Nonlinear Alternative Theorem Research Articles

Fractional order forestry resource conservation model featuring chaos control and simulations for toxin activity and human-caused fire through modified ABC operator

In this work, we proposed a nonlinear mathematical model with fractional-order differential equations employed to illustrate the impacts of depleted forestry resources with the effect of toxin activity and human-caused fire. The numerical and theoretical outcomes are based on the consideration of using a modified ABC-fractional-order depleted forestry resources dynamical system. In the theoretical aspect, we examination of solution positivity, existence, and uniqueness it makes use of Banach’s fixed point and the Leray Schauder nonlinear alternative theorem. The consecutive recursive sequences are purposefully designed to verify the existence of a solution to the depletion of forestry resources as delineated. To showcase the specificity and stability of the solution within the Hyers–Ulam framework, we employ the concepts and findings of functional analysis. Chaos control will stabilize the system following its equilibrium points by applying the regulate for linear responses technique. Using Lagrange polynomials insight of modified ABC-fractional-order, we conduct simulations and present a comparative analysis in graphical form with classical and integer derivatives. Results also demonstrate the impact of different parameters used in a model that is designed on the system, they provide more understanding and a better approach for real-life problems. Our results demonstrate the significant effects of toxic and fire activities produced by humans on forest ecosystems. More accurate management techniques are made possible by the modified ABC operator’s effectiveness in capturing the long-term effects of these disturbances. The findings highlight how crucial it is to use fractional calculus in ecological modeling to comprehend and manage the intricacies of forest preservation in the face of human pressures. To ensure the sustainable management of forest resources in the face of escalating environmental difficulties, this research offers policymakers and environmental managers a fresh paradigm for creating more robust and adaptive conservation policies.

Physiological and chaos effect on dynamics of neurological disorder with memory effect of fractional operator: A mathematical study

Background and objective:To study the dynamical system, it is necessary to formulate the mathematical model to understand the dynamics of various diseases that are spread worldwide. The main objective of our work is to examine neurological disorders by early detection and treatment by taking asymptomatic. The central nervous system (CNS) is impacted by the prevalent neurological condition known as multiple sclerosis (MS), which can result in lesions that spread across time and place. It is widely acknowledged that multiple sclerosis (MS) is an unpredictable disease that can cause lifelong damage to the brain, spinal cord, and optic nerves. The use of integral operators and fractional order (FO) derivatives in mathematical models has become popular in the field of epidemiology. Method:The model consists of segments of healthy or barian brain cells, infected brain cells, and damaged brain cells as a result of immunological or viral effectors with novel fractal fractional operator in sight Mittag Leffler function. The stability analysis, positivity, boundedness, existence, and uniqueness are treated for a proposed model with novel fractional operators. Results:Model is verified the local and global with the Lyapunov function. Chaos Control will use the regulate for linear responses approach to bring the system to stabilize according to its points of equilibrium so that solutions are bounded in the feasible domain. To ensure the existence and uniqueness of the solutions to the suggested model, it makes use of Banach’s fixed point and the Leray Schauder nonlinear alternative theorem. For numerical simulation and results the steps Lagrange interpolation method at different fractional order values and the outcomes are compared with those obtained using the well-known FFM method. Conclusion:Overall, by offering a mathematical model that can be used to replicate and examine the behavior of disease models, this research advances our understanding of the course and recurrence of disease. Such type of investigation will be useful to investigate the spread of disease as well as helpful in developing control strategies from our justified outcomes.

Existence and multiplicity of solutions for boundary value problem of singular two-term fractional differential equation with delay and sign-changing nonlinearity

In this paper, we consider the existence of solutions for a boundary value problem of singular two-term fractional differential equation with delay and sign-changing nonlinearity. By means of the Guo–Krasnosel’skii fixed point theorem and the Leray–Schauder nonlinear alternative theorem, we obtain some results on the existence and multiplicity of solutions, respectively.

Existence and stability analysis of solutions for a new kind of boundary value problems of nonlinear fractional differential equations

This research work is dedicated to an investigation for a new kind of boundary value problem of nonlinear fractional differential equation supplemented with general boundary condition. A full analysis of existence and uniqueness of positive solutions is respectively proved by Leray–Schauder nonlinear alternative theorem and Boyd–Wong’s contraction principles. Furthermore, we prove the Hyers–Ulam (HU) stability and Hyers–Ulam–Rassias (HUR) stability of solutions. An example illustrating the validity of the existence result is also discussed.

Existence of positive solutions for second order impulsive differential equations with integral boundary conditions on the real line

This paper is denoted to study the existence of impulsive differential equations involving integral boundary conditions on the whole line by means of the Leray-Schauder Nonlinear Alternative theorem. An example is demonstrated the effectiveness of the our main result.

EXISTENCE OF POSITIVE SOLUTIONS TO A BOUNDARY VALUE PROBLEM FOR A DELAYED SINGULAR HIGH ORDER FRACTIONAL DIFFERENTIAL EQUATION WITH SIGN-CHANGING NONLINEARITY

In this paper, we discuss the existence of positive solutions to the boundary value problem for a high order fractional differential equation with delay and singularities including changing sign nonlinearity. By using the properties of the Green function, Guo-krasnosel'skii fixed point theorem, Leray-Schauder's nonlinear alternative theorem, some existence results of positive solutions are obtained, respectively.

Positive solutions for m-point p-Laplacian fractional boundary value problem involving Riemann Liouville fractional integral boundary conditions on the half line

This paper investigates the existence of positive solutions for m-point p-Laplacian fractional boundary value problem involving Riemann Liouville fractional integral boundary conditions on the half line via the Leray-Schauder Nonlinear Alternative theorem and the use and some properties of the Green function. As an application, an example is presented to demonstrate our main result.

The Existence of Solutions for Boundary Value Problem of Fractional Functional Differential Equations with Delay

A class of boundary value problem for fractional functional differential equation with delay $ \left\{ {\begin{array}{*{20}c} {^{C} D^{\sigma } \omega (t) = f(t,\omega _{t} ),t \in [0,\zeta ],} \\ {\omega (0) = 0,\,\omega ^{\prime}(0) = 0,\,\omega ^{\prime\prime}(\zeta ) = 1,} \\ \end{array} } \right. $ is studied, where $ 2 < \sigma \le 3,\,\,^{c} D^{\sigma } $ devote standard Caputo fractional derivative. In this article, three new criteria on existence and uniqueness of solution are obtained by Banach contraction mapping principle, Schauder fixed point theorem and nonlinear alternative theorem.

一类无穷区间上的分数阶微分方程边值问题解的存在性

利用Leray-Schauder非线性抉择定理以及Leggett-Williams不动点定理研究了一类无穷区间上的分数阶微分方程积分边值问题，获得了该边值问题至少存在一个无界解和三个正解的充分条件。最后给出了两个例子作为所获结果的应用。 By using the Leray-Schauder nonlinear alternative theorem and the Leggett-Williams fixed point theorem, a class of boundary value problem for fractional differential equations with integral conditions on an infinite interval is investigated. Some sufficient conditions on the existence of at least one unbounded solution and three positive solutions are established. At last, two examples are given to illustrate the results.

A hybrid nonlinear alternative theorem and some hybrid fixed point theorems for multimaps

We present a Krasnosel’skii–Schaefer-type nonlinear alternative theorem and some fixed point theorems of Krasnosel’skii–Sadowskii (or Daher) type for multimaps. Our results improve or extend in a broad sense recent ones obtained in literature.

Study on the existence of solutions for a generalized functional integral equation in L 1 spaces

Using a nonlinear alternative theorem of Krasnosel'skii type proved recently by Smail Djebali and Zahira Sahnoun, we investigate, in this paper, the existence of solutions for a generalized mixed-type functional integral equation in L 1 space. We also present some examples of the integral equation to confirm the efficiency of our results. MSC: 47H10; 45A05

Nonlocal Integrodifferential Boundary Value Problem for Nonlinear Fractional Differential Equations on an Unbounded Domain

This paper investigates the existence of nonnegative solutions for nonlinear fractional differential equations with nonlocal fractional integrodifferential boundary conditions on an unbounded domain by means of Leray-Schauder nonlinear alternative theorem. An example is discussed for the illustration of the main work.

Positive Solutions of a Second‐Order Nonlinear Neutral Delay Difference Equation

The purpose of this paper is to study solvability of the second‐order nonlinear neutral delay difference equation . By making use of the Rothe fixed point theorem, Leray‐Schauder nonlinear alternative theorem, Krasnoselskill fixed point theorem, and some new techniques, we obtain some sufficient conditions which ensure the existence of uncountably many bounded positive solutions for the above equation. Five nontrivial examples are given to illustrate that the results presented in this paper are more effective than the existing ones in the literature.

Bounded Positive Solutions for a Third Order Discrete Equation

This paper studies the following third order neutral delay discrete equation , where τ, l ∈ ℕ, n0 ∈ ℕ ∪ {0}, , , are real sequences with an ≠ 0 for n ≥ n0, with lim n→∞(n − din) = +∞ for i ∈ {1,2, …, l} and . By using a nonlinear alternative theorem of Leray‐Schauder type, we get sufficient conditions which ensure the existence of bounded positive solutions for the equation. Three examples are given to illustrate the results obtained in this paper.

Solvability for a third-order three-point BVP on time scales

We are concerned with the existence of a nontrivial solution to a nonlinear third-order three-point boundary-value problem on general time scales. Using the corresponding Green function, without a non-negative or monotone-type assumption on the nonlinearity, we obtain sufficient conditions for the existence of at least one nontrivial solution, using the Leray–Schauder nonlinear alternative theorem. This paper extends recent results in differential equations to difference equations, quantum equations, and general dynamic equations on time scales.