IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
A Convergence Theorem Associated With a Pair of Second Order Differential Equ...IOSR Journals
We consider the second order matrix differential equation
M 0, 0 x Where M is a second-order matrix differential operator and is a vector having two components. In this
paper we prove a convergence theorem for the vector function 1 2 ( ) ( ) ( ) f x f x f x which is continuous in
0 x and of bounded variation in 0 x , when p(x) and q(x) tend to as x tend to .
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
The Analytical Nature of the Greens Function in the Vicinity of a Simple Poleijtsrd
It is known that the Green function of a boundary value problem is a meromorphic function of a spectral parameter. When the boundary conditions contain integro differential terms, then the meromorphism of the Greens function of such a problem can also be proved. In this case, it is possible to write out the structure of the residue at the singular points of the Greens function of the boundary value problem with integro differential perturbations. An analysis of the structure of the residue allows us to state that the corresponding functions of the original operator are sufficiently smooth functions. Surprisingly, the adjoint operator can have non smooth eigenfunctions. The degree of non smoothness of the eigenfunction of the adjoint operator to an operator with integro differential boundary conditions is clarified. It is indicated that even those conjugations to multipoint boundary value problems have non smooth eigenfunctions. Ghulam Hazrat Aimal Rasa "The Analytical Nature of the Green's Function in the Vicinity of a Simple Pole" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-6 , October 2020, URL: https://www.ijtsrd.com/papers/ijtsrd33696.pdf Paper Url: https://www.ijtsrd.com/mathemetics/applied-mathamatics/33696/the-analytical-nature-of-the-greens-function-in-the-vicinity-of-a-simple-pole/ghulam-hazrat-aimal-rasa
A Study To Locate The Difference Between Active And Passive Recovery After St...IOSR Journals
Abstract: Exercise recovery involves a number of Post-exercise Steps that are essential for any exercise
regime, regardless of fitness level, the type of physical activity or the exercise intensity. Exercise recovery does
not take long or require much effort, but it is often neglected. The exercise recovery is mainly of two types one is
Active Recovery and the other is passive Recovery. Active Recovery is the term used for cooling down (with slow
jogging, stretching etc. ) after a high intensive work out and Passive Recovery is Stopping dead, doing nothing
again until he head out on his next run a couple of days later. The results of a study suggest that an active
recovery of 3 minutes between High intensity, short duration exercise bouts significantly increase peak power
and average power compared to a passive recovery, irrespective of changes in blood lactate levels. Several
research studies have been conducted in this specific area in the international level but the effort is supposed to
be weaker with respect to Indian Sport Scientists. Feeling the importance of recovery in the sport field and
considering its application in a wider range or rather as an effort to make it much more acceptable to the sport
society the investigator under the guidance of her supervisor envisaged to conduct the study entitled “A study to
locate the difference between active and passive recovery after strenuous workout.” The subjects for the present
study were 20 male trained athletes from Howrah and Purba Medinipur districts of West Bengal who were
selected randomly from a coaching camp. In the present study, the investigator has considered the following
things such as their – Age, sex, Height of the subject, Resting Heart Rate, Post Exercise Heart Rate, After Active
Recovery Heart Rate, After Passive Recovery Heart Rate. The scholar made the athletes to run 600 mts. and
recorded their Heart Rate just after completion of the workout and also recorded their HR after one minute of
complete rest the same procedure was repeated but in between the athletes were directed to perform active
recovery i.e. slow jogging. From data analysis it appears that the Active recovery is superior in comparison to
the Passive recovery because the recovery range is greater. Thus the scholar arrived at the explicit conclusion
that active recovery is better than passive recovery between exercise bouts or after strenuous workout.
Key Words: Active recovery, Passive Recovery, Strenuous workout.
Ultimate Behavior of Lightweight High Strength Concrete Filled Steel Tube (LW...IOSR Journals
Strength and ductility of concrete members can be significantly improved with lateral confinement, usually achieved by using a steel tube casing. The concrete confinement can be utilized to make bridge lighter and have longer spans. In addition, a significant portion of the load carried concrete bridge girders is the self-weight of the girders and deck. If all or part of the girder and deck can be made using high strength lightweight concretes, there is a potential for appreciable economic savings since the self-weight could be reduced by as much as 15-20%. The study described herein investigates the static nonlinear behavior of lightweight high strength concrete filled steel tube (LWHCFST) bridges up to failure. The current study had two specific goals. The first was to experimentally determine the static modulus of elasticity of confined high strength lightweight concrete mixture. The second was to develop a nonlinear finite element computer program to study the ultimate behavior of a filled tube (LWHCFST) example bridge. The nonlinear stress-strain behavior of confined high strength lightweight concrete is evaluated experimentally by the authors and is used to help establish a comparison between the ultimate behavior of the bridge using confined normal weight concrete and confined high strength lightweight concrete. The ultimate strength of the bridge is related to the occurrence of an equivalent failure mechanism. The study indicated that the use of (LWHCFST) is beneficial for extending bridge girder lengths
A Convergence Theorem Associated With a Pair of Second Order Differential Equ...IOSR Journals
We consider the second order matrix differential equation
M 0, 0 x Where M is a second-order matrix differential operator and is a vector having two components. In this
paper we prove a convergence theorem for the vector function 1 2 ( ) ( ) ( ) f x f x f x which is continuous in
0 x and of bounded variation in 0 x , when p(x) and q(x) tend to as x tend to .
The peer-reviewed International Journal of Engineering Inventions (IJEI) is started with a mission to encourage contribution to research in Science and Technology. Encourage and motivate researchers in challenging areas of Sciences and Technology.
The Analytical Nature of the Greens Function in the Vicinity of a Simple Poleijtsrd
It is known that the Green function of a boundary value problem is a meromorphic function of a spectral parameter. When the boundary conditions contain integro differential terms, then the meromorphism of the Greens function of such a problem can also be proved. In this case, it is possible to write out the structure of the residue at the singular points of the Greens function of the boundary value problem with integro differential perturbations. An analysis of the structure of the residue allows us to state that the corresponding functions of the original operator are sufficiently smooth functions. Surprisingly, the adjoint operator can have non smooth eigenfunctions. The degree of non smoothness of the eigenfunction of the adjoint operator to an operator with integro differential boundary conditions is clarified. It is indicated that even those conjugations to multipoint boundary value problems have non smooth eigenfunctions. Ghulam Hazrat Aimal Rasa "The Analytical Nature of the Green's Function in the Vicinity of a Simple Pole" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-4 | Issue-6 , October 2020, URL: https://www.ijtsrd.com/papers/ijtsrd33696.pdf Paper Url: https://www.ijtsrd.com/mathemetics/applied-mathamatics/33696/the-analytical-nature-of-the-greens-function-in-the-vicinity-of-a-simple-pole/ghulam-hazrat-aimal-rasa
A Study To Locate The Difference Between Active And Passive Recovery After St...IOSR Journals
Abstract: Exercise recovery involves a number of Post-exercise Steps that are essential for any exercise
regime, regardless of fitness level, the type of physical activity or the exercise intensity. Exercise recovery does
not take long or require much effort, but it is often neglected. The exercise recovery is mainly of two types one is
Active Recovery and the other is passive Recovery. Active Recovery is the term used for cooling down (with slow
jogging, stretching etc. ) after a high intensive work out and Passive Recovery is Stopping dead, doing nothing
again until he head out on his next run a couple of days later. The results of a study suggest that an active
recovery of 3 minutes between High intensity, short duration exercise bouts significantly increase peak power
and average power compared to a passive recovery, irrespective of changes in blood lactate levels. Several
research studies have been conducted in this specific area in the international level but the effort is supposed to
be weaker with respect to Indian Sport Scientists. Feeling the importance of recovery in the sport field and
considering its application in a wider range or rather as an effort to make it much more acceptable to the sport
society the investigator under the guidance of her supervisor envisaged to conduct the study entitled “A study to
locate the difference between active and passive recovery after strenuous workout.” The subjects for the present
study were 20 male trained athletes from Howrah and Purba Medinipur districts of West Bengal who were
selected randomly from a coaching camp. In the present study, the investigator has considered the following
things such as their – Age, sex, Height of the subject, Resting Heart Rate, Post Exercise Heart Rate, After Active
Recovery Heart Rate, After Passive Recovery Heart Rate. The scholar made the athletes to run 600 mts. and
recorded their Heart Rate just after completion of the workout and also recorded their HR after one minute of
complete rest the same procedure was repeated but in between the athletes were directed to perform active
recovery i.e. slow jogging. From data analysis it appears that the Active recovery is superior in comparison to
the Passive recovery because the recovery range is greater. Thus the scholar arrived at the explicit conclusion
that active recovery is better than passive recovery between exercise bouts or after strenuous workout.
Key Words: Active recovery, Passive Recovery, Strenuous workout.
Ultimate Behavior of Lightweight High Strength Concrete Filled Steel Tube (LW...IOSR Journals
Strength and ductility of concrete members can be significantly improved with lateral confinement, usually achieved by using a steel tube casing. The concrete confinement can be utilized to make bridge lighter and have longer spans. In addition, a significant portion of the load carried concrete bridge girders is the self-weight of the girders and deck. If all or part of the girder and deck can be made using high strength lightweight concretes, there is a potential for appreciable economic savings since the self-weight could be reduced by as much as 15-20%. The study described herein investigates the static nonlinear behavior of lightweight high strength concrete filled steel tube (LWHCFST) bridges up to failure. The current study had two specific goals. The first was to experimentally determine the static modulus of elasticity of confined high strength lightweight concrete mixture. The second was to develop a nonlinear finite element computer program to study the ultimate behavior of a filled tube (LWHCFST) example bridge. The nonlinear stress-strain behavior of confined high strength lightweight concrete is evaluated experimentally by the authors and is used to help establish a comparison between the ultimate behavior of the bridge using confined normal weight concrete and confined high strength lightweight concrete. The ultimate strength of the bridge is related to the occurrence of an equivalent failure mechanism. The study indicated that the use of (LWHCFST) is beneficial for extending bridge girder lengths
Some forms of N-closed Maps in supra Topological spacesIOSR Journals
In this paper, we introduce the concept of N-closed maps and we obtain the basic properties and
their relationships with other forms of N-closed maps in supra topological spaces.
Profile of Trace Elements in Selected Medicinal Plants of North East IndiaIOSR Journals
Trace elements like Mn, Fe, Cu, Zn and major elements K and Ca were quantified in ten selected medicinal plants of North East India by using Proton Induced X- ray Emission (PIXE) technique. No toxic heavy metals such as As, Hg, Pb and Cd were detected. The concentration (ppm) of the elements in the studied plants was found to be as follows: manganese(10 to1800 ), iron(27 to 836), copper(6 to140), zinc(10 to 160), potassium(14120 to 76950) and calcium(1660 to 32030). The levels of trace metals present in the plants was found to be beyond the safety standards of WHO in edible plants but around the permissible range for consumed medicinal herbs as defined for different countries.
Congestion Control in Manets Using Hybrid Routing ProtocolIOSR Journals
As the network size increases the probability of congestion occurrence at nodes increases. This is
because of the event driven nature of ad hoc networks that leads to unpredictable network load. As a result
congestion may occur at the nodes which receive more data than that can be forwarded and cause packet losses.
In this paper we propose a hybrid scheme that attempts to avoid packet loss due to congestion as well as reduce
end to end delay in delivering data packets by combining two protocols- Destination sequenced distance vector
routing (DSDV), which is a table driven or proactive protocol and Improved Ad-hoc on demand vector routing
(IAODV) which is an on-demand or reactive protocol that reduces packet loss due to congestion. The strategy
adopted is use DSDV for path selection and if congestion occurs than switch over to IAODV. The routing
performance of this model is then compared with IAODV and DSDV in terms of end to end delay, throughput
and packet delivery fraction
CFD Simulation and Analysis of Fluid Flow Parameters within a Y-Shaped Branch...IOSR Journals
Plumbing system use pipe fittings to connect straight pipe or tubing section for regulating or measuring fluid flow. Y (wye)-shape fitting is one of the important component in the plumbing system. A wye branch allows splitting a branch line equally in two directions. The opening sizes can vary for different situations for instance in situation where a large main line needs to be split into two smaller branches. The wye shape fitting will convert into T shape fitting when the included angle between two pipe branches is 180°. In the present work, effect of angle of turn/bend for a Y-shape pipe will be studied computationally using ANSYS CFX software. For the analysis, all the three pipe branches of 1 inch internal diameter are selected along with equal length so that only the effect of bend angle at 450, 600, 900 and 1800 can be studied. Water as a fluid is selected which flows through the plumbing system. The effect of bend angle, pipe diameter, pipe length, Reynolds number on the resistance coefficient is studied. It was observed that resistance coefficient vary with the change in flow
An efficient incentive of Nitrate and Fluoride on Organic highland cropping s...IOSR Journals
Abstract: Exerting necessitated concentrations of Nitrate and fluoride to the organic highland cropping
systems is a vital management technique. All the chemical elements of the earth’s crust occur in widely differing
omnipresent concentrations, due to their different nuclear chemical formation and geochemical history. The use
of biological nitrogen and fluoride inputs complicates its balancing act due to dandier uncertainty in inorganic
Nitrogen and Fluoride availability. The growers to strike maintain Nitrogen provisioning to support crop
growth and retention of limit pollution followed by fluoride associated soils. Due to various activities of the man
in domestic field, agriculture area and industrial establishment the environment around us consisting soil, water
and air gets polluted. Fluoride inexhaustible concentrations forbid the growth of crop even though nitrates
reposit in the cultivated soils. The purpose of this project was to establish kinetics when nitro fluorides
associated in the highland soils to different crop systems towards environmental pollutions. Cordia Africana
and alfalfa are the plants which make soil to get enrichment of Nitrates and deescalate of concentrations of
fluorides from cultivated soils through its decomposition. This entire study went on its conventionally tilled
crops followed by Cordia Africana and alfalfa stems and leaf particles. Surface soil nitrate concentrations were
measured weekly, biweekly by volumetric analysis and nitrate leaching was estimated from tension Lysimeters
which were buried at the soil bedrock interface. Subsequently by using Orion 720A fluoride ion meter, deescalated
concentrations of fluorides have been measured. The demonstrated concentrations of Soil in NO3
variables, coefficients of variations from the mean concentrations across all samplings have been recorded
sporadically. The total area of the soil bed was maintained the same PH values until project was completed by
weigh Lysimeters. The timing of elevated Nitrate concentrations (10-15PPM) and the concentrations of fluoride
in deeper soil water corresponds with fallow periods. These dynamics will assist growers in adapting the timing
management operations and reduces nitrate departures.
Synthesis, Physicochemical Characterization and Structure Determination of So...IOSR Journals
Some novel nickel(II) complexes with the ligand (z)-4-((2-hydroxy-3-
methoxyphenyl)diazenyl)-1,5-dimethyl-2-phenyl-1H-pyrazol-3-(2H)-one,GAAP,guiacolazoantipyrine, L
having molecular formulae [Ni(L)2X2] and [Ni(L)2(NCS)Cl] where X = Cl-, Br-, NO3
- were synthesized and
characterized. The elemental analysis, Spectral (IR, UV-Visible, EPR, FAB – mass) studies and thermo
gravimetric analysis reveals that the Ni(II) is six coordinated in its complexes. A rhombic symmetry can be
tentatively proposed for the complexes. The magnetic susceptibility measurements show that the complexes are
paramagnetic in nature. The powder XRD study shows its anisotropic nature
Effect of time-of-day specific obese training on body composition and physica...IOSR Journals
The best strategy for management of obese, outside pharmacological interventions, is physical exercise associated to diet. Recent research has discovered that the problem of obesity is largely due to a biological clock and that lipid oxidation is higher in the evening compared to the morning and at night compared to day. The purpose of this study is to investigate the effect of time-of-day specific obese training on body composition and physical capacity in obese following a low calorie diet. 20 sedentary pre-obese and obese with a mean BMI of 34.3 kg/m2 aged 20 to 47 years subjects participated in a concurrent strength and specific endurance training for obese. Subjects were divided into two training groups: a Morning training Group (MG: n = 10) and an Evening training Group (EG: n=10). The specific training associated to lower caloric diet has increased physical capacity (17,7% for EG and 15,6% for MG), decreased body weight (7,3% for EG and 6% for MG) fat percentage (19,5% for EG and 11,3% for MG) and waist circumference (10,2% for EG and 8,2% for MG) in both groups. Afternoon training was more effective than morning training on fat loss (24.9% for EG versus 15.9% for MG) and on lean mass variation (+2.9% for EG versus -0.5% for MG).
Physicochemical and Bacteriological Analyses of Sachets Water Samples in Kano...IOSR Journals
Physicochemical and bacteriological analyses of sachets water samples in Kano metropolis were
carried out using standard procedures to assess the quality of such water consumed in the area. Samples were
collected from four different water depots in different parts of Kano metropolis. The results showed variations in
the concentrations of the analyzed parameters in the water samples. The pH values ranged from 6.97±0.20 to
7.25±0.33; Electrical Conductivity ranged from 176 ±0.02 to 282±0.25μS/cm; Alkalinity ranged from 0.17±0.02
to 0.69±0.28 mg/l; Total solids were in the range of 100.30±0.25 to 157.34±0.30mg/l. Total Dissolved Solids
ranged from 67.80±0.30 to 84.70±0.23mg/l; Total Suspended Solids ranged from 15.60±0.36 to75.84±0.02mg/;
Total Hardness ranged from 85.00±0.03 to 103.00±0.20 mg/ and turbidity ranged from 0.60±0.21 to 2.23±0.32
NTU. Escherichia coli (E.coli) were not detected in all the samples. The levels of some of the anions analyzed
ranged from 0.03±0.00 mg/l NO2
- to 7.06 ±0.02 mg/l SO4
2-. Similarly, the levels of some of the heavy metals
analyzed ranged from 0.12±0.02mg/l Cu to 0.71±0.01mg/l Fe. Accordingly, the water samples were colourless
and odourless. In general, the concentrations of all the parameters analyzed in the samples were below or
within the World Health Organization (WHO) permissible limits, indicating that the sachets water were safe for
human consumption.
An Analysis of Wushu's Theoretical Concept and its Reflection in Wushu's Prac...IOSR Journals
Abstract: Wushu is an essential part of Chinese culture. It is highly valued for its defensive, health-preserving, entertaining and educational functions. It has a rich history and its development can be traced from the dawn of the Chinese civilization till its present status as a sports discipline aiming to enter the Olympic Games. The internationalization of Wushu being one of the main concerns in the context of its present development suggests the existence of comprehensible literature that can provide the non- Chinese reader with a better understanding of Wushu. Therefore, this paper introduces health-preserving, social-regulating and harmonizing functions of Wushu,the way they are reflected in its practice. Key Words:Wushu, traditional Chinese culture, martial arts
Numerical Solution of Nth - Order Fuzzy Initial Value Problems by Fourth Orde...IOSR Journals
In this paper, a numerical method for Nth - order fuzzy initial value problems (FIVP) based on
Seikkala derivative of fuzzy process is studied. The fourth order Runge-Kutta method based on Centroidal Mean
(RKCeM4) is used to find the numerical solution and the convergence and stability of the method is proved. This
method is illustrated by solving second and third order FIVPs. The results show that the proposed method suits
well to find the numerical solution of Nth – order FIVPs.
Developing Design Cracker Anacardium occidentale (mente) for Home IndustrialIOSR Journals
These purpose of my research was to design the peeled of cutting and made its quality and quantity
of the cracker mente. The method for evaluation was designing and testing several peel models, testing crack
effectively, and testing for its quality and quantity cracker. With developing design craker mente, we found the
cracker home industrial. The cracker mente have developed design a cutting knife with follow the mente’s the
top of the ellipse knife. The length effectively was the maximum produce cracker quality is 8 mm with width
mente average 25-27 mm. The result of the desain knife for cracker mente has to be grade mente quality to
produce to be 79,25% each 1 kg with time peeling take 26 menit, 55 detik.
Kata kunci : Peeling mente.
Network Lifetime Analysis of Routing Protocols of Short Network in QualnetIOSR Journals
A Mobile Ad-Hoc Network (MANET) is a collection of wireless mobile nodes that communicates with
each other without using any existing infrastructure, access point or centralized administration. Mobile ad-hoc
network have the attributes such as wireless connection, continuously changing topology, distributed operation
and ease of deployment. In this paper we have compared the energy consumption of reactive, proactive &
hybrid routing protocol AODV,DSR,RIP & ZRP by using different mobility model. We have analyzed the
Network lifetime of protocols by varying pay load, mobility, pause time and type of traffic (CBR). A detailed
simulation has been carried out in qualnet. The metrics used for performance analysis are energy consumed &
battery consumption. It has been observed that RIP has better network lifetime than other
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
We disclose a simple and straightforward method of solving single-order linear partial differential equations. The advantage of the method is that it is applicable to any orders and the big disadvantage is that it is restricted to a single order at a time. As it is very easy compared to classical methods, it has didactic value.
The aim of this paper is to study the existence and approximation of periodic solutions for non-linear systems of integral equations, by using the numerical-analytic method which were introduced by Samoilenko[ 10, 11]. The study of such nonlinear integral equations is more general and leads us to improve and extend the results of Butris [2].
Matrix Transformations on Some Difference Sequence SpacesIOSR Journals
The sequence spaces 𝑙∞(𝑢,𝑣,Δ), 𝑐0(𝑢,𝑣,Δ) and 𝑐(𝑢,𝑣,Δ) were recently introduced. The matrix classes (𝑐 𝑢,𝑣,Δ :𝑐) and (𝑐 𝑢,𝑣,Δ :𝑙∞) were characterized. The object of this paper is to further determine the necessary and sufficient conditions on an infinite matrix to characterize the matrix classes (𝑐 𝑢,𝑣,Δ ∶𝑏𝑠) and (𝑐 𝑢,𝑣,Δ ∶ 𝑙𝑝). It is observed that the later characterizations are additions to the existing ones
Dual Spaces of Generalized Cesaro Sequence Space and Related Matrix Mappinginventionjournals
In this paper we define the generalized Cesaro sequence spaces 푐푒푠(푝, 푞, 푠). We prove the space 푐푒푠(푝, 푞, 푠) is a complete paranorm space. In section-2 we determine its Kothe-Toeplitz dual. In section-3 we establish necessary and sufficient conditions for a matrix A to map 푐푒푠 푝, 푞, 푠 to 푙∞ and 푐푒푠(푝, 푞, 푠) to c, where 푙∞ is the space of all bounded sequences and c is the space of all convergent sequences. We also get some known and unknown results as remarks.
A Non Local Boundary Value Problem with Integral Boundary ConditionIJMERJOURNAL
ABSTRACT: In this article a three point boundary value problem associated with a second order differential equation with integral type boundary conditions is proposed. Then its solution is developed with the help of the Green’s function associated with the homogeneous equation. Using this idea and Iteration method is proposed to solve the corresponding linear problem.
Some Common Fixed Point Results for Expansive Mappings in a Cone Metric SpaceIOSR Journals
The purpose of this work is to extend and generalize some common fixed point theorems for Expansive type mappings in complete cone metric spaces. We are attempting to generalize the several well- known recent results. Mathematical subject classification; 54H25, 47H10
Further Results On The Basis Of Cauchy’s Proper Bound for the Zeros of Entire...IJMER
International Journal of Modern Engineering Research (IJMER) is Peer reviewed, online Journal. It serves as an international archival forum of scholarly research related to engineering and science education.
International Journal of Modern Engineering Research (IJMER) covers all the fields of engineering and science: Electrical Engineering, Mechanical Engineering, Civil Engineering, Chemical Engineering, Computer Engineering, Agricultural Engineering, Aerospace Engineering, Thermodynamics, Structural Engineering, Control Engineering, Robotics, Mechatronics, Fluid Mechanics, Nanotechnology, Simulators, Web-based Learning, Remote Laboratories, Engineering Design Methods, Education Research, Students' Satisfaction and Motivation, Global Projects, and Assessment…. And many more.
Generalised Statistical Convergence For Double SequencesIOSR Journals
Recently, the concept of 𝛽-statistical Convergence was introduced considering a sequence of infinite
matrices 𝛽 = (𝑏𝑛𝑘 𝑖 ). Later, it was used to define and study 𝛽-statistical limit point, 𝛽-statistical cluster point,
𝑠𝑡𝛽 − 𝑙𝑖𝑚𝑖𝑡 inferior and 𝑠𝑡𝛽 − 𝑙𝑖𝑚𝑖𝑡 superior. In this paper we analogously define and study 2𝛽-statistical
limit, 2𝛽-statistical cluster point, 𝑠𝑡2𝛽 − 𝑙𝑖𝑚𝑖𝑡 inferior and 𝑠𝑡2𝛽 − 𝑙𝑖𝑚𝑖𝑡 superior for double sequences.
Existence, Uniqueness and Stability Solution of Differential Equations with B...iosrjce
In this work, we investigate the existence ,uniqueness and stability solution of non-linear
differential equations with boundary conditions by using both method Picard approximation and
Banach fixed point theorem which were introduced by [6] .These investigations lead us to improving
and extending the above method. Also we expand the results obtained by [1] to change the non-linear
differential equations with initial condition to non-linear differential equations with boundary
conditions
A Class of Polynomials Associated with Differential Operator and with a Gener...iosrjce
The object of this paper is to present several classes of linear and bilateral generating relations by
employing operational techniques, which reduces as a special case of (known or new) bilateral generating
relations.At last some of generating functions associated with stirling number of second kind are also discussed.
We present a strong convergence implicit Runge-Kutta method, with four stages, for solution of
initial value problem of ordinary differential equations. Collocation method is used to derive a continuous
scheme; and the continuous scheme evaluated at special points, the Gaussian points of fourth degree Legendre
polynomial, gives us four function evaluations and the Runge-Kutta method for the iteration of the solutions.
Convergent properties of the method are discussed. Experimental problems used to check the quality of the
scheme show that the method is highly efficient, A – stable, has simple structure, converges to exact solution
faster and better than some existing popular methods cited in this paper.
Least Square Plane and Leastsquare Quadric Surface Approximation by Using Mod...IOSRJM
Now a days Surface fitting is applied all engineering and medical fields. Kamron Saniee ,2007 find a simple expression for multivariate LaGrange’s Interpolation. We derive a least square plane and least square quadric surface Approximation from a given N+1 tabular points when the function is unique. We used least square method technique. We can apply this method in surface fitting also.
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
Guy Korland, CEO and Co-founder of FalkorDB, will review two articles on the integration of language models with knowledge graphs.
1. Unifying Large Language Models and Knowledge Graphs: A Roadmap.
https://arxiv.org/abs/2306.08302
2. Microsoft Research's GraphRAG paper and a review paper on various uses of knowledge graphs:
https://www.microsoft.com/en-us/research/blog/graphrag-unlocking-llm-discovery-on-narrative-private-data/
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
UiPath Test Automation using UiPath Test Suite series, part 3DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 3. In this session, we will cover desktop automation along with UI automation.
Topics covered:
UI automation Introduction,
UI automation Sample
Desktop automation flow
Pradeep Chinnala, Senior Consultant Automation Developer @WonderBotz and UiPath MVP
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
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SAP heatmap example with demo
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AI for Every Business: Unlocking Your Product's Universal Potential by VP of ...
B0560508
1. IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728.Volume 5, Issue 6 (Mar. - Apr. 2013), PP 05-08
www.iosrjournals.org
www.iosrjournals.org 5 | Page
Stability of Second Order Equation
P. Shekhar1
,V. Dharmaiah2
and G.Mahadevi3
1
Department of Mathematics, Malla Reddy College of Engineering, Hyderabad-500014, India.
2
Department of Mathematics, Osmania University, Hyderabad-500007 India.
3
Department of Mathematics,St. Martins Engineering College, Hyderabad-500014, India
Abstract: In this paper we discussed the stability of the null solution of the second order differential equation .
Under some unusual assumptions we obtain new stability results for this classical equation.
I. Introduction
Consider the second order ODE
𝑥′′
+ 2𝑓 𝑡 𝑥′
+ 𝛽(𝑡)𝑥 + 𝑔 𝑡 𝑥2
= 0, 𝑡 ∈ 𝑅+
(1.1)
where 𝑅+
= [0, ∞), 𝑓, 𝛽, 𝑔 ∶ 𝑅+
→ 𝑅+
are three given continuous functions.
The most familiar interpretation of this equation is that it describes nonlinear oscillations. Stability problems
for this ODE have been studied intensively so far (see, e.g.,[13]-[15], [16]-[18], and the references therein).
Recently, T.A. Burton and T. Furumochi [5] have introduced a new method to study the stability of the null
solution 𝑥 = 𝑥′
= 0 of equation (1.1), which is based on the Schauder fixed point theorem. They discussed a
particular case of (1.1) (one of their assumptions is 𝛽 𝑡 = 1 ) to illustrate their technique. In [8] Marosamu
and Vladimirescu have proved stability results for the null solution of the same equation by using relatively
classical arguments. Here, we reconsider Eq. (1.1) under more general assumptions, which require more
sophisticated arguments, and prove stability results (see Theorem 2.1 below). In particular, we obtain the
generalized exponential asymptotic stability of the trivial solution. See [20, p. 158] for the definition of this
concept.
II. The main result
The following hypotheses will be required:
(i) 𝑓 ∈ 𝐶′
(𝑅+
) and 𝑓(𝑡) ≥ 0 for all 𝑡 ≥ 0
(ii) 𝑓 𝑡 𝑑𝑡 = ∞
∞
0
(iii) there exist two constants , 𝐾 ≥ 0 such that |
𝑓′
𝑡 + 𝑓2
𝑡 ≤ 𝐾𝑓 𝑡 , 𝑡 ∈ [, ∞); (2.1)
(iv) 𝛽 ∈ 𝐶′
(ℝ+
), 𝛽 decreasing, and
𝛽 𝑡 ≥ 𝛽0 > 𝐾2
, for all 𝑡 ∈ 𝑅+
(2.2)
where 𝛽0 is constant.
(v) 𝑔 is locally Lipschitzian in 𝑥
These assumptions are inspired by those in [5], but are more general.
Notice that (i) and (iii) imply that 𝑓 is uniformly bounded (see [8], Remark 2.2).
The main result of this paper is the following theorem:
Theorem 2.1. If the assumptions (i) , (iii) - (v) are fulfilled, then the null solution of (1.1) is uniformly
stable. If in addition (ii) holds, then the null solution of (1.1) is asymptotically stable
Remark 2.1. Under the assumptions (i) - (v), we cannot expect to have uniform asymptotic stability for
the null solution. Indeed, even in the case 𝑔 = 0 and 𝛽 = constant say 𝛽 𝑡 = 1 ∀ 𝑡 ∈ 𝑅+
one can construct a
fundamental matrix 𝑋(𝑡) for the corresponding first order linear differential system in (𝑥, 𝑦 = 𝑥′
) for which
𝑋 𝑡 𝑋(𝜏)−1
does not converge to zero as 𝑡 − 𝜏 → ∞ (here . denotes a matrix norm)
Proof of Theorem 2.1
we write equation (1.1) to a system
𝒙′
= 𝒚 − 𝒇 𝒕 𝒙
𝒚′
= 𝒇′
𝒕 + 𝒇 𝟐
𝒕 − 𝟏 𝒙 − 𝒇 𝒕 𝒚 − 𝒈(𝒕)𝒙 𝟐
and write it as
𝒛′
=
−𝒇(𝒕) 𝟏
−𝜷(𝒕) −𝒇(𝒕)
𝑿 +
𝟎 𝟎
𝒇′
𝒕 + 𝒇 𝟐
(𝒕) 𝟎
𝑿 +
𝟎
−𝒈(𝒕)𝒙 𝟑
2. Stability of Second Order Equation
www.iosrjournals.org 6 | Page
𝒛′
= 𝑨 𝒕 𝒛 + 𝑩 𝒕 𝒛 + 𝑭(𝒕, 𝒛), (2.3)
where
𝒛 =
𝒙
𝒚 , 𝑨 𝒕 =
−𝒇(𝒕) 𝟏
−𝜷(𝒕) −𝒇(𝒕)
, B 𝒕 =
𝟎 𝟎
𝒇′
𝒕 + 𝒇 𝟐
𝒕 𝟎
𝑭 𝒕, 𝒛 =
𝟎
−𝒈(𝒕)𝒙 𝟐
It is easily seen that our stability question reduces to the stability of the null solution 𝑧 𝑡 = 0 of
system (2.3).
Let 𝑡0 ≥ 0 be arbitrarily fixed and let 𝒛 𝒕, 𝒕 𝟎 =
𝒂(𝒕, 𝒕 𝟎) 𝒃(𝒕, 𝒕 𝟎)
𝒄(𝒕, 𝒕 𝟎) 𝒅(𝒕, 𝒕 𝟎)
, 𝒕 ≥ 𝟎
be the fundamental matrix to the linear system
𝒛′
= 𝑨 𝒕 𝒛 (2.4)
which is equal to the identity matrix for 𝑡 = 𝑡0 then
𝒂′
𝒕, 𝒕 𝟎 = −𝒇 𝒕 𝒂 𝒕, 𝒕 𝟎 + 𝒄 𝒕, 𝒕 𝟎
𝒄′
𝒕, 𝒕 𝟎 = −𝜷 𝒕 𝒂 𝒕, 𝒕 𝟎 − 𝒇(𝒕)𝒄 𝒕, 𝒕 𝟎
𝒃′
𝒕, 𝒕 𝟎 = −𝒇 𝒕 𝒃 𝒕, 𝒕 𝟎 + 𝒅 𝒕, 𝒕 𝟎 (2.5)
𝒅′
𝒕, 𝒕 𝟎 = −𝜷 𝒕 𝒃 𝒕, 𝒕 𝟎 − 𝒇 𝒕 𝒅 𝒕, 𝒕 𝟎
So, since 𝛽 is decreasing (hypothesis (iv)), the first two equations of (2.5) lead us to
𝟏
𝟐
𝜷 𝒕 𝒂(𝒕, 𝒕 𝟎) 𝟐
+ 𝒄(𝒕, 𝒕 𝟎) 𝟐 ′
≤ −𝒇(𝒕) 𝜷 𝒕 𝒂(𝒕, 𝒕 𝟎) 𝟐
+ 𝒄(𝒕, 𝒕 𝟎) 𝟐
and hence
𝜷 𝒕 𝒂(𝒕, 𝒕 𝟎) 𝟐
+ 𝒄(𝒕, 𝒕 𝟎) 𝟐
≤ 𝜷 𝒕 𝟎 𝒆
−𝟐 𝒇 𝒖 𝒅𝒖
𝒕
𝒕 𝟎 , ∀ 𝒕 ≥ 𝒕 𝟎. (2.6)
Similarly, from the last two equations of (2.5), we get
𝜷 𝒕 𝒃(𝒕, 𝒕 𝟎) 𝟐
+ 𝒅(𝒕, 𝒕 𝟎) 𝟐
≤ 𝒆
−𝟐 𝒇 𝒖 𝒅𝒖
𝒕
𝒕 𝟎 , ∀ 𝒕 ≥ 𝒕 𝟎 (2.7)
Consider 𝑧 = (𝑥, 𝑦) 𝑇
∈ 𝑅2
the norm 𝑧 = 𝛽0 𝑥2 + 𝑦2
For 𝑧0 = (𝑥0, 𝑦0) 𝑇
∈ 𝑅2
we obtain from (2.6),(2.7), and hypothesis (iv),
𝒛(𝒕, 𝒕 𝟎)𝒛 𝟎 =
𝒂 𝒕, 𝒕 𝟎 𝒙 𝟎 + 𝒃 𝒕, 𝒕 𝟎 𝒚 𝟎
𝒄 𝒕, 𝒕 𝟎 𝒙 𝟎 + 𝒅 𝒕, 𝒕 𝟎 𝒚 𝟎
≤ 𝒙 𝟎
𝟐 + 𝒚 𝟎
𝟐 × 𝜷 𝒕 𝒂 𝒕, 𝒕 𝟎
𝟐 + 𝒅 𝒕, 𝒕 𝟎
𝟐 + [𝒄(𝒕, 𝒕 𝟎) 𝟐 + 𝒅(𝒕, 𝒕 𝟎) 𝟐
≤ 𝜸 𝟏 + 𝜷(𝒕 𝟎)𝒆
− 𝒇 𝒖 𝒅𝒖
𝒕
𝒕 𝟎 𝒛 𝟎 (2.8)
where 𝛾 = max{1,
1
𝛽0
} Moreover, since
𝒛 𝒕, 𝒕 𝟎 𝒛 𝒔, 𝒕 𝟎
−𝟏 𝟎
𝟏
=
𝜼(𝒕, 𝒔, 𝒕 𝟎)
𝝁(𝒕, 𝒔, 𝒕 𝟎)
, ∀ 𝒕 ≥ 𝒔 ≥ 𝒕 𝟎 ≥ 𝟎
Satisfies system (2.4), we deduce as before
𝒛 𝒕, 𝒕 𝟎 𝒛 𝒔, 𝒕 𝟎
−𝟏 𝟎
𝟏
= 𝜷 𝟎 𝜼(𝒕, 𝒔, 𝒕 𝟎) 𝟐 + 𝝁(𝒕, 𝒔, 𝒕 𝟎) 𝟐
≤ 𝜷(𝒕)𝜼(𝒕, 𝒔, 𝒕 𝟎) 𝟐 + 𝝁(𝒕, 𝒔, 𝒕 𝟎) 𝟐
≤ 𝜷(𝒔)𝜼(𝒕, 𝒔, 𝒕 𝟎) 𝟐 + 𝝁(𝒕, 𝒔, 𝒕 𝟎) 𝟐 𝒆− 𝒇 𝒖 𝒅𝒖
𝒕
𝒔
= 𝒆− 𝒇 𝒖 𝒅𝒖
𝒕
𝒔 , 𝒕 ≥ 𝒔 ≥ 𝒕 𝟎 ≥ 𝟎, (2.9)
Let us prove the first part of Theorem 2.1. Consider 𝑧0 ≠ 0 with 𝑧0 small enough, 𝑡0 ≥ 0, and let us denote
by 𝑧(𝑡, 𝑡0 𝑧0) the unique solution of (2.3) which is equal to 𝑧0 for 𝑡 = 𝑡0. By hypotheses (i) and (v), 𝑧(𝑡, 𝑡0 𝑧0) is
defined on a maximal right interval, say, [𝑡0, 𝑙) and satisfies the following integral equation
𝒛 𝒕, 𝒕 𝟎, 𝒛 𝟎 = 𝐳 𝐭, 𝒕 𝟎 𝒛 𝟎 + 𝒛(𝒕, 𝒕 𝟎)
𝒕
𝒕 𝟎
𝒛(𝒔, 𝒕 𝟎)−𝟏
𝑩 𝒔 𝒛 𝒔, 𝒕 𝟎, 𝒛 𝟎 + 𝑭 𝒔, 𝒛 𝒕, 𝒕 𝟎, 𝒛 𝟎 𝒅𝒔 (2.10)
for all 𝑡 ∈ [𝑡0, 𝑙). From (2.8)-(2.10) we infer that
𝒛 𝒕, 𝒕 𝟎, 𝒛 𝟎 ≤ 𝜸 𝟏 + 𝜷(𝒕 𝟎) 𝒛 𝟎 𝒆
− 𝒇 𝒔 𝒅𝒖
𝒕
𝒕 𝟎
+ 𝒆− 𝒇 𝒖 𝒅𝒖
𝒕
𝒔 𝒇′
𝒔 + 𝒇 𝟐
(𝒔) 𝒙(𝒔, 𝒕 𝟎, 𝒛 𝟎) + 𝒈 𝒔 𝒙 𝟐
(𝒔) 𝒅𝒔
𝒕
𝒕 𝟎
, ∀ 𝒕 ∈ 𝒕 𝟎, 𝒍 (2.11)
we infer from (2.11) that
3. Stability of Second Order Equation
www.iosrjournals.org 7 | Page
𝒛 𝒕, 𝒕 𝟎, 𝒛 𝟎 ≤ 𝜸 𝟏 + 𝜷(𝒕 𝟎) 𝒛 𝟎 + 𝑫 𝒛(𝒔, 𝒕 𝟎, 𝒛 𝟎) 𝒅𝒔
𝒕
𝒕 𝟎
, ∀ 𝒕 ∈ 𝒕 𝟎, 𝒍 (2.12)
with a positive constant D and
𝒛 𝒕, 𝒕 𝟎, 𝒛 𝟎 ≤ 𝜸 𝟏 + 𝜷(𝒕 𝟎)𝒆 𝑫𝒉
𝒛 𝟎 , ∀ 𝒕 ∈ 𝒕 𝟎, 𝒍 (2.13)
Thus 𝑧 𝑡, 𝑡0, 𝑧0 as well as 𝑧′
𝑡, 𝑡0, 𝑧0 are bounded on 𝑡0, 𝑙 and so 𝑧 𝑡, 𝑡0, 𝑧0 can be extended to the right of
𝑙. This fact contradicts the maximality of l. Therefore 𝑧 𝑡, 𝑡0, 𝑧0 exists on 𝑡0, 𝑙 with 𝑙 > .
Now, we assume < 𝑙 < ∞. We are going to find an estimate for 𝑧 𝑡, 𝑡0, 𝑧0 on the interval , 𝑙 . This time,
our hypothesis (iii) comes into play. We have for all 𝑡 ∈ , 𝑙
𝒛 𝒕, 𝒕 𝟎, 𝒛 𝟎 ≤ 𝜸 𝟏 + 𝜷 𝒉 𝒛 𝒉, 𝒕 𝟎, 𝒛 𝟎 𝒆− 𝒇 𝒖 𝒅𝒖
𝒕
𝒉 + 𝒆− 𝒇 𝒖 𝒅𝒖
𝒕
𝒔 𝑲𝒇 𝒔 𝒙 𝒔, 𝒕 𝟎, 𝒛 𝟎 𝒅𝒔
𝒕
𝒉
+ 𝒆− 𝒇 𝒖 𝒅𝒖
𝒕
𝒔 𝒈 𝒔 𝒙 𝟐
(𝒔, 𝒕 𝟎, 𝒛 𝟎)𝒅𝒔
𝒕
𝒉
(2.14)
From (2.14) it follows that
𝒛 𝒕, 𝒕 𝟎, 𝒛 𝟎 ≤ 𝜸 𝟏 + 𝜷 𝒉 𝒛 𝒉, 𝒕 𝟎, 𝒛 𝟎 𝒆− 𝒇 𝒖 𝒅𝒖
𝒕
𝒉 + 𝒆− 𝒇 𝒖 𝒅𝒖
𝒕
𝒔 𝒇′
𝒔 + 𝒇 𝟐
(𝒔) 𝒙(𝒔, 𝒕 𝟎, 𝒛 𝟎) 𝒅𝒔
𝒕
𝒉
+ 𝒆− 𝒇 𝒖 𝒅𝒖
𝒕
𝒔 𝒈 𝒔 𝒙 𝟐
(𝒔, 𝒕 𝟎, 𝒛 𝟎) 𝒅𝒔
𝒕
𝒉
≤ 𝜸 𝟏 + 𝜷 𝒉 𝒛 𝒉, 𝒕 𝟎, 𝒛 𝟎 𝒆− 𝒇 𝒖 𝒅𝒖
𝒕
𝒉
+ 𝒆− 𝒇 𝒖 𝒅𝒖
𝒕
𝒔
𝒕
𝒉
𝑲𝒇 𝒔 𝒙 𝒔, 𝒕 𝟎, 𝒛 𝟎 𝒅𝒔 + 𝑴𝒇 𝒔 𝒙 𝒔, 𝒕 𝟎, 𝒛 𝟎
𝟑
𝒅𝒔
= 𝝑(𝒕) , ∀ 𝒕 ∈ [𝒉, 𝒍) (2.15)
Then by (2.15)
𝝑′
(𝒕) ≤
𝑲+𝑴𝝑 𝒕
𝜷 𝟎
− 𝟏 𝝑(𝒕)𝒇(𝒕) where 𝑴 ∈ 𝟎, 𝜷 𝟎 − 𝑲
𝝑 𝒉 = 𝜸 𝟏 + 𝜷(𝒉) 𝒛 𝒉, 𝒕 𝟎, 𝒛 𝟎
𝒛 𝒕, 𝒕 𝟎, 𝒛 𝟎 ≤ 𝝑 𝒕 ≤ 𝝑 𝒉 𝒆
𝑲+𝑴𝝑 𝒉
𝜷 𝟎
−𝟏 𝒇 𝒖 𝒅𝒖
𝒕
𝒉
, ∀𝒕 ∈ [𝒉, 𝒍] (2.16)
From (2.16) we can see that 𝑧 𝑡, 𝑡0, 𝑧0 is bounded. Since 𝑧′
𝑡, 𝑡0, 𝑧0 is also bounded, it follows that 𝑙 = ∞.
Now, for 𝜀 > 0 we denote
𝜹 = 𝜹 𝜺 =
𝜺𝒆−𝑫𝒉
𝜸 𝟐 (𝟏 + 𝜷(𝟎)(𝟏 + 𝜷(𝒉)
From (2.13) it follows that 𝑧 𝑡, 𝑡0, 𝑧0 ≤
𝜀
𝛾 1+𝛽()
for all 𝑡 ∈ [𝑡0, ] provided that
𝑧0 < 𝛿. Therefore, by 𝑀 ∈ 0, 𝛽0 − 𝐾 and (2.16), 𝑧 𝑡, 𝑡0, 𝑧0 ≤ 𝜗 < 𝜀 for all 𝑡 ≥ .
Summarizing, if 0 ≤ 𝑡0 ≤ , the solution 𝑧 𝑡, 𝑡0, 𝑧0 starting from any point, 𝑧0, with 𝑧0 < 𝛿 exists on
[𝑡0, ∞) and satisfies 𝑧 𝑡, 𝑡0, 𝑧0 < 𝜀 for all 𝑡 ≥ 𝑡0.
If 𝑡0 ≥ , then analogously we obtain that 𝑙 = ∞ and
𝒛 𝒕, 𝒕 𝟎, 𝒛 𝟎 ≤ 𝝑 𝟏 + 𝜷(𝒕 𝟎) 𝒛 𝟎 𝒆
𝑲+𝑴𝝑(𝒉)
𝜷 𝟎
−𝟏 𝒇 𝒖 𝒅𝒖
𝒕
𝒕 𝟎 , ∀𝒕 ∈ [𝒉, ∞) (2.17)
Therefore, with the same . as before, 𝑧0 < 𝛿 implies again that 𝑧 𝑡, 𝑡0, 𝑧0 < 𝜀 for all 𝑡 ≥ 𝑡0 Hence
the null solution is uniformly stable. If in addition (ii) is fulfilled, then by (2.16), and (2.17) it follows
that the null solution to (1.1) is asymptotically stable. The proof of Theorem 2.1 is complete.
Remark 2.2 If 𝑓 satisfies (i) - (iii), then 𝑓 𝑡 > 0 for all 𝑡 ≥ . Let us assume, by contradiction, that 𝑓 𝑡1 =
0 for some 𝑡1 ≥ . Then, one can prove that 𝑓 𝑡 = 0 for 𝑡 ≥ 𝑡1 Indeed, if 𝑓 𝑡2 > 0 for some 𝑡2 > 𝑡1
then the function 𝑢 =
1
𝑓
is well defined on the maximal interval containing 𝑡2 on which 𝑓 > 0 , say
(𝑐, 𝑑), and satisfies the inequality
𝒖′
𝒕 + 𝑲𝒖 𝒕 − 𝟏 ≥ 𝟎, 𝒕 ∈ (𝒄, 𝒅).
This implies that
𝒅
𝒅𝒕
𝒆 𝑲𝒕
𝒖 𝒕 −
𝟏
𝑲
≥ 𝟎, 𝒕 ∈ 𝒄, 𝒕 𝟐
i.e., the function 𝑡 → 𝑒 𝐾𝑡
𝑢 𝑡 −
1
𝐾
decreasing on 𝑐, 𝑡2 , but this is impossible since the limit of this function
is ∞ as 𝑡 → 𝑐+
. Thus, 𝑓 𝑡 = 0 for all 𝑡 ≥ 𝑡1, which contradicts (ii). Therefore, we have proved that indeed
𝑓 𝑡 > 0 for all 𝑡 ≥ Consequently, the function 𝑝 𝑡 = 𝑓 𝑠 𝑑𝑠
∞
0
is strictly increasing, at least for
𝑡 ≥ . By the above proof we have that the null solution is generalized exponentially asymptotically
stable with this 𝑝 𝑡 .
4. Stability of Second Order Equation
www.iosrjournals.org 8 | Page
Remark 2.3. if, 𝛽 𝑡 = 1, 𝑡 ∈ 𝑅+
the fundamental matrix Z(𝑡, 𝑧0) can be determined explicitly
𝒛 𝒕, 𝒛 𝟎 = 𝒆
− 𝒇 𝒖 𝒅𝒖
𝒕
𝒕 𝟎
𝐜𝐨𝐬(𝒕 − 𝒕 𝟎) 𝐬𝐢𝐧(𝒕 − 𝒕 𝟎
−𝐬𝐢𝐧(𝒕 − 𝒕 𝟎 𝐜𝐨𝐬(𝒕 − 𝒕 𝟎
In general, this is not possible, so in our proof we had to get estimates without having an explicit form of 𝑧(𝑡, 𝑡0)
III. Conclusion :
These stability results can be extended to the case when the differential equation is vectorial equation.
More precisely, assume that 𝑓 and 𝛽 are scalar functions satisfying the
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