Monotone Iterative Technique Research Articles

Existence of extremal solutions for a class of fractional integro-differential equations

In the study the existence of solutions of a class of fractional integro-differential equations with boundary conditions was considered. The main tool, we employ, is the conventional monotone iterative technique, which is highly effective method to examine the quantitative and qualitative characteristics of various nonlinear problems. This technique produces monotone sequences whose iterations are unique solutions of the certain linear problems. These bounds converge uniformly to the maximal solutions of the given problems. Some types of coupled solutions are considered to obtain the claim of the main results under suitable conditions.

Qualitative Outcomes on Monotone Iterative Technique and Quasilinearization Method on Time Scale

In this paper, a nonlinear dynamic equation with an initial value problem (IVP) on a time scale is considered. First, applying comparison results with a coupled lower solution (LS) and an upper solution (US), we improved the quasilinearization method (QLM) for the IVP. Unlike other studies, we consider the LS and US pair of the seventh type instead of the natural type. It was determined that the solutions of the dynamic equation converge uniformly and monotonically to the unique solution of the IVP, and the convergence is quadratic. Moreover, we will use the delta derivative (Δγ) instead of the classical derivative (dγ) in the proof because it studies a time scale. In the second part of the paper, we applied the monotone iterative technique (MIT) coupled with the LS and US, which is an effective method, proving a clear analytical representation for the solution of the equation when the relevant functions are monotonically non-decreasing and non-increasing. Then an example is given to illustrate the results obtained.

Monotone iterative technique for a sequential delta-Caputo fractional differential equations with nonlinear boundary conditions

Abstract. In this article, we discuss the existence of extremal solutions for a class of nonlinear sequential δ–Caputo fractional differential equations involving nonlinear boundary conditions. Our results are founded on advanced functional analysis methods. To be more specific, we use the monotone iterative approach in conjunction with the upper and lower solution method to create adequate requirements for the existence of extremal solutions. As an application, we give an example to illustrate our results. Mathematics Subject Classification (2010): 34A08, 26A33. Keywords: Sequential δ–Caputo derivative, nonlinear boundary conditions, monotone iterative technique, upper and lower solutions.

Study on the Solutions of Impulsive Integrodifferential Equations of Mixed Type Based on Infectious Disease Dynamical Systems

Since ancient times, infectious diseases have been a major source of harm to human health. Therefore, scientists have established many mathematical models in the history of fighting infectious diseases to study the law of infection and then analyzed the practicability and effectiveness of various prevention and control measures, providing a scientific basis for human prevention and research of infectious diseases. However, due to the great differences in the transmission mechanisms and modes of many diseases, there are many kinds of infectious disease dynamic models, which make the research more and more difficult. With the continuous progress of infectious disease research technology, people have adopted more ways to prevent and interfere with the derivation and spread of infectious disease, which will make the state of infectious disease system change in an instant. The mutation of this state can be described more scientifically and reasonably by the mathematical impulse dynamic system, which makes the research more practical. Based on this, a time-delay differential system model of infectious disease under impulse effect was established by means of impulse differential equation theory. A class of periodic boundary value problems for impulsive integrodifferential equations of mixed type with integral boundary conditions was studied. The existence of periodic solutions of these equations was obtained by using the comparison theorem, upper and lower solution methods, and the monotone iteration technique. Finally, combined with the practical application, the established time-delay differential system model was applied to the prediction of the stability and persistence of the infectious disease dynamic system, and the correctness of the conclusion was further verified. This study provides some reference for the prevention and treatment of infectious diseases.

Existance of solutions for some class of system quasilinear differentials with nonlocal boundary conditions and with nonlinearity depandant on the first derivate in time scales

This work is conserned with the construction of minimal and maximal solution for some equation quasilinear differentials with nonlocal boundary condition with nonlinearity is continuous function dependent on the first derivate in the time scale. In a previous study, see[11] it was found that there exists the extrimals solutions of a quasilinear elliptic system with intergral boundary conditions in the continuous case, and from here we asked the following question using the upper and lower solutions method coupled with monotone iterative technique and we asked the following question: do we have the existence of extrimals solutions for certain classes of systems of differential equations with nonlocal boundary conditions in time scales by same method. We can classify the problem following the monotonicity of the two functions and in the true types.Type 1: Increasing qasimonotonic systems.Type 2: Decreasing qasimonotonic systems. Type 3: Mexed qasimonotonic systems. In this paper we show the existence minimuls-maximuls solutions if the problem is of the type 1. If the problem is of the type 2 , we show the existence of maximuls-minimuls solutions. If the problem is of the type 3 , we show existence of least quasilear-solution. In the end, we gave examples of the results presented in this work to prove their validity through two cases: the continuous case, represented by the set of real numbers, and the discontinuous case, represented by the set of integers.

Monotone Iterative Technique for a Kind of Nonlinear Fourth-Order Integro-Differential Equations and Its Application

In this paper, we consider the existence and iterative approximation of solutions for a class of nonlinear fourth-order integro-differential equations (IDEs) with Navier boundary conditions. We first prove the existence and uniqueness of analytical solutions for a linear fourth-order IDE, which has rich applications in engineering and physics, and then we establish a maximum principle for the corresponding operator. Based upon the maximum principle, we develop a monotone iterative technique in the presence of lower and upper solutions to obtain iterative solutions for the nonlocal nonlinear problem under certain conditions. Some examples are presented to illustrate the main results.

Monotone iterative technique for multi-term time fractional measure differential equations

Monotone iterative technique for multi-term time fractional measure differential equations

Existence results for a class of four point nonlinear singular BVP arising in thermal explosion in a spherical vessel

In this article, the following class of four-point singular non-linear boundary value problem (NLBVP) is considered which arises in thermal explosion in a spherical vessel −(s2y′(s))′=s2f(s,y,s2y′),s∈(0,1),y′(0)=0,y(1)=δ1y(η1)+δ2y(η2), where Ω=(0,1)×R2, f:Ω→R is continuous on Ω as well as satisfy Lipschitz condition with respect to y and y′ (one sided), δ1, δ2>0 are constants, and 0<η1≤η2<1. We provide an estimation of the region of existence of a solution of above singular NLBVP. We extend the theory of monotone iterative technique (MIT) which provides computable monotone sequences that converge to the solutions of the nonlinear four point BVPs.

Initial value problems of first order fractional differential equations via monotone iterative techniques

Initial value problems of first order fractional differential equations via monotone iterative techniques

An Erdélyi-Kober fractional coupled system: Existence of positive solutions

&lt;abstract&gt;&lt;p&gt;This paper studies an Erdélyi-Kober fractional coupled system where the variable is in an infinite interval, and the existence of positive solutions is considered. We first give proper conditions and then use the Guo-Krasnosel'skii fixed point theorem to discuss our problem in a special Banach space. The monotone iterative technique and the existence results of positive solutions for this system are established naturally. To show the plausibility of our main results, several concrete examples are given at the end.&lt;/p&gt;&lt;/abstract&gt;

A Study of Multi-Term Time-Fractional Delay Differential System with Monotonic Conditions

In this paper, the existence and uniqueness of mild solution for a class of multi-term time-fractional delay differential system have been discussed in ordered Banach space by enforcing monotone iterative technique. The generalized semigroup theory, fractional calculus and measure of noncompactness have been implemented to obtain the required results. A new set of sufficient conditions with the coefficients in the equations satisfying some monotonic properties has been obtained. Finally, an application is given to illustrate the obtained results.

The existence of solutions of Hadamard fractional differential equations with integral and discrete boundary conditions on infinite interval

&lt;abstract&gt;&lt;p&gt;In this article, the properties of solutions of Hadamard fractional differential equations are investigated on an infinite interval. The equations are subject to integral and discrete boundary conditions. A new proper compactness criterion is introduced in a unique space. By applying the monotone iterative technique, we have obtained two positive solutions. And, an error estimate is also shown at the end. This study innovatively uses a monotonic iterative approach to study Hadamard fractional boundary-value problems containing multiple fractional derivative terms on infinite intervals, and it enriches some of the existing conclusions. Meanwhile, it is potentially of practical significance in the research field of computational fluid dynamics related to blood flow problems and in the direction of the development of viscoelastic fluids.&lt;/p&gt;&lt;/abstract&gt;

Monotone iterative technique for delay elastic systems in ordered Banach space

We research extremal S-asymptotically ω-periodic mild solutions to elastic systems by using upper and lower solutions. In our first result, it is assumed that the semigroup is compact. In addition, in our second result, the semigroup does not need to be compact and not assuming lower and upper solutions. Our conclusion is drawn using measures of noncompactness. Finally, the existence of mild solutions for damped equations is discussed as an application.

Monotone iterative technique for evolution equations with delay and nonlocal conditions in ordered Banach space

In this paper, based on monotone iterative method in the presence of the lower and upper solutions, we investigate the existence and uniqueness of the S-asymptotically ω-periodic mild solutions to a class of nonlocal problems of evolution equations with delay in ordered Banach spaces. Firstly, we introduce the concept of lower S-asymptotically ω-periodic solution and upper S-asymptotically ω-periodic solution. Secondly, we construct monotone iterative method in the presence of the lower and upper solutions to evolution equations with delay, and obtain the existence of maximal and minimal S-asymptotically ω-periodic mild solutions for the mentioned system under wide monotone conditions and noncompactness measure condition of nonlinear term. Finally, as the application of abstract results, an example is given to illustrate our main results.

A study on S-asymptotically ω-periodic positive mild solutions for damped elastic systems

The goal of this paper is to consider damped elastic systems with delay and nonlocal conditions in the framework of ordered Banach spaces. Firstly, we investigate the existence of minimal positive S-asymptotically ω-periodic mild solution for structural damped elastic systems with delay and nonlocal conditions on infinite interval. Secondly, based on monotone iterative technique coupled with fixed point theorem, the existence of minimal positive S-asymptotically ω-periodic mild solution is discussed without assuming the existence of upper and lower solutions. Finally, a concrete problem regarding the vibration equation of simply supported beam is given to illustrate the feasibility of our abstract results.

Algorithm for Approximate Solving of a Nonlinear Boundary Value Problem for Generalized Proportional Caputo Fractional Differential Equations

In this paper an algorithm for approximate solving of a boundary value problem for a nonlinear differential equation with a special type of fractional derivative is suggested. This derivative is called a generalized proportional Caputo fractional derivative. The new algorithm is based on the application of the monotone-iterative technique combined with the method of lower and upper solutions. In connection with this, initially, the linear fractional differential equation with a boundary condition is studied, and its explicit solution is obtained. An appropriate integral fractional operator for the nonlinear problem is constructed and it is used to define the mild solutions, upper mild solutions and lower mild solutions of the given problem. Based on this integral operator we suggest a scheme for obtaining two monotone sequences of successive approximations. Both sequences consist of lower mild solutions and lower upper solutions of the studied problem, respectively. The monotonic uniform convergence of both sequences to mild solutions is proved. The algorithm is computerized and applied to a particular example to illustrate the theoretical investigations.

Discrete fuzzy complex-valued function and complex fuzzy Caputo fractional difference equations

In this paper, we introduce a notion of complex compact intervals on which Minkowski addition and scalar multiplication operators are well defined to form the space of complex fuzzy numbers whose α-cuts are complex intervals. Some basic fuzzy partial ordering relations among the complex numbers with “≤C”, the complex intervals with “≪”, the complex fuzzy numbers with “⪯C˜” and the discrete fuzzy complex-valued functions with ≲ are studied. Based on these fundamental properties, we introduce a concept of Caputo fractional sum and difference for the discrete fuzzy complex-valued functions and establish some of their basic results. Moreover, by introducing the complex monotone iterative technique combined with the method of upper and lower solutions, the existence of extremal solutions for discrete complex fuzzy Caputo fractional difference equations is investigated. In each section, several examples are presented to show the feasibility of our obtained results.

Extremal mild solutions to Hilfer evolution equations with non-instantaneous impulses and nonlocal conditions

In this article, we study the existence of extremal mild solutions for a class of Hilfer fractional evolution equation with non-instantaneous impulses and nonlocal conditions in ordered Banach spaces. The definition of mild solutions for our concerned problem was given based on a $$C_0$$ -semigroup $$W(\cdot )$$ generated by the operator $$-A$$ and probability density function. By means of monotone iterative technique and the method of lower and upper solutions, the existence of extremal mild solutions between lower and upper mild solutions for nonlinear Hilfer evolution equations with non-instantaneous impulses and nonlocal conditions is obtained under the situation that the corresponding $$C_0$$ -semigroup $$W(\cdot )$$ and non-instantaneous impulsive function $$\gamma _k$$ are compact, $$W(\cdot )$$ is not compact and $$\gamma _k$$ is compact, $$W(\cdot )$$ and $$\gamma _k$$ are not compact, respectively. At the end, in order to illustrate our results, three concrete examples are given.

Iterative Positive Solutions to a Coupled Riemann-Liouville Fractional q -Difference System with the Caputo Fractional q -Derivative Boundary Conditions

This paper is devoted to the existence of positive solutions for a nonlinear coupled Riemann-Liouville fractional q -difference system, with multistrip and multipoint mixed boundary conditions under Caputo fractional q -derivative. We obtain the existence of positive solutions and initial iterative solutions by the monotone iteration technique. Then, we also calculate the error limits of the numerical approximation solution by induction. In the end, two examples are given to illustrate the above research results, and in the second example, some graphs of the iterative solutions are also drawn to give a more intuitive sense of the iterative process.

Ulam–Hyers stability of Caputo-type fractional fuzzy stochastic differential equations with delay

We explore a new kind of Caputo-type fractional fuzzy stochastic differential equations (FFSDEs) with delay. We establish the existence result of FFSDEs with delay by monotone iterative technique combined with the method of upper and lower solutions, and then the uniqueness is proved. Subsequently, we study the Ulam–Hyers (U–H) stability. Finally, three examples with numerical simulations are provided to illustrate the theoretical results.