Complete CAT Research Articles

Attractive points of further 2-generalized hybrid mappings in a complete CAT(0) space

Attractive points of further 2-generalized hybrid mappings in a complete CAT(0) space

On the proximal point algorithm and demimetric mappings in CAT(0) spaces

AbstractIn this paper, we introduce and study the class of demimetric mappings in CAT(0) spaces.We then propose a modified proximal point algorithm for approximating a common solution of a finite family of minimization problems and fixed point problems in CAT(0) spaces. Furthermore,we establish strong convergence of the proposed algorithm to a common solution of a finite family of minimization problems and fixed point problems for a finite family of demimetric mappings in complete CAT(0) spaces. A numerical example which illustrates the applicability of our proposed algorithm is also given. Our results improve and extend some recent results in the literature.

Strong convergence theorem for monotone inclusion problem in CAT(0) spaces

In this paper, we introduce Halpern-type proximal point algorithm for approximating a common solution of monotone inclusion problem and fixed point problem. We obtain a strong convergence of the proposed algorithm to a common solution of finite family of monotone inclusion problem and fixed point problem for nonexpansive mappings in complete CAT(0) spaces. Nontrivial application and numerical example were given. Our results complement and extend some recent results in literature.

Noncyclic Meir-Keeler contractions and best proximity pair theorems

Abstract In this article, we consider the class of noncyclic Meir-Keeler contractions and study the existence and convergence of best proximity pairs for such mappings in the setting of complete CAT(0) spaces. We also discuss asymptotic pointwise noncyclic Meir-Keeler contractions in the framework of uniformly convex Banach spaces and generalize a main result of Chen [Chen C. M., A note on asymptotic pointwise weaker Meir-Keeler type contractions, Appl. Math. Lett., 2012, 25, 1267-1269]. Examples are given to support our main results.

Strong Convergence of New Two-Step Viscosity Iterative Approximation Methods for Set-Valued Nonexpansive Mappings in CAT(0) Spaces

This paper is for the purpose of introducing and studying a class of new two-step viscosity iteration approximation methods for finding fixed points of set-valued nonexpansive mappings in CAT(0) spaces. By means of some properties and characteristic to CAT(0) space and using Cauchy-Schwarz inequality and Xu’s inequality, strong convergence theorems of the new two-step viscosity iterative process for set-valued nonexpansive and contraction operators in complete CAT(0) spaces are provided. The results of this paper improve and extend the corresponding main theorems in the literature.

A strong convergence theorem for monotone inclusion and minimization problems in complete CAT(0) spaces

ABSTRACTIn this paper, a Halpern-type proximal point algorithm for approximating a common solution of monotone inclusion problem, minimization problem (MP) and fixed point problem is introduced. Using our algorithm, we state and prove a strong convergence theorem for approximating an element in the intersection of the set of solutions of a monotone inclusion problem, the set of solutions of an MP and the set of solutions of a fixed point problem for non-expansive mappings in complete CAT(0) spaces.

Modified Proximal Point Algorithms for Solving Constrained Minimization and Fixed Point Problems in Complete CAT(0) Spaces

In this paper, we propose a new modified proximal point algorithm for a countably infinite family of nonexpansive mappings in complete CAT(0) spaces and prove strong convergence theorems for the proposed process under suitable conditions. We also apply our results to solving linear inverse problems and minimization problems. Several numerical examples are given to show the efficiency of the presented method.

An abstract proximal point algorithm

The proximal point algorithm is a widely used tool for solving a variety of convex optimization problems such as finding zeros of maximally monotone operators, fixed points of nonexpansive mappings, as well as minimizing convex functions. The algorithm works by applying successively so-called “resolvent” mappings associated to the original object that one aims to optimize. In this paper we abstract from the corresponding resolvents employed in these problems the natural notion of jointly firmly nonexpansive families of mappings. This leads to a streamlined method of proving weak convergence of this class of algorithms in the context of complete CAT(0) spaces (and hence also in Hilbert spaces). In addition, we consider the notion of uniform firm nonexpansivity in order to similarly provide a unified presentation of a case where the algorithm converges strongly. Methods which stem from proof mining, an applied subfield of logic, yield in this situation computable and low-complexity rates of convergence.

Correlation of Patient-Reported Outcome Measurement Information System Physical Function Upper Extremity Computer Adaptive Testing, With the American Shoulder and Elbow Surgeons Shoulder Assessment Form and Simple Shoulder Test in Patients With Shoulder Pain

To evaluate the Patient-Reported Outcome Measurement Information System Physical Function Upper Extremity Computer Adaptive Testing (PROMIS PFUE CAT) measurement tool against the already validated American Shoulder and Elbow Surgeons Shoulder Assessment Form (ASES) and the Simple Shoulder Test (SST) in patients presenting with shoulder pain and determine the responder burden for each of the 3 surveys. Ninety patients presenting with shoulder pain were asked to fill out the ASES, SST, and PROMIS PFUE CAT. The time for completion of each survey was measured to determine responder burden, and the Pearson correlation between the 3 instruments was defined as excellent (r > 0.7), excellent-good (0.61 ≤ r ≤ 0.7), good (0.31 ≤ r ≤ 0.6), and poor (0.2 ≤ r ≤ 0.3). The PROMIS PFUE CAT showed an excellent correlation with the SST (r= 0.82, P < .001) and ASES (r= 0.72, P<.001). The average time to complete SST, ASES, and PROMIS PFUE CAT was 92.8 ± 35.8, 142.3 ± 60.1, and 61.3±28.8seconds, respectively. The time to complete the PROMIS PFUE CAT was significantly less than both the SST (P<.001) and ASES (P < .001). The PROMIS PFUE CAT showed an excellent correlation with the previously validated ASES and SST in patients with shoulder pain. The time saving of the PROMIS PFUE CAT was found to be smaller than that of the ASES and SST but shows that moving forward, using the PROMIS PFUE CAT would not place any additional burden on the patient filling out the survey. The lack of ceiling or floor effects with the PROMIS PFUE CAT indicates its ability to differentiate both high and low functioning patients. All of these findings indicate that the PROMIS PFUE CAT is an adequate tool for the evaluation of patients with shoulder pain and should be used in these patients going forward. Level II, diagnostic study.

On the Krein-Milman theorem in CAT(κ) spaces

Let κ &gt; 0 and (X, ρ) be a complete CAT(κ) space whose diameter smaller than ... It is shown that if K is a nonempty compact convex subset of X, then K is the closed convex hull of its set of extreme points. This is an extension of the Krein-Milman theorem to the general setting of CAT(κ) spaces.

Ishikawa iterative process for hemicontractive multi-valued non-self mappings in CAT(0) spaces

Our purpose in this paper is to construct Ishikawa iterative scheme formulti-valued non-self mappings in CAT(0) spaces. Then we obtain strong convergence of the scheme to a fixed point of multi-valued hemicontractive non-self mapping in a complete CAT(0) space. In addition, we define pseudocontractive mapping in CAT(0) spaces and show that a pseudocontractive mapping T with \(F(T)\not =\emptyset \) and \(Tp=\{p\}, \forall p\in F(T)\) is hemicontractive mapping. Furthermore, we give an example of hemicontractive mapping which is not pseudocontarctive to show that the converse is not necessarily true. Our theorems improve and unify most of the results in the literature.

Approximating fixed points of generalized nonexpansive mappings in CAT(k) spaces via modified S-iteration process

In this paper, we approximate common fixed points of two generalized nonexpansive mappings under modified S-iteration process in complete CAT(k) spaces and establish some $$\Delta $$ -convergence theorems. Our results generalize and improve the corresponding known results of the existing literature.

Strong convergence of modified viscosity implicit approximation methods for asymptotically nonexpansive mappings in complete CAT(0) spaces

Strong convergence of modified viscosity implicit approximation methods for asymptotically nonexpansive mappings in complete CAT(0) spaces

Iterative approaches to solving convex minimization problems and fixed point problems in complete CAT(0) spaces

In this paper, we propose a new modified proximal point algorithm for finding a common element of the set of common minimizers of a finite family of convex and lower semi-continuous functions and the set of common fixed points of a finite family of nonexpansive mappings in complete CAT(0) spaces, and prove some convergence theorems of the proposed algorithm under suitable conditions. A numerical example is presented to illustrate the proposed method and convergence result. Our results improve and extend the corresponding results existing in the literature.

Viscosity approximation methods for the implicit midpoint rule of asymptotically nonexpansive mapping in complete CAT(0) spaces

Viscosity approximation methods for the implicit midpoint rule of asymptotically nonexpansive mapping in complete CAT(0) spaces

Fixed point theorems for (L)-type mappings in complete CAT(0) spaces

Fixed point theorems for (L)-type mappings in complete CAT(0) spaces

Translation Lengths of Parabolic Isometries of CAT(0) Spaces and Their Applications

In this article, we provide a sufficient and necessary condition on parabolic isometries of positive translation lengths on complete visibility CAT(0) spaces. One of the consequences is that each parabolic isometry of a complete simply connected visibility manifold of nonpositive sectional curvature has zero translation length. Applications on the geometry of open negatively curved manifolds will also be discussed.

Evaluation of nutritional characteristics of commercial canned cat diets

To evaluate commercial complete canned cat foods according to their composition, labeling and nutritional characteristics. A total of 21 commercial complete canned compound feeds for adult cats were analyzed for crude nutrients, minerals, vitamins, selected amino acids and taurine. The analyzed parameters were compared to the internal set of standards of the European Pet Food Industry Federation (FEDIAF). The energy content was calculated and compared with the labeled recommendations regarding the amounts of diet that should be fed. Analyzed nutrients were compared with the labeled nutrients according to the regulations of the EU food and feed law (directive EU regulation 767/2009). In many cases, the labeled feeding protocols did not match the calculated daily energy requirements. In eight complete foods, the recommended daily feed amounts were underestimated and four recommendations exceeded energy requirements of adult cats. In 12 complete foods, the calcium and phosphorus contents were threefold higher than the respective requirement. In 16 of 21 complete foods, substantial discrepancies were observed between the recommendations and the analyzed trace elements. In particular, selenium contents exceeded the selenium requirement more than threefold. The vitamin, arginine and taurine contents showed no significant discrepancies to the recommendations. With respect to the labeled nutrients, there were only minor deviations from the regulations of the European law. In general, healthy adult cats are adequately supplied with energy and nutrients when feeding commercial canned complete diets for cats. In cases of body weight loss or gain, the labelled feed amounts should be questioned. The high phosphorus contents are an issue of concern, because a high phosphorus intake can potentially increase the risk for urinary stones and particularly for older cats the risk for renal insufficiency. Furthermore, it is recommended to decrease the high selenium levels by the reduction of selenium-rich feed materials such as offal.

Convergence Theorems for Non-Self Mappings in CAT(0) Spaces

ABSTRACTIn this paper, we prove strong convergence of Mann iterative schemes to fixed points of multi-valued demicontractive non-self mappings in complete CAT(0) spaces under appropriate conditions. In addition, △− convergence or strong convergence of Mann iterative scheme to a fixed point of single-valued k-strictly pseudocontractive non-self mapping is obtained. Our theorems improve and unify most of the results in the literature.

Halpern-type proximal point algorithm in complete CAT(0) metric spaces

Abstract First, Halpern-type proximal point algorithm is introduced in complete CAT(0) metric spaces. Then, Browder convergence theorem is considered for this algorithm and also we prove that Halpern-type proximal point algorithm converges strongly to a zero of the operator.