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In this work, we establish the linear convergence estimate for the gradient descent involving the delay τ ∈ ℕ when the cost function is μ -strongly convex and L -smooth. This result improves upon the well-known estimates in [Y. Arjevani, O. Shamir and N. Srebro, A tight convergence analysis for stochastic gradient descent with delayed updates, Proc. Mach. Learn. Res. 117 (2020) 111–132; S. U. Stich and S. P. Karimireddy, The error-feedback framework: Better rates for SGD with delayed gradients and compressed updates, J. Mach. Learn. Res. 21(1) (2020) 9613–9648] in the sense that it is non-ergodic and is still established in spite of weaker constraint of cost function. Also, the range of learning rate η can be extended from η ≤ 1 / (1 0 L τ) to η ≤ 1 / (4 L τ) for τ = 1 and η ≤ 3 / (1 0 L τ) for τ ≥ 2 , where L > 0 is the Lipschitz continuity constant of the gradient of cost function. In a further research, we show the linear convergence of cost function under the Polyak–Łojasiewicz (PL) condition, for which the available choice of learning rate is further improved as η ≤ 9 / (1 0 L τ) for the large delay τ. The framework of the proof for this result is also extended to the stochastic gradient descent with time-varying delay under the PL condition. Finally, some numerical experiments are provided in order to confirm the reliability of the analyzed results. [ABSTRACT FROM AUTHOR]

We fill a gap in the proofs in the previous works (Wu X, Mu, X. Int J. Robust Nonlin Control. 2020;30:3753Ű3772; Zhang Z, Lunze J, Wang L. Int J Control. 2020;93:1005‐1014; Zhang Z, Wang L. J Robust Nonlin Control. 2018;28:4175Ű4187; Dai, M‐Z, Zhang C, Leung H, Dong P, Li B. IEEE Trans Syst, Man, Cybern: Syst. doi:10.1109/TSMC.2021.3119670) for the consensus using the integral‐based event‐triggered controls. More precisely, it was inferred for a Lyapunov function V:[0,∞)→ℝ+$$ V:\left[0,\infty \right)\to {\mathbb{R}}_{+} $$ that V˙(t)$$ \dot{V}(t) $$ is uniformly bounded by showing that V(t)$$ V(t) $$ is uniformly bounded for t≥0$$ t\ge 0 $$. However, this argument may fail without further information while the boundedness of V˙(t)$$ \dot{V}(t) $$ is crucially used for applying Barbalat's lemma. The consequence of Barbalat's lemma is that limt→∞V(t)$$ {\lim}_{t\to \infty }V(t) $$ which corresponds to the desired consensus result. To overcome this gap, Ma and Zhao (Inform Sci. 2018;457‐458:208‐221) put an extra condition about the boundedness of measurement error functions inside the proposed integral‐based event‐triggering protocol. In this article, we propose a new integral‐based event‐triggering protocol for bipartite consensus problems of the multi‐agent systems whose dynamics are described by general linear systems without adding the uniform boundedness of measurement error functions as (Ma Y, Zhao J. Inform Sci. 2018;457‐458:208‐221) did. Via our new integral‐based integral control strategy, we prove that the system achieves the bipartite consensus in asymptotic regime, and provide a complete solution of the freeness of both chattering and genuinely Zeno behaviors. Numerical results are provided supporting the effectiveness of the proposed controller. [ABSTRACT FROM AUTHOR]

In this article, we analyze an operator splitting scheme of the nonlinear heat equation in Ω⊂ℝd$$ \Omega \subset {\mathbb{R}}^d $$ (d≥1$$ d\ge 1 $$): ∂tu=Δu+λ|u|p−1uinΩ×(0,∞),u=0in∂Ω×(0,∞),u(x,0)=ϕ(x)inΩ,$$ \left\{\begin{array}{cc}{\partial}_tu=\Delta u+\lambda {\left|u\right|}^{p-1}u\kern-1em & \mathrm{in}\kern.5em \Omega \times \left(0,\infty \right),\\ {}u=0\kern1em \mathrm{in}\kern.5em \mathrm{\partial \Omega}\times \left(0,\infty \right),\kern1em u\left(\mathbf{x},0\right)=\phi \left(\mathbf{x}\right)\kern0em & \mathrm{in}\kern.5em \Omega, \end{array}\right. $$ where λ∈{−1,1}$$ \lambda \in \left\{-1,1\right\} $$ and ϕ∈W1,q(Ω)∩L∞(Ω)$$ \phi \in {W}^{1,q}\left(\Omega \right)\cap {L}^{\infty}\left(\Omega \right) $$ with 2≤p<∞$$ 2\le p<\infty $$ and d(p−1)/20$$ \tau >0 $$. Finally, we give some numerical examples to confirm the reliability of the analyzed result [ABSTRACT FROM AUTHOR]

In this paper, we establish a convergence result for the operator splitting scheme Zτ introduced by Ignat [12], with initial data in H1, for the nonlinear Schrödinger equation: ∂tu = iΔu + iλ|u|Pu, u(x,0) = φ(x), where p > 0, λ ∈ {−1,1}, and (x,t) ∈ Rd × [0, ∞). We prove the L2 convergence of order O(τ1/2) for the scheme with initial data in the space H1(Rd) for the energy-subcritical range of p. [ABSTRACT FROM AUTHOR]

In this paper, we investigate existence and non-existence of a nontrivial solution to the pseudo-relativistic nonlinear Schrödinger equation $$\left(\sqrt{-c^2\Delta + m^2 c^4}-mc^2\right) u + \mu u = \vert u \vert^{p-1}u\quad {\rm in}~{\open R}^n~(n \ges 2) $$ involving an H1/2-critical/supercritical power-type nonlinearity, that is, p ⩾ ((n + 1)/(n − 1)). We prove that in the non-relativistic regime, there exists a nontrivial solution provided that the nonlinearity is H1/2-critical/supercritical but it is H1-subcritical. On the other hand, we also show that there is no nontrivial bounded solution either (i) if the nonlinearity is H1/2-critical/supercritical in the ultra-relativistic regime or (ii) if the nonlinearity is H1-critical/supercritical in all cases. [ABSTRACT FROM AUTHOR]

We present an optimal a priori error estimates of the elliptic problems with Dirac sources away from the singular point using discontinuous and enriched Galerkin finite element methods. It is widely shown that the finite element solutions for elliptic problems with Dirac source terms converge sub-optimally in classical norms on uniform meshes. However, here we employ inductive estimates and L 2 norm to obtain the optimal order by excluding the small ball regions with the singularities for both two and three dimensional domains. Numerical examples are presented to substantiate our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider nonlocal Schrödinger equations with certain potentials V ∈ RHq (q > n/2s > 1 and 0 < s < 1) of the form LKu + Vu = f in ℝn where LK is an integro-differential operator. We denote the solution of the above equation by SVf := u, which is called the inverse of the nonlocal Schrödinger operator LK + V with potential V; that is, SV = (LK + V)-1. Then we obtain an improved version of the weak Harnack inequality of nonnegative weak subsolutions of the nonlocal equation LKu + Vu = 0 in Ω, u = g in ℝn \ Ω, (1) where g ∈ Hs(ℝn) and Ω is a bounded open domain in ℝn with Lipschitz boundary, and also get an improved decay of a fundamental solution eV for LK + V. Moreover, we obtain Lp and Lp - Lq mapping properties of the inverse SV of the nonlocal Schrödinger operator LK + V. [ABSTRACT FROM AUTHOR]

In this paper, we prove the Malgrange-Ehrenpreis theorem for nonlocal Schrödinger operators LK+V with nonnegative potentials V∈Lqloc(Rn) for q>n/2s with 02s; that is to say, we obtain the existence of a fundamental solution eV for LK+V satisfying (LK+V)eV=δ0 in Rn in the distribution sense, where δ0 denotes the Dirac delta mass at the origin. In addition, we obtain a decay of the fundamental solution eV. [ABSTRACT FROM AUTHOR]

We study the Dirichlet problem for the semilinear equations involving the pseudo-relativistic operator on a bounded domain, ( − Δ + m 2 − m ) u = | u | p − 1 u in Ω , with the Dirichlet boundary condition u = 0 on ∂ Ω . Here, p ∈ ( 1 , ∞ ) and the operator ( − Δ + m 2 − m ) is defined in terms of spectral decomposition. In this paper, we investigate existence and nonexistence of a nontrivial solution, depending on the choice of p , m and Ω . Precisely, we show that ( i ) if p is not H 1 subcritical ( p ≥ n + 2 n − 2 ) and Ω is star-shaped, the equation has no nontrivial solution for all m > 0 ; ( i i ) if p is not H 1 ∕ 2 supercritical ( 1 < p ≤ n + 1 n − 1 ), then there exists a least energy solution for all m > 0 and any bounded domain Ω ; ( i i i ) finally, in the intermediate range ( n + 1 n − 1 < p < n + 2 n − 2 ), the problem has a nontrivial solution, provided that m is sufficiently large and the problem (0.1) − Δ u = | u | p − 1 u in Ω , u = 0 on ∂ Ω admits a non-degenerate nontrivial solution, for example, when Ω is a ball or an annulus. [ABSTRACT FROM AUTHOR]

In this paper we obtain a refined Lp bound for maximal functions of the multiplier operators on stratified groups and maximal functions of the multi-parameter multipliers on product spaces of stratified groups. As an application we find a refined Lp bound for maximal functions of joint spectral multipliers on Heisenberg group. [ABSTRACT FROM AUTHOR]

In this paper we establish sharp L2 and Hp boundedness results for strongly singular operators and oscillating operators on Heisenberg groups. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study the asymptotic behavior of least energy solutions and the existence of multiple bubbling solutions of nonlinear elliptic equations involving the fractional Laplacians and the critical exponents. This work can be seen as a nonlocal analog of the results of Han (1991) [24] and Rey (1990) [35]. [Copyright &y& Elsevier]

In this paper, we study the pseudo-relativistic Lane-Emden equation (− Δ + m 2 − m) u = | u | p − 1 u in Ω , with the Dirichlet boundary condition u = 0 on ∂Ω. Here, Ω ⊂ R n is bounded and the operator (− Δ + m 2 − m) is defined in terms of spectral decomposition. In the authors' previous paper [8] , it is shown the existence and nonexistence results for the least energy solutions, depending on the range of parameter p. This paper is devoted to study the asymptotic behavior of nontrivial solutions when the mass parameter m in the equation either goes to 0 + or diverges to infinity. [ABSTRACT FROM AUTHOR]