Nonlocal Cauchy Problem via a Fractional Operator Involving Power Kernel in Banach Spaces

We investigated existence and uniqueness conditions of solutions of a nonlinear differential equation containing the Caputo–Fabrizio operator in Banach spaces. The mentioned derivative has been proposed by using the exponential decay law and hence it removed the computational complexities arising from the singular kernel functions inherit in the conventional fractional derivatives. The method used in this study is based on the Banach contraction mapping principle. Moreover, we gave a numerical example which shows the applicability of the obtained results.

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On a functional equation arising in mathematical biology and theory of learning

Creative Mathematics and Informatics ◽

10.37193/cmi.2015.01.11 ◽

2015 ◽

Vol 24 (1) ◽

pp. 9-16

Author(s):

VASILE BERINDE ◽

◽

ABDUL RAHIM KHAN ◽

◽

Keyword(s):

Functional Equation ◽

Mathematical Biology ◽

Fixed Point ◽

Contraction Mapping ◽

Contraction Mapping Principle ◽

Approximation Result ◽

Banach Contraction ◽

The Right ◽

Banach Contraction Mapping Principle ◽

Banach Contraction Mapping

V. Istrat¸escu [Istr ˘ at¸escu, V. I., ˘ On a functional equation, J. Math. Anal. Appl., 56 (1976), No. 1, 133–136] used the Banach contraction mapping principle to establish an existence and approximation result for the solution of the functional equation ϕ(x) = xϕ((1 − α)x + α) + (1 − x)ϕ((1 − β)x), x ∈ [0, 1], (0 < α ≤ β < 1), which is important for some mathematical models arising in biology and theory of learning. This equation has been studied by Lyubich and Shapiro [A. P. Lyubich, Yu. I. and Shapiro, A. P., On a functional equation (Russian), Teor. Funkts., Funkts. Anal. Prilozh. 17 (1973), 81–84] and subsequently, by Dmitriev and Shapiro [Dmitriev, A. A. and Shapiro, A. P., On a certain functional equation of the theory of learning (Russian), Usp. Mat. Nauk 37 (1982), No. 4 (226), 155–156]. The main aim of this note is to solve this functional equation with more general arguments for ϕ on the right hand side, by using appropriate fixed point tools.

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Maia type fixed point theorems for Ćirić-Prešić operators

Acta Universitatis Sapientiae Mathematica ◽

10.2478/ausm-2018-0002 ◽

2018 ◽

Vol 10 (1) ◽

pp. 18-31

Author(s):

Margareta-Eliza Balazs

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Contraction Mapping ◽

Fixed Point Theorems ◽

Acta Math ◽

Contraction Mapping Principle ◽

Contraction Condition ◽

Banach Contraction ◽

Banach Contraction Mapping Principle ◽

Banach Contraction Mapping

Abstract The main aim of this paper is to obtain Maia type fixed point results for Ćirić-Prešić contraction condition, following Ćirić L. B. and Prešić S. B. result proved in [Ćirić L. B.; Prešić S. B. On Prešić type generalization of the Banach contraction mapping principle, Acta Math. Univ. Comenian. (N.S.), 2007, v 76, no. 2, 143–147] and Luong N. V. and Thuan N. X. result in [Luong, N. V., Thuan, N. X., Some fixed point theorems of Prešić-Ćirić type, Acta Univ. Apulensis Math. Inform., No. 30, (2012), 237–249]. We unified these theorems with Maia’s fixed point theorem proved in [Maia, Maria Grazia. Un’osservazione sulle contrazioni metriche. (Italian) Rend. Sem. Mat. Univ. Padova 40 1968 139–143] and the obtained results are proved is the present paper. An example is also provided.

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EXISTENCE AND STABILITY ANALYSIS OF SOLUTIONS FOR FRACTIONAL LANGEVIN EQUATION WITH NONLOCAL INTEGRAL AND ANTI-PERIODIC-TYPE BOUNDARY CONDITIONS

Fractals ◽

10.1142/s0218348x2040006x ◽

2020 ◽

Vol 28 (08) ◽

pp. 2040006 ◽

Cited By ~ 1

Author(s):

AMITA DEVI ◽

ANOOP KUMAR ◽

THABET ABDELJAWAD ◽

AZIZ KHAN

Keyword(s):

Boundary Conditions ◽

Contraction Mapping ◽

Contraction Mapping Principle ◽

Krasnoselskii’S Fixed Point Theorem ◽

Banach Contraction ◽

Existence And Stability ◽

Fractional Differential ◽

Type Boundary ◽

Banach Contraction Mapping Principle ◽

Banach Contraction Mapping

In this paper, we deal with the existence and uniqueness (EU) of solutions for nonlinear Langevin fractional differential equations (FDE) having fractional derivative of different orders with nonlocal integral and anti-periodic-type boundary conditions. Also, we investigate the Hyres–Ulam (HU) stability of solutions. The existence result is derived by applying Krasnoselskii’s fixed point theorem and the uniqueness of result is established by applying Banach contraction mapping principle. An example is offered to ensure the validity of our obtained results.

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A Fixed Point Result with a Contractive Iterate at a Point

Mathematics ◽

10.3390/math7070606 ◽

2019 ◽

Vol 7 (7) ◽

pp. 606 ◽

Cited By ~ 2

Author(s):

Badr Alqahtani ◽

Andreea Fulga ◽

Erdal Karapınar

Keyword(s):

Fixed Point ◽

Contraction Mapping ◽

Contraction Mapping Principle ◽

Continuity Condition ◽

Ulam Stability ◽

Banach Contraction ◽

Iteration Number ◽

The Given ◽

Banach Contraction Mapping Principle ◽

Banach Contraction Mapping

In this manuscript, we define generalized Kincses-Totik type contractions within the context of metric space and consider the existence of a fixed point for such operators. Kincses-Totik type contractions extends the renowned Banach contraction mapping principle in different aspects. First, the continuity condition for the considered mapping is not required. Second, the contraction inequality contains all possible geometrical distances. Third, the contraction inequality is formulated for some iteration of the considered operator, instead of the dealing with the given operator. Fourth and last, the iteration number may vary for each point in the domain of the operator for which we look for a fixed point. Consequently, the proved results generalize the acknowledged results in the field, including the well-known theorems of Seghal, Kincses-Totik, and Banach-Caccioppoli. We present two illustrative examples to support our results. As an application, we consider an Ulam-stability of one of our results.

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Banach contraction principle on cone rectangular metric spaces

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm0902236a ◽

2009 ◽

Vol 3 (2) ◽

pp. 236-241 ◽

Cited By ~ 20

Author(s):

Akbar Azam ◽

Muhammad Arshad ◽

Ismat Beg

Keyword(s):

Metric Space ◽

Contraction Mapping ◽

Metric Spaces ◽

Contraction Mapping Principle ◽

Banach Contraction Principle ◽

Space Setting ◽

Banach Contraction ◽

Rectangular Metric Space ◽

Banach Contraction Mapping Principle ◽

Banach Contraction Mapping

We introduce the notion of cone rectangular metric space and prove Banach contraction mapping principle in cone rectangular metric space setting. Our result extends recent known results.

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Existence and Approximation of Fixed Points of Enriched Contractions and Enriched φ-Contractions

Symmetry ◽

10.3390/sym13030498 ◽

2021 ◽

Vol 13 (3) ◽

pp. 498

Author(s):

Vasile Berinde ◽

Mădălina Păcurar

Keyword(s):

Contraction Mapping ◽

Fixed Point Theorems ◽

Metric Spaces ◽

Contraction Mapping Principle ◽

Future Directions ◽

New Approach ◽

Banach Contraction ◽

Contractive Type ◽

Banach Contraction Mapping Principle ◽

Banach Contraction Mapping

We obtain existence and uniqueness fixed point theorems as well as approximation results for some classes of mappings defined by symmetric contractive type conditions in a convex metric space in the sense of Takahashi. By using a new approach, i.e., the technique of enrichment of contractive type mappings, we obtain general results which extend the well known Banach contraction mapping principle from metric spaces as well as other corresponding results for enriched mappings defined on Banach spaces. To indicate the relevance of our new results, we present some important particular cases and future directions of research.

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Solvability for Discrete Fractional Boundary Value Problems with ap-Laplacian Operator

Discrete Dynamics in Nature and Society ◽

10.1155/2013/679290 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8 ◽

Cited By ~ 3

Author(s):

Weidong Lv

Keyword(s):

Contraction Mapping ◽

Boundary Value ◽

Contraction Mapping Principle ◽

Laplacian Operator ◽

Existence And Uniqueness Results ◽

Uniqueness Results ◽

Banach Contraction ◽

Banach Contraction Mapping Principle ◽

Fractional Boundary Value Problems ◽

Banach Contraction Mapping

This paper is concerned with the solvability for a discrete fractionalp-Laplacian boundary value problem. Some existence and uniqueness results are obtained by means of the Banach contraction mapping principle. Additionally, two representative examples are presented to illustrate the effectiveness of the main results.

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Revisiting of some outstanding metric fixed point theorems via E-contraction

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.2478/auom-2018-0034 ◽

2018 ◽

Vol 26 (3) ◽

pp. 73-98

Author(s):

Andreea Fulga ◽

Erdal Karapınar

Keyword(s):

Fixed Point ◽

Existence And Uniqueness ◽

Contraction Mapping ◽

Fixed Point Theorems ◽

Contractive Mapping ◽

Contraction Mapping Principle ◽

Contraction Mappings ◽

Banach Contraction ◽

Banach Contraction Mapping Principle ◽

Banach Contraction Mapping

AbstractIn this paper, we introduce the notion of α-ψ-contractive mapping of type E, to remedy of the weakness of the existing contraction mappings. We investigate the existence and uniqueness of a fixed point of such mappings. We also list some examples to illustrate our results that unify and generalize the several well-known results including the famous Banach contraction mapping principle.

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Weighted norms and Volterra integral equations in LP spaces

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/s1048953391000126 ◽

1991 ◽

Vol 4 (2) ◽

pp. 161-164 ◽

Cited By ~ 1

Author(s):

Jaroslaw Kwapisz

Keyword(s):

Contraction Mapping ◽

Continuous Functions ◽

Contraction Mapping Principle ◽

Weighted Norm ◽

Uniqueness Of Solutions ◽

Lp Spaces ◽

Banach Contraction ◽

Weighted Norms ◽

Banach Contraction Mapping Principle ◽

Banach Contraction Mapping

A new simple proof of existence and uniqueness of solutions of the Volterra integral equation in Lebesque spaces is given. It is shown that the weighted norm technique and the Banach contraction mapping principle can be applied (as in the case of continuous functions space).

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Almost Periodic Solutions for Wilson-Cowan Type Model with Time-Varying Delays

Discrete Dynamics in Nature and Society ◽

10.1155/2013/683091 ◽

2013 ◽

Vol 2013 ◽

pp. 1-7 ◽

Cited By ~ 1

Author(s):

Shasha Xie ◽

Zhenkun Huang

Keyword(s):

Contraction Mapping ◽

Sufficient Conditions ◽

Neuronal Population ◽

Contraction Mapping Principle ◽

Type Model ◽

Time Varying ◽

Almost Periodic ◽

Banach Contraction ◽

Banach Contraction Mapping Principle ◽

Banach Contraction Mapping

Wilson-Cowan model of neuronal population with time-varying delays is considered in this paper. Some sufficient conditions for the existence and delay-based exponential stability of a unique almost periodic solution are established. The approaches are based on constructing Lyapunov functionals and the well-known Banach contraction mapping principle. The results are new, easily checkable, and complement existing periodic ones.

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