In the past few decades, several interesting problems have been solved using fixed point theory. In addition to classical ordinary differential equations and integral equation, researchers also focus ...on fractional differential equations (FDE) and fractional integral equations (FIE). Indeed, FDE and FIE lead to a better understanding of several physical phenomena, which is why such differential equations have been highly appreciated and explored. We also note the importance of distinct abstract spaces, such as quasi-metric, b-metric, symmetric, partial metric, and dislocated metric. Sometimes, one of these spaces is more suitable for a particular application. Fixed point theory techniques in partial metric spaces have been used to solve classical problems of the semantic and domain theory of computer science. This book contains some very recent theoretical results related to some new types of contraction mappings defined in various types of spaces. There are also studies related to applications of the theoretical findings to mathematical models of specific problems, and their approximate computations. In this sense, this book will contribute to the area and provide directions for further developments in fixed point theory and its applications.
In this paper, an iterative algorithm is introduced to solve the split common fixed point problem for asymptotically nonexpansive mappings in Hilbert spaces. The iterative algorithm presented in this ...paper is shown to possess strong convergence for the split common fixed point problem of asymptotically nonexpansive mappings although the mappings do not have semi-compactness. Our results improve and develop previous methods for solving the split common fixed point problem. MSC: 47H09, 47J25.
Let
T
be a self-mapping on a complete metric space (
X
,
d
). In this paper, we obtain new fixed point theorems assuming that
T
satisfies a contractive-type condition of the following form:
ψ
(
d
(
...T
x
,
T
y
)
)
≤
φ
(
d
(
x
,
y
)
)
or
T
satisfies a generalized contractive-type condition of the form
ψ
(
d
(
T
x
,
T
y
)
)
≤
φ
(
m
(
x
,
y
)
)
,
where
ψ
,
φ
:
(
0
,
∞
)
→
R
and
m
(
x
,
y
) is defined by
m
(
x
,
y
)
=
max
d
(
x
,
y
)
,
d
(
x
,
T
x
)
,
d
(
y
,
T
y
)
,
d
(
x
,
T
y
)
+
d
(
y
,
T
x
)
/
2
.
In both cases, the results extend and unify many earlier results. Among the other results, we prove that recent fixed point theorems of Wardowski (2012) and Jleli and Samet (2014) are equivalent to a special case of the well-known fixed point theorem of Skof (1977).
In this paper, inspired by the concept of b-metric space, we introduce the concept of extended b-metric space. We also establish some fixed point theorems for self-mappings defined on such spaces. ...Our results extend/generalize many pre-existing results in literature.
Operator-splitting methods convert optimization and inclusion problems into fixed-point equations; when applied to convex optimization and monotone inclusion problems, the equations given by ...operator-splitting methods are often easy to solve by standard techniques. The hard part of this conversion, then, is to design nicely behaved fixed-point equations. In this paper, we design a new, and thus far, the only nicely behaved fixed-point equation for solving monotone inclusions with three operators; the equation employs resolvent and forward operators, one at a time, in succession. We show that our new equation extends the Douglas-Rachford and forward-backward equations; we prove that standard methods for solving the equation converge; and we give two accelerated methods for solving the equation.
In this paper, we discuss the existence of mild solution of functional integrodifferential equation with nonlocal conditions. To establish this results by using the resolvent operator theory and ...Sadovskii‐Krasnosel'skii type of fixed point theorem and to show the usefulness and the applicability of our results to a broad class of functional integrodifferential equations, an example is given to illustrate the theory.
Fixed Point Strategies in Data Science Combettes, Patrick L.; Pesquet, Jean-Christophe
IEEE transactions on signal processing,
2021, Letnik:
69
Journal Article
Recenzirano
Odprti dostop
The goal of this article is to promote the use of fixed point strategies in data science by showing that they provide a simplifying and unifying framework to model, analyze, and solve a great variety ...of problems. They are seen to constitute a natural environment to explain the behavior of advanced convex optimization methods as well as of recent nonlinear methods in data science which are formulated in terms of paradigms that go beyond minimization concepts and involve constructs such as Nash equilibria or monotone inclusions. We review the pertinent tools of fixed point theory and describe the main state-of-the-art algorithms for provenly convergent fixed point construction. We also incorporate additional ingredients such as stochasticity, block-implementations, and non-Euclidean metrics, which provide further enhancements. Applications to signal and image processing, machine learning, statistics, neural networks, and inverse problems are discussed.
PDF
