Swati Antal, U. C. Gairola
In this paper we introduce the notion of a Caristi-Banach type $\mathcal{Z}_{R}^{b}$- contraction in the framework of $b$- metric space endowed with a transitive relation that combine the ideas of Caristi type contraction and Banach contraction with a help of simulation function. We present an example to clarify the statement of the given result.
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In this paper, we show that several extension of Banach contraction principle, can be easily derived from the Caristi's theorem is one of the useful generalization of Banach contraction principle in the setting of the complete metric spaces. Moreover, some partial answers to some known open problems are given via Caristi's corollaries. Finally, existence of bounded solutions of a functional equation is studied to support our results.
Summable Orbits and the Minimal Caristi Potential
本文提出轨道可和不动点准则，证明其与卡拉蒂定理等价，并恢复Banach收缩原理作为推论，为固定点理论提供统一框架。
In this paper, using a more generalized inequality instead of triangle inequality, the notion of θ-metric space is introduced. Some important properties of induced topology by such spaces are presented. Also, Banach and Caristi type fixed point in such spaces are proved.
Our aim in this paper is to present a new type of the modular space. This space contains the classical modular space. There are some mappings that do not have contractive condition in the usual modular space but become contraction in this new space.
In this paper, we present the generalization of B-contraction and C-contraction due to Sehgal and Hicks respectively. We also study some properties of C-contraction in probabilistic metric space.
In this paper, we investigate the existence and uniqueness of fixed point for partially ordered contraction type operators in Banach Space. We also present applications to integral and differential equations.
We present versions of ergodic theorems for $L_1$--$L_\infty$ contractions in Banach--Kantorovich $L_p$-lattices associated with the Maharam measure taking values in the algebra of measurable functions.
We introduce the concept of weak L-P property for a pair of Banach spaces, in the study of extreme contractions. We give examples of pairs of Banach spaces (not) satisfying weak L-P property and apply the concept to compute the exact number of extreme contractions between a particular pair of polygonal Banach spaces. We also study the optimality of our results on the newly introduced weak L-P property for a pair of Banach spaces, by considering appropriate examples.
Let $X$ be a reflexive Banach space such that for any $x \ne 0$ the set $$ \{x^* \in X^*: \text {$\|x^*\|=1$ and $x^*(x)=\|x\|$}\} $$ is compact. We prove that any unrestricted product of of a finite number of $(W)$ contractions on $X$ converges weakly.
Using the technique of enrichment of contractive type mappings by Krasnoselskij averaging, introduced in [Berinde, V., {\it Approximating fixed points of enriched nonexpansive mappings by Krasnoselskij iteration in Hilbert spaces}, Carpathian J. Math. {\bf 35} (2019), no. 3, 277-288.], we introduce the class of enriched Chatterjea contractions and prove general fixed point theorems for such contractions in the setting of a Banach space. Examples to illustrate the richness of the new class of contractions and the relationship between enriched Banach contractions, enriched Kannan contractions an…
In the present paper we study majorizable operators acting on Banach-Kantorovich $L_p$-lattices, constructed by a measure $m$ with values in the ring of all measurable functions. Then using methods of measurable bundles of Banach-Kantorovich lattices, we prove the strong "zero-two" law for positive contractions of the Banach-Kantorovich $L_p$-lattices.
In the present paper we prove the "zero-two" law for positive contractions of lattices $L^p(\nabla,μ)$ of Banach-Kantorovich, constructed by the measure $μ$ with values in the ring of all measurable functions.
We introduce a large class of mappings, called enriched contractions, which includes, amongst many other contractive type mappings, the Picard-Banach contractions and some nonexpansive mappings. We show that any enriched contraction has a unique fixed point and that this fixed point can be approximated by means of an appropriate Kransnoselskij iterative scheme. Several important results in fixed point theory are shown to be corollaries or consequences of the main results in this paper. We also study the fixed points of local enriched contractions, asymptotic enriched contractions and Maia type…
We introduce a large class of contractive mappings, called Suzuki Berinde type contraction. We show that any Suzuki Berinde type contraction has a fixed point and characterizes the completeness of the underlying normed space. A fixed point theorem for multivalued mapping is also obtained. These results unify, generalize and complement various known comparable results in the literature.
In this paper, by using admissible sets, we give some fixed point results for orbitally contractions which diminish the radius of invariant convex subsets and orbits. Furthermore, a characterization of the weak normal structure by the fixed point property associated with this class of mappings is established.
A Restatement of Fixed Point Existence: From Banach to Singh Contractions
本文重述二阶边值问题的存在唯一性结果，引入Singh压缩条件（允许T^p满足条件），展示广义压缩在Banach压缩原理失效时的应用。
In this paper, we introduce two new types of enriched contractions, viz., enriched $\mathcal{A}$-contraction and enriched $\mathcal{A}'$-contraction. Then we obtain fixed points of mappings satisfying such contractions using the fixed point property of the average operator of the mappings. Further, we study the well-posedness and limit shadowing property of the fixed point problem involving the contractions, and give some examples to validate the results proved. We frame an open question related to the existence of a fixed point of such contractions. We also show that Berinde and Păcurar's rec…
Convergence of C-Class Enriched Hardy-Rogers Contractions in some Banach Spaces
本文利用C类函数推广富集Hardy-Rogers压缩映射，在Banach空间中通过迭代格式建立唯一不动点及公共不动点的收敛性，推广了近期相关结果。
Let $X$ be a complex Banach space and let $J:X \to X^*$ be a duality section on $X$ (i.e. $\langle x,J(x)\rangle=\|J(x)\|\|x\|=\|J(x)\|^2=\|x\|^2$). For any unit vector $x$ and any ($C_0$) contraction semigroup $T=\{e^{tA}:t \geq 0\}$, Goldstein proved that if $X$ is a Hilbert space and if $|\langle T(t) x,J(x)\rangle| \to 1 $ as $t \to \infty$, then $x$ is an eigenvector of $A$ corresponding to a purely imaginary eigenvalue. In this article, we prove the similar result holds if $X$ is a strictly convex complex Banach space.
Extreme contractions on finite-dimensional Banach spaces
本研究聚焦有限维多面体Banach空间的极端压缩算子，受Krein-Milman定理启发，证明二维定义域下及ℓ_∞^n(ℂ)到ℓ_1^n(ℂ)的算子空间中，秩一范一算子可表为秩一极端压缩算子的凸组合，并给出该类算子的几何刻画。
Path--Averaged Contractions: A New Generalization of the Banach Contraction Principle
该研究提出新型自映射PA-压缩（路径平均压缩），证明完备度量空间上连续PA-压缩有唯一不动点且Picard迭代收敛，严格推广Banach压缩原理，验证其与多种经典压缩映射独立，可捕捉逐点压缩失效时的长期压缩行为。
Extreme contractions on finite-dimensional polygonal Banach spaces
本文研究有限维多边形Banach空间之间极端压缩算子的范数可达性，证明其范数在单位球n个线性无关极端点上可达，并引入L-P性质刻画严格凸Banach空间与Hilbert空间的关系。
On extreme contractions between real Banach spaces
本文通过引入相容点对（CPP）和 μ-相容点对（μ-CPP）的新几何概念，首次完全刻画了二维严格凸光滑实 Banach 空间之间的极端压缩算子，并应用于具体空间及无穷维情形，进而用 CPP 刻画了实 Hilbert 空间。
Cyclic p-$ϕ$-contraction mappings
本文提出循环p-φ压缩映射的概念，并研究其在不同空间中的固定点存在性和唯一性。
Characterization of extreme contractions through $k-$smoothness of operators
该研究利用算子$k$-光滑性刻画有限维多面体Banach空间间的极端压缩算子，探讨弱L-P性质并给出其满足的充分条件，还显式计算了特殊Banach空间中极端压缩算子的数量，改进推广了已有结果。
Perimetric contraction on quadrilaterals and related fixed point results
本文提出Banach、Kannan、Chatterjea三类收缩的四点类似形式，推导其在完备度量空间中存在不动点的充分条件，证明经典Banach收缩原理为其特例，并给出非平凡示例验证理论结果。
A converse of the Banach contraction principle for partial metric spaces and the continuum hypothesis
研究将 Banach 压缩原理的 Bessaga 逆命题推广至偏度量空间，并证明其相应版本与连续统假设等价。
We augment the dimension of the Euclidean space by one and the Picard iteration of a contraction by a simple iteration on the real line such that the resulting iteration becomes monotone increasing and bounded with respect to the order defined by the Lorentz cone of the augmented space. This provides a different way of showing the convergence of the Picard iteration of a contraction, exhibiting the strong relationship between the Banach fixed point principle and the ordering structure of the Euclidean space ordered by the Lorentz cone.
It is well known that fixed point problems of contractive-type mappings defined on cone metric spaces over Banach algebras are not equivalent to those in usual metric spaces (see [3] and [10]). In this framework, the novelty of the present paper represents the development of some fixed point results regarding sequences of contractions in the setting of cone metric spaces over Banach algebras. Furthermore, some examples are given in order to strengthen our new concepts. Also, based on the powerful notion of a cone metric space over a Banach algebra, we present important applications to systems …
For any complex Banach space $X$, let $J$ denote the duality mapping of $X$. For any unit vector $x$ in $X$ and any ($C_0$) contraction semigroup $(T_t)_{t>0}$ on $X$, Baillon and Guerre-Delabriere proved that if $X$ is a smooth reflexive Banach space and if there is $x^* \in J(x)$ such that $|\langle T(t) \, x,J(x)\rangle| \to 1 $ as $t \to \infty$, then there is a unit vector $y\in X$ which is an eigenvector of the generator $A$ of $(T_t)_{t>0}$ associated with a purely imaginary eigenvalue. They asked whether this result is still true if $X$ is replaced by $c_o$. In this article, we show th…
There are several extensions of the classical Banach Fixed Point Theorem in technical literature. A branch of generalizations replaces usual contractivity by weaker but still effective assumptions. Our note follows this stream, presenting an elementary proof for a known fixed point result. Some applications are also considered.
Geodesic contraction in vector-valued differential equations is readily verified by linearized operators which are uniformly negative-definite in the Riemannian metric. In the infinite-dimensional setting, however, such analysis is generally restricted to norm-contracting systems. We develop a generalization of geodesic contraction rates to Banach spaces using a smoothly-weighted semi-inner product structure on tangent spaces. We show that negative contraction rates in bijectively weighted spaces imply asymptotic norm-contraction, and apply recent results on asymptotic contractions in Banach s…
The famous Banach Contraction Principle holds in complete metric spaces, but completeness is not a necessary condition -- there are incomplete metric spaces on which every contraction has a fixed point. The aim of this paper is to present various circumstances in which fixed point results imply completeness. For metric spaces this is the case of Ekeland variational principle and of its equivalent - Caristi fixed point theorem. Other fixed point results having this property will be also presented in metric spaces, in quasi-metric spaces and in partial metric spaces. A discussion on topology and…
On the Picard-Lindelöf Argument and the Banach-Caccioppoli Contraction Mapping Principle
本文通过改进收缩映射原理，实现了对皮卡-林德勒夫定理中收敛速率的精确恢复。
Best Proximity Point Results for Cyclic Orbital Contraction Mappings in $CAT_p(0)$ Metric Spaces
本文在CAT_p(0)度量空间中研究了循环轨道收缩映射的最佳接近点存在性，并探讨了其在一致凸巴拿赫空间中的应用。
A branch of generalizations of the Banach Fixed Point Theorem replaces contractivity by a weaker but still effective property. The aim of the present note is to extend the contraction principle in this spirit for such complete semimetric spaces that fulfill an extra regularity property. The stability of fixed points is also investigated in this setting. As applications, fixed point results are presented for several important generalizations of metric spaces.
Contraction rates of time-varying maps induced by dynamical systems illuminate a wide range of asymptotic properties with applications in stability analysis and control theory. In finite-dimensional smoothly varying inner-product spaces such as $\mathbb{R}^n$ and $\mathbb{C}^n$ with Riemannian metrics, contraction rates can be estimated by upper-bounding the real numerical range of the vector field's Jacobian. However, vector spaces with norms other than $L^2$ commonly arise in the stability analysis of infinite-dimensional systems such as those arising from partial differential equations and …
A dilation theoretic approach to Banach spaces
本文通过膨胀理论研究巴拿赫空间，证明了严格压缩可膨胀为等距映射等条件是空间成为希尔伯特空间的充要条件，并刻画了非希尔伯特空间中的等距膨胀及算子范数达到集等性质。
Further results on $p$-adic semigroup of contractions
本文在超度量Banach空间上，给出了算子$A$的预解式条件，使得对所有自然数$m$，$\\\\|A^m\\\\|\leq 1$，即$A$为压缩算子。
Fixed Point Theorems for Multivalued Weak Contractions in Cone Metric Spaces
本文研究了锥度量空间中多值弱压缩的不动点问题，将Berinde弱压缩原理推广至多值设定，利用正规锥结构证明了广义不动点定理，并提供了迭代逼近格式及在微分包含与变分不等式中的应用。
Another Perspective on Chatterjea Contraction
本文从Chatterjea收缩映射的角度出发，探讨了在保持映射自身不等式左边的情况下，该结论的成立条件，并在k-连续性的假设下，证明了新的收缩映射类m-Chatterjea收缩映射在完备度量空间中存在唯一不动点，并通过例子验证了该类严格包含Chatterjea映射类。
In this article, we discuss a new version of metric fixed point theory especially of Banach Contraction Principle, Ran-Reurings Theorem and others.
We prove that, in a large class of Banach lattices, the fixed space of each commuting family of positive linear contractions is a lattice subspace. As consequences, new cyclicity results for the peripheral point spectra of positive operators and semigroups are derived; we also pose an open problem that naturally occurs in this context. Finally, a variety of counterexamples is presented to point out some limits of our results.
On the weakest conditions for the existence of fixed points of Kannan and Chatterjea type contractions
本文研究Kannan和Chatterjea型收缩映射固定点存在的最弱条件，扩展了Suzuki的方法，证明所提条件最优且确保Picard序列收敛。
If $X$ is a separable reflexive Banach space, there are several natural Polish topologies on $\mathcal{B}(X)$, the set of contraction operators on $X$ (none of which being clearly ``more natural'' than the others), and hence several a priori different notions of genericity -- in the Baire category sense -- for properties of contraction operators. So it makes sense to investigate to which extent the generic properties, i.e. the comeager sets, really depend on the chosen topology on $\mathcal{B}(X)$. In this paper, we focus on $\ell_p\,$-$\,$spaces, $1<p\neq 2<\infty$. We show that for some pair…
We introduce a new class of quasi-Banach spaces as an extension of the classical Grand Lebesgue Spaces for small values of the parameter, and we investigate some its properties, in particular, completeness, fundamental function, operators estimates, Boyd indices, contraction principle, tail behavior, dual space, generalized triangle and quadrilateral constants and inequalities.
Let $T$ be a bounded linear operator on a Hilbert space $H$ such that \[ α[T^*,T]:=\sum_{n=0}^\infty α_n T^{*n}T^n\ge 0. \] where $α(t)=\sum_{n=0}^\infty α_n t^n$ is a suitable analytic function in the unit disc $\mathbb{D}$ with real coefficients. We prove that if $α(t) = (1-t) \tildeα (t)$, where $\tildeα$ has no roots in $[0,1]$, then $T$ is similar to a contraction. Operators of this type have been investigated by Agler, Müller, Olofsson, Pott and others, however, we treat cases where their techniques do not apply. We write down an explicit Nagy-Foias type model of an operator in this clas…
This paper shows the usefulness of Perov's contraction principle, which generalizes Banach's contraction principle to a vector-valued metric, for studying dynamic programming problems in which the discount factor can be stochastic. The discounting condition $β<1$ is replaced by $ρ(B)<1$, where $B$ is an appropriate nonnegative matrix and $ρ$ denotes the spectral radius. Blackwell's sufficient condition is also generalized in this setting. Applications to asset pricing and optimal savings are discussed.
Best approximations in metric spaces with property strongly UC
本文引入强UC性质，证明度量空间中几乎循环ψ-压缩映射最佳逼近的存在性，并推广了已有结果至自反Banach空间。
In this paper we study the existence and uniqueness of best proximity points of cyclic contractions as well as the convergence of iterates to such proximity points. We do it from two different approaches, leading each one of them to different results which complete, if not improve, other similar results in the theory. Results in this paper stand for Banach spaces, geodesic metric spaces and metric spaces. We also include an appendix on CAT(0) spaces where we study the particular behavior of these spaces regarding the problems we are concerned with.
Generalized multivariate Fractal Interpolation Function and $α$-Fractal Function
本文提出基于Matkowski和Rakotch压缩映射的新方法，推广多元分形插值函数的已有构造技术，建立非线性迭代函数系统吸引子为插值理论数据点的多元函数图像，并探讨Rakotch压缩下的不变Borel概率测度存在性。
In this article we extend the notion of orthogonal metric space to weak orthogonal metric space. Then we establish fixed point results for a mapping satisfying a more general contraction condition. Several nontrivial examples are given in support of our obtained results. Moreover, we are able to answer of the open question posed by Eshaghi et al. [On orthogonal sets and Banach fixed point theorem, Fixed Point Theory, 18(2017), 569-578].
Existence and uniqueness as well as the iterative approximation of fixed points of enriched almost contractions in Banach spaces are studied. The obtained results are generalizations of the great majority of metric fixed point theorems, in the setting of a Banach space. The main tool used in the investigations is to work with the averaged operator $T_λ$ instead of the original operator $T$. The effectiveness of the new results thus derived is illustrated by appropriate examples. An application of the strong convergence theorems to solving a variational inequality is also presented.
Let $\{T_t\}_{t>0}$ be a strongly continuous semigroup of positive contractions on $L_p(X,μ)$ with $1<p<\infty$. Let $E$ be a UMD Banach lattice of measurable functions on another measure space $(Ω,ν)$. For $f\in L_p(X; E)$ define $$\mathcal M(f)(x, ω)=\sup_{t>0}\frac1t\Big|\int_0^tT_s(f(\cdot,ω))(x)ds\Big|,\quad (x,ω)\in X\timesΩ.$$ Then the following maximal ergodic inequality holds $$\big\|\mathcal M(f)\big\|_{L_p(X; E)}\lesssim \big\|f\big\|_{L_p(X; E)},\quad f\in L_p(X; E).$$ If the semigroup $\{T_t\}_{t>0}$ is additionally assumed to be analytic, then $\{T_t\}_{t>0}$ extends to an analyt…
We consider a large class of geodesic metric spaces, including Banach spaces, hyperbolic spaces and geodesic $\mathrm{CAT}(κ)$-spaces, and investigate the space of nonexpansive mappings on either a convex or a star-shaped subset in these settings. We prove that the strict contractions form a negligible subset of this space in the sense that they form a $σ$-porous subset. For separable metric spaces we show that a generic nonexpansive mapping has Lipschitz constant one at typical points of its domain. These results contain the case of nonexpansive self-mappings and the case of nonexpansive set-…
We prove an implicit function theorem for C^k-maps from arbitrary topological vector spaces over valued fields to Banach spaces (for k at least 2). As a tool, we show the C^k-dependence of fixed points on parameters for suitable families of contractions of a Banach space. Similar results are obtained for k times strictly differentiable maps, and for k times Lipschitz differentiable maps. In the real case, our results subsume an implicit function theorem for Keller C^k_c-maps from arbitrary topological vector spaces to Banach spaces.
In this work we construct certain general bundles $<\mathfrak{M},ρ,X>$ and $<\mathfrak{B},η,X>$ of Hausdorff locally convex spaces associated with a given Banach bundle $<\mathfrak{E},π,X>$. Then we present conditions ensuring the existence of bounded sections $\mathcal{U}\in Γ^{x_{\infty}}(ρ)$ and $\mathcal{P}\in Γ^{x_{\infty}}(η)$ both continuous at a point $x_{\infty}\in X$, such that $\mathcal{U}(x)$ is a $C_{0}-$semigroup of contractions on $\mathfrak{E}_{x}$ and $\mathcal{P}(x)$ is a spectral projector of the infinitesimal generator of the semigroup $\mathcal{U}(x)$, for every $x\in X$.
A unified theory of cone metric spaces and its applications to the fixed point theory
本文提出了一种基于固体向量空间的锥度量空间的统一理论，并应用该理论给出了迭代压缩原理和Banach压缩原理在固体向量空间上的锥度量空间中的完整表述。
Let (e^{tA})_{t \geq 0} be a C_0-contraction semigroup on a 2-smooth Banach space E, let (W_t)_{t \geq 0} be a cylindrical Brownian motion in a Hilbert space H, and let (g_t)_{t \geq 0} be a progressively measurable process with values in the space γ(H,E) of all γ-radonifying operators from H to E. We prove that for all 0<p<\infty there exists a constant C, depending only on p and E, such that for all T \geq 0 we have \E \sup_{0\le t\le T} || \int_0^t e^{(t-s)A} g_s dW_s \ ||^p \leq C \mathbb{E} (\int_0^T || g_t ||_{γ(H,E)}^2 dt)^\frac{p}{2}. For p \geq 2 the proof is based on the observation …
On a minimal Andô dilation for a pair of strict contractions
本文改进安多扩张方法，构造了严格收缩在Hilbert空间上的最小等距扩张，并探讨了Banach空间和更一般规范空间中收缩对的安多扩张条件。
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