Recent zbMATH articles in MSC 45https://zbmath.org/atom/cc/452021-11-25T18:46:10.358925ZWerkzeugA nonlinear integro-differential equation with fractional order and nonlocal conditionshttps://zbmath.org/1472.340162021-11-25T18:46:10.358925Z"Wahash, Hanan A."https://zbmath.org/authors/?q=ai:wahash.hanan-abdulrahman"Abdo, Mohammed S."https://zbmath.org/authors/?q=ai:abdo.mohammed-salem"Panchal, Satish K."https://zbmath.org/authors/?q=ai:panchal.satish-kushabaSummary: This paper deals with a nonlinear integro-differential equation of fractional order \(\alpha\in(0,1)\) with nonlocal conditions involving fractional derivative in the Caputo sense. Under a new approach and minimal assumptions on the function \(f\), we prove the existence, uniqueness, estimates on solutions and continuous dependence of the solutions. The used techniques in analysis rely on fractional calculus, Banach contraction mapping principle, and Pachpatte's inequality. At the end, some numerical examples to justify our results are illustrated.Homogenization of random convolution energieshttps://zbmath.org/1472.350272021-11-25T18:46:10.358925Z"Braides, Andrea"https://zbmath.org/authors/?q=ai:braides.andrea"Piatnitski, Andrey"https://zbmath.org/authors/?q=ai:piatnitski.andrey-lSummary: We prove a homogenization theorem for a class of quadratic convolution energies with random coefficients. Under suitably stated hypotheses of ergodicity and stationarity, we prove that the \(\Gamma\)-limit of such energy is almost surely a deterministic quadratic Dirichlet-type integral functional, whose integrand can be characterized through an asymptotic formula. The proof of this characterization relies on results on the asymptotic behaviour of subadditive processes. The proof of the limit theorem uses a blow-up technique common for local energies, which can be extended to this `asymptotically local' case. As a particular application, we derive a homogenization theorem on random perforated domains.The local behavior of positive solutions for higher order equation with isolated singularitieshttps://zbmath.org/1472.350662021-11-25T18:46:10.358925Z"Li, Yimei"https://zbmath.org/authors/?q=ai:li.yimeiSummary: We use blow up analysis for local integral equations to provide a blow up rates of solutions of higher order Hardy-Hénon equation in a bounded domain with an isolated singularity, and show the asymptotic radial symmetry of the solutions near the singularity. This work generalizes the correspondence results of \textit{T. Jin} and \textit{J. Xiong} (in [``Asymptotic symmetry and local behavior of solutions of higher order conformally invariant equations with isolated singularities'', Preprint, \url{arXiv:1901.01678}]) on higher order conformally invariant equations with an isolated singularity.Stability and uniqueness of self-similar profiles in \(L^1\) spaces for perturbations of the constant kernel in Smoluchowski's coagulation equationhttps://zbmath.org/1472.350842021-11-25T18:46:10.358925Z"Throm, Sebastian"https://zbmath.org/authors/?q=ai:throm.sebastianWhen the coagulation kernel \(K\) is homogeneous with a degree strictly smaller than one, it is expected that solutions to the coagulation equation
\begin{align*}
\partial_t \phi(t,\xi) & = \frac{1}{2} \int_0^\xi K(\xi-\eta,\eta) \phi(t,\xi-\eta) \phi(t,\eta)\ d\eta \\
& \qquad - \int_0^\infty K(\xi,\eta) \phi(t,\xi) \phi(t,\eta)\ d\eta
\end{align*}
where \((t,\xi)\in (0,\infty)\times (0,\infty)\), with non-negative initial condition \(\phi_0\in L^1((0,\infty),\xi d\xi)\), behave in a self-similar way as \(t\to\infty\). This conjecture is up to now known to be true for the constant kernel \(K(\xi,\eta)=2\), see [\textit{G. Menon} and \textit{R. L. Pego}, Commun. Pure Appl. Math. 57, No. 9, 1197--1232 (2004; Zbl 1049.35048)]. An important step in the proof of this conjecture is the uniqueness of finite mass self-similar solutions, and the latter is shown in the paper under review for small perturbations of the constant kernel with homogeneity zero, thereby improving previous results by \textit{B. Niethammer}, the author and \textit{J. J. L. Velázquez} [``A revised proof of uniqueness of self-similar profiles to Smoluchowski's coagulation equation for kernels close to constant'', Preprint, \url{arXiv:1510.03361}; J. Stat. Phys. 164, No. 2, 399--409 (2016; Zbl 1356.82034)], along with providing a new approach. More precisely, let \(W\in C((0,\infty)^2)\) be a symmetric function satisfying
\[
0 \le W(x,y) \le \left( \frac{x}{y} \right)^\alpha + \left( \frac{y}{x} \right)^\alpha \text{ and } W(\lambda x,\lambda y) = W(x,y), \qquad (\lambda,\xi,\eta)\in (0,\infty)^3,
\]
for some \(\alpha\in (0,1)\), and set \(K_\varepsilon = 2 + \varepsilon W\) for \(\varepsilon\ge 0\). Given \(a>-1\), \(b>0\), and \(\delta>0\), it is first shown that, for \(\varepsilon>0\) sufficiently small, any self-similar profile \(u^{(\varepsilon)}\) with unit total mass satisfies
\[
\int_0^\infty \left| u^{(\varepsilon)}(x) - e^{-x} \right| (x^a + x^b)\ dx \le \delta.
\]
We recall that \(u^{(\varepsilon)}\) is a self-similar profile if \((t,\xi) \mapsto (1+t)^{-2} u^{(\varepsilon)}(\xi(1+t)^{-1})\) is a solution to the coagulation equation with kernel \(K_\varepsilon\) and that \((t,\xi)\mapsto (1+t)^{-2} e^{-\xi(1+t)^{-1}}\) is the unique self-similar solution with unit total mass to the coagulation equation with constant kernel \(K_0\). Assuming further that
\[
W(x,y)\ge c_* \left( \frac{x}{y} \right)^\alpha + \left( \frac{y}{x} \right)^\alpha, \qquad (x,y)\in (0,\infty)^2,
\]
for some \(c_*>0\) when \(\alpha\in [1/2,1)\), uniqueness of the self-similar profile with unit total mass is also established, still for \(\varepsilon\) small enough.
We also refer to [\textit{J. A. Cañizo} and the author, J. Differ. Equations 270, 285--342 (2021; Zbl 1456.35065)] for attracting properties of these self-similar solutions under similar assumptions.Method of scaling spheres for integral and polyharmonic systemshttps://zbmath.org/1472.351412021-11-25T18:46:10.358925Z"Le, Phuong"https://zbmath.org/authors/?q=ai:le.phuong-m|le.phuong-quynhSummary: We establish a method of scaling spheres for the integral system
\[\begin{cases}
u ( x ) = \int_{\mathbb{R}^n} \frac{ | y |^a v^p ( y )}{ | x - y |^{n - \alpha}} d y , & x \in \mathbb{R}^n , \\
v ( x ) = \int_{\mathbb{R}^n} \frac{ | y |^b u^q ( y )}{ | x - y |^{n - \beta}} d y , & x \in \mathbb{R}^n ,
\end{cases}\]
where \(0 < \alpha\), \(\beta < n\), \(a > - \alpha\), \(b > - \beta\) and \(p, q > 0\). By using this method, we obtain a Liouville theorem for nonnegative solutions when \(0 < p \leq \frac{ n + \alpha + 2 a}{ n - \beta}\), \(0 < q \leq \frac{ n + \beta + 2 b}{ n - \alpha}\) and \((p, q) \neq(\frac{ n + \alpha + 2 a}{ n - \beta}, \frac{ n + \beta + 2 b}{ n - \alpha})\). As an application, we derive a Liouville theorem for nonnegative solutions of the polyharmonic Hénon-Hardy system
\[\begin{cases}
( - {\Delta} )^m u ( x ) = | x |^a v^p ( x )\quad & \text{in } \mathbb{R}^n , \\
( - {\Delta} )^l v ( x ) = | x |^b u^q ( x ) & \text{in } \mathbb{R}^n ,
\end{cases}\]
where \(m\) and \(l\) are integers in \((0, \frac{ n}{ 2})\).Asymptotic behavior of integral functionals for a two-parameter singularly perturbed nonlinear traction problemhttps://zbmath.org/1472.351872021-11-25T18:46:10.358925Z"Falconi, Riccardo"https://zbmath.org/authors/?q=ai:falconi.riccardo"Luzzini, Paolo"https://zbmath.org/authors/?q=ai:luzzini.paolo"Musolino, Paolo"https://zbmath.org/authors/?q=ai:musolino.paoloSummary: We consider a nonlinear traction boundary value problem for the Lamé equations in an unbounded periodically perforated domain. The edges lengths of the periodicity cell are proportional to a positive parameter \(\delta \), whereas the relative size of the holes is determined by a second positive parameter \(\varepsilon \). Under suitable assumptions on the nonlinearity, there exists a family of solutions \(\{ u (\varepsilon , \delta , \cdot ) \}_{( \varepsilon , \delta ) \in ]0, \varepsilon^{\prime}[ \times ]0, \delta^{\prime}[}\). We analyze the asymptotic behavior of two integral functionals associated to such a family of solutions when the perturbation parameter pair \((\varepsilon , \delta )\) is close to the degenerate value \((0, 0)\).On the generalization of Moyal equation for an arbitrary linear quantizationhttps://zbmath.org/1472.353162021-11-25T18:46:10.358925Z"Borisov, Leonid A."https://zbmath.org/authors/?q=ai:borisov.leonid-a"Orlov, Yuriy N."https://zbmath.org/authors/?q=ai:orlov.yurii-nBorn approximation and sequence for hyperbolic equationshttps://zbmath.org/1472.353182021-11-25T18:46:10.358925Z"Lin, Ching-Lung"https://zbmath.org/authors/?q=ai:lin.ching-lung"Lin, Liren"https://zbmath.org/authors/?q=ai:lin.liren"Nakamura, Gen"https://zbmath.org/authors/?q=ai:nakamura.genSummary: The Born approximation and the Born sequence are considered for hyperbolic equations when we perturb their leading parts. The Born approximation is a finite successive approximation such as the finite terms Neumann series for the solution of a hyperbolic equation in terms of the smallness of the perturbation and if the successive approximation is infinitely many times, then we have the Born series. Due to the so called regularity loss for solutions of hyperbolic equations, we need to assume that data such as the inhomogeneous term of the equation, Cauchy datum and boundary datum are \(\mathit{C}^\infty \), and also they satisfy the compatibility condition of any order in order to define the Born series. Otherwise we need to smooth each term of the Born series. The convergence of the Born series and the Born series with smoothing are very natural questions to be asked. Also giving an estimate of approximating the solution for finite terms Born series is also an important question in practice. The aims of this paper are to discuss about these questions. We would like to emphasize that we found a small improvement in the usual energy estimate for solutions of an initial value problem for a hyperbolic equation, which is very useful for our aims. Since the estimate of approximation is only giving the worst estimate for the approximation, we also provide some numerical studies on these questions which are very suggestive for further theoretical studies on the Born approximation for hyperbolic equations.Large-time behavior of matured population in an age-structured modelhttps://zbmath.org/1472.354032021-11-25T18:46:10.358925Z"Li, Linlin"https://zbmath.org/authors/?q=ai:li.linlin"Ainseba, Bedreddine"https://zbmath.org/authors/?q=ai:ainseba.bedreddineSummary: In this paper, we model a mosquito plasticity problem and investigate the large time behavior of matured population under different control strategies. We prove that when the control is small, then the matured population will become large for large time and when the control is large, then the matured population will become small for large time. In the intermediate case, we derive a time-delayed model for the matured population which can be governed by a sub-equation and a super-equation. We prove the existence of traveling fronts for the sub-equation and use it to prove that the matured population will finally be between the positive states of the sub-equation and super-equation. At last, we present numerical simulations.Mild solutions are weak solutions in a class of (non)linear measure-valued evolution equations on a bounded domainhttps://zbmath.org/1472.354212021-11-25T18:46:10.358925Z"Evers, Joep H. M."https://zbmath.org/authors/?q=ai:evers.joep-h-mSummary: We study the connection between mild and weak solutions for a class of measure-valued evolution equations on the bounded domain \([0,1]\). Mass moves, driven by a velocity field that is either a function of the spatial variable only, \(v=v(x0\), or depends on the solution \(\mu\) itself: \(v=v[\mu](x)\). The flow is stopped at the boundaries of \([0,1]\), while mass is gated away by a certain right-hand side. In previous works \textit{J. H. M. Evers} et al. [J. Differ. Equations 259, No. 3, 1068--1097 (2015; Zbl 1315.35057); SIAM J. Math. Anal. 48, No. 3, 1929--1953 (2016; Zbl 1342.28004)], we showed the existence and uniqueness of appropriately defined mild solutions for \(v=v(x)\) and \(v=v[\mu](x0\), respectively. In the current paper we define weak solutions (by specifying the weak formulation and the space of test functions). The main result is that the aforementioned mild solutions are weak solutions, both when \(v=v(x)\) and when \(v=v[\mu](x)\).Direct linearization approach to discrete integrable systems associated with \(\mathbb{Z}_\mathcal{N}\) graded Lax pairshttps://zbmath.org/1472.370802021-11-25T18:46:10.358925Z"Fu, Wei"https://zbmath.org/authors/?q=ai:fu.weiSummary: \textit{A. P. Fordy} and \textit{P. Xenitidis} [J. Phys. A, Math. Theor. 50, No. 16, Article ID 165205, 30 p. (2017; Zbl 1367.37055)]
recently proposed a large class of discrete integrable systems which include a number of novel integrable difference equations, from the perspective of \(\mathbb{Z}_{\mathcal{N}}\) graded Lax pairs, without providing solutions. In this paper, we establish the link between the Fordy-Xenitidis (FX) discrete systems in coprime case and linear integral equations in certain form, which reveals solution structure of these equations. The bilinear form of the FX integrable difference equations is also presented together with the associated general tau function. Furthermore, the solution structure explains the connections between the FX novel models and the discrete Gel'fand-Dikii hierarchy.Solutions of system of Volterra integro-differential equations using optimal homotopy asymptotic methodhttps://zbmath.org/1472.450012021-11-25T18:46:10.358925Z"Agarwal, Praveen"https://zbmath.org/authors/?q=ai:agarwal.praveen"Akbar, Muhammad"https://zbmath.org/authors/?q=ai:akbar.muhammad-usman"Nawaz, Rashid"https://zbmath.org/authors/?q=ai:nawaz.rashid"Jleli, Mohamed"https://zbmath.org/authors/?q=ai:jleli.mohamedAuthors' abstract: In this paper, a powerful semianalytical method known as optimal homotopy asymptotic method (OHAM) has been formulated for the solution of system of Volterra integro-differential equations. The effectiveness and performance of the proposed technique are verified by different numerical problems in the literature, and the obtained results are compared with Sinc-collocation method. These results show the reliability and effectiveness of the proposed method. The proposed method does not require discretization like other numerical methods. Moreover, the convergence region can easily be controlled. The use of OHAM is simple and straightforward.\(L^q\)-solvability for an equation of viscoelasticity in power type materialhttps://zbmath.org/1472.450022021-11-25T18:46:10.358925Z"de Andrade, Bruno"https://zbmath.org/authors/?q=ai:de-andrade.bruno"Silva, Clessius"https://zbmath.org/authors/?q=ai:silva.clessius"Viana, Arlúcio"https://zbmath.org/authors/?q=ai:viana.arlucioThe authors study the existence, uniqueness, regularity, continuous dependence, unique continuation, a blow-up alternative for mild solutions, and global well-posedness of the nonlinear Volterra equation \[ u_t = \int_0^t dg_\alpha(s) \Delta u(t-s,x)- \nabla p +h - (u\cdot \nabla u),\quad \textrm{div}(u)=0, \] in \((0,\infty)\times \Omega\), where \(u=0\) on \((0,\infty)\times \partial \Omega\) and \(u(0,x)=u_0(x)\) in \(\Omega\). Here the kernel is taken to be \(g_\alpha (t)= t^{\alpha}/\Gamma(\alpha+1)\) with \(0\leq \alpha <1\) and a mild solution is a solution to the equation \[ u(t)= S_\alpha (tA)u_0 + \int_0^t S_\alpha((t-s)A)(F(u)(s)+Ph(s))\, ds, \] where \(P\) is the Leray projection on divergence free functions, \(F(u)= P(u\cdot \nabla)u\), \(A=P\Delta\) and \[S_\alpha(tA)= \frac 1{2\pi i} \int_{Ha}e^{\lambda t}\lambda^\alpha(\lambda^{\alpha+1}I+A)^{-1}\, d\lambda,\] where \(t>0\) and \(Ha\) is a suitable path.
The existence results show that the mild solutions have more spatial regularity in terms of estimates on norms in fractional power spaces when \(\alpha\) is closer to \(0\), the case of the Navier-Stokes equations. The linear estimates needed are stated in an abstract setting for sectorial operators which makes it possible to restate the results for some other equations as well.Oscillatory behavior of solutions of certain integrodynamic equations of second order on time scaleshttps://zbmath.org/1472.450032021-11-25T18:46:10.358925Z"Grace, Said R."https://zbmath.org/authors/?q=ai:grace.said-r"El-Beltagy, Mohamed A."https://zbmath.org/authors/?q=ai:el-beltagy.mohamed-aSummary: This paper deals with the oscillatory behavior of forced second-order integrodynamic equations on time scales. The results are new for the continuous and discrete cases and can be applied to Volterra integral equation on time scale. We also provide a numerical example in the continuous case to illustrate the results.On solutions of a nonlinear Erdélyi-Kober integral equationhttps://zbmath.org/1472.450042021-11-25T18:46:10.358925Z"Ashirbayev, Nurgali K."https://zbmath.org/authors/?q=ai:ashirbayev.nurgali-k"Banaś, Józef"https://zbmath.org/authors/?q=ai:banas.jozef"Bekmoldayeva, Raina"https://zbmath.org/authors/?q=ai:bekmoldayeva.rainaSummary: We conduct some investigations concerning the solvability of a nonlinear integral equation of Erdélyi-Kober type. To facilitate our study we will first consider a nonlinear integral equation of Volterra-Stieltjes type. Since the mentioned Erdélyi-Kober integral equation turns out to be a special case of that of Volterra-Stieltjes type, we can apply the obtained results to the Erdélyi-Kober integral equation. Examples illustrating the obtained results will be also included.Measurable solutions of implicit integral equations with discontinuous right-hand sidehttps://zbmath.org/1472.450052021-11-25T18:46:10.358925Z"Cubiotti, Paolo"https://zbmath.org/authors/?q=ai:cubiotti.paoloThe author investigates the integral equation \(f(t,\int_a^th(s,u(s))\,ds,u(t))=0\), \(a\leq t\leq b\), where \(u:[a,b]\to Y\), \(Y\) is a metric space, and \(h:[a,b]\times Y\to \mathbb R^n\), \(f:[a,b]\times \mathbb R^n\times Y\to \mathbb R\) are given functions. Under some assumptions, the existence of measurable solutions \(u\) of this equation is proved. This result extends the results by \textit{G. Anello} [J. Inequal. Appl. 2006, Article ID 71396, 8 p. (2006; Zbl 1094.45001)], focused on the case \(n=1\).Coupled systems of Hammerstein-type integral equations with sign-changing kernelshttps://zbmath.org/1472.450062021-11-25T18:46:10.358925Z"de Sousa, Robert"https://zbmath.org/authors/?q=ai:de-sousa.robert"Minhós, Feliz"https://zbmath.org/authors/?q=ai:minhos.feliz-manuelUsing the Guo-Krasnoselskii fixed point theorem in the context of expansive and compressive cones theory, the authors obtain an existence result for the solution of a generalized coupled system of integral equations of Hammerstein type. Such integral equations contain sign-changing kernels and nonlinearities depending on several derivatives of both variables in different orders. An application is considered. This refers to a coupled system of beam equations modeling the bending of the road-bed and the cable in suspension bridges.An inverse problem for the integro-differential Dirac system with partial information given on the convolution kernelhttps://zbmath.org/1472.450072021-11-25T18:46:10.358925Z"Bondarenko, Natalia Pavlovna"https://zbmath.org/authors/?q=ai:bondarenko.natalia-pThe author obtains uniqueness results for an inverse problem associated to a Dirac system of linear integro-differential equations with convolution kernel. For this purpose, the system of the direct equations is given and the partial inverse problem is formulated. The uniqueness theorem is proved in terms of the completeness of some system of vector functions associated with the given subspectrum. Moreover, a constructive algorithm for the solution of the specific partial inverse problem is provided. Necessary and sufficient conditions for the unique solvability of the inverse problem in this special case are obtained.Impulsive-integral inequalities for attracting and quasi-invariant sets of neutral stochastic integrodifferential equations with impulsive effectshttps://zbmath.org/1472.450082021-11-25T18:46:10.358925Z"Ramkumar, K."https://zbmath.org/authors/?q=ai:ramkumar.kasinathan"Anguraj, A."https://zbmath.org/authors/?q=ai:anguraj.annamalaiSummary: In this article, we investigate a class of neutral stochastic integrodifferential equations with impulsive effects. The results are obtained by using the new integral inequalities, the attracting and quasi-invariant sets combined with theories of resolvent operators. Moreover, exponential stability of the mild solution is established with sufficient conditions. An example is provided to illustrate the results of this work.Time memory effect in entropy decay of Ornstein-Uhlenbeck operatorshttps://zbmath.org/1472.450092021-11-25T18:46:10.358925Z"Agresti, Antonio"https://zbmath.org/authors/?q=ai:agresti.antonio"Loreti, Paola"https://zbmath.org/authors/?q=ai:loreti.paola"Sforza, Daniela"https://zbmath.org/authors/?q=ai:sforza.danielaSummary: We investigate the effect of memory terms on the entropy decay of the solutions to diffusion equations with Ornstein-Uhlenbeck operators. Our assumptions on the memory kernels include Caputo-Fabrizio operators and, more generally, the stretched exponential functions. We establish a sharp rate decay for the entropy. Examples and numerical simulations are also given to illustrate the results.On abstract Volterra equations in partially ordered spaces and their applicationshttps://zbmath.org/1472.450102021-11-25T18:46:10.358925Z"Burlakov, E. O."https://zbmath.org/authors/?q=ai:burlakov.evgenii"Zhukovskiy, E. S."https://zbmath.org/authors/?q=ai:zhukovskiy.evgeny-sSummary: We introduce the notion of abstract Volterra mapping acting in a partially ordered set. For an equation with such mapping, we define the notions of local, global, and maximally extended solutions and prove a theorem on its solvability. We apply this result to a discontinuous Uryson-type integral equation with respect to a spatiotemporal-dependent phase variable. In particular, such equations generalize a class of ``switching'' models of the electrical activity in the cerebral cortex.
For the entire collection see [Zbl 1467.34001].Shape holomorphy of the Calderón projector for the Laplacian in \(\mathbb{R}^2\)https://zbmath.org/1472.450112021-11-25T18:46:10.358925Z"Henríquez, Fernando"https://zbmath.org/authors/?q=ai:henriquez.fernando"Schwab, Christoph"https://zbmath.org/authors/?q=ai:schwab.christophThe authors establish the holomorphic dependence of the Calderón projector for the Laplace equation on a collection of sufficiently smooth Jordan curves in the Cartesian Euclidean plan. To be precise, they establish holomorphy of the domain-to-operator map associated to the Calderón projector.On inverse boundary value problem for a Fredholm integro-differential equation with degenerate kernel and spectral parameterhttps://zbmath.org/1472.450122021-11-25T18:46:10.358925Z"Yuldashev, T. K."https://zbmath.org/authors/?q=ai:yuldashev.tursun-kamaldinovichThe author considers a nonlocal inverse boundary value problem for a second-order Fredholm integro-differential equation with integral boundary condition at the final endpoint, spectral parameter and degenerate kernel. The unique solvability is proved.A Pólya-Szegő principle for general fractional Orlicz-Sobolev spaceshttps://zbmath.org/1472.460282021-11-25T18:46:10.358925Z"De Nápoli, Pablo"https://zbmath.org/authors/?q=ai:de-napoli.pablo-luis"Fernández Bonder, Julián"https://zbmath.org/authors/?q=ai:fernandez-bonder.julian"Salort, Ariel"https://zbmath.org/authors/?q=ai:salort.ariel-martinSummary: In this article, we prove modular and norm Pólya-Szegő inequalities in general fractional Orlicz-Sobolev spaces by using the polarization technique. We introduce a general framework which includes the different definitions of these spaces in the literature, and we establish some of its basic properties such as the density of smooth functions. As a corollary, we prove a Rayleigh-Faber-Krahn type inequality for Dirichlet eigenvalues under nonlocal nonstandard growth operators.Existence of coexistence states for systems of equations in ordered Banach spaceshttps://zbmath.org/1472.470432021-11-25T18:46:10.358925Z"El Khannoussi, Mohammed Said"https://zbmath.org/authors/?q=ai:khannoussi.mohammed-said-el"Zertiti, Abderrahim"https://zbmath.org/authors/?q=ai:zertiti.abderrahimSummary: In this paper we give some sufficient conditions for the existence of coexistence states to systems of the form
\[
\begin{aligned} x&=F_1(x,y), \\
y&=F_2(x,y), \end{aligned}
\]
where \(F_1\) and \(F_2\) satisfy some conditions.Leray-Schauder-type fixed point theorems in Banach algebras and application to quadratic integral equationshttps://zbmath.org/1472.470442021-11-25T18:46:10.358925Z"Khchine, Abdelmjid"https://zbmath.org/authors/?q=ai:khchine.abdelmjid"Maniar, Lahcen"https://zbmath.org/authors/?q=ai:maniar.lahcen"Taoudi, Mohamed-Aziz"https://zbmath.org/authors/?q=ai:taoudi.mohamed-azizThe authors of the paper under review study some fixed point results in Banach algebras relative to the weak topology under Leray-Schauder-type boundary conditions. They establish existence of fixed points results for nonlinear operators in Banach algebras relative to the weak topology. To illustrate the results obtained, the authors study the existence of continuous solutions of nonlinear quadratic integral equations.A solution of the system of integral equations in product spaces via concept of measures of noncompactnesshttps://zbmath.org/1472.470992021-11-25T18:46:10.358925Z"Nashine, Hemant Kumar"https://zbmath.org/authors/?q=ai:nashine.hemant-kumar"Arab, Reza"https://zbmath.org/authors/?q=ai:arab.reza"Ibrahim, Rabha W."https://zbmath.org/authors/?q=ai:ibrahim.rabha-waellSummary: In this chapter, we present the role of measures of noncompactness and related fixed point results to study the existence of solutions for the system of integral equations of the form
\begin{multline*}
x_i(t) = a_i(t)+f_i(t,x_1(t),x_2(t),\dots ,x_n(t))\\
+g_i(t,x_1(t),x_2(t),\dots ,x_n(t))\int_0^{\alpha (t)} k_i(t,s,x_1(s),x_2(s),\dots ,x_n(s)))\, ds,
\end{multline*}
for all \(t\in\mathbb{R}_+\), \(x_1,x_2,\dots,x_n\in E=BC(\mathbb{R}_+)\) and \(1\leq i\leq n\). We mainly focus on introducing new notion of \(\mu-(F,\varphi,\psi)\)-set contractive operator and establishing some new generalization of Darbo fixed point theorem and Krasnoselskii fixed point result associated with measures of noncompactness. Moreover, we deal with a system of fractional integral equations when \(k_i\) is defined in a fractal space.
For the entire collection see [Zbl 1470.47001].An optimal control problem of forward-backward stochastic Volterra integral equations with state constraintshttps://zbmath.org/1472.490402021-11-25T18:46:10.358925Z"Wei, Qingmeng"https://zbmath.org/authors/?q=ai:wei.qingmeng"Xiao, Xinling"https://zbmath.org/authors/?q=ai:xiao.xinlingSummary: This paper is devoted to the stochastic optimal control problems for systems governed by forward-backward stochastic Volterra integral equations (FBSVIEs, for short) with state constraints. Using Ekeland's variational principle, we obtain one kind of variational inequalities. Then, by dual method, we derive a stochastic maximum principle which gives the necessary conditions for the optimal controls.Viscosity solutions for controlled McKean-Vlasov jump-diffusionshttps://zbmath.org/1472.490542021-11-25T18:46:10.358925Z"Burzoni, Matteo"https://zbmath.org/authors/?q=ai:burzoni.matteo"Ignazio, Vincenzo"https://zbmath.org/authors/?q=ai:ignazio.vincenzo"Reppen, A. Max"https://zbmath.org/authors/?q=ai:reppen.a-max"Soner, H. M."https://zbmath.org/authors/?q=ai:soner.halil-meteThe paper deals with a class of nonlinear integro-differential equations on a subspace of all probability measures on the real line related to the optimal control of McKean-Vlasov jump-diffusions.
The authors investigated an intrinsic notion of viscosity solutions that does not rely on the lifting to a Hilbert space and proved a comparison theorem for these solutions.Reconstruction of low-rank aggregation kernels in univariate population balance equationshttps://zbmath.org/1472.650882021-11-25T18:46:10.358925Z"Ahrens, Robin"https://zbmath.org/authors/?q=ai:ahrens.robin"Le Borne, Sabine"https://zbmath.org/authors/?q=ai:le-borne.sabineSummary: The dynamics of particle processes can be described by population balance equations which are governed by phenomena including growth, nucleation, breakage and aggregation. Estimating the kinetics of the aggregation phenomena from measured density data constitutes an ill-conditioned inverse problem. In this work, we focus on the aggregation problem and present an approach to estimate the aggregation kernel in discrete, low rank form from given (measured or simulated) data. The low-rank assumption for the kernel allows the application of fast techniques for the evaluation of the aggregation integral \((\mathcal{O}(n\log n)\) instead of \(\mathcal{O}(n^2)\) where \(n\) denotes the number of unknowns in the discretization) and reduces the dimension of the optimization problem, allowing for efficient and accurate kernel reconstructions. We provide and compare two approaches which we will illustrate in numerical tests.A fast sparse spectral method for nonlinear integro-differential Volterra equations with general kernelshttps://zbmath.org/1472.650962021-11-25T18:46:10.358925Z"Gutleb, Timon S."https://zbmath.org/authors/?q=ai:gutleb.timon-sSummary: We present a sparse spectral method for nonlinear integro-differential Volterra equations based on the Volterra operator's banded sparsity structure when acting on specific Jacobi polynomial bases. The method is not restricted to convolution-type kernels of the form \(K(x, y)=K(x-y)\) but instead works for general kernels at competitive speeds and with exponential convergence. We provide various numerical experiments based on an open-source implementation for problems with and without known analytic solutions and comparisons with other methods.Uniform \(l^1\) behavior of the first-order interpolant quadrature scheme for some partial integro-differential equationshttps://zbmath.org/1472.651042021-11-25T18:46:10.358925Z"Xu, Da"https://zbmath.org/authors/?q=ai:xu.daIn this paper, the time discrete scheme based on the first-order backward difference method for the following integro-differential problem is studied \[ u_t(t)+\int\limits_0^t\beta(t-s)Au(s)ds=0, \quad t>0, \quad u(0)=u_0. \] Here \(A\) is a positive self-adjoint linear operator defined on a dense subspace \(D(A)\) of the real Hilbert space \(H\), \(u_0\in H\), and \(\beta(t)\) is weakly singular kernel at \(t=0\) such that \(\beta(t)=t^{\alpha-1}/\Gamma(\alpha)\), \(0<\alpha<1\). The paper is organized as follows. Section 1 is an introduction. In this section abovementioned problem is stated and the main theorem of this work is formulated. A review of the suitable papers is also given in this section. The memory term is approximated by the interpolating quadrature in this paper. In Section 2, using the Laplace transform technique is shown that this interpolating quadrature scheme has a convergence rate of \(O(\Delta t)\), where \(\Delta t\) denotes the time step. Special attention is that the uniform \(\ell^1\) convergence property of the discretization in time is given.Analysis of the Euler and trapezoidal discretization methods for the numerical solution of nonlinear functional Volterra integral equations of Urysohn typehttps://zbmath.org/1472.651622021-11-25T18:46:10.358925Z"Bazm, Sohrab"https://zbmath.org/authors/?q=ai:bazm.sohrab"Lima, Pedro"https://zbmath.org/authors/?q=ai:lima.pedro-miguel"Nemati, Somayeh"https://zbmath.org/authors/?q=ai:nemati.somayehSummary: In this paper, we investigate nonlinear functional Volterra-Urysohn integral equations, a class of nonlinear integral equations of Volterra type. The existence and uniqueness of the solution to the equation is proved by a technique based on the Picard iterative method. For the numerical approximation of the solution, the Euler and trapezoidal discretization methods are utilized which result in a system of nonlinear algebraic equations. Using a Gronwall inequality and its discrete version, first order of convergence to the exact solution for the Euler method and quadratic convergence for the trapezoidal method are proved. To prove the convergence of the trapezoidal method, a new Gronwall inequality is developed. Finally, numerical examples show the functionality of the methods.Recovery of high order accuracy in spectral collocation method for linear Volterra integral equations of the third-kind with non-smooth solutionshttps://zbmath.org/1472.651632021-11-25T18:46:10.358925Z"Ma, Xiaohua"https://zbmath.org/authors/?q=ai:ma.xiaohua"Huang, Chengming"https://zbmath.org/authors/?q=ai:huang.chengmingSummary: This paper aims to provide a rigorous analysis of exponential convergence of the Chebyshev collocation method for third kind linear Volterra integral equations. Different from Volterra integral equations of the second kind, the integral operator in third kind equations is noncompact under certain conditions, which brings special challenges to numerical analysis. The key idea of the proposed method is to adopt a smoothing transformation for the Chebyshev collocation method to circumvent the curse of singularity at the beginning of time. Therefore, the solution of the resulting equation will possess better regularity and then the numerical method can achieve the spectral accuracy. Moreover, in order to show the applicability and efficiency of the method, several examples with non-smooth solutions are illustrated.Numerical analysis of asymptotically convolution evolutionary integral equationshttps://zbmath.org/1472.651642021-11-25T18:46:10.358925Z"Messina, Eleonora"https://zbmath.org/authors/?q=ai:messina.eleonora"Vecchio, Antonia"https://zbmath.org/authors/?q=ai:vecchio.antoniaSummary: Asymptotically convolution Volterra equations are characterized by kernel functions which exponentially decay to convolution ones. Their importance in the applications motivates a numerical analysis of the asymptotic behavior of the solution. Here the quasi-convolution nature of the kernel is exploited in order to investigate the stability of \((\rho,\sigma)\) methods for general systems and in some particular cases.Moving least squares and spectral collocation method to approximate the solution of stochastic Volterra-Fredholm integral equationshttps://zbmath.org/1472.651652021-11-25T18:46:10.358925Z"Mirzaee, Farshid"https://zbmath.org/authors/?q=ai:mirzaee.farshid"Solhi, Erfan"https://zbmath.org/authors/?q=ai:solhi.erfan"Samadyar, Nasrin"https://zbmath.org/authors/?q=ai:samadyar.nasrinSummary: In this article, an idea based on moving least squares (MLS) and spectral collocation method is used to estimate the solution of nonlinear stochastic Volterra-Fredholm integral equations (NSVFIEs). The main advantage of the suggested approach is that in some parts where interpolation and integration are necessary, this approach does not require any meshes. Therefore, it is independent of the geometry of the domains, and this advantage helps us to solve the problems on irregular domains with relatively fewer computations. Another advantage of our proposed method is that with a small number of points and base functions, we were able to obtain the results with acceptable accuracy, and this is very attractive and practical. Applying the proposed method leads to the conversion of the problem into a system of algebraic equations. It is worth noting, some examples and error estimations have been provided to illustrate the accuracy and applicability of this technique. Also, we present a convergence analysis of the proposed method.Approximate solution of Abel integral equation in Daubechies wavelet basishttps://zbmath.org/1472.651662021-11-25T18:46:10.358925Z"Mouley, Jyotirmoy"https://zbmath.org/authors/?q=ai:mouley.jyotirmoy"Panja, M. M."https://zbmath.org/authors/?q=ai:panja.madan-mohan"Mandal, B. N."https://zbmath.org/authors/?q=ai:mandal.birendra-nathSummary: This paper presents a new computational method for solving Abel integral equation (both first kind and second kind). The numerical scheme is based on approximations in Daubechies wavelet basis. The properties of Daubechies scale functions are employed to reduce an integral equation to the solution of a system of algebraic equations. The error analysis associated with the method is given. The method is illustrated with some examples and the present method works nicely for low resolution.Mixed Fourier Legendre spectral Galerkin methods for two-dimensional Fredholm integral equations of the second kindhttps://zbmath.org/1472.651672021-11-25T18:46:10.358925Z"Panigrahi, Bijaya Laxmi"https://zbmath.org/authors/?q=ai:panigrahi.bijaya-laxmiSummary: In this article, the mixed Fourier Legendre spectral Galerkin (MFLSG) methods are considered to solve the two-dimensional Fredholm integral equations (\textsc{fie}s) on the Banach spaces with smooth kernel. The same methods are also considered to find the eigenvalues of the eigenvalue problems (\textsc{evp}s) associated with the two-dimensional \textsc{fie}s. Making use of these methods, we establish the error between the approximated solution as well as iterated approximate solution versus exact solution for two-dimensional \textsc{fie}s in both \(L^2\) and \(L^\infty\) norms. We also establish the error between approximated eigen-values, eigen-vectors and iterated eigen-vectors and exact eigen-elements by MFLSG methods in \(L^2\) and \(L^\infty\) norms. The numerical illustrations are introduced for the error of these methods.Smoothing transformation and collocation methods for third-kind linear Volterra integral equationshttps://zbmath.org/1472.651682021-11-25T18:46:10.358925Z"Xu, Xiaoli"https://zbmath.org/authors/?q=ai:xu.xiaoli"Xiao, Yu"https://zbmath.org/authors/?q=ai:xiao.yu"Song, Hui Ming"https://zbmath.org/authors/?q=ai:song.huimingSummary: In 2016, Sonia et al. first considered the convergence order for the third-kind linear Volterra integral equations (VIEs) based on the assumption that solutions are smooth. For the third-kind linear VIEs with nonsmooth solutions, we construct high-order numerical algorithms and discuss the convergence order. By introducing a new suitable independent variable, we obtain a transformed equation with a smooth exact solution. Then the solvability of the transformed equation is investigated on the basis of piecewise polynomial collocation methods. Meanwhile, the convergence order of the collocation solution is given. Furthermore, based on the inverse transformation, we get the convergence order of the original equation. Numerical simulations are finally presented to demonstrate the effectiveness of the theoretical results.Mapped spectral collocation methods for Volterra integral equations with noncompact kernelshttps://zbmath.org/1472.651692021-11-25T18:46:10.358925Z"Yang, Yin"https://zbmath.org/authors/?q=ai:yang.yin"Tang, Zhuyan"https://zbmath.org/authors/?q=ai:tang.zhuyanSummary: This paper is devoted to solve weakly singular Volterra integral equations with noncompact kernels, which differ from the well-known case of Abel-type equations. We consider using the mapped Laguerre spectral method to deal with this type of equations. The construction and analysis of log orthogonal functions collocation method are presented in this paper and some numerical examples are included to show the efficiency of the proposed method.Annular and circular rigid inclusions planted into a penny-shaped crack and factorization of triangular matriceshttps://zbmath.org/1472.741872021-11-25T18:46:10.358925Z"Antipov, Y. A."https://zbmath.org/authors/?q=ai:antipov.yuri-a"Mkhitaryan, S. M."https://zbmath.org/authors/?q=ai:mkhitaryan.s-mSummary: Analytical solutions to two axisymmetric problems of a penny-shaped crack when an annulus-shaped (model 1) or a disc-shaped (model 2) rigid inclusion of arbitrary profile are embedded into the crack are derived. The problems are governed by integral equations with the Weber-Sonine kernel on two segments. By the Mellin convolution theorem, the integral equations associated with models 1 and 2 reduce to vector Riemann-Hilbert problems with \(3 \times 3\) and \(2 \times 2\) triangular matrix coefficients whose entries consist of meromorphic and plus or minus infinite indices exponential functions. Canonical matrices of factorization are derived and the partial indices are computed. Exact representation formulae for the normal stress, the stress intensity factors (SIFs) at the crack and inclusion edges, and the normal displacement are obtained and the results of numerical tests are reported. In addition, simple asymptotic formulae for the SIFs are derived.Erratum to: ``Static and dynamic Green's functions in peridynamics''https://zbmath.org/1472.820092021-11-25T18:46:10.358925Z"Wang, Linjuan"https://zbmath.org/authors/?q=ai:wang.linjuan"Xu, Jifeng"https://zbmath.org/authors/?q=ai:xu.jifeng"Wang, Jianxiang"https://zbmath.org/authors/?q=ai:wang.jianxiangErratum to the authors' paper [ibid. 126, No. 1, 95--125 (2017; Zbl 1366.82013)].