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2014 | Buch

Topics in Mathematical Analysis and Applications

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This volume presents significant advances in a number of theories and problems of Mathematical Analysis and its applications in disciplines such as Analytic Inequalities, Operator Theory, Functional Analysis, Approximation Theory, Functional Equations, Differential Equations, Wavelets, Discrete Mathematics and Mechanics. The contributions focus on recent developments and are written by eminent scientists from the international mathematical community. Special emphasis is given to new results that have been obtained in the above mentioned disciplines in which Nonlinear Analysis plays a central role.

Some review papers published in this volume will be particularly useful for a broader readership in Mathematical Analysis, as well as for graduate students. An attempt is given to present all subjects in this volume in a unified and self-contained manner, to be particularly useful to the mathematical community.

Inhaltsverzeichnis

Frontmatter
Simple Proofs of Some Bernstein–Mordell Type Inequalities

In this paper we give simple proofs of some Bernstein–Mordell type inequalities.

Vandanjav Adiyasuren, Tserendorj Batbold
Hilbert-Type Inequalities Including Some Operators, the Best Possible Constants and Applications: A Survey

The present work is a review article about some recent results dealing with Hilbert-type inequalities including certain operators in both integral and discrete case. A particular emphasis is given to inequalities including classical means operators. The constants appearing in all discussed inequalities are the best possible. For an illustration, some proofs are given, as well as some applications.

Vandanjav Adiyasuren, Tserendorj Batbold, Mario Krnić
A Fixed Point Approach to Stability of the Quadratic Equation

In this paper, by using the fixed point method in Banach spaces, we prove the Hyers–Ulam–Rassias stability for the quadratic functional equation

f

i

=

1

m

x

i

=

i

=

1

m

f

(

x

i

)

+

1

2

1

i

<

j

m

{

f

(

x

i

+

x

j

)

f

(

x

i

x

j

)

}

.

$$\displaystyle{f\left (\sum _{i=1}^{m}x_{ i}\right ) =\sum _{ i=1}^{m}f(x_{ i}) + \frac{1} {2}\sum _{1\leq i<j\leq m}\{f(x_{i} + x_{j}) - f(x_{i} - x_{j})\}.}$$

The concept of the Hyers–Ulam–Rassias stability originated from Rassias’ stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72(2), 297–300 (1978).

M. Almahalebi, A. Charifi, S. Kabbaj, E. Elqorachi
Aspects of Global Analysis of Circle-Valued Mappings

We deal with the minimum number of critical points of circular functions with respect to two different classes of functions. The first one is the whole class of smooth circular functions and, in this case, the minimum number is the so called

circular

φ

$$\varphi$$

-category

of the involved manifold. The second class consists of all smooth circular Morse functions, and the minimum number is the so called

circular Morse–Smale characteristic

of the manifold. The investigations we perform here for the two circular concepts are being studied in relation with their real counterparts. In this respect, we first evaluate the circular

φ

$$\varphi$$

-category of several particular manifolds. In Sect. 5, of more survey flavor, we deal with the computation of the circular Morse–Smale characteristic of closed surfaces. Section 6 provides an upper bound for the Morse–Smale characteristic in terms of a new characteristic derived from the family of circular Morse functions having both a critical point of index 0 and a critical point of index

n

. The minimum number of critical points for real or circle valued Morse functions on a closed orientable surface is the minimum characteristic number of suitable embeddings of the surface in

3

$$\mathbb{R}^{3}$$

with respect to some involutive distributions. In the last section we obtain a lower and an upper bound for the minimum characteristic number of the embedded closed surfaces in the first Heisenberg group with respect to its noninvolutive horizontal distribution.

Dorin Andrica, Dana Mangra, Cornel Pintea
A Remark on Some Simultaneous Functional Inequalities in Riesz Spaces

We study continuous at a point functions that take values in a Riesz space and satisfy some systems of two simultaneous functional inequalities. In this way we obtain in particular generalizations and extensions of some earlier results of Krassowska, Matkowski, Montel, and Popoviciu.

Bogdan Batko, Janusz Brzdȩk
Elliptic Problems on the Sierpinski Gasket

There are treated nonlinear elliptic problems defined on the Sierpinski gasket, a highly non-smooth fractal set. Even if the structure of this fractal differs considerably from that of (open) domains of Euclidean spaces, this note emphasizes that PDEs defined on it may be studied (as in the Euclidean case) by means of certain variational methods. Using such methods, and appropriate abstract multiplicity theorems, there are proved several results concerning the existence of multiple (weak) solutions of Dirichlet problems defined on the Sierpinski gasket.

Brigitte E. Breckner, Csaba Varga
Initial Value Problems in Linear Integral Operator Equations

For some general linear integral operator equations, we investigate consequent initial value problems by using the theory of reproducing kernels. A new method is proposed which—in particular—generates a new field among initial value problems, linear integral operators, eigenfunctions and values, integral transforms and reproducing kernels. In particular, examples are worked out for the integral equations of Lalesco–Picard, Dixon, and Tricomi types.

L. P. Castro, M. M. Rodrigues, S. Saitoh
Extension Operators that Preserve Geometric and Analytic Properties of Biholomorphic Mappings

In this survey we are concerned with certain extension operators which take a univalent function

f

on the unit disc

U

to a univalent mapping

F

from the Euclidean unit ball

B

n

in

n

$$\mathbb{C}^{n}$$

into

n

$$\mathbb{C}^{n}$$

, with the property that

f

(

z

1

)

=

F

(

z

1

,

0

)

$$f(z_{1}) = F(z_{1},0)$$

. This subject began with the Roper–Suffridge extension operator, introduced in 1995, which has the property that if

f

is a convex function of

U

then

F

is a convex mapping of

B

n

. We consider certain generalizations of the Roper–Suffridge extension operator. We show that these operators preserve the notion of

g

-Loewner chains, where

g

(

ζ

)

=

(

1

ζ

)

(

1

+

(

1

2

γ

)

ζ

)

$$g(\zeta ) = (1-\zeta )/(1 + (1 - 2\gamma )\zeta )$$

, | 

ζ

 |  < 1 and

γ

 ∈ (0, 1). As a consequence, the considered operators preserve certain geometric and analytic properties, such as

g

-parametric representation, starlikeness of order

γ

, spirallikeness of type

δ

and order

γ

, almost starlikeness of order

δ

and type

γ

.

We use the method of Loewner chains to generate certain subclasses of normalized biholomorphic mappings on the Euclidean unit ball

B

n

in

n

$$\mathbb{C}^{n}$$

, which have interesting geometric characterizations. We obtain the characterization of

g

-starlike and

g

-spirallike mappings of type

α

(

π

2

,

π

2

)

$$\alpha \in (-\pi /2,\pi /2)$$

, as well as of

g

-almost starlike mappings of order

α

 ∈ [0, 1), by using

g

-Loewner chains. Also, we will show that, under certain assumptions, the mapping

F

(

z

) = 

P

(

z

)

z

,

z

 ∈ 

B

n

, has

g

-parametric representation on

B

n

, where

P

:

B

n

$$P: B^{n} \rightarrow \mathbb{C}$$

is a holomorphic function such that

P

(0) = 1.

Teodora Chirilă
Normal Cones and Thompson Metric

The aim of this paper is to study the basic properties of the Thompson metric

d

T

in the general case of a linear space

X

ordered by a cone

K

. We show that

d

T

has monotonicity properties which make it compatible with the linear structure. We also prove several convexity properties of

d

T

, and some results concerning the topology of

d

T

, including a brief study of the

d

T

-convergence of monotone sequences. It is shown that most results are true without any assumption of an Archimedean-type property for

K

. One considers various completeness properties and one studies the relations between them. Since

d

T

is defined in the context of a generic ordered linear space, with no need of an underlying topological structure, one expects to express its completeness in terms of properties of the ordering with respect to the linear structure. This is done in this paper and, to the best of our knowledge, this has not been done yet. Thompson metric

d

T

and order-unit (semi)norms | ⋅ | 

u

are strongly related and share important properties, as both are defined in terms of the ordered linear structure. Although

d

T

and | ⋅ | 

u

are only topologically (and not metrically) equivalent on

K

u

, we prove that the completeness is a common feature. One proves the completeness of the Thompson metric on a sequentially complete normal cone in a locally convex space. At the end of the paper, it is shown that, in the case of a Banach space, the normality of the cone is also necessary for the completeness of the Thompson metric.

Ştefan Cobzaş, Mircea-Dan Rus
Functional Operators and Approximate Solutions of Functional Equations

In this paper we consider the problem of approximate solutions of functional equations.In the first part of this chapter we present the integral least squares method for functional equations.The second part is devoted to investigations on some functional operators, useful for both theory and applications.Finally, in the last one we present some results on convergence of a sequence of approximate solutions of a functional equation obtained by the integral least squares method as well as some estimations of the errors of approximations.

Stefan Czerwik, Krzysztof Król
Markov-Type Inequalities with Applications in Multivariate Approximation Theory

In this paper, we provide a brief overview of several refinements and applications of the Markov-type inequalities in various contexts.

Nicholas J. Daras
The Number of Prime Factors Function on Shifted Primes and Normal Numbers

In a series of papers, we have constructed large families of normal numbers using the concatenation of the values of the largest prime factor

P

(

n

), as

n

runs through particular sequences of positive integers. A similar approach using the smallest prime factor function also allowed for the construction of normal numbers. Letting

ω

(

n

) stand for the number of distinct prime factors of the positive integer

n

, we show that the concatenation of the successive values of

|

ω

(

n

)

log

log

n

|

$$\vert \omega (n) -\lfloor \log \log n\rfloor \vert $$

, as

n

runs through the integers

n

 ≥ 3, yields a normal number in any given basis

q

 ≥ 2. We show that the same result holds if we consider the concatenation of the successive values of

|

ω

(

p

+

1

)

log

log

(

p

+

1

)

|

$$\vert \omega (p + 1) -\lfloor \log \log (p + 1)\rfloor \vert $$

, as

p

runs through the prime numbers.

Jean-Marie De Koninck, Imre Kátai
Imbedding Inequalities for Composition of Green’s and Potential Operators

In this paper, we prove both local and global imbedding inequalities with

L

φ

$$L^{\varphi }$$

-norms for the composition of the potential operator and Green’s operator applied to differential forms.

Shusen Ding, Yuming Xing
On Approximation Properties of q-King Operators

Based on

q

-integers we introduce the

q

-King operators which approximate each continuous function on [0, 1] and preserve the functions

e

0

(

x

) = 1 and

e

j

(

x

) = 

x

j

. We also construct a

q

-parametric sequence of polynomial bounded positive linear operators possessing similar properties. In both cases the rate of convergence is estimated with the aid of the modulus of continuity.

Zoltán Finta
Certain Szász-Mirakyan-Beta Operators

In the present article we discuss direct estimates of the Durrmeyer type modifications of the well-known Szász-Mirakyan operators. The present article is divided into two sections. In the first section, we mention some of the different integral modifications of the Szász-Mirakyan operators and mention their direct results which were done in ordinary, and specially in the simultaneous approximation.

In the second section for the Szász-Mirakyan-Beta operators, we find the alternate hypergeometric representation and propose their Stancu type generalization based on two parameters. We obtain the moments using confluent Hypergeometric functions. Also it is observed here that the moments are related to the Laguerre polynomials. We study direct approximation results for these Szász-Mirakyan-Beta-Stancu operators, which include a Voronovskaja-type asymptotic formula and error estimations in terms of modulus of continuity.

N. K. Govil, Vijay Gupta, Danyal Soybaş
Extremal Problems and g-Loewner Chains in ℂ n $$\mathbb{C}^{n}$$ and Reflexive Complex Banach Spaces

Let

X

be a reflexive complex Banach space with the unit ball

B

. In the first part of the paper, we survey various growth and coefficient bounds for mappings in the Carathéodory family

$$\mathcal{M}$$

, which plays a key role in the study of the generalized Loewner differential equation. Then we consider recent results in the theory of Loewner chains and the generalized Loewner differential equation on the unit ball of

n

$$\mathbb{C}^{n}$$

and reflexive complex Banach spaces. In the second part of this paper, we obtain sharp growth theorems and second coefficient bounds for mappings with

g

-parametric representation and we present certain particular cases of special interest. Finally, we consider extremal problems related to bounded mappings in

S

g

0

(

B

n

)

$$S_{g}^{0}(B^{n})$$

, where

B

n

is the Euclidean unit ball in

n

$$\mathbb{C}^{n}$$

. To this end, we use ideas from control theory to investigate the (normalized) time-log

M

-reachable family

̃

log

M

(

id

B

n

,

g

)

$$\tilde{\mathcal{R}}_{\log M}(\mathrm{id}_{B^{n}},\mathcal{M}_{g})$$

generated by a subset

g

$$\mathcal{M}_{g}$$

of

$$\mathcal{M}$$

, where

M

 ≥ 1 and

g

is a univalent function on the unit disc

U

such that

g

(0) = 1,

g

(

ζ

)

>

0

$$\mathfrak{R}g(\zeta ) > 0$$

, | 

ζ

 |  < 1, and which satisfies some natural conditions. We characterize this family in terms of univalent subordination chains, and we obtain certain results related to extreme points and support points associated with the compact family

̃

log

M

(

id

B

n

,

g

)

¯

$$\overline{\tilde{\mathcal{R}}_{\log M}(\mathrm{id}_{B^{n}},\mathcal{M}_{g})}$$

. Also, we give some examples of mappings in

̃

log

M

(

id

B

n

,

g

)

$$\tilde{\mathcal{R}}_{\log M}(\mathrm{id}_{B^{n}},\mathcal{M}_{g})$$

and obtain the sharp growth result for this family.

Ian Graham, Hidetaka Hamada, Gabriela Kohr
Different Durrmeyer Variants of Baskakov Operators

The present article deals with the different Durrmeyer type modifications of the well-known Baskakov. These operators came into existence almost 28 years ago when in the year 1985 the Baskakov Durrmeyer operators were introduced. After that several approximation properties of such operators were studied extensively. The present article is an attempt to present some of the results and the approximation properties of the different Durrmeyer type modifications of the classical Baskakov operators. We also give here the alternate form of some of the operators in terms of hypergeometric functions. In the last section, we present some results for mixed operators related to convergence.

Vijay Gupta
Hypergeometric Representation of Certain Summation–Integral Operators

The general sequence of the summation-integral type operators was proposed by Srivastava and Gupta [Math. Comput. Modelling 37(12–13)(2003), 1307–1315]. In the present article we give the alternate forms of such operators in terms of hypergeometric series. We also obtain moments using hypergeometric series. Finally we obtain the rate of convergence for functions having bounded derivatives.

Vijay Gupta, Themistocles M. Rassias
On a Hybrid Fourth Moment Involving the Riemann Zeta-Function

For each integer 1 ≤ 

j

 ≤ 6, we provide explicit ranges for

σ

for which the asymptotic formula

0

T

ζ

1

2

+

i

t

4

|

ζ

(

σ

+

i

t

)

|

2

j

d

t

T

k

=

0

4

a

k

,

j

(

σ

)

log

k

T

$$\displaystyle{\int _{0}^{T}\left \vert \zeta \left (\frac{1} {2} + it\right )\right \vert ^{4}\vert \zeta (\sigma +it)\vert ^{2j}dt \sim T\sum _{ k=0}^{4}a_{ k,j}(\sigma )\log ^{k}T}$$

holds as

T

 → 

, where

ζ

(

s

) is the Riemann zeta-function. The obtained ranges improve on an earlier result of the authors. An application to a weighted divisor problem is also given.

Aleksandar Ivić, Wenguang Zhai
On the Invertibility of Some Elliptic Operators on Manifolds with Boundary and Cylindrical Ends

In this paper we perform several steps towards the layer potential theory for the Brinkman system on manifolds with boundary and cylindrical ends. In addition, we refer to the Dirichlet problem for a Laplace type operator on parallelizable manifolds with cylindrical ends.

Mirela Kohr, Cornel Pintea
Meaned Spaces and a General Duality Principle

We present a new duality principle, in which we do not suppose that the range of the functions to be optimized is a subset of a linear space. The methods used in the proofs of our results are based on the notion of meaned space, which is a generalization of the notion of ordered linear space.

József Kolumbán, József J. Kolumbán
An AQCQ-Functional Equation in Matrix Random Normed Spaces

In this paper, we prove the Hyers–Ulam stability of the following additive-quadratic-cubic-quartic functional equation

f

(

x

+

2

y

)

+

f

(

x

2

y

)

=

4

f

(

x

+

y

)

+

4

f

(

x

y

)

6

f

(

x

)

+

f

(

2

y

)

+

f

(

2

y

)

4

f

(

y

)

4

f

(

y

)

$$\displaystyle\begin{array}{rcl} & & f(x + 2y) + f(x - 2y) {}\\ & & \quad = 4f(x + y) + 4f(x - y) - 6f(x) + f(2y) + f(-2y) - 4f(y) - 4f(-y) {}\\ \end{array}$$

in matrix random normed spaces.

Jung Rye Lee, Choonkil Park, Themistocles M. Rassias
A Planar Location-Allocation Problem with Waiting Time Costs

We study a location-allocation problem where the social planner has to locate some new facilities minimizing the social costs, i.e. the fixed costs plus the waiting time costs, taking into account that the citizens are partitioned in the region according to minimizing the capacity acquisition costs plus the distribution costs in the service regions. In order to find the optimal location of the new facilities and the optimal partition of the consumers, we consider a two-stage optimization model. Theoretical and computational aspects of the location-allocation problem are discussed for a planar region and illustrated with examples.

L. Mallozzi, E. D’Amato, Elia Daniele
The Stability of an Affine Type Functional Equation with the Fixed Point Alternative

In this paper, we consider the following affine functional equation

f

(

3

x

+

y

+

z

)

+

f

(

x

+

3

y

+

z

)

+

f

(

x

+

y

+

3

z

)

+

f

(

x

)

+

f

(

y

)

+

f

(

z

)

=

6

f

(

x

+

y

+

z

)

.

$$\displaystyle{f(3x+y+z)+f(x+3y+z)+f(x+y+3z)+f(x)+f(y)+f(z) = 6f(x+y+z).}$$

We obtain the general solution and establish some stability results by using

direct method

as well as

the fixed point method

. Further we define the stability of the above functional equation by using

the fixed point alternative

.

M. Mursaleen, Khursheed J. Ansari
On Some Integral Operators

Let

P

(

n

,

β

), 0 ≤

β

< 1, be the class of functions

p

:

p

(

z

)

=

1

+

c

n

z

n

+

c

n

+

1

z

n

+

1

+

$$p: p(z) = 1 + c_{n}z^{n} + c_{n+1}z^{n+1}+\ldots$$

analytic in the unit disc

E

such that

Re

{

p

(

z

)} >

β

. The class

P

k

(

n

,

β

),

k

≥ 2 is defined as follows: An analytic function

p

P

k

(

n

,

β

),

k

≥ 2, 0 ≤

β

< 1 if and only if there exist

p

1

,

p

2

P

(

n

,

β

)

$$p_{1},p_{2} \in P(n,\beta )$$

such that

p

(

z

)

=

k

4

+

1

2

p

1

(

z

)

k

4

1

2

p

2

(

z

)

.

$$\displaystyle{p(z) = \left (\frac{k} {4} + \frac{1} {2}\right )p_{1}(z) -\left (\frac{k} {4} -\frac{1} {2}\right )p_{2}(z).}$$

In this paper, we discuss some integral operators for certain classes of analytic functions defined in

E

and related with the class

P

k

(

n

,

β

).

Khalida Inayat Noor
Integer Points in Large Bodies

For a compact body

$$\mathcal{B}$$

in three-dimensional Euclidean space with sufficiently smooth boundary, the number

N

(

;

t

)

$$N(\mathcal{B};t)$$

of points with integer coordinates in a linearly enlarged copy

t

$$t\mathcal{B}$$

is approximated in first order by the volume

v

o

l

(

)

t

3

$$\mathrm{vol}(\mathcal{B})t^{3}$$

. This article provides a survey on the state of art of research on the

lattice discrepancy

D

(

;

t

)

=

N

(

;

t

)

v

o

l

(

)

t

3

$$D(\mathcal{B};t) = N(\mathcal{B};t) -\mathrm{vol}(\mathcal{B})t^{3}$$

, starting from the classic theory and emphasizing recent developments and advances.

Werner Georg Nowak
On the Orderability Problem and the Interval Topology

The class of LOTS (linearly ordered topological spaces, i.e. spaces equipped with a topology generated by a linear order) contains many important spaces, like the set of real numbers, the set of rational numbers and the ordinals. Such spaces have rich topological properties, which are not necessarily hereditary. The Orderability Problem, a very important question on whether a topological space admits a linear order which generates a topology equal to the topology of the space, was given a general solution by van Dalen and Wattel (Gen. Topol. Appl. 3:347–354, 1973). In this article we first examine the role of the interval topology in van Dalen’s and Wattel’s characterization of LOTS, and we then discuss ways to extend this model to transitive relations that are not necessarily linear orders.

Kyriakos Papadopoulos
A Class of Functional-Integral Equations with Applications to a Bilocal Problem

Let

α

 ≤ 

a

 < 

b

 ≤ 

β

be some real numbers,

K

:

[

α

,

β

]

×

[

α

,

β

]

×

[

α

,

β

]

×

[

α

,

β

]

×

m

m

$$K: [\alpha,\beta ] \times [\alpha,\beta ] \times [\alpha,\beta ] \times [\alpha,\beta ] \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{m}$$

and

g

:

[

α

,

β

]

m

$$g: [\alpha,\beta ] \rightarrow \mathbb{R}^{m}$$

be continuous functions. In this work, using the Picard operator technique in a

+

m

$$\mathbb{R}_{+}^{m}$$

-metric space, we study the following functional-integral equation

x

(

t

)

=

a

b

K

(

t

,

s

,

a

,

b

,

x

(

s

)

)

d

s

+

g

(

t

)

,

t

[

α

,

β

]

.

$$\displaystyle{x(t) =\int _{ a}^{b}K(t,s,a,b,x(s))ds + g(t),\ t \in [\alpha,\beta ].}$$

As an application, the following bilocal problem

x

(

t

)

+

p

x

(

t

)

+

q

x

(

t

)

=

f

(

t

,

x

(

t

)

)

,

t

[

α

,

β

]

,

x

(

a

)

=

0

,

x

(

b

)

=

0

.

$$\displaystyle{-x''(t) + px'(t) + qx(t) = f(t,x(t)),\ t \in [\alpha,\beta ],\ \ x(a) = 0,x(b) = 0.}$$

is also discussed.

Adrian Petruşel, Ioan A. Rus
Hyperbolic Wavelets

In the last two decades a number of different types of wavelets transforms have been introduced in various areas of mathematics, natural sciences, and technology. These transforms can be generated by means of a uniform principle based on the machinery of harmonic analysis. In this way we pass from the affine group to the wavelet transforms, from the Heisenberg group to the Gábor transform. Taking the congruences of the hyperbolic geometry and using the same method we introduced the concept of hyperbolic wavelet transforms (HWT). These congruences can be expressed by Blaschke functions, which play an eminent role not only in complex analysis but also in control theory. Therefore we hope that the HWT will become an adequate tool in signal and system theories. In this paper we give an overview on some results and applications concerning HWT.

F. Schipp
One Hundred Years Uniform Distribution Modulo One and Recent Applications to Riemann’s Zeta-Function

We start with a brief account of the theory of uniform distribution modulo one founded by Weyl and others around 100 years ago (which is neither supposed to be complete nor historically depleting the topic). We present a few classical implications to diophantine approximation. However, our main focus is on applications to the Riemann zeta-function. Following Rademacher and Hlawka, we show that the ordinates of the nontrivial zeros of the zeta-function

ζ

(

s

) are uniformly distributed modulo one. We conclude with recent investigations concerning the distribution of the roots of the equation

ζ

(

s

) = 

a

, where

a

is any complex number, and further questions about such uniformly distributed sequences.

Jörn Steuding
On the Energy of Graphs

The energy of a graph,

E

(

G

), is the sum of the absolute values of its eigenvalues. The energy concept has received a high interest over the last decade, at first due to its various applications in chemistry and then in its own right. This paper focuses on some of the most important results on the bounds for the energy of general graphs and the energy of bipartite graphs. Some known bounds for the change in the energy of a graph after deleting a vertex or an edge are also considered.

Irene Triantafillou
Implicit Contractive Maps in Ordered Metric Spaces

In Part 1, we show that most of the implicit contractions introduced by Wardowski [Fixed Point Theory Appl., 2012, 2012:94] are Matkowski type contractions. In Part 2, some limit type extensions are obtained for the fixed point result (involving implicit contractions) due to Altun and Simsek [Fixed Point Theory Appl., Volume 2010, Article ID 621469]. Moreover, the connections with a lot of related statements in the area due to Agarwal, El-Gebeily, and O’Regan [Appl. Anal. 87:109–116, 2008] are also discussed. Finally, in Part 3, a non-limit counterpart of these results is given, under the same general context.

Mihai Turinici
Higher Dimensional Continuous Wavelet Transform in Wiener Amalgam Spaces

Norm convergence and convergence at Lebesgue points of the inverse wavelet transform are obtained for functions from the

L

p

and Wiener amalgam spaces.

Ferenc Weisz
Multidimensional Hilbert-Type Integral Inequalities and Their Operators Expressions

In this chapter, by the use of the methods of weight functions and techniques of Real Analysis, we provide a general multidimensional Hilbert-type integral inequality with a non-homogeneous kernel and a best possible constant factor. The equivalent forms, the reverses and some Hardy-type inequalities are obtained. Furthermore, we consider the operator expressions with the norm, some particular inequalities with the homogeneous kernel and a large number of particular examples.

Bicheng Yang
Metadaten
Titel
Topics in Mathematical Analysis and Applications
herausgegeben von
Themistocles M. Rassias
László Tóth
Copyright-Jahr
2014
Electronic ISBN
978-3-319-06554-0
Print ISBN
978-3-319-06553-3
DOI
https://doi.org/10.1007/978-3-319-06554-0