Nonexpansive Condition Research Articles

New vector transport operators extending a Riemannian CG algorithm to generalized Stiefel manifold with low-rank applications

This paper proposes two innovative vector transport operators, leveraging the Cayley transform, for the generalized Stiefel manifold embedded with a non-standard metric. Specifically, it introduces the differentiated retraction and an approximation of the Cayley transform to the differentiated matrix exponential. These vector transports are demonstrated to satisfy the Ring–Wirth non-expansive condition under non-standard metrics, and one of them is also isometric. Building upon the novel vector transport operators, we extend the modified Polak–Ribière–Polyak (PRP) conjugate gradient method to the generalized Stiefel manifold. Under a non-monotone line search condition, we prove our algorithm globally converges to a stationary point. The efficiency of the proposed vector transport operators is empirically validated through numerical experiments involving generalized eigenvalue problems and canonical correlation analysis.

Common fixed points of fuzzy maps under nonexpansive type condition

Common fixed points of fuzzy maps under nonexpansive type condition

Uniform Value for Some Nonexpansive Optimal Control Problems With General Evaluations

We study the long-run properties of optimal control problems in continuous time, where the running cost of a control problem is evaluated by a probability measure over $[0,+\infty)$ . Li et al. (“Limit value for optimal control with general means,” Discrete Continuous Dyn. Syst.—Series A , vol. 36, pp. 2113–2132, 2016) introduced an asymptotic regularity condition for a sequence of probability measures to be more and more uniform over $[0,+\infty)$ in order to study the limit properties of the value functions with respect to the evaluation. In the particular case of $t$ -horizon Cesaro mean or $\rho$ -discounted Abel mean, this condition implies that the horizon $t$ tends to infinity or the discount factor $\rho$ tends to zero. For the control problem defined on a compact domain, satisfying some nonexpansive condition, and with a running cost function dependent on the state variable only, Li et al. proved the existence of a general limit value, i.e., the value function uniform converges as the evaluation becomes more and more regular. Within the same context, we prove the existence of a general uniform value, i.e., for any $\varepsilon >0$ , there is a robust optimal control that guarantees the general limit value up to $\varepsilon$ for all control problems where the cost is evaluated by a probability measure sufficiently regular. This extends the result presented by Quincampoix and Renault (“On the existence of a limit value in some nonexpansive optimal control problems,” SIAM J. Control Optim. , vol. 49, pp. 2118–2132, 2011) which proved the existence of a uniform value for running costs evaluated by Cesaro means only. Under the compact nonexpansive condition we make, the obtained limit value is in general a function dependent of the initial state, a property not underlined by the traditional ergodic or dissipative approach for the long-run control problems.

Common fixed point theorems for hybrid pair of mappings

In this paper we define the concept of conditionally commuting mappings and establish some common fixed point theorems for hybrid pair of mappings satisfying a nonexpansive type condition.

A Riemannian conjugate gradient method for optimization on the Stiefel manifold

In this paper we propose a new Riemannian conjugate gradient method for optimization on the Stiefel manifold. We introduce two novel vector transports associated with the retraction constructed by the Cayley transform. Both of them satisfy the Ring-Wirth nonexpansive condition, which is fundamental for convergence analysis of Riemannian conjugate gradient methods, and one of them is also isometric. It is known that the Ring-Wirth nonexpansive condition does not hold for traditional vector transports as the differentiated retractions of QR and polar decompositions. Practical formulae of the new vector transports for low-rank matrices are obtained. Dai's nonmonotone conjugate gradient method is generalized to the Riemannian case and global convergence of the new algorithm is established under standard assumptions. Numerical results on a variety of low-rank test problems demonstrate the effectiveness of the new method.

Existence of common fixed point and best proximity point for generalized nonexpansive type maps in convex metric space.

Here, we extend the notion of (E.A.) property in a convex metric space defined by Kumar and Rathee (Fixed Point Theory Appl 1–14, 2014) by introducing a new class of self-maps which satisfies the common property (E.A.) in the context of convex metric space and ensure the existence of common fixed point for this newly introduced class of self-maps. Also, we guarantee the existence of common best proximity points for this class of maps satisfying generalized non-expansive type condition. We furnish an example in support of the proved results.

Convergence Theorems for a Hybrid Pair of Generalized Nonexpansive Mappings in Banach Spaces

In this paper, we study convergence of modified Ishikawa iteration process involving a pair of hybrid mappings satisfying a generalized nonexpansive condition on Banach spaces. In process, several relevant results are generalized and improved.

Generalized nonexpansive mappings in ordered metric spaces and their (common) fixed points with applications

We introduce a generalized nonexpansive condition for a map in an ordered metric space and derive fixed point theorems for a nondecreasing mapping. Moreover, common fixed point results are obtained for a pair of mappings satisfying an appropriate nonexpansive condition. To illustrate our results, we give throughout the paper some examples. Applications of the obtained results to integral equations and first-order periodic boundary value problems are presented.

Nonlinear evolution inclusions: Topological characterizations of solution sets and applications

This paper deals with a nonlinear delay differential inclusion of evolution type involving m-dissipative operator and source term of multi-valued type in a Banach space. Under rather mild conditions, the Rδ-structure of C0-solution set is studied on compact intervals, which is then used to obtain the Rδ-property on non-compact intervals. Secondly, the result about the structure is furthermore employed to show the existence of C0-solutions for the inclusion (mentioned above) subject to nonlocal condition defined on right half-line. No nonexpansive condition on nonlocal function is needed. As samples of applications, we consider a partial differential inclusion with time delay and then with nonlocal condition at the end of the paper.

Common fixed point theorem for noncommuting mappings satisfying a generalized asymptotically nonexpansive condition

We present a common fixed point theorem for generalized asymptotically nonexpansive and noncommuting mappings in normed linear spaces.

Some results on coincidence points

In this paper we prove some coincidence point theorems for nonself single-valued and multivalued maps satisfying a nonexpansive condition. These extend fixed point theorems for multivalued maps of a number of authors.