Symmetry

Research

11 pages, 764 KiB

Open AccessArticle

Multiple Techniques for Studying Asymptotic Properties of a Class of Differential Equations with Variable Coefficients

by Omar Bazighifan and Mihai Postolache

Symmetry 2020, 12(7), 1112; https://doi.org/10.3390/sym12071112 - 03 Jul 2020

Cited by 5 | Viewed by 1451

This manuscript is concerned with the oscillatory properties of 4th-order differential equations with variable coefficients. The main aim of this paper is the combination of the following three techniques used: the comparison method, Riccati technique and integral averaging technique. Two examples are given [...] Read more.

This manuscript is concerned with the oscillatory properties of 4th-order differential equations with variable coefficients. The main aim of this paper is the combination of the following three techniques used: the comparison method, Riccati technique and integral averaging technique. Two examples are given for applying the criteria. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

14 pages, 322 KiB

Open AccessArticle

The Split Equality Fixed Point Problem of Demicontractive Operators with Numerical Example and Application

by Yaqin Wang, Jinzuo Chen and Ariana Pitea

Symmetry 2020, 12(6), 902; https://doi.org/10.3390/sym12060902 - 01 Jun 2020

Cited by 8 | Viewed by 1932

Abstract

This paper aims to propose a new reckoning method for solving the split equality fixed point problem of demicontractive operators in Hilbert spaces, and to establish a theorem with regard to the strong convergence of this new scheme. As an application, we also [...] Read more.

This paper aims to propose a new reckoning method for solving the split equality fixed point problem of demicontractive operators in Hilbert spaces, and to establish a theorem with regard to the strong convergence of this new scheme. As an application, we also consider quasi-pseudo-contractive operators and obtain a result on the solution to the split equality fixed point problem in the framework of Hilbert spaces. A numerical example is also provided. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

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16 pages, 1961 KiB

Open AccessFeature PaperArticle

A Novel Learning Rate Schedule in Optimization for Neural Networks and It’s Convergence

by Jieun Park, Dokkyun Yi and Sangmin Ji

Symmetry 2020, 12(4), 660; https://doi.org/10.3390/sym12040660 - 22 Apr 2020

Cited by 23 | Viewed by 4037

Abstract

The process of machine learning is to find parameters that minimize the cost function constructed by learning the data. This is called optimization and the parameters at that time are called the optimal parameters in neural networks. In the process of finding the [...] Read more.

The process of machine learning is to find parameters that minimize the cost function constructed by learning the data. This is called optimization and the parameters at that time are called the optimal parameters in neural networks. In the process of finding the optimization, there were attempts to solve the symmetric optimization or initialize the parameters symmetrically. Furthermore, in order to obtain the optimal parameters, the existing methods have used methods in which the learning rate is decreased over the iteration time or is changed according to a certain ratio. These methods are a monotonically decreasing method at a constant rate according to the iteration time. Our idea is to make the learning rate changeable unlike the monotonically decreasing method. We introduce a method to find the optimal parameters which adaptively changes the learning rate according to the value of the cost function. Therefore, when the cost function is optimized, the learning is complete and the optimal parameters are obtained. This paper proves that the method ensures convergence to the optimal parameters. This means that our method achieves a minimum of the cost function (or effective learning). Numerical experiments demonstrate that learning is good effective when using the proposed learning rate schedule in various situations. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

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22 pages, 828 KiB

Open AccessFeature PaperArticle

A Filter and Nonmonotone Adaptive Trust Region Line Search Method for Unconstrained Optimization

by Quan Qu, Xianfeng Ding and Xinyi Wang

Symmetry 2020, 12(4), 656; https://doi.org/10.3390/sym12040656 - 21 Apr 2020

Cited by 3 | Viewed by 2270

Abstract

In this paper, a new nonmonotone adaptive trust region algorithm is proposed for unconstrained optimization by combining a multidimensional filter and the Goldstein-type line search technique. A modified trust region ratio is presented which results in more reasonable consistency between the accurate model [...] Read more.

In this paper, a new nonmonotone adaptive trust region algorithm is proposed for unconstrained optimization by combining a multidimensional filter and the Goldstein-type line search technique. A modified trust region ratio is presented which results in more reasonable consistency between the accurate model and the approximate model. When a trial step is rejected, we use a multidimensional filter to increase the likelihood that the trial step is accepted. If the trial step is still not successful with the filter, a nonmonotone Goldstein-type line search is used in the direction of the rejected trial step. The approximation of the Hessian matrix is updated by the modified Quasi-Newton formula (CBFGS). Under appropriate conditions, the proposed algorithm is globally convergent and superlinearly convergent. The new algorithm shows better performance in terms of the Dolan–Moré performance profile. Numerical results demonstrate the efficiency and robustness of the proposed algorithm for solving unconstrained optimization problems. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

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13 pages, 423 KiB

Open AccessArticle

Partially Projective Algorithm for the Split Feasibility Problem with Visualization of the Solution Set

by Andreea Bejenaru and Mihai Postolache

Symmetry 2020, 12(4), 608; https://doi.org/10.3390/sym12040608 - 11 Apr 2020

Cited by 5 | Viewed by 1783

Abstract

This paper introduces a new three-step algorithm to solve the split feasibility problem. The main advantage is that one of the projective operators interferes only in the final step, resulting in less computations at each iteration. An example is provided to support the [...] Read more.

This paper introduces a new three-step algorithm to solve the split feasibility problem. The main advantage is that one of the projective operators interferes only in the final step, resulting in less computations at each iteration. An example is provided to support the theoretical approach. The numerical simulation reveals that the newly introduced procedure has increased performance compared to other existing methods, including the classic CQ algorithm. An interesting outcome of the numerical modeling is an approximate visual image of the solution set. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

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12 pages, 247 KiB

Open AccessArticle

A New Hilbert-Type Inequality with Positive Homogeneous Kernel and Its Equivalent Forms

by Bicheng Yang, Shanhe Wu and Aizhen Wang

Symmetry 2020, 12(3), 342; https://doi.org/10.3390/sym12030342 - 28 Feb 2020

Cited by 13 | Viewed by 1679

Abstract

We establish a new inequality of Hilbert-type containing positive homogeneous kernel

{(\min {m, n})}^{λ}

and derive its equivalent forms. Based on the obtained Hilbert-type inequality, we discuss its equivalent forms and give the operator expressions in some [...] Read more.

We establish a new inequality of Hilbert-type containing positive homogeneous kernel

{(\min {m, n})}^{λ}

and derive its equivalent forms. Based on the obtained Hilbert-type inequality, we discuss its equivalent forms and give the operator expressions in some particular cases. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

26 pages, 345 KiB

Open AccessArticle

Sufficiency for Purely Essentially Bounded Optimal Controls

by Gerardo Sánchez Licea

Symmetry 2020, 12(2), 238; https://doi.org/10.3390/sym12020238 - 04 Feb 2020

Cited by 3 | Viewed by 1356

Abstract

For optimal control problems of Bolza with variable and free end-points, nonlinear dynamics, nonlinear isoperimetric inequality and equality restrictions, and nonlinear pointwise mixed time-state-control inequality and equality constraints, sufficient conditions for strong minima are derived. The algorithm used to prove the main theorem [...] Read more.

For optimal control problems of Bolza with variable and free end-points, nonlinear dynamics, nonlinear isoperimetric inequality and equality restrictions, and nonlinear pointwise mixed time-state-control inequality and equality constraints, sufficient conditions for strong minima are derived. The algorithm used to prove the main theorem of the paper includes a crucial symmetric inequality, making this technique an independent self-contained method of classical concepts such as embedding theorems from ordinary differential equations, Mayer fields, Riccati equations, or Hamilton–Jacobi theory. Moreover, the sufficiency theory given in this article is able to detect discontinuous solutions, that is, solutions which need to be neither continuous nor piecewise continuous but only essentially bounded. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

13 pages, 823 KiB

Open AccessArticle

A New Filter Nonmonotone Adaptive Trust Region Method for Unconstrained Optimization

by Xinyi Wang, Xianfeng Ding and Quan Qu

Symmetry 2020, 12(2), 208; https://doi.org/10.3390/sym12020208 - 02 Feb 2020

Cited by 4 | Viewed by 1906

Abstract

In this paper, a new filter nonmonotone adaptive trust region with fixed step length for unconstrained optimization is proposed. The trust region radius adopts a new adaptive strategy to overcome additional computational costs at each iteration. A new nonmonotone trust region ratio is [...] Read more.

In this paper, a new filter nonmonotone adaptive trust region with fixed step length for unconstrained optimization is proposed. The trust region radius adopts a new adaptive strategy to overcome additional computational costs at each iteration. A new nonmonotone trust region ratio is introduced. When a trial step is not successful, a multidimensional filter is employed to increase the possibility of the trial step being accepted. If the trial step is still not accepted by the filter set, it is possible to find a new iteration point along the trial step and the step length is computed by a fixed formula. The positive definite symmetric matrix of the approximate Hessian matrix is updated using the MBFGS method. The global convergence and superlinear convergence of the proposed algorithm is proven by some classical assumptions. The efficiency of the algorithm is tested by numerical results. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

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14 pages, 544 KiB

Open AccessArticle

Maximum-Entropy-Model-Enabled Complexity Reduction Algorithm in Modern Video Coding Standards

by Xiantao Jiang, Tian Song and Takafumi Katayama

Symmetry 2020, 12(1), 113; https://doi.org/10.3390/sym12010113 - 07 Jan 2020

Cited by 3 | Viewed by 2348

Abstract

Symmetry considerations play a key role in modern science, and any differentiable symmetry of the action of a physical system has a corresponding conservation law. Symmetry may be regarded as reduction of Entropy. This work focuses on reducing the computational complexity of modern [...] Read more.

Symmetry considerations play a key role in modern science, and any differentiable symmetry of the action of a physical system has a corresponding conservation law. Symmetry may be regarded as reduction of Entropy. This work focuses on reducing the computational complexity of modern video coding standards by using the maximum entropy principle. The high computational complexity of the coding unit (CU) size decision in modern video coding standards is a critical challenge for real-time applications. This problem is solved in a novel approach considering CU termination, skip, and normal decisions as three-class making problems. The maximum entropy model (MEM) is formulated to the CU size decision problem, which can optimize the conditional entropy; the improved iterative scaling (IIS) algorithm is used to solve this optimization problem. The classification features consist of the spatio-temporal information of the CU, including the rate–distortion (RD) cost, coded block flag (CBF), and depth. For the case analysis, the proposed method is based on High Efficiency Video Coding (H.265/HEVC) standards. The experimental results demonstrate that the proposed method can reduce the computational complexity of the H.265/HEVC encoder significantly. Compared with the H.265/HEVC reference model, the proposed method can reduce the average encoding time by 53.27% and 56.36% under low delay and random access configurations, while Bjontegaard Delta Bit Rates (BD-BRs) are 0.72% and 0.93% on average. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

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10 pages, 744 KiB

Open AccessArticle

On Quantum Duality of Group Amenability

by Xia Zhang and Ming Liu

Symmetry 2020, 12(1), 85; https://doi.org/10.3390/sym12010085 - 02 Jan 2020

Cited by 5 | Viewed by 1376

Abstract

In this paper, we investigate the co-amenability of compact quantum groups. Combining with some properties of regular C*-norms on algebraic compact quantum groups, we show that the quantum double of co-amenable compact quantum groups is unique. Based on this, this paper proves that [...] Read more.

In this paper, we investigate the co-amenability of compact quantum groups. Combining with some properties of regular C*-norms on algebraic compact quantum groups, we show that the quantum double of co-amenable compact quantum groups is unique. Based on this, this paper proves that co-amenability is preserved under formulation of the quantum double construction of compact quantum groups, which exhibits a type of nice symmetry between the co-amenability of quantum groups and the amenability of groups. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

11 pages, 278 KiB

Open AccessArticle

CQ-Type Algorithm for Reckoning Best Proximity Points of EP-Operators

by Hassan Houmani and Teodor Turcanu

Symmetry 2020, 12(1), 4; https://doi.org/10.3390/sym12010004 - 18 Dec 2019

Cited by 3 | Viewed by 1704

Abstract

We introduce a new class of non-self mappings by means of a condition which is called the (EP)-condition. This class includes proximal generalized nonexpansive mappings. It is shown that the existence of best proximity points for (EP)-mappings is equivalent to the existence of [...] Read more.

We introduce a new class of non-self mappings by means of a condition which is called the (EP)-condition. This class includes proximal generalized nonexpansive mappings. It is shown that the existence of best proximity points for (EP)-mappings is equivalent to the existence of an approximate best proximity point sequence generated by a three-step iterative process. We also construct a CQ-type algorithm which generates a strongly convergent sequence to the best proximity point for a given (EP)-mapping. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

10 pages, 249 KiB

Open AccessArticle

S-Subgradient Projection Methods with S-Subdifferential Functions for Nonconvex Split Feasibility Problems

by Jinzuo Chen, Mihai Postolache and Yonghong Yao

Symmetry 2019, 11(12), 1517; https://doi.org/10.3390/sym11121517 - 14 Dec 2019

Viewed by 1787

Abstract

In this paper, the original

C Q

algorithm, the relaxed

C Q

algorithm, the gradient projection method (

G P M

) algorithm, and the subgradient projection method (

S P M

) algorithm for the convex split feasibility problem are reviewed, and [...] Read more.

In this paper, the original

C Q

algorithm, the relaxed

C Q

algorithm, the gradient projection method (

G P M

) algorithm, and the subgradient projection method (

S P M

) algorithm for the convex split feasibility problem are reviewed, and a renewed

S P M

algorithm with S-subdifferential functions to solve nonconvex split feasibility problems in finite dimensional spaces is suggested. The weak convergence theorem is established. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

18 pages, 458 KiB

Open AccessArticle

Convergence Analysis for a Three-Step Thakur Iteration for Suzuki-Type Nonexpansive Mappings with Visualization

by Gabriela Ioana Usurelu and Mihai Postolache

Symmetry 2019, 11(12), 1441; https://doi.org/10.3390/sym11121441 - 23 Nov 2019

Cited by 14 | Viewed by 2449

Abstract

The class of Suzuki mappings is reanalyzed in connection with a three-steps Thakur procedure. The setting is provided by a uniformly convex Banach space, that is normed space endowed with some symmetric geometric properties and some topological properties. Once more, the fact that [...] Read more.

The class of Suzuki mappings is reanalyzed in connection with a three-steps Thakur procedure. The setting is provided by a uniformly convex Banach space, that is normed space endowed with some symmetric geometric properties and some topological properties. Once more, the fact that property

(C)

holds on as a generalized nonexpansiveness condition is emphasized throughout some examples. One example uses the setting of

R^{2}

with the Taxicab norm. It is further included in a numerical experiment in connection with seven iteration procedures, resulting a visual analysis of convergence. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

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16 pages, 302 KiB

Open AccessArticle

PPF-Dependent Fixed Point Results for Multi-Valued ϕ-F-Contractions in Banach Spaces and Applications

by Mohammed M. M. Jaradat, Babak Mohammadi, Vahid Parvaneh, Hassen Aydi and Zead Mustafa

Symmetry 2019, 11(11), 1375; https://doi.org/10.3390/sym11111375 - 06 Nov 2019

Cited by 2 | Viewed by 1556

Abstract

The solutions for many real life problems is obtained by interpreting the given problem mathematically in the form of

f (x) = x

. One of such examples is that of the famous Borsuk–Ulam theorem, in which using some fixed point [...] Read more.

The solutions for many real life problems is obtained by interpreting the given problem mathematically in the form of

f (x) = x

. One of such examples is that of the famous Borsuk–Ulam theorem, in which using some fixed point argument, it can be guaranteed that at any given time we can find two diametrically opposite places in a planet with same temperature. Thus, the correlation of symmetry is inherent in the study of fixed point theory. In this paper, we initiate

ϕ - F

-contractions and study the existence of PPF-dependent fixed points (fixed points for mappings having variant domains and ranges) for these related mappings in the Razumikhin class. Our theorems extend and improve the results of Hammad and De La Sen [Mathematics, 2019, 7, 52]. As applications of our PPF dependent fixed point results, we study the existence of solutions for delay differential equations (DDEs) which have numerous applications in population dynamics, bioscience problems and control engineering. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

22 pages, 308 KiB

Open AccessArticle

Common Fixed Points Results on Non-Archimedean Metric Modular Spaces

by Wissam Kassab

Symmetry 2019, 11(11), 1355; https://doi.org/10.3390/sym11111355 - 02 Nov 2019

Cited by 1 | Viewed by 1709

Abstract

This paper introduces two new contractive conditions in the setting of non-Archimedean modular spaces, via a C-class function, an altering distance function, and a control function. A non-Archimedean metric modular is shaped as a parameterized family of classical metrics; therefore, for each [...] Read more.

This paper introduces two new contractive conditions in the setting of non-Archimedean modular spaces, via a C-class function, an altering distance function, and a control function. A non-Archimedean metric modular is shaped as a parameterized family of classical metrics; therefore, for each value of the parameter, the positivity, the symmetry, the triangle inequality, or the continuity is ensured. The main outcomes provide sufficient conditions for the existence of common fixed points for four mappings. Examples are provided in order to prove the usability of the theoretical approach. Moreover, these examples use a non-Archimedean metric modular, which is not convex, making the study of nonconvex modulars more appealing. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

13 pages, 287 KiB

Open AccessArticle

Split Systems of Nonconvex Variational Inequalities and Fixed Point Problems on Uniformly Prox-Regular Sets

by Jinzuo Chen, Ariana Pitea and Li-Jun Zhu

Symmetry 2019, 11(10), 1279; https://doi.org/10.3390/sym11101279 - 12 Oct 2019

Cited by 2 | Viewed by 1646

Abstract

In this paper, we studied variational inequalities and fixed point problems in nonconvex cases. By the projection method over prox-regularity sets, the convergence of the suggested iterative scheme was established under some mild rules. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

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10 pages, 259 KiB

Open AccessArticle

gH-Symmetrically Derivative of Interval-Valued Functions and Applications in Interval-Valued Optimization

by Yating Guo, Guoju Ye, Dafang Zhao and Wei Liu

Symmetry 2019, 11(10), 1203; https://doi.org/10.3390/sym11101203 - 25 Sep 2019

Cited by 12 | Viewed by 2242

Abstract

In this paper, we present the gH-symmetrical derivative of interval-valued functions and its properties. In application, we apply this new derivative to investigate the Karush–Kuhn–Tucker (KKT) conditions of interval-valued optimization problems. Meanwhile, some examples are worked out to illuminate the obtained results. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

22 pages, 326 KiB

Open AccessArticle

A Solution of Fredholm Integral Equation by Using the Cyclic η s q -Rational Contractive Mappings Technique in b-Metric-Like Spaces

by Hasanen A. Hammad and Manuel De la Sen

Symmetry 2019, 11(9), 1184; https://doi.org/10.3390/sym11091184 - 19 Sep 2019

Cited by 30 | Viewed by 2280

Abstract

In this article, the notion of cyclic

η_{s}^{q}

-rational contractive mappings is discussed and some fixed point theorems in the context of complete b-metric-like spaces are showed. Here, the obtained consequences unify, extend and generalize various comparable known results. Furthermore, [...] Read more.

In this article, the notion of cyclic

η_{s}^{q}

-rational contractive mappings is discussed and some fixed point theorems in the context of complete b-metric-like spaces are showed. Here, the obtained consequences unify, extend and generalize various comparable known results. Furthermore, new common fixed point outcomes in a directed graph are demonstrated. Moreover, some useful examples are discussed to justify our theoretical results and finding a solution of Fredholm integral equation was discussed as enforcement. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

12 pages, 263 KiB

Open AccessArticle

Generalized Fixed-Point Results for Almost (α,F_σ)-Contractions with Applications to Fredholm Integral Inclusions

by Saleh Abdullah Al-Mezel and Jamshaid Ahmad

Symmetry 2019, 11(9), 1068; https://doi.org/10.3390/sym11091068 - 21 Aug 2019

Cited by 6 | Viewed by 2048

Abstract

The purpose of this article is to define almost

(α, F_{σ})

-contractions and establish some generalized fixed-point results for a new class of contractive conditions in the setting of complete metric spaces. In application, we apply our fixed-point theorem [...] Read more.

The purpose of this article is to define almost

(α, F_{σ})

-contractions and establish some generalized fixed-point results for a new class of contractive conditions in the setting of complete metric spaces. In application, we apply our fixed-point theorem to prove the existence theorem for Fredholm integral inclusions

ϖ (t) \in [f (t) + \int_{0}^{1} K (t, s, x (s)) ϑ s], t \in [0, 1]

where

f \in C [0, 1]

is a given real-valued function and

K : [0, 1]

\times [0, 1]

\times R \to K_{c v} (R)

is a given multivalued operator, where

K_{c v}

represents the family of nonempty compact and convex subsets of

R

and

ϖ \in C [0, 1]

is the unknown function. We also provide a non-trivial example to show the significance of our main result. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

23 pages, 2417 KiB

Open AccessArticle

Conditional Granger Causality and Genetic Algorithms in VAR Model Selection

by Vasile George Marica and Alexandra Horobet

Symmetry 2019, 11(8), 1004; https://doi.org/10.3390/sym11081004 - 03 Aug 2019

Cited by 2 | Viewed by 4089

Abstract

Overcoming symmetry in combinatorial evolutionary algorithms is a challenge for existing niching methods. This research presents a genetic algorithm designed for the shrinkage of the coefficient matrix in vector autoregression (VAR) models, constructed on two pillars: conditional Granger causality and Lasso regression. Departing [...] Read more.

Overcoming symmetry in combinatorial evolutionary algorithms is a challenge for existing niching methods. This research presents a genetic algorithm designed for the shrinkage of the coefficient matrix in vector autoregression (VAR) models, constructed on two pillars: conditional Granger causality and Lasso regression. Departing from a recent information theory proof that Granger causality and transfer entropy are equivalent, we propose a heuristic method for the identification of true structural dependencies in multivariate economic time series. Through rigorous testing, both empirically and through simulations, the present paper proves that genetic algorithms initialized with classical solutions are able to easily break the symmetry of random search and progress towards specific modeling. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

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18 pages, 1941 KiB

Open AccessArticle

Limit Analysis of Progressive Asymmetrical Collapse Failure of Tunnels in Inclined Rock Stratum

by Cheng Lyu, Zhengqiang Zeng and Yucang Dong

Symmetry 2019, 11(7), 904; https://doi.org/10.3390/sym11070904 - 11 Jul 2019

Cited by 6 | Viewed by 2443

Abstract

Tunnels commonly pass through inclined rock stratum, but research on the collapse of the rock surrounding the tunnels in inclined rock strata is currently underdeveloped. The purpose of this study was to predict the progressive asymmetrical collapse failure of deep-buried tunnels in inclined [...] Read more.

Tunnels commonly pass through inclined rock stratum, but research on the collapse of the rock surrounding the tunnels in inclined rock strata is currently underdeveloped. The purpose of this study was to predict the progressive asymmetrical collapse failure of deep-buried tunnels in inclined rock strata to decrease the risk of collapse during tunnel construction. We constructed a new two-dimensional progressive asymmetrical collapse failure mechanism for deep-buried tunnels in inclined rock layers to analyze their collapse failure characteristics with the help of the nonlinear Hoek–Brown yield criterion and the limit analysis theorem. The calculation equations of the range and total weight of the asymmetrical collapsing block in rectangular and circular tunnels were obtained via theoretical derivation. The validity of the proposed method in this work was verified by comparison with existing research. To discuss the impact of different parameters on the range and total weight of an asymmetrical collapsing block of the surrounding rock in inclined rock stratum, the range and total weight of the asymmetrical collapsing block of the most common rectangular and circular tunnels under the varied parameters are provided. The results of this study can provide useful support for practical tunnel construction and design. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

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12 pages, 780 KiB

Open AccessArticle

Common Fixed Point Results of Set Valued Maps for A_φ-Contraction and Generalized ϕ-Type Weak Contraction

by Murchana Neog, Mohammed M. M. Jaradat and Pradip Debnath

Symmetry 2019, 11(7), 894; https://doi.org/10.3390/sym11070894 - 08 Jul 2019

Viewed by 2049

Abstract

The solutions for many real life problems may be obtained by interpreting the given problem mathematically in the form

f (x) = x

. One such example is that of the famous Borsuk–Ulam theorem in which, using some fixed point argument, [...] Read more.

The solutions for many real life problems may be obtained by interpreting the given problem mathematically in the form

f (x) = x

. One such example is that of the famous Borsuk–Ulam theorem in which, using some fixed point argument, it can be guaranteed that at any given time we can find two diametrically opposite places in a planet with same temperature. Thus, the correlation of symmetry is inherent in the study of fixed point theory. In this article, some new results concerning coincidence and a common fixed point for an

A_{φ}

-contraction and a generalized

ϕ

-type weak contraction are established. We prove our results for set valued maps without using continuity of the corresponding maps and completeness of the relevant space. Our results generalize and extend several existing results. Some new examples are given to demonstrate the generality and non-triviality of our results. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

12 pages, 296 KiB

Open AccessArticle

Nonlinear Caputo Fractional Derivative with Nonlocal Riemann-Liouville Fractional Integral Condition Via Fixed Point Theorems

by Piyachat Borisut, Poom Kumam, Idris Ahmed and Kanokwan Sitthithakerngkiet

Symmetry 2019, 11(6), 829; https://doi.org/10.3390/sym11060829 - 22 Jun 2019

Cited by 32 | Viewed by 3689

Abstract

In this paper, we study and investigate an interesting Caputo fractional derivative and Riemann–Liouville integral boundary value problem (BVP): ^c [...] Read more.

In this paper, we study and investigate an interesting Caputo fractional derivative and Riemann–Liouville integral boundary value problem (BVP): ^c

D_{0^{+}}^{q} u (t) = f (t, u (t)), t \in [0, T]

,

u^{(k)} (0) = ξ_{k}, u (T) = \sum_{i = 1}^{m} {β_{i}}_{R L} I_{0^{+}}^{p_{i}} u (η_{i}),

where

n - 1 < q < n, n \geq 2, m, n \in N

,

ξ_{k}, β_{i} \in R

,

k = 0, 1, \dots, n - 2

,

i = 1, 2, \dots, m

, and ^c

D_{0^{+}}^{q}

is the Caputo fractional derivatives,

f : [0, T] \times C ([0, T], E) \to E

, where E is the Banach space. The space E is chosen as an arbitrary Banach space; it can also be

R

(with the absolute value) or

C ([0, T], R)

with the supremum-norm. _RL

I_{0^{+}}^{p_{i}}

is the Riemann–Liouville fractional integral of order

p_{i} > 0

,

η_{i} \in (0, T)

, and

\sum_{i = 1}^{m} β_{i} η_{i}^{p_{i} + n - 1} \frac{Γ (n)}{Γ (n + p_{i})} \neq T^{n - 1}

. Via the fixed point theorems of Krasnoselskii and Darbo, the authors study the existence of solutions to this problem. An example is included to illustrate the applicability of their results. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

13 pages, 291 KiB

Open AccessArticle

Determining Fuzzy Distance between Sets by Application of Fixed Point Technique Using Weak Contractions and Fuzzy Geometric Notions

by Parbati Saha, Shantau Guria, Binayak S. Choudhury and Manuel De la Sen

Symmetry 2019, 11(6), 812; https://doi.org/10.3390/sym11060812 - 19 Jun 2019

Cited by 2 | Viewed by 2013

Abstract

In the present paper, we solve the problem of determining the fuzzy distance between two subsets of a fuzzy metric space. We address the problem by reducing it to the problem of finding an optimal approximate solution of a fixed point equation. This [...] Read more.

In the present paper, we solve the problem of determining the fuzzy distance between two subsets of a fuzzy metric space. We address the problem by reducing it to the problem of finding an optimal approximate solution of a fixed point equation. This approach is well studied for the corresponding problem in metric spaces and is known as proximity point problem. We employ fuzzy weak contractions for that purpose. Fuzzy weak contraction is a recently introduced concept intermediate to a fuzzy contraction and a fuzzy non-expansive mapping. Fuzzy versions of some geometric properties essentially belonging to Hilbert spaces are considered in the main theorem. We include an illustrative example and two corollaries, one of which comes from a well-known fixed point theorem. The illustrative example shows that the main theorem properly includes its corollaries. The work is in the domain of fuzzy global optimization by use of fixed point methods. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

12 pages, 250 KiB

Open AccessArticle

Iterative Algorithms for a System of Variational Inclusions in Banach Spaces

by Lu-Chuan Ceng, Mihai Postolache and Yonghong Yao

Symmetry 2019, 11(6), 811; https://doi.org/10.3390/sym11060811 - 19 Jun 2019

Cited by 16 | Viewed by 2079

Abstract

A system of variational inclusions (GSVI) is considered in Banach spaces. An implicit iterative procedure is proposed for solving the GSVI. Strong convergence of the proposed algorithm is given. Full article

(This article belongs to the Special Issue Advance in Nonlinear Analysis and Optimization)

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