The existence of positive solutions is considered for a fractional q-difference equation with pLaplacian operator in this article. By employing the Avery-Henderson fixed point theorem, a new result is obtained
for the boundary value problems.
Asymptotic Behavior of Solutions of Nonlinear Neutral Delay Forced Impulsive ...IOSR Journals
Sufficient conditions are obtained for every solution of first order nonlinear neutral delay forced
impulsive differential equations with positive and negative coefficients tends to a constant as t ∞.
Mathematics Subject Classification [MSC 2010]:34A37
Positive and negative solutions of a boundary value problem for a fractional ...journal ijrtem
: In this work, we study a boundary value problem for a fractional
q, -difference equation. By
using the monotone iterative technique and lower-upper solution method, we get the existence of positive or
negative solutions under the nonlinear term is local continuity and local monotonicity. The results show that we
can construct two iterative sequences for approximating the solutions
Lyapunov-type inequalities for a fractional q, -difference equation involvin...IJMREMJournal
In this paper, we present new Lyapunov-type inequalities for a fractional boundary value problem of
fractional
q, -difference equation with p-Laplacian operator. The obtained inequalities are used to obtain a
lower bound for the eigenvalues of corresponding equations.
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 .
Asymptotic Behavior of Solutions of Nonlinear Neutral Delay Forced Impulsive ...IOSR Journals
Sufficient conditions are obtained for every solution of first order nonlinear neutral delay forced
impulsive differential equations with positive and negative coefficients tends to a constant as t ∞.
Mathematics Subject Classification [MSC 2010]:34A37
Positive and negative solutions of a boundary value problem for a fractional ...journal ijrtem
: In this work, we study a boundary value problem for a fractional
q, -difference equation. By
using the monotone iterative technique and lower-upper solution method, we get the existence of positive or
negative solutions under the nonlinear term is local continuity and local monotonicity. The results show that we
can construct two iterative sequences for approximating the solutions
Lyapunov-type inequalities for a fractional q, -difference equation involvin...IJMREMJournal
In this paper, we present new Lyapunov-type inequalities for a fractional boundary value problem of
fractional
q, -difference equation with p-Laplacian operator. The obtained inequalities are used to obtain a
lower bound for the eigenvalues of corresponding equations.
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 .
Gamma Function mathematics and history.
Please send comments and suggestions for improvements to solo.hermelin@gmail.com. Thanks.
More presentations on different subjects can be found on my website at http://www.solohermelin.com.
Research Inventy : International Journal of Engineering and Scienceinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
ملزمة الرياضيات للصف السادس التطبيقي الفصل الاول الاعداد المركبة 2022anasKhalaf4
طبعة جديدة ومنقحة
حل تمارين الكتاب
شرح المواضيع الرياضية بالتفصيل وبأسلوب واضح ومفهوم لجميع المستويات
حلول الاسألة الوزارية
اعداد الدكتور أنس ذياب خلف
email: anasdhyiab@gmail.com
Oscillation results for second order nonlinear neutral delay dynamic equation...inventionjournals
In this paper, we establish sufficient conditions for the oscillation of solutions of second order neutral delay dynamic equations [r (t )( x (t ) p (t ) x ( (t ))) ] q (t ) f ( x ( (t ))) = 0 on an arbitrary time scale 핋.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
SOLVING BVPs OF SINGULARLY PERTURBED DISCRETE SYSTEMSTahia ZERIZER
In this article, we study boundary value problems of a large
class of non-linear discrete systems at two-time-scales. Algorithms are given to implement asymptotic solutions for any order of approximation.
On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
Presentation.
9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 12, 2012.
The 24th International Conference on Operator Theory, Timisoara, July 3, 2012.
This presentation gives example of "Calculus of Variations" problems that can be solved analytical. "Calculus of Variations" presentation is prerequisite to this one.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Gamma Function mathematics and history.
Please send comments and suggestions for improvements to solo.hermelin@gmail.com. Thanks.
More presentations on different subjects can be found on my website at http://www.solohermelin.com.
Research Inventy : International Journal of Engineering and Scienceinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
ملزمة الرياضيات للصف السادس التطبيقي الفصل الاول الاعداد المركبة 2022anasKhalaf4
طبعة جديدة ومنقحة
حل تمارين الكتاب
شرح المواضيع الرياضية بالتفصيل وبأسلوب واضح ومفهوم لجميع المستويات
حلول الاسألة الوزارية
اعداد الدكتور أنس ذياب خلف
email: anasdhyiab@gmail.com
Oscillation results for second order nonlinear neutral delay dynamic equation...inventionjournals
In this paper, we establish sufficient conditions for the oscillation of solutions of second order neutral delay dynamic equations [r (t )( x (t ) p (t ) x ( (t ))) ] q (t ) f ( x ( (t ))) = 0 on an arbitrary time scale 핋.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
SOLVING BVPs OF SINGULARLY PERTURBED DISCRETE SYSTEMSTahia ZERIZER
In this article, we study boundary value problems of a large
class of non-linear discrete systems at two-time-scales. Algorithms are given to implement asymptotic solutions for any order of approximation.
On Twisted Paraproducts and some other Multilinear Singular IntegralsVjekoslavKovac1
Presentation.
9th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial, June 12, 2012.
The 24th International Conference on Operator Theory, Timisoara, July 3, 2012.
This presentation gives example of "Calculus of Variations" problems that can be solved analytical. "Calculus of Variations" presentation is prerequisite to this one.
For comments please contact me at solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Quantum Annealing for Dirichlet Process Mixture Models with Applications to N...Shu Tanaka
Our paper entitled “Quantum Annealing for Dirichlet Process Mixture Models with Applications to Network Clustering" was published in Neurocomputing. This work was done in collaboration with Dr. Issei Sato (Univ. of Tokyo), Dr. Kenichi Kurihara (Google), Professor Seiji Miyashita (Univ. of Tokyo), and Prof. Hiroshi Nakagawa (Univ. of Tokyo).
http://www.sciencedirect.com/science/article/pii/S0925231213005535
The preprint version is available:
http://arxiv.org/abs/1305.4325
佐藤一誠さん(東京大学)、栗原賢一さん(Google)、宮下精二教授(東京大学)、中川裕志教授(東京大学)との共同研究論文 “Quantum Annealing for Dirichlet Process Mixture Models with Applications to Network Clustering" が Neurocomputing に掲載されました。
http://www.sciencedirect.com/science/article/pii/S0925231213005535
プレプリントバージョンは
http://arxiv.org/abs/1305.4325
からご覧いただけます。
21st Mediterranean Conference on Control and Automation
The present paper is a survey on linear multivariable systems equivalences. We attempt a review of the most significant types of system equivalence having as a starting point matrix transformations preserving certain types of their spectral structure. From a system theoretic point of view, the need for a variety of forms of polynomial matrix equivalences, arises from the fact that different types of spectral invariants give rise to different types of dynamics of the underlying linear system. A historical perspective of the key results and their contributors is also given.
Partial differentiation, total differentiation, Jacobian, Taylor's expansion, stationary points,maxima & minima (Extreme values),constraint maxima & minima ( Lagrangian multiplier), differentiation of implicit functions.
On Convergence of Jungck Type Iteration for Certain Contractive Conditionsresearchinventy
In this article we prove the strong convergence result for a pair of nonself mappings using Jungck S- iterative scheme in Convex metric spaces satisfying certain contractive condition. The results are the generalization of some existing results in the literature
The effect of functionalized carbon nanotubes on thermalmechanical performanc...IJRTEMJOURNAL
The new approaches for preparing nanocomposite coating by modificated carbon nanonotubes
(CNTs) and epoxy resin was done in the study. thermal-mechanical performance of nanocomposite coating was
investigated and the results were reported in this paper. The physic-chemical techniques such as Differential
scanning calorimetry (DSC) and Thermal gravimetric analysis (TGA) were used to characterize the thermal
performance of Epoxy nanocomposite coating. The test techniques for mechanical properties of paint coating as
adhesion, hardness, impact resistance and bending strength were employed in the work. The results indicated
that CNTs were dispersed in epoxy coating with only ratio of 0.1 wt% enhanced the Glass Transition
Temperature (Tg), decomposition temperature of epoxy coating and improved mechanical properties
significantly. Also functionalized CNTs can be reinforced thermal-mechanical of the epoxy coating better than
neat CNTs.
Study of desalination processes of seawater from the desalination plant of La...IJRTEMJOURNAL
The use of water for food purposes requires excellent physicochemical quality. To contribute to
the control of water quality. Water treated by reverse osmosis is aggressive and demineralize can not be used
directly as a source of drinking water. The objective of this work is to study, physics-chemical analyzes of raw
water, pretreated osmosis and treated (permeate) and produced water (reservoir) at the desalination plant of
seawater Laayoune (SDL), located in southern Morocco. For this, we have followed several qualitative
parameters such as pH, conductivity, turbidity
Multi products storage using randomnessIJRTEMJOURNAL
The following Project shows the benefits of a research established into a multi-products
warehouse belongs to an automotive industry supplier. The main goal was applied a tool recognizing the rules
for distribution and material storage. Once the research was completed, the benefits were, the idle times
reduction per hours/week by the two initial processes. The politics for storage assignment and location, propose
a system to improve the space into this areain order to avoid material management and flow issues. It is
important to mention, the system proposed could be applied into warehouses with storage size and space
restricted by sorting area, also different material types, production settings and physical specifications for
which set warehouses with traditional management of distribution without slack, involves lack of materials,
pieces without records, incorrect location assigned, stock error.
Existence results for fractional q-differential equations with integral and m...IJRTEMJOURNAL
This paper concerns a new kind of fractional q-differential equation of arbitrary order by
combining a multi-point boundary condition with an integral boundary condition. By solving the equation which
is equivalent to the problem we are going to investigate, the Green’s functions are obtained. By defining a
continuous operator on a Banach space and taking advantage of the cone theory and some fixed-point theorems,
the existence of multiple positive solutions for the BVPs is proved based on some properties of Green’s functions
and under the circumstance that the continuous functions f satisfy certain hypothesis. Finally, examples are
provided to illustrate the results.
A study on financial aspect of supply chain managementIJRTEMJOURNAL
The more common approaches used in the SCM consider only the physical logistic operations
and ignore the financial aspects of the supply chain. The main objective to incorporate financial aspects in
supply chain management is to strengthen managerial decisions concerning financial flows in supply chains,
while empirical knowledge about financial supply chain management (FSCM) is in its early stages. This paper
presents a model for FSCM which financial planning in addition to operation planning is decided in it. The
main contribution of this paper is to define two approaches for Financial Supply Chain Management and to
compare them. This financial approaches are: Traditional financial approach and new financial approach.
Traditional financial approach integrates physical goods flows and financial flows. New financial approach
considers in making decisions other financial indicators such as market to book value, liquidity ratios, capital
structure ratios, and return on equity, sales margin, turnover ratios and stock security ratios, among others.
Moreover, the new approach applies the change in equity instead of the traditional approach measures of profit
as the objective function to be maximized in the presented model. To show the attributes of the presented
approaches, the results of the new approach and the traditional approach is compared. The findings indicate
that the traditional approach leads to lower change in equity compared to the financial approach. Also, the
results clearly reveal the better improvement of using the new approach over the traditional approach, and
convince the decision makers to take advantage of the new approach.
Rural Livelihood and Food Security: Insights from Srilanka Tapu of Sunsari Di...IJRTEMJOURNAL
Food security is the foremost need of every human society. It is a fundamental right and
government responsibility but still food insecurity is prevalent in rural areas of least developed nations. To cope
with food insecurity, undertaking diverse income generating activities is common as well as key strategy adopted
by rural people. The objective of this study is to assess rural livelihood and food security status of a remote island
named Srilanka Tapu of Sunsari district. A random sampling technique was used to collect primary data from 40
rural household heads using semi-structured questionnaire. Descriptive methods were used for analyzing. The
findings revealed that the food security situation of the Tapu is insecure. Most basic infrastructures and social
services needed for people livelihood such as road, electricity sufficient food availability, education, healthcare,
sanitation, etc. were found to be extremely poor. Most of the households are small scale farmers involving
themselves in diverse livelihood activities which are mostly temporary, low-skilled and low paying. However,
people are fulfilling their food needs at every cost but are highly vulnerable to food insecurity. Also, their lives
security is equally vulnerable because of disastrous Koshi River flooding which occurs every year in the Tapu.
The findings therefore critically suggest that food security of remote and vulnerable human settlements should be
at top priority in policy formulation and implementation level. The study also recommends a need for an in-depth
research for making evidence based policy interventions for improvement of diversify rural livelihood along with
sustainable environment
With mounting concerns over the state of our planet, there is continuing demand that chemists
and chemical engineers should develop greener chemical processes and products. In the 1990s, with the
growing awareness of the hazardous impacts of the chemical industry, the green chemistry revolution was
launched by American chemists Paul T. Anastas and John Warner. Green chemistry is the kind of chemistry that
seeks to minimize pollution, conserve energy, and promote environmentally friendly production. This paper
provides a brief introduction to green chemistry.
Assessment of Building Failure: The Case of Saint Thomas’s Anglican Church, A...IJRTEMJOURNAL
There have been incessant reports of the collapse of buildings resulting in the loss of lives and
properties globally. However, there has been a dearth of information regarding any findings about the collapse
of building structures. An extensive study of causes of selected building collapse in Nigeria and abroad is carried
out in this work by visiting some locations of building collapse, reading journals and newspaper articles on
structural defects and testing rubbles collected from collapsed areas. This study therefore examined the general
causes of the collapse of some buildings particularly the reasons for the collapse of Saint Thomas’s 2-storey
Church Hall, Akure. Laboratory testing was carried out in this study to investigate the causes of collapse using
samples from the site of the collapsed building. An appraisal of the structural drawings of the collapsed building
was also investigated. Findings revealed that the building collapsed due to poor design, bad construction
materials and inadequate supervision. The paper concludes that buildings collapse can be reduced in Nigeria by
avoiding all. It recommended use of only duly registered professionals in the building industry for construction.
Data warehousing is a technique for collecting and managing data from multiple internal and
external sources to provide meaningful business insights. Data warehouses are designed to give a long-range
view of data over time and provide a decision support system environment. They are a vital component of
business intelligence, which is designed for data analysis and reporting. They are used to provide greater
insight into the performance of a business. This paper provides a brief introduction on data warehousing
Resource recycling and waste-to-energy: The cornerstones of circular economyIJRTEMJOURNAL
"Circular Economy" is the pursued goal of sustainable development of mankind for the 21st
century. In short, the fundamental spirit of circular economy is the concept of "Zero Waste". The example used
in our daily lives means 100% of waste treatment, leaving no trace. At this time, it would be an ideal goal that
the waste could be fully recovered into available raw materials or energies. In particular, "waste-to-energy" is
a key factor, because all the wastes are almost related to energy. Resource recycling of waste metal from the
household garbage is the best example. When smelting metals, the refining industry needs to reduce the metal
oxides (mineral materials) to metals, such as steel, aluminium, copper, etc. The reduction processes consume
considerable portion of energy for the entire smelting process, for example, 70.6% for steel and 77.4% for
aluminium. However, if the waste metallic products can be fully recovered, as long as by melting and reshaping,
the original oxide metal reduction processes that consume a lot of energy can be avoided. On the other hand,
when the general garbage cannot be recovered as a resource, they can be converted into fuel or electricity by
biological or thermal treatment. Another more important human waste utilization is the waste paper recycling.
The production of one tonne of raw pulp emits about 6 tonnes of carbon, consuming about 100 cubic meters of
water, using about 200 kilograms of chemical raw materials, and draining 300 tonnes of toxic waste water. The
entire papermaking process is how terrible environmental pollution! The recycled pulp of one tonne can save
energy 10-13GJ.The proportion of paper waste in Taiwan 2015 is 34.69% and the estimated amount is 2.5
million tonnes. If the paper waste could be fully recycled, it could save energy about 0.725 million kloe (kilolitre oil equivalent). In other words, it virtually reduces Taiwan's oil imports of 4.56 million barrels and CO2
emissions of 2.5 million tonnes annually.
Survivin Immunoreactivity in the Gastric Mucosa of Rats Feedind with Carpet S...IJRTEMJOURNAL
Survivin has been studied many times because of its overexpression in several types of cancer
including lung, kidney, skin, endometrium, stomach, colon, breast, prostate, over, hematologic, head and neck
cancers, histopathology features and polymorphisms in the promoter region which belongs to the inhibitör of
apoptosis gene family by researchers. There is no study of survivin immunoreactivity in the gastric mucosa of the
rats fed with carpet shell clam grown in the Dardanelles. In this study, it was aimed to investigate the effects of
carpet shell clam fed rats on survivin production in the gastric mucosa. The carpet shell clam given as food to the
rats were removed from the Dardanelles Çardak region. Four groups of rats are included in the study, group 1
(n=6), control group fed with standard rat food, group 2 (n=6), 75% carpet shell clam and 25% standard rat
food daily, group 3 (n=6), 75% carpet shell clam and 25% standard rat food every two days, group 4 (n=6), 75%
carpet shell clam and 25% standard rat food every three days. To detect survivin localization in the tissues, the
LAB-SA Detection System was used. Survivin immunoreactivity was detected of epithelial cells in the gastric
mucosa of rats fed with carpet shell clam. After the immunohistochemical staining processing all gastric tissue
samples are evaluated in terms of survivin immunoreactivity with light microscopy and image analysis software.
Survivin immunoreactivity was detected 0% in the first group, 83.33% in the second group, 61.83% in the third
group and 32.67% in the fourth group. There was statistically significant difference between the survivin
immunoreactivity in the gastric gland cells of the rats in the experimental and control groups (p> 0.05). Survivin
production in the gastric mucosa of rats suggests that consumption of carpet shell clam may cause tissue damage.
Security and Crime Management in University Libraries in NigeriaIJRTEMJOURNAL
Security and prevention of crime in university library is very paramount duties of librarian. The
survival of a library depends to a large extend on how secured its collections are, security of library resources
constitutes serious challenge facing university libraries in Nigeria. The paper, therefore, investigates security and
crime management in university libraries in Nigeria using university of Jos and university of Ilorin libraries. The
study adopted a descriptive survey method. The population of the study comprised 108 library personnel and
16,012 registered library users in two university libraries. While the sample size consisted of all the 108 library
personnel, and 2% of the registered users to make a total of 428 respondents. Questionnaire and interview with
the university librarians of the selected university libraries were the instruments used for data collection. Data
were analysed using frequency distribution and percentages. Results revealed that security breaches included
stealing/theft of library materials, mutilation of library materials, and non-return of borrowed items. It also
showed inadequate funding, selfish interest of the culprits and lack of institutional security policy in the library.
Base on the findings, that staff security training, electronic security system should be introduced and improve
funding of university libraries among others. Recommendation orientation of users and staff should be done from
time to time in university libraries to mention but few.
Influence of heat treatment on Vitamin C Levels in Oyster MushroomIJRTEMJOURNAL
The study was conducted to investigate the influence of heat treatment during drying process of
Oyster mushroom in the tropics. Mushroom growing is carried out under carefully controlled conditions mostly
in bulk in specific designed tunnels with aerated floors. There are two main purposes, firstly pasteurization; to
free the compost from undesirable microbes and pests and secondly conditioning; to become mushroom specific
by getting clear of ammonia and free of readily available carbohydrates. Through proper manipulation of
temperature and ventilation these two primary objectives are accomplished. Mushrooms have been identified as
an underutilized crop in Africa, with many nutritive and health benefits. It does not require much land and
investment. However, it is highly perishable and there is need to process it to lengthen its shelf life by drying.
However, there is need to ensure that the nutrients are not lost in the process. It is for this reason that this
project investigated the effect of drying on nutrient levels in mushroom. Vitamin C levels were monitored in the
course of drying at 80⁰C, 60⁰C, 50⁰C, 40⁰C and in direct sunlight. It was concluded that the temperature that
gave the best drying rate with minimal nutrient loss was 60⁰C. In general, more than half the Vitamin C is lost
during the range of drying temperatures investigated.
Optimization of Design Parameters for Crane Hook Using Finite Element AnalysisIJRTEMJOURNAL
The Crane hooks are very at risk segments that are regularly utilized for mechanical purposes.
In this way such segments in an industry must be produced and composed in an approach to convey most extreme
execution without failure. Failure of a crane hook essentially relies upon three central point i.e. measurement,
material, overload. The undertaking is worried towards expanding the safe load by fluctuating the cross-sectional
measurements of the four distinct segments and diverse materials. The chose areas are square, circle, and
trapezoidal. The territory stays consistent while changing the measurements of the four unique segments. The
crane hook is demonstrated utilizing catia programming. The pressure and life investigation is finished utilizing
ANSYS 18.1 workbench. The ordinary worry along add up to misshaping, stress and life’s according to the
materials considered. It is discovered that trapezoidal cross segment yields most extreme load of 4000 KG to 5000
KG for steady cross segment zone among four cross segment.
Macroeconomic stability in the DRC: highlighting the role of exchange rate an...IJRTEMJOURNAL
This study is part of a macroeconomic approach and seeks to identify the role of the rate of
economic growth and the exchange rate in controlling the macroeconomic framework. The approaches adopted
in this paper are part of Keynesian thinking on macroeconomic stability using the macroeconomic stability
index proposed by Burnside and Dollars (2004) and A. Amine (2005). Our results argue that economic growth
is causing macroeconomic stability and that the exchange rate is negatively and significantly accounting for
macroeconomic stability in the Democratic Republic of Congo.
Reserves Estimating Carbon in Forest City District Village Bongohulawa GorontaloIJRTEMJOURNAL
The estimation of Carbon stock and carbon sink in the City Forest of Bongohulawa village,
Regency of Gorontalo (Guided by. The research was aimed to know volume growth of trees planted in the
Village District Bongohulawa Gorontalo, to calculate the volume and content of carbon biomass in the city
forest and green line (left-right path) and average carbon sequestration/tree/species. Research was conducted
in village of Bongohulawa during 4 (fourth) month; started from March until June 2011. The execution of data
collecting [of] was performed within this research area-location through observation and measurement of trees
and forest stand. For green line research area 100% inventory was upllied and for City Forest line plot
sampling was implemented. For city forest sample plots measurement was conducted in 10 sample units (each
unit sampling of 0.25 ha). Tree diameter, tree hight (total and commercial hight) and crown diameter of all tree
species within research line (green line) and research plots (city forest) was measured. Based on the research
data and its calculation, the results show that: Casuarina junghuhiana can store more carbon than other tree
species. From the inventory conducted in 3 km of green line along the road (6 meters width observations) of the
village Bongohulawa, 366 trees (consist of 7 tree species) were measured. Those tree species namely Casuarina
junghuiana 102 trees, sandalwood 46 trees, mango 7 trees, jackfruit 6 trees, Albizia 1 tree, mahogany 202 trees,
headland 2 trees. Crown cover of those tree species is 3032.54 m2
. The result of calculation also indicated that
Casuarina has higher carbon stock than other tree species that is 33.56 tons (equal with 52% of total crbon
stock). Further calculation indicated that during the period of 19 years (since 1992) Casuarina can strocked
carbon average of 1.77 tons/year. The average diameter increment of individual Casuarina tree species is about
1.72 cm/year. Furthermore, for Swietenia magrophilla King, with an average diameter increment of 1.40
cm/year, the leaves of this tree species can absorbed carbon of 18.1233 tons within green line of both sides of
the road. For research plots within City Forest which located in the valey the results of the research show that
the crown cover of 124 trees is about 1,359.67 m2, then carbon absorbtion is about 0.15 ton/tree or about 7.8
kg/tree/year. Within the research area of City Forest (located both in the valey and hill) totally there are 1,353
trees (consist of 13 tree species) and carbon absorption of the canopy is about 25.521 tons. Further calculation
results also indicated that carbon absorption of small trees ( poles) is about 25.521 tons and for sapling is
about 78.163 tons or 39,0815 tons/ha then fionally for mature trees is about 39.813 tons or 15,925 tons/ha.
An Analysis of Tourism Competitiveness Index of Europe and Caucasus: A Study ...IJRTEMJOURNAL
This study aims to find the association-ship between the Regional Rank of the Travel and
Tourism Competitiveness Index and its Indicators in 37 European countries. The cross-sectional data of the 37
European countries are collected from the World Economic Forum report- 2015. The statistical software
package, SPSS v. 20.0 is used to analyze the data. ANOVA (Analysis of Variance), Multi-co-linearity, Multiple
Regression, and Residual Analysis are the tools used to analyze to achieve out the objective of the study. RR:
Regional Rank of the Travel and Tourism Competitiveness Index is used as the dependent variable and TI:
Tourism Services Infrastructure, GP: Ground & Port Infrastructure, BE: Business Environment, PT:
Prioritization of Travel and Tourism, and CR: Cultural resources & business travel are used as the independent
variables. It is found that there is an inverse relationship between the dependent variable and all the
independent variables along with the statistical significance. It is recommended that the governments of the
European countries and the respective agents of these countries should be made aware of learning the findings
of this study to promote their countries which can be victorious in lowering their Regional Rank of the Travel
and Tourism Competitiveness Index.
Translation Errors of Public Signs in English Subtitle: Residents’ Poor Forei...IJRTEMJOURNAL
China is in an Age of Economy Thriving and Technology Advancing. This strike
increasing international visitors. In the foreigners, very few of them are able to communicate in
Chinese, which means that it is significant to provide accurate information to the foreign friends by
their understandable codes. For instance, in a hotel, a foreigner needs to know which way to go for
their daily activities without enquiring at the reception desk. These requirements are served by public
signs, e.g. the location of a canteen. Actually, this service is a challenge of Chinese people’s English
level. In recent years, as a lack of contextually linguistic and cultural knowledge, there are some
errors of translation on public English signs, resulting in some inconvenience to the oversea
travelers. This paper will analyses these problems in root and then advance prospective resolutions.
What are the determinants of the non-reimbursement for SMEs in Central Africa...IJRTEMJOURNAL
This article aims to determine the factors that are the cause of the non-repayment of credits
received from financial institutions by Cameroonian SMEs. This choice is sometimes. This non-repayment is
often caused by factors related to the environment and the functioning of SMEs. It aims to analyze and highlight
the factors that put Cameroonian SMEs in a situation of inability to repay the receivables received from
financial institutions. To achieve this goal, we opted for a mixed approach: Inductive (exploration on the
ground) and hypothetico deductive. To do this, we first analyzed the content of the interviews conducted with 15
SME managers and owners and tested data collected from a questionnaire administered face-to-face with 185
Cameroonian SMEs. . We used descriptive analysis and explanatory analysis. Our results show that the tax rate,
the mismanagement of managers, poor accounting and unforeseen situations have a significant positive
influence on the non-repayment of loans, while the age and size of SMEs exert significant negative influence on
the non-repayment of loans by Cameroonian SMEs.
Multivariate regression methods with infrared spectroscopy to detect the fals...IJRTEMJOURNAL
Recently, food safety and guaranteed of food marks have become more important subjects of
foodstuff production and the marketing of processed foods. This paper demonstrates the ability of Mid Infrared
spectroscopy coupled with multivariate regression tools to detect vegetable butter (as adulterant) in a binary
mixture with traditional cow’s butter. Blends of traditional cow’s butter with different percentages of vegetable
butter were measured using Attenuated Total Reflectance-Fourier Transform Mid Infrared Spectroscopy (ATRFTMIR). Spectral and reference data were firstly analyzed by principal component analysis (PCA) to check
outliers samples; and improve the robustness of the prediction models to be established. Multivariate regression
methods as Principal component regression (PCR) and Partial least square regression (PLSR) were used to
establish calibration model. Excellent correlation between ATR-FTMIR analysis and studied butter blends was
obtained R2 = 0.99; with Root Mean Square Errors of Prediction < 3.04, Limit of Detection 9.12% (By PCR)
and 6.06% (by PLSR), and Relative Prediction Errors as low as 3.13.
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
6th International Conference on Machine Learning & Applications (CMLA 2024)ClaraZara1
6th International Conference on Machine Learning & Applications (CMLA 2024) will provide an excellent international forum for sharing knowledge and results in theory, methodology and applications of on Machine Learning & Applications.
NUMERICAL SIMULATIONS OF HEAT AND MASS TRANSFER IN CONDENSING HEAT EXCHANGERS...ssuser7dcef0
Power plants release a large amount of water vapor into the
atmosphere through the stack. The flue gas can be a potential
source for obtaining much needed cooling water for a power
plant. If a power plant could recover and reuse a portion of this
moisture, it could reduce its total cooling water intake
requirement. One of the most practical way to recover water
from flue gas is to use a condensing heat exchanger. The power
plant could also recover latent heat due to condensation as well
as sensible heat due to lowering the flue gas exit temperature.
Additionally, harmful acids released from the stack can be
reduced in a condensing heat exchanger by acid condensation. reduced in a condensing heat exchanger by acid condensation.
Condensation of vapors in flue gas is a complicated
phenomenon since heat and mass transfer of water vapor and
various acids simultaneously occur in the presence of noncondensable
gases such as nitrogen and oxygen. Design of a
condenser depends on the knowledge and understanding of the
heat and mass transfer processes. A computer program for
numerical simulations of water (H2O) and sulfuric acid (H2SO4)
condensation in a flue gas condensing heat exchanger was
developed using MATLAB. Governing equations based on
mass and energy balances for the system were derived to
predict variables such as flue gas exit temperature, cooling
water outlet temperature, mole fraction and condensation rates
of water and sulfuric acid vapors. The equations were solved
using an iterative solution technique with calculations of heat
and mass transfer coefficients and physical properties.
We have compiled the most important slides from each speaker's presentation. This year’s compilation, available for free, captures the key insights and contributions shared during the DfMAy 2024 conference.
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
HEAP SORT ILLUSTRATED WITH HEAPIFY, BUILD HEAP FOR DYNAMIC ARRAYS.
Heap sort is a comparison-based sorting technique based on Binary Heap data structure. It is similar to the selection sort where we first find the minimum element and place the minimum element at the beginning. Repeat the same process for the remaining elements.
Heap Sort (SS).ppt FOR ENGINEERING GRADUATES, BCA, MCA, MTECH, BSC STUDENTS
Existence of positive solutions for fractional q-difference equations involving integral boundary conditions with pLaplacian operator
1. Invention Journal of Research Technology in Engineering & Management (IJRTEM)
ISSN: 2455-3689
www.ijrtem.com Volume 2 Issue 7 ǁ July 2018 ǁ PP 05-15
|Volume 2| Issue 7 | www.ijrtem.com | 5 |
Existence of positive solutions for fractional q-difference
equations involving integral boundary conditions with p-
Laplacian operator
Yizhu Wang1
, Chengmin Hou1*
(Department of Mathematics, Yanbian University, Yanji, 133002, P.R. China)
ABSTRACT: The existence of positive solutions is considered for a fractional q-difference equation with p-
Laplacian operator in this article. By employing the Avery-Henderson fixed point theorem, a new result is obtained
for the boundary value problems.
KEY WORDS: positive solutions; Riemann-Liouville fractional q-derivatives; Caputo fractional q-derivatives;
p-Laplacian; Avery-Henderson fixed point theorem
I. INTRODUCTION
Fractional calculus is the extension of integer order calculus to arbitrary order calculus. With the development of
fractional calculus, fractional differential equations have wide applications in the modeling of different physical
and natural science fields, such as fluid mechanics, chemistry, control system, heat conduction, etc. There are
many papers concerning fractional differential equations with the p-Laplacian operator [1-5] and fractional
differential equations with integral boundary conditions [6-10].
In [11], Yang studied the following fractional q-difference boundary value problem with p-Laplacian operator
where 1 , 2. The existence results for the above boundary value problem were obtained by using the
upper and lower solutions method associates with the Schauder fixed point theorem.
In [12], Yuan and Yang considered a class of four-point boundary value problems of fractional q-difference
equations with p-Laplacian operator
Where is the fractional q-derivative of the Riemann-Liouville type with 1 , 2. By applying the
upper and lower solutions method associated with the Schauder fixed point theorem, the existence results of at
least one positive solution for the above fractional q-difference boundary value problem with p-Laplacian operator
are established. Motivated by the aforementioned work, this work discusses the existence of positive solutions for
this fractional q-difference equation:
( ( ( ))) ( , ( )),0 1,2 3,
(0) (1) 0,
c c
q p qD D x t f t u t t
x x
=
= = (1) (0) 0,c c
q qD x D x
= =
( ( ( ))) ( , ( )),0 1,2 3,
(0) 0, (1) ,
q P qD D x t f t u t t
x x ax
=
= = (0) 0, (1) ( ),q q qD x D x bD x
= =
,q qD D
2. Existence of positive solutions for fractional q-difference…
|Volume 2| Issue 7 | www.ijrtem.com | 6 |
(1.1)
where 2 3, 2 3, and 5 6. +
2
( ) ,
p
p u u u
−
= 1.p is the Caputo fractional q-
derivative, is the Riemann-Liouville fractional q-derivative.
We will always suppose the following conditions are satisfied:
1( )H ( ):[0,1] [0, )g t → + with 1
( ) [0,1],g t L and
2( )H , (0, ),a b + and ;b a
3( )H ( , ) :[0,1] (0, ) (0, )f t u + → + is continuous.
II. BACKGROUND AND DEFINITIONS
To show the main result of this work, we give in the following some basic definitions and theorems, which can be
found in [13,14].
Definition 2.1 [13] Let 0 and f be a function defined on [0,1] . The fractional q-integral of Riemann-
Liouville type is and
Lemma 2.2 [13] Let 0, then the following equality holds:
Lemma 2.3 [14]Let 0, If then
where 1.n = +
Theorem 2.4 (Avery-Henderson fixed point theorem [15]) Let ( , )E be a Banach space, and be a
cone. Let and be increasing non-negative, continuous functionals on ,P and be a non-negative
continuous functionals on P with such that, for some 3 0r and 0,M ( ) ( ) ( ),u u u
( 1)
,0 0
( )
( )( ) ( ) ,
( )
t
qq
q
t qs
I f t f s d s
+
−
−
=
0, [0,1].t
1
,0 ,0 ,0
0
( ) ( ) ( )(0 ).
( 1)
k
c k
q q q
k q
t
I D f t f t D f
k
+ + +
−
+
=
= −
+
0
,0
( )( ) ( )q
I f t f t+ =
,0 ,0
'
,0 ,0 ,0
2
,0 ,0
1
,0 0
[ ( ( ))] ( , ( )) 0, (0,1),
( (0)) [ ( (0))] ( (1)) 0,
(0) (1) 0,
(0) (0) ( ) ( ) ,
c
pq q
c c c
p p pq q q
q q
qq
D D u t f t u t t
D u D u D u
D u D u
au bD u g t u t d t
+ +
+ + +
+ +
+
+ =
= = =
= =
+ =
,0
c
q
D
+
,0q
D
+
1
0
( ) 0,qg t d t
1
0
( ) 0;qtg t d t
1
0
( ) qa g t d t
,0
[0,1],q
D u C
+
( )
,0 ,0
1
( ) ( )
n
i
iq q
i
I D f t f t k t
+ +
−
=
= +
P E
(0) 0, =
3. Existence of positive solutions for fractional q-difference…
|Volume 2| Issue 7 | www.ijrtem.com | 7 |
and ( ),u M u for all 3( , ),u p r where 3 3( , ) { : ( ) }.p r u P u r = Suppose that there exist
positive numbers 1 2 3,r r r such that
( ) ( )lu l u for 0 1,l and 2( , ).u P r
If 3: ( , )T p r P → is a completely continuous operator satisfying:
1( )C 3( )Tu r for all 3( , );u P r
2( )C 2( )Tu r for all 2( , );u P r
3( )C 1( , ) ,P r and 1( )Tu r for all 1( , ),u P r
then T has at least two fixed points 1u and 2u such that 1 1( )r u with 1 2( )u r and 2 2( )r u
with 2 3( ) .u r
III. PRELIMINARY LEMMAS
Lemma 3.1 The boundary value problem (1.1) is equivalent to the following equation:
(3.1)
where
(3.2)
(3.3)
(3.4)
(3.5)
is the inverse function of a.e.,
Proof From we get
( 1)
0 1 0
1
( ) ( ) ( ) ,
( )
t
q
q
u t d d t t qs v s d s
−
= + + −
1
( 1)
0 1 0 0
0
1
( ) ( ) ( )
[ ( ) ] ( )
t
q q
q q
d g t t qs v s d sd t
a g t d t
−
= −
−
1
1
( 2)0
1 0
0
[ ( ) ]
(1 ) ( ) ,
[ ( ) ] ( 1)
q
q
q q
b tg t d t
qs v s d s
a g t d t
−
−
+ −
− −
1
( 2)
1 0
1
(1 ) ( ) ,
( 1)
q
q
d qs v s d s
−
= − −
−
( )1
0
( ) ( , ) ( , ( )) ,q qv s H s q f u d =
( 1) ( 1)
( 1)
( (1 )) ( ) ,1
( , )
( ) ( (1 )) ,q
s s
H s
s
− −
−
− − −
=
−
0 1,s
0 1.s
( )q s
2
( ) ,
q
q s s s
−
=
1 1
1.
p q
+ =
,0 ,0
[ ( ( ))] ( , ( )) 0,c
pq q
D D u t f t u t
+ + + =
( 1) 1 2 3
1 2 3,0 0
1
( ( )) ( ) ( , ( )) .
( )
t
c
p qq
q
D u t t q f u d c t c t c t
+
− − − −
= − − + + +
( ),p s
4. Existence of positive solutions for fractional q-difference…
|Volume 2| Issue 7 | www.ijrtem.com | 8 |
In view of we get 2 3 0,c c= = i.e.,
(3.6)
Conditions imply that
(3.7)
By use of (3.6) and (3.7), we get
(3.8)
In view of (3.8), we obtain
(3.9)
Let
by use of (3.9), we get
Conditions imply that 2 0,d = i.e.,
then we have
Conditions imply that
From we get
Therefore, we can obtain
'
,0 ,0
( (0)) [ ( (0))] 0,c c
p pq q
D u D u
+ += =
( 1) 1
1,0 0
1
( ( )) ( ) ( , ( )) .
( )
t
c
p qq
q
D u t t q f u d c t
+
− −
= − − +
,0
( (1)) 0c
p q
D u
+ =
1
( 1)
1 0
1
(1 ) ( , ( )) .
( )
q
q
c q f u d
−
= − −
1
,0 0
( ( )) ( , ) ( , ( )) .c
p qq
D u t H t q f u d
+ =
( )1
,0 0
( ) ( , ) ( , ( )) .c
q qq
D u t H t q f u d
+ =
( )1
0
( ) ( , ) ( , ( )) ,q qv t H t q f u d =
( 1) 2
0 1 20
1
( ) ( ) ( ) .
( )
t
q
q
u t t qs v s d s d d t d t
−
= − + + +
2
,0
(0) 0q
D u+ =
( 1)
0 10
1
( ) ( ) ( ) ,
( )
t
q
q
u t t qs v s d s d d t
−
= − + +
' ( 2)
10
1
( ) ( ) ( ) .
( 1)
t
q
q
u t t qs v s d s d
−
= − +
−
,0
(1) 0q
D u+ =
1
( 2)
1 0
1
(1 ) ( ) .
( 1)
q
q
d qs v s d s
−
= − −
−
1
,0 0
(0) (0) ( ) ( ) ,qq
au bD u g t u t d t++ =
1
( 1)
0 1 0 0
0
1
( ) ( ) ( )
[ ( ) ] ( )
t
q q
q q
d g t t qs v s d sd t
a g t d t
−
= −
−
1
1
( 2)0
1 0
0
[ ( ) ]
(1 ) ( ) .
[ ( ) ] ( 1)
q
q
q q
b tg t d t
qs v s d s
a g t d t
−
−
+ −
− −
( 1)
0 1 0
1
( ) ( ) ( )
( )
t
q
q
u t d d t t qs v s d s
−
= + + −
5. Existence of positive solutions for fractional q-difference…
|Volume 2| Issue 7 | www.ijrtem.com | 9 |
The proof is complete. □
Lemma 3.2 ([16]) The function ( , )H s defined by (3.5) is continuous on [0,1] [0,1] and satisfy
for , [0,1].s
Let E be the real Banach space [0,1]C with the maximum norm, define the operator :T E E→ by
Lemma 3.3 For with is non-increasing and non-negative.
Proof Since
so we get
So ( )( )Tu t is non-increasing, then we have We have
1
( 1)
1 0 0
0
1
( ) ( ) ( )
[ ( ) ] ( )
t
q q
q q
g t t qs v s d sd t
a g t d t
−
= −
−
1
1
( 2)0
1 0
0
[ ( ) ]
(1 ) ( )
[ ( ) ] ( 1)
q
q
q q
b tg t d t
qs v s d s
a g t d t
−
−
+ −
− −
1
( 2) ( 1)
0 0
1
(1 ) ( ) ( ) ( ) .
( 1) ( )
t
q q
q q
t
qs v s d s t qs v s d s
− −
− − + −
−
1 ( 1) ( 1)
(1 ) (1 ) (1 )
( , )
( ) ( 1)q q
s s
H s
− − −
− − −
−
( 1)
0 1 0
1
( ) ( ) ( )
( )
t
q
q
Tu t d d t t qs v s d s
−
= + + −
1
( 1)
1 0 0
0
1
( ) ( ) ( )
[ ( ) ] ( )
t
q q
q q
g t t qs v s d sd t
a g t d t
−
= −
−
1
1
( 2)0
1 0
0
[ ( ) ]
(1 ) ( )
[ ( ) ] ( 1)
q
q
q q
b tg t d t
qs v s d s
a g t d t
−
−
+ −
− −
1
( 2) ( 1)
0 0
1
(1 ) ( ) ( ) ( ) .
( 1) ( )
t
q q
q q
t
qs v s d s t qs v s d s
− −
− − + −
−
[0,1]u C ( )( )Tu t( ) 0,u t
( 1)
0 1 0
1
( ) ( ) ( ) ,
( )
t
q
q
Tu t d d t t qs v s d s
−
= + + −
( )
' ( 2)
1 0
1
( ) ( ) ( )
( 1)
t
q
q
Tu t d t qs v s d s
−
= + −
−
1
( 2) ( 2)
0 0
1 1
(1 ) ( ) ( ) ( )
( 1) ( 1)
t
q q
q q
qs v s d s t qs v s d s
− −
= − − + −
− −
0.
[0,1]
min ( ) (1).
t
Tu t Tu
=
1
( 1)
0 1 0
1
(1) (1 ) ( )
( )
q
q
Tu d d qs v s d s
−
= + + −
6. Existence of positive solutions for fractional q-difference…
|Volume 2| Issue 7 | www.ijrtem.com | 10 |
The proof is complete. □
IV. MAIN RESULTS
Theorem 4.1 Suppose that there exist numbers 1 2 30 r r r such that f satisfies the following conditions:
1( )H 3( , ) ,f t u M for [0,1],t
2( )H 2( , ) ,f t u M for [0,1],t 2[0, ];u r
3( )H 1( , ) ,f t u M for [0,1],t 1[0, ],u r
where
Then the problem (1.1) has at least two positive solutions 1u and 2u such that 1 1( )r u with 1 2( )u r
1
( 1)
1 0 0
0
1
( ) ( ) ( )
[ ( ) ] ( )
t
q q
q q
g t t qs v s d sd t
a g t d t
−
= −
−
1
1
( 2)0
1 0
0
[ ( ) ]
(1 ) ( )
[ ( ) ] ( 1)
q
q
q q
b tg t d t
qs v s d s
a g t d t
−
−
+ −
− −
1 1
( 2) ( 1)
0 0
1 1
(1 ) ( ) (1 ) ( )
( 1) ( )
q q
q q
qs v s d s qs v s d s
− −
− − + −
−
1
( 1)
1 0 0
0
1
( ) ( ) ( )
[ ( ) ] ( )
t
q q
q q
g t t qs v s d sd t
a g t d t
−
−
−
1 1
1
( 2)0 0
1 0
0
[ ( ) ] [ ( ) ]
(1 ) ( )
[ ( ) ] ( 1)
q q
q
q q
b tg t d t a g t d t
qs v s d s
a g t d t
−
− − −
+ −
− −
1
( 1)
0
1
(1 ) ( )
( )
q
q
qs v s d s
−
+ −
0.
3
3[ , ];
r
u r
k
1
3
3
3
( )
,
(2, )
p
q
q
r
M
qB L
−
=
1
2
2
2
( 1)
,
(2, )
p
q
q
r
M
qB L
−
−
=
1
1
1
1
( )
,
(2, )
p
q
q
r
M
qB L
−
=
( )
1 11
3 1 0
0
1
(1 ) ,
( 1)[ ( ) ] ( )
q
q
q
q q
b a
L s s d s
a g t d t
−−
−
= + −
+−
1 1
0 0
2 1 1
0 0
( ) ( )
,
[ ( ) ] ( 1) [ ( ) ] ( )
q q
q q q q
g t d t b tg t d t
L
a g t d t a g t d t
−
= +
− + −
( )
1
1 110
1 1 0
0
( )
(1 ) .
[ ( ) ] ( )
qq
q
q q
b tg t d t
L s s d s
a g t d t
−−
−
= −
−
7. Existence of positive solutions for fractional q-difference…
|Volume 2| Issue 7 | www.ijrtem.com | 11 |
and 2 2( )r u with 2 3( ) .u r
Proof Define the cone P E by
where
For any ,u P in view of Lemma 3.3, we get
Therefore, :T P P→ . In view of the Arzela-Ascoli theorem, we have :T P P→ is completely continuous.
We define the functions on the cone P:
Obviously, we have (0) 0, = ( ) ( ) ( ).u u u
For any we get that is, therefore we obtain
[0,1]
,min ( ) , [0,1] ,
t
P u u E u t k u t
=
1 1
0 0
1
0
[ ( ) ] [ ( ) ]
,
( )
q q
q
b tg t d t a g t d t
k
b tg t d t
− − −
=
−
0 1.k
1
( 1)
0 1 0[0,1]
1
min ( ) (1) (1 ) ( )
( )
q
t
q
Tu t Tu d d qs v s d s
−
= = + + −
1
( 1)
1 0 0
0
1
( ) ( ) ( )
[ ( ) ] ( )
t
q q
q q
g t t qs v s d sd t
a g t d t
−
= −
−
1
1
( 2)0
1 0
0
[ ( ) ]
(1 ) ( )
[ ( ) ] ( 1)
q
q
q q
b tg t d t
qs v s d s
a g t d t
−
−
+ −
− −
1 1
( 2) ( 1)
0 0
1 1
(1 ) ( ) (1 ) ( )
( 1) ( )
q q
q q
qs v s d s qs v s d s
− −
− − + −
−
1
( 1)
1 0 0
0
1
( ) ( ) ( )
[ ( ) ] ( )
t
q q
q q
g t t qs v s d sd t
a g t d t
−
−
−
1 1
1
( 2)0 0
1 0
0
[ ( ) ] [ ( ) ]
(1 ) ( )
[ ( ) ] ( 1)
q q
q
q q
b tg t d t a g t d t
qs v s d s
a g t d t
−
− − −
+ −
− −
1
( 1)
1 0 0
0
1
( ) ( ) ( )
[ ( ) ] ( )
t
q q
q q
k g t t qs v s d sd t
a g t d t
−
−
−
1
1
( 2)0
1 0
0
[ ( ) ]
(1 ) ( )
[ ( ) ] ( 1)
q
q
q q
b tg t d t
qs v s d s
a g t d t
−
−
+ −
− −
(0) .kTu k Tu= =
[0,1]
( ) min ( ) (1),
t
u u t u
= =
[0,1]
( ) max ( ) (0),
t
u u t u
= =
[0,1]
( ) max ( ) (0).
t
u u t u
= =
[0,1]
min ( ) ,
t
u t k u
1
( ).u u
k
( ) ,u k u 3( , ),u p r
8. Existence of positive solutions for fractional q-difference…
|Volume 2| Issue 7 | www.ijrtem.com | 12 |
For any 2( , ),u P r we get ( ) ( )lu l u = for 0 1.l
In the following, we prove that the conditions of Theorem 2.1 hold.
Firstly, let 3( , ),u P r that is, for [0,1].t By means of 1( ),H we have
where So we get
3
3[ , ]
r
u r
k
( )1
0
( ) ( , ) ( , ( ))q qv s H s q f u d =
1 ( 1)
1
3 0
(1 ) (1 )
( )
q q
q
s s q q
M d
− − − −
1
1
3 (1 ) (2, )
,
( )
q
q
q
M s s qB
−
−
−
=
1
( 1)
0
(2, ) (1 ) .q qB q d
−
= −
1
( 1)
0 1 0[0,1]
1
( ) min ( ) (1) (1 ) ( )
( )
q
t
q
Tu Tu t Tu d d qs v s d s
−
= = = + + −
1
( 1)
1 0 0
0
1
( ) ( ) ( )
[ ( ) ] ( )
t
q q
q q
g t t qs v s d sd t
a g t d t
−
= −
−
1
1
( 2)0
1 0
0
[ ( ) ]
(1 ) ( )
[ ( ) ] ( 1)
q
q
q q
b tg t d t
qs v s d s
a g t d t
−
−
+ −
− −
1 1
( 2) ( 1)
0 0
1 1
(1 ) ( ) (1 ) ( )
( 1) ( )
q q
q q
qs v s d s qs v s d s
− −
− − + −
−
1 1
1
( 2)0 0
1 0
0
[ ( ) ] [ ( ) ]
(1 ) ( )
[ ( ) ] ( 1)
q q
q
q q
b tg t d t a g t d t
qs v s d s
a g t d t
−
− − −
−
− −
1
( 1)
0
1
(1 ) ( )
( )
q
q
qs v s d s
−
+ −
1
1
1 3( 2)
1 0
0
(1 ) (2, )
(1 )
( )[ ( ) ] ( 1)
q
q
q
q
q q
M s s qBb a
qs d s
a g t d t
−
−
−
−−
− − −
1
1
1 3( 1)
0
(1 ) (2, )1
(1 )
( ) ( )
q
q
q
q q
M s s qB
qs d s
−
−
−
−
+ −
( )
1
1 13 1
0
(2, )1
(1 )
( 1) ( )
q
qq
q
q q
M qB
s s d s
−
−−
+ − +
( )
1
1 13 1
1 0
0
(2, )
(1 )
( )[ ( ) ] ( )
q
qq
q
q
q q
M qBb a
s s d s
a g t d t
−
−−
−
= − −
1
3
3 3
(2, )
.
( )
q
q
q
M qB
L r
−
= =
9. Existence of positive solutions for fractional q-difference…
|Volume 2| Issue 7 | www.ijrtem.com | 13 |
Secondly, let 2( , ),u P r that is, 2[0, ]u r for [0,1].t By means of 2( ),H we get
So, we have
Finally, let 1( , ),u P r that is, 1[0, ]u r for [0,1].t By means of 3( ),H we get
So we get
( )1
0
( ) ( , ) ( , ( ))q qv s H s q f u d =
1
( 1)
1 2
2 0
(2, )(1 )
.
( 1) ( 1)
q
q
q q
q q
M qBq q
M d
−
− −
= − −
0
[0,1]
( ) max ( ) (0)
t
Tu Tu t Tu d
= = =
1
( 1)
1 0 0
0
1
( ) ( ) ( )
[ ( ) ] ( )
t
q q
q q
g t t qs v s d sd t
a g t d t
−
= −
−
1
1
( 2)0
1 0
0
[ ( ) ]
(1 ) ( )
[ ( ) ] ( 1)
q
q
q q
b tg t d t
qs v s d s
a g t d t
−
−
+ −
− −
1
1 2( 1)
1 0 0
0
(2, )1
( ) ( )
( 1)[ ( ) ] ( )
q
t q
q q
q
q q
M qB
g t t qs d sd t
a g t d t
−
−
− −−
1 1
1 2( 2)0
1 0
0
[ ( ) ] (2, )
(1 )
( 1)[ ( ) ] ( 1)
q
q q
q
q
q q
b tg t d t M qB
qs d s
a g t d t
−
−
−
+ − −− −
1 1
20
1
0
( ) (2, )
( 1)[ ( ) ] ( 1)
q
q q
q
q q
g t d t M qB
a g t d t
−
−− +
1 1
20
1
0
[ ( ) ] (2, )
( 1)[ ( ) ] ( )
q
q q
q
q q
b tg t d t M qB
a g t d t
−
−
+ −−
1
2
2 2
(2, )
.
( 1)
q
q
q
M qB
L r
−
= = −
( )1
0
( ) ( , ) ( , ( ))q qv s H s q f u d =
1
11 ( 1)
1 1
1 0
(1 ) (2, )(1 ) (1 )
.
( ) ( )
q
q
q q
q q
M s s qBs s q q
M d
−
−− − −− −
=
0
[0,1]
( ) max ( ) (0)
t
Tu Tu t Tu d
= = =
1
( 1)
1 0 0
0
1
( ) ( ) ( )
[ ( ) ] ( )
t
q q
q q
g t t qs v s d sd t
a g t d t
−
= −
−
10. Existence of positive solutions for fractional q-difference…
|Volume 2| Issue 7 | www.ijrtem.com | 14 |
Therefore, in view of Theorem 2.1, we see that the problem (1.1) has at least two positive solutions 1u and 2u
such that 1 1( )r u with 1 2( )u r and 2 2( )r u with 2 3( ) .u r □
VI. EXAMPLE
In this section, we give a simple example to explain the main result.
Example 5.1 For the problem (1.1), let 2.8, = 2.3, = 4,a = 10,b = 2,p = ( ) ,g t t= then we get 2,q =
Let
From a direct calculation, we get
for
for
for
In view of Theorem 4.1, we see that the aforementioned problem has at least two positive solutions 1u and 2u
such that 10.5 ( )u with 1( ) 9u and 29 ( )u with 2( ) 10.u
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t u 3( , ) 583.266938f t u M
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[0,1], [9,10],t u
[0,1], [10, ).t u +
[0,1], [0,9],t u
[0,1], [0,0.5].t u
11. Existence of positive solutions for fractional q-difference…
|Volume 2| Issue 7 | www.ijrtem.com | 15 |
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