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We study the following fourth order quasilinear Schrodinger equations { Δ 2 u − Δ u + V ( x ) u − 1 2 u Δ ( u 2 ) = f ( u ) , u ∈ H 2 ( R N ) , where Δ 2 u = Δ ( Δ u ) and 3 ≤ N ≤ 6 . For some suitable assumptions on V ∈ C 1 ( R N ) and f ∈ C ( R ) , by developing a new Brezis-Lieb type convergence result, we prove the existence of Nehari-Pohozaev type ground state solutions. Our main results generalize and improve the existing ones, such as Chen et al. (2014) [10] , Liu and Zhao (2019) [26] , and some other related literature.

We consider a hemivariational inequality of elliptic type (HVI, for short) in a reflexive Banach space, prove its solvability and the compactness of its set of solutions. To this end we employ a surjectivity theorem for multivalued mappings that we use for the sum of a maximal monotone operator and a bounded pseudomonotone operator. Next, we introduce the concepts of strongly and weakly well-posedness in the generalized sense for the HVI and provide two characterizations for the strongly well-posedness, under different assumptions on the data. These characterizations are formulated in terms of the metric properties of the approximating sets. We also provide sufficient conditions which guarantee the weakly and strongly well-posedness in the generalized sense of the HVI. Finally, we consider two perturbations of the HVI for which we obtain convergence results in the sense of Kuratowski.

In the setting of hyperbolic spaces, we show that the convergence of Browder-type sequences and Halpern iterations respectively entail the convergence of their viscosity version with a Rakotch map. We also show that the convergence of a hybrid viscosity version of the Krasnoselskii-Mann iteration follows from the convergence of the Browder type sequence. Our results follow from proof-theoretic techniques (proof mining). From an analysis of theorems due to T. Suzuki, we extract a transformation of rates for the original Browder type and Halpern iterations into rates for the corresponding viscosity versions. We show that these transformations can be applied to earlier quantitative studies of these iterations. From an analysis of a theorem due to H.-K. Xu, N. Altwaijry and S. Chebbi, we obtain similar results. Finally, in uniformly convex Banach spaces we study a strong notion of accretive operator due to Brezis and Sibony and extract an uniform modulus of uniqueness for the property of being a zero point. In this context, we show that it is possible to obtain Cauchy rates for the Browder type and the Halpern iterations (and hence also for their viscosity versions).

We consider the convergence of solutions and cost functional in some optimal control problems arising in the study of the adsorption chemical phenomenon in which some microscopic reactant particles are placed over an internal manifold γ of the chemical reactor Ω. The chemical reaction is given by some Robin-type boundary condition on the boundary of the periodic set of particles. We consider the special case in which there is a critical relation between the coefficient of the reaction, the size of the particles and the dimension of the space. This gives rise to a “strange term”, which is not occurring for other scales, and thus the limit cost functional must be suitably defined. In a last section, we use this type of technique to prove a similar “energy convergence” result (improving the H 0 1 ( Ω ) -weak convergence) for the problem without control for the critical scale case.

In this present article we set forth with the new notion of I -αβ-statistical convergence which becomes more generalized version of I -statistical convergence. Successively to compare with the following important results of Balcerzak et al. (2020) [6] (i): Let I be an ideal such that I ⊆ Z = { A ⊆ N : lim n → ∞ ⁡ | A ∩ [ 1 , n ] | n = 0 } . Then I -statistical convergence coincides with statistical convergence, (ii): I -statistical convergence coincides with statistical convergence if I = I 1 n = { S ⊆ N : ∑ n ∈ S 1 n ∞ } , (iii): Let I be a maximal ideal. Then I -statistical convergence does not coincide with statistical convergence, we produce significant results that elucidate incongruity between I -αβ-statistical convergence and I -statistical convergence. Also, following the line of works of Aytar (2008) [4] , Savas et al. (2015) [40] and Ghosal et al. (2020) [23] , we propose two new notions, namely rough weighted I -αβ-statistical convergence and weighted I -αβ-statistical cluster points set over locally solid Riesz spaces. The core factor “degree of roughness” had been conventionally treated as a non-negative real number where as we invent the idea of neighborhood approximation. Finally, topological properties and relationships of the above mentioned notions are investigated.

Our goal is to study fractional powers of a cascade system of partial differential equations. We explicitly calculate the fractional powers of linear operators associated to this type of system and we discuss local solvability of the fractional equation with subcritical nonlinearity. As an example, a cascade system of Schrodinger equation is analyzed. A connection between the fractional system and the original system is established and we prove the convergence of the linear semigroups obtained by the fractional power operator to the original linear semigroup, as the power α approaches 1.

In this study, we investigate the convergence behavior as α → 2 of the solutions to stochastic differential equations (SDEs) with Lipschitz and Holder drifts, and driven by α-stable Levy processes. First, we prove that the solution of an SDE driven by an α-stable Levy process converges weakly to that of an SDE forced by Brownian motion in the Skorokhod space. We mainly employ Aldous' criterion and martingale characterization. Then, by using an abstract result given by Gyongy and Krylov, we prove that the solution of an SDE driven by the subordinated Brownian motion via an α / 2 -stable subordinator strongly converges to that of an SDE forced by Brownian motion. Based on uniform estimates of the mean exit times and related probabilistic representations, we also present an α-dependent result for fractional diffusion equations. Finally, given α ⁎ in ( 1 , 2 ) , we discuss the α-continuity of the solution to an SDE driven by an α-stable Levy process at α ⁎ .

We prove the continuity of logarithmic capacity under Hausdorff convergence of uniformly perfect planar sets. The continuity holds when the Hausdorff distance to the limit set tends to zero at sufficiently rapid rate, compared to the decay of the parameters involved in the uniformly perfect condition. The continuity may fail otherwise., Comment: 13 pages

In this article we apply proper splittings of matrices to develop an iterative process to approximate solutions of matrix equations of the form T X = W . Moreover, by using the partial order induced by positive semidefinite matrices, we obtain equivalent conditions to the convergence of this process. We also include some speed comparison results of the convergence of this method. In addition, for all matrix T we propose a proper splitting based on the polar decomposition of T.

Although fractional powers of non-negative operators have received much attention in recent years, there is still little known about their behavior if real-valued exponents are greater than one. In this article, we define and study the discrete fractional Laplace operator of arbitrary real-valued positive order. A series representation of the discrete fractional Laplace operator for positive non-integer powers is developed. Its convergence to a series representation of a known case of positive integer powers is proven as the power tends to the integer value. Furthermore, we show that the new representation for arbitrary real-valued positive powers of the discrete Laplace operator is consistent with existing theoretical results.

Maximal estimates for Schrodinger means and convergence almost everywhere of sequences of Schrodinger means are studied.

We investigate the asymptotic behavior of solutions to a nonhomogeneous second order difference inclusion of monotone type and prove new results on the existence and approximation of zeroes of the operator. In particular, we introduce new weighted averages of the solutions and prove their convergence to a zero of the operator under different control conditions on the parameters.

In this paper, the state reconstruction of the wave equation with general viscosity and non-collocated observation and control are given from both the theoretical aspect and the numerical aspect. The forward-backward observers-based algorithm is utilized to solve this problem. The formula for calculating the initial value is derived. Moreover, the iterative sequence is also built for any given guess value and it is showed that it strongly converges to initial value. However, because the exponential stabilities of the error systems between the original systems and the forward∖backward observers play important roles in the involved algorithm, the exponential stability of some related system is firstly discussed. Furthermore, combining the theory of port-Hamiltonian system and the finite difference, the semi-discretization scheme of the finite difference with order reduction is given and the uniform exponential stability of the semi-discretization system is verified by the method paralleling to the continuous system. Finally, the convergence analysis of the finite difference scheme is given and the convergence of the solution of the mixed finite element scheme induced by the finite difference scheme with order reduction to the solution of the continuous counterpart is also presented.

We consider the 3D magnetohydrodynamic-alpha model driven by a multiplicative noise. The model arises in the modelling of turbulent flows in magneto fluids. We prove the existence and uniqueness of the probabilistic strong solution. We also propose and analyze a numerical scheme for the approximation of the unique strong solution. We prove the convergence in mean square of the scheme, which is a linear evolution equation with additive noise.

For scalar conservation laws posed on bounded domains in R d , the convergence of quasilinear parabolic viscous approximations to the entropy solution in the sense of Bardos-Leroux-Nedelec is established.