The aim of this work is to investigate a class of boundary Cauchy problems with infinite delay. We give some sufficient conditions ensuring the uniqueness, existence, and regularity of solutions. For illustration, we apply the result to an age dependent population equation, which covers some special cases considered in some recent papers.

We present a model based on dynamics of electrons in a plasma using a simplified Boltzmann equation coupled with Poisson’s equation. The motivation arose from simulating active plasma resonance spectroscopy, which is used for plasma diagnostic techniques; see Braithwaite and Franklin (2009), Lapke et al. (2010), and Oberrath et al. (2011). Mathematically, we are interested in designing splitting methods for the model problem. While the full Boltzmann equation is delicate to solve, we decouple it into a transport and collision part, which are then solved in different ways. First we reduce it to a simplified transport-collision equation and start to analyse the abstract Cauchy problem using semigroup methods. Second, we pass to the coupled transport and collision model and apply the splitting ideas, resecting the different discretization schemes. The results are discussed first with numerical experiments and then we verify the underlying theoretical novelties.

By developing a new comparison result and using the monotone iterative technique, we are able to obtain existence of minimal and maximal solutions of periodic boundary value problems for first-order impulsive functional integrodifferential equations with integral-jump conditions. An example is also given to illustrate our results.

The matrix Riccati equation that must be solved to obtain the solution to stochastic optimal control problems known as LQG homing is linearized for a class of processes. The results generalize a theorem proved by Whittle and the one-dimensional case already considered by the authors. A particular two-dimensional problem is solved explicitly.

Based on iterative methods without memory of eighth-order convergence proposed by Thukral (2012), some iterative methods with memory and high efficiency index are presented. We show that the order of convergence is increased without any additional function evaluations. Numerical comparisons are made to show the performance of the presented methods.

We consider an initial-boundary value problem to a nonlinear string equations with linear damping term. It is proved that under suitable conditions the solution is global in time and the solution with a negative initial energy blows up in finite time.

The main objective of the paper is to study new integral inequality on time scales which is used for the study of some partial dynamic equations. Some applications of our results are also given.

We have studied the periodicity of solutions of some nonlinear time-varying differential models by using the theory of reflecting functions. We have established a new relationship between the linear differential system and the Riccati equations and applied the obtained results to discuss the behavior of periodic solutions of the Riccati equations.

We establish sufficient conditions for the oscillation of solutions of even order neutral type differential equations of the form ${[r(t){[x(t)+p(t)x(\tau (t))]}^{(n-1)}]}^{\prime }+q(t)f(x(\sigma (t)))=0$.

The Dirichlet resonant boundary value problems are considered. If the respective nonlinear equation can be reduced to a quasilinear one with a nonresonant linear part and both equations are equivalent in some domain $\mathrm{\Omega }$ and if solutions of the quasilinear problem are in $\mathrm{\Omega }$, then the original problem has a solution. We say then that the original problem allows for quasilinearization. If quasilinearization is possible for essentially different linear parts, then the original problem has multiple solutions. We give conditions for Emden-Fowler type resonant boundary value problem solvability and consider examples.

We give a consistent discretization of a continuous model of HIV infection, with distributed time delays to express the lag between the times when the virus enters a cell and when the cell becomes infected. The global stability of the steady states of the model is determined and numerical simulations are presented to illustrate our theoretical results.

This paper is concerned with the asymptotic stability analysis for a class of neutral systems with time-delay and saturating actuators, which is further to reduce the conservatism of neutral system. Based on the model transformation and the delay-dividing approach, a new type of augmented Lyapunov functional is constructed, which has fully exploited the information on the lower bound of the delay. Then the delay-dependent conditions for asymptotic stability are derived by applying some integral inequalities and Lyapunov theory. Finally, numerical examples are provided to show that the proposed results significantly improve the allowed upper bounds of the delay size over some existing ones in the literature.

We consider an impulsive neutral fractional integrodifferential equation with infinite delay in an arbitrary Banach space $X$. The existence of mild solution is established by using solution operator and Hausdorff measure of noncompactness.

The substantiation of a possibility of application of partial averaging method on finite interval for differential inclusions with the fuzzy right-hand side with a small parameter is considered.

We consider the approximate solution of the coupled Schrödinger-KdV equation by using the extended optimal homotopy asymptotic method (OHAM). We obtained the extended OHAM solution of the problem and compared with the exact, variational iteration method (VIM) and homotopy perturbation method (HPM) solutions. The obtained solution shows that extended OHAM is effective, simpler, easier, and explicit and gives a suitable way to control the convergence of the approximate solution.

This paper establishes the existence of at least three positive solutions for a coupled system of $p$-Laplacian fractional order two-point boundary value problems, $D_{0^{+}}^{{\beta }_1}({\varphi }_p(D_{0^{+}}^{{\alpha }_1}u(t)))=f_1(t,u(t),v(t))$, $t\in (\mathrm{0,1})$, $D_{0^{+}}^{{\beta }_2}({\varphi }_p(D_{0^{+}}^{{\alpha }_2}v(t)))=f_2(t,u(t),v(t))$, $t\in (\mathrm{0,1})$, $u(0)=D_{0^{+}}^{q_1}u(0)=0$, $\gamma u(1)+\delta D_{0^{+}}^{q_2}u(1)=0$, $D_{0^{+}}^{{\alpha }_1}u(0)=D_{0^{+}}^{{\alpha }_1}u(1)=0$, $v(0)=D_{0^{+}}^{q_1}v(0)=0$, $\gamma v(1)+\delta D_{0^{+}}^{q_2}v(1)=0$, $D_{0^{+}}^{{\alpha }_2}v(0)=D_{0^{+}}^{{\alpha }_2}v(1)=0$, by applying five functionals fixed point theorem.

In this work we study existence results for mixed Volterra-Fredholm neutral functional integrodifferential equations with infinite delay in Banach spaces. To obtain a priori bounds of solutions required in Krasnoselski-Schaefer type fixed point theorem, we have used an integral inequality established by B. G. Pachpatte. The variants for obtained results are given. An example is considered to illustrate the obtained results.

We investigate the oscillatory behavior of solutions of the $m$th order half-linear functional difference equations with damping term of the form $\mathrm{\Delta }{\mathrm{[p}}_{n}Q({\mathrm{\Delta }}^{m-1}{y}_n)]+q_{n}Q({\mathrm{\Delta }}^{m-1}{y}_n)+r_{n}Q(y_{{\tau }_n})=0$, $n\ge n_0$, where $m$ is even and $Q(s)={\left|s\right|}^{\alpha -2}s$, $\alpha >1$ is a fixed real number. Our main results are obtained via employing the generalized Riccati transformation. We provide two examples to illustrate the effectiveness of the proposed results.

We study the asymptotic behaviour of quasi-stationary solutions of a free boundary problem which had been discussed by Bueno (2005). Using a simpler method we prove that the quasi-steady solutions of the problem converge uniformly to the unique nontrivial steady solution.

This paper studies Volterra integral evolution equations of convolution type from the point of view of complex inversion formula and the admissibility in the Salamon-Weiss sens. We first present results on the validity of the inverse formula of the Laplace transform for the resolvent families associated with scalar Volterra integral equations of convolution type in Banach spaces, which extends and improves the results in Hille and Philllips (1957) and Cioranescu and Lizama (2003, Lemma 5), respectively, including the stronger version for a class of scalar Volterra integrodifferential equations of convolution type on unconditional martingale differences UMD spaces, provided that the leading operator generates a $C_0$-semigroup. Next, a necessary and sufficient condition for $L^p$-admissibility $\left(p\in \left[1,\infty \right[\right)$ of the system's control operator is given in terms of the UMD-property of its underlying control space for a wider class of Volterra integrodifferential equations when the leading operator is not necessarily a generator, which provides a generalization of a result known to hold for the standard Cauchy problem (Bounit et al., 2010, Proposition 3.2).

We study the existence of mild and classical solutions are proved for a class of impulsive integrodifferential equations with nonlocal conditions in Banach spaces. The main results are obtained by using measure of noncompactness and semigroup theory. An example is presented.

We use the generalized Riccati transformation and the inequality technique to establish some new oscillation criteria for the second-order nonlinear delay dynamic equation with damping on a time scale $\mathbb{T}$$(r(t)\mathrm{g}(x(t)$, $x^{\mathrm{\Delta }}(t)){)}^{\mathrm{\Delta }}+p(t)g(x(t)$, $x^{\mathrm{\Delta }}(t))\hspace{0.17em}+\hspace{0.17em}q(t)f(x(\tau (t)))=0,$ where $r(t)$, $p(t)$, and $q(t)$ are positive right dense continuous (rd-continuous) functions on $\mathbb{T}$. Our results improve and extend some results established by Zhang et al., 2011. Also, our results unify the oscillation of the second-order nonlinear delay differential equation with damping and the second-order nonlinear delay difference equation with damping. Finally, we give some examples to illustrate our main results.

In this paper, optimal homotopy asymptotic method (OHAM) for the semianalytic solutions of nonlinear singular two-point boundary value problems has been applied to several problems. The solutions obtained by OHAM have been compared with the solutions of another method named as modified adomain decomposition (MADM). For testing the success of OHAM, both of the techniques have been analyzed against the exact solutions in all problems. It is proved by this paper that solutions of OHAM converge rapidly to the exact solution and show most effectiveness as compared to MADM.

We introduce a new subclass $N_n(\lambda ,\alpha ,\beta ,A,B)$ of Non-Bazilevič functions of order $\alpha +i\beta$. Some subordination relations and inequality properties are discussed. The results obtained generalize the related work of some authors. In addition, some other new results are also obtained.

We construct and analyze a nonlocal global pandemic model that comprises a system of two nonlocal integrodifferential equations (functional differential equations) and an ordinary differential equation. This model was constructed by considering a spherical coordinate transformation of a previously established epidemiology model that can be applied to insect-borne diseases, like yellow fever. This transformation amounts to a nonlocal boundary value problem on the unit sphere and therefore can be interpreted as a global pandemic model for insect-borne diseases. We ultimately show that a weak solution to the weak formulation of this model exists using a fixed point argument, which calls upon the construction of a weak formulation and the existence and uniqueness of an auxiliary problem.

We establish the existence results for two-point boundary value problem of fractional differential equations at resonance by means of the coincidence degree theory. Furthermore, a result on the uniqueness of solution is obtained. We give an example to demonstrate our results.

The purpose of the present paper is to investigate the mixed Dirichlet-Neumann boundary value problems for the anisotropic Laplace-Beltrami equation ${\text{div}}_{\mathcal{C}}(A{\nabla }_{\mathcal{C}}\phi )=f$ on a smooth hypersurface $\mathcal{C}$ with the boundary $\mathrm{\Gamma }=\partial \mathcal{C}$ in ${\mathbb{R}}^n$. $A(x)$ is an $n\times n$ bounded measurable positive definite matrix function. The boundary is decomposed into two nonintersecting connected parts $\mathrm{\Gamma }={\mathrm{\Gamma }}_D\cup {\mathrm{\Gamma }}_N$ and on ${\mathrm{\Gamma }}_D$ the Dirichlet boundary conditions are prescribed, while on ${\mathrm{\Gamma }}_N$ the Neumann conditions. The unique solvability of the mixed BVP is proved, based upon the Green formulae and Lax-Milgram Lemma. Further, the existence of the fundamental solution to ${\text{div}}_{\mathcal{S}}(A{\nabla }_{\mathcal{S}})$ is proved, which is interpreted as the invertibility of this operator in the setting ${\mathbb{H}}_{p,\#}^s(\mathcal{S})\to {\mathbb{H}}_{p,\#}^{s-2}(\mathcal{S})$, where ${\mathbb{H}}_{p,\#}^s(\mathcal{S})$ is a subspace of the Bessel potential space and consists of functions with mean value zero.

We investigate the large time behavior of the global weak entropy solutions to the symmetric Keyfitz-Kranzer system with linear damping. It is proved that as $t\to \infty$ the entropy solutions tend to zero in the $L^p$ norm.