1,720,963 research outputs found
Integrability and L1-Convergence of Certain Cosine SUMS
No full textM.Sc. (Mathematics and Computing)The work presented in this dissertation has been divided into four chapters. The
first chapter is introductory. In this chapter, apart from setting up the notations and
terminology to be used in sequel, we have presented some known results interrelated to
our results along with a brief plan of our results presented in the subsequent chapters. The
purpose of chapter II is to study the integrability and 1 L -convergence of Rees and
Stanojevic cosine sum under the Class C of coefficient sequences. In chapter III, I studied
the results concerning the 1 L -convergence of cosine trigonometric series under the class
2 S of coefficient sequences which is equivalent to class S of Sidon.
In chapter IV, I have studied the generalization of the results of Garrett and
Stanojevic by considering the class (BV )m instead of class BV .School of Mathematics and Computer Applications, Thapar University, Patial
Fixed Point Theorems for Different Contraction Mappings
No full textThe present dissertation entitled, "FIXED POINT THEOREMS FOR DIFFERENT CONTRACTION MAPPINGS", contains a study about Fixed point
theory by me on existence of fixed points of self mappings in metric space under the supervision of Dr. Jatinderdeep Kaur, Associate Professor, School of
Mathematics, Thapar Institute of Engineering and Technology, Patiala.
The aim of this work is to study and obtain some result on existence
and uniqueness of fixed points. Fixed point theory has been revealed as a very
powerful and important tool in the study of non-linear analysis. Various problems
in physics, chemistry, biology, economics etc. can be solved by maling use of
fixed point theorems.
The work presented in this dissertation has been divided into four chapters. Chapter I is introduction which includes brief account of definitions and
results which will be required for the later chapters. In Chapter II, we have studied Banach fixed point theorem which guarantees the existence and uniqueness of
fixed points of certain self-maps of metric spaces. Also, we present some fixed
point theorems in compact metric space.
The purpose of the Chapter III is to study fixed point theorems for generalized contraction mappings on a S-orbitally complete metric space and studied
the existence and uniqueness of fixed points. In the Chapter IV, we have studied
some fixed point theorems under quasi contraction condition. The purpose of this
chapter is to extend the result presented in chapter III.
At the end of the present dissertation, we have added bibliography
On L1-Convergence of Trigonometric Sine Series with Special Coefficients
No full textThe present dissertation entitled \L1Convergence of trigonometric sine series
with special coe cients"contains a brief account of study carried out by me on
L1convergence of Trigonometric Sine Sums under the supervision of Dr. Jatin-
derdeep Kaur, Assistant Professor, School of Mathematics, Thapar Institute of
Engineering and Technology, Patiala.
In the literature so far available, very few work has been done concerning the con-
vergence of trigonometric sine series in L1norm. Keeping this in view, The L1
convergence of trigonometric sine series under di erent conditions on coe cients have
been studied. Also, several authors introduced modi ed trigonometric cosine and sine
sums.
The rst chapter is introductory. In this chapter, apart from setting up the notations
and terminology to be used in sequel, we have presented some known results. The
purpose of chapter II is to study the L1convergence of trigonometric sine series us-
ing class eS1.
In the chapter III, the result on L1convergence of trigonometric sine series has been
obtained using Ram and Kumari Modi ed sine sums
Xn
k=1
Xn
j=k
4
aj
j
k sin kx
!
un-
der the classes e BV
T e C.
In chapter IV, L1convergence of r times di erentiable trigonometric sine series has
been studied using the classes e BVr(r = 0; 1; 2; 3; : : : ) and e Cr(r = 0; 1; 2; 3; : : : ).
In the end, references of various publications cited in the present dissertation have
been reported.
Fixed Point Theorems on Contraction and Commuting Mappings
No full textThe aim of this work is to study and obtain some result on existence and uniqueness of fixed points. Fixed point theory has wide ranging application in many areas of mathematics. For example, in finding the solution of the system of linear equations, in proving the existence of solutions of ordinary and partial differential equation, integral equations, analysis and many other discipline
A Study of New Topologies on R.
No full textM.Sc (Mathematics and Computing) Thesis.The present dissertation entitled, “A Study of New Topologies on R” comprises
certain investigations carried out by me at School of Mathematics, Thapar University,
Patiala, under the supervision of Dr. Jatinderdeep Kaur, Assistant Professor, School of Mathematics, Thapar University, Patiala.
The aim of this work is to find new topologies with this new set of basis B−
and B+. The topologies generated from this new set of basis are different from standard, lower limit, upper limit and k-topologies. We compared well known topologies
with new topologies on R.
The whole work is divided into four chapters. Chapter 1 is introduction which
includes brief account of definitions and their examples which will be required for
the later chapters. In chapter 2, we have studied some main theorems which guarantees how topology can be generated from given base and vice-versa. The aim of chapter 3 is to study well known topologies on R and their topological properties.
In chapter 4, we introduced new topologies on real line and compared them with
well known topologies on R.
At the end of this dissertation, we have bibliography of research papers and
books cited in the dissertation
Iterative Solutions for Non-Linear Systems
No full textPhD ThesisThe research work presented in this thesis deals with the study of the “Iterative solutions for non-linear systems”.
A non-linear system is a set of simultaneous ‘n’ non-linear equations in which each equation is a function in ‘n’ unknown variables. Such type of non-linear systems have perceived several significant contributions in mathematics and allied engineering areas, for example, electrical circuits, chemical reactions, physical law, biological phenomena, computational economics etc. Due to the wider variety of behavior, finding solutions of
non-linear systems is much more tedious than a scalar case. Moreover it is extremely hard to solve non-linear systems analytically. In this context, numerical techniques provide a fruitful way to solve non-linear systems. On account of this reality, one needs to rely on numerical techniques for solving non-linear systems. Therefore, a reliably expanding extent of present-day numerical research is focused on the analysis of the approximate solutions of non-linear systems. Among all numerical techniques, Newton’s technique is the most basic and outstanding iterative method for solving non-linear systems. In literature, numerous adjustments have been incorporated in Newton’s technique (known as Newton’s variants), which have either equivalent or better efficiency over Newton’s technique. In the present thesis, an endeavor has been made to the construct more variants of Newton’s method to solve non-linear systems. The essential standard of numerical algorithms has
been followed to attain computational efficiency, which is always proportional to the quality of an algorithm and inversely proportional to its computational cost. The quality of an algorithm concerns with the convergence speed of algorithm along with its structure. Computational cost concerns with the amount of calculation work required to evaluate functions, derivatives, matrix inversions during the entire process. The primary focus of present research work is to address the construction of iterative schemes to propose solutions for systems of non-linear equations arising in different disciplines of science and engineering. The present work also sheds light on the development of iterative schemes for systems of non-linear equations associated with ordinary and partial differential equations. The development of iterative schemes consist of two parts: the first one is the ‘construction part’ and the second part establishes the proof of local convergence in Banach settings. Finally, a variety of problems involving non-linear systems have been numerically tested in order to demonstrate the exactness and the computational efficiency of the proposed iterative algorithms
Existence of Fixed Points for Some Mappings in Various Spaces
No full textPhD-Mathematics-ThesisThe present thesis entitled “Existence of Fixed Points for Some Mappings in Various
Spaces” comprises certain investigations carried out by me at the School of Mathematics
and Computer Applications (SMCA), Thapar University, Patiala, under the supervision of
Dr. S. S. Bhatia, Professor, SMCA, Thapar University, Patiala and Dr. Jatinderdeep Kaur,
Assistant Professor, SMCA, Thapar University, Patiala.
A very important mechanism for the progress in the field of science and technology is
to generalize the existing ideas. The present thesis has been written on the same basis. In
this thesis, various fixed point results in various abstract spaces such as: metric spaces, G-
metric spaces, complex valued metric spaces, partial Hausdorff metric spaces and partially
ordered metric spaces have been discussed and thereby many existing results have been
extended and generalized.SMCA, Thapar University, Patial
Existence of Fixed Points for Various Mappings in Abstract Spaces
Full text linkFixed point theory is an important branch of non-linear analysis. Many problems, occurring in different branches of mathematics, such as differential equations, optimization theory and variational analysis, can be converted into the equation T x = x, where T is some non-linear operator defined on a certain space X. Solutions of this equation are called fixed point of T. Fixed point theory can be classified into three major areas: Metric fixed point theory, Topological fixed point theory and Discrete fixed point theory. The principal findings in these areas are Banach’s fixed point theorem, Brouwer’s fixed point theorem, and Tarski’s fixed point theorem respectively.
Abstract space is a set of elements satisfying certain axioms. In 1906, the French mathematician Fr´echet introduced the first abstract space, called metric space. In 1922, Polish mathematician Stefan Banach gave the first fixed point theorem for contraction mappings in metric spaces, and this theorem is famous as the Banach
contraction principle. This principle states that every contraction self-mapping defined on a complete metric space has a unique fixed point. This result has become one of the most popular and effective tools in solving existence problem in many branches of mathematics.
Banach contraction principle has been generalized in several directions. There are two ways to extend or improve this principle. One way is to extend/improve the condition of contraction mappings, and the second approach is to replace complete metric space with a more general abstract space. In the first direction, there are numerous results in the literature proved by Kannan, Chatterjea, Reich, Hardy and Rogers, C´iri´c, Wang et al., Alber and Delabriae, Samet et al., Shahi et al., Wardowski and many more. In the second direction, we have several abstract spaces in the literature such as partial metric spaces, b-metric spaces, cone metric spaces, metric-like spaces, partial b-metric spaces, b-metric-like spaces, generalized metric spaces, F-metric spaces, 2-metric spaces, D-metric spaces, G-metric spaces, GPmetric spaces, generalized b-metric spaces etc.
In the present thesis, we will work in both directions. The present thesis consists of six chapters. Chapter 1 is about the introduction related to our work. From the literature, a brief about various mappings related to fixed point theory and different abstract spaces are discussed in this chapter. At the end of the chapter, a brief plan
of the results presented in the subsequent chapters is given. In Chapter 2, inspired by the concept of b-metric space, G-metric space, and generalized b-metric space, a new abstract space (named generalized Gb-metric
space) has been introduced. Some basic concepts and properties of new space have been studied. Various fixed point theorems in the framework of generalized Gbmetric space has been proved. Multiple examples have been presented for the authenticity of the main results. After that, an application of one of our main results
has also been given.
In Chapter 3, another new class of abstract spaces (named G∗ -metric space) has been introduced as a generalization of generalized Gb-metric spaces and GP-metric spaces. Some basic concepts in G∗ -metric space have been studied. Some new types of Cauchy sequences have been noticed in this new abstract space. Various examples have also been presented for these new concepts. Chapter 4 deals with fixed point results for contraction and quasi-contraction type mappings in G∗ -metric space. As a consequence of these results, some fixed point theorems have been deduced in the framework of generalized Gb-metric space.
Some examples have also been presented in support of the main results and consequences.
In Chapter 5, a new class of functions has been introduced. With the help of this new class of functions, some new contractive mappings in b-metric spaces have been introduced. We establish some fixed point results for these new contractive mappings in b-metric spaces. As a consequence of the main results, various fixed
point theorems have also been presented. An example has also been illustrated in support of our results. At the end of the chapter, an application has been provided to prove the uniqueness of the solution to a system of simultaneous linear equations. The aim of Chapter 6 is to extend the main results of Chapter 5 in the framework of b-metric-like spaces. Also, some common fixed point theorems have presented for weakly compatible mappings. Suitable examples have been provided for the main results. Related to the main results, some corollaries have also been presented at the end of this chapter
Integrability and L1-convergence of trigonometric series with real and complex coefficients
No full textThe present thesis entitled \lq \lq \textbf{Integrability and -convergence of trigonometric series with real and complex coefficients}" comprises certain investigations carried out by me at the School of Mathematics, Thapar Institute of Engineering and Technology, Patiala (TIET), under the supervision of Dr. Jatinderdeep Kaur, Assistant Professor, School of Mathematics, TIET, Patiala and Dr . S.S. Bhatia, Professor, School of Mathematics, TIET, Patiala.
The existence of sine and cosine series as a Fourier series, their integrability and -convergence seems to be one of the difficult question in theory of convergence of trigonometric series in -metric norm. It is well known that if a trigonometric series converges in -metric to a function , then it is a Fourier series of the function but the converse of the said result does not hold good. The present thesis has been written on the same basis. In this thesis, Integrability and -convergence of trigonometric cosine and sine series have been studied by imposing different conditions on coefficient sequences as well as by introducing new modified trigonometric cosine and sine sums \textit{as these sums approximate their limit better than the classical trigonometric series in the sense that they converge in -metric to the sum of trigonometric series whereas the classical series itself may not.}
The thesis embodies six chapters. \textbf{Chapter I,} is introductory. In this chapter, apart from setting up the notations and terminologies to be used in the sequel, some known results interrelated with the work done in the present thesis have been presented. Further a brief plan of the results presented in the subsequent chapters is given in this chapter.
In \textbf{Chapter II,} a new modified trigonometric cosine sum has been introduced and criterion for the summability and -convergence of modified cosine sum has been obtained under the class of generalized semi-convex sequences. Also, the -convergence of the cosine series has been deduced as corollary in this chapter.
In the literature so far available, most of the authors have studied the integrability and -convergence of trigonometric cosine series using different classes of coefficient sequences. However, very few of them have studied the -convergence of the trigonometric sine series. In \textbf{Chapter III,} a new modified trigonometric sine sum has been introduced and its -convergence has been studied under the class . Further, in this chapter we have obtained -convergence of derivative of modified trigonometric sine sum under a new extended class of coefficient sequences. Also, an application has been given to illustrate the main result.
\textbf{Chapter IV,} is devoted to the study of -convergence of complex form of the new modified trigonometric cosine and sine sums introduced in chapter II and III under the new generalized classes and of complex coefficient sequences.
In \textbf{Chapter V,} we have introduced a new modified double trigonometric sine sum and have studied the integrability and -convergence of double trigonometric cosine series under the new class of double coefficient sequences.
The objective of \textbf{Chapter VI} is to study the integrability and -convergence of Fourier series of periodic fuzzy valued functions with new class of fuzzy coefficients
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