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Let T be an injective bounded linear operator on a complex Hilbert space. We characterize the complex numbers λ , μ for which (I + λ T) (I + μ T) − 1 is a contraction, the characterization being expressed in terms of the numerical range of the possibly unbounded operator T − 1. When T = V , the Volterra operator on L 2 [ 0 , 1 ] , this leads to a result of Khadkhuu, Zemánek and the second author, characterizing those λ , μ for which (I + λ V) (I + μ V) − 1 is a contraction. Taking T = V n , we further deduce that (I + λ V n) (I + μ V n) − 1 is never a contraction if n ≥ 2 and λ ≠ μ. [ABSTRACT FROM AUTHOR]

In this article, we propose a numerical method for a general form of nonlinear cordial Volterra integral equations. We discuss conditions that under them the problem has solutions. Since the existence of solutions for the problem depends on the solvability of a scalar equation and also a linear form of the problem, then we employ quasilinearization technique in which solving a nonlinear problem is reduced to solve a sequence of linear equations. The existence of solutions of linear equations and their quadratically convergence to the solutions of the nonlinear problem is considered. For the numerical solution of the produced linear equations we apply Legendre-collocation method along with a regularization technique for the quadrature formulas. We discuss the error analysis of the collocation method considering that the cordial Volterra integral operators are noncompact. To test the efficiency and accuracy of the proposed method, the solution of different cases of numerical examples are reported. [ABSTRACT FROM AUTHOR]

In this work we consider a class of singular fractional differential equations (SFDEs). Using a suitable variable transformation we rewrite the SFDE as a cordial Volterra integral equation and propose a polynomial collocation method to find an approximate solution of the original problem. We provide the error analysis of the numerical method and we illustrate its feasibility and accuracy through some numerical examples. [ABSTRACT FROM AUTHOR]

In the present paper, we consider a convex combination of non‐Volterra quadratic stochastic operators defined on a finite‐dimensional simplex depending on a parameter θ$$ \theta $$ and study their trajectory behaviors. We showed that in the case m=3$$ m=3 $$ depending on the parameter θ∈[0,1]$$ \theta \in \left[0,1\right] $$ and an initial point, the trajectories of such operator converge to a fixed point. For m≥4$$ m\ge 4 $$, any trajectory of the operator converges to the center of the simplex. [ABSTRACT FROM AUTHOR]

We investigate the boundedness and compactness properties of integral operators, Volterra operators, and generalized Volterra operators between the Dales–Davie algebras. Additionally, we study the analogous properties of these operators when applied to the Lipschitz versions of Dales–Davie algebras. [ABSTRACT FROM AUTHOR]

We study certain Volterra integral equations that extend and recover first order ordinary differential equations (ODEs). We formulate the former equations from the latter by replacing classical derivatives with nonlocal integral operators with anti-symmetric kernels. Replacements of spatial derivatives have seen success in fracture mechanics, diffusion, and image processing. In this paper, we consider nonlocal replacements of time derivatives which contain future data. To account for the nonlocal nature of the operators, we formulate initial "volume" problems (IVPs) for these integral equations; the initial data is prescribed on a time interval rather than at a single point. As a nonlocality parameter vanishes, we show that the solutions to these equations recover those of classical ODEs. We demonstrate this convergence with exact solutions of some simple IVPs. However, we find that the solutions of these nonlocal models exhibit several properties distinct from their classical counterparts. For example, the solutions exhibit discontinuities at periodic intervals. In addition, for some IVPs, a continuous initial profile develops a measure-valued singularity in finite time. At subsequent periodic intervals, these solutions develop increasingly higher order distributional singularities. [ABSTRACT FROM AUTHOR]

The aim of this paper is to emphasize the role of the fibre contraction principle in the study of the solution of integral equations with maxima in connection with the weakly Picard operator technique. The results complement and extend some known results given in the paper: I.A. Rus, Some variants of contraction principle in the case of operators with Volterra property: step by step contraction principle, Advances in the Theory of Nonlinear Analysis and its Applications, 3(2019), no. 3, 111-120. The last section is devoted to Gronwall lemma type results and comparison theorems. [ABSTRACT FROM AUTHOR]

Let M(n+1) be the group of all isometric transformations of an n+1-dimensional Euclidean space Rn+1 , SM(n + 1) is the group of all Euclidean motions of Rn+1 , U is the domain {(u1) 2 + · · · + (un) 2 < 1} in Rn and x(u) is an U-hyper-surface (that is a C ∞-mapping x : U → Rn+1) in Rn+1. Let x(u) be a regular hyper-surface, I(x) = Σn i,j=1 gij (x)duiduj and II(x) = PΣn i,j=1 Lij (x)duiduj are the first and second fundamental forms of x. Put K = {(i, j) : 1 ≤ i ≤ j ≤ n}. We consider the following problem: find a proper (minimal, it is desirable) subset T of K such that equalities gij (x)(u) = gij (y)(u), Lij (x)(u) = Lij (y)(u) for all (i, j) ∈ T and u ∈ U implies an existence of F ∈ SM(n + 1) such that y(u) = F x(u) for all u ∈ U. For a solution of this problem, we have obtained a generating system of the differential field of all G-invariant differential rational functions of a hyper-surface in Rn+1 for groups G = M(n + 1), SM(n + 1). In our result, |T| = n(n+1) /2 + n, where |T| is the number of elements of T. [ABSTRACT FROM AUTHOR]

The compositions of Lotka - Volterra operators with corresponding non-isomorphic partially oriented graphs are investigated in the paper. Fixed points of composite operators are found and fixed points cards are constructed. Topologically equivalent compositional operators are classified according to the character of fixed point card. [ABSTRACT FROM AUTHOR]

In this work, we consider genetic algebras, generated by Volterra quadratic stochastic operators, and establish relation between regularity of Volterra operators and associativity of corresponding genetic algebra. [ABSTRACT FROM AUTHOR]

We consider a class of fractional integro-differential equations with certain type of singularities at the origin. Using a change of variables we reformulate the original problem as a cordial Volterra integral equation and study the existence, uniqueness and regularity of the exact solution. With the help of the obtained information we construct a collocation based numerical method for finding the approximate solution of the original problem and analyse the convergence and the convergence order of the proposed method. We also give some numerical examples. [ABSTRACT FROM AUTHOR]

We give a new proof of the Ostapenko–Tarasov unicellularity theorem for the classical Volterra integration operator on the space . [ABSTRACT FROM AUTHOR]

Debonding of an elastic layer with a circular cylindrical cavity 0 ≤ ζ ≤ h , ρ ≥ 1 , from a rigid substrate under action of a rigid cylinder is the object of this study. The annular debonding zone 1 ≤ ρ < 1 + η is caused by rotation of a cylinder bonded to the cavity surface. The problem is reformulated as dual integral equations with Weber integral transforms kernels. A Volterra operator transforming a Weber transforms kernel into a Bessel function of the first kind and Hankel integral transforms allow us to reduce dual equations to a Fredholm integral equation of the second kind and then, by some transformation, to another Fredholm integral equation which is more suitable for approximate methods. As h ≥ 0. 5 and 0 < η ≤ 1 , a highly accurate analytic approximate solution of the problem is suggested. The asymptotic solution of the problem is obtained as the width of debonding zone is very small while the thickness is not small. When the thickness is small, the Fredholm integral equations are computationally inefficient. A new method based on an operator transforming a Bessel function of the first kind into the kernel of Mehler–Fock integral transforms enabled us to convert one of the above-mentioned Fredholm equations into an equivalent Fredholm integral equation of the second kind that is effective for a small thickness. The asymptotic solution of the problem is obtained when both the layer thickness and the debonding zone width are small. Accurate mathematical methods, in particular investigations, and transformations of equations, developed in this study can be interesting to researchers employing dual integral equations technique in problems of mechanics and mathematical physics with mixed boundary conditions. [ABSTRACT FROM AUTHOR]

We completely characterize the boundedness of area operators from the Bergman spaces A α p (B n) to the Lebesgue spaces L q (S n) for all 0 < p,q < ∞. For the case n = 1, some partial results were previously obtained by Wu in [Wu, Z.: Volterra operator, area integral and Carleson measures, Sci. China Math., 54, 2487–2500 (2011)]. Especially, in the case q < p and q < s, we obtain some characterizations for the area operators to be bounded. We solve the cases left open there and extend the results to n-complex dimension. [ABSTRACT FROM AUTHOR]

Dealing with the inverse source problem for the scalar wave equation, we have shown recently that we can reconstruct the space-time dependent source function from the measurement of the wave, collected at a single point $ x $ for a large enough interval of time, generated by a small scaled droplets, enjoying large contrasts of its bulk modulus, injected inside the domain to image. Here, we extend this result to reconstruct not only the source function but also the variable wave speed. Indeed, from the measured waves, we first localize the internal values of the travel-time function by looking at the behavior of this collected wave in terms of time. Then from the Eikonal equation, we recover the wave speed. Second, we recover the internal values of the wave generated only by the background (in the absence of the small droplets) from the same measured data by inverting a Volterra integral operator of the second kind. From this reconstructed wave, we recover the source function at the expense of a numerical differentiation. [ABSTRACT FROM AUTHOR]

Let φ be an analytic selfmap of the open unit disk D and g be an analytic function on D. The Volterra-type composition operators induced by the maps g and φ are defined as (Ig φ f) (z) = ∫0 z f' (φ(ζ))g(ζ)dζ and (Tg φf) (z) = ∫0 z f(φ(ζ))g′ (ζ)dζ. For 1 ≤ p < ∞, S p (D) is the space of all analytic functions on D whose first derivative f ′ lies in the Hardy space Hp (D), endowed with the norm ∥f∥Sp = |f(0)| + ∥f′ ∥Hp . Let µ : (0, 1] → (0, ∞) be a positive continuous function on D such that for z ∈ D we define µ(z) = µ(|z|). The weighted Zygmund space Zµ(D) is the space of all analytic functions f on D such that supz∈D µ(z)|f ′′(z)| is finite. In this paper, we characterize the boundedness and compactness of the Volterra-type composition operators that act between S p spaces and weighted Zygmund space [ABSTRACT FROM AUTHOR]

In this paper, we investigate the behavior of infinite-dimensional Volterra quadratic stochastic operators corresponding to the negative upper triangular skew-symmetric matrix. It is proved that the trajectory of such operators cannot consist of convergent subsequences. This property immediately implies non-ergodicity of the operator. Moreover, it is shown that the operator’s dynamics with respect to pointwise convergence are regular. [ABSTRACT FROM AUTHOR]

In the present paper, we consider cubic stochastic operators defined on a finite-dimensional simplex. These operators are called separable cubic stochastic operators and depend on a permutation and four parameters. We show that for any permutation, except the identity permutation, any trajectory of corresponding operators converges to a periodic trajectory. Any trajectory of the operator corresponding to the identity permutation converges to a fixed point. [ABSTRACT FROM AUTHOR]

Let V be the classical Volterra operator, and let S be the operator of multiplication by an exponential function on L2(0, 1). We introduce the numerical range technique for the operators S-1V S and (I+V)−1 on L2(0, 1). We present the operator norm, numerical range, numerical radius, and accretive properties of the real and imaginary parts for such polynomials. [ABSTRACT FROM AUTHOR]

The present paper deals to a family of non-Volterra quadratic stochastic operators depending on a parameter and a permutation and their trajectory behaviours. We find all fixed and periodic points for such non-Volterra quadratic stochastic operator on a finite-dimensional simplex. We proved that the ω- limit set of any trajectory of such non-Volterra quadratic stochastic operator consists of either single point or finite points. [ABSTRACT FROM AUTHOR]

We characterize the integration operators V g with symbol g for which V g acts as an absolutely summing operator on weighted Bloch spaces B β and on weighted Bergman spaces A α p . We show that V g is r-summing on A α p , 1 ≤ p < ∞ , if and only if g belongs to a suitable Besov space. We also show that there is no non trivial nuclear Volterra operators V g on Bloch spaces and on Bergman spaces. [ABSTRACT FROM AUTHOR]

The response of a body described by a quasi-linear viscoelastic constitutive relation, whose material moduli depend on the mechanical pressure (that is one-third the trace of stress) is studied. The constitutive relation stems from a class of implicit relations between the histories of the stress and the relative deformation gradient. A-priori thresholding is enforced through the pressure that ensures that the displacement gradient remains small. The resulting mixed variational problem consists of an evolutionary equation with the Volterra convolution operator; this equation is studied for well-posedness within the theory of maximal monotone graphs. For isotropic extension or compression, a semi-analytic solution of the quasi-linear viscoelastic problem is constructed under stress control. The equations are studied numerically with respect to monotone loading both with and without thresholding. In the example, the thresholding procedure ensures that the solution does not blow-up in finite time. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the general stability results of a fully dynamic piezoelectric beam system with internal fractional delays under memory boundary controls. Firstly, by introducing two new equations and using two Volterra's inverse operators to deal with fractional delay terms and boundary memory terms, respectively, a new equivalent system is obtained. Secondly, by combining Faedo–Galerkin approximations with the compactness argument, we prove the local existence and uniqueness of weak solution. Finally, under a broader class of kernel functions, by constructing Lyapunov functionals and new equations and applying some technical lemmas, we establish the optimal and explicit energy decay results for two cases, namely, initial values (v0,p0)=(0,0)$$ \left({v}_0,{p}_0\right)&#x0003D;\left(0,0\right) $$ and without imposing initial values equal to 0, that is, (v0,p0)≠(0,0)$$ \left({v}_0,{p}_0\right)\ne \left(0,0\right) $$. It is worth pointing out that we focus more on the analysis of (v0,p0)≠(0,0)$$ \left({v}_0,{p}_0\right)\ne \left(0,0\right) $$, because the stability results of (v0,p0)=(0,0)$$ \left({v}_0,{p}_0\right)&#x0003D;\left(0,0\right) $$ are special cases of (v0,p0)≠(0,0)$$ \left({v}_0,{p}_0\right)\ne \left(0,0\right) $$. This is the first time to analyze the stability of strong coupling problem by combining internal fractional delay and boundary viscoelastic damping. The obtained general decay results are not necessarily exponential or polynomial, which generalize the relevant results in the existing literature. [ABSTRACT FROM AUTHOR]

As is known [3], each quadratic stochastic operator of Volterra type acting on a finite-dimensional simplex defines a certain tournament, the properties of which make it possible to study the asymptotic behavior of the trajectories of this Volterra operator. In this paper, we introduce the concept of a homogeneous tournament and study the dynamic properties of Volterra operators corresponding to homogeneous tournaments in the simplex S4. [ABSTRACT FROM AUTHOR]

In the first part of the article [1], a physically and geometrically nonlinear boundary-value problem, that describes the compression of a fiber-reinforced plastic rod with [0]s layup, was formulated. The rod had a rectangular cross-section and thin elastic side tabs. The boundary-value problem was reduced to a system of integral-algebraic equilibrium equations containing Volterra integral operators of the second type. To find its numerical solution, the method of finite sums in the variant of integrating matrices was used. The advantage of the method is the possibility of a strong local thickening of the computational grid in the region of large gradients of solutions. Based on the algorithm constructed, an application software package was developed. The results of computational experiments showed that the test specimens under compression according to one of the most commonly used test schemes predominantly failed when the localized transverse shear stresses reached their ultimate values. Failure was also possible according to the shear buckling mode in stress concentration zones. The identification of such modes was possible by using a proposed refined geometrically and physically nonlinear deformation model built in the quadratic approximation with account of transverse shear strains and transverse compression. To verify the numerical method developed, physical experiments were carried out on unidirectional carbon-fiber-reinforced specimens with [0]s layup. They showed a good agreement between the theoretical and experimental results of the research. [ABSTRACT FROM AUTHOR]

In this paper, we consider the inverse Laplace transform of the Volterra μ-function (the Bromwich integral) and evaluate it with respect to the Hankel, Schläfli and Bourguet complex contours. In this sense, we establish the generalized Ramanujan's integral representations of the Volterra μ-function for general variations of the parameters. We also discuss the asymptotic analysis of this function with large parameters using the steepest descent method. Further, we show that the solution of Volterra integral equation with differentiated-order fractional integral operator is the Volterra μ-function. [ABSTRACT FROM AUTHOR]

We consider a class of discrete time prey–predator models with three interacting species defined on the two-dimensional simplex. For some choices of parameters of the operator describing the evolution of the relative frequencies, we show that the trajectories approach a heteroclinic cycle and that the ergodic hypothesis does not hold. Moreover, we prove that any order Cesàro mean of the trajectories diverges. For another class of parameters, we show that all orbits starting from the interior of the simplex converge to the unique fixed point of the operator while for the remaining choices of parameters all orbits converge to one of the vertices of the simplex. Contrary to many authors, we study discrete time models but we include a speed functionf in the dynamics which allows us to approximate the continuous-time case arbitrarily well when f is small. [ABSTRACT FROM AUTHOR]

This work deals with a higher order numerical approximation for analyzing a class of multi-term time fractional partial integro-differential equations involving Volterra integral operators. The solutions to these problems have a mild singularity at the initial time, due to which an initial layer appears, which becomes more sharper as the highest order time fractional derivative decreases. This behaviour reduces the rate of convergence by standard approaches. We start the present work by considering the existence and uniqueness of a class of generalized partial integro-differential equations and then, present the L1 discretization on a graded mesh in time which is adapted towards the initial time level. This discretization leads to a higher order accuracy than the solutions obtained on a time uniform mesh. The convergence analysis corresponding to the Volterra integral operator is nontrivial as it uses a repeated quadrature rule. This analysis can also be extended for weakly singular kernels. The stability analysis of the present scheme with a sharp error estimation is also provided. The analysis with extensive experiments shows that a higher rate of accuracy can be attained for several suitable choices of the grading parameters for solving several classes of time fractional integro-differential equations. [ABSTRACT FROM AUTHOR]

Finite strict co-singularity of Volterra operators Va: L¹[0, 1+a] → C[0, 1], where 0 ≤ a < ∞, defined by (Vaf)(x) =∫0 (1+a)x f(t)dt, x ∈ [0, 1], t ∈ [0, 1 + a] is explored by calculating Mityagin numbers. Also, Approximation, Gelfand, Kolmogorov, Bernstein, and Isomorphism s-numbers are obtained for these operators. [ABSTRACT FROM AUTHOR]

We first study the Volterra operator V acting on spaces of Dirichlet series. We prove that V is bounded on the Hardy space ℋ 0 p for any 0 < p ≤ ∞, and is compact on ℋ 0 p for 1

A family of Dirichlet-Morrey spaces Dλ,K of functions analytic in the open unit disk D are defined in this paper. We completely characterize the boundedness of the Volterra integral operators Tg, Ig and the multiplication operator Mg on the space Dλ,K. In addition, the compactness and essential norm of the operators Tg and Ig on Dλ,K are also investigated. [ABSTRACT FROM AUTHOR]

We characterize power bounded generalized Volterra companion operator on the Fock space, in terms of functional-analytic properties of the inducing functions. Moreover, we describe Ritt's resolvent growth condition for a bounded generalized Volterra companion operator. [ABSTRACT FROM AUTHOR]

In this paper, we investigate weighted composition, Volterra and Integral operators on second derivative Hardy spaces. Some equivalent conditions for boundedness of the operators will be given using the boundedness on the Hardy spaces. Also we give a criteria for compactness of weighted composition operators. [ABSTRACT FROM AUTHOR]

We consider Volterra multidimensional integral operators with continuous coefficients in Lebesgue spaces. The kernel of an integral operator is assumed homogeneous of degree , invariant under the rotation group , and satisfying some summability condition that ensures the boundedness of the operator. The main object of research is the noncommutative Banach algebra generated by all operators of the above type and the identity operator. To study , we turn to the quotient algebra , with the set of all compact operators. We prove that is commutative, which allows us to apply the general methods of commutative Banach algebras. In particular, we describe the space of maximal ideals of and find some invertibility criterion for the elements of . We then construct some symbolic calculus for on assigning to each operator in a continuous function, the symbol of the operator. Using the symbols, we obtain the necessary and sufficient conditions for the Noetherian property of an operator in , as well as an index formula. [ABSTRACT FROM AUTHOR]

The problem of the steady activation of a skeletal muscle is one of the applicable phenomena in real life that can be modeled by a Volterra integral equation. The current research aims to investigate this problem by using an effective operational matrix-based method. For this purpose, the operational matrix of integration is derived for the barycentric rational cardinal basis functions. Then, by utilizing the obtained operational matrix and without using any collocation points, the governing integral equation is reduced to a system of nonlinear algebraic equations. Convergence analysis of the proposed numerical method is studied thoroughly. Moreover, the obtained numerical results based on the proposed method with acceptable computational times support the theoretical results and reveal the accuracy and efficiency of the method. [ABSTRACT FROM AUTHOR]

In this paper, a generalized version of the auto-convolution Volterra integral equation of the first kind as an ill-posed problem is studied. We apply the piecewise polynomial collocation method to reduce the numerical solution of this equation to a system of algebraic equations. According to the proposed numerical method, for n = 0 and n = 1,. .. ,N - 1, we obtain a nonlinear and linear system, respectively. We have to distinguish between two cases, nonlinear and linear systems of algebraic equations. A double iteration process based on the modified Tikhonov regularization method is considered to solve the nonlinear algebraic equations. In this process, the outer iteration controls the evolution path of the unknown vector Uδ 0 in the selected direction ˜u0, which is determined from the inner iteration process. For the linear case, we apply the Lavrentiev ˜m times iterated regularization method to deal with the ill-posed linear system. The validity and efficiency of the proposed method are demonstrated by several numerical experiments. [ABSTRACT FROM AUTHOR]

The Cauchy problem for singular functional differential equations is considered. The sufficient conditions of the unique solvability are established. [ABSTRACT FROM AUTHOR]

In this paper we suggest an enhanced Geraghty-type contractive mapping for examining the existence properties of classical nonlinear operators with or without prior degenerates. The nonlinear operators are proved to exist with the imposition of the Geraghty-type condition in a non-empty closed subset of complete metric spaces. To showcase some efficacies of the Geraghty-type condition, convergent rate and stability are deduced. The results are used to study some asymptotic properties of perturbed integral and hypergeometric operators. The results also extend and generalize some existing Geraghty-type conditions. [ABSTRACT FROM AUTHOR]

Triangular operators are an essential tool in the study of nonselfadjoint operators that appear in different fields with a wide range of applications. Although the development of a quaternionic counterpart for this theory started at the beginning of this century, the lack of a proper spectral theory combined with problems caused by the underlying noncommutative structure prevented its real development for a long time. In this paper, we give criteria for a quaternionic linear operator to have a triangular representation, namely, under which conditions such operators can be represented as a sum of a diagonal operator with a Volterra operator. To this effect, we investigate quaternionic Volterra operators based on the quaternionic spectral theory arising from the S‐spectrum. This allow us to obtain conditions when a non‐selfadjoint operator admits a triangular representation. [ABSTRACT FROM AUTHOR]

Integro-differential equations involving Volterra and Fredholm operators (VFIDEs) are used to model many phenomena in science and engineering. Nonlocal boundary conditions are more effective, and in some cases necessary, because they are more accurate measurements of the true state than classical (local) initial and boundary conditions. Closed-form solutions are always desirable, not only because they are more efficient, but also because they can be valuable benchmarks for validating approximate and numerical procedures. This paper presents a direct operator method for solving, in closed form, a class of Volterra–Fredholm–Hammerstein-type integro-differential equations under nonlocal boundary conditions when the inverse operator of the associated Volterra integro-differential operator exists and can be found explicitly. A technique for constructing inverse operators of convolution-type Volterra integro-differential operators (VIDEs) under multipoint and integral conditions is provided. The proposed methods are suitable for integration into any computer algebra system. Several linear and nonlinear examples are solved to demonstrate the effectiveness of the method. [ABSTRACT FROM AUTHOR]

It is a classical theorem of Sarason that an analytic function of bounded mean oscillation (BMOA) is of vanishing mean oscillation if and only if its rotations converge in norm to the original function as the angle of the rotation tends to zero. In a series of two papers, Blasco et al. have raised the problem of characterizing all semigroups of holomorphic functions (φt) that can replace the semigroup of rotations in Sarason's theorem. We give a complete answer to this question, in terms of a logarithmic vanishing oscillation condition on the infinitesimal generator of the semigroup (φt). In addition, we confirm the conjecture of Blasco et al. that all such semigroups are elliptic. We also investigate the analogous question for the Bloch and the little Bloch spaces, and surprisingly enough, we find that the semigroups for which the Bloch version of Sarason's theorem holds are exactly the same as in the BMOA case. [ABSTRACT FROM AUTHOR]

In [1] it is shown that the Bloch space B in the unit disc has the following radicality property: if an analytic function g satisfies that g n ∈ B , then g m ∈ B , for all m ≤ n. Since B coincides with the space T (A α p) of analytic symbols g such that the Volterra-type operator T g f (z) = ∫ 0 z f (ζ) g ′ (ζ) d ζ is bounded on the classical weighted Bergman space A α p , the radicality property was used to study the composition of paraproducts T g and S g f = T f g on A α p. Motivated by this fact, we prove that T (A ω p) also has the radicality property, for any radial weight ω. Unlike the classical case, the lack of a precise description of T (A ω p) for a general radial weight, induces us to prove the radicality property for A ω p from precise norm-operator results for compositions of analytic paraproducts. [ABSTRACT FROM AUTHOR]

We prove the existence of a Stackelberg equilibrium (in the style of M.S. Nikol'skii) for a nonlinear Volterra functional operator equation controlled by two players with the help of finite-dimensional program controls with integral objective functionals. On this way, we use our formerly proved results on the continuous dependence of the state and functionals on finite-dimensional controls and also the classical Weierstrass theorem. The property of being a singleton for the minimizer set of the first player is proved by M.S. Nikol'skii's scheme applied earlier to a linear ordinary differential equation. [ABSTRACT FROM AUTHOR]