In this article, we consider optimization problems in which the sums of the largest eigenvalues of symmetric matrices are involved. Considered as functions of a symmetric matrix, the eigenvalues are not smooth once the multiplicity of the function is not single; this brings some difficulties to solve. For this, the function of the sums of the largest eigenvalues with affine matrix-valued mappings is

The damped nonlinear wave equation, also known as the nonlinear telegraph equation, is studied within the framework of semigroups and eigenfunction approximation. The linear semigroup assumes a central role: it is bounded on the domain of its generator for all time t ≥ 0 . This permits eigenfunction approximation within the semigroup framework, as a tool for the study of weak solutions. The semigroup

For a class of classical vector-valued sequence spaces in Banach spaces, this article introduces a class of important subsets, which contain all the fully bounded sets and many non-bounded sets in the sequence spaces. Using this collection class, we obtain a convergence theorem for the assignment of series of operator series, and apply the conclusion to the characterization of infinite matrix transformations

In this article, by using the bounded quasi-linear projection generalized inverse, we prove the transcritical and pitchfork bifurcation theorem for multiple eigenvalues, which is a generalized result of the famous transcritical and pitchfork bifurcation theorem for simple eigenvalue.

Microscopic analysis of mineral distributions in shale receives much attention in recent years. Traditional methods such as optical and scanning electron microscopy are common tools for providing valuable information of microstructures; however, they belong to surface observations, which damage the shale sample and only generate a 2D image. We consider using the X-ray computerized tomography (CT) based

In this article, we study the existence criterion and formulae of Drazin inverses of elements in a certain finite-dimensional algebra generated by two idempotents, under some prescribed conditions.

Let T be a bounded linear operator from a Banach space to a Banach space with closed range and let T ¯ = T + δ T . Nashed’s condition is that ( I + δ T T + ) − 1 T ¯ maps the null space of T into the range of T. The stable perturbation means that the intersection of the range of T ¯ and the null space of the generalized inverse of T is {0}. We show that the stable perturbation is the same as Nashed’s

In this paper, we introduce the notion of stable perturbation of a subspace in Banach space. Utilizing this notion and the gap between subspaces, we develop a perturbation analysis for the oblique projection generalized inverse when the operator T, the range and null space of the projectors have perturbations simultaneously and estimate an upper bound of | | T ¯ P ¯ , Q ¯ + − T P , Q + | | . Our results

This work is devoted to establish the optimal solutions of impulsive optimal controls for a class of Lagrange problem. Using the approximate method and fixed point theorem, the corresponding semilinear system is solvable under some appropriate assumptions. From the derived results, we propose to construct two kinds of minimizing sequences of approximating solutions to obtain the optimal solutions.

Let B   ( X ) be the set of bounded linear operators on a Banach space X, and a liner operator A ∈ B   ( X ) be Drazin invertible. A linear operator B ∈ B   ( X ) is said to be a stable perturbation of A if B is Drazin invertible and I − A π − B π is invertible, where I is the identity operator on X, A π ( = I − A A D ) and B π ( = I − B B D ) are the spectral projectors of A and B respectively. We

For any singular matrix A ∈ C n × n , let AD , A π = I n − A A D be its Drazin inverse and spectral projector, respectively. Let | | · | | be any norm on C n × n and A ¯ ∈ C n × n be any stable perturbation of A. In this note a sufficient condition is given under which | | A ¯ π − A π | | < 1 .

Let P and Q be a pair of orthogonal projections. In this note we will prove that there exists a unitary U ∈ B ( H ) such that UP = QU and UQ = PU if and only if dim [ R ( P ) ∩ N ( Q ) ] = dim [ N ( P ) ∩ R ( Q ) ] .

In this paper, we propose a Padé-type approximation based on truncated total least squares (P – TTLS) and compare it with three commonly used smoothing methods: Penalized spline, Kernel smoothing and smoothing spline methods that have become very powerful smoothing techniques in the nonparametric regression setting. We consider the nonparametric regression model, y i = g ( x i ) + ε i , and discuss

We consider a one dimensional elliptic distributed optimal control problem with pointwise constraints on the derivative of the state. By exploiting the variational inequality satisfied by the derivative of the optimal state, we obtain higher regularity for the optimal state under appropriate assumptions on the data. We also solve the optimal control problem as a fourth order variational inequality

Newton’s method is usually preferred when solving optimization problems due to its superior convergence properties compared to gradient-based or derivative-free optimization algorithms. However, deriving and computing second-order derivatives needed by Newton’s method often is not trivial and, in some cases, not possible. In such cases quasi-Newton algorithms are a great alternative. In this paper

We consider a quadratic optimal control problem for the 3D Navier-Stokes-Voigt equations with periodic inputs. We prove the existence of optimal solutions, then establish necessary and sufficient optimality conditions. We also define semidiscrete-in-time approximations for the optimal control problem and then prove the existence of a subsequence that converges to an optimal solution.

The Peterlin viscoelastic model describes the motion of certain incompressible polymeric fluids. It employs a nonlinear dumbbell model with a nonlinear spring force law making it more nonlinear than other viscoelastic models. In this paper, we propose and study a fully implicit stabilized Crank-Nicolson time stepping scheme for finite element spatial discretization of the non-stationary Peterlin viscoelastic

Let G be an LCA group, H a closed subgroup, Γ the dual group of G and μ be a regular finite non-negative Borel measure on Γ. Motivated by the problem of sampling and reconstruction of a stationary random process over G from the values of m different linear measurements on H, we give some necessary or sufficient conditions for the density of certain sets of functions in L 2 ( Γ , μ ) , which arise in

In this article, we consider optimal control problems for the one-dimensional Frémond model for shape memory alloys. This model is constructed in terms of basic functionals like free energy and pseudo-potential of dissipation. The state problem is expressed by a system of partial differential equations involving the balance equations for energy and momentum. We prove the existence of an optimal control

In this work, the boundedness and essential norm of weighted composition operators from the Zygmund space to nth weighted-type spaces are characterized. As a corollary, some characterizations for the compactness of weighted composition operators from the Zygmund space to nth weighted-type spaces are also presented.

In this paper, we present a numerical verification method of solutions for nonlinear parabolic initial boundary value problems. Decomposing the problem into a nonlinear part and an initial value part, we apply Nakao’s projection method, which is based on the full-discrete finite element method with constructive error estimates, to the nonlinear part and use the theoretical analysis for the heat equation

The approximation of functions using linear positive operators is affected by saturation. The quality of approximation offered by iterated Boolean sums increases with the regularity of the function. We present some qualitative and quantitative results concerning the approximation by such Boolean sums. The general results are illustrated by examples.

The influence of errors on the convergence of infinite products of weak quasi-contraction mappings in b-metric spaces is explored. An example demonstrating the necessity of convergence of the sequence of computational errors to zero is also provided. Moreover, we discuss weak ergodic theorems in the setting of b-metric spaces.

We study a source identification problem for a parabolic equation with multipoint nonlocal boundary condition. Stability estimates for the solution of abstract source identification problem are established. In application, stability estimates for the solution of the overdetermined mixed boundary value problem for multidimensional parabolic equation are obtained. The first and second order of accuracy

We consider an optimal control problem Q governed by an elliptic quasivariational inequality with unilateral constraints. We associate to Q a new optimal control problem Q˜, obtained by perturbing the state inequality (including the set of constraints and the nonlinear operator) and the cost functional, as well. Then, we provide sufficient conditions which guarantee the convergence of solutions of

Abstract In this paper, we consider convex Tikhonov regularization for the solution of linear operator equations on Hilbert spaces. We show that standard fractional source conditions can be employed in order to derive convergence rates in terms of the Bregman distance, assuming some stronger convexity properties of either the regularization term or its convex conjugate. In the special case of quadratic

The goal of this paper is to develop a framework that would allow one to solve the problems of optimal recovery of the monotone operators that act on a possibly most extensive class of monotone functions. The results of this work allow us to include the operators that act on the spaces of multi- and fuzzy- valued functions, as well as on the spaces of functions with values in partially ordered normed

In this paper, we study a Thakur three-step iterative process in the new context of generalized nonexpansive mappings enriched with property (E). A uniformly convex Banach space is used as underlying setting for our approach. In order to emphasize our results we offer an example of a mapping on R2 endowed with property (E) that exceeds the well known class of Suzuki mappings. By connecting the Thakur

The Peterlin viscoelastic model describes the motion of certain incompressible polymeric fluids. It employs a nonlinear dumbbell model with a nonlinear spring force law making it more nonlinear than other viscoelastic models. In this paper, we propose and study a fully implicit stabilized Crank-Nicolson time stepping scheme for finite element spatial discretization of the non-stationary Peterlin viscoelastic

We study the proximal point method for solving quasi-equilibrium problems in Banach spaces, which generalizes the proximal point method for equilibrium problems and quasi-variational inequalities. We propose a regularization procedure which ensures strong convergence of the generated sequence to a solution of the quasi-equilibrium problem, under standard assumptions on the problem without assuming

The current article intends to study some elementary constrained approximation aspects of the bivariate fractal functions. To this end, firstly the construction of bivariate fractal interpolation functions available in the literature is revisited with a focus to obtain a parameterized family of fractal functions corresponding to a prescribed bivariate continuous function on a rectangular region in

The goal of this paper is to develop a framework that would allow one to solve the problems of optimal recovery of the monotone operators that act on a possibly most extensive class of monotone functions. The results of this work allow us to include the operators that act on the spaces of multi- and fuzzy- valued functions, as well as on the spaces of functions with values in partially ordered normed

The approximation of functions using linear positive operators is affected by saturation. The quality of approximation offered by iterated Boolean sums increases with the regularity of the function. We present some qualitative and quantitative results concerning the approximation by such Boolean sums. The general results are illustrated by examples.

The article presents an explicit form for the second moment of cubic Schoenberg operators with arbitrary knots and in the particular case of equidistant knots. Applications are given for quantitative direct results in approximation and in construction of special classes of Schoenberg operators.

In this paper, we use the upward power preorder relation on the power sets of ordered sets to construct a mathematical social benefit maximization model in a fiscal year with respect to the initially planned fiscal budgetary policies and the practically implementation of the fiscal budgetary policies. We only consider the uncertainty case for the social benefit functions. By applying the Fan-KKM Theorem

In this paper, we study the sampling and average sampling problem in a quasi shift-invariant space VX(φ), where X is a discrete subset of R and φ is a continuously differentiable positive definite function satisfying certain decay conditions. We show that any f belonging to VX(φ) can be uniquely and stably reconstructed from its samples {f(yk):k∈Z} as well as from its average samples provided sampling

In this article, it is concerned existence of solutions to a class of nonlinear system of fractional differential equations. It is proved the existence of at least three different weak solutions through the critical point theory and the variational method. In this way, it is applied well-known theorem proved by Bonanno and Marano on the construction of the critical set of functionals with a weak compactness

This note presents ad-nonprojection method for approximation to infinite-dimensional Moore-Penrose inverse, which includes generalized projection method as a special case. The main result (Theorem 4) gives characteristics of perfect convergence, weak perfect convergence, and uniform perfect convergence of ad-nonprojection method. This result is a generalization of several fundamental theorems known

We derive and analyze a new variant of the iteratively regularized Landweber iteration, for solving linear and nonlinear ill-posed inverse problems. The method takes into account training data, which are used to estimate the interior of a black box, which is used to define the iteration process. We prove convergence and stability for the scheme in infinite dimensional Hilbert spaces. These theoretical

In this paper, using the hybrid method defined by Nakajo and Takahashi, we first obtain a strong convergence theorem for two noncommutative generic skew generalized nonspreading mappings in a Banach space. Next, using the shrinking projection method defined by Takahashi, Takeuchi and Kubota, we prove another strong convergence theorem for the mappings in a Banach space. Using these results, we get

In this article, we will present some results relating sharp ordering (a≤♯b) and investigate the properties of one-sided cyclic ideals in rings. Necessary and sufficient conditions for the invertibility and the group invertibility of the linear combination c1a+c2b are given, where a is below b under the sharp order, a and b are elements of an algebra, and c1, c2 are elements of an arbitrary field.

The Zhang neural network (ZNN), a recurrent neural network, proposed in 2001, is particularly effective in solving time-varying problems. It has shown high efficiency and excellent performance in various applications. The weighted pseudoinverse is a useful tool in solving and analyzing the constrained least-squares problems. In this paper, we propose a ZNN model for computing the weighted pseudoinverse

Here, we study the numerical solution of singularly perturbed system of two-point boundary-value problems of reaction-diffusion type. The solutions of these problems exhibit twin overlapping exponential boundary layers. To obtain the numerical solutions of these problems, we apply the nonsymmetric discontinuous Galerkin FEM with interior penalties (NIPG method) on layer-adapted piecewise-uniform Shishkin

We study variational inequalities which are governed by a strongly monotone and Lipschitz continuous operator F over a closed and convex set S. We assume that S=C∩A−1(Q) is the nonempty solution set of a (multiple-set) split convex feasibility problem, where C and Q are both closed and convex subsets of two real Hilbert spaces H1 and H2, respectively, and the operator A acting between them is linear

In a recent paper of Cuenya and Ferreyra, a condition, namely, the Cp-condition in Lp-spaces was introduced that is weaker than the notion of Lp-derivative given by Calderón-Zygmund. In the present article, we define the Legendre derivative for functions in L2 generalizing both notions, the Cp-condition and Lp-derivative in the case p = 2. As a consequence, we give a necessary and sufficient condition

In this paper, we introduce a novel algorithm for finding a common point of the solution set of a class of equilibrium problems involving pseudo-monotone bifunctions and satisfying the Lipschitz-type condition and the set of fixed points of a quasi-nonexpansive in a real Hilbert space. This algorithm can be considered as a combination of the subgradient extragradient method for equilibrium problems

This article mainly studies the nonuniform and semi-average sampling of time-varying shift-invariant signals living in a mixed Lebesgue space Lp,q(Rd+1), under the condition that the generator φ of the shift-invariant subspace belongs to a hybrid-norm space of mixed form depending on the orders p and q. First, the sampling stability for two kinds of semi-average sampling functionals is established

In this paper, we consider the nonconvex quadratic optimization problem with a single quadratic constraint. First we give a theoretical characterization of the local non-global minimizers. Then we extend the recent characterization of the global minimizer via a generalized eigenvalue problem to the local non-global minimizers. Finally, we use these results to derive an efficient algorithm that finds

In the current paper, we discuss sufficient and necessary conditions for the existence of best proximity points for cyclic f- contractive mappings in the setting of strictly convex Banach spaces. Extensions of Edelstein’s theorem are considered as well as an extension of a main result in Park [Park, S. (1978). Fixed points of f-contractive maps. Rocky Mountain J. Math. 8:743–750]. Another existence

Given a positive weight function and an isometry map on a Hilbert spaces H, we study a class of linear maps which is a g-frame, g-Riesz basis, and a g-orthonormal basis for H with respect to C in terms of the weight function. We apply our results to study the frame for shift-invariant subspaces on the Heisenberg group.

In this paper, we apply, to the case of approximate minimizers for vector optimization problems with set-valued maps, some techniques developed previously in literature for the situation of standard vector efficiency. The approach we propose here allows us, on one hand, to extend some known results, and, on the other hand, to emphasize some numerical aspects that could be of certain importance for

In this paper, we extend the study of an iteration process introduced by Thakur-Thakur-Postolache [J. Inequal. Appl., (2014), 147–155] from the class of nonexpansive operators to the more general class of Suzuki operators. We prove weak and strong convergence theorems regarding the iteration method for mappings satisfying the condition (C) of Suzuki in uniformly convex Banach spaces. We study the stability

Several logarithmic-barrier interiors-point methods are now available for solving two-stage stochastic optimization problems with recourse in the finite-dimensional setting. However, despite the genuine need for studying such methods in general spaces, there are no infinite-dimensional analogs of these methods. Inspired by this evident gap in the literature, in this paper, we propose logarithmic-barrier

We construct some generalized Hermite-type interpolation operators for the case of a triangle with one curved side, and we consider their product and Boolean sum operators. We study the interpolation properties of these operators, the orders of accuracy and the remainders of the interpolation formulas. Finally, we give some numerical examples.

In this paper, the equivalence between multi-valued maps satisfying the Mizoguchi-Takahashi’s uniformly locally contractive condition and multi-valued maps satisfying the Nadler’s uniformly locally contractive condition is obtained on metrically convex space. We have provided examples to illustrate that this equivalence need not be true on any arbitrary metric space.

(2019). Correction. Numerical Functional Analysis and Optimization: Vol. 40, No. 16, pp. 1972-1976.

Strong geodesic convex function and strong monotone vector field of order m on Riemannian manifolds are established. A characterization of strong geodesic convex function of order m for the continuously differentiable functions is discussed. The relation between the solution of a new variational inequality problem and the strict minimizers of order m for a multiobjective programing problem is also

In this paper, we introduce a new discontinuous operator and investigate the existence and uniqueness of fixed points for the operators in complete metric spaces. We also provide rate of convergence and data dependency of S-iterative scheme for a fixed point of the discontinuous operators in Banach spaces. Moreover, we prove the estimation Collage theorems and compare error estimate between data dependency

In this article, we first discuss the subduality and orthogonality of the cones and the dual cones when the norm is monotone in Banach spaces. Then, under different assumptions, the necessary and sufficient conditions for the ordering increasing property of the metric projection onto cones and order intervals are studied. Moreover, representations of the metric projection onto cones and order intervals

In the present paper, we summarize the recent literature concerning the spectrum and fine spectrum with respect to Goldberg’s classification of some operators represented by an infinite matrix over certain sequence spaces.