Félix Cabello Sánchez, Javier Cabello Sánchez
Motivated by some recent twaddles on Mazur rotations problem, we study the "dynamics" of the semigroup of contractive automorphisms of Banach spaces, mostly in finite-dimensional spaces. We focus on the metric aspects of the "action" of such semigroups, the size of the orbits and semitransitivity properties, and their impact on the geometry of the unit ball of the underlying space.
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Bounds for contractive semigroups and second order systems
如果两个压缩半群的生成元的差是由生成元的厄米特部分形式有界的，则得到了它们的差的一致有界性。我们构造了具有相当一般的算子系数的二阶系统的半群动力学，并将我们的界应用于阻尼项的扰动。结果示出了一个耗散波动方程。从而证明了某些二阶系统的指数衰减性。
On pure contractive semigroups
本文研究了 Hilbert 空间上的纯压缩半群，求出其换位子，并证明任意双重交换纯压缩半群元组可扩张为双重交换纯等距半群元组，进而得到等距半群元组的完整模型。
Dilation theorems for contractive semigroups
这篇笔记记录了关于希尔伯特空间上的压缩半群的一些膨胀定理--所有这些都属于“已知”或“可能已知”的范畴--这是我在攻读数学博士学位时（在巴鲁克·索莱尔的指导下）证明的。把它们记录下来以供参考是很方便的。
A result by Liskevich and Perelmuter from 1995 yields the optimal angle of analyticity for symmetric submarkovian semigroups on $L_p$, $1<p<\infty$. C.~Kriegler showed in 2011 that the result remains true without the assumption of positivity of the semigroup. Here we give an elementary proof of Kriegler's result.
We analyse $C_0$-semigroups of contractive operators on real-valued $L^p$-spaces for $p \not= 2$ and on other classes of non-Hilbert spaces. We show that, under some regularity assumptions on the semigroup, the geometry of the unit ball of those spaces forces the semigroup's generator to have only trivial (point) spectrum on the imaginary axis. This has interesting consequences for the asymptotic behaviour as $t \to \infty$. For example, we can show that a contractive and eventually norm continuous $C_0$-semigroup on a real-valued $L^p$-space automatically converges strongly if $p \not\in \{1,…
A Representation Theorem for Completely Contractive Dual Banach Algebras
本文证明了在自反算子空间上的完全有界线性算子的算子空间中，每个完全压缩的对偶巴拿赫代数都与一个$w^\ast$ -闭子代数完全等距.
Weak and Strong Convergence of an Implicit Iteration for Hemi-Contractive Semigroups in Banach Spaces
针对Banach空间中Lipschitzian半压缩和α-半压缩半群，提出隐式迭代格式逼近公共不动点，在满足Opial条件的均匀凸Banach空间中建立弱收敛，在一般Banach空间中获得强收敛，并推广到伪压缩和半压缩映射。
The space of unitary $C_{0}$-semigroups on separable infinite dimensional Hilbert space, when viewed under the topology of uniform weak convergence on compact subsets of $\mathbb{R}_{+}$, is known to admit various interesting residual subspaces. Before treating the contractive case, the problem of the complete metrisability of this space was raised in [Eisner, 2010]. Utilising Borel complexity computations and automatic continuity results for semigroups, we obtain a general result, which in particular implies that the one-/multiparameter contractive $C_{0}$-semigroups constitute Polish spaces …
Based on the technique of enriching contractive type mappings, a technique that has been used successfully in some recent papers, we introduce the concept of {\it saturated} class of contractive mappings. We show that, from this perspective, the contractive type mappings in the metric fixed point theory can be separated into two distinct classes, unsaturated and saturated, and that, for any unsaturated class of mappings, the technique of enriching contractive type mappings provides genuine new fixed point results. We illustrate the concept by surveying some significant fixed point results obta…
On the Approximation of Contractive Semigroups of Operators in Discretizable Hilbert Spaces
研究可离散化希尔伯特空间中由逼近方案诱导的离散压缩半群的性质，并应用于量子动力学半群的Heisenberg表示。
Abel averages and holomorphically pseudo-contractive maps in Banach spaces
研究了复Banach空间中的一类映射，它既包含无界线性算子，又包含非线性全纯映射。定义的性质，我们称之为{\sl伪压缩}，介绍了通过阿贝尔平均这样的地图。我们表明，所研究的地图是耗散的精神，经典的Lumer-Phillips定理。对于伪压缩全纯映射，我们建立了Abel平均到全纯收缩的幂收敛。
Working over infinite dimensional separable Hilbert spaces, residual results have been achieved for the space of contractive $C_{0}$-semigroups under the topology of uniform weak operator convergence on compact subsets of $\mathbb{R}_{+}$. Eisner and Serény raised in 2009 the open problem: Does this space constitute a Baire space? Observing that the subspace of unitary semigroups is completely metrisable and appealing to known density results, we solve this problem positively by showing that certain topological properties can in general be transferred from dense subspaces to larger spaces. The…
A comparison of the weakest contractive conditions for Banach and Kannan mappings
本文比较Banach和Kannan映射的最弱收敛型条件，通过Picard序列分析，证明Kannan型映射在完备度量空间中若干最弱收敛条件等价，而Banach收缩映射则不成立，且在G-完备度量空间中差异消失。
On unconventional limit sets of contractive functions on $\mathbb Z_p$
研究了$p$ -adic数的单位球$\mathbb Z_p$上的压缩函数$\{f_{i}\}_{i=1}^N$生成的半群$G$的非常规极限集的度量性质.也就是说，我们证明了非常规极限集是紧的、完美的和一致不连通的。此外，我们提供了一个例子，相应的非常规限制集是拟对称等价的符号康托尔集的两个压缩。
Generalized Evolution Semigroups and $h-$Dichotomies for Evolution Families on Banach Spaces
本文针对Banach空间中发展过程的指数衰减模式推广问题，通过构造递增同胚$h$诱导的群结构定义广义半群，证明其与经典发展半群等价，并确立广义半群双曲性、发展过程二分性和生成元谱条件的等价性，扩展了二分性理论。
On quasi-contractivity of $C_0$-semigroups on Banach spaces
半群理论中的一个基本结果是：每个$C_0$ -半群关于某个适当选择的等价范数是拟压缩的。本文包含了这一众所周知的事实的对应物。也就是说，通过检查Troter型公式$(e^{\frac{t}{n}A}P)^n$（其中$P$表示有界投影）的收敛性，我们证明了，只要生成元$A$是无界的，就可以在半群是{ \it{not} }拟压缩的空间上引入等价范数。
Contractive representations of odometer semigroup
本文研究了里程计半群 $O_n$ 的有界线性算子表示，证明了其压缩表示可导出更优表示，并完整描述了 Fock 空间上的表示及对 Nica 协变表示进行了分类。
Choose an arbitrary but fixed set of $n\times n$ matrices $A_1, \ldots, A_m$ and let $Ω_\mathbf A\subset \mathbb C^m$ be the unit ball with respect to the norm $\|\cdot\|_{\mathbf A},$ where $\|(z_1,\ldots ,z_m)\|_{\mathbf A}=\|z_1A_1+ \cdots+z_mA_m\|_{\rm op}.$ It is known that if $m\geq 3$ and $\mathbb B$ is any ball in $\mathbb C^m$ with respect to some norm, say $\|\cdot\|_{\mathbb B},$ then there exists a contractive linear map $L:(\mathbb C^m,\|\cdot\|^*_{\mathbb B})\to \mathcal M_k$ which is not completely contractive. The characterization of those balls in $\mathbb C^2$ for which contr…
Twisted crossed products of Banach algebras
本文研究了Banach代数在局部紧群作用下的扭交叉积构造，证明了当表示族由合同表示组成时，该结构是一个具有合同近似单位的Banach代数，并给出了其等距universal性质。同时，文章将结果推广到L^p算子代数领域，定义了L^p扭交叉积及其缩减版本，并推广了Packer-Raeburn技巧到L^p情形，证明了任何L^p扭交叉积都能等距同构于一个未扭的交叉积。
In this paper, we continue our previous research studies of exponential ultradistribution semigroups in Banach spaces. The existence and uniqueness of analytical solutions of abstract fractional relaxation equations associated with the generators of exponential ultradistribution semigroups have been considered. Some other results are also proved.
Counterexample to the Trotter product formula for projections
在希尔伯特空间上构造了一个酉半群$(e^{tA})_{t \geq 0}$和一个正交投影$P$，使得极限$\lim_{n \to \infty} [ e^{\frac{t}{n}A}P ]^n$不强存在.构造了一个正压缩半群和$L_p$上正压缩投影的类似例子。
Evolutionary Semigroups and Lyapunov Theorems in Banach Spaces
本文给出了巴拿赫空间$E$上连续算子半群的一个谱映射定理。利用演化半群在$E$ -值函数空间中的生成元给出了$E$上半群双曲的条件。研究了$E$中非自治微分方程的传播子生成的演化半群。发展了一种研究演化半群的“离散”技术，并应用于描述非线性方程的双曲性（指数二分性）.文件长度：68 K
In this paper, we prove the existence of a common fixed point in $C(K)$, the Chebyshev center of K, for a left reversible semigroup of isometry mappings. This existence result improves the results obtained by Lim et al. and Brodskii and Milman.
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.
A modular theory of contractive projections on Banach spaces with applications to noncommutative $\mathrm{L}^p$-spaces
本文提出基于 Levi 形式的模理论，刻画 Banach 空间压缩投影，并应用于非交换 L^p 空间，证明 2-压缩投影等价于完全压缩，且值域具有三元结构。
On Necessary and Sufficient Conditions for Fixed Point Convergence: A Contractive Iteration Principle
本研究提出了一种在完整度量空间中基于收缩迭代原理的固定点收敛条件，提供了必要和充分条件，并通过实例展示了其在传统定理无法适用的情况下的应用。
Multiplication Semigroups on Banach Function Spaces
本文刻画了Banach函数空间上由算子值映射诱导的乘法算子。我们还研究了乘法半群和这些算子的稳定性。
We generalize results concerning $\mathrm{C}_0$-semigroups on Banach lattices to a setting of ordered Banach spaces. We prove that the generator of a disjointness preserving $\mathrm{C}_0$-semigroup is local. Some basic properties of local operators are also given. We investigate cases where local operators generate local $\mathrm{C}_0$-semigroups, by using Taylor series or Yosida approximations. As norms we consider regular norms and show that bands are closed with respect to such norms. Our proofs rely on the theory of embedding pre-Riesz spaces in vector lattices and on corresponding extens…
In this paper we obtain new characterizations of the uniformly convex and smooth Banach spaces. These characterizations are established by using Lp-boundedness properties of Littlewood-Paley functions and area integrals associated with heat semigroups and involving fractional derivatives. Our results apply for instance by considering the heat semigroups defined by Hermite and Laguerre operators that do not satisfy the Markovian property.
The first part of this article is a brief survey of the properties of so-called almost interior points in ordered Banach spaces. Those vectors can be seen as a generalization of ``functions which are strictly positive almost everywhere'' on $L^p$-spaces and of ``quasi-interior points'' in Banach lattices. In the second part we study the long--term behaviour of strongly positive operator semigroups on ordered Banach spaces; these are semigroups which, in a sense, map every non-zero positive vector to an almost interior point. Using the Jacobs--de Leeuw--Glicksberg decomposition together with th…
Let $A$ be the generator of a $C_0$-semigroup $T$ on a Banach space of analytic functions on the open unit disc. If $T$ consists of composition operators, then there exists a holomorphic function $G:{\mathbb D}\to{\mathbb C}$ such that $Af=Gf'$ with maximal domain. The aim of the paper is the study of the reciprocal implication.
The purpose of this paper is to revisit previous works of the author with J. Sjöstrand (2010--2021) proved in the Hilbert case by considering the Banach case at the light of a paper by Y.~Latushkin and V.~Yurov (2013).
Regularity of Semigroups via the Asymptotic Behaviour at Zero
T. Kato和A. Pazy说一致凸空间X上的压缩半群（T（t））是全纯的当且仅当limsup_{t \downarrow 0}|| T（t）-Id|| < 2。我们研究了这一结果的推广，这是有效的任意巴拿赫空间的半群不一定是压缩的。这使得我们可以证明半群在（0，1）中指数为θ的插值空间上的全纯性的一个一般外推结果。作为应用，我们用零-二律刻画了UMD-空间上余弦族生成元的有界性。此外，我们的方法可以应用到R-扇区性：我们得到的最大正则性的半群在零的行为的表征，并显示外推结果。
The paper deals with multidimensional Bochner-Phillips functional calculus. In the previous paper by the author bounded perturbations of Bernstein functions of several commuting semigroup generators on Banach spaces where considered, conditions for Lipschitzness and estimates for the norm of commutators of such functions where proved. Also in the one-dimensional case the Frechet differentiability of Bernstein functions of semigroup generators on Banach spaces where proved and a generalization of Livschits-Kreĭn trace formula derived. The aim of the present paper is to prove the Frechet differe…
Weighted Semigroup Algebras as Dual Banach algebras
本文研究了加权半群代数$\ell^1(S,ω)$是关于预对偶$c_0(S)$的对偶巴拿赫代数的条件。一些有用的例子，说明的结果，也包括在内。
In this paper, we prove that the generator of any bounded analytic semigroup in $(θ,1)$-type real interpolation of its domain and underlying Banach space has maximal $L^1$-regularity, using a duality argument combined with the result of maximal continuous regularity. As an application, we consider maximal $L^1$-regularity of the Dirichlet-Laplacian and the Stokes operator in inhomogeneous $B^s_{q,1}$-type Besov spaces on domains of $\mathbb R^n$, $n\geq 2$.
A remark on contraction semigroups on Banach spaces
设$X$是一个复Banach空间，$J:X \to X^*$是$X$上的一个对偶截面（即$\langle x,J(x)\rangle=\|J(x)\|\|x\|=\|J(x)\|^2=\|x\|^2$）. Goldstein证明了：对于任意单位向量$x$和任意（$C_0$）压缩半群$T=\{e^{tA}:t \geq 0\}$，如果$X$是Hilbert空间且$|\langle T(t) x,J(x)\rangle| \to 1 $是$t \to \infty$，则$x$是$A$的一个特征向量，对应于一个纯虚特征值.本文证明了当$X$是严格凸复Banach空间时，类似的结果也成立.
Dilating Semigroup Representations to the Boundary Quotient
本文证明可嵌入或右LCM消去半群的表示可扩张到其约化边界商C*-代数当且仅当它可扩张为约化半群算子代数的完全压缩表示，并推广了Sz.-Nagy和Ando的扩张定理。
Tensor products of recurrent hypercyclic semigroups
研究了巴拿赫空间上满足超循环性判据、回归超循环性判据或混沌的强连续半群的张量积.
Smooth Contractive Embeddings and Application to Feynman Formula for Parabolic Equations on Smooth Bounded Domains
我们证明了Ya.A.在一篇文章中提出的两个假设。Butko，M. Grothaus，O.G. Schwaranov关于有界区域上椭圆型微分算子的Cauchy-狄利克雷问题的强连续算子半群的存在性以及该半群的生成元的核在空间$C_c^{2,α}(\R^n)$中的光滑压缩嵌入的存在性.在此基础上，本文构造了柯西-狄利克雷问题解的费曼公式。在这篇文章中，我们表明，假设满足域与$C^{4,α}$ -光滑边界和系数在$C^{2,α}$。
A semigroup analogue of the Fonf--Lin--Wojtaszczyk ergodic characterization of reflexive Banach spaces with a basis
本文通过半群类比的方法，研究了具有基的Banach空间的反射性和有限维性特征。
Let X be a real or complex Banach space and let F in X be a non-empty set. F is called an existence set of best coapproximation (existence set for brevity), if for any x in X, $R_F(x)$ is not the empty set, where $$ R_F (x) = \{ d \in F : \|d-c\| \leq \|x-c\| \hbox{ for any } c \in F \}.$$ It is clear that any existence set is a contractive subset of X. The aim of this paper is to present some conditions on F and X under which the notions of exsistence set and contractive set are equivalent.
A semigroup approach to the numerical range of operators on Banach spaces
在Banach空间$X$上引入了（无界）线性算子$A$的数值谱$σ_n(A)\subset \mathbb{C}$，并研究了它的性质.我们的定义与$A$的数值域$W(A)$密切相关，并且总是产生$W(A)$的超集。在希尔伯特空间上的有界算子的情况下，这两个概念是一致的。然而，与数值域不同，$σ_n(A)$总是封闭的、凸的并且包含$A$的谱。在本文中，我们着重强调我们的方法与$C_0$ -半群理论的联系。
Quasi-compactness and uniform stabilization on general Banach spaces under $θ$-subordinate perturbations of semigroup generators
本文证明在一般Banach空间上，解析半群在沿轨道紧的$\ heta$-从属扰动下保持拟紧性和一致指数稳定性，并推广至Crandall-Pazy半群及立即可微半群，最后通过反例说明$A$-有界扰动下稳定性可能缺失。
Let $G$ be a locally compact group. We prove that the augmentation ideal in $L^1(G)$ is (algebraically) finitely-generated as a left ideal if and only if $G$ is finite. We then investigate weighted versions of this result, as well as a version for semigroup algebras. Weighted measure algebras are also considered. We are motivated by a recent conjecture of Dales and Żelazko, which states that a unital Banach algebra in which every maximal left ideal is finitely-generated is necessarily finite-dimensional. We prove that this conjecture holds for many of the algebras considered. Finally, we use t…
Positive semigroups and algebraic Riccati equations in Banach spaces
将关于代数算子Riccati方程可解性的Wonham定理推广到巴拿赫空间，即当（A，B）指数可镇定且（C，A）指数可检测时，方程A*P+PA-PBB*P+C*C=0存在唯一的稳定解.证明是基于一个新的方法，把方程的线性部分作为一个正半群的生成元的空间上的对称算子从巴拿赫空间，其对偶，和二次部分作为一个秩序凹映射。然后用一个直接类似于凹函数的整体牛顿迭代逼近解，逼近收敛于强算子拓扑，且收敛是单调的。线性化后的方程是著名的形式为A*P+PA=-Q的李雅普诺夫方程，并推广了相应的半群稳定性判据。
We study, to certain Banach spaces $X$, families of weighted composition operators. Notably, we show that if this family form a strongly continuous semigroup, then its infinitesimal generator ($Γ, D(Γ)$) is given by $Γf = gf+Gf^\prime$ with $D(Γ) = \{ f\in X \ | \ gf+Gf^\prime\in X \}$ where $g, G$ are holomorphic functions. Moreover, our second maim result is to study the reciprocal implication. That is if $(Γ, D(Γ))$, define like above, generate a strongly continuous semigroup, then this one is a family of weighted composition operators.
We give a relation between the exponential stability of $ C_{0}- $semigroup $ \textbf{T}=\left\lbrace T(t) \right\rbrace_{t\geq 0} $ and the solutions of Lyapunov inequality \( \left\langle QAx,x\right\langle +\left\langle Qx,Ax\right\langle \leq -||x||^{2}, \) in $ B^{+}(X,X^{*}) $, with $ X $ is a Banach space. The solutions of this inequality characterizes, the boundedness of the resolvent $ R(λ,A) $ inside and outside of the left half-plane $ \Reλ\geq 0 $, and also the left invertibility of the $ C_{0}- $semigroup T.
Module super-amenability for semigroup algebras
设$S$是具有幂等元集$E$的逆半群.本文定义了一个巴拿赫代数的模超顺从性，并证明了当$E$是向上定向的且从左到右作用于$S$时，半群代数$ \ell ^{1}(S)$是$\ell^{1}(E)$ -模超顺从的当且仅当$S$的适当群同态像是有限的.
We characterize strong continuity of general operator semigroups on some Lebesgue spaces. In particular, a characterization of strong continuity of weighted composition semigroups on classical Hardy spaces and weighted Bergman spaces with regular weights is given. As applications, our result improves the results of Siskakis, A. G. \cite{AG1} and König, W. \cite{K} and answers a question of Siskakis, A. G. proposed in \cite{AG4}. We also characterize strongly continuous semigroups of weighted composition operators on weighted Bergman spaces in terms of abelian intertwiners of multiplication ope…
Banach algebra crossed products by inverse semigroup actions
本文提出了一种对逆半群作用下的巴拿赫代数交叉积进行普遍描述的方法，允许对交叉积的所有表示进行分解。
In this paper, we analyze recurrent $C_{0}$-semigroups of bounded operators on Banach spaces. We also introduce the notion of a (uniformly) $C_{0}$-rigid semigroups of bounded operators and give a structural characterization of them. A characterization of a Li-Yorke chaoticity of the translation semigroup $(T(t))_{t\geq 0}$ on weighted spaces of integrable functions and continuous functions in terms of admissible weight function is given. The recurrent $C_0$-semigroups induced by semiflows are characterized on the spaces of integrable functions and of spaces of continuous functions.
Multiplication Operator Semigroups on Banach lattice valued continuous function spaces
本文研究Banach格值连续函数空间$C_0(Ω,E)$上生成强连续半群的乘法算子，并将标量情形$C_0(Ω)$的刻画推广到向量值情形。
Banach representations and affine compactifications of dynamical systems
对于每个巴拿赫空间V，我们都对应一个紧右拓扑仿射半群E（V）.我们证明了可分巴拿赫空间V是Asplund当且仅当E（V）是可度量化的，并且它是Rosenthal（即它不包含$l_1$的同构拷贝）当且仅当E（V）是Rosenthal紧的.研究了紧右拓扑半群在E（V）中的表示.特别地，tame和HNS-半群的表示分别自然地产生为tame和HNS（遗传非敏感）动力系统的包络半群。作为应用，我们得到了R.埃利斯我们研究的一个主题是动力系统X的包络半群与它的各种仿射紧化Q（X）的包络半群之间的关系。当两者重合时，我们说仿射紧化Q（X）是E相容的。这是内射性概念的一个改进。我们证明了远端非等度连续系统不允许任何E-相容紧化。我们提出了几个新的非内射动力系统的例子，并检查内射性和E-相容性之间的关系。
We develop a theory of eventually positive $C_0$-semigroups on Banach lattices, that is, of semigroups for which, for every positive initial value, the solution of the corresponding Cauchy problem becomes positive for large times. We give characterisations of such semigroups by means of spectral and resolvent properties of the corresponding generators, complementing existing results on spaces of continuous functions. This enables us to treat a range of new examples including the square of the Laplacian with Dirichlet boundary conditions, the bi-Laplacian on $L^p$-spaces, the Dirichlet-to-Neuma…
Transition Semigroups of Banach Space Valued Ornstein-Uhlenbeck Processes
本文研究了随机发展方程$dX(t) = AX(t)dt +dW_H(t)$，$t\ge 0,$的解的转移半群，其中$A$是可分实巴拿赫空间$E$上$C_0$ -半群$S$的生成元，而$W_H$是连续嵌入$E$中的实Hilbert空间$H$中的圆柱白噪声.利用$S$在$H$中的行为刻画了这些半群的各种性质，如强Feller性质、谱隙性质和解析性.特别是，我们调查之间的相互作用的解析性的过渡半群，$S$ -不变性的$H$，和解析性的限制半群$S_H$。
On the structure of semigroups on $L_p$ with a bounded $H{^\infty}$-calculus
证明了$L_p$ -空间上的有界解析半群对某些$φ< \fracπ{2}$有界$H^{\infty}(Σ_φ)$ -演算当且仅当该半群可由某些较大的在实直线上为正压缩的$L_p$ -空间上的有界解析半群在限制到不变子空间后，通过不变子空间和相似变换进行因子分解而得到.
We prove a Desch-Schappacher type perturbation theorem for one-parameter semigroups on Banach spaces which are not strongly continuous for the norm, but possess a weaker continuity property. In this paper we chose to work in the framework of bi-continuous semigroups. This choice has the advantage that we can treat in a unified manner two important classes of semigroups: implemented semigroups on the Banach algebra $\mathscr{L}(E)$ of bounded, linear operators on a Banach space $E$, and semigroups on the space of bounded and continuous functions over a Polish space induced by jointly continuous…
Stability radius and internal versus external stability in Banach spaces: an evolution semigroup approach
本文发展了发展半群理论，并为研究一般线性控制系统的稳定性提供了框架。这些包括时变系统建模与无界状态空间运营商的巴拿赫空间。这种方法允许人们将强连续半群的经典理论应用于时变系统。特别地，复稳定半径可以用（演化）半群的生成元来明确表示。给出的例子表明，经典公式的稳定半径的自治希尔伯特空间系统失败，在更一般的设置。对这些一般系统的稳定半径的上限和下界。此外，它表明，理论的发展半群允许一个简单的算子理论分析的内部稳定性所确定的经典频域和输入输出算子，即使是非自治的Banach空间系统
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.
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