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In this paper we construct two limit cycles with a heteroclinic polycycle for the three-dimensional continuous model of the rock–scissors–paper (RSP) game between bacteriocin producing bacteria. Our construction gives a partial answer to an open question posed by Neumann and Schuster (2007) concerning how many limit cycles can coexist for the RSP game. [ABSTRACT FROM AUTHOR]

In this short note we indicate some improvement of an article published in this journal by Crespo, Ivorra and Ramos 6. The techniques used are connected with several smoothing effects associated with linear partial differential operators which give rise to some accretive operators in L¹(Ω), as well as with some H²(Ω) estimates independent on time. [ABSTRACT FROM AUTHOR]

The filled function method is a deterministic algorithm for finding a global minimizer of global optimization problems, and its effectiveness is closely related to the form of the constructed filled function. Currently, the filled functions mainly have three drawbacks in form, namely, parameter adjustment and control (if any), inclusion of exponential or logarithmic functions, and properties that are discontinuous and non-differentiable. In order to overcome these limitations, this paper proposed a parameter-free filled function that does not include exponential or logarithmic functions and is continuous and differentiable. Based on the new filled function, a filled function method for solving unconstrained global optimization problems was designed. The algorithm selected points in the feasible domain that were far from the global minimum point as initial points, and improved the setting of the step size in the stage of minimizing the filled function to enhance the algorithm’s global optimization capability. In addition, tests were conducted on 14 benchmark functions and compared with existing filled function algorithms. The numerical experimental results showed that the new algorithm proposed in this paper was feasible and effective. [ABSTRACT FROM AUTHOR]

Abstract: This paper provides a generalization of the main lemma and a correct proof of an extended version of Theorem 2, in [G. Vidossich, Differential inequalities for evolution equations, Nonlinear Anal. TMA 25 (1995) 1063–1069]. [Copyright &y& Elsevier]

In this paper several tensorial norm inequalities for continuous functions of selfadjoint operators in Hilbert spaces have been obtained. Multiple inequalities are obtained with variations due to the convexity properties of the mapping f || 1/6 [f(A) x 1 + 4f - A - 1 + 1 B 2 - + 1 f(B) - - Z 1 0 f((1 - k)A < 1 + k1 B)dk || 5\36 1 B-A 1 f'I,+8. [ABSTRACT FROM AUTHOR]

The paper investigates uniformly closed subspaces, sublattices, and ideals of finite codimension in Archimedean vector lattices. It is shown that every uniformly closed subspace (or sublattice) of finite codimension may be written as an intersection of uniformly closed subspaces (respectively, sublattices) of codimension one. Every uniformly closed sublattice of codimension n contains a uniformly closed ideal of codimension at most $ 2n $ 2 n. If the vector lattice is uniformly complete then every ideal of finite codimension is uniformly closed. Results of the paper extend (and are motivated by) results of Abramovich Y.A., Lipecki Z. [On ideals and sublattices in linear lattices and F-lattices. Math Proc Cambridge Philos Soc. 1990;108(1):79–87.; On lattices and algebras of simple functions. Comment Math Univ Carolin. 1990;31(4):627–635.], as well as Kakutani's characterization of closed sublattices of $ C(K) $ C (K) spaces. [ABSTRACT FROM AUTHOR]

The traditional Satcom-on-the-Move (SOTM) mechanical structure consists of a dual-axis configuration with an azimuth axis and a pitch axis. In this structure, when the pitch angle is 90 degrees, the rotation of the azimuth axis cannot change the antenna's direction. To solve this issue, a three-axis SOTM mechanical structure has been developed. The traditional three-axis SOTM servo control system adopts a closed-loop control scheme. In this scheme, due to the difficulty in directly obtaining the antenna's rotation angle, the angles of rotation for each axis are typically selected to represent the antenna's rotation angle. The closed-loop feedback includes the angles and angular velocities of the axes, which cannot completely capture the antenna's motion state, essentially constituting an indirect closed-loop control. Addressing the shortcomings of this indirect closed-loop control, this paper first establishes the kinematic relations between the axes of the three-axis SOTM antenna using the Denavit–Hartenberg (DH) method. Subsequently, the relationship between antenna pointing and the rotational states of the three axes was derived using the Jacobian operator. Building upon this foundation, a direct closed-loop control structure for a three-axis SOTM antenna was designed. To enable the control system to achieve rapid convergence with minimal overshoot, an Active Disturbance Rejection Control (ADRC) algorithm based on smooth continuous functions is introduced as the inner and outer loop controller algorithms within the direct closed-loop control structure. To address the nonlinearity in the design scheme, a piecewise linearization method is proposed to reduce the demands on the microprocessor's performance and enhance the engineering feasibility of the solution. Finally, the effectiveness of the proposed approach is validated through experiments. The experimental results demonstrate that compared to traditional indirect closed-loop control methods, utilizing the direct closed-loop control method for the three-axis SOTM antenna presented in this paper can lead to higher precision in pointing the antenna towards satellites and enhance communication effectiveness. [ABSTRACT FROM AUTHOR]

In this paper several tensorial norm inequalities for continuous functions of selfadjoint operators in Hilbert space have been obtained. The recent progression of the Hilbert space inequalities following the definition of the convex operator inequality has lead researchers to explore the concept of Hilbert space inequalities even further. The motivation for this paper stems from the recent development in the theory of tensorial and Hilbert space inequalities. Multiple inequalities are obtained with variations due to the convexity properties of the mapping f... 1 6 A01 + 10 B 2) -exp(A) 0 1 + 4 exp +10exp(B) 1-4 + ∣ exp((--~-a 0 1 1 C + k} 1 0 b) k-1 dk ∕ exp-1 -- A 0 1 + k 1 0 (1 -- k)- 2dk-|| 47 ≤ "1 0 B -- A 0 1∣∣2 (Hexp(A)H + l∣exp(B)∣∣)∙ 360 Tensorial version of a Lemma given by Hezenci is derived and utilized to obtain the desired inequalities. in the introduction section is given a brief history of the inequalities, while in the preliminary section we give necessary Lemmas and results in order to understand the paper. Structure and novelty of the paper are discussed at the end of the introduction section. [ABSTRACT FROM AUTHOR]

Abstract: In this note, we will modify several gaps in Chidume and Ofoedu [C.E. Chidume, E.U. Ofoedu, A new iteration process for generalized Lipschitz pseudo-contractive and generalized Lipschitz accretive mappings, Nonlinear Anal. (2006), in press (doi:10.1016/j.na.2006.05.012)]. [Copyright &y& Elsevier]

In order to research uniform continuity of fractal interpolation surface function on a closed rectangular area, the accumulation principle was applied to prove uniform continuity of fractal interpolation surface function on a closed rectangular area. First, fractal interpolation surface function was constructed by affine mapping. Second, the continuous concept of fractal interpolation surface function at a planar point in a three-dimensional cartesian coordinate space system and uniform continuity of fractal interpolation surface function on a closed rectangular area were defined in the paper. Finally, the uniformly continuous theorem of fractal interpolation surface function was proven through accumulation principle in the paper. The conclusion showed that fractal interpolation surface was uniformly continuous function on a closed rectangular area. [ABSTRACT FROM AUTHOR]

This paper investigates models of working memory in which memory traces evolve according to stochastic attractor dynamics. These models have previously been shown to account for response-biases that are manifest across multiple trials of a visual working memory task. Here we adapt this approach by making the stable fixed points correspond to the multiple items to be remembered within a single-trial, in accordance with standard dynamical perspectives of memory, and find evidence that this multi-item model can provide a better account of behavioural data from continuous-report tasks. Additionally, the multi-item model proposes a simple mechanism by which swap-errors arise: memory traces diffuse away from their initial state and are captured by the attractors of other items. Swap-error curves reveal the evolution of this process as a continuous function of time throughout the maintenance interval and can be inferred from experimental data. Consistent with previous findings, we find that empirical memory performance is not well characterised by a purely-diffusive process but rather by a stochastic process that also embodies error-correcting dynamics. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of functions called strongly θ(Λ, p)-continuous functions. Moreover, we investigate several characterizations and some properties concerning strongly θ(Λ, p)-continuous functions. [ABSTRACT FROM AUTHOR]

In the subject paper, a new set of sensitivity indices for multivariate outputs are defined when the uncertainty on the input factors is either discrete or continuous and when the dynamic model is either discrete or functional. Admittedly, these indices can quantify the contribution of each input factor to each principal component and the total inertia reasonably, and provide rich information on dynamic model behaviour. However, Propositions 2–4 in the subject paper which claim that the generalized sensitivity index is equal to the sensitivity index defined on the sum of the principal components are not correct. These incorrect propositions are concluded by the incorrect derivation in the Appendices B and C of the subject paper. This discussion intends to clarify this issue associated with the paper. [ABSTRACT FROM AUTHOR]

This paper presents a family of new continuous wavelet function and describes few properties of it. A method of estimating parameters for a Gaussians using this new continuous wavelet transform is described. The robustness of this method for estimating the parameters in the presence of noise is also studied by simulating different possible cases. [ABSTRACT FROM AUTHOR]

The antidisturbance control problem of autonomous vehicle path tracking considering lateral stability is studied in this paper. This paper proposes an improved active disturbance rejection control (IADRC) control method including an improved extended state observer (IESO) and an error compensator based on LQR, where a new continuous nonlinear function is proposed in the IESO instead of the classical piecewise function. Based on the IADRC, an autonomous vehicle path-tracking controller considering lateral stability is designed. Using the output wheel steering angle and external yaw moment, the IESO estimates the disturbance value and compensates for the disturbance in the feedback to meet the goal of antidisturbance control. Based on the concept of control allocation (CA), the control distributor is designed to distribute the external yaw moment to the four wheels in a reasonable and optimal way to achieve differential braking. Finally, the control scheme is evaluated in the form of CarSim/Simulink cosimulation; the results show that the proposed autonomous vehicle path-tracking control scheme has better path-tracking effect and higher antidisturbance robustness. [ABSTRACT FROM AUTHOR]

In this paper, we study a method of polynomial interpolation that lies in-between Lagrange and Hermite methods. The novelty is that we use very general nodal systems on the unit circle as well as on the bounded interval only characterized by a separation property. The way in which we interpolate consists in considering all the nodes for the prescribed values and only half for the derivatives. Firstly, we develop the theory on the unit circle, obtaining the main properties of the nodal polynomials and studying the convergence of the interpolation polynomials corresponding to continuous functions with some kind of modulus of continuity and with general conditions on the prescribed values for half of the derivatives. We complete this first part of the paper with the study of the convergence for smooth functions obtaining the rate of convergence, which is slightly slower than that when equidistributed nodal points are considered. The second part of the paper is devoted to solving a similar problem on the bounded interval by using nodal systems having good properties of separation, generalizing the Chebyshev–Lobatto system, and well related to the nodal systems on the unit circle studied before. We obtain an expression of the interpolation polynomials as well as results about their convergence in the case of continuous functions with a convenient modulus of continuity and, particularly, for differentiable functions. Finally, we present some numerical experiments related to the application of the method with the nodal systems dealt with. [ABSTRACT FROM AUTHOR]

The famous Ascoli-Arzelà theorem served as a springboard for research into compactness in function spaces, particularly spaces of continuous functions. This paper investigates compact subsets of spaces of continuous functions endowed with topologies between the topology of pointwise convergence and the topology of uniform convergence. More precisely, this paper studies necessary and sufficient conditions for a subset to be compact in Cλ,u(X) for a locally-λ space X when λ ⊇ 퓕(X), for a hemi- λ λf-space X when λ ⊆ 퓟 퓢(X), and for a k-space X when λ ⊇ 퓚(X). This paper also studies that every bounded subset of Cλ,u(X) has compact closure for some classes of topological spaces X. [ABSTRACT FROM AUTHOR]

In this paper several tensorial norm inequalities for continuous functions of selfadjoint operators in Hilbert spaces have been obtained. Multiple inequalities of the form.. are obtained with variations due to the convexity properties of the mapping f. [ABSTRACT FROM AUTHOR]

In this paper, absolute g*ωα-continuous functions in bigeneralized topological spaces is introduced and characterized. [ABSTRACT FROM AUTHOR]

In this paper, various tensorial inequalities of trapezoid type were obtained. Identity from classical analysis is utilized to obtain the tensorial version of the said identity which in turn allowed us to obtain tensorial inequalities in Hilbert space. The continuous functions of self-adjoint operators in Hilbert spaces have several tensorial norm inequalities discovered in this study. The convexity features of the mapping f lead to the variation in several inequalities of the trapezoid type. [ABSTRACT FROM AUTHOR]

In this paper, I will answer a question that Albrecht asked me when preparing a paper on spectral instabilities: Is the ε -pseudospectrum of a Toeplitz operator T (a) , say with a continuous generating function a , always contained in the ε -pseudospectrum of the compactly perturbed operator T (a) + K ? [ABSTRACT FROM AUTHOR]

In this paper, the fixed-/preassigned-time anti-synchronisation problem is studied for delayed memristive neural networks (DMNNs) with discontinuous activation functions and external disturbance. First of all, several simple controllers are structured for the fixed-time anti-synchronisation (FTAS) of DMNNs with discontinuous activation functions. Moreover, an effective and convenient control scheme for preassigned-time anti-synchronisation (PTAS) is presented by modifying the controller parameters. The proposed control scheme is also available for DMNNs with continuous activation functions or without time delays. In addition, sufficient conditions for FTAS and PTAS of DMNNs are derived and the accurate estimation for settling-time is given. Finally, simulation examples are presented to verify the validity of the main results. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the study of the following double-phase equation - div (| ∇ u | p - 2 ∇ u + μ ε (x) | ∇ u | q - 2 ∇ u) + V ε (x) (| u | p - 2 u + μ ε (x) | u | q - 2 u) = f (u) in R N , where N ≥ 2 , 1 < p < q < N , q < p ∗ with p ∗ = Np N - p , μ : R N → R is a continuous non-negative function, μ ε (x) = μ (ε x) , V : R N → R is a positive potential satisfying a local minimum condition, V ε (x) = V (ε x) , and the nonlinearity f : R → R is a continuous function with subcritical growth. Under natural assumptions on μ , V and f, by using penalization methods and Lusternik–Schnirelmann theory we first establish the multiplicity of solutions, and then, we obtain concentration properties of solutions. [ABSTRACT FROM AUTHOR]

Considering that the existing path planning algorithms for mobile robots under rugged terrain do not consider the ground flatness and the lack of optimality, which leads to the instability of the center of mass of the mobile robot, this paper proposes an improved A* algorithm for mobile robots under rugged terrain based on the ground accessibility model and the ground ruggedness model. Firstly, the ground accessibility and ruggedness models are established based on the elevation map, expressing the ground flatness. Secondly, the elevation cost function that can obtain the optimal path is designed based on the two types of models combined with the characteristics of the A* algorithm, and the continuous cost function is established by connecting with the original distance cost function, which avoids the center-of-mass instability caused by the non-optimal path. Finally, the effectiveness of the improved algorithm is verified by simulation and experiment. The results show that compared with the existing commonly used path planning algorithms under rugged terrain, the enhanced algorithm improves the smoothness of paths and the optimization degree of paths in the path planning process under rough terrain. [ABSTRACT FROM AUTHOR]

In this paper, we study a higher order Kirchhoff problem with variable exponent of type M∫Ω|Dru|m(x)m(x)dxΔm(x)ru=f(x,u)inΩ,Dαu=0,on∂Ω,for eachα∈ℝNwith|α|≤r−1,$$ \left\{\begin{array}{ll}M\left({\int}_{\Omega}\frac{{\left&#x0007C;{\mathcal{D}}_ru\right&#x0007C;}&#x0005E;{m(x)}}{m(x)} dx\right){\Delta}_{m(x)}&#x0005E;ru&#x0003D;f\left(x,u\right)& \mathrm{in}\kern0.30em \Omega, \\ {}{D}&#x0005E;{\alpha }u&#x0003D;0,\kern0.30em & \mathrm{on}\kern0.30em \mathrm{\partial \Omega },\kern0.30em \mathrm{for}\ \mathrm{each}\kern0.4em \alpha \in {\mathrm{\mathbb{R}}}&#x0005E;N\kern0.4em \mathrm{with}\kern0.4em \mid \alpha \mid \le r-1,\end{array}\right. $$where Ω⊂ℝN$$ \Omega \subset {\mathrm{\mathbb{R}}}&#x0005E;N $$ is a smooth bounded domain, r∈ℕ∗,m∈C(Ω‾),1

This paper uses the Galerkin method to investigate the existence of positive solution to a class of singular elliptic problems given by { − Δ u = λ 0 u β 0 + Λ 0 | ∇ u | γ 0 + f 0 (u) | x | α 0 + h 0 (x) , u > 0 in Ω , u = 0 on ∂ Ω , where Ω ⊂ R 2 is a bounded smooth domain, 0 < β 0 , γ 0 ≤ 1 , α 0 ∈ [ 0 , 2) , h 0 (x) ≥ 0 , h 0 ≠ 0 , h 0 ∈ L ∞ (Ω) , 0 < ∥ h 0 ∥ ∞ < λ 0 < Λ 0 , and f 0 are continuous functions. More precisely, f 0 has a critical exponential growth, that is, the nonlinearity behaves like exp (ϒ ‾ s 2) as | s | → ∞ , for some ϒ ‾ > 0 . [ABSTRACT FROM AUTHOR]

The classical Rolle's theorem establishes the existence of (at least) one zero of the derivative of a continuous one-variable function on a compact interval in the real line, which attains the same value at the extremes, and it is differentiable in the interior of the interval. In this paper, we generalize the statement in four ways. First, we provide a version for functions whose domain is in a locally convex topological Hausdorff vector space, which can possibly be infinite-dimensional. Then, we deal with the functions defined in a real interval: we consider the case of unbounded intervals, the case of functions endowed with a weak derivative, and, finally, we consider the case of distributions over an open interval in the real line. [ABSTRACT FROM AUTHOR]

This paper attempts to explore possible box dimension of two added fractal continuous functions with the same dimension. Two interesting and meaningful results are obtained. Let g(x) and h(x) have the same box dimension t (1 < t ≤ 2), the box dimension of g(x) + h(x) may or may not exist. If it exists, it can take an arbitrary real number γ satisfying 1 < γ ≤ t. If it does not exist, its lower and upper box dimensions can reach arbitrary different real numbers t1 and t2 that satisfy 1 < t1 < t2 < t ≤ 2. These unexpected conclusions drive us to probe into the characteristics of collection of all fractal continuous functions with the same box dimension under ordinary linear operations (scalar multiplication and addition). Following the known fractal features of some typical fractal functions such as the Weierstrass function Wt(x), we classify the fractal functions into three types: consistent fractal functions, non-consistent fractal functions, and simple fractal functions. By utilizing these classifications and fractal feature descriptions, the causality of the box dimension of two added fractal functions can be partially revealed. We hope that these initial superficial discussions will lead deeper consideration on the essence of variants of fractal dimension under linear combinations of fractal functions. Moreover, these fractal features may be applied further in other fields of fractals. [ABSTRACT FROM AUTHOR]

Through galleries of graphs and short filmstrips, this paper aims to sharpen students' eyes for visually recognizing continuous functions. It seeks to develop intuition for what visual features of a graph continuity does and does not allow for. We have found that even students who can work correctly with rigorous definitions may not be able to recognize a continuous function from its graph. To help mitigate this problem, these visuals can be used alongside the definitions in analysis courses, and in calculus courses where the same definitions are used. The content is arranged into a series of pictures accompanied by questions, with suggestions for the instructor on how to guide students in their learning. The design allows students to make their own discoveries and connections. The investigations crescendo to a surprising fact: that almost all continuous functions are differentiable nowhere! [ABSTRACT FROM AUTHOR]

This paper discusses the existence of positive radial symmetric solutions of the nonlinear biharmonic equation ▵ 2 u = f (u , ▵ u) on an annular domain Ω in R N with the Navier boundary conditions u | ∂ Ω = 0 and ▵ u | ∂ Ω = 0 , where f : R + × R − → R + is a continuous function. We present some some inequality conditions of f to obtain the existence results of positive radial symmetric solutions. These inequality conditions allow f (ξ , η) to have superlinear or sublinear growth on ξ , η as | (ξ , η) | → 0 and ∞. Our discussion is mainly based on the fixed-point index theory in cones. [ABSTRACT FROM AUTHOR]

A function g\colon \mathbb {R}^n\to \mathbb {R} is linearly continuous provided its restriction g\restriction \ell to every straight line \ell \subset \mathbb {R}^n is continuous. It is known that the set D(g) of points of discontinuity of any linearly continuous g\colon \mathbb {R}^n\to \mathbb {R} is a countable union of isometric copies of (the graphs of) f\restriction P, where f\colon \mathbb {R}^{n-1}\to \mathbb {R} is Lipschitz and P\subset \mathbb {R}^{n-1} is compact nowhere dense. On the other hand, for every twice continuously differentiable function f\colon \mathbb {R}\to \mathbb {R} and every nowhere dense perfect P\subset \mathbb {R} there is a linearly continuous g\colon \mathbb {R}^2\to \mathbb {R} with D(g)=f\restriction P. The goal of this paper is to show that this last statement fails, if we do not assume that f'' is continuous. More specifically, we show that this failure occurs for every continuously differentiable function f\colon \mathbb {R}\to \mathbb {R} with nowhere monotone derivative, which includes twice differentiable functions f with such property. This generalizes a recent result of professor Luděk Zajíček [On sets of discontinuities of functions continuous on all lines, arxiv.org/abs/2201.00772v1, 2022] and fully solves a problem from a 2013 paper of the first author and Timothy Glatzer [Real Anal. Exchange 38 (2012/13), pp. 377–389]. [ABSTRACT FROM AUTHOR]

The main object of this paper is to study the basic properties of somewhat gp-continuous functions using the concept of gp-open sets in topological spaces. Also, the concept of somewhat gp-open functions and somewhat gpirresolute functions with some counter examples are explained in this paper. Further, the author establishes the relationship between the new classes of functions with other classes of functions by giving examples, counterexamples, properties and characterizations. [ABSTRACT FROM AUTHOR]

The paper is devoted to the investigation of problem of approximation of continuous periodic functions by trigonometric polynomials, which are generated by linear methods of summation of Fourier series. The simplest example of a linear approximation of periodic functions is the approximation of functions by partial sums of their Fourier series. However, the sequences of partial Fourier sums are not uniformly convergent over the class of continuous periodic functions. Therefore, many studies devoted to the research of the approximative properties of approximation methods, which are generated by transformations of the partial sums of Fourier series and allow us to construct sequences of trigonometrical polynomials that would be uniformly convergent for the whole class of continuous functions. Particularly, Fej'er sums have been widely studied recently. One of the important problems in this area is the study of asymptotic behavior of the sharp upper bounds over a given class of functions of deviations of the trigonometric polynomials. In the paper, we study upper asymptotic estimates for deviations between a function and the Fej'er means for the Fourier series of the function. The asymptotic behavior is considered for the functions represented by the Poisson integrals of periodic functions of a real variable. The mentioned classes consist of analytic functions of a real variable. These functions can be regularly extended into the corresponding strip of the complex plane. An asymptotic equality for the upper bounds of Fej'er means deviations on classes of Poisson integrals was obtained. [ABSTRACT FROM AUTHOR]

The paper presents a novel approach to generalize the two-point weighted Ostrowski's formula for Riemann-Stieltjes integrals by utilizing a unique class of functions of bounded r - variation. The proposed approach yields several results that exhibit sharp and better bounds compared to existing established results by using parameters and weights. Additionally, the paper also captures many of the known results as special cases. [ABSTRACT FROM AUTHOR]

According to the Weierstrass Approximation Theorem, any continuous function on the closed and bounded interval can be approximated by polynomials. A constructive proof of this theorem uses the so-called Bernstein polynomials. For the approximation of integrable functions, we may consider Kantorovich operators as certain modifications for Bernstein polynomials. In this paper, we investigate the behaviour of Kantorovich operators in Lebesgue spaces. We first give an alternative proof of the uniform boundedness of Kantorovich operators in Lebesgue spaces by using the Riesz-Thorin Interpolation Theorem. In addition, we examine the convergence of Kantorovich operators in the space of essentially bounded functions. We also give an example related to the rate of convergence of Kantorovich operators in a subspace of Lebesgue spaces. [ABSTRACT FROM AUTHOR]

Let I ⊆ R be an interval and β : I → I a strictly increasing continuous function with a unique fixed point s0 ∈ I satisfying (t − s0)(β(t) −t) ≤ 0 for all t ∈ I. Hamza et al. introduced the general quantum difference operator Dβ by Dβ f(t) := f(β(t))−f(t) /β(t)−t if t 6= s0 and Dβ f(t) := f 0 (s0) if t = s0. In this paper, we establish results concerning Taylor’s formula associated with Dβ . For this, we define two types of monomials and then present our main results. The obtained results are new in the literature and are useful for further research in the field. [ABSTRACT FROM AUTHOR]

This paper probes into change of box dimension for an arbitrary fractal continuous function after Hadamard fractional integration. For classic calculus, we know that a function's differentiability increases one-order after integration, and decreases one-order after differentiation. But for fractional integro-differentiation, this similar fundamental problem is still open [F. B. Tatom, The relationship between fractional calculus and fractals, Fractals2 (1995) 217–229; M. Zähle and H. Ziezold, Fractional derivatives of Weierstrass-type functions, J. Comput. Appl. Math.76 (1996) 265–275; Y. S. Liang and W. Y. Su, Riemann–Liouville fractional calculus of 1-dimensional continuous functions, Sci. Sin. Math.4 (2016) 423–438 (in Chinese); Y. S. Liang, Progress on estimation of fractal dimensions of fractional calculus of continuous functions, Fractals26 (2019), doi:10.1142/S0218348X19500841] (see conjecture 1.1), although it is believed that the roughness of a fractal function is increased or decreased accounts α after α -order fractional differentiation or integration. This paper partly answers this problem. Particularly, the estimation of box dimension of Hadamard fractional integral is of methodology for considering similar fractional integration. In this paper, it is proved that the upper box dimension of the graph of f (x) does not increase after α -order Hadamard fractional integral D H − α f (x) , i.e. dim ¯ B Υ (D H − α f , I) ≤ dim ¯ B Υ (f , I). Particularly, while dim B Υ (f , I) = 1 , for example, a function is of unbounded variation and/or infinite length, it holds dim B Υ (D H − α f , I) = dim B Υ (f , I) = 1 , which answers completely this conjecture for one-dimensional fractal function. The published paper could verify Conjecture 1.1 under Riemann–Liouville or Wyel fractional integral, for only constructed functions with one-dimensional fractal [Y. S. Liang and W. Y. Su, Riemann–Liouville fractional calculus of 1-dimensional continuous functions, Sci. Sin. Math.4 (2016) 423–438 (in Chinese); Y. Li and W. Xiao, Fractal dimensions of Riemann–Liouville fractional integral of certain unbounded variational continuous function, Fractals25 (2017), doi:10.1142/S0218348X17500475; Y. S. Liang, Fractal dimension of Riemann–Liouville fractional integral of 1-dimensional continuous functions, Fract. Calc. Appl. Anal.21 (2018) 1651–1658; Q. Zhang, Some remarks on one-dimensional functions and their Riemann–Liouville fractional calculus, Acta Math. Sin.3 (2014) 517–524; J. Wang and K. Yao, Construction and analysis of a special one-dimensional continuous function, Fractals25 (2017), doi:10.1142/S0218348X17500207; X. Liu, J. Wang and H. L. Li, The classification of one-dimensional continuous functions and their fractional integral, Fractals26 (2018), doi:10.1142/S0218348X18500639]. [ABSTRACT FROM AUTHOR]

In this paper we get error bounds for fully discrete approximations of infinite horizon problems via the dynamic programming approach. It is well known that, considering a time discretization with a positive step size h, an error bound of size h can be proved for the difference between the value function (viscosity solution of the Hamilton-Jacobi-Bellman equation corresponding to the infinite horizon) and the value function of the discrete time problem. However, including also a spatial discretization based on elements of size k, an error bound of size O(k/h) can be found in the literature for the error between the value functions of the continuous problem and the fully discrete problem. In this paper we revise the error bound of the fully discrete method and prove, under assumptions similar to those of the time discrete case, that the error of the fully discrete case is in fact O(h + k), which gives first order in time and space for the method. This error bound matches the numerical experiments of many papers in the literature in which the behavior 1/h from the bound O(k/h) has not been observed. [ABSTRACT FROM AUTHOR]

The paper defines a new contractive condition within k-orbitally complete fuzzy metric spaces (Θ;M;T), as well as fixed point theorems for single-valued and multi-valued function on Θ which is not necessarily continuous. The contractive condition is motivated by an idea proposed in C' iric's paper "On some maps with a nonunique fixed points". Continuity of mapping k is replaced by k-orbitally continuity property which provides the existence of the fixed point, but not necessarily uniqueness. [ABSTRACT FROM AUTHOR]

In this paper, we obtain conditions of the solvability of the Riemann boundary value problem for sectionally analytic functions in multiply connected domains in Besov spaces embedded into the class of continuous functions. We indicate a new class of Cauchy-type integrals, which are continuous on a closed domain with continuous (not Hölder) density in terms of Besov spaces, and for which the Sokhotski–Plemelj formulas are valid. [ABSTRACT FROM AUTHOR]

In this paper, we focus our attention on an outer Lebesgue measure and density-type generalized topologies connected with this measure and with nondecreasing and unbounded sequences of positive reals. Some properties of such generalized topologies and continuous functions connected with this space are presented. [ABSTRACT FROM AUTHOR]

In this paper, we study the following critical nonlinear Schrödinger–Kirchhoff equation: ($P$) $$\begin{align*} \left \{ \begin{array}{@{}l@{}} \displaystyle -\left(a+b\int_{R^{N}}|\nabla u|^{2}\,{\rm d}x\right)\Delta u + V(x)u =P(x)|u|^{2^*-2}u+\mu|u|^{q-2}u, \ {\rm in}\ \mathbb{R}^{N},\\ u\in H^1(\mathbb{R}^N) \end{array} \right. \end{align*}$$ { − (a + b ∫ R N | ∇u | 2 d x) Δu + V (x) u = P (x) | u | 2 ∗ − 2 u + μ | u | q − 2 u , in R N , u ∈ H 1 (R N) where $ a, b, \mu \gt 0 $ a , b , μ > 0 , $ N\geq 3 $ N ≥ 3 , $ \max \{2^*-1, 2\} \lt q \lt 2^*=\frac {2N}{N-2} $ max { 2 ∗ − 1 , 2 } < q < 2 ∗ = 2 N N − 2 , $ V(x) \gt 0 $ V (x) > 0 and $ P(x)\geq 0 $ P (x) ≥ 0 are two continuous functions. By using the variational method and truncation technique, we prove the multiplicity of solutions for Equation (P). [ABSTRACT FROM AUTHOR]

Global optimization is widely adopted presently in a variety of practical and scientific problems. In this context, a group of widely used techniques are evolutionary techniques. A relatively new evolutionary technique in this direction is that of Giant-Armadillo Optimization, which is based on the hunting strategy of giant armadillos. In this paper, modifications to this technique are proposed, such as the periodic application of a local minimization method as well as the use of modern termination techniques based on statistical observations. The proposed modifications have been tested on a wide series of test functions available from the relevant literature and compared against other evolutionary methods. [ABSTRACT FROM AUTHOR]