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Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013, Article ID 158358, 3 pageshttp://dx.doi.org/10.1155/2013/158358

EditorialNonlinear Functional Analysis of Boundary Value Problems:Novel Theory, Methods, and Applications

Yong Hong Wu,1 Lishan Liu,2 Benchawan Wiwatanapataphee,3 and Shaoyong Lai4

1 Curtin University of Technology, Perth, WA 6845, Australia2 Qufu Normal University, Qufu, Shandong 273165, China3Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand4 Southwestern University of Finance and Economics, Chengdu, Sichuan 610074, China

Correspondence should be addressed to Benchawan Wiwatanapataphee; [email protected]

Received 24 February 2013; Accepted 24 February 2013

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

Nonlinear boundary value problems often arise from sci-entific research, modelling of nonlinear phenomena, andoptimal control of complex systems. In some cases, it is notpossible to solve the underlying nonlinear boundary valueproblems in a closed form, and, for these cases, nonlinearfunctional analysis can play its role as a tool for obtainingqualitative information about the solution. The primaryproblems concerned in functional analysis are as follows:does a solution of the boundary value problem exist? If so,is it unique and stable? If the answers to these questions areaffirmative, the problem is well posed and the analysis canproceed in a firm base to the next step—construct a methodfor finding approximate solutions and evaluate the quality ofthe approximation.

Over the last few decades, intensive research has beencarried out worldwide to develop functional analysis theoriesand methods to tackle complex boundary value problemsarising continually from scientific research and modelling ofreal world phenomena. The papers selected for this specialissue represent a typical set of contributions in this fieldof research. Of course, the selected topics and papers arenot an exhaustive representation of the recent developmentin the field. Nevertheless, they provide novel functionaltheories and methods as well as rich analysis for manycomplex boundary value problems with different applicationbackgrounds and we have the pleasure of sharing theseresearch results with the readers. Here, wewould like to thankthe authors and reviewers of the papers for their excellent

contributions. The efforts made by staff of the editorial officeof the publishing corporation are also greatly acknowledged.

This special issue contains twenty-nine papers, coveringfunctional analysis of various types of complex bound-ary value problems for different differential equations andboundary conditions. The particular focus is on fractionalorder differential equations and partial differential equations,and the boundary conditions include Riemann-Stieltjes inte-gral boundary conditions and nonlocal boundary conditions.

The first set of papers, including nine papers, focus onthe study of various complex fractional order boundary valueproblems.

(i) In the paper titled “Existence of positive solution forsemipositone fractional differential equations involvingRiemann-Stieltjes integral conditions,” based on thefixed-point theorem in a cone, the existence of atleast one positive solution is established for a classof semipositone fractional differential equations withRiemann-Stieltjes integral boundary condition.

(ii) In the paper titled “Multiple solutions for a class of frac-tional boundary value problems,” the authors establishconditions for the existence of multiple solutionsfor a class of fractional bocundary value problemsinvolving left and right Riemann-Liouville fractionalintegrals of order 0 ≤ 𝛽 < 1 by the variationalmethod.

2 Abstract and Applied Analysis

(iii) In the paper titled “Positive solutions of eigenvalueproblems for a class of fractional differential equationswith derivatives,” the existence of positive solutions forthe eigenvalue problem of a class of fractional differ-ential equations is established through establishing amaximal principles and constructing upper and lowersolutions.

(iv) In the paper titled “Positive solutions of nonlinearfractional differential equations with integral boundaryvalue conditions,” by using the fixed-point theorem inpartially ordered set, the authors investigate the exis-tence and uniqueness of the solutions of a boundaryvalue problem involvingCaputo fractional derivativesand integral boundary conditions.

(v) In the paper titled “On impulsive boundary value prob-lems of fractional differential equations with irregularboundary conditions,” the existence and uniquenessof solutions are established for a class of boundaryvalue problems with irregular boundary conditionsand impulsive jumps in the system state and its first-order derivative.

(vi) In the paper titled “Existence of solutions for nonlinearimpulsive fractional differential equations of order𝛼 ∈ (2, 3] with nonlocal boundary conditions,” theauthors established the existence and uniqueness ofthe solution to a nonlocal boundary value problem fornonlinear impulsive fractional differential equationsof order 𝛼 ∈ (2, 3].

(vii) In the paper titled “Positive solutions of a fractionalboundary value problem with changing sign nonlinear-ity,” by investigating the properties of the associatedGreen function and utilizing the Krasnoselskii fixed-point theorem in a cone, the authors establish theexistence of positive solutions of a boundary valueproblem for nonlinear fractional differential equa-tions with changing sign nonlinearities.

(viii) In the paper titled “Multiple solutions for a fractionaldifference boundary value problem via variationalapproach,” the existence of multiple solutions for afractional difference boundary value problem witha parameter is established within the variationalframework.

(ix) In the paper titled “Solutions of sign-changing frac-tional differential equation with the fractional deriva-tives,” the existence of nontrivial solutions is estab-lished for a singular fractional order boundaryvalue problem with a sign changing nonlinear term,through computing the topological degree of a com-pletely continuous field.

The second set of papers, including thirteen papers, focuson the study of various complex partial differential equationboundary value problems.

(i) In the paper titled “On the study of local solutionsfor a generalized Camassa-Holm equation,” the localwell-posedness of strong solutions for a nonlinear

dispersive model is established in the Sobolev space𝐻𝑠

(𝑅) with 𝑠 > 3/2 by using the pseudoparabolicregularization technique.

(ii) In the paper titled “Thewell-posedness of solutions for ageneralized shallow water wave equation,” the authorsinvestigate a generalized shallow water wave equationand discuss the local well-posedness of solutions inthe Sobolev space.

(iii) In the paper titled “Blow-up analysis for a quasilineardegenerate parabolic equation with strongly nonlinearsource,” the authors investigate the properties of thepositive solution of the Cauchy problem for a quasi-linear degenerate parabolic equation with a stronglynonlinear source and show the existence of a single-point blow-up for a large class of radial decreasingsolutions.

(iv) In the paper titled “The cauchy problem to a shallowwater wave equation with a weakly dissipative term,” ashallowwater wave equationwith a weakly dissipativeterm is investigated, and the sufficient conditions forthe existence of global strong solution are established.

(v) In the paper titled ”The local strong and weak solutionsfor a generalized pseudo-parabolic equation,” the well-posedness of local strong solutions for the Cauchyproblem of a nonlinear generalized pseudoparabolicequation is established in the Sobolev space.

(vi) In the paper titled “The local strong and weak solutionsfor a generalized Novikov equation,” the Kato theoremfor abstract differential equations is applied to estab-lish the local well-posedness of the strong solution fora nonlinear generalized Novikov equation.

(vii) In the paper titled “The asymptotic solution of theinitial boundary value problem to a generalized Boussi-nesq equation,” the 𝐿2 solution of an initial boundaryproblem for a generalized damped Boussinesq equa-tion is constructed, and the large time asymptoticsolution is also obtained.

(viii) In the paper titled “Strong global attractors for 3Dwaveequations with weakly damping,” the authors prove theexistence of a global attractor for a 3Dweakly dampedwave equation in a strong topological space.

(ix) In the paper titled “Regularity and exponential growthof pullback attractors for semilinear parabolic equa-tions involving the Grushin operator,” the authorsconsider a first order nonlinear initial boundary valueproblem for a semilinear degenerate parabolic equa-tion involving the Grushin operator in a boundeddomain and prove the regularity and exponentialgrowth of a pullback attractor in certain functionspace for the nonautonomous dynamical system asso-ciated to the problem.

(x) In the paper titled “A generalization of Mahadevan’sversion of the Krein-Rutman theorem and applicationsto 𝑝-Laplacian boundary value problems,” the authorsestablish a generalization of Mahadevan’s version ofthe Krein-Rutman theorem for a compact, positively

Abstract and Applied Analysis 3

1-homogeneous operator in a Banach space and thendemonstrate its application in establishing the exis-tence of positive solutions for 𝑝-Laplacian boundaryvalue problems under certain conditions.

(xi) In the paper titled “Existence of solutions for nonhomo-geneous A-harmonic equations with variable growth,”the authors establish a theorem for the existenceof weak solutions for nonhomogeneous A-harmonicequations in subspace and then give three examplesto demonstrate its application.

(xii) In the paper titled “Multiple solutions for degenerateelliptic systems near resonance at higher eigenvalues,”the authors study the degenerate semilinear ellipticsystem in an open bounded domain with smoothboundary, and some multiplicity results of solutionsare obtained for the system near resonance at certaineigenvalues by the classical saddle point theorem anda local saddle point theorem in critical point theory.

(xiii) In the paper titled “A regularity criterion for theNavier-Stokes equations in the multiplier spaces,” theauthors establish a regularity criterion in terms of thepressure gradient for weak solutions to the Navier-Stokes equations in a special class.

The third set of papers, including four papers, deal withseveral boundary value problems for highly nonlinear ordi-nary differential equations.

(i) In the paper titled “Positive solutions for second-ordersingular semipositone differential equations involvingStieltjes integral conditions,” the authors investigate theexistence of positive solutions for second-order singu-lar differential equations with a negatively perturbedterm, by means of the fixed-point theory in cones.

(ii) In the paper titled “Positive solutions for Sturm-Liou-ville boundary value problems in a Banach Space,” thesufficient conditions for the existence of single andmultiple positive solutions for a second-order Sturm-Liouville boundary value problem are established ina Banach space, by using the fixed-point theorem ofstrict set contraction operators in the frame of theODE technique.

(iii) In the paper titled “Positive solutions of a nonlinearfourth-order dynamic eigenvalue problem on timescales,” the authors study a nonlinear fourth-orderdynamic eigenvalue problem on time scales andobtain the existence and nonexistence of positivesolutions when 0 < 𝜆 ≤ 𝜆∗ and 𝜆 > 𝜆∗, respectively,for some 𝜆∗, by using the Schauder fixed-pointtheorem and the upper and lower solution method.

(iv) In the paper titled “Bifurcation analysis for a predator-prey model with time delay and delay-dependentparameters,” a class of stage-structured predator-preymodel with time delay and delay-dependent parame-ters is considered. By using the normal form theoryand center manifold theory, some explicit formulaefor determining the stability and the direction of the

Hopf bifurcation periodic solutions bifur-cating fromHopf bifurcations are obtained.

The fourth set of papers focus on finding the approxi-mate and numerical solutions of various complex nonlinearboundary value problems.

(i) In the paper titled “On spectral homotopy analy-sis method for solving linear Volterra and Fredholmintegrodifferential equations,” a spectral homotopyanalysis method (SHAM) is proposed to solve lin-ear Volterra integrodifferential equations, and someexamples are given to test the efficiency and theaccuracy of the proposed method.

(ii) In the paper titled “The solution of a class of singu-larly perturbed two-point boundary value problems bythe iterative reproducing kernel method,” the authorsestablish an iterative reproducing kernel method(IRKM) for solving singular perturbation problemswith boundary layers and give two numerical exam-ples to demonstrate the effectiveness of the method.

(iii) In the paper titled “A Galerkin solution for Burgers’equation using cubic B-spline finite elements,” a Gal-erkin method using cubic B-splines is set up to findthe numerical solutions of Burgers’ equation, and themethod is shown to be capable of solving Burgers’equation accurately for values of viscosity rangingfrom very small to very large.

(iv) In the paper titled “Forward-backward splitting meth-ods for accretive operators in Banach spaces,” theauthors introduce two iterative forward-backwardsplitting methods with relaxations to find zeros ofthe sum of two accretive operators in Banach spacesand prove the weak and strong convergence of thesemethods under mild conditions, and also discussapplications of these methods to variational inequal-ities, the split feasibility problem, and a constrainedconvex minimization problem.

Yong Hong WuLishan Liu

Benchawan WiwatanapatapheeShaoyong Lai

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Editorial Nonlinear Functional Analysis of Boundary Value ...downloads.hindawi.com/journals/aaa/2013/158358.pdftional di erential equation with the fractional deriva-tives, the existence