Editorial Nonlinear Functional Difference Equations with ...downloads.hindawi.com/journals/ddns/2013/360860.pdf · Nonlinear Functional Difference Equations with Applications HuaSu,

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2013, Article ID 360860, 2 pageshttp://dx.doi.org/10.1155/2013/360860

EditorialNonlinear Functional Difference Equations with Applications

Hua Su,1 Yuriy V. Rogovchenko,2 Youssef Raffoul,3 Yanbin Sang,4 and Fuyi Xu5

1 School of Mathematics and Quantitative Economics, Shandong University of Finance and Economics, Jinan 250014, China2University of Agder, Kristiansand, Norway3Department of Mathematics, University of Dayton, Dayton, OH 45469-2316, USA4Department of Mathematics, North University of China, Taiyuan 030051, China5 School of Science, Shandong University of Technology, Zibo 255049, China

Correspondence should be addressed to Hua Su; [email protected]

Received 20 May 2013; Accepted 20 May 2013

Copyright © 2013 Hua Su et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

It was a pleasure being responsible for the coordination ofseveral proposals for research articles focusing on the themenonlinear functional difference equations with applications.Studying the papers, we were happy to see that manyresearchers tend to focus on the study of the problems that wehave been searching for years.

We appreciated the depth of ideas and the intention togive them a scientific approach.Therefore, we considered that17 of the articles have to be considered in wider mathematicalcircles, being published in this journal in a special volume.It remains to be revealed in further comments, that willundoubtedly appear, how do they bring more knowledge.

We open the opportunity for the journal readers to makecomments, that perhaps we were not able to put out, in rela-tion to the following presentations.

In the paper titled “On the nonhomogeneous fourth-orderp-Laplacian generalized Sturm-Liouville nonlocal boundaryvalue problems” the authors study the nonlinear nonhomo-geneous n-point generalized Sturm-Liouville fourth-order p-Laplacian boundary value problem by using Leray-Schaudernonlinear alternative and Leggett-Williams fixed-point theo-rem.

The paper titled “Chaos in a discrete delay populationmodel” is concerned with chaos in a discrete delay populationmodel. The map of the model is proved to be chaotic in thesense of both Devaney and Li-Yorke under some conditions,by employing the snap-back repeller theory. Some computersimulations are provided to visualize the theoretical result.

The paper titled “A posteriori error estimates for a semidis-crete parabolic integrodifferential control on multimeshes”

extends the existing techniques to study semidiscrete adap-tive finite element approximation schemes for a constrainedoptimal control problem governed by parabolic integrod-ifferential equations. They first prove the uniqueness andexistence of the solution of this optimal control problem. Andthen they derive the upper a posteriori error estimators forboth the state and the control approximation, which are use-ful indicators in adaptive multimesh finite element approxi-mation schemes.

The paper titled “On the expected discounted penalty func-tion for a Markov regime-switching insurance risk model withstochastic premium income” proposed a Markovian regime-switching risk model (also called Markov-modulated riskmodel) with stochastic premium income, in which the pre-mium income and the claim occurrence are driven byMarko-vian regime-switching process. The purpose of this paperis to study the integral equations satisfied by the expecteddiscounted penalty function. In particular, the discount inter-est force process is also regulated by Markovian regime-switching process. Applications of the integral equations aregiven to be Laplace transform of the time of ruin, the deficitat ruin, and the surplus immediately before ruin occurs. Forexponential distribution, the explicit expressions for thesequantities are obtained. Finally, numerical example is alsogiven to illustrate the effect of the related parameters on thesequantities.

In the paper titled “Global regularity criterion for themagneto-micropolar fluid equations” the authors are con-cerned with the magneto-micropolar fluid equations in

2 Discrete Dynamics in Nature and Society

R3. Using Littlewood-Paley decomposition, they obtain anOsgood-type global regularity criterion for the system.

In the paper titled “Eigenvalue of fractional differentialequations with p-Laplacian operator,” by constructing upperand lower solutions, the authors obtained the existence ofpositive solutions for the fractional order eigenvalue problemwith p-Laplacian operator.The authors results can be appliedto the forefront of research of real estate asset securitization.

In the paper titled “Generalized antiperiodic boundaryvalue problems for the fractional differential equation with p-Laplacian operator” the existence of solutions about general-ized antiperiodic boundary value problems for the fractionaldifferential equation with p-Laplacian operator is obtainedunder some suitable hypotheses. The author proof is basedon fixed point theorem and contraction mapping principle.Furthermore, three examples are also given to illustrate theresults.

In the paper titled “Studies on a double Poisson-Geometricinsurance risk model with interference” the authors mainlystudy a generalized double Poisson-Geometric insurance riskmodel. By martingale and stopping time approach, theyobtain adjustment coefficient equation, the Lundberg ine-quality, and the formula for the ruin probability. Also theLaplace transformation of the time when the surplus reachesa given level for the first time is discussed, and the expectationand its variance are obtained. Finally, we give the numericalexamples.

In the paper titled “Finite unions of D-spaces and appli-cations of nearly good relation” the authors give some resultson finite unions of D-spaces. It is proved that if a space is theunion of finitelymany locally compact D-subspaces, then it isa D-space. It follows that a space is a D-space if it is the unionof finitely many locally compact submetacompact subspaces.And a space is a D-space if it is the union of a D-subspacewith a locally compact D-subspace. At last, some examplesare given to exhibit the applications of nearly good relation todiscover D-classes.

In the paper titled “The solutions of mixed monotoneFredholm-type integral equations in Banach spaces,” by meansof the monotone iterative techniques without any compact-ness conditions, the author obtains the iterative unique solu-tion of nonlinear mixed monotone Fredholm-type integralequations in Banach spaces. They results are even new toconvex and concave quasi operator. And then the authorapplies these results to the two-point boundary value problemof second-order nonlinear ordinary differential equations inordered Banach spaces.

In the paper titled “A regularity criterion for the magneto-micropolar fluid equations in ̇𝐵−1

∞,∞

” the authors study theCauchy problem for the magneto-micropolar fluid equationsin three dimensions spaces. A new logarithmically improvedregularity criterion for the magneto-micropolar fluid equa-tions is established in terms of the pressure in the homoge-neous Besov space.

In the paper titled “Stabilization of discrete-time planarswitched linear systems with impulse” the authors study thestabilization problem of discrete-time planar switched linearsystems with impulse. When all subsystems are controllable,

based on an explicit estimation on the state transitionmatrix,they establish a sufficient condition such that the switchedimpulsive system is stabilizable under arbitrary switchingsignal with given switching frequency. When there existsat least one uncontrollable subsystem, a sufficient conditionis also given to guarantee the stabilization of the switchedimpulsive system under appropriate switching signal.

In the paper titled “Characterizations of strongly paracom-pact spaces” the author proves that a countably paracompactnormal space (a perfectly normal space or a monotonicallynormal space) is strongly paracompact if and only if everyincreasing open cover of the space has a star-countable openrefinement. Moreover, it is shown that a space is linearly D,provided that every increasing open cover of the space has apoint-countable open refinement.

In the paper titled “Positive solutions for three-pointboundary value problem of fractional differential equationwithp-Laplacian operator” the authors investigate the existenceof multiple positive solutions for three-point boundary valueproblem of fractional differential equation with p-Laplacianoperator. By applying monotone iterative technique, somesufficient conditions for the existence of multiple positivesolutions are established; moreover iterative schemes forapproximating these solutions are also obtained, which startoff a known simple linear function. In the end, an example isworked out to illustrate the main results.

In the paper titled “Blow-up criteria for three-dimensionalBoussinesq equations in Triebel-Lizorkin spaces” the authorestablishes a new blow-up criteria for solution of the three-dimensional Boussinesq equations in Triebel-Lizorkin spacesby using Littlewood-Paley decomposition.

In the paper titled “Bifurcation analysis in a delayed dif-fusive Leslie-Gower model” the authors deal with the stabilityanalysis of the equilibria of the model and show the existenceof Hopf bifurcation at the positive equilibrium under someconditions. Furthermore, they investigate the stability anddirection of bifurcating periodic orbits by using normal formtheorem and the center manifold theorem.

In the paper titled “Existence and uniqueness of positivesolutions to nonlinear second order impulsive differentialequations with concave or convex nonlinearities,” by using anew fixed point theorem of generalized concave operators,the authors present criteria which guarantee the existenceand uniqueness of positive solutions to nonlinear two-pointboundary value problems for second-order impulsive differ-ential equations with concave or convex nonlinearities.

Acknowledgments

Wewould like to express appreciation to the authors for theirexcellent contributions and patience in assisting us. Finally,the fundamental work of all reviewers on these papers is alsovery greatly acknowledged.

Hua SuYuriy V. Rogovchenko

Youssef RaffoulYanbin Sang

Fuyi Xu

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Editorial Nonlinear Functional Difference Equations with ...downloads.hindawi.com/journals/ddns/2013/360860.pdf · Nonlinear Functional Difference Equations with Applications HuaSu,