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In this paper, we propose an iteration algorithm for finding a split common fixed point of an asymptotically nonexpansive mapping in the frameworks of two real Banach spaces. Under some suitable conditions imposed on the sequences of parameters, some strong convergence theorems are proved, which also solve some variational inequalities that are closely related to optimization problems. The results here generalize and improve the main results of other authors. [ABSTRACT FROM AUTHOR]

An inertial iterative algorithm for approximating a point in the set of zeros of a maximal monotone operator which is also a common fixed point of a countable family of relatively nonexpansive operators is studied. Strong convergence theorem is proved in a uniformly convex and uniformly smooth real Banach space. This theorem extends, generalizes and complements several recent important results. Furthermore, the theorem is applied to convex optimization problems and to J-fixed point problems. Finally, some numerical examples are presented to show the effect of the inertial term in the convergence of the sequence of the algorithm. [ABSTRACT FROM AUTHOR]

Let B be a uniformly convex and uniformly smooth real Banach space with dual space B ∗ . Let F : B → B ∗ , K : B ∗ → B be maximal monotone mappings. An iterative algorithm is constructed and the sequence of the algorithm is proved to converge strongly to a solution of the Hammerstein equation u + K F u = 0 . This theorem is a significant improvement of some important recent results which were proved in real Hilbert spaces under the assumption that F and K are maximal monotone continuous and bounded. The continuity and boundedness restrictions on K and F have been dispensed with, using our new method, even in the more general setting considered in our theorems. Finally, numerical experiments are presented to illustrate the convergence of the sequence of our algorithm. [ABSTRACT FROM AUTHOR]

Let E be a real Banach space with dual space E ∗ . A new class of relatively weakJ-nonexpansive maps, T : E → E ∗ , is introduced and studied. An algorithm to approximate a common element of J-fixed points for a countable family of relatively weak J-nonexpansive maps and zeros of a countable family of inverse strongly monotone maps in a 2-uniformly convex and uniformly smooth real Banach space is constructed. Furthermore, assuming existence, the sequence of the algorithm is proved to converge strongly. Finally, a numerical example is given to illustrate the convergence of the sequence generated by the algorithm. [ABSTRACT FROM AUTHOR]

We apply the topological degree theory for condensing maps to study approximation of solutions to a fractional-order semilinear differential equation in a Banach space. We assume that the linear part of the equation is a closed unbounded generator of a $C_{0}$ -semigroup. We also suppose that the nonlinearity satisfies a regularity condition expressed in terms of the Hausdorff measure of noncompactness. We justify the scheme of semidiscretization of the Cauchy problem for a differential equation of a given type and evaluate the topological index of the solution set. This makes it possible to obtain a result on the approximation of solutions to the problem. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of solutions for systems of random semilinear impulsive differential equations. The existence results are established by means of a new version of Perov's, a nonlinear alternative of Leray-Schauder's fixed point principles combined with a technique based on vector-valued metrics and convergent to zero matrices. Also, we give a random abstract formulation to Sadovskii's fixed point theorem in a vector-valued Banach space. Examples illustrating the results are included. [ABSTRACT FROM AUTHOR]

A dislocated cone metric space over Banach algebra is introduced as a generalisation of a cone metric space over Banach algebra as well as a dislocated metric space. Fixed point theorems for Perov-type α-quasi contraction mapping, Kannan-type contraction as well as Chatterjee-type contraction mappings are proved in a dislocated cone metric space over Banach algebra. Proper examples are provided to establish the validity of our claims. [ABSTRACT FROM AUTHOR]

In this article, a Krasnoselskii-type and a Halpern-type algorithm for approximating a common fixed point of a countable family of totally quasi- ϕ-asymptotically nonexpansive nonself multi-valued maps and a solution of a system of generalized mixed equilibrium problem are constructed. Strong convergence of the sequences generated by these algorithms is proved in uniformly smooth and strictly convex real Banach spaces with the Kadec-Klee property. Several applications of our theorems are also presented. Finally, our theorems are a significant improvement of several important recent results. [ABSTRACT FROM AUTHOR]

Recently, Wardowski (Fixed Point Theory Appl. 2012:94, 2012) introduced a new concept of F-contraction and proved a fixed point theorem which generalizes the Banach contraction principle. Following this direction of research, in this paper, we present some new fixed point results for F-expanding mappings, especially on a complete G-metric space. [ABSTRACT FROM AUTHOR]

We consider a new type of monotone nonexpansive mappings in an ordered Banach space X with partial order ⪯. This new class of nonlinear mappings properly contains nonexpansive, firmly-nonexpansive and Suzuki-type generalized nonexpansive mappings and partially extends α-nonexpansive mappings. We obtain some existence theorems and weak and strong convergence theorems for the Mann iteration. Some useful examples are presented to illustrate the facts. [ABSTRACT FROM AUTHOR]

We present the random version in partially ordered metric spaces of the classical Banach contraction principle and some of its generalizations to ordered metric spaces. The results are used to prove the existence of solutions for random differential equations with boundary conditions. [ABSTRACT FROM AUTHOR]

Let E be a real normed space with dual space $E^{*}$ and let $A:E\rightarrow2^{E^{*}}$ be any map. Let $J:E\rightarrow2^{E^{*}}$ be the normalized duality map on E. A new class of mappings, J- pseudocontractive maps, is introduced and the notion of J- fixed points is used to prove that $T:=(J-A)$ is J-pseudocontractive if and only if A is monotone. In the case that E is a uniformly convex and uniformly smooth real Banach space with dual $E^{*}$ , $T: E\rightarrow2^{E^{*}}$ is a bounded J- pseudocontractive map with a nonempty J- fixed point set, and $J-T :E\rightarrow2^{E^{*}}$ is maximal monotone, a sequence is constructed which converges strongly to a J-fixed point of T. As an immediate consequence of this result, an analog of a recent important result of Chidume for bounded m-accretive maps is obtained in the case that $A:E\rightarrow2^{E^{*}}$ is bounded maximal monotone, a result which complements the proximal point algorithm of Martinet and Rockafellar. Furthermore, this analog is applied to approximate solutions of Hammerstein integral equations and is also applied to convex optimization problems. Finally, the techniques of the proofs are of independent interest. [ABSTRACT FROM AUTHOR]

In this paper, we propose an iteration method for finding a split common fixed point of asymptotically nonexpansive semigroups in the setting of two Banach spaces, and we obtain some weak and strong convergence theorems of the iteration scheme proposed. The results presented in the paper are new and improve and extend some recent corresponding results. [ABSTRACT FROM AUTHOR]

The aim of this work is to establish some new fixed point theorems for generalized F-Suzuki-contraction mappings in complete b-metric spaces. [ABSTRACT FROM AUTHOR]

In this paper, we prove the weak and strong convergence of a sequence $\{x_{n}\}$ generated by the Ishikawa iteration to some common fixed points of two G-nonexpansive mappings defined on a Banach space endowed with a graph. [ABSTRACT FROM AUTHOR]

In this paper, we prove some existence theorems of fixed points of a monotone nonexpansive mapping T in a Banach space E with the partial order '≤', where a such mapping may be discontinuous. In particular, in finite dimensional spaces, such a mapping T has a fixed point in E if and only if the sequence $\{T^{n}0\}$ is bounded in E. In order to find a fixed point of such a mapping T, we prove the weak convergence of the Mann iteration scheme under the condition $\sum_{n=1}^{\infty}\beta_{n}(1-\beta_{n})=\infty$, which entails $\beta _{n}=\frac{1}{n+1}$ as a special case. [ABSTRACT FROM AUTHOR]

A fixed point theorem for weakly inward A-proper maps defined on cones in Banach spaces is established using a fixed point index for such maps. The result generalizes a theorem in Deimling (Nonlinear Functional Analysis, ) for weakly inward maps defined on a cone in $\mathbb{R}^{n}$. We then apply the theorem to a Picard boundary value problem and obtain the existence of a positive solution. [ABSTRACT FROM AUTHOR]

This paper provides new common fixed point theorems for pairs of multivalued and single-valued mappings operating between ordered Banach spaces. Our results lead to new existence theorems for a system of integral inclusions. [ABSTRACT FROM AUTHOR]

This paper aims to study the strong convergence of generalized modified Krasnoselskii iterative process for finding the minimum norm solutions of certain nonlinear equations with generalized strictly pseudocontractive, demiclosed, coercive, bounded, and potential mappings in uniformly convex Banach spaces. An application to nonlinear pseudomonotone equations is provided. The results extend and improve recent work in this direction. [ABSTRACT FROM AUTHOR]

In this paper, we study an iterative scheme for two different types of resolvents of a monotone operator defined on a Banach space. These resolvents are generalizations of resolvents of a monotone operator in a Hilbert space. We obtain iterative approximations of a zero point of a monotone operator generated by the shrinking projection method with errors in a Banach space. Using our result, we discuss some applications. [ABSTRACT FROM AUTHOR]

In this paper we develop various new fixed point theorems for a class of operator equations with three general mixed monotone operators,namely $A(x,x)+B(x,x) +C(x,x)=x$ on ordered Banach spaces, where A, B, C are the mixed monotone operators. A is such that for any $t\in(0,1)$, there exists $\varphi(t)\in(t,1]$ such that for all $x,y\in P$, $A(tx,t^{-1}y)\geq\varphi(t)A(x,y)$; B is hypo-homogeneous, i.e. B satisfies that for any $t\in(0,1)$, $x,y\in P$, $B(tx,t^{-1}y)\geq tB(x,y)$; C is concave-convex, i.e. C satisfies that for fixed y, $C(\cdot,y):P\rightarrow P$ is concave; for fixed x, $C(x,\cdot)$: $P\rightarrow P$ is convex. Also we study the solution of the nonlinear eigenvalue equation $A(x,x)+B(x,x)+C(x,x)=\lambda x$ and discuss its dependency to the parameter. Our work extends many existing results in the field of study. As an application, we utilize the results obtained in this paper for the operator equation to study the existence and uniqueness of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of the existence of solution to the following system of fractional hybrid differential equations: where $D^{\alpha}$ is the R-L fractional derivative of order α, $J=[0,T]$, $T>0$, and the functions $f :J\times\mathbb{R} \times\mathbb{R} \rightarrow\mathbb{R}$, $f(0,0)=0$ and $g:J \times\mathbb{R} \times\mathbb{R}\rightarrow\mathbb{R}$ satisfy certain conditions. The proof of the existence theorem is based on a coupled fixed point theorem of Krasnoselskii type, which extends a fixed point theorem of Burton (Appl. Math. Lett. 11:85-88, ). Finally, our results are illustrated by a concrete example. [ABSTRACT FROM AUTHOR]

Let $\mathcal{A}$ be an index set, and $C=\{C_{\alpha}\}_{\alpha\in \mathcal{A}}\in{}[1;\infty)^{\mathcal{A}}$. Fuzzy quasi-triangular space is defined to be $(X,\mathcal{M}_{C;\mathcal{A}},\ast)$, where X is a nonempty set, a fuzzy family $\mathcal{M}_{C;\mathcal{A}}=\{M_{\alpha }:X\times X\times(0;\infty)\rightarrow(0;1],\alpha\in\mathcal{A}\}$ satisfies $\forall_{\alpha\in\mathcal{A}}\forall_{x,y,z\in X}\forall _{t,s\in(0;\infty)}\{M_{\alpha}(x,y,t)\ast M_{\alpha}(y,z,s)\leq M_{\alpha}(x,z,C_{\alpha}(t+s))\}$, and ∗ is the continuous t-norm $\ast:[0;1]\times{}[0;1]\rightarrow{}[0;1]$. In $(X,\mathcal{M}_{C;\mathcal{A}},\ast)$, left (right) $\mathcal{G}$-families and $\mathcal{W}$-families $\mathcal{K}_{C;\mathcal{A}}$ generated by $\mathcal{M}_{C;\mathcal{A}}$ ( $\mathcal{K}_{C;\mathcal{A}}$ generalize $\mathcal {M}_{C;\mathcal{A}}$) are defined and described. Using families $\mathcal {K}_{C;\mathcal{A}}$, three kinds of left (right) fuzzy sets of Pompeiu-Hausdorff type on $2^{X}\times2^{X}\times(0;\infty)$ are introduced. Using these fuzzy sets, three kinds of left (right) set-valued fuzzy contractions $T:X\rightarrow2^{X}$ are constructed, and for such fuzzy contractions, conditions guaranteeing the existence of periodic points and left (right) $\mathcal {M}_{C;\mathcal{A}}$-convergence to these periodic points of dynamic processes $(w^{m}:m\in\{0\}\cup\mathbb{N})$, $w^{m}\in T(w^{m-1})$ for $m\in \mathbb{N}$, starting at $w^{0}\in X$, are established. Moreover, in $(X,\mathcal {M}_{C;\mathcal{A}},\ast)$, using left (right) $\mathcal{G}$-families and $\mathcal{W}$-families $\mathcal{K}_{C;\mathcal{A}}$ generated by $\mathcal{M}_{C;\mathcal{A}}$, two kinds of left (right) single-valued fuzzy contractions $T:X\rightarrow X$ are constructed, and for such fuzzy contractions, the convergence, existence, approximation, uniqueness, periodic point, and fixed point result is also obtained. Examples are provided. [ABSTRACT FROM AUTHOR]

Let X be a Banach space or a complete hyperbolic metric space. Let C be a nonempty, bounded, closed, and convex subset of X and $T: C \rightarrow C$ be a monotone nonexpansive mapping. In this paper, we show that if X is a Banach space which is uniformly convex in every direction or a uniformly convex hyperbolic metric space, then T has a fixed point. This is the analog to Browder and Göhde's fixed point theorem for monotone nonexpansive mappings. [ABSTRACT FROM AUTHOR]

We first present the concepts of proximal contraction and proximal nonexpansive mappings on star-shaped sets in probabilistic Banach (Menger) spaces. We derive some results about the best proximity points for these mappings in probabilistic Banach (Menger) spaces. Next, we bring some examples that defend our main results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce the notions of Θ-type contractions and Θ-type Suzuki contractions and to establish some new fixed point theorems for these two kinds of mappings in the setting of complete metric spaces. The results presented in the paper are an extension of the Banach contraction principle, the Suzuki contraction theorem, and the Jleli and Samet fixed point theorem. As an application, we utilize our results to study the existence problem of solutions of nonlinear Hammerstein integral equations. [ABSTRACT FROM AUTHOR]

We introduce the notion of a $C^{*}$-algebra-valued b-metric space. We generalize the Banach contraction principle in this new setting. As an application of our result, we establish an existence result for an integral equation in a $C^{*}$-algebra-valued b-metric space. [ABSTRACT FROM AUTHOR]

By processing the problem through fixed point theory and propagator theory, we investigate the periodicity of solutions to a class of impulsive evolution equations in Hilbert spaces and establish some existence theorems for periodic solutions. Moreover, the asymptotic stability of periodic solutions is obtained under suitable conditions. As one will see, the concept of an impulsive propagator is introduced for the first time in the paper. [ABSTRACT FROM AUTHOR]

In this work, we discuss the concept of Banach operator pairs in modular vector spaces. We prove the existence of common fixed points for these type of operators which satisfy a modular continuity in modular compact sets. On the basis of our result, we are able to give an analog of DeMarr's common fixed point theorem for a family of symmetric Banach operator pairs in modular vector spaces. [ABSTRACT FROM AUTHOR]

In this paper we study the existence of best proximity points of a nonself upper semicontinuous multivalued mapping $T:A\rightarrow2^{B}$ in a strictly convex Banach space. This multivalued mapping commutes with affine, noncyclic, and relatively u-continuous single-valued mapping $f:A \cup B\rightarrow A\cup B$. Also, we study the case when T commutes with a family of commuting, affine, noncyclic, and relatively u-continuous single-valued mappings on $A\cup B$. Moreover, we present some examples to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper we introduce a family of weakly contractive maps on the space $B(S)$ of bounded real valued functions and use it to show that a fundamental step in the proof of the well-known Stone-Weierstrass approximation theorem can be achieved via Rakotch's fixed point theorem for weakly contractive maps. With the same technique, we obtain Zemanek's theorem on the existence of square roots in certain Banach subalgebras of $B(S)$, and, finally, in the context of abstract Banach algebras, we exhibit some relationship between weakly contractive maps on the closed unit ball and the geometry of the spheres. [ABSTRACT FROM AUTHOR]

In this paper, first, we introduce the condition (BP) which is weaker than the completely continuous mapping in Banach spaces. Second, we consider a simple iteration and prove some strong convergence theorems of the proposed iteration for an asymptotically nonexpansive nonself-mapping with the condition (BP). Finally, we give two examples to illustrate the main result in this paper. Our results improve and extend the corresponding results given by some authors. [ABSTRACT FROM AUTHOR]

We construct a relaxed hybrid shrinking iteration algorithm for approximating common fixed points of a countable family of totally quasi- ϕ-asymptotically nonexpansive multi-valued mappings. A strong convergence theorem for solving generalized mixed equilibrium problems is established in the framework of Banach spaces under relaxed conditions. Since there is no need to impose a uniformity assumption on the involved mappings and no need to compute complex series in the iteration process, the results improve those of the authors with related interests. [ABSTRACT FROM AUTHOR]

A general viscosity iterative method for a finite family of generalized asymptotically quasi-nonexpansive mappings in a convex metric space is introduced. Special cases of the new iterative method are the viscosity iterative method of Chang et al. (Appl. Math. Comput. 212:51-59, 2009), an analogue of the viscosity iterative method of Fukhar-ud-din et al. (J. Nonlinear Convex Anal. 16:47-58, 2015) and an extension of the multistep iterative method of Yildirim and Özdemir (Arab. J. Sci. Eng. 36:393-403, 2011). Our results generalize and extend the corresponding known results in uniformly convex Banach spaces and $\operatorname{CAT}(0)$ spaces simultaneously. [ABSTRACT FROM AUTHOR]

In this paper we give Banach fixed point theorems and explicit estimates on the rates of convergence of the transition function to the stationary distribution for a class of exponential ergodic Markov chains. Our results are different from earlier estimates using coupling theory, and from estimates using stochastically monotone one. Our estimates show a noticeable improvement on existing results if Markov chains contain instantaneous states or nonconservative states. The proof uses existing results of discrete time Markov chains together with h-skeleton. At last, we apply this result, Ray-Knight compactification and Itô excursion theory to two examples: a class of singular Markov chains and Kolmogorov matrix. [ABSTRACT FROM AUTHOR]

As an extension of the Leray-Schauder degree, we introduce a topological degree theory for a class of demicontinuous operators of generalized $(S_{+})$ type in real reflexive Banach spaces, based on the recent Berkovits degree. Using the degree theory, we show that the Borsuk theorem holds true for this class. Moreover, we study the Dirichlet boundary value problem involving the p-Laplacian by way of an abstract Hammerstein equation. [ABSTRACT FROM AUTHOR]

In this paper, we propose a class of generalized proximal-type method by the virtue of Bregman functions to solve weak vector variational inequality problems in Banach spaces. We carry out a convergence analysis on the method and prove the weak convergence of the generated sequence to a solution of the weak vector variational inequality problems under some mild conditions. Our results extend some known results to more general cases. [ABSTRACT FROM AUTHOR]

In this paper, we present a new iterative algorithm with errors to solve the problems of finding zeros of the sum of finitely many m-accretive operators and finitely many α-inversely strongly accretive operators in a real smooth and uniformly convex Banach space. Strong convergence theorems are established, which extend the corresponding works given by some authors. Moreover, the relationship among the zero of the sum of m-accretive operator and α-inversely strongly accretive operator, the solution of one kind variational inequality, and the solution of the capillarity equation is investigated. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to introduce a new iteration by the combination of the viscosity approximation with Meir-Keeler contractions and proximal point algorithm for finding common zeros of a finite family of accretive operators in a Banach space with a uniformly Gâteaux differentiable norm. The results of this paper improve and extend corresponding well-known results by many others. [ABSTRACT FROM AUTHOR]

In this paper, we consider an iterative algorithm for finding the common element of the set of solutions for the generalized mixed equilibrium problems, the common fixed points set of two generalized nonexpansive type mappings, and the set of solutions of the variational inequality for an inverse-strongly skew-monotone operator in Banach spaces. Under mild conditions, the weak convergence theorem is established by using the sunny generalized nonexpansive retraction in Banach spaces. Our results refine, supplement, and extend the corresponding results in (Saewan et al. in Optim. Lett. 8:501-518, 2014), and other results announced by many other authors. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new algorithm for solving split equality mixed equilibrium problems in the framework of infinite-dimensional real Hilbert spaces. The strong and weak convergence theorems are obtained. As application, we shall utilize our results to study the split equality mixed variational inequality problem and the split equality convex minimization problem. Our results presented in this paper improve and extend some recent corresponding results. [ABSTRACT FROM AUTHOR]

In this paper, a new modified Ishikawa iterative algorithm with errors by a shrinking projection method for generalized mixed equilibrium problems and a countable family of uniformly Bregman totally quasi- D-asymptotically nonexpansive mappings is introduced and investigated in the framework of a real Banach space. Strong convergence of the sequence generated by the proposed algorithm is derived under some suitable assumptions. These results are new and develop some recent results in this field. [ABSTRACT FROM AUTHOR]

In this paper, we construct an iterative process involving a hybrid pair of a finite family of generalized asymptotically nonexpansive single-valued mappings and a finite family of generalized nonexpansive multi-valued mappings and prove weak and strong convergence theorems of the proposed iterative process in Banach spaces. Our main results extend and generalize many results in the literature. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce a general iterative algorithm for finding a common element of the set of common fixed points of an infinite family of strict pseudo-contractions and the set of solutions of the variational inequalities for finite family of strongly accretive mappings in a q-uniformly smooth Banach space. Furthermore, we prove strong convergence of the iterative sequence under suitable conditions. Our results generalize some recent results. [ABSTRACT FROM AUTHOR]

In this paper we use a matrix approach to approximate solutions of variational inequalities in Hilbert spaces. The methods studied combine new or well-known iterative methods (as the original Mann method) with regularized processes that involve regular matrices in the sense of Toeplitz. We obtain ergodic type results and convergence. [ABSTRACT FROM AUTHOR]

In this paper, we establish dislocated quasi-b-metric spaces and introduce the notions of Geraghty type dqb-cyclic-Banach contraction and dqb-cyclic-Kannan mapping and derive the existence of fixed point theorems for such spaces. Our main theorem extends and unifies existing results in the recent literature. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of fixed points for a class of self-mappings satisfying certain rational expressions on closed, bounded and convex subsets with normal structures in reflexive Banach spaces. We show that, in particular, this class extends that introduced by Ray and Singh (Indian J. Pure Appl. Math. 9:216-221, 1978). As an application, we give an investigation of the convergence and stability of some iterative processes associated to these mappings. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study stability and strong convergence of asymptotically pseudocontractive mappings by using a new composite implicit iteration process in an arbitrary real Banach space. The results in this paper improve and extend the corresponding results in the literature. [ABSTRACT FROM AUTHOR]

In 1994, Matthews obtained an extension of the celebrated Banach fixed point theorem to the partial metric framework (Ann. N.Y. Acad. Sci. 728:183-197, 1994). Motivated by the Matthews extension of the Banach theorem, we present a Nemytskii-Edelstein type fixed point theorem for self-mappings in partial metric spaces in such a way that the classical one can be retrieved as a particular case of our new result. We give examples which show that the assumed hypothesis in our new result cannot be weakened. Moreover, we show that our new fixed point theorem allows one to find fixed points of mappings in some cases in which the Matthews result and the classical Nemytskii-Edelstein one cannot be applied. Furthermore, we provide a negative answer to the question about whether our new result can be retrieved as a particular case of the classical Nemytskii-Edelstein one whenever the metrization technique, developed by Hitzler and Seda (Mathematical Aspects of Logic Programming Semantics, 2011), is applied to partial metric spaces. [ABSTRACT FROM AUTHOR]

We give a generalization of the Banach contraction principle on a modular metric space endowed with a graph. The notion of a modular metric on an arbitrary set and the corresponding modular spaces, generalizing classical modulars over linear spaces like Orlicz spaces, were recently introduced. This paper can be seen as the modular metric version of Jachymski's fixed point result for mappings on a metric space with a graph. [ABSTRACT FROM AUTHOR]