Sturm–Liouville properties for Atkinson's semi‐definite p‐Laplacian eigenvalue problems

We define Atkinson's semi‐definite p‐Laplacian eigenvalue problems, which include the regular p‐Laplacian eigenvalue problems with L ¹ coefficient functions. Then we show that the Sturm oscillation theorem also holds for this eigenvalue problem.     

Inverse nodal problem for Dirac operator with integral type nonlocal boundary conditions

In this paper, Dirac operator with some integral type nonlocal boundary conditions is studied. We show that the coefficients of the problem can be uniquely determined by a dense set of nodal points. Moreover, we give an algorithm for the reconstruction of some coefficients of the operator.     

Inverse problems for Sturm–Liouville operators on a compact equilateral graph by partial nodal data

Partial inverse nodal problems for Sturm–Liouville operators on a compact equilateral star graph are investigated in this paper. Uniqueness theorems from partial twin‐dense nodal subsets in interior subintervals or arbitrary interior subintervals having the central vertex are proved. In particular, we posed and solved a new type partial inverse nodal problems for the Sturm–Liouville operator on the compact equilateral star graph.     

On a boundary value problem for a p‐Dirac equation

The p‐Laplace equation is a nonlinear generalization of the Laplace equation. This generalization is often used as a model problem for special types of nonlinearities. The p‐Laplace equation can be seen as a bridge between very general nonlinear equations and the linear Laplace equation. The aim of this paper is to solve the p‐Laplace equation for 1 < p < 2 and to find strong solutions. The idea is to apply a hypercomplex integral operator and spatial function theoretic methods to transform the p‐Laplace equation into the p‐Dirac equation. This equation will be solved iteratively by using a fixed‐point theorem. Applying operator‐theoretical methods for the p‐Dirac equation and p‐Laplace equation, the existence and uniqueness of solutions in certain Sobolev spaces will be proved. Copyright © 2016 John Wiley & Sons, Ltd.     

Inverse nodal problems for the Sturm-Liouville differential operators on a star-type graph

We study inverse nodal problems for the second order differential operators on a star-type graph satisfying the standard matching conditions at the interior vertex. We prove uniqueness theorems and obtain a constructive solution to the inverse problems of this class. Original Russian Text Copyright ? 2009 Yurko V. A. The author was supported by the Russian Foundation for Basic Research (Grant 07-01-00003) and the National Science Council of Taiwan (Grant 07-01-92000-NSC-a). __________ Saratov. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 50, No. 2, pp. 469–475, March–April, 2009.     

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.     

Inverse problems for the boundary value problem with the interior nodal subsets

In this paper, inverse nodal problems for Sturm–Liouville equations with boundary conditions polynomially dependent on the spectral parameter were studied. The authors showed that some uniqueness theorems on the potential function hold by the Weyl function, respectively.     

Existence of positive solutions for nonlinear singular boundary value problem with p‐Laplacian

In this paper, we study the existence of positive solutions for the following Sturm–Liouville‐like four‐point singular boundary value problem (BVP) with p‐Laplacian where ?_p(s)=|s|^p?2 s, p>1, f is a lower semi‐continuous function. Using the fixed‐point theorem of cone expansion and compression of norm type, the existence of positive solution and infinitely many positive solutions for Sturm–Liouville‐like singular BVP with p‐Laplacian are obtained. Copyright © 2009 John Wiley & Sons, Ltd.     

Nonlocal boundary value problem for fractional differential equations with p‐Laplacian

In this paper, we are concerned with the existence of positive solutions for the following nonlocal BVP of fractional DEs with p‐Laplacian operator By using the fixed point theorem in a cone, multiplicity solutions of the BVP are obtained. An example is also given to show the effectiveness of the obtained result. Copyright © 2013 John Wiley & Sons, Ltd.     

Identification problem for a one‐dimensional vibrating system

We discuss an inverse problem of determining a coefficient matrix and an initial value for a one‐dimensional non‐symmetric hyperbolic system of the first order by means of boundary values over a time interval. Provided that a time interval is sufficiently long and a given initial value satisfies some non‐degeneracy condition, we characterize coefficient matrices and initial values realizing the same boundary values. In the case where the initial value is fixed, we can prove the uniqueness in determining all the components of the coefficient matrices. The proof is based on a transformation formula and spectral properties. Copyright © 2005 John Wiley & Sons, Ltd.     

Attractors for the strongly damped wave equation with p‐Laplacian

This paper is concerned with the initial‐boundary value problem for one‐dimensional strongly damped wave equation involving p‐Laplacian. For p > 2 , we establish the existence of weak local attractors for this problem in . Under restriction 2 < p < 4, we prove that the semigroup, generated by the considered problem, possesses a strong global attractor in , and this attractor is a bounded subset of . Copyright © 2017 John Wiley & Sons, Ltd.     

Positive solutions for a fourth order discrete p‐Laplacian boundary value problem

In this paper, we study the existence and multiplicity of positive solutions for the following fourth order nonlinear discrete p‐Laplacian boundary value problem where φ_p(s) = | s | ^{p ? 2}s, p > 1, is continuous, T is an integer with T ≥ 5 and . By virtue of Jensen's discrete inequalities, we use fixed point index theory to establish our main results. Copyright © 2013 John Wiley & Sons, Ltd.     

A note on blow‐up for fast diffusive p‐Laplacian with sources

In this note we improve the result of Theorem 3.1 in Yin and Jin (Math. Meth. Appl. Sci. 2007; 30 (10):1147–1167) and establish a blow‐up result for certain solution with positive initial energy. Copyright © 2008 John Wiley & Sons, Ltd.     

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.     

Inverse spectral problem for the operators with non‐local potential

The main object under consideration in the paper is the second derivative operator on a finite interval with zero boundary conditions perturbed by a self‐adjoint integral operator with the degenerate kernel (non‐local potential). The inverse problem, i.e., the reconstruction of the perturbation from the spectral data, is solved by means of the step‐by‐step procedure based on the n‐interlacing property of the spectrum.     

Oscillation of solutions of a boundary value problem for Dirac canonical system

A theorem is proved on oscillation of the components of the eigenvector-functions of a boundary value problem for the canonical one-dimensional Dirac system.     

Inverse problem with a time‐integral condition for a fractional diffusion equation

We find the conditions for the unique solvability of the inverse problem for a time‐fractional diffusion equation with Schwarz‐type distributions in the right‐hand sides. This problem is to find a generalized solution of the Cauchy problem and an unknown space‐dependent part of an equation's right‐hand side under a time‐integral overdetermination condition.     

Inverse problem for upper asymptotic density

For a set of natural numbers, the structural properties are described when the upper asymptotic density of achieves the infimum of the upper asymptotic densities of all sets of the form , where the upper asymptotic density of is greater than or equal to the upper asymptotic density of . As a corollary, we prove that if the upper asymptotic density of is less than and the upper asymptotic density of achieves the infimum, then the lower asymptotic density of must be .

     

Inverse source problem for a diffusion equation involving the fractional spectral Laplacian

In this paper, we consider an inverse problem related to a fractional diffusion equation. The model problem is governed by a nonlinear partial differential equation involving the fractional spectral Laplacian. This study is focused on the reconstruction of an unknown source term from a partial internal measured data. The considered ill‐posed inverse problem is formulated as a minimization one. The existence, uniqueness, and stability of the solution are discussed. Some theoretical results are established. The numerical reconstruction of the unknown source term is investigated using an iterative process. The proposed method involves a denoising procedure at each iteration step and provides a sequence of source term approximations converging in norm to the actual solution of the minimization problem. Some numerical results are presented to show the efficiency and the accuracy of the proposed approach.