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IntroductionThe autonomy-supportive coaching style is recognized for its positive impact on athletes’ well-being and performance. However, the transition of excessive autonomy into a laissez-faire coaching style has not been thoroughly examined within the context of coach evaluation scales. Existing scales focus predominantly on the positive dimensions of autonomy support, and do not possess the capabilities to measure outcomes which may be viewed as negative or other outcomes. This study aims to integrate the autonomy-supportive and laissez-faire coaching styles within the same measurement framework.MethodsOur study developed a comprehensive scale to assess both the autonomy-supportive and laissez-faire coaching styles, drawing on items from the Sport Climate Questionnaire for autonomy support and adapting items from leadership research for laissez-faire coaching. We conducted two studies: the first with 148 athletes to refine the laissez-faire items and the second with 460 athletes to validate the full scale, utilizing exploratory factor analysis, confirmatory factor analysis, and correlation analysis. We also measured internal consistency and split-half reliability.ResultsThe finalized scale includes a 6-item autonomy-supportive subscale and a 5-item laissez-faire subscale. Validation processes confirmed the scale’s construct and criterion validity, alongside its reliability.DiscussionThe Chinese Coaches’ Autonomy-Supportive—Laissez-Faire Coaching Style Scale effectively captures both the beneficial and potentially detrimental aspects of coaching styles, addressing a critical gap in the literature and providing a reliable tool for evaluating coaching approaches.

PurposeCoaches’ behaviors and coaching styles play a critical role in influencing athletes’ psychological experiences and performance. According to the self-determination theory (SDT), coaches’ interpersonal behaviors are commonly categorized as autonomy-supportive and controlling. Due to less focus on the unique behaviors of Chinese coaches, this study incorporated coaches’ parental care for athletes, referred to as paternalistic benevolence, in their interpersonal styles in the context of the Chinese culture.MethodsExploratory factor analyses were used in studies 1 and 2 to find items associated with benevolent coaching behaviors and items to create the Chinese Coaches’ Interpersonal Style Scale. Study 3 used the constructed scale, as well as the Subjective Vitality Scale and Athlete Burnout Questionnaire, with a sample of athletes to examine scale reliability. The 15-item Chinese Coaches’ Interpersonal Style Scale contained three dimensions: benevolent, autonomy-supportive, and controlling coaching styles.ResultsThe findings showed that: (1) benevolent coaching behaviors held significant explanatory weight in the Chinese cultural context; (2) controlling and autonomy-supportive coaching styles were culturally congruent among both Eastern and Western athletes; and (3) benevolent and autonomy-supportive coaching behaviors positively impacted athletes, whereas controlling coaching behaviors had a negative impact.ConclusionThe measure showed strong validity and reliability, making it useful for future practice and research on the interpersonal style of Chinese coaches.

This article is mainly concerned with the approximate controllability for some semi-linear fractional integro-differential impulsive evolution equations of order 1<α<2 with delay in Banach spaces. Firstly, we study the existence of the PC-mild solution for our objective system via some characteristic solution operators related to the Mainardi’s Wright function. Secondly, by using the spatial decomposition techniques and the range condition of control operator B, some new results of approximate controllability for the fractional delay system with impulsive effects are obtained. The results cover and extend some relevant outcomes in many related papers. The main tools utilized in this paper are the theory of cosine families, fixed-point strategy, and the Grönwall-Bellman inequality. At last, an example is given to demonstrate the effectiveness of our research results.

Abstract The aim of this paper is to establish some results about the existence of multiple solutions for the following singular semipositone boundary value problem of fourth-order differential systems with parameters: { u ( 4 ) ( t ) + β 1 u ″ ( t ) − α 1 u ( t ) = f 1 ( t , u ( t ) , v ( t ) ) , 0 < t < 1 ; v ( 4 ) ( t ) + β 2 v ″ ( t ) − α 2 v ( t ) = f 2 ( t , u ( t ) , v ( t ) ) , 0 < t < 1 ; u ( 0 ) = u ( 1 ) = u ″ ( 0 ) = u ″ ( 1 ) = 0 ; v ( 0 ) = v ( 1 ) = v ″ ( 0 ) = v ″ ( 1 ) = 0 , $$ \textstyle\begin{cases} u^{(4)}(t)+\beta _{1}u''(t)-\alpha _{1}u(t)=f_{1}(t,u(t),v(t)),\quad 0< t< 1; \\ v^{(4)}(t)+\beta _{2}v''(t)-\alpha _{2}v(t)=f_{2}(t,u(t),v(t)),\quad 0< t< 1; \\ u(0)=u(1)=u''(0)=u''(1)=0; \\ v(0)=v(1)=v''(0)=v''(1)=0, \end{cases} $$ where f 1 , f 2 ∈ C [ ( 0 , 1 ) × R 0 + × R , R ] $f_{1},f_{2}\in C[(0,1)\times \mathbb{R}^{+}_{0}\times \mathbb{R}, \mathbb{R}]$ , R 0 + = ( 0 , + ∞ ) $\mathbb{R}_{0}^{+}=(0,+\infty )$ . By constructing a special cone and applying fixed point index theory, some new existence results of multiple solutions for the considered system are obtained under some suitable assumptions. Finally, an example is worked out to illustrate the main results.

Coaching style is key to athletes’ performance and mental well-being. However, few attempts have examined the effects of coaching style on athletes in a Chinese cultural context. Based on previous literature and 23 expert interviews (11 with athletes and 12 with coaches), we rebuilt a 16-items on the Controlling Coach Behaviors Scale. In study 1, 130 provincial team athletes completed a 16-items questionnaire on controlling coach behaviors. The questionnaire items were then screened using exploratory factor analysis and transformed into a formal scale. In study 2, another 560 provincial athletes completed several measures related to coaching style, motivation, subjective vitality, and burnout, and systematic tests were carried out to validate the scale. Study 3 examined the test-retest reliability of the Controlling Coach Behaviors Scale over a 2-week interval. Finally, the present study yielded a nine-item Controlling Coach Behaviors Scale with three dimensions (controlling use of reward, negative conditional regard, and excessive personal control). It suggests that cultural differences played an important role in the communication between athletes and coaches. The new Controlling Coach Behaviors Scale shows good validity and can be used in future research.

This manuscript mainly discusses the approximate controllability for certain fractional delay evolution equations in Banach spaces. We introduce a suitable complete space to deal with the disturbance due to the time delay. Compared with many related papers on this issue, the major tool we use is a set of differentiable properties based on resolvent operators, rather than the theory of C0-semigroup and the properties of some associated characteristic solution operators. By implementing an iterative method, some new controllability results of the considered system are derived. In addition, the system with non-local conditions and a parameter is also discussed as an extension of the original system. An instance is proposed to support the theoretical results.

Under a new generalized definition of exact controllability we introduced and with a appropriately constructed time delay term in a special complete space to overcome the delay-induced-difficulty, we establish the sufficient conditions of the exact controllability for a class of impulsive fractional nonlinear evolution equations with delay by using the resolvent operator theory and the theory of nonlinear functional analysis. Nonlinearity in the system is only supposed to be continuous rather than Lipschitz continuous by contrast. The results obtained in the present work are generalizations and continuations of the recent results on this issue. Further, an example is presented to show the effectiveness of the new results.

The present work addresses some new controllability results for a class of fractional integrodifferential dynamical systems with a delay in Banach spaces. Under the new definition of controllability , first introduced by us, we obtain some sufficient conditions of controllability for the considered dynamic systems. To conquer the difficulties arising from time delay, we also introduce a suitable delay item in a special complete space. In this work, a nonlinear item is not assumed to have Lipschitz continuity or other growth hypotheses compared with most existing articles. Our main tools are resolvent operator theory and fixed point theory. At last, an example is presented to explain our abstract conclusions.

In the present paper, sufficient conditions ensuring the complete controllability for a class of semilinear fractional nonlocal evolution systems with finite delay in Banach spaces are derived. The new results are obtained under a weaker definition of complete controllability we introduced, and then the Lipschitz continuity and other growth conditions for the nonlinearity and nonlocal item are not required in comparison with the existing literatures. In addition, an appropriate complete space and a corresponding time delay item are introduced to conquer the difficulties caused by time delay. Our main tools are properties of resolvent operators, theory of measure of noncompactness, and Mönch fixed point theorem.

This paper is mainly concerned with the exact controllability for a class of impulsive ψ-Caputo fractional evolution equations with nonlocal conditions. First, by generalized Laplace transforms, a mild solution for considered problems is introduced. Next, by the Mönch fixed point theorem, the exact controllability result for the considered systems is obtained under some suitable assumptions. Finally, an example is given to support the validity of the main results.

This paper is concerned with a class of implicit-type coupled system with integral boundary conditions involving Caputo fractional derivatives. First, the existence result of solutions for the considered system is obtained by means of topological degree theory. Next, Ulam–Hyers stability and generalized Ulam–Hyers stability are studied under some suitable assumptions. Finally, one example is worked out to illustrate the main results.

In this paper, sufficient conditions ensuring existence and multiplicity of positive solutions for a class of nonlinear singular fractional differential systems are derived with Riemann–Stieltjes coupled integral boundary value conditions in Banach Spaces. Nonlinear functions f(t,u,v) and g(t,u,v) in the considered systems are allowed to be singular at every variable. The boundary conditions here are coupled forms with Riemann–Stieltjes integrals. In order to overcome the difficulties arising from the singularity, a suitable cone is constructed through the properties of Green’s functions associated with the systems. The main tool used in the present paper is the fixed point theorem on cone. Lastly, an example is offered to show the effectiveness of our obtained new results.

This paper is concerned with multiple solutions for a class of nonlinear fourth-order boundary value problems with parameters. By constructing a special cone and applying fixed point index theory, the multiple solutions for the considered systems are obtained under some suitable assumptions. The main feature of obtained solutions (u(t),v(t)) is that the solution u(t) is positive, and the other solution v(t) may change sign. Finally, two examples with continuous function f1 being positive and f2 being semipositone are worked out to illustrate the main results.

By using the theory of fixed-point index on cone for differentiable operators, spectral radii of some related linear integral operators, and properties of Green’s function, the existence of multiple positive solutions to a nonlinear fractional differential system with integral boundary value conditions and a parameter is established. At last, some examples are also provided to illustrate the validity of our main results.

This paper is devoted to investigating the eigenvalue problems of a class of nonlinear impulsive singular boundary value problem in Banach spaces: μx′′+f(t,x)=0,t∈(0,1),t≠ti; Δx|t=ti=αix(ti-0),i=1,2,…,k;ax(0)-bx′(0)=θ;cx(1)+dx′(1)=θ, where θ denotes the zero element of Banach space, Δx|t=ti=x(ti+0)-x(ti-0), αi>-1, a,b,c,d∈R+,γ=ac+ad+bc>0, μ is a parameter, and f(t,x) may be singular at t=0,1 and x=θ. The arguments are mainly based upon the theory of fixed point index, measure of noncompactness, and the special cone, which is constructed to overcome the singularity.

This paper is devoted to the existence of multiple positive solutions for fractional boundary value problem DC0+αu(t)=f(t,u(t),u′(t)), 0

This paper investigates the existence of at least two positive solutions for the following high-order fractional semipositone boundary value problem (SBVP, for short) with coupled integral boundary value conditions:.. where n - 1 < α ≤ n, n ≥ 3, 0 < η1, η2 ≤ 1, λ, λ1, λ2 are parameters and satisfy λ1λ2(η1η2) α < Γ²(α + 1), Dα0+ is the standard Riemann-Liouville derivative, and f, g are continuous and semipositone. By using the nonlinear alternative of Leray-Schauder type, Krasnoselskii's fixed point theorems, and the theory of fixed point index on cone, we establish some existence results of multiple positive solutions to the considered fractional SBVP. As applications, two examples are presented to illustrate our main results. [ABSTRACT FROM AUTHOR]

This paper investigates the existence of positive solutions for the following high-order nonlinear fractional differential boundary value problem (BVP, for short) {Dα0+u(t)+f(t,v(t)) = 0, t ∊ (0,1), Dα0+v(t)+g(t,u(t)) = 0, t ∊ (0,1), u(j) (0) = v(j) (0) = 0, 0≤j≤n-1, j ≠ 1, u' (1) = λ ∫10 u(t)dt, v' (1) = λ ∫10 v(t)dt, where n - 1 < α ≤ n, n ≥ 3; 0 ≤ λ < 2, Dα0+ is the Caputo fractional derivative. By using the monotone method, the theory of fixed point index on cone for di erentiable operators and the properties of Green's function, some new uniqueness and existence criteria for the considered fractional BVP are established. As applications, some examples are worked out to demonstrate the main results. [ABSTRACT FROM AUTHOR]

This study aims to import multi-source information fusion (MSIF) into structural damage diagnosis to improve its validity. Two structural damage identification methods based on MSIF are put forward, one of which is to fuse two or more structural damage detection methods by MSIF and another of which is the improved modal strain energy method by multi-mode information processing based on MSIF. Through a concrete plate experiment it is proved that, if two methods are integrated by character-level information fusion, structural initial damages can be more accurately identified than by a single method. In a simulation of a concrete box beam bridge, it is indicated that the improved modal strain energy method has a nice sensitivity to structural initial damages and a favorable robusticity to noise. These two structural damage diagnosis methods based on MSIF have good effects on structural damage identification and good practicability to actual structures. [ABSTRACT FROM AUTHOR]