On implicit function theorems for set-valued maps and their application to mathematical programming under inclusion constraints

In this paper we establish some implicit function theorems for a class of locally Lipschitz set-valued maps and then apply them to investigate some questions concerning the stability of optimization problems with inclusion constraints. In consequence we have an extension of some of the corresponding results of Robinson, Aubin, and others.     

A Strong Law for Mixing Random Variables

A sharp almost sure bound is derived for limit points of average sum of weakly dependent random variables, which ensures strong laws of large numbers for and -mixing random variables, without assumptions on rate of tending to zero of and -mixing parameters _n and _n.     

Self-adaptive gradient projection algorithms for variational inequalities involving non-Lipschitz continuous operators

Numerical Algorithms - In this paper, we introduce a self-adaptive inertial gradient projection algorithm for solving monotone or strongly pseudomonotone variational inequalities in real Hilbert...     

On exponential stability of nonlinear Volterra difference equations in phase spaces

Using a novel approach, we present some new explicit criteria for global exponential stability of the zero solution of general nonlinear Volterra difference equations in phase spaces. In particular, this gives a solution to an open problem posed very recently by E. Braverman and I. M. Karabash in Journal of Difference Equations and Applications 18 , 909–939 (2012). As an application, we apply the obtained results to study asymptotic behavior of equilibriums of discrete time neural networks.     

An Initial-Boundary Value Problem for the Korteweg–de Vries Equation with Dominant Surface Tension

We consider the initial-boundary value problem (IBVP) for the Korteweg–de Vries equation with zero boundary conditions at x=0 and arbitrary smooth decreasing initial data. We prove that the solution of this IBVP can be found by solving two linear inverse scattering problems (SPs) on two different spectral planes. The first SP is associated with the KdV equation. The second SP is self-conjugate and its scattering function is found in terms of entries of the scattering matrix s(k) for the first SP. Knowing the scattering function, we solve the second inverse SP for finding the potential self-conjugate matrix. Consequently, the unknown object entering coefficients in the system of evolution equations for s(k,t) is found. Then, the time-dependent scattering matrix s(k,t) is expressed in terms of s(k)=s(k,0) and of solutions of the self-conjugate SP. Knowing s(k,t), we find the solution of the IBVP in terms of the solution of the Gelfand–Levitan–Marchenko equation in the first inverse SP.     

Halpern projection methods for solving pseudomonotone multivalued variational inequalities in Hilbert spaces

In this paper, we introduce new approximate projection and proximal algorithms for solving multivalued variational inequalities involving pseudomonotone and Lipschitz continuous multivalued cost mappings in a real Hilbert space. The first proposed algorithm combines the approximate projection method with the Halpern iteration technique. The second one is an extension of the Halpern projection method to variational inequalities by using proximal operators. The strongly convergent theorems are established under standard assumptions imposed on cost mappings. Finally we introduce a new and interesting example to the multivalued cost mapping, and show its pseudomontone and Lipschitz continuous properties. We also present some numerical experiments to illustrate the behavior of the proposed algorithms.

     

Existence and Exponential Decay of Solutions for a Wave Equation with Integral Nonlocal Boundary Conditions of Memory Type

In this paper, we consider a wave equation with integral nonlocal boundary conditions of memory type. First, we establish two local existence theorems by using Faedo–Galerkin method and standard arguments of density. Next, we give a su?cient condition to guarantee the global existence and exponential decay of weak solutions. Finally, we present numerical results.     

Monotonic optimization for sensor cover energy problem

We study the sensor cover energy problem (SCEP) in wireless communication—a difficult nonconvex problem with nonconvex constraints. A local approach based on DC programming called DCA was proposed by Astorino and Miglionico (Optim Lett 10(2):355–368, 2016) for solving this problem. In the present paper, we propose a global approach to (SCEP) based on the theory of monotonic optimization. By using an appropriate reformulation of (SCEP) we propose an algorithm for finding quickly a local optimal solution along with an efficient algorithm for computing a global optimal solution. Computational experiments are reported which demonstrate the practicability of the approach.     

A splitting algorithm for equilibrium problem given by the difference of two bifunctions

In this paper, we introduce a splitting algorithm for solving equilibrium problems given by the difference of two bifunctions in a real Hilbert space. Under suitable assumptions on component bifunctions, we prove strong convergence of the proposed algorithm. In contrast to most existing projection-type methods for equilibrium problems, our algorithm does not require any convexity or monotonicity conditions on the resulting bifunction. Some numerical experiments and comparisons are given to illustrate the efficiency of the proposed algorithm.