An iterative process for nonlinear monotonic nonexpansive operators in Hilbert space

We present two algorithms for finding a zero of the sum of two monotone operators and a fixed point of a nonexpansive operator in Hilbert spaces. We show that these two algorithms converge strongly to the minimum norm common element of the zero of the sum of two monotone operators and the fixed point of a nonexpansive operator.

Abstract. The aim of this paper is to present some properties of firmly nonexpansive operators in Hilbert spaces. As a consequence of the principal result, a fixed point theorem about nonexpansive operators is obtained.

Convergence of composite iterative methods for finding zeros of accretive operators

In the first part of this paper we present a representation theorem for the directional derivative of the metric projection operator in an arbitrary Hilbert space. As a consequence of the representation theorem, we present in the second part the development of the theory of projected dynamical systems in infinite dimensional Hilbert space. We show that this development is possible if we use the viable solutions of differential inclusions. We use also pseudomonotone operators.

Remark on the decay for damped string and beam equations

Strong convergence of the CQ method for fixed point iteration processes

We investigate a class of new <svg style="vertical-align:-2.3205pt;width:85.949997px;" id="M2" height="15.0875" version="1.1" viewBox="0 0 85.949997 15.0875" width="85.949997" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D43B"/></g><g transform="matrix(.017,-0,0,-.017,14.97,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,20.852,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,26.734,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,30.609,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,37.307,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,41.182,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,47.064,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,53.779,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,59.66,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,63.536,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,70.233,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,74.109,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,79.991,12.138)"><use xlink:href="#x29"/></g> </svg>-mixed cocoercive operators in Hilbert spaces. We extend the concept of resolvent operators associated with <svg style="vertical-align:-2.3205pt;width:41.25px;" id="M3" height="15.0875" version="1.1" viewBox="0 0 41.25 15.0875" width="41.25" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D43B"/></g><g transform="matrix(.017,-0,0,-.017,14.97,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,20.852,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,24.728,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,31.425,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,35.301,12.138)"><use xlink:href="#x29"/></g> </svg>-cocoercive operators to the <svg style="vertical-align:-2.3205pt;width:85.949997px;" id="M4" height="15.0875" version="1.1" viewBox="0 0 85.949997 15.0875" width="85.949997" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D43B"/></g><g transform="matrix(.017,-0,0,-.017,14.97,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,20.852,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,26.734,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,30.609,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,37.307,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,41.182,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,47.064,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,53.779,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,59.66,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,63.536,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,70.233,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,74.109,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,79.991,12.138)"><use xlink:href="#x29"/></g> </svg>-mixed cocoercive operators and prove that the resolvent operator of <svg style="vertical-align:-2.3205pt;width:85.949997px;" id="M5" height="15.0875" version="1.1" viewBox="0 0 85.949997 15.0875" width="85.949997" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D43B"/></g><g transform="matrix(.017,-0,0,-.017,14.97,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,20.852,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,26.734,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,30.609,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,37.307,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,41.182,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,47.064,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,53.779,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,59.66,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,63.536,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,70.233,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,74.109,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,79.991,12.138)"><use xlink:href="#x29"/></g> </svg>-mixed cocoercive operator is single valued and Lipschitz continuous. Some examples are given to justify the definition of <svg style="vertical-align:-2.3205pt;width:85.949997px;" id="M6" height="15.0875" version="1.1" viewBox="0 0 85.949997 15.0875" width="85.949997" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D43B"/></g><g transform="matrix(.017,-0,0,-.017,14.97,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,20.852,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,26.734,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,30.609,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,37.307,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,41.182,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,47.064,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,53.779,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,59.66,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,63.536,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,70.233,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,74.109,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,79.991,12.138)"><use xlink:href="#x29"/></g> </svg>-mixed cocoercive operators. Further, by using resolvent operator technique, we discuss the approximate solution and suggest an iterative algorithm for the generalized mixed variational inclusions involving <svg style="vertical-align:-2.3205pt;width:85.949997px;" id="M7" height="15.0875" version="1.1" viewBox="0 0 85.949997 15.0875" width="85.949997" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,12.138)"><use xlink:href="#x1D43B"/></g><g transform="matrix(.017,-0,0,-.017,14.97,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,20.852,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,26.734,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,30.609,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,37.307,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,41.182,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,47.064,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,53.779,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,59.66,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,63.536,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,70.233,12.138)"><use xlink:href="#x22C5"/></g><g transform="matrix(.017,-0,0,-.017,74.109,12.138)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,79.991,12.138)"><use xlink:href="#x29"/></g> </svg>-mixed cocoercive operators in Hilbert spaces. We also discuss the convergence criteria for the iterative algorithm under some suitable conditions. Our results can be viewed as a generalization of some known results in the literature.

From the viewpoint of the theory of orthomodular lattices of elementary propositions, Quantum Theories can be formulated in real, complex or quaternionic Hilbert spaces as established in Sol\'er's theorem. The said lattice eventually coincides with the lattice of all orthogonal projectors on a separable Hilbert space over R, C, or over the algebra of quaternions H. Quantum states are $\sigma$-additive probability measures on that non-Boolean lattice. Gleason's theorem proves that, if the Hilbert space is separable with dimension >2 and the Hilbert space is either real or complex, then states are one-to-one with standard density matrices (self-adjoint, positive, unit-trace, trace-class operators). The extension of this result to quaternionic Hilbert spaces was obtained by Varadarajan in 1968. Unfortunately, even if the hard part of the proof is correct, the formulation of this extension is mathematically incorrect. This is due to some peculiarities of the notion of trace in quaternionic Hilbert spaces, e.g., basis dependence, making the theory of trace-class operators in quaternionic Hilbert spaces different from the standard theory in real and complex Hilbert spaces. A minor issue also affects Varadarajan's statement for real Hilbert space formulation. This paper is mainly devoted to present Gleason-Varadarajan's theorem into a technically correct form valid for the three types of Hilbert spaces. After having develped part of the general mathematical technology of trace-class operators in (generally non-separable) quaternionic Hilbert spaces, we prove that only the {\em real part} of the trace enters the formalism of quantum theories (also dealing with unbounded observables and symmetries) and it can be safely used to formulate and prove a common statement of Gleason's theorem.

We propose a variable metric extension of the forward–backward-forward algorithm for finding a zero of the sum of a maximally monotone operator and a monotone Lipschitzian operator in Hilbert spaces. In turn, this framework provides a variable metric splitting algorithm for solving monotone inclusions involving sums of composite operators. Monotone operator splitting methods recently proposed in the literature are recovered as special cases.

We obtained sharp "multiplicative" and sharp "additive" inequalities of Hardy-Littlewood-Polya type for operators in Hilbert space.

Some new inequalities for the Cebysev functional of two functions of selfadjoint linear operators in Hilbert spaces, under suitable assumptions for the involved functions and operators, are given. 2010 Mathematics Subject Classifications. 47A63; 47A99. .

In a real Hilbert space, let the CFPP, VIP, and HFPP denote the common fixed-point problem of countable nonexpansive operators and asymptotically nonexpansive operator, variational inequality problem, and hierarchical fixed point problem, respectively. With the help of the Mann iteration method, a subgradient extragradient approach with a linear-search process, and a hybrid deepest-descent technique, we construct two modified Mann-type subgradient extragradient rules with a linear-search process for finding a common solution of the CFPP and VIP. Under suitable assumptions, we demonstrate the strong convergence of the suggested rules to a common solution of the CFPP and VIP, which is only a solution of a certain HFPP.

We consider a two-point boundary value problem for the equation $u_{tt} - A(t)u = f(t,u)$ where, for each t, $A(t)$ is an unbounded operator in Hilbert space whose domain varies with t. We establish the existence (but not the uniqueness) of “weak” solutions of this problem by using finite difference methods. When $A(t)$ represents a differential operator in the space variables, the method of proof can be converted into a convergent numerical procedure.

In this paper, we introduce an efficient subgradient extragradient (SE) based method for solving variational inequality problems with monotone operator in Hilbert space. In many existing SE methods, two values of operator are needed over each iteration and the Lipschitz constant of the operator or linesearch is required for estimating step sizes, which are usually not practical and expensive. To overcome these drawbacks, we present an inertial SE based algorithm with adaptive step sizes, estimated by using an approximation of the local Lipschitz constant without running a linesearch. Each iteration of the method only requires a projection on the feasible set and a value of the operator. The numerical experiments illustrate the efficiency of the proposed algorithm.

Modelling water infiltration in soils.- Brief overview of unsaturated flow concepts.- Settlement of the mathematical models of nonhysteretic infiltration.- Analysis of infiltration models.- Basic existence theorems for evolution equations with monotone operators in Hilbert spaces.- Functional approach to the quasi-unsaturated infiltration model.- Functional approach to the saturated-unsaturated infiltration model.- Specific problems in infiltration.- Inverse problems in infiltration.- Identification of the boundary conditions from recorded observations.- Background tools.