In this paper, we define and introduce some new concepts of the higher order strongly general biconvex functions involving arbitrary bifunction a function. Some relationships among various have been established. It is shown that parallelogram laws for Banach spaces can be obtained as applications affine functions, which itself an novel application. proved optimality conditions are characterized...

We study the complexity of proving the Pigeon Hole Principle (PHP) in a monotone variant of the Gentzen Calculus, also known as Geometric Logic. We show that the standard encoding of the PHP as a monotone sequent admits quasipolynomial-size proofs in this system. This result is a consequence of deriving the basic properties of certain quasipolynomial-size monotone formulas computing the boolean...

and Applied Analysis 3 The generalized mixed equilibrium problems with perturbation is very general in the sense that it includes fixed point problems, optimization problems, variational inequality problems, Nash equilibrium problems, and equilibrium problems as special cases see, e.g., 4, 5 . Numerous problems in physics, optimization, and economics reduce to find a solution of problem 1.2 . S...

We consider the forward-backward splitting method for nding a zero of the sum of two maximal monotone mappings. This method is known to converge when the inverse of the forward mapping is strongly monotone. We propose a modiication to this method, in the spirit of the extragradient method for monotone variational inequalities, under which the method converges assuming only the forward mapping i...

In this paper, we study the existence of generalized solutions for the infinite dimensional nonlinear stochastic differential inclusions $dx(t) in F(t,x(t))dt +G(t,x(t))dW_t$ in which the multifunction $F$ is semimonotone and hemicontinuous and the operator-valued multifunction $G$ satisfies a Lipschitz condition. We define the It^{o} stochastic integral of operator set-valued stochastic pr...

In this paper, a monotone positive solution is studied for buckling of a distributed model of multi walled carbon nanotube (MWCNT) cantilevers in the vicinity of thin and thick graphite sheets subject to intermolecular forces. In the modeling of intermolecular forces, Van der Waals forces are taken into account. A hybrid nano-scale continuum model based on Lennard–Jones potential is applied to ...

This paper is concerned with solving nonlinear monotone difference schemes of the parabolic type. The monotone Jacobi and monotone Gauss– Seidel methods are constructed. Convergence rates of the methods are compared and estimated. The proposed methods are applied to solving nonlinear singularly perturbed parabolic problems. Uniform convergence of the monotone methods is proved. Numerical experi...

We consider several questions about monotone AC-tree automata, a class of equational tree automata [21] whose transition rules correspond to rules in Kuroda normal form of context-sensitive grammars. Whereas it is known that this class has a decision procedure to determine if a given monotone AC-tree automaton accepts no term [23], other decidability and complexity results have not been well-in...

We show that if a Boolean function f : {0, 1}n → {0, 1} can be computed by a monotone real circuit of size s using k-ary monotone gates then f can be computed by a monotone real circuit of size O(snk−2) which uses unary or binary monotone gates only. This partially solves an open problem presented in [2]. In fact, in size O(snk−1), the circuit uses only unary monotone gates and binary addition....