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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 result of best proximity points will be obtained for asymptotically relatively nonexpansive mappings under different conditions with respect to the recent paper of Rajesh and Veeramani [Rajesh, S., Veeramani, P. (2016). Best Proximity point theorems for asymptotically relatively nonexpansive mappings. Numer. Funct. Anal. Optim. 37:80–91]. [ABSTRACT FROM AUTHOR]

In the present paper, we discuss the approximation properties of Durrmeyer-Stancu type variant of Jain operators with the modified forms of the Beta basis functions. We establish some direct results, which include the asymptotic formula, the error estimation in terms of the modulus of continuity and weighted approximation. Also, we construct a King modification of these operators which preserves the test functions e0 and e1. [ABSTRACT FROM AUTHOR]

The goal of this paper is to further the investigation begun in Chalmers and Prophet, Numer. Funct. Anal. Optimiz. 1997; 18:507–520. With the benefit of nearly 10 years of work, we begin by indicating how several proofs from Chalmers and Prophet, Numer. Funct. Anal. Optimiz. 1997; 18:507–520, can be substantially improved. We show that the problem of preserving k-convexity onto Πn is one part of a larger shape-preserving problem (multiconvex preservation) relative to Πn, and we completely solve this expanded problem. And finally, we demonstrate that multiconvex preserving projections constructed in this paper are in fact of minimal operator norm in a large class of Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the uniform approximation of functions by Meyer-König and Zeller operators of max-product type in some weighted spaces. We estimate the rate of approximation in terms of a suitable modulus of continuity. [ABSTRACT FROM AUTHOR]

In this paper, we present two main results about best proximity points for multivalued mappings. First, taking into account Klim and Wardowski's approach in fixed-point theory, we obtain a new result for multivalued mappings which includes the main theorem of Kamran [Kamranm, T. (2009). Mizoguchi–Takahashi's type fixed point theorem. Comput. Math. Appl. 57(3):507–511]. Second, combining this approach with cyclic contraction, we give a general best proximity point result, and so we obtain many well-known fixed-point theorems such as Mizoguchi–Takahashi, Feng–Liu and Klim–Wardowski [Mizoguchi, N., Takahashi, W. (1989). Fixed point theorems for multivalued mappings on complete metric spaces. J. Math. Anal. Appl. 141(1):177–188; Feng, Y., Liu, S. (2006). Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings. J. Math. Anal. Appl. 317(1):103–112; Klim, D., Wardowski, D. (2007). Fixed point theorems for set-valued contractions in complete metric spaces. J. Math. Anal. Appl. 334(1):132–139]. Also, we support our results by providing some nontrivial, illustrative and comparative examples. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study the convergence of finite element approximation to the exact solution of general self-adjoint elliptic equations with discontinuous coefficients. Due to low global regularity of the solution, it is difficult to achieve optimal order of convergence with classical finite element methods [ Numer. Math. 1998; 79:175–202]. In this paper, an isoparametric type of discretization is used to prove optimal order error estimates in L 2 and H 1 norms when the global regularity of the solution is low. The interface is assumed to be of arbitrary shape and is smooth for our purpose. Further, for the purpose of numerical computations, we discuss the effect of numerical quadrature on finite element solution, and the related optimal order estimates are also established. [ABSTRACT FROM AUTHOR]

In this paper, we attempt to investigate a class of constrained nonsmooth convex optimization problems, that is, piecewise C² convex objectives with smooth convex inequality constraints. By using the Moreau-Yosida regularization, we convert these problems into unconstrained smooth convex programs. Then, we investigate the second-order properties of the Moreau-Yosida regularization n. By introducing the (GAIPCQ) qualification, we show that the gradient of the regularized function n is piecewise smooth, thereby, semismooth. [ABSTRACT FROM AUTHOR]

This paper deals with a fully implicit time discretization scheme with variable time-step for a nonlinear system modelling phase transition and mechanical deformations in shape memory alloys. The model is studied in the non-stationary case and accounts for local microscopic interactions between the phases introducing the gradients of the phase parameters. The resulting initial-boundary value problem has already been studied by the author who proved existence, uniqueness and continuous dependence on data for a suitable weak solution along some regularity results. A careful and detailed investigation of the variable time-step discretization is the goal of this paper. Thus, we deduce some estimates for the discretization error. These estimates depend only on data, impose no constraints between consecutive time-steps and show an optimal order of convergence. Finally, we prove another regularity result for the solution under stronger regularity assumptions on data. [ABSTRACT FROM AUTHOR]

On a vector space, one can define an order by using a cone in the vector space. In this way, Huang and Zhang reviewed cone metric spaces as a generalization of metric spaces with a different view (J. Math. Anal. Appl. 2007; 332:1468-1476). Most known cones are normal with normal constant M = 1. In this paper, we shall give some results about fixed point of multifunctions on the cone metric spaces with normal constant M = 1. In this way, we shall provide a generalization for main results of Kunze et al. (J. Math. Anal. Appl. 2007; 330:159-173). [ABSTRACT FROM AUTHOR]

Finding verifiable conditions for wavelet systems to be wavelet frames is among the core problems in wavelet analysis. In this paper, we give some simple and sufficient conditions that ensure a multidimensional irregular wavelet system to be a frame or a weighted frame. Quantitative results are provided, and explicit frame bounds are given. [ABSTRACT FROM AUTHOR]

In this paper, we study two types of approximate solutions for a vector optimization problem in Banach space setting. Our main concern is to define two new concepts of approximate saddle points and relate them to the above solution concepts. As a result, a dual is formulated, and duality results are established. [ABSTRACT FROM AUTHOR]

In this paper, we prove that the ordinary core of a sequence x = (xi) of real numbers is equal to its 2[image omitted]-limit set, where [image omitted]: = inf {r ≥ 0:LIMrx ≠ Ø}. Defining the sets r-limit inferior and r-limit superior of a sequence, we show that the r-limit set of the sequence is equal to the intersection of these sets and that r-core of the sequence is equal to the union of these sets. Finally, we prove an ordinary convergence criterion that says a sequence is convergent iff its rough core is equal to its rough limit set for the same roughness degree. [ABSTRACT FROM AUTHOR]

In this paper, we propose an inexact smoothing continuation method for mathematical problem with complementarity constraints. Under suitable conditions, we establish the convergence of the proposed method by showing that any accumulation point of the generated sequence is a B -stationary point of the problem. [ABSTRACT FROM AUTHOR]

In this paper, a mathematical program with complementarity constraints (MPCC) is reformulated as a nonsmooth constrained mathematical program via the Fischer–Burmeister function. Smooth penalty functions are used to treat this nonsmooth constrained program. Under linear independence constraint qualification, and upper level strict complementarity condition, together with some other mild conditions, we prove that the limit point of stationary points satisfying second-order necessary conditions of unconstrained penalized problems is a strongly stationary point, hence a B-stationary point of the original MPCC. Furthermore, this limit point also satisfies a second-order necessary condition of the original MPCC. Numerical results are presented to test the performance of this method. [ABSTRACT FROM AUTHOR]

In this paper, we consider an obstacle control problem where the state satisfies a quasilinear elliptic bilateral variational inequality and the control functions are the upper and the lower obstacles. Existence and necessary conditions for the optimal control are established. [ABSTRACT FROM AUTHOR]

Fractional multistep methods were introduced by C. Lubich for the quadrature of Abel integral operators and the solution of weakly singular Volterra integral equations of the first kind with exactly given right-hand sides. In the current paper, we consider the regularizing properties of these methods to solve the mentioned integral equations of the first kind for perturbed right-hand sides. Finally, numerical results are presented. [ABSTRACT FROM AUTHOR]

In this paper, we study the stability of the tower level set ... of a finite valued increasing convex-along-rays (ICAR) function f defined on ... In monotonic analysis, ICAR functions play the role of usual convex functions in classical convex analysis. We show that each ICAR function f is locally Lipschitz on int dom f and that the pointwise convergence of a sequence of ICAR functions implies its uniform convergence on each compact subset of ... The latter allows us to establish stability results for ICAR functions in some sense similar to those for convex functions. [ABSTRACT FROM AUTHOR]

The Rayleigh conjecture on the representation of the scattered field in the exterior of an obstacle D is widely used in applications. However, this conjecture is false for some obstacles. A.G.R. introduced the modified Rayleigh conjecture (MRC), and in this paper we present successful numerical algorithms based on the MRC for various 2D and 3D obstacle scattering problems. The 3D obstacles include a cube and an ellipsoid. The MRC method is easy to implement for both simple and complex geometries. It is shown to be a viable alternative for other obstacle scattering methods. [ABSTRACT FROM AUTHOR]

In this second part of the paper, we state some general conditions that guarantee that the C0-semigroups generated by special classes of one-dimensional second-order differential operators acting on weighted spaces of continuous functions on an arbitrary real interval can be represented as limits of iterates of the positive linear operators constructively defined according to the method developed in the first part. The particular case where the interval is compact is treated in full detail. Finally, an application to the Black-Scholes equation, which appears in some models arising from mathematical finance, is discussed as well. [ABSTRACT FROM AUTHOR]

In the paper, a new trust region algorithm is given for nonlinear inequality constrained optimization problems. Motivated by a dual problem introduced by Han and Mangasarian [Han, S. P., Mangasarian, O. L. (1983). A dual differentiable exact penalty function. Math. Programming 25:293-306], which is a nonnegatively constrained maximization problem, we construct a trust region algorithm for solving the dual problem. At each iteration, we only need to minimize a quadratic subproblem with simple bound constraints. Under the condition that the iterate sequence generated by the algorithm is contained in some bounded closed set, any accumulation point of the sequence is a Karush- Kuhn-Tucker point of the original problem. [ABSTRACT FROM AUTHOR]

The aim of this paper is to generalize the results of Yamada et al. [Yamada, I., Ogura, N., Yamashita, Y., Sakaniwa, K. (1998). Quadratic approximation of fixed points of nonexpansive mappings in Hilbert spaces. Numer. Funct. Anal. Optimiz. 19(l):165-190], and to provide complementary results to those of Deutsch and Yamada [Deutsch, F., Yamada, I. (1998). Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings. Numer. Funct. Anal. Optim. 19(1&2):33-56] in which they consider the minimization of some function d over a closed convex set F, the nonempty intersection of N fixed point sets. We start by considering a quadratic function θ and providing a relaxation of conditions of Theorem 1 of Yamada et al. (1998) to obtain a sequence of fixed points of certain contraction maps, converging to the unique minimizer of θ over F. We then extend Theorem 2 and obtain a complementary result to Theorem 3 of Yamada et al. (1998) by replacing the condition lim n → ∞ (λn - λn+1)/λ²n+1 = 0 on the parameters by the more general condition lim n → ∞ λn / λn+1 = 1. We next look at minimizing a more general function θ than a quadratic function which was proposed by Deutsch and Yamada (1998) and show that the sequence of fixed points of certain maps converge to the unique minimizer of 9 over F. Finally, we prove a complementary result to that of Deutsch and Yamada (1998) by using the alternate condition on the parameters. [ABSTRACT FROM AUTHOR]

In many wavelets applications, a scalar product of given function with the scaling function has to be calculated. For deriving effective one point quadrature formulas, the relation among the first scaling moment and the second one is crucial. In this paper, new relations among scaling moments are deduced. [ABSTRACT FROM AUTHOR]

In this paper, we study stability radii of linear systems under multi-perturbation of the coefficient matrices. Formulas for complex stability radius are derived. We then consider linear positive systems and prove that for this class of systems, the complex stability radius is equal to the real stability radius which can be computed via a simple formula. We illustrated the obtained results by two examples. [ABSTRACT FROM AUTHOR]

In this paper we consider a control-constrained optimal control problem governed by a system of semilinear parabolic reaction–diffusion equations. The optimal solutions are subject to perturbations of the dynamics and of the objective. We prove that local optimal solutions, as a function of the perturbation parameter, are Lipschitz continuous and directionally differentiable. We characterize the directional derivatives, also known as parametric sensitivities, as the solutions of auxiliary quadratic programming problems, i.e., linear-quadratic optimal control problems. Parametric sensitivities provide valuable information, e.g., in realtime optimal control environments. [ABSTRACT FROM AUTHOR]

In this work, we investigate the class of integral operators whose generating kernels have expansion series in terms of fundamental monomials. We study the reproducing kernel Hilbert space associated with such kernels, specifically when defined on balls about the origin of . Consequently, spectral properties, including the asymptotic behavior of the eigenvalues of these operators will be obtained. We also revisit our main results in the case in which the generating kernel is the Bergman kernel. [ABSTRACT FROM AUTHOR]

In this paper, generalized mth-order contingent epiderivative and generalized mth-order epiderivative of set-valued maps are introduced, respectively. By virtue of the generalized mth-order epiderivatives, generalized necessary and sufficient optimality conditions are obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a fixed set. Generalized Kuhn-Tucker type necessary and sufficient optimality conditions are also obtained for Henig efficient solutions to a set-valued optimization problem whose constraint set is determined by a set-valued map. [ABSTRACT FROM AUTHOR]

In this paper, new classes of generalized invex functions called B-(p, r)-pseudo-invex and B-(p, r)-quasi-invex (with respect to η and b) are introduced. These classes of functions are defined by relaxing the definitions of the classes of quasi-invex and pseudo-invex functions earlier defined by Hanson (J. Math. Anal. Appl. 1981, 80:545-550), and also the concept of B-(p, r)-invexity (with respect to η and b) earlier introduced by Antczak (J. Math. Anal. Appl. 2003, 286:187-206). Further, mathematical programming problems involving generalized B-(p, r)-invex functions with respect to η and b are considered. The sufficient optimality conditions and various duality results are established for such optimization problems. [ABSTRACT FROM AUTHOR]

This paper is concerned with the analysis of the discontinuous Galerkin finite element method applied to the space semidiscretization of nonlinear nonstationary convection-diffusion problems. The attention is paid to the derivation of the method and the investigation of error estimates for schemes with a symmetric as well as nonsymmetric discretization of diffusion terms and with the interior and boundary penalty. The theoretical results are illustrated by numerical experiments. [ABSTRACT FROM AUTHOR]

Let p be a continuous seminorm on a real locally convex space X. The current paper is concerned with the problem of restricted p-centers in X. Characterizations of restricted p-centers and strongly unique restricted p-centers with respect to a “sun” are provided, and then some recent results due to Laurent and Pai [9] are extended and improved. [ABSTRACT FROM AUTHOR]

In this paper, we study the problem of whether the best approximation to any x in a Hilbert space X from the set where C is a closed convex subset of X , S is a closed convex cone that does not necessarily have nonempty interior, Y is a Banach space, and g : X ? Y is a continuous S -convex function, can be characterized by the best approximation to a perturbation x - l of x from the set C for some l ? X . We provide a global approach to this problem by presenting a dual global constraint qualification, which is less restrictive than the Slater type (or interior-point) conditions, guaranteeing the strong conical hull intersection property (CHIP). We then show that the strong CHIP characterizes the perturbation property under a mild closure condition. The closure condition, for instance, holds whenever the explicit constraint set is described by finitely many linear inequality constraints. We also establish easily verifiable dual conditions that are equivalent to the best approximation from the set K . We finally demonstrate that our results recapture the corresponding known results in the particular case where Y is a finite dimensional space. [ABSTRACT FROM AUTHOR]

The goal of this paper is to establish the rates of convergence of Newton-like methods corresponding to a Holder continuity assumption on the derivative of the associated nonlinear operator. This generalizes the classical convergence results. Applications are worked out on different kinds of nonlinear integral and differential problems. The abstract general framework is a real or complex Banach space. Applications and computations include a nonlinear boundary value problem and a nonlinear integral equation appearing in mathematical models of radiative transfer. Computations have been done with the only purpose of illustrating how practical convergence respects the theoretical error bounds. [ABSTRACT FROM AUTHOR]

This paper describes some properties and applications of Steklov eigenproblems for prototypical second-order elliptic operators on bounded regions in R″. Results are described for Schroedinger and weighted harmonic equations. A vari-ational description of the least eigenvalue leads to optimal L2-trace inequalities. It is shown that the eigenfunctions provide complete orthonormal bases of certain closed subspaces of H1(Ω) and also of L2(∂ &Omega, d σ). This allows the description, and representation, of solution operators for homogeneous elliptic equations subject to inhomogeneous Dirichlet, Neumann or Robin boundary data. They are also used to describe Robin to Dirichlet and Neumann to Dirichlet operators for these equations, and to describe the spectrum of these operators. The allowable regions are quite general; in particular classes of bounded regions with a finite number of disjoint Lipschitz components for the boundary are allowed. [ABSTRACT FROM AUTHOR]

In this paper we introduce a boundary coercive condition on the regularizing function of the proximal point method in Banach spaces, which has a penalization effect and is useful for solving certain variational inequality problems. As its finite dimensional counterpart (proximal point method with Bregman distances) it avoids explicit consideration of the constrains, because the feasibility is taken care of by the regularizing function, whose derivative diverges on the boundary of the feasible set. The algorithm is therefore an interior point one. We show that this effect is guaranteed even for an inexact version of the method, namely the hybrid extragradient proximal point method proposed by Solodov and Svaiter for finite dimensional spaces. We give a full convergence analysis, and examples of regularizing functions satisfying the required conditions, for the cases of the feasible set being a closed ball or a polyhedron. [ABSTRACT FROM AUTHOR]

In this paper we investigate an optimal control problem for semilinear hyperbolic systems with unknown initial data. We first give some general existence results and then derive optimality conditions via a penalization method. Finally, we apply our abstract results to an example. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study viability results for mutational equations with delay. First a notion of tangent set in a mutational space is introduced. Then, an existence theorem is proved and stability results of the solutions are given. An example is presented in the power set of . [ABSTRACT FROM AUTHOR]

In this paper our propose is to find a common term which is included in the assumptions of theorems proving existence of zeros, implicit functions, fixed points or coincidence points. This new point of view allows us to weaken the assumptions which guarantee the solvability of nonlinear equations and to recommend a possible unified treatment of several classes of operators which appear in the theory of nonlinear equations. [ABSTRACT FROM AUTHOR]

In this paper we construct upper and lower solutions for a class of parabolic initial-boundary value problems in terms of the solution of the St-Venant problem and first eigenvalue problem. These bounds are sharp in the sense that they coincide with the exact solution in particular situations. [ABSTRACT FROM AUTHOR]

The boundedness and compactness of the generalized weighted composition operator on weighted Bergman spaces are discussed. [ABSTRACT FROM AUTHOR]

Compactness of embeddings in discrete counterparts of Sobolev spaces is considered. We study the embeddings in spaces of cell-centered grid functions, in one- and two-dimensional domains. No restrictions are made on the mesh-ratios of the underlying meshes. [ABSTRACT FROM AUTHOR]

In this work, using the concept of natural density, we introduce the notion of rough statistical convergence. We define the set of rough statistical limit points of a sequence and obtain two statistical convergence criteria associated with this set. Later, we prove that this set is closed and convex. Finally, we examine the relations between the set of statistical cluster points and the set of rough statistical limit points of a sequence. [ABSTRACT FROM AUTHOR]

We consider nonlinear discrete optimal control problems with variable end points and with inequality and equality type constraints on trajectories and control. We derive first- and second-order necessary optimality conditions that are meaningful without a priori normality assumptions. [ABSTRACT FROM AUTHOR]

The convergence criterion of Newton's method for underdetermined system of equations under the γ-condition is established and the radius of the convergence ball is obtained. Applications to analytic operator are provided and some results due to Shub and Smale (SIAM J. Numer. Anal. 1996, 33:128-148) are extended and improved. [ABSTRACT FROM AUTHOR]

Poincaré-Friedrichs inequalities are derived for piecewise H² functions on two dimensional domains. These inequalities can be applied to classical nonconforming finite element methods, mortar methods and discontinuous Galerfcin methods. [ABSTRACT FROM AUTHOR]

The nonlocal boundary-value problem for hyperbolic equations d²u(t)/dt² + Au(t) = f(t) (0 ≤ t ≤ 1), u(0) = xu(1) +φ, u'(0) = βu'(1) + ψ in a Hilbert space H with the self-adjoint positive definite operator A is considered. Applying the operator approach, we establish the stability estimates for solution of this nonlocal boundary-value problem. In applications, the stability estimates for the solution of the nonlocal boundary value problems for hyperbolic equations are obtained. The first and second order of accuracy difference schemes generated by the integer power of A for approximately solving this abstract nonlocal boundary-value problem are presented. The stability estimates for the solution of these difference schemes are obtained. The theoretical statements for the solution of this difference schemes are supported by the results of numerical experiments. [ABSTRACT FROM AUTHOR]

A multi resolution transform corresponding to interpolatory techniques is used for fast application of second order Taylor's approximations. In designing this algorithm we apply data compression to the linear and the bilinear forms that appear on the approximation. Analysis of the error is performed. Finally, some numerical results are presented. [ABSTRACT FROM AUTHOR]

A new approach to a solution of a nonlinear constrained mathematical programming problem and its Mond-Weir duals is introduced. An n-approximated problem associated with a primal nonlinear programming problem is presented that involves n-approximated functions constituting the primal problem. The equivalence between the original mathematical programming problem and its associated n-approximated optimization problem is established under invexity assumption. Furthermore, n-approximated dual problems in the sense of Mond-Weir are introduced for the obtained n-approximated optimization problem in this method. By the help of n-approximated dual problems some duality results are established for the original mathematical programming problem and its original duals. [ABSTRACT FROM AUTHOR]

The notation of I-convergence was introduced and studied by Kostyrko, Macaj, Salat, and Wilczynski. Recently, the concept of I-convergent for a sequence of bounded linear operators has been studied by Khan and Shafiq. This has motivated us to introduce and study some new spaces of double sequences of bounded linear operators and their basic topological and algebraic properties of these spaces. And we study some of their basic topological and algebraic properties of these spaces. We prove some inclusion relations on these spaces. [ABSTRACT FROM AUTHOR]

In this article, the relationship between absolute projection constants with dimensionnand with codimensionnis given. In particular, bounds from below are found for absolute projection constants with codimension 2. Also, exact formulas for minimal projections in this case are given. It is shown that these minimal projections are not unique. [ABSTRACT FROM AUTHOR]

We investigate continuous dependence on initial data for solutions of a nonlinear parabolic problem, when Robin conditions are prescribed on the boundary ∂ Ω × (t > 0), Ω a bounded R2 domain. [ABSTRACT FROM AUTHOR]

In this article, we consider the approximate controllability for a class of second-order nonlinear functional differential equations with state-dependent delay. Assuming the approximate controllability of the corresponding linearized equation, we obtain sufficient conditions for the approximate controllability of the nonlinear second-order equations. Examples are presented to show the application of the proposed result. [ABSTRACT FROM AUTHOR]