Analytical and Qualitative Study of Some Families of FODEs via Differential Transform Method

This current work is devoted to develop qualitative theory of existence of solution to some families of fractional order differential equations (FODEs). For this purposes we utilize fixed point theory due to Banach and Schauder. Further using differential transform method (DTM), we also compute analytical or semi-analytical results to the proposed problems. Also by some proper examples we demonstrate the results.

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EXISTENCE OF SOLUTION FOR A VOLTERRA TYPE INTEGRAL EQUATION USING DARBO-TYPE F-CONTRACTION

Advances in Mathematics: Scientific Journal ◽

10.37418/amsj.10.6.2 ◽

2021 ◽

Vol 10 (6) ◽

pp. 2687-2710

Author(s):

F. Akutsah ◽

A. A. Mebawondu ◽

O. K. Narain

Keyword(s):

Integral Equation ◽

Fixed Point ◽

Metric Spaces ◽

Fixed Point Theory ◽

Point Theory ◽

Existence Of Solution ◽

Volterra Type ◽

Complete Metric Spaces ◽

Type Integral ◽

New Type

In this paper, we provide some generalizations of the Darbo's fixed point theorem and further develop the notion of $F$-contraction introduced by Wardowski in (\cite{wad}, D. Wardowski, \emph{Fixed points of a new type of contractive mappings in complete metric spaces,} Fixed Point Theory and Appl., 94, (2012)). To achieve this, we introduce the notion of Darbo-type $F$-contraction, cyclic $(\alpha,\beta)$-admissible operator and we also establish some fixed point and common fixed point results for this class of mappings in the framework of Banach spaces. In addition, we apply our fixed point results to establish the existence of solution to a Volterra type integral equation.

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Approximate solutions and Hyers–Ulam stability for a system of the coupled fractional thermostat control model via the generalized differential transform

Advances in Difference Equations ◽

10.1186/s13662-021-03563-x ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Sina Etemad ◽

Brahim Tellab ◽

Jehad Alzabut ◽

Shahram Rezapour ◽

Mohamed Ibrahim Abbas

Keyword(s):

Fixed Point ◽

Fixed Point Theory ◽

Numerical Technique ◽

Coupled System ◽

Control Model ◽

Ulam Stability ◽

Transform Method ◽

Krasnoselskii’S Fixed Point Theorem ◽

Differential Transform ◽

Generalized Differential

AbstractIn this paper, we consider a new coupled system of fractional boundary value problems based on the thermostat control model. With the help of fixed point theory, we investigate the existence criterion of the solution to the given coupled system. This property is proved by using the Krasnoselskii’s fixed point theorem and its uniqueness is proved via the Banach principle for contractions. Further, the Hyers–Ulam stability of solutions is investigated. Then, we find the approximate solution of the coupled fractional thermostat control system by using a numerical technique called the generalized differential transform method. To show the consistency and validity of our theoretical results, we provide two illustrative examples.

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A new modeling and existence–uniqueness analysis for Babesiosis disease of fractional order

Modern Physics Letters B ◽

10.1142/s0217984921504431 ◽

2021 ◽

pp. 2150443

Author(s):

Rajarama Mohan Jena ◽

Snehashish Chakraverty ◽

Mehmet Yavuz ◽

Thabet Abdeljawad

Keyword(s):

Fixed Point ◽

Fractional Derivative ◽

Numerical Simulations ◽

Arbitrary Order ◽

Fixed Point Theory ◽

Point Theory ◽

Qualitative Properties ◽

Diseased Animal ◽

Transform Method ◽

Homotopy Perturbation

In this study, we consider the dynamics of the Babesiosis transmission on bovine populations and ticks. The most important role in the transmission of the parasite is the ticks from the Ixodidae family. The vector tick takes factors (merozoites in erythrocytes) from the diseased animal while sucking the blood. To model and investigate the transmissions of this parasite and address this important issue, we have considered the disease in a fractional epidemiological model. This paper, therefore, discusses the mechanisms of transmission of Babesiosis defined in the fractional derivative sense. The Caputo–Fabrizio (CF) derivative is considered to study the propagation mechanisms of Babesiosis. First, the important characteristics of the model have been presented, and then the transmission of the Babesiosis model defined in CF is discussed. The application of fixed-point theory is used to derive the concept of the qualitative properties of the mentioned model. The solution is obtained by using the Homotopy perturbation Elzaki transform method (HPETM). Numerical simulations are performed, and the effects of the arbitrary-order derivatives are investigated graphically.

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Some Notes on the Existence of Solution for Ordinary Differential Equations via Fixed Point Theory

Abstract and Applied Analysis ◽

10.1155/2014/309613 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6 ◽

Cited By ~ 1

Author(s):

Fei He

Keyword(s):

Fixed Point ◽

Metric Spaces ◽

Fixed Point Theory ◽

Point Theory ◽

Lower Solution ◽

Existence Of Solution ◽

Order Ordinary Differential Equation ◽

Complete Metric Spaces ◽

First Order ◽

Contractive Mappings

We establish a fixed point theorem withw-distance for nonlinear contractive mappings in complete metric spaces. As applications of our results, we derive the existence and uniqueness of solution for a first-order ordinary differential equation with periodic boundary conditions. Here, we need not assume that the equation has a lower solution.

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Investigation of a Class of Implicit Anti-Periodic Boundary Value Problems

Journal of Mathematical Analysis and Modeling ◽

10.48185/jmam.v2i1.169 ◽

2021 ◽

Vol 2 (1) ◽

pp. 47-61

Author(s):

Laila Hashtamand

Keyword(s):

Fixed Point ◽

Differential Equations ◽

Boundary Conditions ◽

Periodic Boundary ◽

Fixed Point Theory ◽

Periodic Boundary Conditions ◽

Point Theory ◽

Fractional Order Differential Equations ◽

Periodic Boundary Value Problems ◽

Stability Results

This research is devoted to studying a class of implicit fractional order differential equations ($\mathrm{FODEs}$) under anti-periodic boundary conditions ($\mathrm{APBCs}$). With the help of classical fixed point theory due to $\mathrm{Schauder}$ and $\mathrm{Banach}$, we derive some adequate results about the existence of at least one solution. Moreover, this manuscript discusses the concept of stability results including Ulam-Hyers (HU) stability, generalized Hyers-Ulam (GHU) stability, Hyers-Ulam Rassias (HUR) stability, and generalized Hyers-Ulam- Rassias (GHUR)stability. Finally, we give three examples to illustrate our results.

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The Meir–Keeler Fixed Point Theorem for Quasi-Metric Spaces and Some Consequences

Symmetry ◽

10.3390/sym11060741 ◽

2019 ◽

Vol 11 (6) ◽

pp. 741 ◽

Cited By ~ 2

Author(s):

Salvador Romaguera ◽

Pedro Tirado

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Point Theorem ◽

Difference Equations ◽

Metric Spaces ◽

Fixed Point Theory ◽

Point Theory ◽

Existence Of Solution ◽

Nonlinear Difference Equations ◽

Recurrence Equations

We obtain quasi-metric versions of the famous Meir–Keeler fixed point theorem from which we deduce quasi-metric generalizations of Boyd–Wong’s fixed point theorem. In fact, one of these generalizations provides a solution for a question recently raised in the paper “On the fixed point theory in bicomplete quasi-metric spaces”, J. Nonlinear Sci. Appl. 2016, 9, 5245–5251. We also give an application to the study of existence of solution for a type of recurrence equations associated to certain nonlinear difference equations.

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