Equation Tx Research Articles

Expansive Type Rational Contraction in Metric Space and Common Fixed Point Theorems

The field of expansive mappings in fixed-point theory is one of the most fascinating areas in mathematics. In this theory, contraction is one of the main tools used to prove a fixed point's existence and uniqueness. For all of the analyses, the fixed point theorem proposed by Banach's contraction theory is highly popular and widely used to prove that a solution to the operator equation Tx=x exists and is unique. Through the present article, we utilize rational expressions in metric spaces to deliver unique common stable (fixed) point results in expansive mapping. The main outcomes of numerous relevant innovations in the newest research are built upon them.

Perturbation of the Moore–Penrose Metric generalized inverse with applications to the best approximate solution problem in Lp(Ω, μ)

ABSTRACTLet (), let with closed range. In this paper, utilizing the gap between closed subspaces and the perturbation bounds of metric projections, we present some new perturbation results of the Moore–Penrose metric generalized inverse. As applications of our results, we also investigate the best approximate solution problem for the ill-posed operator equation Tx=y under some conditions. The main results have three parts, part one covers the null space preserving case, part two covers the range preserving case, and part three covers the general case. Examples in connection with the theoretical results will be also presented.

A unified treatment for Tikhonov regularization using a general stabilizing operator

While dealing with the problem of solving an ill-posed operator equation Tx = y, where T : X → Y is a bounded linear operator between Hilbert spaces X and Y, one looks for a stable method for approximating [Formula: see text], a least-residual norm solution which minimizes a seminorm x ↦ ‖Lx‖, where L : D(L) ⊆ X → X is a (possibly unbounded) closed densely defined operator in X. If the operators T and L satisfy a completion condition ‖Tx‖2 + ‖Lx‖2 ≥ γ‖x‖2 for all x ∈ D(L*L) for some constant γ > 0, then Tikhonov regularization is one of the simple and widely used of such procedures in which the regularized solution is obtained by solving a well-posed equation [Formula: see text] where yδ is a noisy data and α > 0 is the regularization parameter to be chosen appropriately. We prescribe a condition on (T, L) which unifies the analysis for ordinary Tikhonov regularization, that is, L = I, and also the case of L = Bs with B being a strictly positive closed densely defined unbounded operator which generates a Hilbert scale {Xt}t>0. Under the new framework, we provide estimates for the best possible worst error and order optimal error estimates for the regularized solutions under certain general source condition which incorporates in its fold many existing results as special cases, by choosing regularization parameter using a Morozov-type discrepancy principle.

On Best Proximity Points In b-Metric Space

Eldred and Veeramani proved a theorem which ensures the existence of a best proximity point of cyclic contraction in metric space. In this paper we prove the existence of best proximity points of cyclic contractions and generalize proximal contractions of first and second kinds in the setting of complete b-metric space, thereby ascertaining an optimal approximate solution to the equation Tx = x in a b-metric space.

Extended spectrum, extended eigenspaces and normal operators

We say that a complex number λ is an extended eigenvalue of a bounded linear operator T on a Hilbert space H if there exists a nonzero bounded linear operator X acting on H, called extended eigenvector associated to λ, and satisfying the equation TX=λXT. In this paper we describe the sets of extended eigenvalues and extended eigenvectors for the product of a positive and a self-adjoint operator which are both injective. We also treat the case of normal operators.

Some Convergence Theorems for the Fixed Point in Banach Spaces

Let X be a uniformly smooth Banach space, T:X  X be -strongly quasi accretive (-hemi contractive) mappings. It is shown under suitable conditions that the Ishikawa iteration sequence converges strongly to the unique solution of the equation Tx = f. Our main results is to improve and extend some results about Ishikawa iteration for type from contractive, announced by many others.

Best proximity point theorems: An exploration of a common solution to approximation and optimization problems

Given a non-self mapping T:A→B in the setting of a metric space, this work concentrates on the resolution of the non-linear programming problem of globally minimizing the real valued function x→d(x,Tx), thereby yielding an optimal approximate solution to the equation Tx=x. An iterative algorithm is also presented to compute a solution of such problems. As a sequel, it is possible to compute an optimal approximate solution to some non-linear equations.

Best proximity point theorems for generalized proximal contractions

Best proximity point theorems unravel the techniques for determining an optimal approximate solution, designated as a best proximity point, to the equation Tx = x which is likely to have no solution when T is a non-self mapping. This article presents best proximity point theorems for new classes of non-self mappings, known as generalized proximal contractions, in the setting of metric spaces. Further, the famous Banach's contraction principle and some of its generalizations and variants are realizable as special cases of the aforesaid best proximity point theorems.

A Quadratic Convergence Yielding Iterative Method for Nonlinear Ill-posed Operator Equations

AbstractIn this paper, we consider an iterative method for the approximate solution of the nonlinear ill-posed operator equation Tx = y. The iteration procedure converges quadratically to the unique solution of the equation for the regularized approximation. It is known that (Tautanhahn (2002)) this solution converges to the solution of the given ill-posed operator equation. The convergence analysis and the stopping rule are based on a suitably constructed majorizing sequence. We show that the adaptive scheme considered by Perverzev and Schock (2005) for choosing the regularization parameter can be effectively used here for obtaining an optimal order error estimate.

Best proximity point theorems

Let us assume that A and B are non-empty subsets of a metric space. In view of the fact that a non-self mapping T : A ⟶ B does not necessarily have a fixed point, it is of considerable significance to explore the existence of an element x that is as close to T x as possible. In other words, when the fixed point equation T x = x has no solution, then it is attempted to determine an approximate solution x such that the error d ( x , T x ) is minimum. Indeed, best proximity point theorems investigate the existence of such optimal approximate solutions, known as best proximity points, of the fixed point equation T x = x when there is no solution. Because d ( x , T x ) is at least d ( A , B ) , a best proximity point theorem ascertains an absolute minimum of the error d ( x , T x ) by stipulating an approximate solution x of the fixed point equation T x = x to satisfy the condition that d ( x , T x ) = d ( A , B ) . This article establishes best proximity point theorems for proximal contractions, thereby extending Banach’s contraction principle to the case of non-self mappings.

Extensions of Banach's Contraction Principle

Let A and B be nonempty subsets of a metric space. As a non-self mapping T: A → B does not necessarily have a fixed point, it is of considerable interest to find an element x that is as close to Tx as possible. In other words, if the fixed point equation Tx = x has no exact solution, then it is contemplated to find an approximate solution x such that the error d(x, Tx) is minimum. Indeed, best proximity point theorems investigate the existence of such optimal approximate solutions, called best proximity points, of the fixed point equation Tx = x when there is no exact solution. As d(x, Tx) is at least d(A, B), a best proximity point theorem achieves an absolute minimum of the error d(x, Tx) by stipulating an approximate solution x of the fixed point equation Tx = x to satisfy the condition that d(x, Tx) = d(A, B). This article furnishes extensions of Banach's contraction principle to the case of non-self mappings. On account of the preceding argument, the proposed generalizations are formulated as best proximity point theorems for non-self contractions.

Best proximity points: global optimal approximate solutions

Let A and B be non-empty subsets of a metric space. As a non-self mapping $${T:A\longrightarrow B}$$ does not necessarily have a fixed point, it is of considerable interest to find an element x in A that is as close to Tx in B as possible. In other words, if the fixed point equation Tx = x has no exact solution, then it is contemplated to find an approximate solution x in A such that the error d(x, Tx) is minimum, where d is the distance function. Indeed, best proximity point theorems investigate the existence of such optimal approximate solutions, called best proximity points, to the fixed point equation Tx = x when there is no exact solution. As the distance between any element x in A and its image Tx in B is at least the distance between the sets A and B, a best proximity pair theorem achieves global minimum of d(x, Tx) by stipulating an approximate solution x of the fixed point equation Tx = x to satisfy the condition that d(x, Tx) = d(A, B). The purpose of this article is to establish best proximity point theorems for contractive non-self mappings, yielding global optimal approximate solutions of certain fixed point equations. Besides establishing the existence of best proximity points, iterative algorithms are also furnished to determine such optimal approximate solutions.

Sur le développement en fraction continue d’une généralisation de la cubique de Baum et Sweet

In 1976, Baum and Sweet gave the first example of an algebraic power series of degree 3 over the field 𝔽 2 (T) and whose continued fraction expansion has partial quotients with degree at most 2. This power series is the unique solution in the field 𝔽 2 ((T -1 )) at the equation TX 3 +X-T=0. In 1986, Mills and Robbins described an algorithm that allows one to compute the continued fraction expansion of the Baum and Sweet power series.

The Structure of the Set of Extremal Solutions of Ill-Posed Operator Equation Tx = y with codim R(T) = 1

In this paper, we study the problem of the ill-posed operator equation Tx = y with codim R(T) = 1 in normed linear spaces. The structure of the set of extremal solutions of the equation has been obtained by the maximal elements of the ones in N(T*) and the generalized inverse T + of T. Furthermore, the representation of the set of the extremal solutions of the equation is given formally.

Solutions to operator equations on Hilbert [formula omitted]-modules

Let A be a C ∗ -algebra, E , F and G be Hilbert A -modules, T ∈ L A ( E , F ) , and T ′ ∈ L A ( G , F ) . We generalize the Douglas theorem about the operator equation TX = T ′ from Hilbert space to Hilbert C ∗ -module. To the equation TX = T ′ and to the system of two equations TX = T ′ and XS = S ′ , we get the forms of general solutions (in the case that there exists a solution), and give some sufficient and necessary conditions for the existence of solutions, and the existence of hermitian solutions and positive solutions (in the case G = E ). In addition, the forms of general hermitian solution and general positive solution (in the case that there exists a solution and G = E ) to the equation TX = T ′ are given too.

Best proximity pair theorems for relatively nonexpansive mappings

Let A, B be nonempty closed bounded convex subsets of a uniformly convex Banach space and T : A∪B → A∪B be a map such that T(A) ⊆ B, T(B) ⊆ A and ǁTx − Tyǁ ≤ ǁx − yǁ, for x in A and y in B. The fixed point equation Tx = x does not possess a solution when A ∩ B = Ø. In such a situation it is natural to explore to find an element x0 in A satisfying ǁx0 − Tx0ǁ = inf{ǁa − bǁ : a ∈ A, b ∈ B} = dist(A,B). Using Zorn’s lemma, Eldred et.al proved that such a point x0 exists in a uniformly convex Banach space settings under the conditions stated above. In this paper, by using a convergence theorem we attempt to prove the existence of such a point x0 (called best proximity point) without invoking Zorn’s lemma.

Computability of solutions of operator equations

AbstractWe study operator equations within the Turing machine based framework for computability in analysis. Is there an algorithm that maps pairs (T,u) (whereTis given in form of a program) to solutions ofTx=u? Here we consider the case whenTis a bounded linear mapping between Hilbert spaces. We are in particular interested in computingthe generalized inverse T†u, which is the standard concept of solution in the theory of inverse problems. Typically,T†is discontinuous (i. e. the equationTx=uisill‐posed) and hence no computable mapping. However, we will use effective versions of theorems from the theory ofregularizationto show that the mapping (T,T*,u, ‖T†u‖) ↦T†uis computable. We then go on to study the computability ofaverage‐case solutionswith respect to Gaussian measures which have been considered ininformation based complexity. Here,T†is considered as an element of anL2‐space. We define suitable representations for such spaces and use the results from the first part of the paper to show that (T,T*, ‖T†‖) ↦ T†is computable. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Computability of Solutions of Operator Equations

We study operator equations within the Turing machine based framework for computability in analysis. Is there an algorithm that maps pairs (T,u) (where T is given in form of a program) to good approximate solutions of Tx=u? Here we consider the case when T is a bounded linear mapping of Hilbert spaces. We are in particular interested in computing the generalized inverseT†u which is the standard concept of solution in the theory of inverse problems. Typically, T† is discontinuous (i.e. the equation Tx=u is ill-posed) and hence no computable mapping. However, we will use effective versions of theorems from the theory of regularization to show that the mapping (T,T∗,u,‖T†u‖)↦T†u is computable. We then go on to study the computability of average-case solutions with respect to Gaussian measures which have been considered in information based complexity. Here T† is considered as an element of an L2-space. We define suitable representations for such spaces and use the results from the first part of the paper to show that (T,T∗,‖T†‖L2)↦T† is computable.

Derivatives of the Stretch, Rotation and Exponential Tensors in n-Dimensional Vector Spaces

We present a solution for the tensor equation TX + XTT =H, where T is a diagonalizable ( in particular, symmetric tensor, which is valid for any arbitrary underlying vector space dimension n. This solution is then used to derive compact expressions for the derivatives of the stretch and rotation tensors, which in turn are used to derive expressions for the material time derivatives of these tensors. Some existing expressions for n = 2 and n = 3 are shown to follow from the presented solution as special cases. An alternative methodology for finding the derivatives of diagonalizable tensor-valued functions that is based on differentiating the spectral decomposition is also discussed. Lastly, we also present a method for finding the derivatives of the exponential of an arbitrary tensor for arbitrary n.

Solution of a tridiagonal operator equation

Let H be a separable Hilbert space with an orthonormal basis { e n / n ∈ N } , T be a bounded tridiagonal operator on H and T n be its truncation on span ({ e 1, e 2, … , e n }). We study the operator equation Tx = y through its finite dimensional truncations T n x n = y n . It is shown that if { ‖ T n - 1 e n ‖ } and { ‖ T n ∗ - 1 e n ‖ } are bounded, then T is invertible and the solution of Tx = y can be obtained as a limit in the norm topology of the solutions of its finite dimensional truncations. This leads to uniform boundedness of the sequence { T n - 1 } . We also give sufficient conditions for the boundedness of { ‖ T n - 1 e n ‖ } and { ‖ T n ∗ - 1 e n ‖ } in terms of the entries of the matrix of T.