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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 work is devoted to the mathematical and numerical study of a framework handling a system of coupled variational inequalities. We prove both the existence and the uniqueness of a weak solution to the problem. Then, we introduce a convergent iterative scheme. Using this latter, we decouple the problem into further subproblems and derive their corresponding minimization problems. As a practical application of this class of coupled abstract variational inequalities, we consider a class of problems that model an electroelastic body coming into frictional contact with a rigid electrically conductive foundation. Both electrical and mechanical contacts are of Signorini type. In other words, our model prescribes the mechanical response produced by the foundation and the outflow of the free charges across the contact zone. The last part of this paper is mainly reserved for the numerical resolution of the problem at hand. For this purpose, we have developed an alternating direction method of multipliers and convex dualities to compute and illustrate the solutions. [ABSTRACT FROM AUTHOR]

In this study, we suggest a new class of forward-backward (FB) algorithms designed to solve convex minimization problems. Our method incorporates a linesearch technique, eliminating the need to choose Lipschitz assumptions explicitly. Additionally, we apply double inertial extrapolations to enhance the algorithm’s convergence rate. We establish a weak convergence theorem under some mild conditions. Furthermore, we perform numerical tests, and apply the algorithm to image restoration and data classification as a practical application. The experimental results show our approach’s superior performance and effectiveness, surpassing some existing methods in the literature. [ABSTRACT FROM AUTHOR]

Colors are not seen in the same way among all people. For example, to dichromats such as protanope and deuteranope, red and green appear similar ochre-like colors. Therefore, it is difficult for dichromats to distinguish such color combinations. The purpose of this study is to propose a color conversion method that produces images that are easy for dichromats to discriminate colors, without significantly changing the impression of the original image for both dichromats and trichromats. The proposed method modifies the lightness of pixels to take into account colors that are difficult for dichromats to discriminate, thus enabling color discrimination. Furthermore, by also modifying the saturation of the pixels, the image quality degradation caused by overemphasis, which occurs when only modifying the lightness, is suppressed. To demonstrate the effectiveness of the proposed method, experiments using various images are conducted. From the resultant images, it can be confirmed that the proposed method achieves the color conversion that serves its purpose. The main results of quantitative evaluation are as follows. (1) In the index evaluating the degree of contrast improvement for colors which are difficult for dichromats to discriminate, the average value of the proposed method is about 0.84, and indicates that the proposed method is comparable to the conventional methods. (2) In the index represented by the ratio of the number of pixels clipped with white or black, the average of the proposed method is 4.38×10−4 and indicates that there are almost no clipped white and black pixels.

We consider a history-dependent variational inequality in a real Hilbert space, for which we recall an existence and uniqueness result. We associate this inequality with a gap function, together with two additional problems: a nonlinear equation and a minimization problem. Then, we prove that solving these problems is equivalent to solving the original history-dependent variational inequality. Next, we state and prove a convergence criterion, i.e., we provide necessary and sufficient conditions which guarantee the convergence of a sequence of functions to the solution of the considered inequality. Based on the equivalence above, we deduce various consequences that present some interest on their own, and, moreover, we obtain convergence results for the two additional problems considered. Finally, we apply our abstract results to the study of an inequality problem in solid mechanics. It concerns the study of a viscoelastic constitutive law with long memory and unilateral constraints, for which we deduce a convergence result and provide the corresponding mechanical interpretations. [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]

The Harmony Search (HS) algorithm, inspired by the improvisational process of musicians, offers a novel and effective approach to optimization problems. This paper presents an application designed to solve a wide range of minimization problems using the HS algorithm. By simulating the creative exploration and refinement seen in musical harmonies, the HS algorithm efficiently navigates complex solution landscapes, delivering high accuracy and computational efficiency. Illustrated through various optimization examples, this tool showcases the versatility and power of the HS algorithm in addressing linear, non-linear, and discrete models. Our work highlights the practical utility of bio-inspired algorithms in solving real-world problems, providing a user-friendly platform for researchers and engineers to harness the potential of the HS algorithm in diverse fields.

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]

We consider a history-dependent variational inequality in a real Hilbert space, for which we recall an existence and uniqueness result. We associate this inequality with a gap function, together with two additional problems: a nonlinear equation and a minimization problem. Then, we prove that solving these problems is equivalent to solving the original history-dependent variational inequality. Next, we state and prove a convergence criterion, i.e., we provide necessary and sufficient conditions which guarantee the convergence of a sequence of functions to the solution of the considered inequality. Based on the equivalence above, we deduce various consequences that present some interest on their own, and, moreover, we obtain convergence results for the two additional problems considered. Finally, we apply our abstract results to the study of an inequality problem in solid mechanics. It concerns the study of a viscoelastic constitutive law with long memory and unilateral constraints, for which we deduce a convergence result and provide the corresponding mechanical interpretations.

In this paper, we introduce an inertial forward-backward splitting method together with a Halpern iterative algorithm for approximating a common solution of a finite family of split minimization problem involving two proper, lower semicontinuous and convex functions and fixed point problem of a nonexpansive mapping in real Hilbert spaces. Under suitable conditions, we proved that the sequence generated by our algorithm converges strongly to a solution of the aforementioned problems. The stepsizes studied in this paper are designed in such a way that they do not require the Lipschitz continuity condition on the gradient and prior knowledge of operator norm. Finally, we illustrate a numerical experiment to show the performance of the proposed method. The result discussed in this paper extends and complements many related results in literature.

In this work, we present a new proximal gradient algorithm based on Tseng’s extragradient method and an inertial technique to solve the convex minimization problem in real Hilbert spaces. Using the stepsize rules, the selection of the Lipschitz constant of the gradient of functions is avoided. We then prove the weak convergence theorem and present the numerical experiments for image recovery. The comparative results show that the proposed algorithm has better efficiency than other methods.

Proximal point algorithm is one of the most popular technique to find either zero of monotone operator or minimizer of a lower semi-continuous function. In this paper, we propose a new modified proximal point algorithm for solving minimization problems and common fixed point problems in CAT(0) spaces. We prove Δ and strong convergence of the proposed algorithm. Our results extend and improve the corresponding recent results in the literature.

In this article, we present some new fixed point existence results for nonexpansive-type mappings in geodesic spaces. We also give a number of illustrative examples to settle our claims. We study the asymptotic behavior of Picard iterates generated by these class of mappings under different conditions. Finally, we approximate the solutions of the constrained minimization problem in the setting of Cartan, Alexandrov, and Toponogov (CAT(0)) spaces.

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]

In this article, we consider certain minimization problems. If d 1 , d 2 and d 3 are the distances of a boundary or inner point to the sides of a given triangle, find the point which minimizes d 1 n + d 2 n + d 3 n , for positive integer n. These problems can be afforded easily with GeoGebra. We consider two examples, the first concerns an isosceles triangle and the second a scalene triangle. In both cases, we divide the triangle horizontally into line segments parallel to the base and look at a family of polynomial functions. Using GeoGebra, we observe in the case of isosceles triangle that the minimum point of each member of the family lies on the y-axis. Tracking these points vertically we discover the critical point which minimizes d 1 n + d 2 n + d 3 n , n ≥ 2. In particular, we show that the sequence of these critical points converges to the incenter of the triangle. In the case of a scalene, we observe that the minima points of the polynomials lie on a curve, the minimum of which can be traced with GeoGebra and computed with basic calculus. Finally, we consider a discussion with some references concerning general solutions of 'minimal sums of distances' and 'minimal sums of squared distances'. [ABSTRACT FROM AUTHOR]

In this paper, we propose and study new inertial viscosity Tseng's extragradient algorithms with self-adaptive step size to solve the variational inequality problem (VIP) and the fixed point problem (FPP) in Hilbert spaces. Our proposed methods involve a projection onto a half-space and self-adaptive step size. We prove that the sequence generated by our proposed methods converges strongly to a common solution of the VIP and FPP of an infinite family of strict pseudo-contractive mappings in Hilbert spaces under some mild assumptions when the underlying operator is monotone and Lipschitz continuous. Furthermore, we apply our results to find a common solution of VIP and zero-point problem (ZPP) for an infinite family of maximal monotone operators. Finally, we provide some numerical experiments of the proposed methods in comparison with other existing methods in the literature. [ABSTRACT FROM AUTHOR]

Proximal point algorithm is one of the most popular technique to find either zero of monotone operator or minimizer of a lower semi-continuous function. In this paper, we propose a new modified proximal point algorithm for solving minimization problems and common fixed point problems in CAT(0) spaces. We prove Δ and strong convergence of the proposed algorithm. Our results extend and improve the corresponding recent results in the literature. [ABSTRACT FROM AUTHOR]

We consider an abstract problem P in a metric space X which has a unique solution u ∈ X. Our aim in this current paper is two folds: first, to provide a convergence criterion to the solution of Problem P, that is, to give necessary and sufficient conditions on a sequence {un} ⊂ X which guarantee the convergence un → u in the space X; second, to find a Tyknonov triple T such that a sequence {un} ⊂ X is a T-approximating sequence if and only if it converges to u. The two problems stated above, associated to the original Problem P, are closely related. We illustrate how they can be solved in three particular cases of Problem P: a variational inequality in a Hilbert space, a fixed point problem in a metric space and a minimization problem in a reflexive Banach space. For each of these problems we state and prove a convergence criterion that we use to define a convenient Tykhonov triple T which requires the condition stated above. We also show how the convergence criterion and the corresponding T -well posedness concept can be used to deduce convergence and classical well-posedness results, respectively. [ABSTRACT FROM AUTHOR]

In this article, we present a new modified proximal point algorithm in the framework of CAT(1) spaces which is utilized for solving common fixed point problem and minimization problems. Also, we prove convergence results of the obtained process under some mild conditions. Our results extend and improve several corresponding results of the existing literature.

The article introduce a generalized viscosity method which includes a sequence of contractive mappings for approximating solutions of fixed point problem of finitely many resolvents of a lower semi-continuous function together with a fixed point problem of countable infinite family of L-Lipschitzian and quasi-pseudocontractive mappings in Hadamard spaces.

Abstract 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.

In this paper, we establish a stochastic maximum principle for a stochastic minimization problem under partial information. With the Backward stochastic differential equations (in short BSDE's), we establish a sufficient condition of optimality to characterize and determine an optimal control. This is done instead of using the Hamiltonian which is a deterministic function. The equations translating the dynamics of the state variables of the controlled system contain an BSPDE (Backward stochastic partial differential equation) which can be the unnormalized conditional density like the Zakai equation born from a problem of passage from partial to full information. [ABSTRACT FROM AUTHOR]

In this paper, we study a model of deformations of a beam with elastic supports and nonlinear boundary conditions under the influence of an external force. The case when the force can be concentrated at separate points is considered. The minimization problem of the potential energy functional with a constraint on the displacement of the beam end is investigated. The correctness of the model is established, and the necessary and sufficient conditions for the minimum of the potential energy functional are proved. [ABSTRACT FROM AUTHOR]

Pollution of aquifers by liquid radioactive waste is especially hazardous due to high permeability of aquifer rocks. Pollution can take place as a result of deep injection disposal of liquid waste or due to its leakage from a surface reservoir. The ecological hazard depends on distribution of aquifer permeability, which can be highly heterogeneous. The only source of data for determination of such distribution is measurements in exploratory boreholes at the polluted site. A technique is proposed for determination of spatial distribution of aquifer transmissibility on the basis of hydrological measurements in a limited number of boreholes. The problem is reduced to a minimization problem which was solved numerically. [ABSTRACT FROM AUTHOR]

In this paper we obtain the Lipschitz continuity of nonnegative local minimizers of the functional $J(v)=\int_\Omega\big( F(x, v, \nabla v)+\lambda(x)\chi_{\{v>0\}}\big)\,dx$, under nonstandard growth conditions of the energy function $F(x,s,\eta)$ and $0

Since polar codes are capacity-achieving codes, there have been various research works devoted to devising efficient implementation methods as well as improving error-correction performance. Since quantization is a critical implementation issue, in this paper, a nonuniform quantization method is proposed for the successive cancellation (SC) decoder of polar codes, which finds quantization boundary values based on the analysis of various quantization levels over the additive white Gaussian noise channel. Since low computational complexity, high reliability, and efficient memory management are required in the next-generation communication and memory systems, 2–4 bit precision levels are mainly considered in the proposed nonuniform quantization method. Depending on the presence of erasure, quantization levels are divided into three types, and the message alphabets and update rules are derived for each type. Also, a construction method of polar codes suitable for the proposed nonuniform-quantized SC decoder is proposed, which simultaneously determines the information set and the quantization boundary values based on the density evolution analysis and an upper bound of the block error probability. To determine quantization boundary values, a multivariate objective function is defined and an iterative coarse-to-fine search algorithm to minimize this objective function is proposed. In addition, a scaling method of quantizer output values is proposed when the number of quantization levels of quantizer is smaller than the number of quantization levels of decoder. Finally, simulation results confirm that the proposed nonuniform-quantized SC decoder shows better error-correction performance, lower decoding complexity, and higher memory efficiency compared to the best known uniform-quantized SC decoder.

We consider the nonexistence of minimizers for the energy containing a nonlocal perimeter with a general kernel K, a Riesz potential, and a background potential in RN with N ≥ 2 under the volume constraint. We show that the energy has no minimizer for a sufficiently large volume under suitable assumptions on K. The proof is based on the partition of a minimizer and the comparison of the sum of the energy for each part with the energy for the original configuration. [ABSTRACT FROM AUTHOR]

This paper is aimed at the derivation of a discrete data smoothing function for the discrete Dirichlet condition in a regular grid on the surface of a spheroid. The method employed here is the local L2−semi-norm minimization, through Euler-Lagrange method, for the Beltrami operator. The method results in a weighted average of the surrounding points in a template based on the first order Taylor expansion of the unknown function under consideration. The coefficients of the weighted average are calculated and used to smooth the Geoid height data in Iran, derived from the EGM2008 geopotential model.

Let $N\geq 2$ be an integer. For each real number $s\in(0,1)$ we denote by $(-\Delta)^s$ the corresponding fractional Laplace operator. First, we investigate the eigenvalue problem $(-\Delta)^s u=\lambda V(x)u$ on $\mathbb{R}^N$, where $V:\mathbb{R}^N\rightarrow\mathbb{R}$ is a given function. Under suitable conditions imposed on $V$ we show the existence of an unbounded, increasing sequence of positive eigenvalues. Next, we perturb the above eigenvalue problem with a fractional $(t,p)$-Laplace operator, when $t\in(0,1)$ and $p\in(1,\infty)$ are such that $t

In this paper, we show the existence of solutions of the convex minimization problems and common fixed problems in CAT(1) spaces under some mild conditions. For this, we propose the new modified the proximal point algorithm. Further, we give some applications for the convex minimization problem and the fixed point problem in CAT(κ) spaces with the bounded positive real number κ. Our results improve and generalize many recent important results in the literature. [ABSTRACT FROM AUTHOR]

In this paper the solution for hesitant fuzzy system as AX = B is introduced where A is an n×n known hesitant fuzzy matrix, B is an n×1 known hesitant fuzzy vector and X is an n×1 unknown hesitant fuzzy vector. First, L∞-norm and L1-norm of a hesitant fuzzy vector are introduced. Then, the concepts of hesitant fuzzy zero, 'almost equal' and 'less than' and 'equal' are defined for two hesitant fuzzy numbers. Finally, using a minimization problem; the hesitant fuzzy system is solved. At the end, some numerical examples are presented to show the effectiveness of the proposed method. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to solve the variational problem containing real‐ and complex‐order fractional derivatives. We define a new version of the complex‐order derivative based on the ψ‐Riemann‐Liouville fractional derivative, and get the Euler–Lagrange equation for the variational problem. By introducing the approximated expansion formula of the complex‐order fractional derivative to the variational problem, we derive the corresponding approximated Euler–Lagrange equation. It is proved that the approximated Euler–Lagrange equation converges to the original one in the weak sense. At the same time, the minimal value of the approximated action integral tends to the minimal value of the original one. We also conduct a stress relaxation experiment and discuss the feasibility of the complex‐order derivative in real problem modeling. [ABSTRACT FROM AUTHOR]

• Effective accumulated temperature (EAT) model is effective in pest forecasting. • Parameters of EAT model are usually determined by rearing experiments in laboratory. • We developed a machine-learning-based method to estimate EAT parameters. • This method enables us to use historical occurrence data on insects in forecasting. • The improved EAT model will contribute to better forecasting of insect pests. To improve the prediction accuracy of pest forecasting, we developed a machine-learning-based method of estimating parameters of the effective accumulated temperature (EAT) model on the basis of numerous historical occurrence data on target pests and meteorological data in the field. The parameters were estimated by using past occurrence data of the white-backed planthopper (WBPH), Sogatella furcifera , collected in light traps at 20 sites on Shikoku Island from 1980 to 2000. When the accuracy of the estimated parameters was compared with that of those derived from traditional approaches using field data from 2001 to 2022, the parameters estimated by using the novel method had the best accuracy, with a predictive lower threshold temperature (T p) = 12.29 (°C) and an effective accumulated temperature (K T p ) = 372.23 (degree-days) for the first generation of WBPH and T p = –5.00 and K T p = 800.31 for the second generation. Use of these parameters improved the prediction accuracy by approximately 3 days compared with the traditional approach. We also found that prediction of parameters on the basis of the coefficient of variation of the predicted EATs resulted in better forecasting accuracy than prediction based on the standard deviation. Our novel method of estimating parameters for the EAT model will contribute to better forecasting of insect pests. [ABSTRACT FROM AUTHOR]

In the article we have obtained some estimates of the rate of convergence for the recently proposed by Yu. E.Nesterov method of minimization of a convex Lipschitz-continuous function of two variables on a square with a fixed side. The idea of the method is to divide the square into smaller parts and gradually remove them so that in the remaining sufficiently small part. The method consists in solving auxiliary problems of one-dimensional minimization along the separating segments and does not imply the calculation of the exact value of the gradient of the objective functional. The main result of the paper is proved in the class of smooth convex functions having a Lipschitz-continuous gradient. Moreover, it is noted that the property of Lipschitzcontinuity for gradient is sufficient to require not on the whole square, but only on some segments. It is shown that the method can work in the presence of errors in solving auxiliary one-dimensional problems, as well as in calculating the direction of gradients. Also we describe the situation when it is possible to neglect or reduce the time spent on solving auxiliary one-dimensional problems. For some examples, experiments have demonstrated that the method can work effectively on some classes of non-smooth functions. In this case, an example of a simple non-smooth function is constructed, for which, if the subgradient is chosen incorrectly, even if the auxiliary one-dimensional problem is exactly solved, the convergence property of the method may not hold. Experiments have shown that the method under consideration can achieve the desired accuracy of solving the problem in less time than the other methods (gradient descent and ellipsoid method) considered. Partially, it is noted that with an increase in the accuracy of the desired solution, the operating time for the Yu. E. Nesterovs method can grow slower than the time of the ellipsoid method.

In this paper, we consider an eigenvalue problem defined on a two-patch domain. Our first aim is to show that the two-patch eigenvalue problem is equivalent to the eigenvalue problem of a measure differential equation defined on an one-patch domain. Our second aim is to study the existence of principal eigenvalue of the measure differential equation, and we will prove the principal eigenvalue is continuously depending on the weight measure in the weak⁎ topology of the measure space. Our third aim is to solve a minimization problem on principal eigenvalues. Some main results of this paper have interesting relations with population dynamics. We will interpret these results in terms of survival chances and optimal distribution of resources. [ABSTRACT FROM AUTHOR]

We consider an optimization problem with volume constraint for an energy functional associated to an inhomogeneous operator with nonstandard growth. By studying an auxiliary penalized problem, we prove existence and regularity of solution to the original problem: every optimal configuration is a solution to a one phase free boundary problem—for an operator with nonstandard growth and non-zero right hand side—and the free boundary is a smooth surface. [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]

In this paper, we present an optimization model consisting of a dynamic supply chain network related to Personal Protective Equipment (PPE). We suppose that the variables in the model, namely flows on arcs and additional capacities on arcs, depend both on time and on a delay function. The aim of the firm is to find the optimal flows and the optimal additional capacities on arcs to satisfy the huge and immediate increasing request in the demand markets due to the spread of the COVID-19 disease, minimizing, simultaneously, its total costs. We obtain a minimization problem and the related "retarded" evolutionary variational inequality (rEVI). We introduce the associated infinite-dimensional projected dynamical system to obtain a computational procedure to find the optimal solution to the rEVI associated with our minimization problem and, finally, we propose some numerical examples based on real scenarios. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a modified Ishikawa-type proximal point algorithm for approximating a common solution of minimization problem, mono-tone inclusion problem and fixed point problem. We obtain a strong convergence of the proposed algorithm to a common solution of finite family of minimization problem, finite family of monotone inclusion problem and fixed point problem for asymptotically demicontractive mapping in Hadamard spaces. Numerical example is given to illustrate the applicability of our main result. Our results complement and extend some recent results in literature. [ABSTRACT FROM AUTHOR]

In this paper, we study fuzzy top-down tree automata over lattices (LTA s , for short). The purpose of this contribution is to investigate the minimization problem for LTA s. We first define the concept of statewise equivalence between two LTA s. Thereafter, we show the existence of the statewise minimal form for an LTA. To this end, we find a statewise irreducible LTA which is equivalent to a given LTA. Then, we provide an algorithm to find the statewise minimal LTA and by a theorem, we show that the output statewise minimal LTA is statewise equivalent to the given input. Moreover, we compute the time complexity of the given algorithm. The proposed algorithm can be applied to any given LTA and, unlike some minimization algorithms given in the literature, the input doesn't need to be a complete, deterministic, or reduced lattice-valued tree automaton. Finally, we provide some examples to show the efficiency of the presented algorithm. [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]

In this paper, we consider an inverse problem related to a fractional diffusion equation. The model problem is governed by a nonlinear partial differential equation involving the fractional spectral Laplacian. This study is focused on the reconstruction of an unknown source term from a partial internal measured data. The considered ill‐posed inverse problem is formulated as a minimization one. The existence, uniqueness, and stability of the solution are discussed. Some theoretical results are established. The numerical reconstruction of the unknown source term is investigated using an iterative process. The proposed method involves a denoising procedure at each iteration step and provides a sequence of source term approximations converging in norm to the actual solution of the minimization problem. Some numerical results are presented to show the efficiency and the accuracy of the proposed approach. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to introduce a modified viscosity-type proximal point algorithm and prove its strong convergence to a zero of a monotone operator, which is also a minimizer of a proper convex and lower semi-continuous function and a fixed point of a demicontractive-type (or demimetric) mapping in an Hadamard space. We apply the results obtained to solve variational inequality problems in Hadamard spaces. To further show the applicability and the advantages of the obtained results, several numerical experiments of our algorithm in comparison with that studied by Ranjbar and Khatibzadeh [Mediterranean Journal of Mathematics 14(2017), 15 pp.] are carried out in the framework of Hadamard spaces. Our results improve and generalize some recent important results in the literature. [ABSTRACT FROM AUTHOR]

Abstract 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.

In this paper, we introduce and study the class of demimetric mappings in CAT(0) spaces.We then propose a modified proximal point algorithm for approximating a common solution of a finite family of minimization problems and fixed point problems in CAT(0) spaces. Furthermore,we establish strong convergence of the proposed algorithm to a common solution of a finite family of minimization problems and fixed point problems for a finite family of demimetric mappings in complete CAT(0) spaces. A numerical example which illustrates the applicability of our proposed algorithm is also given. Our results improve and extend some recent results in the literature.