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In this work we study strong spectral properties of Ruelle transfer operators related to a large family of Gibbs measures for contact Anosov flows. The ultimate aim is to establish exponential decay of correlations for Hölder observables with respect to a very general class of Gibbs measures. The approach invented in 1997 by Dolgopyat in “On decay of correlations in Anosov flows” and further developed in Stoyanov (2011) is substantially refined here, allowing to deal with much more general situations than before, although we still restrict ourselves to the uniformly hyperbolic case. A rather general procedure is established which produces the desired estimates whenever the Gibbs measure admits a Pesin set with exponentially small tails, that is a Pesin set whose preimages along the flow have measures decaying exponentially fast. We call such Gibbs measures regular. Recent results in Gouëzel and Stoyanov (2019) prove existence of such Pesin sets for hyperbolic diffeomorphisms and flows for a large variety of Gibbs measures determined by Hölder continuous potentials. The strong spectral estimates for Ruelle operators and well-established techniques lead to exponential decay of correlations for Hölder continuous observables, as well as to some other consequences such as: (a) existence of a non-zero analytic continuation of the Ruelle zeta function with a pole at the entropy in a vertical strip containing the entropy in its interior; (b) a Prime Orbit Theorem with an exponentially small error.

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, we present new findings on F-contraction in bipolar p-metric spaces. We establish a covariant Banach-type fixed-point theorem and a contravariant Reich-type fixed-point theorem based on F-contraction in these spaces. Additionally, we include an example that demonstrates the applicability of our results. Our results non-trivially extend this covariant Banach-type fixed-point theorem and contravariant Reich type theorem via the concept of F-contraction.

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.

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.

This paper presents a new coupled fixed-point theorem for a pair of set-valued mappings acting on the Cartesian product of (m1, m2)- and (n1, n2)-quasi-metric spaces. Within the general, non-symmetric quasi-metric setting, we establish the existence of an approximate coupled fixed point. Moreover, under the additional assumption of q0-symmetry, we guarantee the existence of a coupled fixed point. Together, these results extend and unify several known theorems in fixed-point theory for quasi-metric and asymmetric spaces. We illustrate the obtained results regarding fixed points when the underlying space is equipped with a graph structure and, thus, sufficient conditions are found to guarantee the existence of a subgraph with a loop with a length greater than or equal to 2.

This paper establishes existence and uniqueness theorems for fixed points in complete G-metric spaces under new contractive conditions. Our approach utilizes a parameter α∈[0,2] and a function mapping into a finite subset of [0,1/2), incorporating various G-metric distances between points and their images. A primary contribution of this work is the identification and rectification of critical logical flaws and gaps in the proofs presented in previous studies. By providing a more rigorous analytical framework, we re-establish the fixed-point results and extend them to include the iterates of the mapping. These results strengthen the theoretical foundation of fixed-point theory in generalized metric structures.

By using the fractional Caputo–Fabrizio derivative, we investigate a new version for the mathematical model of HIV. In this way, we review the existence and uniqueness of the solution for the model by using fixed point theory. We solve the equation by a combination of the Laplace transform and homotopy analysis method. Finally, we provide some numerical analytics and comparisons of the results.

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.

The aim of this study is to prove the existence and uniqueness of fixed point and common fixed point theorems for self-mappings in modular ultrametric spaces. These theorems are proved under varying contractive circumstances and without the property of spherical completeness. As a consequence, the examples of fixed point and common fixed point problems are correctly formulated. As an application, the well-posedness of a common fixed point problem is proved. This study expands on prior research in modular ultrametric space to provide a more comprehensive understanding of such spaces using generalized contraction.