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Journal articles on the topic 'Banach's fixed point theorem'

Author: Grafiati

Published: 4 June 2021

Last updated: 19 February 2022

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1

Brzdęk, Janusz, Liviu Cădariu, and Krzysztof Ciepliński. "Fixed Point Theory and the Ulam Stability." Journal of Function Spaces 2014 (2014): 1–16. http://dx.doi.org/10.1155/2014/829419.

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The fixed point method has been applied for the first time, in proving the stability results for functional equations, by Baker (1991); he used a variant of Banach's fixed point theorem to obtain the stability of a functional equation in a single variable. However, most authors follow the approaches involving a theorem of Diaz and Margolis. The main aim of this survey is to present applications of different fixed point theorems to the theory of stability of functional equations, motivated by a problem raised by Ulam in 1940.

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2

Subrahmanyam, P. V., and I. L. Reilly. "Some fixed point theorems." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 53, no. 3 (December 1992): 304–12. http://dx.doi.org/10.1017/s144678870003648x.

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AbstractBanach's contraction principle guarantees the existence of a unique fixed point for any contractive selfmapping of a complete metric space. This paper considers generalizations of the completeness of the space and of the contractiveness of the mapping and shows that some recent extensions of Banach's theorem carry over to spaces whose topologies are generated by families of quasi-pseudometrics.

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3

Zima, M. "A certain fixed point theorem and its applications to integral-functional equations." Bulletin of the Australian Mathematical Society 46, no. 2 (October 1992): 179–86. http://dx.doi.org/10.1017/s0004972700011813.

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In this paper a variant of Banach's contraction principle is established. By using the properties of the spectral radius of a bounded linear operator A defined in a suitable Banach space, we conclude that another operator A has exactly one fixed point in this space. In the second part of this paper some applications are given.

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4

Hicks, Troy L. "Fixed point theorems ford-complete topological spaces I." International Journal of Mathematics and Mathematical Sciences 15, no. 3 (1992): 435–39. http://dx.doi.org/10.1155/s0161171292000589.

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Generalizations of Banach's fixed point theorem are proved for a large class of non-metric spaces. These included-complete symmetric (semi-metric) spaces and complete quasi-metric spaces. The distance function used need not be symmetric and need not satisfy the triangular inequality.

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5

Bouzaroura, Asma, and Saïd Mazouzi. "An Alternative Method for the Study of Impulsive Differential Equations of Fractional Orders in a Banach Space." International Journal of Differential Equations 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/191060.

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This paper is concerned with the existence, uniqueness, and stability of the solution of some impulsive fractional problem in a Banach space subjected to a nonlocal condition. Meanwhile, we give a new concept of a solution to impulsive fractional equations of multiorders. The derived results are based on Banach's contraction theorem as well as Schaefer's fixed point theorem.

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6

Sangurlu, S. M., and D. Turkoglu. "Some fixed point results in complete generalized metric spaces." Carpathian Mathematical Publications 9, no. 2 (January 2, 2018): 171–80. http://dx.doi.org/10.15330/cmp.9.2.171-180.

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The Banach contraction principle is the most important result. This principle has many applications and some authors was interested in this principle in various metric spaces as Brianciari. The author initiated the notion of the generalized metric space as a generalization of a metric space by replacing the triangle inequality by a more general inequality, $d(x,y)\leq d(x,u)+d(u,v)+d(v,y)$ for all pairwise distinct points $x,y,u,v$ of $X$. As such, any metric space is a generalized metric space but the converse is not true. He proved the Banach fixed point theorem in such a space. Some authors proved different types of fixed point theorems by extending the Banach's result. Wardowski introduced a new contraction, which generalizes the Banach contraction. He using a mapping $F: \mathbb{R}^{+} \rightarrow \mathbb{R}$ introduced a new type of contraction called $F$-contraction and proved a new fixed point theorem concerning $F$-contraction. In this paper, we have dealt with $F$-contraction and $F$-weak contraction in complete generalized metric spaces. We prove some results for $F$-contraction and $F$-weak contraction and we show that the existence and uniqueness of fixed point for satisfying $F$-contraction and $F$-weak contraction in complete generalized metric spaces. Some examples are supplied in order to support the useability of our results. The obtained result is an extension and a generalization of many existing results in the literature.

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7

Yan, Zuomao. "On solutions of semilinear evolution integrodifferential equations with nonlocal conditions." Tamkang Journal of Mathematics 40, no. 3 (September 30, 2009): 257–69. http://dx.doi.org/10.5556/j.tkjm.40.2009.505.

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In this paper, by using the theory of evolution families, Banach's contraction principle and Schauder's fixed point theorem, we prove the existence of mild solutions of a class of semilinear evolution integrodifferential equations with nonlocal conditions in Banach space. An example is provided to illustrate the obtained results.

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8

DECK, THOMAS. "CONTINUOUS DEPENDENCE ON INITIAL DATA FOR SOLUTIONS OF NONLINEAR STOCHASTIC EVOLUTION EQUATIONS." Infinite Dimensional Analysis, Quantum Probability and Related Topics 05, no. 03 (September 2002): 333–50. http://dx.doi.org/10.1142/s0219025702000870.

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We consider stochastic evolution equations in the framework of white noise analysis. Contraction operators on inductive limits of Banach spaces arise naturally in this context and we first extend Banach's fixed point theorem to this type of spaces. In order to apply the fixed point theorem to evolution equations, we construct a topological isomorphism between spaces of generalized random fields and the corresponding spaces of U-functionals. As an application we show that the solutions of some nonlinear stochastic heat equations depend continuously on their initial data. This method also applies to stochastic Volterra equations, stochastic reaction–diffusion equations and to anticipating stochastic differential equations.

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9

de Bakker, J. W., and E. P. de Vink. "Denotational models for programming languages: applications of Banach's Fixed Point Theorem." Topology and its Applications 85, no. 1-3 (May 1998): 35–52. http://dx.doi.org/10.1016/s0166-8641(97)00140-5.

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10

Adamek, J., and J. Reiterman. "Banach's Fixed-Point Theorem as a base for data-type equations." Applied Categorical Structures 2, no. 1 (1994): 77–90. http://dx.doi.org/10.1007/bf00878504.

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11

Agarwal, Ravi P., Bashir Ahmad, Ahmed Alsaedi, and Hana Al-Hutami. "Existence Theory forq-Antiperiodic Boundary Value Problems of Sequentialq-Fractional Integrodifferential Equations." Abstract and Applied Analysis 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/207547.

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We discuss the existence and uniqueness of solutions for a new class of sequentialq-fractional integrodifferential equations withq-antiperiodic boundary conditions. Our results rely on the standard tools of fixed-point theory such as Krasnoselskii's fixed-point theorem, Leray-Schauder nonlinear alternative, and Banach's contraction principle. An illustrative example is also presented.

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12

CVETKOVIC, MARIJA, and VLADIMIR RAKOCEVIC. "Extensions of Perov theorem." Carpathian Journal of Mathematics 31, no. 2 (2015): 181–88. http://dx.doi.org/10.37193/cjm.2015.02.05.

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[Perov, A. I., On Cauchy problem for a system of ordinary diferential equations, (in Russian), Priblizhen. Metody Reshen. Difer. Uravn., 2 (1964), 115-134] used the concept of vector valued metric space and obtained a Banach type fixed point theorem on such a complete generalized metric space. In this article we study fixed point results for the new extensions of Banach’s contraction principle to cone metric space, and we give some generalized versions of the fixed point theorem of Perov. As corollaries some results of [Zima, M., A certain fixed point theorem and its applications to integral-functional equations, Bull. Austral. Math. Soc., 46 (1992), 179–186] and [Borkowski, M., Bugajewski, D. and Zima, M., On some fixed-point theorems for generalized contractions and their perturbations, J. Math. Anal. Appl., 367 (2010), 464–475] are generalized for a Banach cone space with a non-normal cone. The theory is illustrated with some examples.

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13

Pongarm, Nittaya, Suphawat Asawasamrit, and Jessada Tariboon. "Sequential Derivatives of Nonlinearq-Difference Equations with Three-Pointq-Integral Boundary Conditions." Journal of Applied Mathematics 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/605169.

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This paper studies sufficient conditions for the existence of solutions to the problem of sequential derivatives of nonlinearq-difference equations with three-pointq-integral boundary conditions. Our results are concerned with several quantum numbers of derivatives and integrals. By using Banach's contraction mapping, Krasnoselskii's fixed-point theorem, and Leray-Schauder degree theory, some new existence results are obtained. Two examples illustrate our results.

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14

Czerwik, Stefan, and Krzysztof Król. "Fixed point theorems in generalized metric spaces." Asian-European Journal of Mathematics 10, no. 02 (July 26, 2016): 1750030. http://dx.doi.org/10.1142/s1793557117500309.

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In the paper, we shall prove the results on the existence of fixed points of mapping defined on generalized metric space satisfying a nonlinear contraction condition, which is a generalization of Diaz and Margolis theorem (see [A fixed point theorem of the alternative, for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968) 305–309]). We also present local fixed point theorems both in generalized and ordinary metric spaces. Our results are generalizations of Banach fixed point theorem and many other results.

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15

Riyas, P., and KT Ravindran. "Unique Fixed Point Theorem for Weakly Inward Contractions and Fixed Point Curve in 2-Banach Space." Mathematical Journal of Interdisciplinary Sciences 3, no. 2 (March 30, 2015): 173–82. http://dx.doi.org/10.15415/mjis.2015.32015.

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16

Salem, Ahmed, Faris Alzahrani, and Mohammad Alnegga. "Coupled System of Nonlinear Fractional Langevin Equations with Multipoint and Nonlocal Integral Boundary Conditions." Mathematical Problems in Engineering 2020 (February 21, 2020): 1–15. http://dx.doi.org/10.1155/2020/7345658.

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This research paper is about the existence and uniqueness of the coupled system of nonlinear fractional Langevin equations with multipoint and nonlocal integral boundary conditions. The Caputo fractional derivative is used to formulate the fractional differential equations, and the fractional integrals mentioned in the boundary conditions are due to Atangana–Baleanu and Katugampola. The existence of solution has been proven by two main fixed-point theorems: O’Regan’s fixed-point theorem and Krasnoselskii’s fixed-point theorem. By applying Banach’s fixed-point theorem, we proved the uniqueness result for the concerned problem. This research paper highlights the examples related with theorems that have already been proven.

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17

Tariboon, Jessada, Sotiris K. Ntouyas, and Chatthai Thaiprayoon. "Nonlinear Langevin Equation of Hadamard-Caputo Type Fractional Derivatives with Nonlocal Fractional Integral Conditions." Advances in Mathematical Physics 2014 (2014): 1–15. http://dx.doi.org/10.1155/2014/372749.

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We study existence and uniqueness of solutions for a problem consisting of nonlinear Langevin equation of Hadamard-Caputo type fractional derivatives with nonlocal fractional integral conditions. A variety of fixed point theorems are used, such as Banach’s fixed point theorem, Krasnoselskii’s fixed point theorem, Leray-Schauder’s nonlinear alternative, and Leray-Schauder’s degree theory. Enlightening examples illustrating the obtained results are also presented.

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18

Cvetkovic, Marija. "On the equivalence between Perov fixed point theorem and Banach contraction principle." Filomat 31, no. 11 (2017): 3137–46. http://dx.doi.org/10.2298/fil1711137c.

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There are many results in the fixed point theory that were presented as generalizations of Banach theorem and other well-known fixed point theorems, but later proved equivalent to these results. In this article we prove that Perov?s existence result follows from Banach theorem by using renormization of normal cone and obtained metric. The observed estimations of approximate point given by Perov, could not be obtained from consequences of Banach theorem on metric spaces.

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19

Shukla, Satish, Sriram Balasubramanian, and Mirjana Pavlović. "A Generalized Banach Fixed Point Theorem." Bulletin of the Malaysian Mathematical Sciences Society 39, no. 4 (October 30, 2015): 1529–39. http://dx.doi.org/10.1007/s40840-015-0255-5.

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20

MICU, SORIN, JAIME H. ORTEGA, and ADEMIR F. PAZOTO. "ON THE CONTROLLABILITY OF A COUPLED SYSTEM OF TWO KORTEWEG–DE VRIES EQUATIONS." Communications in Contemporary Mathematics 11, no. 05 (October 2009): 799–827. http://dx.doi.org/10.1142/s0219199709003600.

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This paper proves the local exact boundary controllability property of a nonlinear system of two coupled Korteweg–de Vries equations which models the interactions of weakly nonlinear gravity waves (see [10]). Following the method in [24], which combines the analysis of the linearized system and the Banach's fixed point theorem, the controllability problem is reduced to prove a nonstandard unique continuation property of the eigenfunctions of the corresponding differential operator.

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21

Sitthiwirattham, Thanin, Jessada Tariboon, and Sotiris K. Ntouyas. "Three-Point Boundary Value Problems of Nonlinear Second-Orderq-Difference Equations Involving Different Numbers ofq." Journal of Applied Mathematics 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/763786.

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We study a new class of three-point boundary value problems of nonlinear second-orderq-difference equations. Our problems contain different numbers ofqin derivatives and integrals. By using a variety of fixed point theorems (such as Banach’s contraction principle, Boyd and Wong fixed point theorem for nonlinear contractions, Krasnoselskii’s fixed point theorem, and Leray-Schauder nonlinear alternative) and Leray-Schauder degree theory, some new existence and uniqueness results are obtained. Illustrative examples are also presented.

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22

Islam, Muhammad N. "Three Fixed Point Theorems: Periodic Solutions of a Volterra Type Integral Equation with Infinite Heredity." Canadian Mathematical Bulletin 56, no. 1 (March 1, 2013): 80–91. http://dx.doi.org/10.4153/cmb-2011-123-5.

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AbstractIn this paper we study the existence of periodic solutions of a Volterra type integral equation with infinite heredity. Banach fixed point theorem, Krasnosel'skii's fixed point theorem, and a combination of Krasnosel'skii's and Schaefer's fixed point theorems are employed in the analysis. The combination theorem of Krasnosel'skii and Schaefer requires an a priori bound on all solutions. We employ Liapunov's direct method to obtain such an a priori bound. In the process, we compare these theorems in terms of assumptions and outcomes.

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23

FUKHAR-UD-DIN, HAFIZ. "Existence and approximation of fixed points in convex metric spaces." Carpathian Journal of Mathematics 30, no. 2 (2014): 175–85. http://dx.doi.org/10.37193/cjm.2014.02.11.

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A fixed point theorem for a generalized nonexpansive mapping is established in a convex metric space introduced by Takahashi [A convexity in metric spaces and nonexpansive mappings, Kodai Math. Sem. Rep., 22 (1970), 142–149]. Our theorem generalizes simultaneously the fixed point theorem of Bose and Laskar [Fixed point theorems for certain class of mappings, Jour. Math. Phy. Sci., 19 (1985), 503–509] and the well-known fixed point theorem of Goebel and Kirk [A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35 (1972), 171–174] on a nonlinear domain. The fixed point obtained is approximated by averaging Krasnosel’skii iterations of the mapping. Our results substantially improve and extend several known results in uniformly convex Banach spaces and CAT(0) spaces.

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24

Yao, Si-sheng, and Nan-jing Huang. "A CLASS OF QUASISTATIC CONTACT PROBLEMS FOR VISCOELASTIC MATERIALS WITH NONLOCAL COULOMB FRICTION AND TIME-DELAY." Mathematical Modelling and Analysis 19, no. 4 (September 1, 2014): 491–508. http://dx.doi.org/10.3846/13926292.2014.956354.

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In this paper, a mathematical model which describes the explicit time dependent quasistatic frictional contact problems is introduced and studied. The material behavior is described with a nonlinear viscoelastic constitutive law with time-delay and the frictional contact is modeled with nonlocal Coulomb boundary conditions. A variational formulation of the mathematical model is given, which is called a quasistatic integro-differential variational inequality. Using the Banach's fixed point theorem, an existence and uniqueness theorem of the solution for the quasistatic integro-differential variational inequality is proved under some suitable assumptions. As an application, an existence and uniqueness theorem of the solution for the dual variational formulation is also given.

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25

Mecheraoui, Rachid, Zoran D. Mitrović, Vahid Parvaneh, Hassen Aydi, and Naeem Saleem. "On Some Fixed Point Results in E − Fuzzy Metric Spaces." Journal of Mathematics 2021 (September 17, 2021): 1–6. http://dx.doi.org/10.1155/2021/9196642.

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In the existing literature, Banach contraction theorem as well as Meir-Keeler fixed point theorem were extended to fuzzy metric spaces. However, the existing extensions require strong additional assumptions. The purpose of this paper is to determine a class of fuzzy metric spaces in which both theorems remain true without the need of any additional condition. We demonstrate the wide validity of the new class.

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26

Ansari, Arslan Hojat, Mohammad Saeed Khan, and Vladimir Rakočević. "Maia type fixed point results via C-class function." Acta Universitatis Sapientiae, Mathematica 12, no. 2 (November 1, 2020): 227–44. http://dx.doi.org/10.2478/ausm-2020-0015.

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AbstractIn 1968, M. G. Maia [16] generalized Banach’s fixed point theorem for a set X endowed with two metrics. In 2014, Ansari [2]introduced the concept of C-class functions and generalized many fixed point theorems in the literature. In this paper, we prove some Maia’s type fixed point results via C-class function in the setting of two metrics space endowed with a binary relation. Our results, generalized and extended many existing fixed point theorems, for generalized contractive and quasi-contractive mappings, in a metric space endowed with binary relation.

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27

CHUENSUPANTHARAT, NANTAPORN, and DHANANJAY GOPAL. "On Caristi’s fixed point theorem in metric spaces with a graph." Carpathian Journal of Mathematics 36, no. 2 (2020): 259–68. http://dx.doi.org/10.37193/cjm.2020.02.09.

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We generalize the Caristi’s fixed point theorem for single valued as well as multivalued mappings defined on ametric space endowed with a graph andw-distance. Particularly, we modify the concept of the (OSC)-propertydue to Alfuraidan and Khamsi (Alfuraidan M. R. and Khamsi, M. A.,Caristi fixed point theorem in metric spaceswith graph, Abstr. Appl. Anal., (2014) Art. ID 303484, 5.) which enable us to reformulated their stated graphtheory version theorem (Theorem 3.2 in Alfuraidan M. R. and Khamsi, M. A.,Caristi fixed point theorem in metricspaces with graph, Abstr. Appl. Anal., (2014) Art. ID 303484, 5. ) to the case ofw-distance. Consequently,we extend and improve some recent works concerning extension of Banach Contraction Theorem tow-distancewith graph e.g. (Jachymski, J.,The contraction principle for mappings on a metric space with graph, Proc. Amer. Math.Soc.,136(2008), No. 4, 1359–1373; Nieto, J. J., Pouso, R. L. and Rodriguez-Lopez R.,Fixed point theorems in orderedabstract spaces, Proc. Amer. Math. Soc.,135(2007), 2505–2517 and Petrusel, A. and Rus, I.,Fixed point theorems inorderedL−spaces endowed with graph, Proc. Amer, Math. Soc.,134(2006), 411–418.

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VETRO, FRANCESCA. "A generalization of Nadler fixed point theorem." Carpathian Journal of Mathematics 31, no. 3 (2015): 403–10. http://dx.doi.org/10.37193/cjm.2015.03.18.

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Jleli and Samet gave a new generalization of the Banach contraction principle in the setting of Branciari metric spaces [Jleli, M. and Samet, B., A new generalization of the Banach contraction principle, J. Inequal. Appl., 2014:38 (2014)]. The purpose of this paper is to study the existence of fixed points for multivalued mappings, under a similar contractive condition, in the setting of complete metric spaces. Some examples are provided to illustrate the new theory.

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Czerwik, Stefan, and Krzysztof Król. "Fixed point theorems for a system of transformations in generalized metric spaces." Asian-European Journal of Mathematics 08, no. 04 (November 17, 2015): 1550068. http://dx.doi.org/10.1142/s1793557115500680.

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In this paper we present the results on the existence of fixed points of system of mappings in generalized metric spaces generalizing the result of Diaz and Margolis. Also the “local fixed point theorems” of a system of such mappings both in generalized and ordinary metric spaces are stated. Banach fixed point theorem and many others are consequences of our results.

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Manaka, Hiroko. "Fixed Point Theorems for an Elastic Nonlinear Mapping in Banach Spaces." Abstract and Applied Analysis 2015 (2015): 1–9. http://dx.doi.org/10.1155/2015/760671.

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LetEbe a smooth Banach space with a norm·. LetV(x,y)=x2+y2-2 x,Jyfor anyx,y∈E, where·,·stands for the duality pair andJis the normalized duality mapping. We define aV-strongly nonexpansive mapping byV(·,·). This nonlinear mapping is nonexpansive in a Hilbert space. However, we show that there exists aV-strongly nonexpansive mapping with fixed points which is not nonexpansive in a Banach space. In this paper, we show a weak convergence theorem and strong convergence theorems for fixed points of this elastic nonlinear mapping and give the existence theorem.

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Hannabou, Mohamed, Khalid Hilal, and Ahmed Kajouni. "Existence and Uniqueness of Mild Solutions to Impulsive Nonlocal Cauchy Problems." Journal of Mathematics 2020 (November 12, 2020): 1–9. http://dx.doi.org/10.1155/2020/5729128.

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In this paper, a class of nonlocal impulsive differential equation with conformable fractional derivative is studied. By utilizing the theory of operators semigroup and fractional derivative, a new concept on a solution for our problem is introduced. We used some fixed point theorems such as Banach contraction mapping principle, Schauder’s fixed point theorem, Schaefer’s fixed point theorem, and Krasnoselskii’s fixed point theorem, and we derive many existence and uniqueness results concerning the solution for impulsive nonlocal Cauchy problems. Some concrete applications to partial differential equations are considered. Some concrete applications to partial differential equations are considered.

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Zhang, Jichao, Lingxin Bao, and Lili Su. "On Fixed Point Property under Lipschitz and Uniform Embeddings." Journal of Function Spaces 2018 (October 21, 2018): 1–6. http://dx.doi.org/10.1155/2018/4758546.

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We first present a generalization of ω⁎-Gâteaux differentiability theorems of Lipschitz mappings from open sets to those closed convex sets admitting nonsupport points and then show that every nonempty bounded closed convex subset of a Banach space has the fixed point property for isometries if it Lipschitz embeds into a super reflexive space. With the application of Baudier-Lancien-Schlumprecht’s theorem, we finally show that every nonempty bounded closed convex subset of a Banach space has the fixed point property for continuous affine mappings if it uniformly embeds into the Tsirelson space T⁎.

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Ege, Ozgur, and Ismet Karaca. "Banach fixed point theorem for digital images." Journal of Nonlinear Sciences and Applications 08, no. 03 (May 28, 2016): 237–45. http://dx.doi.org/10.22436/jnsa.008.03.08.

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Vivek, Devaraj, Kuppusamy Kanagarajan, and Seenith Sivasundaram. "On the behavior of solutions of fractional differential equations on time scale via Hilfer fractional derivatives." Fractional Calculus and Applied Analysis 21, no. 4 (August 28, 2018): 1120–38. http://dx.doi.org/10.1515/fca-2018-0060.

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Abstract In this paper, we study the existence and stability of Hilfer-type fractional differential equations (dynamic equations) on time scales. We obtain sufficient conditions for existence and uniqueness of solutions by using classical fixed point theorems such as Schauder's fixed point theorem and Banach fixed point theorem. In addition, Ulam stability of the proposed problem is also discussed. As in application, we provide an example to illustrate our main results.

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35

Elekes, Márton. "On a converse to Banach’s Fixed Point Theorem." Proceedings of the American Mathematical Society 137, no. 09 (September 1, 2009): 3139. http://dx.doi.org/10.1090/s0002-9939-09-09904-3.

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Hussain, N., M. A. Kutbi, and P. Salimi. "Best Proximity Point Results for Modified --Proximal Rational Contractions." Abstract and Applied Analysis 2013 (2013): 1–14. http://dx.doi.org/10.1155/2013/927457.

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We first introduce certain new concepts of --proximal admissible and ---rational proximal contractions of the first and second kinds. Then we establish certain best proximity point theorems for such rational proximal contractions in metric spaces. As an application, we deduce best proximity and fixed point results in partially ordered metric spaces. The presented results generalize and improve various known results from best proximity point theory. Several interesting consequences of our obtained results are presented in the form of new fixed point theorems which contain famous Banach's contraction principle and some of its generalizations as special cases. Moreover, some examples are given to illustrate the usability of the obtained results.

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Karapınar, Erdal, Farshid Khojasteh, and Zoran Mitrović. "A Proposal for Revisiting Banach and Caristi Type Theorems in b-Metric Spaces." Mathematics 7, no. 4 (March 27, 2019): 308. http://dx.doi.org/10.3390/math7040308.

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In this paper, we revisit the renowned fixed point theorems belongs to Caristi and Banach. We propose a new fixed point theorem which is inspired from both Caristi and Banach. We also consider an example to illustrate our result.

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38

Gordon, Joseph Frank. "Fixed-Point Theorem for Isometric Self-Mappings." International Journal of Mathematics and Mathematical Sciences 2020 (October 13, 2020): 1–4. http://dx.doi.org/10.1155/2020/3640539.

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In this paper, we derive a fixed-point theorem for self-mappings. That is, it is shown that every isometric self-mapping on a weakly compact convex subset of a strictly convex Banach space has a fixed point.

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Nuchpong, Cholticha, Sotiris K. Ntouyas, and Jessada Tariboon. "Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions." Open Mathematics 18, no. 1 (January 1, 2020): 1879–94. http://dx.doi.org/10.1515/math-2020-0122.

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Abstract In this paper, we study boundary value problems of fractional integro-differential equations and inclusions involving Hilfer fractional derivative. Existence and uniqueness results are obtained by using the classical fixed point theorems of Banach, Krasnosel’skiĭ, and Leray-Schauder in the single-valued case, while Martelli’s fixed point theorem, nonlinear alternative for multi-valued maps, and Covitz-Nadler fixed point theorem are used in the inclusion case. Examples illustrating the obtained results are also presented.

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Wongcharoen, Athasit, Bashir Ahmad, Sotiris K. Ntouyas, and Jessada Tariboon. "Three-Point Boundary Value Problems for the Langevin Equation with the Hilfer Fractional Derivative." Advances in Mathematical Physics 2020 (May 4, 2020): 1–11. http://dx.doi.org/10.1155/2020/9606428.

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We discuss the existence and uniqueness of solutions for the Langevin fractional differential equation and its inclusion counterpart involving the Hilfer fractional derivatives, supplemented with three-point boundary conditions by means of standard tools of the fixed-point theorems for single and multivalued functions. We make use of Banach’s fixed-point theorem to obtain the uniqueness result, while the nonlinear alternative of the Leray-Schauder type and Krasnoselskii’s fixed-point theorem are applied to obtain the existence results for the single-valued problem. Existence results for the convex and nonconvex valued cases of the inclusion problem are derived via the nonlinear alternative for Kakutani’s maps and Covitz and Nadler’s fixed-point theorem respectively. Examples illustrating the obtained results are also constructed. (2010) Mathematics Subject Classifications. This study is classified under the following classification codes: 26A33; 34A08; 34A60; and 34B15.

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41

Fisher, Brian, and Salvatore Sessa. "On a fixed point theorem of Greguš." International Journal of Mathematics and Mathematical Sciences 9, no. 1 (1986): 23–28. http://dx.doi.org/10.1155/s0161171286000030.

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We consider two selfmapsTandIof a closed convex subsetCof a Banach spaceXwhich are weakly commuting inX, i.e.‖TIx−ITx‖≤‖Ix−Tx‖ for any x in X,and satisfy the inequality‖Tx−Ty‖≤a‖Ix−Iy‖+(1−a)max{‖Tx−Ix‖,‖Ty−Iy‖}for allx,yinC, where0<a<1. It is proved that ifIis linear and non-expansive inCand such thatICcontainsTC, thenTandIhave a unique common fixed point inC.

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Ahmad, Naveed, Zeeshan Ali, Kamal Shah, Akbar Zada, and Ghaus ur Rahman. "Analysis of Implicit Type Nonlinear Dynamical Problem of Impulsive Fractional Differential Equations." Complexity 2018 (2018): 1–15. http://dx.doi.org/10.1155/2018/6423974.

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We study the existence, uniqueness, and various kinds of Ulam–Hyers stability of the solutions to a nonlinear implicit type dynamical problem of impulsive fractional differential equations with nonlocal boundary conditions involving Caputo derivative. We develop conditions for uniqueness and existence by using the classical fixed point theorems such as Banach fixed point theorem and Krasnoselskii’s fixed point theorem. For stability, we utilized classical functional analysis. Also, an example is given to demonstrate our main theoretical results.

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Phuangthong, Nawapol, Sotiris K. Ntouyas, Jessada Tariboon, and Kamsing Nonlaopon. "Nonlocal Sequential Boundary Value Problems for Hilfer Type Fractional Integro-Differential Equations and Inclusions." Mathematics 9, no. 6 (March 15, 2021): 615. http://dx.doi.org/10.3390/math9060615.

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In the present research, we study boundary value problems for fractional integro-differential equations and inclusions involving the Hilfer fractional derivative. Existence and uniqueness results are obtained by using the classical fixed point theorems of Banach, Krasnosel’skiĭ, and Leray–Schauder in the single-valued case, while Martelli’s fixed point theorem, a nonlinear alternative for multivalued maps, and the Covitz–Nadler fixed point theorem are used in the inclusion case. Examples are presented to illustrate our results.

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Bisht, R. K. "Common Fixed Points of Generalized Meir-Keeler Type Condition and Nonexpansive Mappings." International Journal of Mathematics and Mathematical Sciences 2012 (2012): 1–12. http://dx.doi.org/10.1155/2012/786814.

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The aim of the present paper is to obtain common fixed point theorems by employing the recently introduced notion of weak reciprocal continuity. The new notion is a proper generalization of reciprocal continuity and is applicable to compatible mappings as well as noncompatible mappings. We demonstrate that weak reciprocal continuity ensures the existence of common fixed points under contractive conditions, which otherwise do not ensure the existence of fixed points. Our results generalize and extend Banach contraction principle and Meir-Keeler-type fixed point theorem.

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GUILLÉN-GONZÁLEZ, F., and M. A. RODRÍGUEZ-BELLIDO. "CONVERGENCE AND ERROR ESTIMATES OF TWO ITERATIVE METHODS FOR THE STRONG SOLUTION OF THE INCOMPRESSIBLE KORTEWEG MODEL." Mathematical Models and Methods in Applied Sciences 19, no. 09 (September 2009): 1713–42. http://dx.doi.org/10.1142/s0218202509003929.

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We show the existence of strong solutions for a fluid model with Korteweg tensor, which is obtained as limit of two iterative linear schemes. The different unknowns are sequentially decoupled in the first scheme and in parallel form in the second one. In both cases, the whole sequences are bounded in strong norms and convergent towards the strong solution of the system, by using a generalization of Banach's fixed point theorem. Moreover, we explicit a priori and a posteriori error estimates (respect to the weak norms), which let us to compare both schemes.

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Beg, Ismat. "Random fixed points of non-self maps and random approximations." Journal of Applied Mathematics and Stochastic Analysis 10, no. 2 (January 1, 1997): 127–30. http://dx.doi.org/10.1155/s1048953397000154.

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In this paper we prove random fixed point theorems in reflexive Banach spaces for nonexpansive random operators satisfying inward or Leray-Schauder condition and establish a random approximation theorem.

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Suzuki, Tomonari. "Common fixed points of two nonexpansive mappings in Banach spaces." Bulletin of the Australian Mathematical Society 69, no. 1 (February 2004): 1–18. http://dx.doi.org/10.1017/s0004972700034213.

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In this paper, we discuss a necessary and sufficient condition for common fixed points of two nonexpansive mappings. We then prove a convergence theorem to a common fixed point. Finally, we discuss the existence of a nonexpansive retraction onto the set of common fixed points of nonexpansive mappings. In these theorems, we do not assume the strict (uniform) convexity of the norm of the Banach space.

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EHRNSTRÖM, MATS, CHRISTOPHER C. TISDELL, and ERIK WAHLÉN. "ASYMPTOTIC INTEGRATION OF SECOND-ORDER NONLINEAR DIFFERENCE EQUATIONS." Glasgow Mathematical Journal 53, no. 2 (December 8, 2010): 223–43. http://dx.doi.org/10.1017/s0017089510000650.

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AbstractIn this work we analyse a nonlinear, second-order difference equation on an unbounded interval. We present new conditions under which the problem admits a unique solution that is of a particular linear asymptotic form. The results concern the general behaviour of solutions to the initial-value problem, as well as solutions with a given asymptote. Our methods involve establishing suitable complete metric spaces and an application of Banach's fixed-point theorem. For the solutions found in our two main theorems—fixed initial data and fixed asymptote, respectively—we establish exact convergence rates to solutions of the differential equation related to our difference equation. It turns out that for the asymptotic case there is uniform convergence for both the solution and its derivative, while in the other case the convergence is somewhat weaker. Two different techniques are utilized, and for each one has to employ ad-hoc methods for the unbounded interval. Of particular importance is the exact form of the operators and metric spaces formulated in the earlier sections.

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ishihara, Hajime. "Fixed Point Theorems for Lipspchitzian Semigroups." Canadian Mathematical Bulletin 32, no. 1 (March 1, 1989): 90–97. http://dx.doi.org/10.4153/cmb-1989-013-3.

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AbstractLet U be a nonempty subset of a Banach space, S a left reversible semitopological semigroup, a continuous representation of S as lipschitzian mappings on U into itself, that is for each s ∊ S, there exists ks > 0 such that for x, y ∊ U. We first show that if there exists a closed subset C of U such that then S with lim sups has a common fixed point in a Hilbert space. Next, we prove that the theorem is valid in a Banach space E if lim sups

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Jachymski, Jacek R., and James D. Stein. "A minimum condition and some realted fixed-point theorems." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 66, no. 2 (April 1999): 224–43. http://dx.doi.org/10.1017/s144678870003932x.

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AbstractThe classic Banach Contraction Principle assumes that the self-map is a contraction. Rather than requiring that a single operator be a contraction, we weaken this hypothesis by considering a minimum involving a set of iterates of that operator. This idea is a central motif for many of the results of this paper, in which we also study how this weakended hypothesis may be applied in Caristi's theorem, and how combinatorial arguments may be used in proving fixed-point theorems.

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