Nonlinear Functional Analysis Research Articles

Modeling and numerical analysis of fractional order hepatitis B virus model with harmonic mean type incidence rate

A mathematical epidemiological model for the transmission of Hepatitis B virus in the frame of fractional derivative with harmonic mean type incidence rate is proposed in this article. The proposed mathematical model is then fictionalized by utilizing the Atangana–Baleanu–Capotu () operator with vaccination effects. The threshold number R0 is calculated by utilizing the next-generation matrix approach. The existence and uniqueness of solution of the proposed model are proved by utilizing the well-known fixed point theory. For the numerical solution of the proposed model with derivative the well-known Adams–Bashforth–Molton (ABM) method is utilized. Likewise, stability is required in regard of the numerical arrangement. In this manner, Ulam–Hyers stability utilizing nonlinear functional analysis is utilized for the proposed model.

Extended Newton–Kantorovich Theorem for Solving Nonlinear Equations

The Newton–Kantorovich theorem for solving Banach space-valued equations is a very important tool in nonlinear functional analysis. Several versions of this theorem have been given by Adley, Argyros, Ciarlet, Ezquerro, Kantorovich, Potra, Proinov, Wang, et al. This result, e.g., establishes the existence and uniqueness of the solution. Moreover, the Newton sequence converges to the solution under certain conditions of the initial data. However, the convergence region in all of these approaches is small in general; the error bounds on the distances involved are pessimistic, and information about the location of the solutions appears improvable. The novelty of our study lies in the fact that, motivated by optimization concerns, we address all of these. In particular, we introduce a technique that extends the convergence region; provides weaker sufficient semi-local convergence criteria; offers tighter error bounds on the distances involved and more precise information on the location of the solution. These advantages are achieved without additional conditions. This technique can be used to extend other iterative methods along the same lines. Numerical experiments illustrate the theoretical results.

ANALYSIS OF HIDDEN ATTRACTORS OF NON-EQUILIBRIUM FRACTAL-FRACTIONAL CHAOTIC SYSTEM WITH ONE SIGNUM FUNCTION

In 2017, Atangana proposed more generalized operators depending on two parameters: one is fractional order (FO) and other is fractal dimension. The novel operators are defined with three different kernels. These operators produced excellent dynamics of the chaotic systems. In this paper, the Caputo fractal-fractional operator is used to explore a chaotic system which contains only one signum function. The existence theory is developed by using the fixed-point result of Leray–Schauder to prove that the considered chaotic system possesses at least one solution. The proposed chaotic system has a unique solution, according to Banach’s fixed-point theorem. We demonstrate that the suggested chaotic system is Ulam–Hyres (UH) stable under the novel operator of power law kernel by employing nonlinear functional analysis. The Adams–Bashforth technique is used to evaluate the numerical outcomes of the considered model. We show the complex structure of numerical solutions for different FO and fractal dimension values.

ON THE ANALYSIS OF FRACTAL-FRACTIONAL ORDER MODEL OF MIDDLE EAST RESPIRATION SYNDROME CORONAVIRUS (MERS-CoV) UNDER CAPUTO OPERATOR

In this paper, the dynamical behavior of Middle East respiration syndrome coronavirus (MERS-CoV) via a sense of Caputo fractal-fractional order system of differential equation is established. A novel approach of fractional operator known as fractal-fractional Riemann–Liouville derivative is applied to the model considered. Moreover, fractal-fractional derivative is applied to the said problem. Furthermore, the existence and uniqueness of the solution of the considered model are verified by the fixed-point theory approach. The local and global stability of developed system are investigated with the help of the Ulam–Hyers stability technique from nonlinear functional analysis. The fractal-fractional type Adams–Bashforth iterative method is used to establish the numerical solution of said problem. The proposed model is simulated by considering different fractal dimensions [Formula: see text] and fractional order [Formula: see text], converging to the integer order. Hence, it is evident that all the compartmental quantities possess convergence and stability in fractal-fractional form. Moreover, the Fractal-fractional techniques may also be used as a powerful technique/tool to investigate the global dynamics of the disease. However, it can be concluded from the results that quick recovery is possible for human population in the pandemic if there is no animal interaction with humans. In future, we are planning to extend the reported analysis to other fractional (fractal-fractional) operators. Furthermore, the behavior of different infectious models can also be analyzed with the help of newly developed scheme at different fractional orders lying between 0 and 1.

Convergence analysis and applications of the inertial algorithm solving inclusion problems

Nonlinear operator theory is an important area of nonlinear functional analysis. This area encompasses diverse nonlinear problems in many areas of mathematics, the physical sciences and engineering such as monotone operator equations, fixed point problems and more.In this work we are concern with the problem of finding a common solution of a monotone operator equation and fixed point of a nonexpansive mapping in real Hilbert spaces. Derived from dynamical systems, a simple inertial forward-backward splitting method for solving the problem is presented and analyzed under mild and standard assumptions. Some numerical examples in real-world and comparisons with related works, illustrate the theoretical advantages as well the potential applicability of the proposed scheme.

Controllability of Sobolev type fuzzy differential equation with non-instantaneous impulsive condition

<p style='text-indent:20px;'>In this manuscript, we investigate the existence, uniqueness and controllability results of a Sobolev type fuzzy differential equation with non-instantaneous impulsive conditions. Non-linear functional analysis, Banach fixed point theorem and fuzzy theory are the main techniques used to establish these results. In support, an example is given to validate the obtained analytical findings.</p>

Existence of local and global solutions to fractional order fuzzy delay differential equation with non-instantaneous impulses

<abstract><p>The main concern of this manuscript is to examine some sufficient conditions under which the fractional order fuzzy delay differential system with the non-instantaneous impulsive condition has a unique solution. We also study the existence of a global solution for the considered system. Fuzzy set theory, Banach fixed point theorem and Non-linear functional analysis are the major tools to demonstrate our results. In last, an example is given to illustrate these analytical results.</p></abstract>

A Special Issue on Nonlinear Functional Analysis and Its Applications in Memory of Professor Ronald E. Bruck

A Special Issue on Nonlinear Functional Analysis and Its Applications in Memory of Professor Ronald E. Bruck

Analysis of dengue transmission using fractional order scheme

<abstract> <p>In this paper, we will check the existence and stability of the dengue internal transmission model with fraction order derivative as well as analyze it qualitatively. The solution has been determined using Atangana-Baleanu in Caputo sense (ABC) with the help of Sumudu transform (ST). Atangana-Toufik (AT) and fractal fractional operator are used to analyze the dengue transmission which is an advanced approach for such types of biological models. Existence theory and uniqueness for the equilibrium solution are provided via nonlinear functional analysis and fixed point theory. Global stability of the system was also proved by using the Lyapunov function. Such kind of study helps us to analyze dengue transmission which shows the actual effect of dengue transmission in society, also will be helpful in future analysis and control strategies.</p> </abstract>

Optimal harvesting for a periodic n-dimensional food chain model with size structure in a polluted environment.

This study examines an optimal harvesting problem for a periodic n-dimensional food chain model that is dependent on size structure in a polluted environment. This is closely related to the protection of biodiversity, as well as the development and utilization of renewable resources. The model contains state variables representing the density of the ith population, the concentration of toxicants in the ith population, and the concentration of toxicants in the environment. The well-posedness of the hybrid system is proved by using the fixed point theorem. The necessary optimality conditions are derived by using the tangent-normal cone technique in nonlinear functional analysis. The existence and uniqueness of the optimal control pair are verified via the Ekeland variational principle. The finite difference scheme and the chasing method are used to approximate the nonnegative T-periodic solution of the state system corresponding to a given initial datum. Some numerical tests are given to illustrate that the numerical solution has good periodicity. The objective functional here represents the total profit obtained from harvesting n species.

Existence theory and numerical solution of leptospirosis disease model via exponential decay law

<abstract><p>We investigated the leptospirosis epidemic model by using Caputo and Fabrizio fractional derivatives. Picard's successive iterative method and Sumudu transform are taken into consideration for developing the iterative solutions for the leptospirosis disease. Employing nonlinear functional analysis, the stability and uniqueness of the proposed model are established. Sensitivity analysis is taken into account to highlight the most sensitive parameters corresponding to the basic reproductive number. Various solutions to the proposed system have been interpolated by graphs with the application of Matlab software.</p></abstract>

Ulam-Hyers stability for conformable fractional integro-differential impulsive equations with the antiperiodic boundary conditions

<abstract><p>This paper focuses on the stability for a class of conformable fractional impulsive integro-differential equations with the antiperiodic boundary conditions. Firstly, the existence and uniqueness of solutions of the integro-differential equations are studied by using the fixed point theorem under the condition of nonlinear term increasing at most linearly. And then, the Ulam-Hyers stability and Ulam-Hyers-Rassias stability for the boundary value problems are discussed by using the nonlinear functional analysis method and constraining related parameters. Finally, an example is given out to illustrate the applicability and feasibility of our main conclusions. It is worth mentioning that the stability studied in this paper highlights the role of boundary conditions. This method of studying stability is effective and can be applied to the study of stability for many types of differential equations.</p></abstract>

Utilizing fixed point approach to investigate piecewise equations with non-singular type derivative

<abstract><p>In this research work, we establish some new results about piecewise equation involving Caputo Fabrizio derivative (CFD). The concerned class has been recently introduced and these results are fundamental for investigation of qualitative theory and numerical interpretation. We derive some necessary results for the existence, uniqueness and various form of Hyers-Ulam (H-U) type stability for the considered problem. For the required results, we need to utilize usual classical fixed point theorems due to Banach and Krasnoselskii's. Moreover, results devoted to H-U stability are derived by using classical tools of nonlinear functional analysis. Some pertinent test problems are given to demonstrate our results.</p></abstract>

Qualitative analysis of nonlinear impulse langevin equation with helfer fractional order derivatives

<abstract><p>In this manuscript, a class of impulsive Langevin equation with Hilfer fractional derivatives is considered. Using the techniques of nonlinear functional analysis, we establish appropriate conditions and results to discuss existence, uniqueness and different types of Ulam-Hyers stability results of our proposed model, with the help of Banach's fixed point theorem. An example is provided at the end to illustrate our results.</p></abstract>

Ulam-Hyers-Rassias stability for a class of nonlinear implicit Hadamard fractional integral boundary value problem with impulses

<abstract><p>This paper considers a class of nonlinear implicit Hadamard fractional differential equations with impulses. By using Banach's contraction mapping principle, we establish some sufficient criteria to ensure the existence and uniqueness of solution. Furthermore, the Ulam-Hyers stability and Ulam-Hyers-Rassias stability of this system are obtained by applying nonlinear functional analysis technique. As applications, an interesting example is provided to illustrate the effectiveness of main results.</p></abstract>

On a novel impulsive boundary value pantograph problem under Caputo proportional fractional derivative operator with respect to another function

<abstract><p>In this manuscript, we study the existence and Ulam's stability results for impulsive multi-order Caputo proportional fractional pantograph differential equations equipped with boundary and integral conditions with respect to another function. The uniqueness result is proved via Banach's fixed point theorem, and the existence results are based on Schaefer's fixed point theorem. In addition, the Ulam-Hyers stability and Ulam-Hyers-Rassias stability of the proposed problem are obtained by applying the nonlinear functional analysis technique. Finally, numerical examples are provided to supplement the applicability of the acquired theoretical results.</p></abstract>

Corrigendum to “The existence of solutions for integral boundary value problems with p-Laplacian operator on infinite interval” [Journal of Nonlinear Functional Analysis, Vol. 2021 (2021), Article ID 13

Corrigendum to “The existence of solutions for integral boundary value problems with p-Laplacian operator on infinite interval” [Journal of Nonlinear Functional Analysis, Vol. 2021 (2021), Article ID 13

Study of multi term delay fractional order impulsive differential equation using fixed point approach

<abstract><p>This manuscript is devoted to investigate a class of multi terms delay fractional order impulsive differential equations. Our investigation includes existence theory along with Ulam type stability. By using classical fixed point theorems, we establish sufficient conditions for existence and uniqueness of solution to the proposed problem. We develop some appropriate conditions for different kinds of Ulam-Hyers stability results by using tools of nonlinear functional analysis. We demonstrate our results by an example.</p></abstract>

A Study on ψ‐Caputo‐Type Hybrid Multifractional Differential Equations with Hybrid Boundary Conditions

In this research paper, we investigate the existence, uniqueness, and Ulam–Hyers stability of hybrid sequential fractional differential equations with multiple fractional derivatives of ψ‐Caputo with different orders. Using an advantageous generalization of Krasnoselskii’s fixed point theorem, we establish results of at least one solution, whereas the uniqueness of solution is derived via Banach’s fixed point theorem. Besides, the Ulam–Hyers stability for the proposed problem is investigated by applying the techniques of nonlinear functional analysis. In the end, we provide an example to illustrate the applicability of our results.

Results on Implicit Fractional Pantograph Equations with Mittag‐Leffler Kernel and Nonlocal Condition

In this study, the main focus is on an investigation of the sufficient conditions of existence and uniqueness of solution for two‐classess of nonlinear implicit fractional pantograph equations with nonlocal conditions via Atangana–Baleanu–Riemann–Liouville (ABR) and Atangana–Baleanu–Caputo (ABC) fractional derivative with order σ ∈ (1,2]. We introduce the properties of solutions as well as stability results for the proposed problem without using the semigroup property. In the beginning, we convert the given problems into equivalent fractional integral equations. Then, by employing some fixed‐point theorems such as Krasnoselskii and Banach’s techniques, we examine the existence and uniqueness of solutions to proposed problems. Moreover, by using techniques of nonlinear functional analysis, we analyze Ulam–Hyers (UH) and generalized Ulam–Hyers (GUH) stability results. As an application, we provide some examples to illustrate the validity of our results.