For those semigroups, which may have power type singularities and whose generators are abstract multivalued linear operators, we characterize the behaviour with respect to a certain set of intermediate and interpolation spaces. The obtained results are then applied to provide maximal time regularity for the solutions to a wide class of degenerate integro- and non-integro-differential evolution equations in Banach spaces.

A numerical technique based on reproducing kernel methods for the exact solution of linear Volterra integral equations system of the second kind is given. The traditional reproducing kernel method requests that operator a satisfied linear operator equation $Au=f$, is bounded and its image space is the reproducing kernel space $W_2^1[a,b]$. It limits its application. Now, we modify the reproducing kernel method such that it can be more widely applicable. The n-term approximation solution obtained by the modified method is of high accuracy. The numerical example compared with other methods shows that the modified method is more efficient.

The aim of this paper is to show the use of the coupled quasisolutions method as a useful technique when dealing with ordinary differential equations with functional arguments of bounded variation. We will do this by looking for solutions for a first-order ordinary differential equation with an advanced argument of bounded variation. The main trick is to use the Jordan decomposition of this argument in a nondecreasing part and a nonincreasing one. As a necessary step, we will also talk about coupled fixed points of multivalued operators.

For solving Laplace’s equation in circular domains with circular holes, the null field method (NFM) was developed by Chen and his research group (see Chen and Shen (2009)). In Li et al. (2012) the explicit algebraic equations of the NFM were provided, where some stability analysis was made. For the NFM, the conservative schemes were proposed in Lee et al. (2013), and the algorithm singularity was fully investigated in Lee et al., submitted to Engineering Analysis with Boundary Elements, (2013). To target the same problems, a new interior field method (IFM) is also proposed. Besides the NFM and the IFM, the collocation Trefftz method (CTM) and the boundary integral equation method (BIE) are two effective boundary methods. This paper is devoted to a further study on NFM and IFM for three goals. The first goal is to explore their intrinsic relations. Since there exists no error analysis for the NFM, the second goal is to drive error bounds of the numerical solutions. The third goal is to apply those methods to Laplace’s equation in the domains with extremely small holes, which are called actually punctured disks. By NFM, IFM, BIE, and CTM, numerical experiments are carried out, and comparisons are provided. This paper provides an in-depth overview of four methods, the error analysis of the NFM, and the intriguing computation, which are essential for the boundary methods.

Some new oscillation criteria are given for first order neutral delay differential equations with variable coefficients. Our results generalize and extend some of the well-known results in the literature. Some examples are considered to illustrate the main results.

Numerical algorithms for solving first-order fuzzy differential equations and hybrid fuzzy differential equations have been investigated. Sufficient conditions for stability and convergence of the proposed algorithms are given, and their applicability is illustrated with some examples.

We consider the homogenization of the linear parabolic problem $\rho (x{/\epsilon }_{\mathrm{2}}){\partial }_t{u}^{\epsilon }(x,t)-\nabla ·(a(x/{\epsilon }_{\mathrm{1}},t/{\epsilon }_{\mathrm{1}}^{\mathrm{2}})\nabla u^{\epsilon }(x,t))=f(x,t)$ which exhibits a mismatch between the spatial scales in the sense that the coefficient $a(x{/\epsilon }_{\mathrm{1}},t/{\epsilon }_{\mathrm{1}}^{\mathrm{2}})$ of the elliptic part has one frequency of fast spatial oscillations, whereas the coefficient $\rho (x/{\epsilon }_{\mathrm{2}})$ of the time derivative contains a faster spatial scale. It is shown that the faster spatial microscale does not give rise to any corrector term and that there is only one local problem needed to characterize the homogenized problem. Hence, the problem is not of a reiterated type even though two rapid scales of spatial oscillation appear.

We deal with a constrained quasivariational inequality under a general form. We study existence of solutions in two situations depending on whether the set of constraints is bounded or possibly unbounded.

We investigate a mathematical model introduced by Shikhmurzaev to remove singularities that arise when classical hydrodynamic models are applied to certain physical phenomena. The model is described as a free boundary problem consisting of the Navier-Stokes equations and a surface mass balance equation. We prove the local-in-time solvability in Hölder spaces.

This paper investigates the inverse problem of finding a time-dependent diffusion coefficient in a parabolic equation with the periodic boundary and integral overdetermination conditions. Under some assumption on the data, the existence, uniqueness, and continuous dependence on the data of the solution are shown by using the generalized Fourier method. The accuracy and computational efficiency of the proposed method are verified with the help of the numerical examples.

We establish general conditions for the unique solvability of nonlinear measure functional differential equations in terms of properties of suitable linear majorants.

A novel modification of the variational iteration method is proposed by means of Laplace transform and homotopy perturbation method. The fractional lagrange multiplier is accurately determined by the Laplace transform and the nonlinear one can be easily handled by the use of He’s polynomials. Several fractional nonlinear nonhomogeneous equations are analytically solved as examples and the methodology is demonstrated.

We present the existence of extremal solution and relaxation problem for fractional differential inclusion with initial conditions.

This paper is concerned with the asymptotical behavior of solutions to the reaction-diffusion system under homogeneous Neumann boundary condition. By taking food ingestion and species' moving into account, the model is further coupled with Michaelis-Menten type functional response and nonlocal delay. Sufficient conditions are derived for the global stability of the positive steady state and the semitrivial steady state of the proposed problem by using the Lyapunov functional. Our results show that intraspecific competition benefits the coexistence of prey and predator. Furthermore, the introduction of Michaelis-Menten type functional response positively affects the coexistence of prey and predator, and the nonlocal delay is harmless for stabilities of all nonnegative steady states of the system. Numerical simulations are carried out to illustrate the main results.

We study the final state problem for the Dirac-Klein-Gordon equations (DKG) in two space dimensions. We prove that if the nonresonance mass condition is satisfied, then the wave operator for DKG is well defined from a neighborhood at the origin in lower order weighted Sobolev space to some Sobolev space.

We prove the existence and uniqueness of a positive continuous solution to the following singular semilinear fractional Dirichlet problem ${(-\mathrm{\Delta })}^{\alpha /\mathrm{2}}u=a_{\mathrm{1}}(x)u^{{\sigma }_{\mathrm{1}}}+a_{\mathrm{2}}(x)u^{{\sigma }_{\mathrm{2}}}$, in $\mathrm{\hspace{0.17em}D\hspace{0.17em}}{\text{\hspace{0.17em}lim}}_{x\to z\in \partial D}(\delta (x){)}^{\mathrm{1}-(\mathrm{\alpha }/\mathrm{2})}u(x)=0,$ where ${\sigma }_{\mathrm{1}},{\sigma }_{\mathrm{2}}\in (-\mathrm{1,1}),\hspace{0.17em}$ $D$ is a bounded $C^{\mathrm{1,1}}$-domain in ${ℝ}^n,$ $n\ge \mathrm{2},$ and $\delta (x)$ denotes the Euclidian distance from $x$ to the boundary of $D.$ The nonnegative weight functions $a_{\mathrm{1}},a_{\mathrm{2}}$ are required to satisfy certain hypotheses related to the Karamata class. We also investigate the global behavior of such solution.

Neutral differential equations have been used to describe the systems that not only depend on the present and past states but also involve derivatives with delays. This paper considers hybrid nonlinear neutral stochastic functional differential equations (HNSFDEs) without the linear growth condition and examines the boundedness and exponential stability. Two illustrative examples are given to show the effectiveness of our theoretical results.

The motion of dilute charged particles can be modeled by Vlasov-Poisson-Boltzmann system. We study the large time stability of the VPB system. To be precise, we prove that when time goes to infinity, the solution of VPB system tends to global Maxwellian state in a rate $O\left(t^{-\infty }\right)$, by using a method developed for Boltzmann equation without force in the work of Desvillettes and Villani (2005). The improvement of the present paper is the removal of condition on parameter $\lambda$ as in the work of Li (2008).

The global asymptotic behavior of a nonautonomous competitor-competitor-mutualist model is investigated, where all the coefficients are time-dependent and asymptotically approach periodic functions, respectively. Under certain conditions, it is shown that the limit periodic system of this asymptotically periodic model admits two positive periodic solutions $(u_{\mathrm{1}}^T,u_{\mathrm{2}T},u_{\mathrm{3}}^T),(u_{\mathrm{1}T},u_{\mathrm{2}}^T,u_{\mathrm{3}T})$ such that $u_{iT}\le u_i^T(i=\mathrm{1,2},\mathrm{3})$, and the sector $\{(u_{\mathrm{1}},u_{\mathrm{2}},u_{\mathrm{3}}):u_{iT}\le u_i\le u_i^T,i=\mathrm{1,2},\mathrm{3}\}$ is a global attractor of the asymptotically periodic model. In particular, we derive sufficient conditions that guarantee the existence of a positive periodic solution which is globally asymptotically stable.

The main feature of the boundary layer flow problems is the inclusion of the boundary conditions at infinity. Such boundary conditions cause difficulties for any of the series methods when applied to solve such problems. To the best of the authors’ knowledge, two procedures were used extensively in the past two decades to deal with the boundary conditions at infinity, either the Padé approximation or the direct numerical codes. However, an intensive work is needed to perform the calculations using the Padé technique. Regarding this point, a new idea is proposed in this paper. The idea is based on transforming the unbounded domain into a bounded one by the help of a transformation. Accordingly, the original differential equation is transformed into a singular differential equation with classical boundary conditions. The current approach is applied to solve a class of the Blasius problem and a special class of the Falkner-Skan problem via an improved version of Adomian’s method (Ebaid, 2011). In addition, the numerical results obtained by using the proposed technique are compared with the other published solutions, where good agreement has been achieved. The main characteristic of the present approach is the avoidance of the Padé approximation to deal with the infinity boundary conditions.

We use the reproducing kernel Hilbert space method to solve the fifth-order boundary value problems. The exact solution to the fifth-order boundary value problems is obtained in reproducing kernel space. The approximate solution is given by using an iterative method and the finite section method. The present method reveals to be more effective and convenient compared with the other methods.

A class of nonlinear elliptic problems driven by $p(x)$-Laplacian-like with a nonsmooth locally Lipschitz potential was considered. Applying the version of a nonsmooth three-critical-point theorem, existence of three solutions of the problem is proved.

We study a slow diffusive $p$-Laplace equation in a bounded domain with the Neumann boundary conditions. A natural energy is associated to the equation. It is shown that the solution blows up in finite time with the nonpositive initial energy, based on an energy technique. Furthermore, under some assumptions of initial data, we prove that the solutions with bounded initial energy also blow up.

We give an overview on some recent results concerning the study of the Dirichlet problem for second-order linear elliptic partial differential equations in divergence form and with discontinuous coefficients, in unbounded domains. The main theorem consists in an $L^p$-a priori bound, $P>1$. Some applications of this bound in the framework of non-variational problems, in a weighted and a non-weighted case, are also given.

We firstly study the existence of PC-mild solutions for impulsive fractional semilinear integrodifferential equations and then present controllability results for fractional impulsive integrodifferential systems in Banach spaces. The method we adopt is based on fixed point theorem, semigroup theory, and generalized Bellman inequality. The results obtained in this paper improve and extend some known results. At last, an example is presented to demonstrate the applications of our main results.

We give some sufficient conditions for $\mathrm{\Psi }$-uniform stability of the trivial solutions of a nonlinear differential system and of nonlinear Volterra integro-differential systems with time delay.

Inverse problem for the Bessel operator is studied. A set of values of eigenfunctions at some internal point and parts of two spectra are taken as data. Uniqueness theorems are obtained. The approach that was used in investigation of problems with partially known potential is employed.