Existence of Nonstationary Fixed Point Results on Multivalued Maps and Fractal Construction Using Trajectories

Duality in fixed point theory of multivalued mappings with applications

We prove ∆-convergence and strong convergence theorems of an iterative sequence generated by the Ishikawa’s method to a fixed point of a single-valued quasi-nonexpansive mappings in p-uniformly convex metric spaces without assuming the metric convexity assumption. As a consequence of our single-valued version, we obtain a result for multi-valued mappings by showing that every multi-valued quasi-nonexpansive mapping taking compact values admits a quasi-nonexpansive selection whose fixed-point set of the selection is equal to the strict fixed-point set of the multi-valued mapping. In particular, we immediately obtain all of the convergence theorems of Laokul and Panyanak [Laokul, T.; Panyanak, B. A generalization of the (CN) inequality and its applications. Carpathian J. Math. 36 (2020), no. 1, 81–90] and we show that some of their assumptions are superfluous.

The vital goals of this manuscript are to combine metric-like spaces with S-metric spaces under a control function to obtain a new space called the controlled S-metric-like spaces (CSMLSs, for short). Under this name, many fixed-point (FP) results have been obtained for multi-valued mappings (MVMs). In addition, we present several non-trivial examples to back up our statements. The results obtained generalize and unify many results in the same direction. Finally, to support and test the adequacy of the theoretical results, the existence of the solution to the differential inclusion problem (DIP) was studied as a type of application.

In the present paper, we introduced the concept of generalized multi-valued contraction mappings, via the class functions \(\Phi\) and \(\Psi\) Also we proved some fixed point results for (\(\phi\),\(\mathfrak{F}\))- multi-valued mappings on cone b-metric spaces over Banach algebra \(\mathfrak{V}\). The conditions for existence and uniqueness of the fixed point are investigated. We give an example to support our main result.

In this paper, a new type of graph contractive multi-valued mappings in a metric space with a directed graph is introduced and studied. A common fixed point theorem of those two multi-valued mappings is established under some appropriate conditions. Moreover, some examples illustrating our main result are also given. The obtained result extends and generalizes several fixed point results of multivalued mappings in the literature. We apply our main result to obtain common fixed point results for two multi-valued mappings in ?-chainable complete metric spaces and two cyclic contraction multi-valued mappings.

The purpose of this work is to investigate the existence of fixed points of some mappings in fixed point theory by combining some important concepts which are F-weak contractions, multivalued mappings, integral transformations and α-admissible mappings. In fixed point theory, it is important to find fixed points of some classess under F- or F-weak contractions. Also multivalued mappings is the other important classes. Along with that, α-admissible mapping is a different approach in the fixed point theory. According to this method, a single or multivalued mapping does not have a fixed point in general. But, under some restriction on the mapping, a fixed point can be obtained. In this article, we combine four significant notions and also establish fixed point theorem for this mappings in complete metric spaces. Moreover, we give an example to show the interesting of our results according to earlier results in literature.

In this paper, we introduce an iteration scheme for two multivalued maps in Kohlenbach hyperbolic spaces. This extends the single-valued iteration process due to Agarwal et al. (J. Nonlinear Convex Anal. 8(1):61-79, 2007). Using this new algorithm, we approximate common fixed points of two multivalued mappings through � -convergence and strong convergence under some weaker conditions. A necessary and sufficient condition is given for strong convergence. MSC: Primary 47A06; 47H09; 47H10; secondary 49M05

The purpose of this paper is that iteration scheme of multivalued non-expansive mappings in Banach spaces is extended to hyperbolic spaces and to prove some Δ−convergence theorems of the mixed type iteration process to approximating a common fixed point for two multivalued non-expansive mappings and two non-expansive mappings in hyperbolic spaces. The results presented in the paper extend and improve some recent results announced in the current literature.

A class of demicontractive mappings was first introduced in [Hicks, T. L. and Kubicek, J. D.,On the Mann ite-ration process in a Hilbert space, J. Math. Anal. Appl.,59(1977) 498–504 and M ̆arus ̧ter, S ̧ .,The solution by iterationof nonlinear equations in Hilbert spaces, Proc. Amer. Math. Soc.,63(1977), 69–73] and was first mentioned in thecase of multi-valued mappings in [Chidume, C. E., Bello, A. U. and Ndambomve, P.,Strong and∆-convergencetheorems for common fixed points of a finite family of multivalued demicontractive mappings in CAT(0) spaces, Abstr.Appl. Anal.,2014(2014), https://doi.org/10.1155/2014/805168 and Isiogugu, F. O. and Osilike, M. O.,Conver-gence theorems for new classes of multivalued hemicontractive-type mappings, Fixed Point Theory Appl.,2014(2014),https://doi.org/10.1186/1687-1812-2014-93]. The demicontractivity with some weak smoothness conditionsensures only weak convergence of Mann iteration. In 2015, M ̆arus ̧ter and Rus [Kannan contractions and stronglydemicontractive mappings, Creat. Math. Inform.,24(2015), No. 2, 173–182], introduced a class of strongly de-micontractive mappings, and also discussed some relationships between strongly demicontractive mappingsand Kannan contractions. In this paper, we introduce a new class of strongly demicontractive multi-valuedmappings in Hilbert spaces. Strong convergence theorems of Picard and Mann iterative methods for stronglydemicontractive multi-valued mappings are established under some suitable coefficients and control sequences.

We establish the existence of fixed point for F-contractive multivalued mappings of Hardy–Rogers type in metric-like spaces. Then, we introduce a new notion called bilateral approximate fixed points for multivalued mappings and establish a common bilateral approximate fixed point result for two [Formula: see text]-dominated multivalued contractive mappings in the sense of [M. Arshad, Z. Kadelburgb, S. Radenovic, A. Shoaibe and S. Shukla, Filomat. 31(11) (2017) 3041–3056] on a closed ball of a K-sequentially complete quasi metric-like space. Finally, one can observe that the common bilateral approximate fixed point results coincide with common fixed point results, whenever we reduce the quasi metric-like spaces to metric-like spaces.

Fixed point theory is a crucial branch of mathematical analysis that investigates the conditions under which a function returns a point to itself, symbolizing stability and equilibrium. This paper delves into various fixed point theorems within the contexts of metric spaces, Banach spaces, and Hilbert spaces, emphasizing their foundational importance and wide-ranging applications. The Banach Fixed Point Theorem guarantees the existence and uniqueness of fixed points for contraction mappings in complete metric spaces, while Brouwer's Fixed Point Theorem states that any continuous function mapping a compact convex set to it will have at least one fixed point. Recent developments have expanded these classical results to encompass new types of contraction mappings and generalized distance functions, enhancing their relevance in dynamic systems, control theory, and optimization challenges. Additionally, the paper discusses the Schauder Fixed Point Theorem in Banach spaces, highlighting its significance in analysing nonlinear operators. In Hilbert spaces, fixed point results are examined in relation to nonlinear integral equations and optimization methods, showcasing their practical implications in engineering and variation techniques. Emerging trends include the study of fixed point results in fuzzy and probabilistic environments, as well as the integration of computational approaches with traditional fixed point methods. This paper illustrates the continuous evolution of fixed point theory, connecting abstract mathematical principles with practical problem-solving across various fields. Finally, it proposes future research directions to further explore the potential of fixed point theory in modern mathematics.

It is proved that if a multivalued k-strictly pseudocontractive mapping S of Chidume et al. (Abstr. Appl. Anal. 2013:629468, 2013) is of type one, then $I-S$ I − S is demiclosed at zero. Also, under this condition, the Mann (respectively, Ishikawa) sequence weakly (respectively, strongly) converges to a fixed point of a multivalued k-strictly pseudocontractive (respectively, pseudocontractive) mapping S without the condition that the fixed point set of S is strict, where S is of type one if for any pair $r,g \in D(S)$ r , g ∈ D ( S ) , $$\|u-v\|\leq\Phi(Sr,Sg) \quad\mbox{for all } u\in P_{S}r, v\in P_{S}g, $$ ∥ u − v ∥ ≤ Φ ( S r , S g ) for all u ∈ P S r , v ∈ P S g , and Φ denotes the Hausdorff metric. The results obtained give a partial answer to the problem of the removal of the strict fixed point set condition, which is usually imposed on multivalued mappings. Thus, the results extend, complement, and improve the results on multivalued and single-valued mappings in the contemporary literature.

Introduction.Let I bea reflexive real Banach space, X* its conjugate space, (w, u) the pairing between w in X* and u in X.We consider multi-valued mappings T of X into X* (i.e, mappings in the ordinary sense of X into 2**) which are monotone, i.e., if veT(u), Vy eT(ux) for u and ux in X, then (V -Vy, U -Uy) ^ 0.It is our object in the present paper to generalize to the multi-valued case the results obtained in a number of recent papers by the author and G. J. Minty for single-valued mappings T (cf.[2]-[14]).The first results for multi-valued mappings for X a Hubert space have been obtained in an unpublished paper of Minty [15].The methods of [15] are not directly extendable to more general spaces, but our discussion of the finite-dimensional case (Lemma 2.1) has been very much influenced by the manuscript of [15] which Minty has recently transmitted to the author.(The basic result of [15] is stated at the end of §2 below.)Our results for general multi-valued monotone mappings have an interesting specific application given in §3 below to the generalization of a theorem of Beurling and Livingston [1] on duality mappings in Banach spaces.In a previous paper [12], we showed that for strictly convex reflexive spaces, this theorem could be obtained from results on single-valued monotone mappings.In §3 below we give a generalization of this theorem to general reflexive Banach spaces which runs as follows: Let X be a reflexive Banach space, ej)(r) a non-negative nondecreasing function on P1 with ej)(0) =0.The duality map Tof X with respect to c¡> is defined by T{u) m \v\veX*> M-*M>.> \(V,U) = \\v\\ • || M I .

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.

A new metric in the space clos(X) of all closed subsets of a metric space X is proposed. This metric, unlike the generalized Hausdorff metric, takes finite values only, and the convergence of a sequence of closed sets H i , i=1,2,… , with respect to this metric is equivalent to the convergence (in the sense of Hausdorff) for any r⩾0 of the unions of H i with a closed ‘exterior ball’ of radius r. Using this metric allows one to investigate multi-valued maps that have images in clos(X) and are not continuous in the Hausdorff metric. In the work, the necessary and sufficient conditions for a multi-valued map to be continuous and Lipschitz with respect to the metric presented are studied, a connection of these properties with their analogues in the Hausdorff metric is derived, and a generalization of the Nadler fixed point theorem is obtained.MSC:47H04, 47H10, 54E35.