We consider the discrete Safronov-Dubovskiĭ aggregation equation associated with the physical condition, where particle injection and extraction take place in the dynamical system. In application, this model is used to describe the aggregation of particle-monomers in combination with sedimentation of particle-clusters. More precisely, we prove well-posedness of the considered model for a large class of aggregation kernel with source and efflux coefficients. Furthermore, over a long time period, we prove that the dynamical model attains a unique equilibrium solution with an exponential rate under a suitable condition on the forcing coefficient.

An emerging technique in image segmentation, semi-supervised learning and general classification problems concerns the use of phase-separating flows defined on finite graphs. This technique was pioneered in Bertozzi and Flenner (2012, Multiscale Modeling and Simulation10(3), 1090–1118), which used the Allen–Cahn flow on a graph, and was then extended in Merkurjev et al. (2013, SIAM J. Imaging Sci.6(4), 1903–1930) using instead the Merriman–Bence–Osher (MBO) scheme on a graph. In previous work by the authors, Budd and Van Gennip (2020, SIAM J. Math. Anal.52(5), 4101–4139), we gave a theoretical justification for this use of the MBO scheme in place of Allen–Cahn flow, showing that the MBO scheme is a special case of a ‘semi-discrete’ numerical scheme for Allen–Cahn flow. In this paper, we extend this earlier work, showing that this link via the semi-discrete scheme is robust to passing to the mass-conserving case. Inspired by Rubinstein and Sternberg (1992, IMA J. Appl. Math.48, 249–264), we define a mass-conserving Allen–Cahn equation on a graph. Then, with the help of the tools of convex optimisation, we show that our earlier machinery can be applied to derive the mass-conserving MBO scheme on a graph as a special case of a semi-discrete scheme for mass-conserving Allen–Cahn. We give a theoretical analysis of this flow and scheme, proving various desired properties like existence and uniqueness of the flow and convergence of the scheme, and also show that the semi-discrete scheme yields a choice function for solutions to the mass-conserving MBO scheme.

We assume that human carrying capacity is determined by food availability. We propose three classes of human population dynamical models of logistic type, where the carrying capacity is a function of the food production index. We also employ an integration-based parameter estimation technique to derive explicit formulas for the model parameters. Using actual population and food production index data, numerical simulations of our models suggest that an increase in food availability implies an increase in carrying capacity, but the carrying capacity is “self-limiting” and does not increase indefinitely.

In this paper, we are concerned with unbounded solutions of the singularly perturbed forced pendulum equation in the presence of friction, namely

$$\varepsilon ^2u_\varepsilon ^{{\prime}{\prime}} + \sin u_\varepsilon = \varepsilon ^2\alpha (t)u_\varepsilon + \varepsilon ^2\beta (t)u_\varepsilon ^{\prime} \quad {\rm in}\;(-L,L).{\rm }$$

We consider relatively Meir–Keeler condensing operators to study the existence of best proximity points (pairs) by using the notion of measure of noncompactness, and extend a result of Aghajani et al. [‘Fixed point theorems for Meir–Keeler condensing operators via measure of noncompactness’, Acta Math. Sci. Ser. B35 (2015), 552–566]. As an application of our main result, we investigate the existence of an optimal solution for a system of integrodifferential equations.

In this paper, we discuss the dependence of the solutions on the parameters (order, initial function, right-hand function) of fractional neutral delay differential equations (FNDDEs). The corresponding theoretical results are given respectively. Furthermore, we present some numerical results that support our theoretical analysis.

The Lagrange-d'Alembert equations with constraints belonging to H1,∞ have been considered. A concept of weak solutions to these equations has been built. A global existence theorem for Cauchy problem has been obtained.

In this paper, we investigate the dependence of the solutions on the parameters (order, initial function, right-hand function) of fractional delay differential equations (FDDEs) with the Caputo fractional derivative. Some results including an estimate of the solutions of FDDEs are given respectively. Theoretical results are verified by some numerical examples.

We establish that the elliptic equation defined in an exterior domain of ℝn, n≥3, has a positive solution which decays to 0 as under quite general assumptions upon f and g.

Starting from results of Dubé and Mingarelli, Wahlén, and Ehrström, who give conditions that ensure the existence and uniqueness of nonnegative nondecreasing solutions asymptotically constant of the equation we have been able to reduce their hypotheses in order to obtain the same existence results, at the expense of losing the uniqueness part. The main tool they used is the Banach Fixed Point Theorem, while ours has been the Schauder Fixed Point Theorem together with one version of the Arzelà-Ascoli Theorem.

We study the semilinear differential equation u″ + F(t,u,u′)=0 on a half-line. Under different growth conditions on the function F, equations with globally defined solutions asymptotic to lines are characterized. Both fixed initial data and fixed asymptote are considered.

The main result ensures that the scalar problem $x''\,{=}\,f(x)$, $x(0)\,{=}\,x_0$, $x'(0)\,{=}\,x_1,$ has a nonconstant locally $W^{2,1}$ solution if and only if there exists a nontrivial interval $J$ such that $x_0 \in J$, $f \in L^1_{\rm loc}(J)$, $x_1^2+2\int_{x_0}^{y}{f(s)\,ds} > 0$ for almost all $y \in J$ and \[\frac{\max\{1,|f|\}}{ \sqrt{x_1^2+2\int_{x_0}^{\cdot}{f(s)\,ds}}} \in L^1_{\rm loc}(J).\] Necessary and sufficient conditions for local and global uniqueness and for existence of periodic solutions are also established.

Let X be a Banach space and . Let Π and Π0 be two subspaces of , the Banach space of bounded continuous functions from 𝕁 to X. We seek conditions under which Π + Π0 is closed in . This led to introduce a general space, which contains many classes of almost periodic type functions as subspaces. We prove some recent results on indefinite integral for the elements of these classes. We apply certain results on harmonic analysis to investigate solutions of differential equations. As an application we study specific spaces: the spaces of asymptotic and pseudo almost automorphic functions and their solutions of some ordinary quasi-linear and a non-linear parabolic partial differential equations.

Existence principles are given for systems of differential equations with reflection of the argument. These are derived using fixed point analysis, specifically the Nonlinear Alternative. Then existence results are deduced for certain classes of first and second order equations with reflection of the argument.