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This paper is concerned with the following minimization problem e p (M) = inf { E p (u) : u ∈ H 1 (R N) , ‖ u ‖ L 2 2 = M 2 } , where energy functional E p (u) is defined by E p (u) = ‖ ∇ u ‖ L 2 2 + ∫ R N V (x) | u | 2 d x - 2 p + 2 ∫ R N | x | - h | u | p + 2 d x and V is a bounded potential. For 0 < p < p ∗ : = 4 - 2 h N (0 < h < min { 2 , N }) , it is shown that there exists a constant M 0 ≥ 0 , such that the minimization problem exists at least one minimizer if M > M 0 . When p = p ∗ , the minimization problem exists at least one minimizer if M ∈ (M ∗ , ‖ Q p ∗ ‖ L 2) , where constant M ∗ ≥ 0 and Q p ∗ is the unique positive radial solution of - Δ u + u - | x | - h | u | p ∗ u = 0 , and under some assumptions on V, there is no minimizer if M ≥ ‖ Q p ∗ ‖ L 2 . Moreover, when 0 < p < p ∗ , for fixed M > ‖ Q p ∗ ‖ L 2 , we analyze the concentration behavior of minimizers as p ↗ p ∗ . [ABSTRACT FROM AUTHOR]

This paper is concerned with an inverse source problem for a space-time fractional diffusion equation. We aim to identify an unknown source term from partially observed data. The employed model involves the Caputo fractional derivative in time and the non-local fractional Laplacian operator in space. The well-posedness of the forward problem is discussed. The considered ill-posed inverse source problem is formulated as a minimization one. The existence, uniqueness and stability of the solution of the minimization problem are examined. An iterative process is developed for identifying the unknown source term. A numerical implementation of the proposed approach is performed. The convergence of the discretized fractional derivatives is analyzed. The efficiency and accuracy of the proposed identification algorithm are confirmed by some numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new algorithm of multi-step inertial form for approximating a common solution of Split Generalized Mixed Equilibrium Problem (SGMEP) and Minimization Problem (MP) in real Hilbert spaces. Motivated by the multi-step inertial technique, we incorporate the multi-step inertial term to accelerate the convergence of the proposed method. Under standard and mild assumption of monotonicity and lower semicontinuity of the SGMEP associated mappings, we establish the strong convergence of the scheme. Some numerical examples are presented to illustrate the performance of our method as well as comparing it with the non-inertial version. [ABSTRACT FROM AUTHOR]

The aim of this article is to study a minimization problem involving fractional Sobolev spaces on Nehari manifold via the variational methods. We show the existence of minimizers by analyzing the fibering maps associated with the energy functional. [ABSTRACT FROM AUTHOR]

The forward–backward algorithm is a splitting method for solving convex minimization problems of the sum of two objective functions. It has a great attention in optimization due to its broad application to many disciplines, such as image and signal processing, optimal control, regression, and classification problems. In this work, we aim to introduce new forward–backward algorithms for solving both unconstrained and constrained convex minimization problems by using linesearch technique. We discuss the convergence under mild conditions that do not depend on the Lipschitz continuity assumption of the gradient. Finally, we provide some applications to solving compressive sensing and image inpainting problems. Numerical results show that the proposed algorithm is more efficient than some algorithms in the literature. We also discuss the optimal choice of parameters in algorithms via numerical experiments. [ABSTRACT FROM AUTHOR]

We consider an abstract minimization problem in reflexive Banach spaces together with a specific family of approximating sets, constructed by perturbing the cost functional and the set of constraints. For this problem, we state and prove various well-posedness results in the sense of Tykhonov, under different assumptions on the data. The proofs are based on arguments of lower semicontinuity, compactness and Mosco convergence of sets. Our results are useful in the study of various mathematical models in contact mechanics. To provide examples, we introduce 2 models, which describe the equilibrium of an elastic body in contact with a rigid body covered by a rigid-plastic and an elastic material, respectively. The weak formulation of each model is in the form of a minimization problem for the displacement field. We use our abstract well-posedness results in the study of these minimization problems. In this way, we obtain existence, uniqueness and convergence results, and moreover, we provide their mechanical interpretations. [ABSTRACT FROM AUTHOR]

Inverse problems of magnetostatics for continuity defects can be formulated in different ways. One can set the task of determining defects with high accuracy and resolution or limit the task to revealing several specified types of ("hazardous") defects with varying degrees of probability. In this case, other real ("minor") defects will be either skipped by the program or "replaced" by defects from among the ones specified. Minimizing the residual functional between experimental and calculated data is the main task when solving any inverse problems, both in real time and off-line. The paper considers an approach to calculating a simplified inverse problem in real time without accumulating experimental data bases. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new modification of the Gradient Projection Algorithm and the Forward–Backward Algorithm. Using our proposed algorithms, we establish two strong convergence theorems for solving convex minimization problem, monotone variational inclusion problem and fixed point problem for demicontractive mappings in a real Hilbert space. Furthermore, we apply our results to solve split feasibility and optimal control problems. We also give two numerical examples of our algorithm in real Euclidean space of dimension 4 and in an infinite dimensional Hilbert space, to show the efficiency and advantage of our results. [ABSTRACT FROM AUTHOR]

We introduce the notion of quasimonotone polar of a multivalued operator, in a similar way as the well-known monotone polar due to Martínez-Legaz and Svaiter. We first recover several properties similar to the monotone polar, including a characterization in terms of normal cones. Next, we use it to analyze certain aspects of maximal (in the sense of graph inclusion) quasimonotonicity, and its relation to the notion of maximal quasimonotonicity introduced by Aussel and Eberhard. Furthermore, we study the connections between quasimonotonicity and Minty Variational Inequality Problems and, in particular, we consider the general minimization problem. We conclude by characterizing the maximal quasimonotonicity of operators defined in the real line. [ABSTRACT FROM AUTHOR]

We study the convergence of exact and inexact versions of the proximal point method with a generalized regularization function in Hadamard manifolds for solving scalar and vectorial optimization problems involving Lipschitz functions. We consider a local dominance property of the directional derivative of the objective function over the regularization term in order to obtain that cluster points of the sequence are stationary points. Under an additional assumption, we prove that every cluster point of the sequence is a minimizer in the scalar case and a weak efficient point in the vectorial case. Our results extend some of the existing ones in the literature about optimization on manifolds. [ABSTRACT FROM AUTHOR]

A mathematical processing technique and an algorithm for the numerical correction of experimental displacement fields on the specimen surface in the vicinity of a crack tip with consideration of the displacement of a body as a rigid whole under its loading and the actual position of a crack tip were proposed. The initial experimental displacement fields in the vicinity of a fatigue crack tip of a compact specimen were determined by digital image correlation. This approach was successfully used for a compact specimen and the possibility of its expansion to obtain the distributions of the stress intensity factors and the nonsingular T stresses along the spatial crack front was shown. [ABSTRACT FROM AUTHOR]

We study minimization problems on Hardy–Sobolev type inequality. We consider the case where singularity is in interior of bounded domain Ω ⊂ R N . The attainability of best constants for Hardy–Sobolev type inequalities with boundary singularities have been studied so far, for example Ghoussoub and Kang (Ann Inst Henri Poincare Anal Non Lineaire 21(6):767–793, 2004), Ghoussoub and Robert (IMRP 21867:1–85, 2006), Ghoussoub and Robert (Trans Am Math Soc 361(9):4843–4870, 2009) etc.... According to their results, the mean curvature of ∂ Ω at singularity affects the attainability of the best constants. In contrast with boundary singularity case, in interior singularity case it is well known that the best Hardy–Sobolev constant μ s (Ω) : = ∫ Ω | ∇ u | 2 d x | u ∈ H 0 1 (Ω) , ∫ Ω | u | 2 ∗ (s) | x | s d x = 1 is never achieved for all bounded domain Ω . We can see that the position of singularity on domain is related to the existence of minimizer. In this paper, we consider the attainability of the best constant for the embedding H 1 (Ω) ↪ L 2 ∗ (s) (Ω , | x | - s d x) for bounded domain Ω with 0 ∈ Ω . In this problem, scaling invariance doesn’t hold and we can not obtain information of singularity like mean curvature. [ABSTRACT FROM AUTHOR]

Compressive sensing has emerged as a promising technique for signal processing. Most of the existing reconstruction methods cannot describe the sparsity of an image in different transform domains simultaneously, which results in an instability performance for images. In this paper, a new model based on compressive sensing is developed to describe the sparsity of an image in different transform domains simultaneously. The parameters of the a priori regularization terms, including the structural redundant information of wavelet coefficient, gradient coefficient and nonlocal mean, are adjusted adaptively to balance the influence of each corresponding constraint in the model. Some approximate substitutions are introduced to reduce the computational complexity. In addition, a two-step solution is proposed, in which an approximate image is reconstructed rapidly by picking parts of the constraints of the model. Then, an enhanced reconstruction performance can be achieved by adding the constraint of nonlocal mean. Simulation results demonstrate that the proposed method provides better performance in both peak signal-to-noise ratio and visual quality with a relatively fast convergence rate, especially at the low sample ratio. [ABSTRACT FROM AUTHOR]

The color of an object appears to be different for elderly and young persons. This is because human visual features vary with on age. In general, the spectral transmittance of the crystalline lens of an elderly person is lower than that of a young person. The low spectral transmittance decreases the brightness of the visual field. This phenomenon affects the color appearance of an image in human visual perception. In this paper, a lightness transform method is proposed to improve the color appearance of images for elderly persons. In the proposed method, colors that are difficult for an elderly person to see are transformed into colors that are easy to see by adding lightness contrast. The performance of the proposed method is confirmed in experiments using digital images. [ABSTRACT FROM AUTHOR]

We give a simplified proof of the nonexistence of large nuclei in the liquid drop model and provide an explicit bound. Our bound is within a factor of 2.3 of the conjectured value and seems to be the first quantitative result. [ABSTRACT FROM AUTHOR]

We first use the Schwarz rearrangement to solve a minimization problem on eigenvalues of the one-dimensional p-Laplacian with integrable potentials. Then we construct an optimal class of non-degenerate potentials for the one-dimensional p-Laplacian with the Dirichlet boundary condition. Such a class of non-degenerate potentials is a generalization of many known classes of non-degenerate potentials and will be useful in many problems of nonlinear differential equations. [ABSTRACT FROM AUTHOR]

We consider the problem of finding the optimal constant for the embedding of the space into the space $$L^1(\Omega )$$ , where $$\Omega \subset \mathbb {R}^n$$ is a bounded domain with boundary of class $$C^{1,1}$$ . This is equivalent to find the first eigenvalue $$\Lambda _{1,1}^c(\Omega )$$ of the clamped 1-biharmonic operator. In this paper, we identify the correct relaxation of the problem on $$BL_0(\Omega )$$ , the space of functions whose distributional Laplacian is a finite Radon measure, we obtain the associated Euler-Lagrange equation, and we give lower bounds for $$\Lambda _{1,1}^c(\Omega )$$ , investigating the validity of an inequality of Faber-Krahn type. [ABSTRACT FROM AUTHOR]

We propose a modified proximal point algorithm for solving minimization problems in CAT(0) spaces. We then prove that the sequence converges to a minimizer of convex objective functions. We finally provide the numerical examples for supporting our main result. [ABSTRACT FROM AUTHOR]

This paper deals with maximization and minimization of quasiconvex functions in a finite dimensional setting. Firstly, some existence results on closed convex sets, possibly containing lines, are presented. This is given via a careful study of reduction to the boundary and/or extremality of the feasible set. Necessary or sufficient optimality conditions are derived in terms of radial epiderivatives. Then, the problem of minimizing quasiconvex functions are analyzed via asymptotic analysis. Finally, some attempts to define asymptotic functions under quasiconvexity are also outlined. Several examples illustrating the applicability of our results are shown. [ABSTRACT FROM AUTHOR]

The aim of this paper is to study the convergence of solutions of the following second order difference inclusion { exp-1ui ui+1 + θi exp-1ui ui-1 ∊ ciA(ui), i ≥ 1 u0 = x ∊ M; sup i≥0 d(ui, x) < +∝, to a singularity of a multi-valued maximal monotone vector field A on a Hadamard manifold M, where {ci} and {θi} are sequences of positive real numbers and x is an arbitrary fixed point in M. The results of this paper extend previous results in the literature from Hilbert spaces to Hadamard manifolds for general maximal monotone, strongly monotone multi-valued vector fields and subdifferentials of proper, lower semicontinuous and geodesically convex functions f : M →] - ∝,+∝]. In the recent case, when A = ϐf, we show that the sequence {ui}, given by the equation, converges to a point of the solution set of the following constraint minimization problem Min x∊M f(x). [ABSTRACT FROM AUTHOR]

Some numerical algorithm, basing on the methods of constrained minimization, is proposed for the solution of an inverse problem for the stationary equations of heat convection. This algorithm is applied for the reconstruction of the heat flux on a part of the boundary from the measured values of temperature or the velocity vector in the flow domain. [ABSTRACT FROM AUTHOR]

In this paper, we introduce and study a triple hierarchical variational inequality (THVI) with constraints of minimization and equilibrium problems. More precisely, let Fix(T) be the fixed point set of a nonexpansive mapping, let MEP(Θ,ϕ) be the solution set of a mixed equilibrium problem (MEP), and let Γ be the solution set of a minimization problem (MP) for a convex and continuously Frechet differential functional in Hilbert spaces. We want to find a solution x* ∈ Fix(T) ∩ MEP(Θ,ϕ) ∩ Γ of a variational inequality with a variational inequality constraint over the intersection of Fix(T), MEP(Θ,ϕ), and Γ . We propose a hybrid iterative algorithm with regularization to compute approximate solutions of the THVI, and we present the convergence analysis of the proposed iterative algorithm. [ABSTRACT FROM AUTHOR]

In this paper we introduce an iterative algorithm for finding a common element of the fixed point set of an asymptotically strict pseudocontractive mapping S in the intermediate sense and the solution set of the minimization problem (MP) for a convex and continuously Frechet differentiable functional in Hilbert space. The iterative algorithm is based on several well-known methods including the extragradient method, CQ method, Mann-type iterative method and hybrid gradient projection algorithm with regularization. We obtain a strong convergence theorem for three sequences generated by our iterative algorithm. In addition, we also prove a new weak convergence theorem by a modified extragradient method with regularization for the MP and the mapping S. [ABSTRACT FROM AUTHOR]

According to a class of constrained minimization problems, the Schwartz symmetrization process and the compactness lemma of Strauss, we prove that there is a nontrivial ground state solution for a class of p-Laplace equations without the Ambrosetti-Rabinowitz condition. [ABSTRACT FROM AUTHOR]

In this paper, an implicit net with perturbations for solving the mixed equilibrium problems and fixed point problems has been constructed and it is shown that the proposed net converges strongly to a common solution of the mixed equilibrium problems and fixed point problems. Also, as applications, some corollaries for solving the minimum-norm problems are also included. [ABSTRACT FROM AUTHOR]

In this paper, we use the penalty approach in order to study a class of constrained vector minimization problems on complete metric spaces. A penalty function is said to have the generalized exact penalty property iff there is a penalty coefficient for which approximate solutions of the unconstrained penalized problem are close enough to approximate solutions of the corresponding constrained problem. For our class of problems, we establish the generalized exact penalty property and obtain an estimation of the exact penalty. [ABSTRACT FROM AUTHOR]

In this paper we consider the following nonhomogeneous gradient system on a Hadamard manifold M where φ: M → ℝ is a geodesically convex function of class C with argmin φ ≠ Ø. We prove global convergence of solutions of the gradient systemto a minimum point of φ. We also discuss on the rate of convergence of φ( x( t)) to the minimum value of φ as well as the rate of convergence of | x′( t)| and d( x( t), p) to zero, where p is the minimum point of φ. Finally, we present some problems and future directions to study. [ABSTRACT FROM AUTHOR]

An experimental calculation method for the determination of parameters of the stress-strain state of full-scale objects, namely, loads, residual stresses, imperfection, and parameters of fracture mechanics, is considered. The method is based on the calculation evaluation of these parameters from the minimum condition for the efficiency function reflecting the disagreement of significant experimental data arrays and results of numerical solution of model problems with the use of the finite element method. The efficiency of the method and the corresponding program complex is illustrated by results of solving a series of routine practical problems. [ABSTRACT FROM AUTHOR]

This paper investigates an infinite-horizon problem in the one-dimensional calculus of variations, arising from the Ramsey model of endogeneous economic growth. Following Chichilnisky, we introduce an additional term, which models concern for the well-being of future generations. We show that there are no optimal solutions, but that there are equilibrium strateges, i.e. Nash equilibria of the leader-follower game between successive generations. To solve the problem, we approximate the Chichilnisky criterion by a biexponential criterion, we characterize its equilibria by a pair of coupled differential equations of HJB type, and we go to the limit. We find all the equilibrium strategies for the Chichilnisky criterion. The mathematical analysis is difficult because one has to solve an implicit differential equation in the sense of Thom. Our analysis extends earlier work by Ekeland and Lazrak. [ABSTRACT FROM AUTHOR]

In this paper we use the penalty approach in order to study a class of constrained minimization problems on complete metric spaces. A penalty function is said to have the generalized exact penalty property if there is a penalty coefficient for which approximate solutions of the unconstrained penalized problem are close enough to approximate solutions of the corresponding constrained problem. For our class of problems we establish the generalized exact penalty property and obtain an estimation of the exact penalty. [ABSTRACT FROM AUTHOR]

This paper presents a new approach to reinitialization in finite element methods for the level set transport equation. The proposed variational formulation is derived by solving a minimization problem. A penalty term is introduced to preserve the shape of the free interface in the process of redistancing. In contrast to hyperbolic PDE reinitialization, the resulting boundary value problem is elliptic and can be solved using a simple fixed-point iteration method. The minimization-based approach makes it possible to define the desired geometric properties in terms of a suitable potential function. In particular, truncated distance functions can be generated using a double-well potential. The results of a numerical study indicate that the new methodology is a promising alternative to conventional reinitialization techniques. [ABSTRACT FROM AUTHOR]

In this paper, we will use the variational method and limiting approach to solve the minimization problems of the Dirichlet/Neumann eigenvalues of the one-dimensional p-Laplacian when the L norm of integrable potentials is given. Combining with the results for the corresponding maximization problems, we have obtained the explicit results for these eigenvalues. [ABSTRACT FROM AUTHOR]

Least-squares technique is well-known and widely used to determine the coefficients of a explanatory model from observations based on a concept of distance. Traditionally, the observations consist of pairs of numeric values. However, in many real-life problems, the independent or explanatory variable can be observed precisely (for instance, the time) and the dependent or response variable is usually described by approximate values, such as 'about $$\pounds300$$' or 'approximately $500', instead of exact values, due to sources of uncertainty that may affect the response. In this paper, we present a new technique to obtain fuzzy regression models that consider triangular fuzzy numbers in the response variable. The procedure solves linear and non-linear problems and is easy to compute in practice and may be applied in different contexts. The usefulness of the proposed method is illustrated using simulated and real-life examples. [ABSTRACT FROM AUTHOR]

Let A and X be nonempty, bounded and closed subsets of a geodesic metric space ( E, d). The minimization (resp. maximization) problem denoted by min( A, X) (resp. max( A, X)) consists in finding $${(a_0,x_0) \in A \times X}$$ such that $${d(a_0,x_0) = \inf\left\{d(a,x) : a \in A, x \in X\right\}}$$ (resp. $${d(a_0,x_0) = \sup\left\{d(a,x) : a \in A, x \in X\right\}}$$). We give generic results on the well-posedness of these problems in different geodesic spaces and under different conditions considering the set A fixed. Besides, we analyze the situations when one set or both sets are compact and prove some specific results for CAT(0) spaces. We also prove a variant of the Drop Theorem in Busemann convex geodesic spaces and apply it to obtain an optimization result for convex functions. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two iterative schemes (one implicit and one explicit) for finding a common element of the set of solutions of the generalized equilibrium problems and the set of all common fixed points of a nonexpansive semigroup in the framework of a real Hilbert space. We prove that both approaches converge strongly to a common element of such two sets. Such common element is the unique solution of a variational inequality, which is the optimality condition for a minimization problem. Furthermore, we utilize the main results to obtain two mean ergodic theorems for nonexpansive mappings in a Hilbert space. The results of this paper extend and improve the results of Li et al. (J Nonlinear Anal 70:3065-3071, ), Cianciaruso et al. (J Optim Theory Appl 146:491-509, ) and many others. [ABSTRACT FROM AUTHOR]

We use the penalty approach in order to study inequality-constrained minimization problems in infinite dimensional spaces. A penalty function is said to have the exact penalty property if there is a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. In this paper we consider a large class of inequality-constrained minimization problems for which a constraint is a mapping with values in a normed ordered space. For this class of problems we introduce a new type of penalty functions, establish the exact penalty property and obtain an estimation of the exact penalty. Using this exact penalty property we obtain necessary and sufficient optimality conditions for the constrained minimization problems. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an iterative method for finding a common element of the set of fixed points of a nonexpansive mapping and the set of common fixed points of a countable family of nonexpansive mappings in Hilbert spaces. Using the result we consider a strong convergence theorem in variational inequalities and equilibrium problems. The result present in this paper extend and improve the corresponding result of Qin et al. (Nonlinear Anal 69:3897-3909, 2008), Plubtieng and Punpaeng (J Math Anal Appl 336:455-469, 2007) and many others. [ABSTRACT FROM AUTHOR]

In this paper, we devote to find the solution of the following quadratic minimization problem where Ω is the intersection set of the solution set of some equilibrium problem, the fixed points set of a nonexpansive mapping and the solution set of some variational inequality. In order to solve the above minimization problem, we first construct an implicit algorithm by using the projection method. Further, we suggest an explicit algorithm by discretizing this implicit algorithm. Finally, we prove that the proposed implicit and explicit algorithms converge strongly to a solution of the above minimization problem. [ABSTRACT FROM AUTHOR]

This paper proposes a modified iterative algorithm using a viscosity approximation method with a weak contraction. The purpose is to find a common element of the set of common fixed points of an infinite family of nonexpansive mappings and the set of a finite family of equilibrium problems that is also a solution to a variational inequality. Under suitable conditions, some strong convergence theorems are established in the framework of Hilbert spaces. The results presented in the paper improve and extend the corresponding results of Colao et al. (Colao, V., Acedo, G. L., and Marino, G. An implicit method for finding common solutions of variational inequalities and systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings. Nonlinear Anal. 71, 2708-2715 (2009)), Plubtieng and Punpaeng (Plubtieng, S. and Punpaeng, R. A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces. J. Math. Anal. Appl. 336, 455-469 (2007)), Colao et al. (Colao, V., Marino, G., and Xu, H. K. An iterative method for finding common solutions of equilibrium problem and fixed point problems. J. Math. Anal. Appl. 344, 340-352 (2008)), Yao et al. (Yao, Y., Liou, Y. C., and Yao, J. C. Convergence theorem for equilibrium problems and fixed point problems of infinite family of nonexpansive mappings. Fixed Point Theory Application 2007, Article ID 64363 (2007) DOI 10.1155/2007/64363), and others. [ABSTRACT FROM AUTHOR]

In this paper, we first establish an existence theorem related with intersection theorem, maximal element theorem and common fixed point theorem for multivalued maps by applying an abstract maximal element principle proved by Lin and Du. Some new stationary point theorems, minimization problems, new fixed point theorems and a system of nonconvex equilibrium theorem are also given. [ABSTRACT FROM AUTHOR]

The article focuses on the lightness modification method for people with color vision deficiency. It notes that colors play a vital function in expressing visual information. However, color information is not uniformly perceived by all people wherein 5-8% males and 0.8 females have anomalous color vision. It proposes a lightness modification method which allows barrier-free color vision for people with dichromacy while keeping the color information in the original image.

In this paper, we use the penalty approach for constrained minimization problems in infinite dimensional Banach spaces. A penalty function is said to have the exact penalty property if there is a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. We establish a simple sufficient condition for exact penalty property for two large classes of constrained minimization problems. [ABSTRACT FROM AUTHOR]

We use the penalty approach in order to study constrained minimization problems. A penalty function is said to have the exact penalty property if there is a penalty coefficient for which a solution of an unconstrained penalized problem is a solution of the corresponding constrained problem. In this paper we establish the exact penalty property for a large class of inequality-constrained minimization problems. [ABSTRACT FROM AUTHOR]

We pose the problem of minimization for the relative preimage problem and prove a minimization theorem. The relative common preimage problem for a finite system of maps, whose particular cases are the relative coincidence and common root problems for finitely many maps, is reduced to the relative preimage problem. As corollaries, results concerning the relative coincidence problems for two maps, problems of fixed points and roots are obtained; these results, with certain distinctions, can be found in known papers by other authors. [ABSTRACT FROM AUTHOR]

We prove theorems characterizing the minimizers for the Cahn-Hilliard free energy functional, which is used to describe the liquid vapor phase transition (or the 2 state magnetization transition). In particular, we exactly determine the critical density for droplet formation, and the geometry of the droplets. [ABSTRACT FROM AUTHOR]

This paper studies convergence properties of regularized Newton methods for minimizing a convex function whose Hessian matrix may be singular everywhere. We show that if the objective function is LC², then the methods possess local quadratic convergence under a local error bound condition without the requirement of isolated nonsingular solutions. By using a backtracking line search, we globalize an inexact regularized Newton method. We show that the unit stepsize is accepted eventually. Limited numerical experiments are presented, which show the practical advantage of the method. [ABSTRACT FROM AUTHOR]

In this paper we first improve the concentration-compactness lemma by proving that the vanishing case is a special case of dichotomy, then we apply this improved concentration-compactness lemma to a typical restcted minimization problem, and get some new results. [ABSTRACT FROM AUTHOR]

In this paper, we will find an explicit optimal lower bound for the first eigenvalue of a fourth order measure differential equation, motivated by physical problems when some physically meaningful measurement is fixed. We establish a variational characterization for the first eigenvalue of the fourth order measure differential equation and find the explicit optimal lower bound for the first eigenvalue. As an application, we will find the explicit optimal lower bound for the first eigenvalue of a vibrating beam which represents the smallest axial compressive force necessary to cause the beam to buckle. [ABSTRACT FROM AUTHOR]