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We continue our study of a new boundedness condition for affine permutations, motivated by the fruitful concept of periodic boundary conditions in statistical physics. We focus on bounded affine permutations of size $N$ that avoid the monotone decreasing pattern of fixed size $m$. We prove that the number of such permutations is asymptotically equal to $(m-1)^{2N} N^{(m-2)/2}$ times an explicit constant as $N\to\infty$. For instance, the number of bounded affine permutations of size $N$ that avoid $321$ is asymptotically equal to $4^N (N/4\pi)^{1/2}$. We also prove a permuton-like result for the scaling limit of random permutations from this class, showing that the plot of a typical bounded affine permutation avoiding $m\cdots1$ looks like $m-1$ random lines of slope $1$ whose $y$ intercepts sum to $0$.

In this work, we develop a systematic framework for computing the resolvent of the sum of two or more monotone operators which only activates each operator in the sum individually. The key tool in the development of this framework is the notion of the “strengthening” of a set-valued operator, which can be viewed as a type of regularisation that preserves computational tractability. After deriving a number of iterative schemes through this framework, we demonstrate their application to best approximation problems, image denoising and elliptic PDEs. FJAA and RC were partially supported by the Ministry of Science, Innovation and Universities of Spain and the European Regional Development Fund (ERDF) of the European Commission, Grant PGC2018-097960-B-C22. MKT is supported in part by ARC grant DE200100063.

We compute the free energy and surface tension function for the five-vertex model, a model of non-intersecting monotone lattice paths on the grid in which each corner gets a positive weight. We give a variational principle for limit shapes in this setting, and show that the resulting Euler-Lagrange equation can be integrated, giving explicit limit shapes parameterized by analytic functions., 37 pages, 21 figures

We consider the propagation dynamics of a general heterogeneous reaction-diffusion system under a shifting environment. By developing the fixed-point theory for second order non-autonomous differential system and constructing appropriate upper and lower solutions, we show there exists a nondecreasing wave front with the speed consistent with the habitat shifting speed. We further show the uniqueness of forced waves by the sliding method and some analytical skills, and we obtain the global stability of forced waves by applying the dynamical systems approach. Moreover, we establish the spreading speed of the system by appealing to the abstract theory of monotone semiflows. Applications and numerical simulations are also given to illustrate the analytical results.

The paper presents two inertial viscosity-type extragradient algorithms for finding a common solution of the variational inequality problem involving a monotone and Lipschitz continuous operator and of the fixed point problem with a demicontractive mapping in real Hilbert spaces. Our algorithms use a simple step size rule which is generated by some calculations at each iteration. Two strong convergence theorems are obtained without the prior knowledge of the Lipschitz constant of the operator. The numerical behaviors of the proposed algorithms in some numerical experiments are reported and compared with previously known ones.

In this paper, we introduce two algorithms for finding a common solution of the monotone inclusion problem and the fixed point problem for a relatively nonexpansive mapping in reflexive Banach spaces. The weak convergence results for both algorithms are established without the prior knowledge of the Lipschitz constant of the mapping. An application to the variational inequality problem is considered. Finally, some numerical experiments of the proposed algorithms including comparisons with other algorithms are provided.

We investigate qualitative properties of solutions of special classes of convolution type nonlinear integral equations on the whole line. We study the asymptotic properties, continuity, and monotonicity of arbitrary nontrivial bounded solutions. Depending on the properties of the kernel of the equation, we find out whether there exist nontrivial bounded solutions with a finite limit at ±∞. Based on the obtained results, we establish uniqueness theorems for large classes of bounded functions. The results obtained are illustrated by examples from applications.

In this article, the notions of partially continuous mappings, partially bounded, partially compact and partially closed sets of an ordered E-metric space are defined. These notions are used to introduce partial E-measure of noncompactness and prove the fixed point results of monotone mappings and sum of two monotone mappings. In this way we generalize many results including the well known results of Schauder, Darbo and Krasnoselskii, in the settings of ordered E-metric and ordered Banach spaces. We also provide nontrivial examples and existence results for a class of integral equations to validate the significance of our theory and results.

We prove that if $$X^{*}$$ is strictly convex, a convex function $$f$$ is coercive and b-Lipschitzian iff there exist two convex function sequences $$\{f_{n}\}_{n=1}^{\infty}$$ and $$\{g_{n}\}_{n=1}^{\infty}$$ such that (1) $$f_{n}\leq f_{n+1}\leq f$$ and $$f\leq g_{n+1}\leq g_{n}$$ for all integers $$n \geq 1$$ ; (2) $$f_{n}$$ and $$g_{n}$$ are continuous and Gâteaux differentiable on $$X$$ ; (3) $$f_n \to f$$ and $${g_n \to f}$$ uniformly on $$X$$ ; (4) $$f_n$$ and $$g_n$$ are coercive and b-Lipschitzian. Moreover, we also prove that if $$X^{*}$$ is strictly convex, then a convex function f is Lipschitzian iff conditions (1)-(3) are true and there exists $$m>0$$ such that $$|f_{n}(x)-f_{n}(y)|\leq m\|x-y\|$$ and $$|g_{n}(x)-g_{n}(y)|\leq m\|x-y\|$$ whenever $$x, y \in X$$ and $$n\in N$$ .

It is well known that the algorithms with using a proximal operator can be not convergent for monotone variational inequality problems in the general case. Malitsky (Optim. Methods Softw. 33 (1) 140–164, ??) proposed a proximal extrapolated gradient algorithm ensuring convergence for the problems, where the constraints are a finite-dimensional vector space. Based on this proximal extrapolated gradient techniques, we propose a new subgradient proximal iteration method for solving monotone multivalued variational inequality problems with the closed convex constraint. At each iteration, two strongly convex subprograms are required to solve separately by using proximal operators. Then, the algorithm is convergent for monotone and Lipschitz continuous cost mapping. We also use the proposed algorithm to solve a jointly constrained Cournot-Nash equilibirum model. Some numerical experiment and comparison results for convex nonlinear programming confirm efficiency of the proposed modification.

Monotone inclusion problems are crucial to solve engineering problems and problems arising in different branches of science. In this paper, we propose a novel two-step inertial Douglas-Rachford algorithm to solve the monotone inclusion problem of the sum of two maximally monotone operators based on the normal S-iteration method (Sahu, D.R.: Applications of the S-iteration process to constrained minimization problems and split feasibility problems. Fixed Point Theory 12(1), 187–204 (2011)). We have studied the convergence behavior of the proposed algorithm. Based on the proposed method, we develop an inertial primal-dual algorithm to solve highly structured monotone inclusions containing the mixtures of linearly composed and parallel-sum type operators. Finally, we apply the proposed inertial primal-dual algorithm to solve a highly structured minimization problem. We also perform a numerical experiment to solve the generalized Heron problem and compare the performance of the proposed inertial primal-dual algorithm to already known algorithms.

In a recent paper, Bauschke et al. study $$\rho $$ ρ -comonotonicity as a generalized notion of monotonicity of set-valued operators A in Hilbert space and characterize this condition on A in terms of the averagedness of its resolvent $$J_A.$$ J A . In this note we show that this result makes it possible to adapt many proofs of properties of the proximal point algorithm PPA and its strongly convergent Halpern-type variant HPPA to this more general class of operators. This also applies to quantitative results on the rates of convergence or metastability (in the sense of T. Tao). E.g. using this approach we get a simple proof for the convergence of the PPA in the boundedly compact case for $$\rho $$ ρ -comonotone operators and obtain an effective rate of metastability. If A has a modulus of regularity w.r.t. $$zer\, A$$ z e r A we also get a rate of convergence to some zero of A even without any compactness assumption. We also study a Halpern-type variant HPPA of the PPA for $$\rho $$ ρ -comonotone operators, prove its strong convergence (without any compactness or regularity assumption) and give a rate of metastability.

Setting a proper minimum capital requirement is one of the most fundamental problems in the insurance industry. Ruin theory proposes a solution to this problem by identifying the minimum capital that a company needs to hold in order to stay solvent with a high probability. In this note we discuss the ruin theory risk consistency. More precisely we show that the ruin-consistent Value-at-Risk (VaR) is not continuous in probability, in $$L^{p},0\le p

The paper concerns with three iterative regularization methods for solving a variational inclusion problem of the sum of two operators, the one is maximally monotone and the another is monotone and Lipschitz continuous, in a Hilbert space. We first describe how to incorporate regularization terms in the methods of forward-backward types, and then establish the strong convergence of the resulting methods. With several new stepsize rules considered, the methods can work with or without knowing previously the Lipschitz constant of cost operator. Unlike known hybrid methods, the strong convergence of the proposed methods comes from the regularization technique. Several applications to signal recovery problems and optimal control problems together with numerical experiments are also presented in this paper. Our numerical results have illustrated the fast convergence and computational effectiveness of the new methods over known hybrid methods.

Monotone modal logics have emerged in several application areas such as computer science and social choice theory. Since many of the most studied selfextensional logics have a conjunction, in this paper we study some distributive extensions obtained from a semilattice based deductive system with monotonic modal operators, and we give them neighborhood and algebraic semantics. For each logic defined our main objective is to prove completeness with respect to its characteristic class of monotonic frames.

In this paper, we study global bifurcation diagrams and the exact multiplicity of positive solutions for Minkowski-curvature problem $$\begin{aligned} \left\{ \begin{array}{l} -\left( u^{\prime }/\sqrt{1-{u^{\prime }}^{2}}\right) ^{\prime }=\lambda \exp u,\text { in }\left( -L,L\right) , \\ u(-L)=u(L)=0, \end{array} \right. \end{aligned}$$ where $$\lambda >0$$ is a bifurcation parameter and $$L>0$$ is an evolution parameter. It can be viewed as a variant of the one-dimensional Liouville–Bratu–Gelfand problem. We prove that there exists $$L_{0}>0$$ such that the bifurcation curve $$S_{L}$$ is S-shaped for $$L>L_{0}$$ and is monotone increasing for $$0

We provide a novel expression of the scale function for a Lévy process with negative phase-type jumps. It is in terms of a certain transition rate matrix which is explicit up to a single positive number. A monotone iterative scheme for the calculation of the latter is presented and it is shown that the error decays exponentially fast. Our numerical examples suggest that this algorithm allows us to employ phase-type distributions with hundreds of phases, which is problematic when using the known formula for the scale function in terms of roots. Extensions to other distributions, such as matrix-exponential and infinite-dimensional phase-type, can be anticipated.

In this paper, we investigate the asymptotic dynamics of a time-periodic parasitic–mutualistic model of mistletoes and birds in heterogeneous environment with the especial concerns over the spreading–vanishing scenarios, in which the Stefan class free boundary is introduced as the spreading frontier. By defining the ecological reproduction number and generalizing it as the spatial-temporal risk index, a considerably universal spreading–vanishing dichotomy and the sharp criteria are first established in birds world in the absence of mistletoes, and some estimates of the asymptotic spreading speed of the free boundary provided that spreading occurs are also obtained. Furthermore, the comprehensive considerations containing the spreading frontiers, asymptotic profiles and estimates of the asymptotic spreading speed are exhibited in mistletoes-birds world by the monotone iteration technique with the proper upper and lower solutions. The results suggest that even for the spreading case, the mistletoes population will eventually persist in long term provided that its own risk index is larger than 1, otherwise it may be eradicated.

In this paper, we provide a new application of the Douglas–Rachford splitting method for finding a zero of the sum of two maximal monotone operators where one of them depends implicitly on the state variable. Our proposed algorithms are much simpler with better rate of convergence than existing results and can be implemented under general conditions. Applications to generalized Nash games and quasivariational inequalities are provided. Numerical experiments are also given to illustrate the convergence rate of the proposed algorithms.

The aim of this paper is to present a monotone numerical method on uniform mesh for solving singularly perturbed three-point reaction-diffusion boundary value problems. Firstly, properties of the exact solution are analyzed. Difference schemes are established by the method of the integral identities with the appropriate quadrature rules with remainder terms in integral form. It is then proved that the method is second-order uniformly convergent with respect to singular perturbation parameter, in discrete maximum norm. Finally, one numerical example is presented to demonstrate the efficiency of the proposed method.

Many iterative schemes have already been developed to solve the equilibrium problems, one of which is the most efficient two-step extragradient method. The objective of this research is to propose two new iterative methods with inertial effect to solve equilibrium problems. These iterative methods are based on an extra-gradient method and a Mann-type iterative method. Two strong convergence theorems have been proved in the setting of real Hilbert space, with mild assumptions that the underlying bi-function is Lipschitz-type continuous and pseudo-monotone. The primary advantage of the second method is that it does not require the information of the Lipschitz-type bi-functional constants. We have also studied the applications of our research results to solve particular classes of equilibrium problems. Numerical studies are carried out to show the behaviour of proposed methods and to compare them with the existing ones in the literature.

International audience; Existence of solutions of reaction-diffusion systems of equations in unbounded domains is studied by the Leray-Schauder (LS) method based on the topological degree for elliptic operators in unbounded domains and on a priori estimates of solutions in weighted spaces. We identify some reaction-diffusion systems for which there exist two subclasses of solutions separated in the function space, monotone and nonmonotone solutions. A priori estimates and existence of solutions are obtained for monotone solutions allowing to prove their existence by the LS method. Various applications of this method are given.

Many studies have been devoted to develop and improve the iterative methods for solving convex constraint nonlinear equations problem (CCP). Based on the projection technique, we introduce a derivative-free method for approximating the solution of CCP. The proposed method is suitable for solving large-scale nonlinear equations due to its lower storage requirements. The directions generated by the proposed method at every iteration are bounded. Under some mild conditions, we establish the global convergence result of the proposed method. Numerical experiments are provided to show the efficiency of the method in solving CCP. Moreover, we tested the capability of the method in solving the monotone nonlinear operator equation equivalent to the $$\ell _1$$ -norm regularized minimization problem.

This paper focuses on the problem of convex constraint nonlinear equations involving monotone operators in Euclidean space. A Fletcher and Reeves type derivative-free conjugate gradient method is proposed. The proposed method is designed to ensure the descent property of the search direction at each iteration. Furthermore, the convergence of the proposed method is proved under the assumption that the underlying operator is monotone and Lipschitz continuous. The numerical results show that the method is efficient for the given test problems.

Positive monotone modal logic is the negation- and implication-free fragment of monotone modal logic, i.e., the fragment with connectives and . We axiomatise positive monotone modal logic, give monotone neighbourhood semantics based on posets, and prove soundness and completeness. The latter follows from the main result of this paper: a (categorical) duality between so-called $$M^+$$ -spaces (poset-based monotone neighbourhood frames with extra structure) and the algebraic semantics of positive monotone modal logic. The main technical tool is the use of coalgebra.

In this paper we characterize the functions that map the positive general monotone sequences to themselves, i.e., we obtain a necessary and a sufficient condition for the functions φ to satisfy $${\sum\limits_{k = n}^{2n} {\left| {{a_{k + 1}} - {a_k}} \right| \le C{a_n} \Rightarrow \sum\limits_{k = n}^{2n} {\left| {\phi \left( {{a_{k + 1}}} \right)} \right| \le {C^\prime }\phi \left( {{a_n}} \right).} } }$$ for all $${\left\{ {{a_n}} \right\}_{n = 1}^\infty \subset {\mathbb{R}^ + }}$$ .

Unified approach (Chen and Chen in J R Stat Soc B 62(3):449–460, 2000) uses a working regression model to extract information from auxiliary variables in two-stage study for computing an efficient estimator of regression parameter. As far as we know, the method is limited to deal with missing complete at random data in a simple monotone missing data pattern. In this research, we extend the unified approach to estimate regression models with nonmonotone missing at random data. We describe an inverse probability weighting estimator condition on estimators from a set of working regression models which contains information from incomplete data and auxiliary variables. The proposed method is flexible and can easily accommodate incomplete data and auxiliary variables. We investigate the finite-sample performance of the proposed estimators using simulation studies and further illustrate the estimation method on a case–control study investigating the risk factors of hip fractures.

Harer and Zagier proved a recursion to enumerate gluings of a 2d-gon that result in an orientable genus g surface, in their work on Euler characteristics of moduli spaces of curves. Analogous results have been discovered for other enumerative problems, so it is natural to pose the following question: how large is the family of problems for which these so-called 1-point recursions exist? In this paper, we prove the existence of 1-point recursions for a class of enumerative problems that have Schur function expansions. In particular, we recover the Harer–Zagier recursion, but our methodology also applies to the enumeration of dessins d’enfant, to Bousquet-Melou–Schaeffer numbers, to monotone Hurwitz numbers, and more. On the other hand, we prove that there is no 1-point recursion that governs single Hurwitz numbers. Our results are effective in the sense that one can explicitly compute particular instances of 1-point recursions, and we provide several examples. We conclude the paper with a brief discussion and a conjecture relating 1-point recursions to the theory of topological recursion.

In this paper, building upon auxiliary principle technique and using proximal operator, we introduce a new explicit algorithm for solving monotone hierarchical equilibrium problems. The considered problem is a monotone equilibrium problem, where the constraint is the solution set of a set-valued variational inequality problem. The strong convergence of the proposed algorithm is studied under strongly monotone and Lipschitz-type assumptions of the bifunction. By combining with parallel techniques, the convergence result is also established for the equilibrium problem involving a finite system of demicontractive mappings. Several fundamental experiments are provided to illustrate the numerical behavior of the proposed algorithm and comparison with other known algorithms is studied.

We consider a special class of mean field SDEs with common noise which depends on the image of the solution (i.e., the conditional distribution given noise). The strong well-posedness is derived under a monotone condition which is weaker than those used in the literature of mean field games; the Feynman–Kac formula is established to solve Schrordinegr type PDEs on $$\mathscr {P}_2$$ , and the ergodicity is proved for a class of measure-valued diffusion processes.

This article concerns the existence of positive solutions for elliptic (p, q)-Kirchhoff type systems with multiple parameters. Our approach is based on the method of sub and super-solutions. The concepts of sub- and super-solution were introduced by Nagumo (Proc Phys-Math Soc Jpn19:861–866, 1937) in 1937 who proved, using also the shooting method, the existence of at least one solution for a class of nonlinear Sturm-Liouville problems. In fact, the premises of the sub- and super-solution method can be traced back to Picard. He applied, in the early 1880s, the method of successive approximations to argue the existence of solutions for nonlinear elliptic equations that are suitable perturbations of uniquely solvable linear problems. This is the starting point of the use of sub- and super-solutions in connection with monotone methods. Picard’s techniques were applied later by Poincare (J Math Pures Appl 4:137–230, 1898) in connection with problems arising in astrophysics. We refer to Radulescu (Qualitative analysis ofnonlinear elliptic partial differential equations: monotonicity, analytic, and variational methods, contemporary mathematics and its applications, Hindawi Publishing Corporation, New York, 2008).

In this paper, we propose a new viscosity extragradient algorithm for solving variational inequality problems of pseudo-monotone and non-Lipschitz continuous operator in real Hilbert spaces. We prove a strong convergence theorem under some appropriate conditions imposed on the parameters. Finally, we give some numerical experiments to illustrate the advantages of our proposed algorithms. The main results obtained in this paper extend and improve some related works in the literature.

In the paper, we consider bilevel problems: variational inequality problems over the set of solutions of the equilibrium problem. Finding normal Nash equilibrium is an example of such a problem. To solve these problems, an iterative algorithm is proposed that combines the ideas of the two-stage proximal method, adaptability, and iterative regularization. In contrast to the previously used rules for choosing the step size, the proposed algorithm does not calculate bifunction values at additional points and does not require knowledge of information on bifunction’s Lipschitz constants and operator’s Lipschitz and strong monotonicity constants. For monotone bifunctions of Lipschitz type and strongly monotone Lipschitz operators, the theorem on strong convergence of sequences generated by the algorithm is proved. The proposed algorithm is shown to be applicable to monotone bilevel variational inequalities in Hilbert spaces.

We study the sums of nondegenerate sine series with monotone coefficients and consider the sets of positivity of such functions. We obtain the sharp lower estimate of the measure of such a set on $$[\pi/2, \pi]$$ and a new lower bound on its measure on $$[0,\pi]$$ . It is shown that the latter measure is at least $$\pi/2 + 0.24$$ and in the case of fulfilling special conditions it is at least $$2\pi/3$$ , which is an unimprovable estimate.

In this paper we establish a multiplicity result for a class of unilateral, nonlinear, nonlocal problems with nonsmooth potential (variational-hemivariational inequalities), using the degree map of multivalued perturbations of fractional nonlinear operators of monotone type, the fact that the degree at a local minimizer of the corresponding Euler functional is equal one, and controlling the degree at small balls and at big balls. published

International audience; We consider discrete Schrödinger operators on ${\mathbb{Z}}^d$ for which the perturbation consists of the sum of a long-range type potential and a Wigner-von Neumann type potential. Still working in a framework of weighted Mourre theory, we improve the limiting absorption principle (LAP) that was obtained in [Ma1]. To our knowledge, this is a new result even in the one-dimensional case. The improvement consists in a weakening of the assumptions on the long-range potential and better LAP weights. The improvement relies only on the fact that the generator of dilations (which serves as conjugate operator) is bounded from above by the position operator. To exploit this, Loewner's theorem on operator monotone functions is invoked.

In this work, we study the split null point problem and the fixed point problem in Hilbert spaces. We introduce a self-adaptive algorithm based on the viscosity approximation method without prior knowledge of the operator norm for finding a common solution of the considered problem for maximal monotone mappings and demicontractive multivalued mappings. A strong convergence result of our proposed algorithm is established under some suitable conditions. Some convergence results for the split feasibility problem and the split minimization problem are consequences of our main result. Finally, we also give numerical examples for supporting our main result.

We consider a class of nonlocal viscous Cahn–Hilliard equations with Neumann boundary conditions for the chemical potential. The double-well potential is allowed to be singular (e.g. of logarithmic type), while the singularity of the convolution kernel does not fall in any available existence theory under Neumann boundary conditions. We prove well-posedness for the nonlocal equation in a suitable variational sense. Secondly, we show that the solutions to the nonlocal equation converge to the corresponding solutions to the local equation, as the convolution kernels approximate a Dirac delta. The asymptotic behaviour is analyzed by means of monotone analysis and Gamma convergence results, both when the limiting local Cahn–Hilliard equation is of viscous type and of pure type.

This work is concerned with the classical problem of finding a zero of a sum of maximal monotone operators. For the projective splitting framework recently proposed by Combettes and Eckstein, we show how to replace the fundamental subproblem calculation using a backward step with one based on two forward steps. The resulting algorithms have the same kind of coordination procedure and can be implemented in the same block-iterative and highly flexible manner, but may perform backward steps on some operators and forward steps on others. Prior algorithms in the projective splitting family have used only backward steps. Forward steps can be used for any Lipschitz-continuous operators provided the stepsize is bounded by the inverse of the Lipschitz constant. If the Lipschitz constant is unknown, a simple backtracking linesearch procedure may be used. For affine operators, the stepsize can be chosen adaptively without knowledge of the Lipschitz constant and without any additional forward steps. We close the paper by empirically studying the performance of several kinds of splitting algorithms on a large-scale rare feature selection problem.

In this paper, two effective derivative-free methods are proposed for solving large-scale nonlinear monotone equations, in which the search directions are sufficiently descent and independent of the line search. The methods are the extensions of the conjugate gradient methods proposed by Bojari and Eslahchi (Numer. Algorithms 83, pp. 901–933, 2020) combined with the hyperplane projection technique. Our approaches are low storage memory and derivative-free, which makes them suitable for large-scale nonsmooth monotone nonlinear equations. Under proper assumptions, we analyze the global convergence property of the proposed methods. Finally, numerical experiments show that the proposed methods outperform some existing ones.

Let $$h:V\subset {\mathbb {R}}^{2}\longrightarrow {\mathbb {R}}^{2}$$ be an embedding. The aim of this paper is to analyze the dynamical behavior of h depending on the number of fixed points and 2-cycles, their local behaviors and the features of V. Our approach allows us to extend some celebrated results of the theory of monotone flows, namely the order interval trichotomy, for non-monotone maps. Moreover, we discuss several applications in classical models. In the particular case of the Ricker system, we recover some recent results deduced from computer assistance.

In this paper, the preimage branch t-entropy and entropy dimension for nonautonomous systems are studied and some systems with preimage branch t-entropy zero are introduced. Moreover, formulas calculating the s-topological entropy of a sequence of equi-continuous monotone maps on the unit circle are given. Finally, examples to show that the entropy dimension of non-autonomous systems can be attained by any positive number s are constructed.

In this paper, we investigate the existence of monotone positive solutions for a fourth order boundary value problem with dependence on the derivative in nonlinearity under integral and multi-point boundary conditions. By applying the fixed point theorem in a cone, some criteria on the existence of positive solutions are acquired. These criteria are given by explicit conditions which are generally weaker than those derived by using the classical norm-type expansion and compression theorem. As applications, three examples are presented to illustrate the validity of our mains results.

Fondecyt Postdoc Project 3180080 Basal Program from CONICYT-Chile CMM-AFB 170001 National Foundation for Science & Technology Development (NAFOSTED) 101.01-2017.325

In this paper, we first introduce a new class of structured tensors, named strictly positive semidefinite tensors, and show that a strictly positive semidefinite tensor is not an $$R_0$$ tensor. We focus on the stochastic tensor complementarity problem (STCP), where the expectation of the involved tensor is a strictly positive semidefinite tensor. We denote such an STCP as a monotone STCP. Based on three popular NCP functions, the min function, the Fischer–Burmeister (FB) function and the penalized FB function, as well as the special structure of the monotone STCP, we introduce three new NCP functions and establish the expected residual minimization (ERM) formulation of the monotone STCP. We show that the solution set of the ERM problem is nonempty and bounded if the solution set of the expected value (EV) formulation for such an STCP is nonempty and bounded. Moreover, an approximate regularized model is proposed to weaken the conditions for nonemptiness and boundedness of the solution set of the ERM problem in practice. Numerical results indicate that the performance of the ERM method is better than that of the EV method.

We prove that almost every level set of a Sobolev function in a planar domain consists of points, Jordan curves, or homeomorphic copies of an interval. For monotone Sobolev functions in the plane we have the stronger conclusion that almost every level set is an embedded $1$-dimensional topological submanifold of the plane. Here monotonicity is in the sense of Lebesgue: the maximum and minimum of the function in an open set are attained at the boundary. Our result is an analog of Sard's theorem, which asserts that for a $C^2$-smooth function in a planar domain almost every value is a regular value. As an application we show that monotone Sobolev functions in planar domains can be approximated uniformly and in the Sobolev norm by smooth monotone functions., 27 pages

Inspired by the idea of Colding-Minicozzi in [CM1], we define (mean curvature flow) entropy for submanifolds in a general ambient Riemannian manifold. In particular, this entropy is equivalent to area growth of a closed submanifold in a closed ambient manifold with non-negative Ricci curvature. Moreover, this entropy is monotone along the mean curvature flow in a closed Riemannian manifold with non-negative sectional curvatures and parallel Ricci curvature. As an application, we show the partial regularity of the limit of mean curvature flow of surfaces in a three dimensional Riemannian manifold with non-negative sectional curvatures and parallel Ricci curvature., Comment: 16 pages, comments are welcomed! The paper is revised according to the comments of the journal referees, and some references are added. Accepted by The Journal of Geometric Analysis

Let E be a real reflexive Banach space with its dual $$E^*$$ and f be a proper, convex and lower-semi-continuous function on E. The purpose of this paper is to introduce and study a new class of mappings from E into $$E^*$$ called f-pseudocontractive mappings with the notion of f-fixed points. In the case that E is a real reflexive Banach space and f is a strongly coercive, bounded and uniformly Frechet differentiable Legendre function which is strongly convex on bounded subsets of E, a sequence is constructed which converges strongly to a common f-fixed point of two f-pseudocontractive mappings. As a consequence, we obtain a scheme which converges strongly to a common zero of monotone mappings. Furthermore, this analog is applied to approximate solutions to convex optimization problems. Our results improve and generalize many of the results in the literature.

In this paper, a full-Newton step Interior-Point Method for solving monotone Weighted Linear Complementarity Problem is designed and analyzed. This problem has been introduced recently as a generalization of the Linear Complementarity Problem with modified complementarity equation, where zero on the right-hand side is replaced with the nonnegative weight vector. With a zero weight vector, the problem reduces to a linear complementarity problem. The importance of Weighted Linear Complementarity Problem lies in the fact that it can be used for modelling a large class of problems from science, engineering and economics. Because the algorithm takes only full-Newton steps, the calculation of the step size is avoided. Under a suitable condition, the algorithm has a quadratic rate of convergence to the target point on the central path. The iteration bound for the algorithm coincides with the best iteration bound obtained for these types of problems.