Showing papers on "Fixed-point theorem published in 2020"
TL;DR: In this paper, a new fractional model for human liver involving Caputo-Fabrizio derivative with the exponential kernel was proposed, and the existence of a unique solution was explored by using the Picard-Lindelof approach and the fixed-point theory.
Abstract: In this research, we aim to propose a new fractional model for human liver involving Caputo–Fabrizio derivative with the exponential kernel. Concerning the new model, the existence of a unique solution is explored by using the Picard–Lindelof approach and the fixed-point theory. In addition, the mathematical model is implemented by the homotopy analysis transform method whose convergence is also investigated. Eventually, numerical experiments are carried out to better illustrate the results. Comparative results with the real clinical data indicate the superiority of the new fractional model over the pre-existent integer-order model with ordinary time-derivatives.
460 citations
TL;DR: In this paper, the authors provide an extension for the second-order differential equation of a thermostat model to the fractional hybrid equation and inclusion versions, where boundary value conditions of this problem in the form of the hybrid conditions are considered.
Abstract: We provide an extension for the second-order differential equation of a thermostat model to the fractional hybrid equation and inclusion versions. We consider boundary value conditions of this problem in the form of the hybrid conditions. To prove the existence of solutions for our hybrid fractional thermostat equation and inclusion versions, we apply the well-known Dhage fixed point theorems for single-valued and set-valued maps. Finally, we give two examples to illustrate our main results.
211 citations
TL;DR: In this article, the authors presented a mathematical model for the transmission of COVID-19 by the Caputo fractional-order derivative and calculated the equilibrium points and reproduction number for the model and obtained the region of the feasibility of system.
Abstract: We present a mathematical model for the transmission of COVID-19 by the Caputo fractional-order derivative. We calculate the equilibrium points and the reproduction number for the model and obtain the region of the feasibility of system. By fixed point theory, we prove the existence of a unique solution. Using the generalized Adams-Bashforth-Moulton method, we solve the system and obtain the approximate solutions. We present a numerical simulation for the transmission of COVID-19 in the world, and in this simulation, the reproduction number is obtained as R 0 = 1 : 610007996 , which shows that the epidemic continues.
206 citations
TL;DR: The fixed point theorems of Krasnoselskii and Banach are hired to present the existence, uniqueness as well as stability of the model and the behavior of the approximate solution is presented in terms of graphs through various fractional orders.
Abstract: The major purpose of the presented study is to analyze and find the solution for the model of nonlinear fractional differential equations (FDEs) describing the deadly and most parlous virus so-called coronavirus (COVID-19). The mathematical model depending of fourteen nonlinear FDEs is presented and the corresponding numerical results are studied by applying the fractional Adams Bashforth (AB) method. Moreover, a recently introduced fractional nonlocal operator known as Atangana-Baleanu (AB) is applied in order to realize more effectively. For the current results, the fixed point theorems of Krasnoselskii and Banach are hired to present the existence, uniqueness as well as stability of the model. For numerical simulations, the behavior of the approximate solution is presented in terms of graphs through various fractional orders. Finally, a brief discussion on conclusion about the simulation is given to describe how the transmission dynamics of infection take place in society.
150 citations
TL;DR: In this article, the existence of solutions for an initial value problem of a fractional differential equation is obtained by means of Monch's fixed point theorem and the technique of measures of noncompactness.
Abstract: In this paper, the existence of solutions for an initial value problem of a fractional differential equation is obtained by means of Monch's fixed point theorem and the technique of measures of noncompactness. AMS (MOS) Subject Classification. 26A33, 34B10, 34G20
108 citations
TL;DR: A qualitative analysis of the mathematical model of novel corona virus named COVID-19 under nonsingular derivative of fractional order is considered and the results of stability of Ulam's type are presented by using the tools of nonlinear analysis.
Abstract: In this article, a qualitative analysis of the mathematical model of novel corona virus named COVID-19 under nonsingular derivative of fractional order is considered. The concerned model is composed of two compartments, namely, healthy and infected. Under the new nonsingular derivative, we, first of all, establish some sufficient conditions for existence and uniqueness of solution to the model under consideration. Because of the dynamics of the phenomenon when described by a mathematical model, its existence must be guaranteed. Therefore, via using the classical fixed point theory, we establish the required results. Also, we present the results of stability of Ulam's type by using the tools of nonlinear analysis. For the semianalytical results, we extend the usual Laplace transform coupled with Adomian decomposition method to obtain the approximate solutions for the corresponding compartments of the considered model. Finally, in order to support our study, graphical interpretations are provided to illustrate the results by using some numerical values for the corresponding parameters of the model.
86 citations
TL;DR: In this article, the existence of Hilfer fractional integro-differential equations with nonlocal conditions is discussed and an application is presented to validate the theoretical results, using M o ¼ nch fixed point theorem and techniques of noncompactness.
Abstract: In this work, the existence of Hilfer fractional integro-differential equations with nonlocal conditions are discussed. To obtain such result, we use M o ¨ nch fixed point theorem and the techniques of noncompactness. An application is presented to validate the theoretical results.
85 citations
TL;DR: The present study can confirm the applicability of the new generalized Caputo type fractional operator to mathematical epidemiology or real-world problems.
Abstract: In this manuscript, we solve a model of the novel coronavirus (COVID-19) epidemic by using Corrector-predictor scheme. For the considered system exemplifying the model of COVID-19, the solution is established within the frame of the new generalized Caputo type fractional derivative. The existence and uniqueness analysis of the given initial value problem are established by the help of some important fixed point theorems like Schauder’s second and Weissinger’s theorems. Arzela-Ascoli theorem and property of equicontinuity are also used to prove the existence of unique solution. A new analysis with the considered epidemic COVID-19 model is effectuated. Obtained results are described using figures which show the behaviour of the classes of projected model. The results show that the used scheme is highly emphatic and easy to implementation for the system of non-linear equations. The present study can confirm the applicability of the new generalized Caputo type fractional operator to mathematical epidemiology or real-world problems. The stability analysis of the projected scheme is given by the help of some important lemma or results.
83 citations
TL;DR: In this article, it was shown that fixed point theorems of Wardowski and Jleli and Samet (2014) are equivalent to a special case of the well-known fixed point theorem of Skof.
Abstract: Let T be a self-mapping on a complete metric space (X, d). In this paper, we obtain new fixed point theorems assuming that T satisfies a contractive-type condition of the following form: $$\begin{aligned} \psi (d(Tx,Ty)) \le \varphi (d(x,y)) \end{aligned}$$or T satisfies a generalized contractive-type condition of the form $$\begin{aligned} \psi (d(Tx,Ty)) \le \varphi (m(x,y)), \end{aligned}$$where $${\psi ,\varphi :(0,\infty ) \rightarrow {\mathbb {R}}}$$ and m(x, y) is defined by $$\begin{aligned} m(x,y) = \max \left\{ d(x,y), d(x,Tx), d(y,Ty), [d(x,Ty)+d(y,Tx)] / 2 \right\} . \end{aligned}$$In both cases, the results extend and unify many earlier results. Among the other results, we prove that recent fixed point theorems of Wardowski (2012) and Jleli and Samet (2014) are equivalent to a special case of the well-known fixed point theorem of Skof (1977).
63 citations
TL;DR: In this paper, the approximate controllability of non-densely defined Hilfer fractional differential systems with infinite delay has been studied, where the authors employ fractional calculus and Bohnenblust-Karlin's fixed point theorem.
Abstract: This manuscript is mainly focusing on approximate controllability for non-densely defined Hilfer fractional differential system with infinite delay. We study our primary outcomes by employing fractional calculus and Bohnenblust-Karlin’s fixed point theorem. Then, we continue our study to prove the approximate controllability of the Hilfer fractional system with nonlocal conditions. Lastly, we give two applications to support the validity of the study.
61 citations
TL;DR: In this article, the existence, uniqueness and data dependence of solutions of an Atangana-Baleanu-Caputo (ABC)-fractional order differential impulsive system are investigated.
Abstract: The study of existence of solution ensures the essential conditions required for a solution. Keeping the importance of the study, we initiate the existence, uniqueness and data dependence of solutions an Atangana-Baleanu-Caputo (ABC)-fractional order differential impulsive system. For this purpose, the suggested ABC-fractional order differential impulsive system is transferred into equivalent fixed point problem via integral operator. The operator is then analyzed for boundedness, continuity and equicontinuity. Then Arzela-Ascolli theorem ensures the relatively compactness of the operator and the Schauder’s fixed point theorem and Banach’s fixed point theorem are utilized for the existence and uniqueness of solution. Data dependence and expressive application are also provided.
TL;DR: In this paper, the authors solved a time delay fractional COVID-19 SEIR epidemic model via Caputo fractional derivatives using a predictor-corrector method and provided numerical simulations to show the nature of the diseases for different classes.
Abstract: Novel coronavirus (COVID-19), a global threat whose source is not correctly yet known, was firstly recognised in the city of Wuhan, China, in December 2019. Now, this disease has been spread out to many countries in all over the world. In this paper, we solved a time delay fractional COVID-19 SEIR epidemic model via Caputo fractional derivatives using a predictor-corrector method. We provided numerical simulations to show the nature of the diseases for different classes. We derived existence of unique global solutions to the given time delay fractional differential equations (DFDEs) under a mild Lipschitz condition using properties of a weighted norm, Mittag-Leffler functions and the Banach fixed point theorem. For the graphical simulations, we used real numerical data based on a case study of Wuhan, China, to show the nature of the projected model with respect to time variable. We performed various plots for different values of time delay and fractional order. We observed that the proposed scheme is highly emphatic and easy to implementation for the system of DFDEs.
TL;DR: In this paper, the authors investigated the existence of solution of non-autonomous fractional differential equations with integral impulse condition by the measure of noncompactness (MNC), fixed point theorems, and k-set contraction.
Abstract: In this paper, we investigate the existence of solution of non-autonomous fractional differential equations with integral impulse condition by the measure of non-compactness (MNC), fixed point theorems, and k-set contraction. The obtained results are verified via a supporting example.
TL;DR: In this article, the authors examined the existence and Ulam stability of solutions for a class of boundary value problems for nonlinear implicit fractional differential equations with instantaneous impulses in Banach spaces.
Abstract: In this manuscript, we examine the existence and the Ulam stability of solutions for a class of boundary value problems for nonlinear implicit fractional differential equations with instantaneous impulses in Banach spaces. The results are based on fixed point theorems of Darbo and Monch associated with the technique of measure of noncompactness. We provide some examples to indicate the applicability of our results.
TL;DR: In this paper, a fractional order epidemic model was proposed to describe the dynamics of COVID-19 under nonsingular kernel type of fractional derivative. And the existence of the model using the fixed point theorem of Banach and Krasnoselskii's type was discussed.
Abstract: In the current article, we studied the novel corona virus (2019-nCoV or COVID-19) which is a threat to the whole world nowadays. We consider a fractional order epidemic model which describes the dynamics of COVID-19 under nonsingular kernel type of fractional derivative. An attempt is made to discuss the existence of the model using the fixed point theorem of Banach and Krasnoselskii’s type. We will also discuss the Ulam-Hyers type of stability of the mentioned problem. For semi analytical solution of the problem the Laplace Adomian decomposition method (LADM) is suggested to obtain the required solution. The results are simulated via Matlab by graphs. Also we have compare the simulated results with some reported real data for Commutative class at classical order.
TL;DR: A new discrete Caputo fractional difference equation is proposed in complex field based on the theory of discrete fractional calculus, which generalizes the fractional-order neural networks in the real domain, and some sufficient criteria of delay-dependent are deduced to ensure the existence and finite-time stability of solutions for proposed networks.
TL;DR: In this paper, a compartmental mathematical model for the transmission dynamics of the novel Coronavirus-19 under Caputo fractional order derivative was proposed by using fixed point theory of Schauder's and Banach and established some necessary conditions for existence of at least one solution to model under investigation and its uniqueness.
Abstract: This article is devoted to study a compartmental mathematical model for the transmission dynamics of the novel Coronavirus-19 under Caputo fractional order derivative. By using fixed point theory of Schauder’s and Banach we establish some necessary conditions for existence of at least one solution to model under investigation and its uniqueness. After the existence a general numerical algorithm based on Haar collocation method is established to compute the approximate solution of the model. Using some real data we simulate the results for various fractional order using Matlab to reveal the transmission dynamics of the current disease due to Coronavirus-19 through graphs.
TL;DR: In this paper, an inverse problem of parameter identification in a nonlinear quasi-hemivariational inequality posed in a Banach space is studied, and an existence result for the inverse problem is provided.
Abstract: This paper is devoted to studying an inverse problem of parameter identification in a nonlinear quasi-hemivariational inequality posed in a Banach space. We employ the Kluge's fixed point theorem for the set-valued selection map, use the Minty approach and some properties of the Clarke subgradient to prove that the quasi-hemivariational inequality associated to the inverse problem has a nonempty, bounded, and weakly compact solution set. We develop a general regularization framework to provide an existence result for the inverse problem. As an illustrative application, we study an identification inverse problem in a complicated mixed elliptic boundary value problem with p-Laplace operator and an implicit obstacle.
TL;DR: In this paper, a new fractional derivative with Mittag Leffler kernels (AB-derivative) was investigated for the fractional neutral integro-differential equations in Banach spaces.
Abstract: In this paper, we investigate a new fractional derivative with Mittag Leffler kernels ( AB-derivative ) to the fractional neutral integro-differential equations in Banach spaces. The results are based on fixed point theorems, then we implement a suitable examples and illustrate it by graphical methods with variation of fractional order.
TL;DR: Quasi-pinning synchronization and β-exponential pinning stabilization for a class of fractional order BAM neural networks with time-varying delays and discontinuous neuron activations (FBAMNNDDAs) are concerns.
Abstract: This manuscript concerns quasi-pinning synchronization and β-exponential pinning stabilization for a class of fractional order BAM neural networks with time-varying delays and discontinuous neuron activations (FBAMNNDDAs). Firstly, under the framework of Filippov solution and fractional-order differential inclusions analysis for the initial value problem of FBAMNNDDAs is presented. Secondly, two kinds of novel pinning controllers according to pinning control technique are designed. By means of fractional order Lyapunov method and designed pinning control strategy, the sufficient criteria is given first to ensure the quasi-synchronization for the dynamic behavior of FBAMNNDDAs. Furthermore, the error bound of pinning synchronization is explicitly evaluated. Thirdly, via Kakutani s fixed point theorem of set-valued map analysis, Razumikhin condition, and a nonlinear pinning controller, the existence and β-exponential stabilization of FBAMNNDDAs equilibrium point is obtained in the voice of linear matrix inequality (LMI) technique. Fourthly, based on as well as Mittag-Leffler function and growth condition, the global existence of a solution in the Filippov sense of such system is guaranteed with detailed proof. At last, a numerical example with computer simulations are performed to illustrate the effectiveness of proposed theoretical consequences.
TL;DR: In this article, a class of impulsive implicit fractional integrodifferential equations having the boundary value problem with mixed Riemann-Liouville fractional integral boundary conditions was considered.
Abstract: This paper is concerned with a class of impulsive implicit fractional integrodifferential equations having the boundary value problem with mixed Riemann–Liouville fractional integral boundary conditions. We establish some existence and uniqueness results for the given problem by applying the tools of fixed point theory. Furthermore, we investigate different kinds of stability such as Ulam–Hyers stability, generalized Ulam–Hyers stability, Ulam–Hyers–Rassias stability, and generalized Ulam–Hyers–Rassias stability. Finally, we give two examples to demonstrate the validity of main results.
TL;DR: The proposed q-homotopy analysis transform method is hired to find the solution for the time-fractional Klein–Fock–Gordon (FKFG) equation and the obtained results elucidate that, the proposed technique is easy to implement and very effective to analyse the behaviour complex problems arise in science and technology.
05 Feb 2020
TL;DR: In this article, a common fixed point theorem via extended Z-contraction with respect to ψ -simulation function over an auxiliary function was investigated in the setting of b-metric space.
Abstract: In this article, we aim to evaluate and merge the as-scattered-as-possible results in fixed point theory from a general framework. In particular, we considered a common fixed point theorem via extended Z-contraction with respect to ψ -simulation function over an auxiliary function ξ in the setting of b-metric space. We investigated both the existence and uniqueness of common fixed points of such mappings. We used an example to illustrate the main result observed. Our main results cover several existing results in the corresponding literature.
07 Jan 2020
TL;DR: In this article, a class of implicit pantograph fractional differential equation with more general Riemann-Liouville fractional integral condition is presented, where a certain class of generalized fractional derivative is used to set the problem.
Abstract: This paper presents a class of implicit pantograph fractional differential equation with more general Riemann-Liouville fractional integral condition. A certain class of generalized fractional derivative is used to set the problem. The existence and uniqueness of the problem is obtained using Schaefer’s and Banach fixed point theorems. In addition, the Ulam-Hyers and generalized Ulam-Hyers stability of the problem are established. Finally, some examples are given to illustrative the results.
TL;DR: In this paper, the authors introduced a large class of contractive mappings, called enriched contractions, a class which includes, amongst many other contractive type mappings including the Picard-Banach contractions and some non-non-convex mappings.
Abstract: We introduce a large class of contractive mappings, called enriched contractions, a class which includes, amongst many other contractive type mappings, the Picard–Banach contractions and some nonexpansive mappings. We show that any enriched contraction has a unique fixed point and that this fixed point can be approximated by means of an appropriate Krasnoselskij iterative scheme. Several important results in fixed point theory are shown to be corollaries or consequences of the main results of this paper. We also study the fixed points of local enriched contractions, asymptotic enriched contractions and Maia-type enriched contractions. Examples to illustrate the generality of our new concepts and the corresponding fixed point theorems are also given.
TL;DR: In this paper, the authors considered a fractional thermostat model with convex and concave source terms and provided an iterative algorithm to approximate the solutions based on a fixed point theorem on cones.
Abstract: We consider a fractional thermostat model involving $$\psi $$-Caputo fractional derivatives. Two cases are discussed: the case when the source term is concave and the case when the source term is convex. For each case, the existence and uniqueness of positive solutions are investigated. Moreover, an iterative algorithm is provided to approximate the solutions. Our approach is based on a fixed point theorem on cones.
TL;DR: In this paper, the authors obtain fixed point theorems under contractive conditions which admit discontinuity at the fixed point, which is a generalization of the fixed-circle problem on metric spaces.
Abstract: We obtain some fixed point theorems under contractive conditions which admit discontinuity at the fixed point. Our results subsume all the known results of similar type, provide new answers to the question of continuity of contractive mappings at their fixed points, and also extend some recent results. Furthermore, we consider the fixed-circle problem on metric spaces. We obtain new fixed-circle results as the generalizations of some known fixed-circle theorems. Some applications of obtained results to discontinuous activation functions are also given.
TL;DR: In this article, the numerical solution of fractional nonlinear system of equations representing the one-dimensional Cauchy problem arising in thermoelasticity is studied. The proposed technique is...
Abstract: In this paper, we study on the numerical solution of fractional nonlinear system of equations representing the one-dimensional Cauchy problem arising in thermoelasticity. The proposed technique is ...
TL;DR: In this article, sufficient conditions for the existence and uniqueness of solutions for a class of nonlinear fractional integrodifferential equations with boundary conditions involving α > 0, α < 1.
Abstract: We establish sufficient conditions for the existence and uniqueness of solutions for a class of nonlinear fractional integrodifferential equations with boundary conditions involving
$$\psi $$
-Hilfer fractional derivative of order
$$0<\alpha <1 $$
and type
$$0\le \beta \le 1$$
. Different types of Ulam–Hyers stability for solutions of the given problem are also discussed. The desired results are proved in weighted spaces with the aid of fixed point theorems due to Schauder, Schaefer and Banach together with generalized Gronwall inequality. Examples illustrating the obtained results are presented.
TL;DR: In this article, the Cauchy problem for a class of nonlinear time fractional non-autonomous integro-differential evolution equation of mixed type via measure of noncompactness in infinite-dimensional Banach spaces was studied.
Abstract: This paper deals with the Cauchy problem to a class of nonlinear time fractional non-autonomous integro-differential evolution equation of mixed type via measure of noncompactness in infinite-dimensional Banach spaces. Combining the theory of fractional calculus and evolution families, the fixed point theorem with respect to convex-power condensing operator and a new estimation technique of the measure of noncompactness, we obtained the existence of mild solutions under the situation that the nonlinear function satisfy some appropriate local growth condition and a noncompactness measure condition. Our results generalize and improve some previous results on this topic, since the condition of uniformly continuity of the nonlinearity is not required, and also the strong restriction on the constants in the condition of noncompactness measure is completely deleted. As samples of applications, we consider the initial value problem to a class of time fractional non-autonomous partial differential equation with homogeneous Dirichlet boundary condition at the end of this paper.