Mathematics, Volume 10, Issue 7 (April-1 2022) – 195 articles

Complex-variable chaotic systems (CVCSs) have numerous advantages over real-variable chaotic systems in chaos communication due to their increased unpredictability, confidentiality, and the ease of implementation. Synchronization between the master and slave systems in CVCSs is key to achieving encryption and decryption. However, existing synchronization schemes for CVCSs require the amplitude of the chaotic signal to be much larger than that of the plaintext. Moreover, traditional chaotic masking of complete synchronization (CS) requires uniformity between the transmitter and receiver ends. Therefore, we propose a complex modified projective difference function synchronization (CMPDFS) of CVCSs to address these issues, where the modified projective matrix helps address the issues with the amplitude. The receiver end is reconstructed without uniformity of the transmitter. We design the CMPDFS controller and propose a new secure communication scheme for wireless sensor networks (WSNs). The basic principle is fundamentally different from traditional chaotic masking. Simulation results and security analysis demonstrate that the CMPDFS communication scheme has a large key space, high sensitivity to encryption keys, high security, and an acceptable encryption speed. Hence, the proposed scheme can improve the security of WSNs. Moreover, it also can be applied to similar communication systems. Full article

This paper proposes an economical-environmental-technical dispatch (EETD) model for adjusted IEEE 30-bus and IEEE 57-bus systems, including thermal and high penetration of renewable energy sources (RESs). Total fuel costs, emissions level, power losses, voltage deviation, and voltage stability are the five objectives addressed in this work. A large set of equality and inequality constraints are included in the problem formulation. Metaheuristic optimization approaches—Coronavirus herd immunity optimizer (CHIO), salp swarm algorithm (SSA), and ant lion optimizer (ALO)—are used to identify the optimal cost of generation, emissions, voltage deviation, losses, and voltage stability solutions. Several scenarios are reviewed to validate the problem-solving competency of the defined optimisation model. Numerous scenarios are studied to verify the proficiency of the optimisation model in problem-solving. The multi-objective problem is converted into a normalized one-objective issue through a weighted sum-approach utilizing the analytical hierarchy process (AHP). Additionally, the technique for order preference by similarity to ideal solution (TOPSIS) is presented for identifying the optimal value of Pareto alternatives. Ultimately, the results achieved reveal that the proposed CHIO performs the other approaches in the EETD problem-solving. Full article

In the game industry, tardiness is an important issue. Unlike a unifunctional machine, a developer may excel in programming but be mediocre in scene modeling. His/her processing speed varies with job type. To minimize tardiness, we need to schedule these developers carefully. Clearly, traditional scheduling algorithms for unifunctional machines are not suitable for such versatile developers. On the other hand, in an unrelated machine scheduling problem, n jobs can be processed by m machines at n × m different speeds, i.e., its solution space is too wide to be simplified. Therefore, a tardiness minimization problem considering three job types and versatile developers is presented. In this study, a branch-and-bound algorithm and a lower bound based on harmonic mean are proposed for minimizing the total tardiness. Theoretical analyses ensure the correctness of the proposed method. Computational experiments also show that the proposed method can ensure the optimality and efficiency for n ≤ 18. With the exact algorithm, we can fairly evaluate other approximate algorithms in the future. Full article

In this paper, we propose a linear regression model in which the error term follows a log-gamma-normal (LGN) distribution. The assumption of LGN distribution gives flexibility to accommodate skew forms to the left and to the right. Kurtosis greater or smaller than the normal model can also be accommodated. The regression model for censored asymmetric data is also considered (censored LGN model). Parameter estimation is implemented using the maximum likelihood approach and a small simulation study is conducted to evaluate parameter recovery. The main conclusion is that the approach is very much satisfactory for moderate and large sample sizes. Results for two applications of the proposed model to real datasets are provided for illustrative purposes. Full article

The rising global importance of global value chains was enabled by developing information and communication technologies, ICT. A correct understanding of ICT roles determines a country’s global competitiveness. The study aims to examine the role of intangible ICT assets in creating domestic and foreign value added in export. Based on a sample of available longitudinal data from the EU-KLEMS database, for the period 2000–2015, 10 EU countries have been selected and analysed. We applied several panel regression models to confirm the important role of ICT capital, specific to intangible ICT, in creating domestic added value in exports and participation in the global value chains. Our results show that intangible ICT assets have a higher impact on the global value chain participation than tangible ICT assets. Moreover, the analysis at the sectoral level reveals a stronger effect of total ICT assets in the case of total business sectors. Full article

In this work, a novel fuzzy neural network (NFNN) with a long short-term memory (LSTM) structure was derived and an adaptive sliding mode controller, using NFNN (ASMC-NFNN), was developed for a class of nonlinear systems. Aimed at the unknown uncertainties in nonlinear systems, an NFNN was designed to estimate unknown uncertainties, which combined the advantages of fuzzy systems and neural networks, and also introduced a special LSTM recursive structure. The special three gating units in the LSTM structure enabled it to have selective forgetting and memory mechanisms, which could make full use of historical information, and have a stronger ability to learn and estimate unknown uncertainties than general recurrent neural networks. The Lyapunov stability rule guaranteed the parameter convergence of the neural network and system stability. Finally, research into a simulation of an active power filter system showed that the proposed new algorithm had better static and dynamic properties and robustness compared with a sliding controller that uses a recurrent fuzzy neural network (RFNN). Full article

Flight Maneuver Recognition (FMR) refers to the automatic recognition of a series of aircraft flight patterns and is a key technology in many fields. The chaotic nature of its input data and the professional complexity of the identification process make it difficult and expensive to identify, and none of the existing models have general generalization capabilities. A general framework is proposed in this paper, which can be used for all kinds of flight tasks, independent of the aircraft type. We first preprocessed the raw data with unsupervised clustering method, segmented it into maneuver sequences, then reconstructed the sequences in phase space, calculated their approximate entropy, quantitatively characterized the sequence complexity, and distinguished the flight maneuvers. Experiments on a real flight training dataset have shown that the framework can quickly and correctly identify various flight maneuvers for multiple aircraft types with minimal human intervention. Full article

Human health is reflected in all spheres of life and the economy. One of the main causes of morbidity and early mortality is polluted air. Ambient air pollution is a serious source of disease and mortality across the world. Cities are notorious for their high levels of air pollution and sickness. However, the precise degree of the health impacts of air pollution at the municipal level are still largely unclear. One of the main reasons for increased morbidity is the presence of particulate matter. The aim of our study is to show the relationship between elevated levels of particulate matter in the air and certain diseases. In this paper, we apply InterCriteria Analysis (ICrA) to find the correlation between the level of air pollution and the number of people seeking medical help. This is a new approach for the problem. The results show the affect of air pollution on certain diseases with a short exposure on polluted air and when the exposure is prolonged. We observed that some diseases are exacerbated by brief exposure to polluted air, while in others, exacerbation occurs after prolonged exposure. Full article

We consider in this article the stochastic fractional Zakharov system derived by the multiplicative Wiener process in the Stratonovich sense. We utilize two distinct methods, the Riccati–Bernoulli sub-ODE method and Jacobi elliptic function method, to obtain new rational, trigonometric, hyperbolic, and elliptic stochastic solutions. The acquired solutions are helpful in explaining certain fascinating physical phenomena due to the importance of the Zakharov system in the theory of turbulence for plasma waves. In order to show the influence of the multiplicative Wiener process on the exact solutions of the Zakharov system, we employ the MATLAB tools to plot our figures to introduce a number of 2D and 3D graphs. We establish that the multiplicative Wiener process stabilizes the solutions of the Zakharov system around zero. Full article

In this paper, we investigate the controllability of first order linear fuzzy differential systems. We use the direct construction method to derive the controllability results for three types of first order linear fuzzy controlled systems via $\left(c1\right)$-solution and $\left(c2\right)$-solution, respectively. An example is presented to illustrate our theoretical results. Full article

E-learning is a popular area in terms of learning from social media websites in various terms and contents for every group of people in this world with different knowledge backgrounds and jobs. E-learning sites help users such as students, business workers, instructors, and those searching for different educational institutions. Excluding the benefits of this system, there are various challenges that the users face in online platforms. One of the important challenges is the true information and right content based on these resources, search results and quality. This research proposes virtual and intelligent agent-based recommendation, which requires users’ profile information and preferences to recommend the proper content and search results based on their search history. We applied Natural Language Processing (NLP) techniques and semantic analysis approaches for the recommendation of course selection to e-learners and tutors. Moreover, machine learning performance analysis applied to improve the user rating results in the e-learning environment. The system automatically learns and analyzes the learner characteristics and processes the learning style through the clustering strategy. Compared with the recent state-of-the-art in this field, the proposed system and the simulation results show the minimizing number of metric errors compared to other works. The achievements of the presented approach are providing a comfortable platform to the user for course selection and recommendations. Similarly, we avoid recommending the same contents and courses. We analyze the user preferences and improving the recommendation system performance to provide highly related content based on the user profile situation. The prediction accuracy of the proposed system is 98% compared to hybrid filtering, self organization systems and ensemble modeling. Full article

Critical, creative, and flexible thinking is one of the educational priorities in the 21st Century, at a time in which the world is showing its most ephemeral and uncertain side. In this sense, visible thinking is a fundamental tool for achieving more significant and meaningful learning. The VESS (Meaningful Life with Balance and Wisdom) model, considering how we learn, provides different tools, such as the aforementioned, which facilitate learning. Thus, thinking, and, more specifically, the development of mathematical thinking through the VESS model, provides an open door towards the development of basic cognitive skills, such as attention, metacognition, and memory, and has a direct effect on the three areas of the early childhood education curriculum. The aim of the present work is to discover, in a descriptive manner, how the learning of the VESS model influences future teachers. In the present article, a series of descriptive, correlational, and survey-based models conducted at the University of Cordoba (Córdoba, Spain) are described. Beforehand, a validation of the instrument was performed, which obtained high goodness-of-fit indices, with an adequate validity and reliability. Full article

Fast and accurate smoke detection is very important for reducing fire damage. Due to the complexity and changeable nature of smoke scenes, existing smoke detection technology has the problems of a low detection rate and a high false negative rate, and the robustness and generalization ability of the algorithms are not high. Therefore, this paper proposes a smoke detection model based on the improved YOLOv5. First, a large number of real smoke and synthetic smoke images were collected to form a dataset. Different loss functions (GIoU, DIoU, CIoU) were used on three different models of YOLOv5 (YOLOv5s, YOLOv5m, YOLOv5l), and YOLOv5m was used as the baseline model. Then, because of the problem of small numbers of smoke training samples, the mosaic enhancement method was used to randomly crop, scale and arrange nine images to form new images. To solve the problem of inaccurate anchor box prior information in YOLOv5, a dynamic anchor box mechanism is proposed. An anchor box was generated for the training dataset through the k-means++ clustering algorithm. The dynamic anchor box module was added to the model, and the size and position of the anchor box were dynamically updated in the network training process. Aiming at the problem of unbalanced feature maps in different scales of YOLOv5, an attention mechanism is proposed to improve the network detection performance by adding channel attention and spatial attention to the original network structure. Compared with the traditional deep learning algorithm, the detection performance of the improved algorithm in this paper was is 4.4% higher than the mAP of the baseline model, and the detection speed reached 85 FPS, which is obviously better and can meet engineering application requirements. Full article

The aim of this study is to develop an automatic segmentation algorithm based on paranasal sinus CT images, which realizes automatic identification and segmentation of the sinus boundary and its inflamed proportions, as well as the reconstruction of normal sinus and inflamed site volumes. Our goal is to overcome the current clinical dilemma of manually calculating the inflammatory sinus volume, which is objective and ineffective. A semi-supervised learning algorithm using pseudo-labels for self-training was proposed to train convolutional neural networks, which consisted of SENet, MobileNet, and ResNet. An aggregate of 175 CT sets was analyzed, 50 of which were from patients who subsequently underwent sinus surgery. A 3D view and volume-based modified Lund-Mackay score were determined and compared with traditional scores. Compared to state-of-the-art networks, our modifications achieved significant improvements in both sinus segmentation and classification, with an average pixel accuracy of 99.67%, an MIoU of 89.75%, and a Dice coefficient of 90.79%. The fully automatic nasal sinus volume reconstruction system was successfully obtained the relevant detailed information by accurately acquiring the nasal sinus contour edges in the CT images. The accuracy of our algorithm has been validated and the results can be effectively applied to actual clinical medicine or forensic research. Full article

We investigate the Hyers–Ulam stability of an equation involving a single variable of the form $\parallel f\left(x\right)-\alpha f\left(k^n\left(x\right)\right)-\beta f\left(k^{n+1}\left(x\right)\right)\parallel ⩽u\left(x\right)$ where f is an unknown operator from a nonempty set X into a Banach space Y, and it preserves the addition operation, besides other certain conditions. The theory is employed and stability theorems are proven for various functional equations involving several variables. By comparing this method with the available techniques, it was noticed that this method does not require any restriction on the parity, on the domain, and on the range of the function. Our findings suggest that it is very much easy and more appropriate to apply the proposed method while investigating the stability of functional equations, in particular for several variables. Full article

In order-of-addition experiments, the full order-of-addition designs are often unaffordable due to their large run sizes. The problem of finding efficient fractional OofA designs arises. The order-of-addition orthogonal arrays are a class of optimal fractional order-of-addition designs for the prevalent pair-wise ordering model, under a variety of widely used design criteria. In the literature, the studies on order-of-addition orthogonal arrays focused on strength 2 while the order-of-addition orthogonal arrays of higher strength have not been investigated yet. In this paper, we focus on order-of-addition orthogonal arrays of strength 3. First, the method of constructing order-of-addition orthogonal arrays of strength 3 is proposed. Second, a theoretical result that states that the order-of-addition orthogonal arrays of strength 3 have better balance properties than those of strength 2 is developed. Third, we provide thorough simulation studies which show that the constructed order-of-addition orthogonal arrays of strength 3 have desirable performance for estimating optimal orders of addition. Full article

This work provides an analytical approach to characterize and determine solutions to a porous medium system of equations with views in applications to invasive-invaded biological dynamics. Firstly, the existence and uniqueness of solutions are proved. Afterwards, profiles of solutions are obtained making use of the self-similar structure that permits showing the existence of a diffusive front. The solutions are then studied within the Travelling Waves (TW) domain showing the existence of potential and exponential profiles in the stable connection that converges to the stationary solutions in which the invasive species predominates. The TW profiles are shown to exist based on the geometry perturbation theory together with an analytical-topological argument in the phase plane. The finding of an exponential decaying rate (related with the advection and diffusion parameters) in the invaded species TW is not trivial in the nonlinear diffusion case and reflects the existence of a TW trajectory governed by the invaded species runaway (in the direction of the advection) and the diffusion (acting in a finite speed front or support). Full article

This mathematical model explains the behavior of sinusoidal radius activity in stagnation point three-dimensional flow of hybrid nanoparticles through a circular cylinder. The energy equation of heat source/sink effect and the mass equation of Arrhenius energy of activation and chemical reaction effects are incorporated. Self-relation transformations are adopted to reduce the PDEs to ODEs, then the RKF-45 method is solved with shooting proficiency. The nodal and saddle point action is studied in pertinent parameters for thermal, mass, and velocity curves. Further statistical values of skin friction, Nusselt number, and Sherwood number of both nodal and saddle points are portrayed in tables format. It is ascertained that higher values of activation energy and reaction rate enhance the concentration curve. In addition, the nodal point curves are always less than saddle point curves. Full article

A fundamental problem in the design of a classroom is to identify what characteristics it should have in order to optimize learning. This is a complex problem because learning is a construct related to several cognitive processes. The aim of this study is to maximize learning, represented by the processes of attention, memory, and preference, depending on six classroom parameters: height, width, color hue, color saturation, color temperature, and illuminance. Multi-objective integer linear programming with three objective functions and 56 binary variables was used to solve this optimization problem. Virtual reality tools were used to gather the data; novel software was used to create variations of virtual classrooms for a sample of 112 students. Using an interactive method, more than 4700 integer linear programming problems were optimally solved to obtain 13 efficient solutions to the multi-objective problem, which allowed the decision maker to analyze all the information and make a final choice. The results showed that achieving the best cognitive processing performance involves using different classroom configurations. The use of a multi-objective interactive approach is interesting because in human behavioral studies, it is important to consider the judgement of an expert in order to make decisions. Full article

Currently, coronavirus disease 2019 (COVID-19) has become the most severe infectious disease affecting the world, which has spread around the world to more than 200 countries in 2020. Until the number of COVID-19 vaccines is insufficient, nucleic acid testing is considered as an effective way to screen virus carriers and control the spread of the virus. Considering that the medical resources and infection rates are different across various countries and regions, if all infected areas adopt the traditional individual nucleic acid testing method, the workload will be heavy and time-consuming. Therefore, this will not lead to the control of the pandemic. After Wuhan completed a citywide nucleic acid testing in May 2020, China basically controlled the spread of COVID-19 and entered the post-epidemic period. Since then, although some cities in China, such as Qingdao, Xinjiang, Beijing, and Dalian, have experienced a local epidemic resurgence, the pandemic was quickly suppressed through wide-scale pooled nucleic acid testing methods. Combined with the successful experience of mass nucleic acid testing in China, this study introduces two main pooled testing methods used in two cities with a population of more than ten million people, Wuhan’s “five-in-one” and Qingdao’s “ten-in-one” rapid pooled testing methods. This study proposes an improved method for optimising the second round of “ten-in-one” pooled testing, known as “the pentagram mini-pooled testing method”, which speeds up the testing process (as a result of reducing the numbers of testing by 40%) and significantly reduces the cost. Qingdao’s optimised “ten-in-one” pooled testing method quickly screens out the infections by running fewer testing samples. This study also mathematically examines the probabilistic principles and applicability conditions for pooled testing of COVID-19. Herein, the study theoretically determines the optimal number of samples that could successfully be combined into a pool under different infection rates. Then, it quantitatively discusses the applicability and principles for choosing the pooled testing instead of individual testing. Overall, this research offers a reference for other countries with different infection rates to help them in implementing the mass testing for COVID-19 to reduce the spread of coronavirus. Full article

In this article, a sliding mode control (SMC) is proposed on the basis of an adaptive neural network (NN) for a class of Single-Input–Single-Output (SISO) nonlinear systems containing unknown dynamic functions. Since the control objective is to steer the system states to track the given reference signals, the SMC method is considered by employing the adaptive neural network (NN) strategy for dealing with the unknown dynamic problem. In order to compress the impaction coming from chattering phenomenon (which inherently exists in most SMC methods because of the discontinuous switching term), the boundary layer technique is considered. The basic design idea is to introduce a continuous proportional function to replace the discontinuous switching control term inside the boundary layer so that the chattering can be effectively alleviated. Finally, both Lyapunov theoretical analysis and computer numerical simulation are used to verify the effectiveness of the proposed SMC method. Full article

One finds several q-differential equations of a higher order for q-Euler polynomials and q-Genocchi polynomials. Additionally, we have a few q-differential equations of a higher order, which are mixed with q-Euler numbers and q-Genocchi polynomials. Moreover, we investigate some symmetric q-differential equations of a higher order by applying symmetric properties of q-Euler polynomials and q-Genocchi polynomials. Full article

We study numerical approaches to computation of spectral properties of composition operators. We provide a characterization of Koopman Modes in Banach spaces using Generalized Laplace Analysis. We cast the Dynamic Mode Decomposition-type methods in the context of Finite Section theory of infinite dimensional operators, and provide an example of a mixing map for which the finite section method fails. Under assumptions on the underlying dynamics, we provide the first result on the convergence rate under sample size increase in the finite-section approximation. We study the error in the Krylov subspace version of the finite section method and prove convergence in pseudospectral sense for operators with pure point spectrum. Since Krylov sequence-based approximations can mitigate the curse of dimensionality, this result indicates that they may also have low spectral error without an exponential-in-dimension increase in the number of functions needed. Full article

The present work focuses on the attributes of flow, heat, and mass transfer together with double diffusive Cattaneo–Christov mechanism with regards to their applications. The aim of this study is to investigate the non-Newtonian Powell–Eyring fluid flow, taking into account the twofold impact of the heat generation mechanism and the viscous dissipation due to an extensible sheet. The chemical reaction between the fluid particles and the fluid variable properties is assumed in this study. The motive behind this study is the continuous and great interest in the utilization of non-Newtonian liquids in organic and technical disciplines. This model is administered and governed by the momentum equation, energy equation, and concentration, all of which are in the form of partial differential equations. With the help of the shooting technique, the numerical solution is obtained. Graphs show the characteristics of flow, heat, and mass transfer mechanisms for various governing parameters. Additionally, significant physical non-dimensional quantities have been presented in a tabular form. The outcomes detect that increasing the Deborah number, which is connected with the mass transfer field and the chemical reaction parameter, decreases the concentration distribution. Full article

In this paper, we develop two new theorems relating to the series of floor and ceiling functions. We then use these two theorems to develop more than forty distinct novel results. Furthermore, we provide specific cases for the theorems and corollaries which show that our results constitute a generalisation of the currently available results such as the summation of first n Fibonacci numbers and Pascal’s identity. Finally, we provide three miscellaneous examples to showcase the vast scope of our developed theorems. Full article

Predicates and rules are usually enclosed as built-in functions in automated geometry reasoning systems, meaning users cannot add any predicate or rule, thus resulting in a limited reasoning capability of the systems. A method for expanding predicates and rules in automated geometry reasoning systems is, thus, proposed. Specifically, predicate and rule descriptions are transformed to knowledge trees and forests based on formal representations of geometric knowledge, and executable codes are dynamically and automatically generated by using “code templates”. Thus, a transformation from controlled natural language descriptions to mechanization algorithms is completed, and finally, the dynamic expansion of predicates and rules in the reasoning system is achieved. Moreover, the method has been implemented in an automated geometry reasoning system for Chinese college entrance examination questions, and the practicality and effectiveness of the method were tested. In conclusion, the enclosed setting, which is a shortcoming of traditional reasoning systems, is avoided, the user-defined dynamic expansion of predicates and rules is realized, the application scope of the reasoning system is extended, and the reasoning capability is improved. Full article

Here we describe a toolkit for presenting physiologically based pharmacokinetic (PBPK) models in a modular graphical view in the BioUML platform. Firstly, we demonstrate the BioUML capabilities for PBPK modeling tested on an existing model of nanoparticles delivery to solid tumors in mice. Secondly, we provide guidance on the conversion of the PBPK model code from a text modeling language like Berkeley Madonna to a visual modular diagram in the BioUML. We give step-by-step explanations of the model transformation and demonstrate that simulation results from the original model are exactly the same as numerical results obtained for the transformed model. The main advantage of the proposed approach is its clarity and ease of perception. Additionally, the modular representation serves as a simplified and convenient base for in silico investigation of the model and reduces the risk of technical errors during its reuse and extension by concomitant biochemical processes. In summary, this article demonstrates that BioUML can be used as an alternative and robust tool for PBPK modeling. Full article

Based on a successive convex programming method, an alternating iteration algorithm is proposed for solving a parameter-dependent distributionally robust optimization. Under the Slater-type condition, the convergence analysis of the algorithm is obtained. When the objective function is convex, a modified algorithm is proposed and a less-conservative solution is obtained. Lastly, some numerical tests results are illustrated to show the efficiency of the algorithm. Full article

Obtaining a dynamic model of the robotic manipulator is a complex task. With the growing application of machine learning (ML) approaches in modern robotics, a question arises of using ML for dynamic modeling. Still, due to the large amounts of data necessary for this approach, data collection may be time and resource-intensive. For this reason, this paper aims to research the possibility of synthetic dataset creation by using pre-existing dynamic models to test the possibilities of both applications of such synthetic datasets, as well as modeling the dynamics of an industrial manipulator using ML. Authors generate the dataset consisting of 20,000 data points and train seven separate multilayer perceptron (MLP) artificial neural networks (ANN)—one for each joint of the manipulator and one for the total torque—using randomized search (RS) for hyperparameter tuning. Additional MLP is trained for the total torsion of the entire manipulator using the same approach. Each model is evaluated using the coefficient of determination ($R^2$) and mean absolute percentage error (MAPE), with 10-fold cross-validation applied. With these settings, all individual joint torque models achieved $R^2$ scores higher than 0.9, with the models for first four joints achieving scores above 0.95. Furthermore, all models for all individual joints achieve MAPE lower than 2%. The model for the total torque of all joints of the robotic manipulator achieves weaker regression scores, with the $R^2$ score of 0.89 and MAPE slightly higher than 2%. The results show that the torsion models of each individual joint, and of the entire manipulator, can be regressed using the described method, with satisfactory accuracy. Full article

The main crux of this work is to study the existence of extremal solutions for a new class of nonlinear sequential fractional differential equations (NSFDEs) with nonlinear boundary conditions (NBCs) under the $\psi$-Caputo operator. The obtained outcomes of the proposed problem are derived by means of the monotone iterative technique (MIT) associated with the method of upper and lower solutions. Lastly, the desired findings are well illustrated by an example. Full article