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We will show in this paper that if λ is very close to 1, then can be attained, where M is a compact–manifold with boundary. This result gives a counter–example to the conjecture of de Figueiredo and Ruf in their paper titled "On an inequality by Trudinger and Moser and related elliptic equations" ( Comm. Pure. Appl. Math., 55, 135–152, 2002). [ABSTRACT FROM AUTHOR]

In this paper, it is proved that with at most $$ O{\left( {N^{{\frac{{65}} {{66}}}} } \right)} $$ exceptions, all even positive integers up to N are expressible in the form $$ p^{2}_{2} + p^{3}_{3} + p^{4}_{4} + p^{5}_{5} $$ . This improves a recent result $$ O{\left( {N^{{\frac{{19193}} {{19200}} + \varepsilon }} } \right)} $$ due to C. Bauer. [ABSTRACT FROM AUTHOR]

In this paper, a class of global optimization problems is considered. Corresponding to each local minimizer obtained, we introduced a new modified function and construct a corresponding optimization subproblem with one constraint. Then, by applying a local search method to the one-constraint optimization subproblem and using the local minimizer as the starting point, we obtain a better local optimal solution. This process is continued iteratively. A termination rule is obtained which can serve as stopping criterion for the iterating process. To demonstrate the efficiency of the proposed approach, numerical examples are solved. [ABSTRACT FROM AUTHOR]

Let u be a solution to a second order elliptic equation with singular potentials belonging to Kato–Fefferman–Phong’s class in Lipschitz domains. An elementary proof of the doubling property for u 2 over balls is presented, if the balls are contained in the domain or centered at some points near an open subset of the boundary on which the solution u vanishes continuously. Moreover, we prove the inner unique continuation theorems and the boundary unique continuation theorems for the elliptic equations, and we derive the ℬ p weight properties for the solution u near the boundary. [ABSTRACT FROM AUTHOR]

We investigate some properties related to the generalized Newton method for the Fischer-Burmeister (FB) function over second-order cones, which allows us to reformulate the second-order cone complementarity problem (SOCCP) as a semismooth system of equations. Specifically, we characterize the B-subdifferential of the FB function at a general point and study the condition for every element of the B-subdifferential at a solution being nonsingular. In addition, for the induced FB merit function, we establish its coerciveness and provide a weaker condition than Chen and Tseng (Math. Program. 104:293–327, ) for each stationary point to be a solution, under suitable Cartesian P-properties of the involved mapping. By this, a damped Gauss-Newton method is proposed, and the global and superlinear convergence results are obtained. Numerical results are reported for the second-order cone programs from the DIMACS library, which verify the good theoretical properties of the method. [ABSTRACT FROM AUTHOR]

The article examines the existence of solutions for weighted p(x)-Laplacian ordinary boundary value problem. The method used in this study is based on Leray-Schauder degree. As an application, the existence of weak solutions for p(x)-Laplacian partial differential equations is given. On the p(x)-Laplacian problems, there is a possibility that it does not have the first eigenvalue and the first eigenfunction. Several sub-super-solution theorems for the existence of solutions for weighted p(x)-Laplacian equation with Dirichlet, Robin, and Periodic boundary value conditions are established.

The article analyzes the boundary asymptotic behavior of solutions for weighted p(x)-Laplacian equations that take infinite value on a bounded domain. It was found that the boundary asymptotic of solutions of the Laplacian function is continuous and positive. The p(x)-Laplacian possesses more complicated inhomogeneous nonlinearities, thus, some special techniques are needed. Its main difficulty arises from the lack of compactness. The uniqueness and asymptotic behavior of solutions for problem is a nonnegative function which is allowed to vanish on the boundary.

Based on the Orlov and Shulman’s M operator, the additional symmetries and the string equation of the CKP hierarchy are established, and then the higher order constraints on L l are obtained. In addition, the generating function and some properties are also given. In particular, the additional symmetry flows form a new infinite dimensional algebra $$W^C_{1+\infty}$$ , which is a subalgebra of W 1+∞. [ABSTRACT FROM AUTHOR]

Let R be an exchange ring with primitive factors artinian. We prove that there exists a u∈ U( R) such that 1 R ± u ∈ U( R), if and only if for any a∈ R, there exists a u ∈ U( R) such that a ± u ∈ U( R). Furthermore, we prove that, for any A ∈ M n ( R)( n ≥ 2), there exists a U ∈ GL n ( R) such that A ± U ∈ GL n ( R). [ABSTRACT FROM AUTHOR]

The article presents a study on the mathematical equation of two-dimensional linear differential systems with deviating arguments using Kneser type solutions. It presents an equation with positive coefficient and two linearly independent solutions. The article also evaluates the existence and uniqueness of these types of solutions. It proves that Kneser type solution is equivalent to nonvanishing of the Wronskian fundamental system and positivity of Green's function of the problem.

The article presents a research on the iterative methods for generalized von Foerster equations with functional dependence. It examines the meeting of a natural iterative method to the exact solution of a differential-functional von Foerster-type equation that notes a single population depending on its total size, past time and states densities. It also assumes either Perron comparison conditions and some monotonicity. It also investigates on the natural conditions which ensures convergence of iterative methods.

The classical proofs of the De Giorgi–Nash–Moser Theorem are based on the iteration of some inequality through countably many concentric balls. In this note, we present a new approach to the Hölder continuity of solutions to elliptic equations in divergence form, which avoids any form of discrete iteration. In particular, we prove that a suitable energy function satisfies a differential inequality, whose integration yields a new proof of the crucial step in the regularity result. [ABSTRACT FROM AUTHOR]

Based on a continuation theorem of Mawhin, the existence of a positive periodic solution for a nonlinear difference system is studied. [ABSTRACT FROM AUTHOR]

Focuses on multivalued feedback with semicontinuity property. Unknown disturbance in equations; Mathematical models of controlled objects; Mutivalued initial conditions and multivalued boundary conditions; Differential games with functionals.