Nonlinear Contractive Mappings Research Articles

Applying periodic and anti-periodic boundary conditions in existence results of fractional differential equations via nonlinear contractive mappings

We introduce a notion of nonlinear cyclic orbital (xi -mathscr{F})-contraction and prove related results. With these results, we address the existence and uniqueness results with periodic/anti-periodic boundary conditions for:1. The nonlinear multi-order fractional differential equation L(D)θ(ς)=σ(ς,θ(ς)),ς∈J=[0,A],A>0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{L}(\\mathcal{D})\ heta (\\varsigma )=\\sigma \\bigl(\\varsigma , \ heta ( \\varsigma ) \\bigr), \\quad \\varsigma \\in \\mathscr{J}=[0,\\mathscr{A}], \\mathscr{A}>0, $$\\end{document} where L(D)=γwcDδw+γw−1cDδw−1+⋯+γ1cDδ1+γ0cDδ0,γ♭∈R(♭=0,1,2,3,…,w),γw≠0,0≤δ0<δ1<δ2<⋯<δw−1<δw<1;\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned} &\\mathcal{L}(\\mathcal{D})=\\gamma _{w} \\,{}^{c} \\mathcal{D}^{\\delta _{w}}+ \\gamma _{w-1} \\,{}^{c} \\mathcal{D}^{\\delta _{w-1}}+\\cdots+\\gamma _{1} \\,{}^{c} \\mathcal{D}^{\\delta _{1}}+\\gamma _{0} \\,{}^{c} \\mathcal{D}^{\\delta _{0}},\\\\ &\\gamma _{\\flat}\\in \\mathbb{R}\\quad (\\flat =0,1,2,3,\\ldots,w), \\qquad \\gamma _{w} \ eq 0, \\\\ &0\\leq \\delta _{0}< \\delta _{1}< \\delta _{2}< \\cdots< \\delta _{w-1}< \\delta _{w}< 1; \\end{aligned}$$ \\end{document}2. The nonlinear multi-term fractional delay differential equation L(D)θ(ς)=σ(ς,θ(ς),θ(ς−τ)),ς∈J=[0,A],A>0;θ(ς)=σ¯(ς),ς∈[−τ,0],\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned} &\\mathcal{L}(\\mathcal{D})\ heta (\\varsigma ) =\\sigma \\bigl(\\varsigma , \ heta ( \\varsigma ),\ heta (\\varsigma -\ au ) \\bigr), \\quad \\varsigma \\in \\mathscr{J}=[0, \\mathscr{A}], \\mathscr{A}>0; \\\\ &\ heta (\\varsigma ) =\\bar{\\sigma}(\\varsigma ),\\quad \\varsigma \\in [-\ au ,0], \\end{aligned}$$ \\end{document} where L(D)=γwcDδw+γw−1cDδw−1+⋯+γ1cDδ1+γ0cDδ0,γ♭∈R(♭=0,1,2,3,…,w),γw≠0,0≤δ0<δ1<δ2<⋯<δw−1<δw<1;\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document} $$\\begin{aligned} &\\mathcal{L}(\\mathcal{D})=\\gamma _{w} \\,{}^{c} \\mathcal{D}^{\\delta _{w}}+ \\gamma _{w-1} \\,{}^{c} \\mathcal{D}^{\\delta _{w-1}}+\\cdots+\\gamma _{1} \\,{}^{c} \\mathcal{D}^{\\delta _{1}}+\\gamma _{0} \\,{}^{c} \\mathcal{D}^{\\delta _{0}},\\\\ &\\gamma _{\\flat}\\in \\mathbb{R}\\quad (\\flat =0,1,2,3,\\ldots,w), \\qquad \\gamma _{w} \ eq 0, \\\\ &0\\leq \\delta _{0}< \\delta _{1}< \\delta _{2}< \\cdots< \\delta _{w-1}< \\delta _{w}< 1; \\end{aligned}$$ \\end{document} moreover, here {}^{c}mathcal{D}^{delta} is predominantly called Caputo fractional derivative of order δ.

Common fixed point of nonlinear contractive mappings

&lt;abstract&gt;&lt;p&gt;The purpose of this paper is to study the existence of a common fixed point for a pair of mappings without assumption of the contractive coefficient being fixed and less than 1. By replacing the fixed contractive coefficient with a nonlinear contractive function, we establish a unique common fixed point theorem for a pair of asymptotically regular self-mappings with either orbital continuity or $ q $-continuity in a metric space. Moreover, by the asymptotical regularity of two approximate mappings, we prove that a pair of nonexpansive and continuous self-mappings, which are defined on a nonempty closed convex subset of a Banach space, have a common fixed point. Some examples are given to illustrate that our results are extensions of a recent result in the existing literature.&lt;/p&gt;&lt;/abstract&gt;

Coincidence point results of nonlinear contractive mappings in partially ordered metric spaces

In this paper, we proved some coincidence point results for \(f\)- nondecreasing self-mapping satisfying certain rational type contractions in the context of a metric space endowed with a partial order. Moreover, some consequences of the main result are given by involving integral type contractions in the space. Some numerical examples are illustrated to support our results. As an application, we have discussed the existence of a unique solution of integral equation.

Convergence of iterates of nonlinear contractive mappings

Convergence of iterates of nonlinear contractive mappings

A tripled fixed point technique for solving a tripled-system of integral equations and Markov process in CCbMS

We prove the existence of tripled fixed points (TFPs) of a new generalized nonlinear contraction mapping in complete cone b-metric spaces (CCbMSs). Also, we present some exciting consequences as corollaries and three nontrivial examples. Finally, we find a solution for a tripled-system of integral equations (TSIE) and discussed a unique stationary distribution for the Markov process (SDMP).

Fractal image compression with adaptive quadtree partitioning and non-linear affine map

Fractal image compression techniques are now very popular for its high compression rates and resolution independence property. However, the qualities of decoded images of the existing techniques are not satisfactory. An adaptive partitioning scheme can improve the image quality significantly. These existing adaptive techniques use linear affine maps during encoding that have limited pixel intensity approximation ability. In order to increase the image quality further, non-linear affine maps can be used that generalizes the pixel intensity approximation and generates much better approximation. Here, a fractal based technique for image compression using non-linear contractive affine maps has been proposed that applies adaptive quadtree partitioning to partition image in a context dependent way to enhance decoded image quality. The technique partitions twice an image to be compressed to obtain collection of ranges and domains and finds the highest matching non-linear affine transformed domain of each range. The corresponding affine parameters are kept in the compressed file. However, a range may be broken into sub-ranges using adaptive quadtree partitioning for unavailability of enough matching domains and repeat the same on those. The comparative results show that the proposed technique greatly improves the decoded image quality than existing techniques and also maintains the high compression ratios. Two variants have also been proposed that improve compression ratio of the proposed technique without any degradation of image quality using loss-less coding.

Best Proximity Point for the Sum of Two Non-Self-Operators

In the present paper, we focus our attention on the existence of the fixed point for the sum of the cyclic contraction and the noncyclic accretive operator. Also, we study the best proximity point for the sum of two non-self-mappings. Moreover, we provide the existence of the best proximity point for the cyclic contraction through the notion of the nonlinear D-set contraction. Finally, we give the existence of the best proximity point for the sum of the nonlinear D-set contraction mapping and partially completely continuous mapping in the setting of the partially ordered complete normed linear space.

Coupled fixed points theorems for non-linear contractions in partially ordered modular spaces

The aim of this paper is to determine some coupled coincidence and coupled common fixed point theorems for mixed $g$-monotone nonlinear contractive mappings in partially ordered modular spaces.

An extension of Lakzian-Rhoades results in the structure of ordered b-metric spaces via wt-distance with an application

In this present article, we establish two new kinds of nonlinear contraction mappings to obtain fixed point results in the structure of ordered b - metric space via wt-distance. In fact, our presented results are extensions of recent theorems due to Lakzian-Rhoades [2019. Appl. Math. Comput.] and other existing classical results of fixed point theory. Furthermore, we provide examples to show the validity of our new investigations. As an application we apply our new findings to obtain solution of a matrix equation. Finally, we verify the accuracy of our new results numerically.

Coupled coincidence point results for mappings without mixed monotone property in partially ordered `G`-metric spaces

In this paper, we prove some coupled fixed point theorems for nonlinear contractive mappings which doesn't have the mixed monotone property, in the context of partially ordered `G`-metric spaces. Hence, these results can be applied in a much wider class of problems. Our results improve the result of D. Dori'{c}, Z. Kadelburg and S. Radenovi'{c} [Appl. Math. Lett. (2012)]. We also present two examples to support these new results.

Some fixed point theorems on nonlinear contractive mappings in Hilbert space

Abstract The purpose of this paper is to establish the existence and uniqueness of a common fixed point for a pair of continuous self mappings over a closed subset of Hilbert space satisfying certain nonlinear rational type contraction condition. Further this result is extended and generalized for some positive integers powers of a pair of continuous self mappings and then is devolved for a sequence of continuous self mappings in the Hilbert space. Our results generalize and extend many well known results in the literature.

Iterating nonlinear contractive mappings in Banach spaces

We introduce a new class of nonlinear contractive mappings in Banach spaces, study their iterates and establish a fixed point theorem for them.

On the Nonlinear Neutral Conformable Fractional Integral-Differential Equation

In this paper, we investigate the nonlinear neutral fractional integral-differential equation involving conformable fractional derivative and integral. First of all, we give the form of the solution by lemma. Furthermore, existence results for the solution and sufficient conditions for uniqueness solution are given by the Leray-Schauder nonlinear alternative and Banach contraction mapping principle. Finally, an example is provided to show the application of results.

Random Common Fixed Point Theorems for Two Pairs of Nonlinear Contractive Maps in Polish Spaces

This research work proves the random common fixed point theorem for two pairs of random weakly compatible mappings fulfilling certain generalized random nonlinear contractive conditions in Polish spaces. An example is given to support the validity of our results. Our results generalize and extend some works in literature.

Existence of a Unique Fixed Point for Nonlinear Contractive Mappings

In a recent work, we established the existence of a unique fixed point for nonlinear contractive self-mappings of a bounded and closed set in a Banach space. In the present paper we extend this result to the case of unbounded sets.

Fixed point theorems via w-distance in relational metric spaces with an application

In this paper, we establish fixed point theorems for generalized nonlinear contractive mappings using the concept of w-distance on metric spaces endowed with an arbitrary binary relation. Our fixed point theorems generalize recent results of Senapati and Dey [ J. Fixed Point Theory Appl., 19, 2945-2961, (2017)] and many other important results of the existing literature. Moreover, in order to revel the usability of our findings an example and an application to first order periodic boundary value problem are given.

Common fixed point theorems for non-self mappings of nonlinear contractive maps in convex metric spaces

In this paper, we introduce a class of nonlinear contractive mappings in metric space. We also establish common fixed point theorems for these pair of non-self mappings satisfying the new contractive conditions in the convex metric space . An example is given to validate our results. The results generalize and extend some results in literature.

Common fixed point for two mapping in N-cone metric spaces

In this paper, we obtain some new coincidence and common fixed point theorems for two nonlinear contractive mappings in the N-cone metric space without the assumption of normal cone. Our main results improve and generalize the corresponding results of Fan et al (J. Math. Comput. Sci. 5 (2015), No. 6, 811-821).

Best proximity coincidence point theorems for generalized non-linear contraction mappings

In this paper, weobtained best proximity coincidence point theorems for ?-Geraghty contractions in the setting of complete metric spaces by using weak P-property. Also we presented some examples to prove the validity of our results. Our results extended and unify many existing results in the literature. Moreover, in the last section as applications of our main results, we can apply some coincidence best proximity point and coupled coincidence proximity point on metric spaces endowed with an arbitrary binary relation.

Coupled fixed point theorem and t-stability for nonlinear contractive mappings in cone metric spaces over Banach algebras

In this paper, we establish the existence of coupled coincidence point and prove coincidence point theorem for nonlinear contractive mappings in cone metric space over Banach algebras. Our results generalize some known results in cone metric space. Moreover, we verify the T-stability of iteration sequence.