New Fixed Point Tools in Non-metrizable Spaces

The iterates of the Cesaro operator were studied recently (see Fontes, F. G. and Sol ` ´ıs, F. J., Iterating the Ces´aro operators, Proc. Amer. Math. Soc., 136 (2008), No. 6, 2147–2153) on some subsets of s(C), on (C[0, 1], C) and on C([0, ∞[, C). They proved that under suitable conditions the sequence of iterates converges to a constant function. In [Andras, Sz. and Rus, I. A., ´ Iterates of Ces`aro operators, Fixed Point Theory, 11 (2010), No. 2, 171–178] the authors gave some more general results regarding the convergence of the iterates by proving that the Cesaro operator is a contraction on a ` dense subset of (C[0, 1], B), equipped with a well chosen norm, where B is a Banach space. The convergence of iterates for some general averaging operators involving one variable functions was also investigated by Sz. Andras and I. A. Rus in [ ´ Iterates of Ces`aro operators, Fixed Point Theory, 11 (2010), No. 2, 171–178]. The aim of this paper is to prove similar results involving Cesaro operators and general averaging operators for several ` variable functions. The proofs are suggested by the characterization theorem of weakly Picard operators on an L-space (see Rus, I. A., Picard operators and applications, Sci. Math. Jpn., 58 (2003), 191–219) and the method can be applied also in the study of some singular integral equations.

Let (X, d) be a metric space, G be a graph associated with X and f : X → X be an operator which satisfies two main assumptions: (1) f is generalized G-monotone; (2) f is a G-contraction with respect to d. In the above framework, we will present sufficient conditions under which: (i) f is a Picard operator; (ii) the fixed point problem x = f(x), x ∈ X is well-posed in the sense of Reich and Zaslavski; (iii) the fixed point problem x = f(x), x ∈ X has the Ulam-Hyers stability property; (iv) f has the Ostrowski stability property; (v) f satisfies to some Gronwall type inequalities. Some open questions are presented.KeywordsMetric spaceDirected graph G-contractionGeneralized contractionIncreasing operatorDecreasing operatorGeneralized G-monotone operatorFixed point(Weakly) Picard operatorStabilityGronwall lemmaOpen problem2010 Mathematics Subject Classification47H1034G2045N0506A0647H0947H0754E3554H25

In order to better understand the wave properties of the Timoshenko equation, the derivation of the refined equation from the equations of the plane elasticity problem for a long band is carried out. The simple iterations method is used for the derivation. It includes known methods: the semi-inverse method of Saint-Venant and Picard operator. In accordance with the semi-inverse method, a part of the unknowns is defined, which are interpreted as the values of the initial (zero) approximation. Proceeding from them, a sequential computation is carried out using a sequence of the four Picard operators in such a way that the outputs of the one operator are the inputs for the next. Calculating in this way all the unknowns in the zeroth approximation by the direct integration over the transverse coordinate, the values of the initial approximation are calculated in the first approximation. These quantities are small of the second order with respect to the dimensionless thickness. Expressions for the unknowns are obtained as power functions of the transverse coordinate and as a function of the deriva-tives along the longitudinal coordinate. By the Banach fixed point theorem, the computation process is asymptot-ically convergent one. After this, boundary conditions on the long sides are satisfied by means of the derivatives of the arbitrariness, depending only on the longitudinal coordinate. This gives us the ordinary differential equations for the determination of these arbitrary functions. In turn, the integration constants of the last equations can be found from the conditions on the short sides of the strip. The ordinary differential equations are split into equations for slowly varying and quickly varying quantities. The slowly changing values give the classical solu-tion of the beam oscillations. The quickly varying solutions give the perturbed solutions describing high-frequency oscillations and singularly perturbed wave solutions for time-concentrated effects. Some of these soluions are absent in the Timoshenko equation. It is assumed that the selected shear waves provoke in the buildings subjected to the rapid impacts (shock by airplane, explosions, and seismic movements of the base) the interruptions of interlayers between the floors and subsequent progressive collapse.

In this paper, we introduce the concept of Eilenberg–Jachymski collection on a nonempty set. Then, we establish three results equivalent to Bourbaki–Kneser’s fixed point theorem, and, therefore, to the axiom of choice. As consequences, we present new fixed point theorems in compact topological spaces, which extend and unify those of Nemytskii–Edelstein, Liepins and Suzuki.

In this article, we studied the existence, uniqueness, stability and controllability of solution of non-linear Volterra–Fredholm–Hammerstein adjoint integro-delay dynamic system with impulses on time scales. The Banach fixed point theorem is employed to demonstrate both the existence and uniqueness of solutions. The Hyers–Ulam stability was verified by applying Grönwall’s inequality on time scale. To address controllability within the presented framework, control function and Picard operator are utilized. To deal with the difficulties in getting the desired results, some assumptions are made. Finally, our theoretical results were supported through two numerical examples and simulations on different time scales.

Converses to fixed point theorems of Zermelo and Caristi

<abstract><p>We introduce a Ćirić type contraction principle in a vector-valued $ b $-metric space that generalizes Perov's contraction principle. We investigate the possible conditions on the mappings $ W, E:G\rightarrow G $ ($ G $ is a non-empty set), for which these mappings admit a unique common fixed point in $ G $ subject to a nonlinear operator $ {\bf F}:\mathbb{P}^{m} \rightarrow \mathbb{R}^{m} $. We illustrate the hypothesis of our findings with examples. We consider an infectious disease model represented by the system of delay integro-differential equations and apply the obtained fixed point theorem to show the existence of a solution to this model.</p></abstract>

In this paper, we investigate the existence of fixed points that are not necessarily unique in the setting of extended b-metric space. We state some examples to illustrate our results.

This chapter pursues three lines of investigation in the subject of functional analysis—one involving smooth functions and distributions, one involving fixed-point theorems, and one involving spectral theory.Section 1 introduces topological vector spaces. These are real or complex vector spaces with a Hausdorff topology in which addition and scalar multiplication are continuous. Examples include normed linear spaces, spaces given by a separating family of countably many seminorms, and weak and weak-star topologies in the context of Banach spaces. Various general properties of topological vector spaces are proved, and it is proved that the quotient of a topological vector space by a closed vector subspace is Hausdorff and is therefore a topological vector space.Section 2 introduces a topology on the space C∞(U) of smooth functions on an open subset of ℝN. The support of a continuous linear functional on C∞(U) is defined and shown to be a compact subset of U. Accordingly, the continuous linear functionals are called distributions of compact support.Section 3 studies weak and weak-star topologies in more detail. The main result is Alaoglu’s Theorem, which says that the closed unit ball in the weak-star topology on the dual of a normed linear space is compact. In an earlier chapter a preliminary form of this theorem was used to construct elements in a dual space as limits of weak-star convergent subsequences.Section 4 follows Alaoglu’s Theorem along a particular path, giving what amounts to a first example of the Gelfand theory of Banach algebras. The relevant theorem, known as the Stone Representation Theorem, says that conjugate-closed uniformly closed subalgebras containing the constants in B(S) are isomorphic via a norm-preserving algebra isomorphism to the space of all continuous functions on some compact Hausdorff space. The compact space in question is the space of multiplicative linear functionals on the subalgebra, and the proof of compactness uses Alaoglu’s Theorem.Sections 5–6 return to the lines of study toward distributions and fixed-point theorems. Section 5 studies the relationship between convexity and the existence of separating linear functionals. The main theorem makes use of the Hahn-Banach Theorem. Section 6 introduces locally convex topological vector spaces. Application of the basic separation theorem from the previous section shows the existence of many continuous linear functionals on such a space.Section 7 specializes to the line of study via smooth functions and distributions. The topic is the introduction of a certain locally convex topology on the space C ∞com (U) of smooth functions of compact support on U. This is best characterized by a universal mapping property introduced in the section.Sections 8–9 pursue locally convex spaces along the other line of study that split off in Section 5. Section 8 gives the Krein-Milman Theorem, which asserts the existence of a supply of extreme points for any nonempty compact convex set in a locally convex topological vector space. Section 9 relates compact convex sets to the subject of fixed-point theorems.Section 10 takes up the abstract theory of Banach algebras, with particular attention to commutative C* algebras with identity. Three examples are the algebras characterized by the Stone Representation Theorem, any L∞ space, and any adjoint-closed commutative Banach algebra consisting of bounded linear operators on a Hilbert space and containing the identity.Section 11 continues the investigation of the last of the examples in the previous section and derives the Spectral Theorem for bounded self-adjoint operators and certain related families of operators. Powerful applications follow from a functional calculus implied by the Spectral Theorem. The section concludes with remarks about the Spectral Theorem for unbounded self-adjoint operators.KeywordsBanach AlgebraTopological Vector SpaceNormed Linear SpaceInternal PointCompact Hausdorff SpaceThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper, we construct a deterministic fractal in fuzzy metric space using generalized fuzzy contraction mapping and its fixed-point theorem in hyperspace of non-empty compact sets. Moreover, we present the self-similar group of H-contraction in fuzzy metric space and prove some familiar results of self-similar group for fuzzy metric space.

A nonlinear discrete model for approximating a conservative multi-fractional Zakharov system: Analysis and computational simulations

The aim of this manuscript is to analyze the stability for fractional neutral stochastic integro-differential delay systems (FNSIDDSs) with impulses driven by Poisson jumps. Sufficient conditions are established for a class of FNSIDDSs with impulses driven by Poisson jumps with the help of fractional calculus, stochastic theory, semigroup theory, and Mönch fixed point theorem in an infinite-dimensional space followed by the existence of a mild solution. The main reason for using Mönch fixed point theorem is its relation to Hausdorff measure of noncompactness (HMNC) adopted to prove the relative compactness conditions. Finally, an example is demonstrated to illustrate the obtained theoretical results.

<p>We explored a class of quantum calculus boundary value problems that include fractional $ q $-difference integrals. Sufficient and necessary conditions for demonstrating the existence and uniqueness of positive solutions were stated using fixed point theorems in partially ordered spaces. Moreover, the existence of a positive solution for a boundary value problem with a Riemann-Liouville fractional derivative and an integral boundary condition was examined by utilizing a novel fixed point theorem that included a $ \mathfrak{a} $-$ \eta $-Geraghty contraction. Several examples were provided to demonstrate the efficacy of the outcomes.</p>

This paper investigates the existence, uniqueness, and monotonicity of positive solutions for a new class of hybrid nonlinear fractional integro-differential equations involving the Caputo–Fabrizio derivative. Unlike the classical Caputo operator, the Caputo–Fabrizio derivative is characterized by a non-singular exponential kernel, which leads to improved analytical tractability and better modeling of memory effects. Using Dhage's fixed point theorem, sufficient conditions for the well-posedness of the problem are derived. A concrete example is provided to demonstrate the effectiveness of the theoretical results, showing the relevance of the Caputo–Fabrizio approach in fractional modeling.

In this article, we first establish a series of user-friendly versions of fixed point theorems in cones for sum of two operators with one being either contractive or expansive and the other being compact, one of which covers the classical Krasnoselskii’s fixed point theorem concerning cone expansion and compression of norm type. Along the way, we also offer some sufficient conditions which slightly relax the compactness requirement on the one summand operator. Second, as applications to some of our main results, we consider the eigenvalue problems of Krasnoselskii type in critical case and the existence of one positive solution to one parameter operator equations. Finally, to illustrate the usefulness and the applicability of our fixed point results, we study the existence of one nontrivial positive solution to certain integral equations of Hammerstein type and of perturbed Volterra type.