Non-self Mappings Research Articles

Numerical Algorithm for Coupled Fixed Points in Normed Spaces with Applications to Fractional Differential Equations and Economics

This paper introduces interpolative enriched cyclic Reich–Rus–Ćirić operators in normed spaces, expanding existing contraction principles by integrating interpolation and cyclic conditions. This class of operators addresses mappings with discontinuities or non-self mappings, enhancing the applicability of fixed-point theory to more complex problems. This class of operators expands on existing cyclic contractions, including interpolative Kannan mappings, interpolative Reich–Rus–Ćirić contractions, and other known contractions in the literature. We demonstrate the existence and uniqueness of fixed points for these operators and provide an example to illustrate our findings. Moreover, we discuss the applications of our results in solving nonlinear integral equations. Furthermore, we introduce the idea of a coupled interpolative enriched cyclic Reich–Rus–Ćirić operator and establish the existence of a strongly coupled fixed-point theorem for this contraction. Finally, we provide an application to fractional differential equations to show the validity of the main result.

New best proximity point results on orthogonal [formula omitted]-proximal contractions with applications

In this work, we propose the ideas of orthogonal F-proximal contraction mappings, generalized orthogonal F-proximal contraction mappings and establish several best proximity point theorems in an orthogonally complete metric space (OCMS) through a non-self mapping. Hence, utilizing these recently discovered results, numerous of the existing findings in the literature are generalized or expanded. An illustration is provided to highlight the utility of our findings. Finally, for illustrative applications, we discuss the qualitative properties of the solutions for a fractional boundary value problem in the Caputo sense, and we study the dynamic economic equilibrium problem.

Existence Results of Some Nonlinear Minimization Problems with Application to a System of PDE

A new class of non-self mappings, called proximal condensing operators, is introduced and used to investigate the existence of best proximity points for non-self mappings in reflexive Banach spaces by applying an appropriate measure of noncompactness. In this way, we present generalizations of well-known Schauder and Darbo fixed point problems. We then define the notion of best proximity points for multiplication of two non-self mappings in the setting of Banach algebras and present some sufficient conditions to ensure the existence of such points. Finally, we renorm a Banach space consists of all real valued and continuous functions with two variables to obtain the strict convexity condition and then we survey the existence of an optimal solution for a system of partial differential equations.

Best approximation theorems via relatively u-continuous and Jaggi nonexpansive mappings

In this paper, we introduced a new type of cyclic (noncyclic) mappings, called Jaggi relatively nonexpansive mappings and use to investigate the existence of best proximity points (pairs) in the framework of reflexive (strictly convex) Banach spaces by using a geometric concept of proximal normal structure. We also present a best proximity version of Schauder’s fixed point problem for non-self mappings. Finally, we prove the existence of a common best proximity point for an arbitrary family of cyclic relatively u-continuous mappings which are compact and then we obtain a generalization of Markov-Kakutani’s theorem.

A Fixed Point Theorem for Non-Self Mappings of Rational Type in 𝔟−Multiplicative Metric Spaces with Application to Integral Equations

Objectives: To prove fixed point theorem for non-self mappings of rational type in -multiplicative metric spaces and apply the theorem to solve integral equations. Method: Multiplicative metrically convex is defined in the context of - multiplicative metric spaces to prove fixed point theorems for pairs of non-self mappings using rational type contraction. Findings: The theorem demonstrates that under certain rational type contraction, a unique fixed point exists for non-self mappings in -multiplicative metric spaces. Applications of integral equations shown improved convergence and solution accuracy. Novelty: First fixed point theorem for non-self mappings in -multiplicative metric spaces, introducing a new approach to solving Volterra-Hammerstein integral equations. 2020 Mathematics Subject Classification: 47H10, 54H25. Keywords: Coincidence Point, Nonself Mappings, Rational Type contraction, b-Multiplicative Metric Spaces (b-m ́ms), Volterra-Hammerstein integral equations

Common best proximity point theorems in Hausdorff topological spaces

In the present paper, we have obtained common best proximity point theorems of nonself maps in Hausdorff topological space. Further, our results extend the results due to Gerald F. Jungck, thereby proving a generalized version of Kirk’s theorem (J. London Math. 1(1):107–111, 1969).

Fixed Point Theorems for Non-self Mappings Using Disconnected Graphs

Objectives: To prove the fixed point theorems for non-self mappings using disconnected graphs. Method: Graph theoretical approach is adopted to prove the fixed point theorems for non-self mappings. In all the previous works, connected graphs were used for establishing the results, but it is demonstrated in this work that disconnected graphs are best suited, and this new approach simplifies the proofs to a greater extent. Findings: The fixed point theorems by Banach, Kannan, Chatterjea, and Bianchini are proved using the new methodology. Novelty: An important part of the results concerning fixed point theorems is proving the iterated sequence to be a Cauchy sequence, and this is amalgamated with the edge sequence of the disconnected graph. Subject Classification: 54H25, 47H10 Keywords: Non-self mapping, Iterated sequence, Disconnected graph, Edge sequence, Fixed point

Finding the Best Proximity Point of Generalized Multivalued Contractions with Applications

This paper introduces new kind of algorithms for multivalued non-self mapping to obtain the best proximity point without assuming the continuity of involved mapping. Some non-trivial examples are presented to illustrate the facts. Consequently, an application to finding an optimal approximate solution for the homotopy theory is also discussed.

Comments on the paper ‘A new contribution to best proximity point theory on quasi metric spaces and an application to nonlinear integral equations’

Just recently M. Aslantas [A new contribution to best proximity point theory on quasi metric spaces and an application to nonlinear integral equations. Optimization. 2022] established a new best proximity point theorem multivalued non-self mappings which satisfy an appropriate contractive condition in the framework of quasi metric spaces. Then he applied the existing result to present an application to the solvability of a class of nonlinear integral equations. In this short note, we show that the best proximity point theorem proved in the aforesaid article is a special case of a fixed point theorem for multivalued contraction type mappings which was studied by J. Marin et al. [Q-functions on quasimetric spaces and fixed points for multivalued maps. Fixed Point Theory Appl. 2011:Article ID 603861].

A common fixed point theorem for pairs of non-self mappings in a length space

A common fixed point theorem for pairs of non-self mappings in a length space

Strong Convergence Theorem for Finite Family of Asymptotically Nonexpansive in the Intermediate Sense Nonself Maps

Let K be a nonexpansive retract of a uniformly convex Banach space X with retraction P. Let Ti: K → X (i= 1,...,m) be a finite family of uniformly continuous asymptotically nonexpansive in the intermediate sense maps with a nonempty common fixed points set F. Sufficient conditions for the strong convergence of a sequence of successive approximations generated by an m-step algorithm to a point of F are proved.MSC(2010): 47H10, 47J25

Mann–Dotson’s algorithm for a countable family of non-self strict pseudo-contractive mappings

The aim of this paper to present some weak and strong convergence results for countable family of non-self mappings. More precisely, we employ the Mann–Dotson’s algorithm to approximate common fixed points of a countable family of non-self k-strict Pseudocontractive mappings in q-uniformly smooth Banach spaces.

Mann-Dotson's algorithm for a countable family of non-self Lipschitz mappings in hyperbolic metric space

Abstract The aim of this article is to present some Δ \Delta -convergence and strong convergence results for a countable family of non-self mappings. More precisely, we employ Mann-Dotson’s algorithm to approximate, common fixed points of a countable family of non-self L n {L}_{n} -Lipschitz mappings in hyperbolic metric spaces.

Stability and convergence of Jungck-modified three-step iteration scheme using contractive condition

Abstract The purpose of this paper is to establish convergence and stability of Jungck -modified three-step iterations for three nonself mappings in a Banach space. The results obtained in this paper extend and improve the recent ones announced by Khan et al., Olatinwo, Hussain et al. and many papers.

A new iterative method for approximating common fixed points of two non-self mappings in a CAT(0) space

In this paper, we introduce a new iterative process for approximating common fixed points of two non-self mappings in the setting of CAT(0) spaces. Then we establish Delta -convergence and strong convergence results for two nonexpansive non-self mappings under appropriate conditions. Moreover, we establish strong convergence theorems for approximating common fixed points of two Lipschitz quasi-nonexpansive non-self mappings under some additional conditions. Our results improve, complement and unify most of the results in the literature.

A contribution to best proximity point theory and an application to partial differential equation

In this work, the main discussion centres on avoiding the use of triangle inequality while proving Cauchy sequence to establish best proximity point theorems for single valued as well as multivalued non-self mappings. We prove such best proximity point theorems in the setting of non-triangular metric spaces and elaborate through examples. In this process host of the existing best proximity results are generalized and improved. To arouse further interest in the subject, we connect this work in solving a specific type of partial differential equation problem.

A New Best Proximity Point Results in Partial Metric Spaces Endowed with a Graph

For a given mapping f in the framework of different spaces, the fixed-point equations of the form fx=x can model several problems in different areas, such as differential equations, optimization, and computer science. In this work, the aim is to find the best proximity point and to prove its uniqueness on partial metric spaces where the symmetry condition is preserved for several types of contractive non-self mapping endowed with a graph. Our theorems generalize different results in the literature. In addition, we will illustrate the usability of our outcomes with some examples. The proposed model can be considered as a theoretical foundation for applications to real cases.

Common fixed-point theorems for non-linear non-self contractive mappings in convex metric spaces

Abstract In this article, two pairs of non-self mappings satisfying a collection of non-linear contractive conditions in convex metric space are considered to prove a common fixed-point theorem. An appropriate example is provided to support the results proved herein. In addition, we have proved a theorem as an application of our main result. Our results generalize and extend several existing theorems in the literature.

Best Proximity Point Theorem for α ̃–ψ ̃-Contractive Type Mapping in Fuzzy Normed Space

The best proximity point is a generalization of a fixed point that is beneficial when the contraction map is not a self-map. On other hand, best approximation theorems offer an approximate solution to the fixed point equation . It is used to solve the problem in order to come up with a good approximation. This paper's main purpose is to introduce new types of proximal contraction for nonself mappings in fuzzy normed space and then proved the best proximity point theorem for these mappings. At first, the definition of fuzzy normed space is given. Then the notions of the best proximity point and - proximal admissible in the context of fuzzy normed space are presented. The notion of α ̃–ψ ̃- proximal contractive mapping is introduced. After that, the best proximity point theorem for such type of mapping in a fuzzy normed space is state and prove. In addition, the idea of α ̃–ϕ ̃-proximal contractive mapping is presented in a fuzzy normed space and under specific conditions, the best proximity point theorem for such type of mappings is proved. Furthermore, some examples are offered to show the results' usefulness.

Fixed point theorems for non-self mappings on strictly star-shaped sets in generalized Banach spaces

Fixed point theorems for non-self mappings on strictly star-shaped sets in generalized Banach spaces