A modified Kacanov iteration scheme with application to quasilinear diffusion models

The classical Kacanov scheme for the solution of nonlinear variational problems can be interpreted as a fixed point iteration method that updates a given approximation by solving a linear problem in each step. Based on this observation, we introduce a modified Kacanov method, which allows for (adaptive) damping, and, thereby, to derive a new convergence analysis under more general assumptions and for a wider range of applications. For instance, in the specific context of quasilinear diffusion models, our new approach does no longer require a standard monotonicity condition on the nonlinear diffusion coefficient to hold. Moreover, we propose two different adaptive strategies for the practical selection of the damping parameters involved.

The Causal Interpretation of Two-Stage Least Squares with Multiple Instrumental Variables

Empirical researchers often combine multiple instrumental variables (IVs) for a single treatment using two-stage least squares (2SLS). When treatment effects are heterogeneous, a common justification for including multiple IVs is that the 2SLS estimand can be given a causal interpretation as a positively weighted average of local average treatment effects (LATEs). This justification requires the well-known monotonicity condition. However, we show that with more than one instrument, this condition can only be satisfied if choice behavior is effectively homogeneous. Based on this finding, we consider the use of multiple IVs under a weaker, partial monotonicity condition. We characterize empirically verifiable sufficient and necessary conditions for the 2SLS estimand to be a positively weighted average of LATEs under partial monotonicity. We apply these results to an empirical analysis of the returns to college with multiple instruments. We show that the standard monotonicity condition is at odds with the data. Nevertheless, our empirical checks reveal that the 2SLS estimate retains a causal interpretation as a positively weighted average of the effects of college attendance among complier groups. (JEL C26, I23, I26, J24, J31, R23)

AN EFFICIENT HYBRID DERIVATIVE-FREE PROJECTION ALGORITHM FOR CONSTRAINT NONLINEAR EQUATIONS

In this paper, by combining the Solodov and Svaiter projection technique with the conjugate gradient method for unconstrained optimization proposed by Mohamed et al. (2020), we develop a derivative-free conjugate gradient method to solve nonlinear equations with convex constraints. The proposed method involves a spectral parameter which satisfies the sufficient descent condition. The global convergence is proved under the assumption that the underlying mapping is Lipschitz continuous and satisfies a weaker monotonicity condition. Numerical experiment shows that the proposed method is efficient.

Ground state solutions to a class of critical Schrödinger problem

We consider the following critical nonlocal Schrödinger problem with general nonlinearities − ε 2 a + ε b ∫ R 3 | ∇ u | 2 Δ u + V ( x ) u = f ( u ) + u 5 , x ∈ R 3 , u ∈ H 1 ( R 3 ) , $$\begin{array}{} \displaystyle \left\{\begin{array}{} &-\left(\varepsilon^{2}a+\varepsilon b\displaystyle\int\limits_{\mathbb{R}^{3}}|\nabla u|^{2}\right){\it\Delta} u+V(x)u=f(u)+u^{5}, &x \in \mathbb{R}^{3},\\ &u\in H^{1}(\mathbb{R}^{3}), \end{array}\right. \end{array}$$ (SKε ) and study the existence of semiclassical ground state solutions of Nehari-Pohožaev type to (SK ε ), where f(u) may behave like |u| q–2 u for q ∈ (2, 4] which is seldom studied. With some decay assumption on V, we establish an existence result which improves some exiting works which only handle q ∈ (4, 6). With some monotonicity condition on V, we also get a ground state solution v̄ ε and analysis its concentrating behaviour around global minimum x ε of V as ε → 0. Our results extend some related works.

Existence and Uniqueness of Weak Solutions for Novel Anisotropic Nonlinear Diffusion Equations Related to Image Analysis

This paper establishes the existence and uniqueness of weak solutions for the initial-boundary value problem of anisotropic nonlinear diffusion partial differential equations related to image processing and analysis. An implicit iterative method combined with a variational approach has been applied to construct approximate solutions for this problem. Then, under some a priori estimates and a monotonicity condition, the existence of unique weak solutions for this problem has been proven. This work has been complemented by a consistent and stable approximation scheme showing its great significance as an image restoration technique.

On the Existence of a Unique Solution for a Class of Fractional Differential Inclusions in a Hilbert Space

We obtained results on the existence and uniqueness of a mild solution for a fractional-order semi-linear differential inclusion in a Hilbert space whose right-hand side contains an unbounded linear monotone operator and a Carathéodory-type multivalued nonlinearity satisfying some monotonicity condition in the phase variables. We used the Yosida approximations of the linear part of the inclusion, the method of a priori estimates of solutions, and the topological degree method for condensing vector fields. As an example, we considered the existence and uniqueness of a solution to the Cauchy problem for a system governed by a perturbed subdifferential inclusion of a fractional diffusion type.

General time interval multidimensional BSDEs with generators satisfying a weak stochastic-monotonicity condition

<p style='text-indent:20px;'>This paper establishes an existence and uniqueness result for the adapted solution of a general time interval multidimensional backward stochastic differential equation (BSDE), where the generator <inline-formula> <tex-math id="M1">\begin{document}$ g $\end{document}</tex-math> </inline-formula> satisfies a weak stochastic-monotonicity condition and a general growth condition in the state variable <inline-formula> <tex-math id="M2">\begin{document}$ y $\end{document}</tex-math> </inline-formula>, and a stochastic-Lipschitz condition in the state variable <inline-formula> <tex-math id="M3">\begin{document}$ z $\end{document}</tex-math> </inline-formula>. This unifies and strengthens some known works. In order to prove this result, we develop some ideas and techniques employed in Xiao and Fan [<xref ref-type="bibr" rid="b25">25</xref>] and Liu et al. [<xref ref-type="bibr" rid="b15">15</xref>]. In particular, we put forward and prove a stochastic Gronwall-type inequality and a stochastic Bihari-type inequality, which generalize the classical ones and may be useful in other applications. The martingale representation theorem, Itô’s formula, and the BMO martingale tool are used to prove these two inequalities. </p>

Boundary layer solutions to singularly perturbed quasilinear systems

<p style='text-indent:20px;'>We consider weak boundary layer solutions to the singularly perturbed ODE systems of the type <inline-formula><tex-math id="M1">\begin{document}$ \varepsilon^2\left(A(x, u(x), \varepsilon)u'(x)\right)' = f(x, u(x), \varepsilon) $\end{document}</tex-math></inline-formula>. The new features are that we do not consider one scalar equation, but systems, that the systems are allowed to be quasilinear, and that the systems are spatially non-smooth. Although the results about existence, asymptotic behavior, local uniqueness and stability of boundary layer solutions are similar to those known for semilinear, scalar and smooth problems, there are at least three essential differences. First, the asymptotic convergence rates valid for smooth problems are not true anymore, in general, in the non-smooth case. Second, a specific local uniqueness condition from the scalar case is not sufficient anymore in the vectorial case. And third, the monotonicity condition, which is sufficient for stability of boundary layers in the scalar case, must be adjusted to the vectorial case.</p>

Ground state sign-changing solutions and infinitely many solutions for fractional logarithmic Schrödinger equations in bounded domains

We consider a class of fractional logarithmic Schrödinger equation in bounded domains. First, by means of the constraint variational method, quantitative deformation lemma and some new inequalities, the positive ground state solutions and ground state sign-changing solutions are obtained. These inequalities are derived from the special properties of fractional logarithmic equations and are critical for us to obtain our main results. Moreover, we show that the energy of any sign-changing solution is strictly larger than twice the ground state energy. Finally, we obtain that the equation has infinitely many nontrivial solutions. Our result complements the existing ones to fractional Schrödinger problems when the nonlinearity is sign-changing and satisfies neither the monotonicity condition nor Ambrosetti-Rabinowitz condition.