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The aim of this paper is to present existence and uniqueness results for the solution of a general system of Fredholm integral equations by applying the vectorial form of Maia's fixed‐point theorem. A Gronwall‐type lemma and comparison theorems are also obtained. [ABSTRACT FROM AUTHOR]

The operators T on a Hilbert space H which have 3-isometric liftings S on Hilbert spaces K containing H are investigated. Here, we deal with the liftings S which are 2-isometries on their invariant subspace K ⊖ H and also have this subspace invariant for S ∗ S . Several characterizations for such operators T are obtained, including the case when the lifting S is expansive. As an application, we refer to the class of (A, 2)-contractions for a positive operator A on H , and particularly to the expansive 3-concave operators. Also, we obtain an upper triangulation for T induced by S, where the operators on the main diagonal are close to contractions. Finally, we study such liftings S which are minimal, in the sense that K is generated by { S n H , n ≥ 0 } . [ABSTRACT FROM AUTHOR]

The bounded linear operators T on a Hilbert space which have 2-isometric liftings S on are investigated in this paper. We refer to the structure of those operators in the case of general liftings, as well as for a more special type of such liftings S, namely for which is invariant to the orthogonal projection onto the kernel of S∗S − I. In this special case we describe the canonical triangulation of T induced by S, with entries expressed by operators close to contractions or isometries. Also, in this case we refer to some asymptotic properties of T and of its adjoint T∗ obtained by means of asymptotic limits associated to T∗ and T, relative to a lifting S for T. The canonical triangulations of T in this setting are described in detail, and as an application, the Wold type decomposition of a concave operator is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we study the a -contraction property of small extremal shocks for 1-d systems of hyperbolic conservation laws endowed with a single convex entropy, when subjected to large perturbations. We show that the weight coefficient a can be chosen with amplitude proportional to the size of the shock. The main result of this paper is a key building block in the companion paper [G. Chen, S. G. Krupa and A. F. Vasseur, Uniqueness and weak-BV stability for 2 × 2 conservation laws, Arch. Ration. Mech. Anal. 246(1) (2022) 299–332] in which uniqueness and BV-weak stability results for 2 × 2 systems of hyperbolic conservation laws are proved. [ABSTRACT FROM AUTHOR]

There are certain contractive conditions (and contractions) available in the literature that ensure the existence of common fixed points of a couple and a family of mappings. However, to verify the validity of these conditions, each result must be checked separately. Thus, it becomes legitimate to obtain some contraction or contractive conditions that can bypass the computational difficulties of checking contraction and contractive conditions via individual results and ensure the existence of common fixed points simultaneously. In this article, by the virtue of implicit relations, we acquire some contraction and contractive conditions, christened A-contraction and A-contractive conditions, which serve the desired purpose. The utility of the established conditions is exhibited through some typical fixed point results and concrete examples. [ABSTRACT FROM AUTHOR]

We investigate the operators T on a Hilbert space H which have 2-isometric liftings S with the property S ∗ S H ⊂ H . We show that such liftings are closely related to some extensions of T, which have 2 × 2 upper triangular block matrices having contractive entries. We describe these extensions by the corresponding 2-isometric liftings, and respectively as regular A-contractions for some positive operators A. Also, we show that some extensions have Brownian isometric (or unitary) liftings and, in particular, we refer to expansive operators with such liftings. [ABSTRACT FROM AUTHOR]

Certain bounded linear operators T on a complex Hilbert space H which have 2-isometric liftings S on another space K ⊃ H are being investigated. We refer also to a more special type of liftings, as well as to those which additionally meet the condition S ∗SH ⊂ H. Furthermore we describe other types of dilations for T, which are close to 2-isometries. Among these we refer to expansive (concave) operators and also to Brownian unitary dilations. Different matrix representations for such operators are obtained, where matrix entries involve contractive operators. [ABSTRACT FROM AUTHOR]

We investigate the bounded linear operators T on a Hilbert space H which have expansive m-isometric liftings for some integer m ≥ 2 . We refer to such liftings which are analytic, and we characterize the minimal liftings S under the condition S ∗ S H ⊂ H . We also describe the operators which have liftings satisfying this condition and are induced by intertwining relations with compressions of m-isometries, or by complete polynomial domination with m-isometries. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce the concept of a class of generalizedcontraction mappings called A-contraction on S-metric space and investigate theexistence of fixed points over such spaces. Analogue result has been formulatedin integral setting over such an S-metric space. Moreover, the result is applied tohomotopy theory. -

The aim of this paper is to study the existence and uniqueness of solutions for some Fredholm integral equation systems by applying the vectorial form of Maia's fixed point theorem. Some abstract Gronwall lemmas and an abstract comparison lemma are also obtained. [ABSTRACT FROM AUTHOR]

The paper deals with bounded linear operators T on a complex Hilbert space H which have 2-isometric liftings S on a Hilbert space K containing H as a closed subspace. We investigate three types of such liftings, and for each type we describe in detail the corresponding operators T by the 2 × 2 upper triangular block matrices. The entries of those matrices are expressed by operators which belong to certain classes of A-contractions for some positive operators A on H . We also refer to the expansive operators with such liftings, as well as to Brownian isometric or quasi-Brownian unitary liftings. [ABSTRACT FROM AUTHOR]

This paper deals with an interesting open problem of B.E. Rhoades (Contemporary Math. (Amer. Math. Soc.) 72(1988), 233-245) on the existence of general contractive conditions which have fixed points, but are not necessarily continuous at the fixed points. We propose some more solutions to this problem by introducing two new types of contractive mappings, that is, A-contractive and A'-contractive, which are, in some sense, more appropriate than those of the important previous attempts. We establish some new fixed point results involving these two contractive mappings in compact metric spaces and also in complete metric spaces and show that these contractive mappings are not necessarily continuous at their fixed points. Finally, we suggest an applicable area, where our main results may be employed. [ABSTRACT FROM AUTHOR]

In this paper we prove some common fixed points for self mappings in a complete 2-metric space using A-contraction and weakly compatible mappings which is a generalization of many results and improve of earlier results in this literature. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to extend Krasnoselskii’s fixed point theorem to the case of generalized Banach spaces for multivalued operators. As application, we will give an existence result for a system of Fredholm-Volterra type differential inclusions.

The purpose of this paper is to extend Krasnoselskii's fixed point theorem to the case of generalized Banach spaces for singlevalued and multivalued operators. As applications, we will give some existence results for abstract system of Fredholm-Volterra type differential equations and inclusions.

We prove the time-asymptotic stability of composite waves consisting of the superposition of a viscous shock and a rarefaction for the one-dimensional compressible barotropic Navier-Stokes equations. Our result solves a long-standing problem first mentioned in 1986 by Matsumura and Nishihara in [28]. The same authors introduced it officially as an open problem in 1992 in [29] and it was again described as very challenging open problem in 2018 in the survey paper [26]. The main difficulty is due to the incompatibility of the standard anti-derivative method, used to study the stability of viscous shocks, and the energy method used for the stability of rarefactions. Instead of the anti-derivative method, our proof uses the a -contraction with shifts theory recently developed by two of the authors. This method is energy based, and can seamlessly handle the superposition of waves of different kinds. [ABSTRACT FROM AUTHOR]

Abstract: We study some generalized Toeplitz operators associated to operators T on a Hilbert space , for which there exists the limit of for every . We refer to the asymptotic limit of such a T, in the sense of [L. Kerchy, Operators with regular norm-sequences, Acta Sci. Math. (Szeged) 63 (1997) 571–605; L. Kerchy, Generalized Toeplitz operators, Acta Sci. Math. (Szeged) 68 (2002) 373–400; G. Cassier, Generalized Toeplitz operators, restrictions to invariant subspaces and similarity problems, J. Operator Theory 53 (1) (2005) 101–140; C.S. Kubrusly, An Introduction to Models and Decompositions in Operator Theory, Birkhäuser, Boston, 1997], and we give some conditions of ergodicity for T. Also, certain results of Douglas [R.G. Douglas, On the operator equation and related topics, Acta Sci. Math. (Szeged) 30 (1969) 19–32] involving generalized Toeplitz operators are extended in our more general setting, and we apply these results to ρ-contractions. [Copyright &y& Elsevier]

The current article pleads for the possibility to obtain an orthogonal decomposition of a Hilbert space $${\mathcal{H}}$$ which is induced by a regular A-contraction defined in [9, 10], A being a positive operator on $${\mathcal{H}}$$. The decomposition generalizes the well-known decomposition related to a contraction T of $$ {\mathcal{H}} $$, which gives the ergodic character of T. This decomposition is being used to prove certain versions for regular A-contractions of the mean ergodic theorem, as well as a version of Patil’s theorem from [8]. Also, we characterize the solutions of corresponding functional equations in the range of A1/2, by analogy with the result of Lin-Sine in [7]. [ABSTRACT FROM AUTHOR]