisher equation with delay.