Monotone Iterative Technique Research Articles

Positive Solutions for a Class of p-Laplacian Hadamard Fractional-Order Three-Point Boundary Value Problems

In this paper, using the Avery–Henderson fixed point theorem and the monotone iterative technique, we investigate the existence of positive solutions for a class of p-Laplacian Hadamard fractional-order three-point boundary value problems.

The Green\u2019s function of a class of two-term fractional differential equation boundary value problem and its applications

In this paper, we consider a Riemann–Liouville type two-term fractional differential equation boundary value problem. Some positive properties of the Green’s function are deduced by using techniques of analysis. As application, we obtain the existence and multiplicity of positive solutions for a fractional boundary value problem under conditions that the nonlinearity f(t,x) may change sign and may be singular at t = 0,1 and x=0, and we also obtain the uniqueness results of positive solution for a singular problem by means of the monotone iterative technique.

Explicit iterative solution of a Caputo‐Hadamard–type fractional turbulent flow model

In this article, by using the method of upper and lower solutions combined with the monotone iterative technique, we investigate a Caputo‐Hadamard–type fractional turbulent flow model. We prove new comparison results and obtain explicit monotone iterative sequences converging to the extremal solutions and unique solution of the model problem. Examples illustrating the main results are also presented.

Constant Sign Solutions to Linear Fractional Integral Problems and Their Applications to the Monotone Method

This manuscript provides some results concerning the sign of solutions for linear fractional integral equations with constant coefficients. This information is later used to prove the existence of solutions to some nonlinear problems, together with underestimates and overestimates. These results are obtained after applying suitable modifications in the classical process of monotone iterative techniques. Finally, we provide an example where we prove the existence of solutions, and we compute some estimates.

Qualitative analysis of fractional relaxation equation and coupled system with Ψ-Caputo fractional derivative in Banach spaces

<abstract> <p>The aim of the reported results in this manuscript is to handle the existence, uniqueness, extremal solutions, and Ulam-Hyers stability of solutions for a class of $ \Psi $-Caputo fractional relaxation differential equations and a coupled system of $ \Psi $-Caputo fractional relaxation differential equations in Banach spaces. The obtained results are derived by different methods of nonlinear analysis like the method of upper and lower solutions along with monotone iterative technique, Banach contraction principle, and Mönch's fixed point theorem concerted with the measures of noncompactness. Furthermore, the Ulam-Hyers stability of the proposed system is studied. Finally, two examples are presented to illustrate our theoretical findings. Our acquired results are recent in the frame of a $ \Psi $-Caputo derivative with initial conditions in Banach spaces via the monotone iterative technique. As a results, we aim to fill this gap in the literature and contribute to enriching this academic area.</p> </abstract>

Monotone iterative schemes for positive solutions of a fractional differential system with integral boundary conditions on an infinite interval

In this paper, using the monotone iterative technique and the Banach contraction mapping principle, we study a class of fractional differential system with integral boundary on an infinite interval. Some explicit monotone iterative schemes for approximating the extreme positive solutions and the unique positive solution are constructed.

A class of impulsive vibration equation with fractional derivatives

In this paper, we study a class of second order impulsive vibration equation containing fractional derivatives. By using monotone iterative technique, some new results on the multiplicity for solutions of the equations under nonlinear boundary conditions are obtained, and the properties of the solutions are discussed. Finally, the practicability of our results is discussed through a concrete example.

Periodic boundary value problem involving sequential fractional derivatives in Banach space

In this paper, by the method of upper and lower solutions coupled with the monotone iterative technique, we investigate the existence and uniqueness results of solutions for a periodic boundary value problem of nonlinear fractional differential equation involving conformable sequential fractional derivatives in Banach space. An example is given to illustrate our main result.

MIXED MONOTONE ITERATIVE TECHNIQUE FOR HILFER FRACTIONAL EVOLUTION EQUATIONS WITH NONLOCAL CONDITIONS

The purpose of this paper is concerned with the existence of mild $ L $-quasi-solutions for Hilfer fractional evolution equations with nonlocal conditions in an ordered Banach spaces $ E $. By employing mixed monotone iterative technique, measure of noncompactness and Sadovskii's fixed point theorem, we obtain the existence of mild $ L $-quasi-solutions for Hilfer fractional evolution equations with noncompact semigroups. Finally, an example is provide to illustrate the feasibility of our main results.

Dynamical analysis and control strategies of an SIVS epidemic model with imperfect vaccination on scale-free networks.

Vaccination is an effective method to prevent the spread of infectious diseases. In this paper, we develop an SIVS epidemic model with degree-related transmission rates and imperfect vaccination on scale-free networks. Firstly, we derive two threshold parameters and existence conditions of multiple endemic equilibria. Secondly, not only the global asymptotical stability of disease-free equilibrium and the persistence of the disease are derived, but also the global attractivity of the unique endemic equilibrium is proved using the monotone iterative technique. Thirdly, the effects of various immunization schemes including uniform immunization, targeted immunization and acquaintance immunization are studied, and the optimal vaccination strategy is analyzed by Pontryagin’s maximum principle. Finally, we perform numerical simulations to verify these theoretical results.

Monotone Iterative Technique for Nonlocal Impulsive Finite Delay Differential Equations of Fractional Order

The paper is concerned with the extension of a monotone iterative technique to impulsive finite delay differential equations of fractional order with a nonlocal initial condition in an ordered Banach space. We study the existence of extremal mild solutions with or without assuming the compactness of a semigroup and also prove the uniqueness of the mild solution of the system. The results are obtained with the help of fractional calculus, a measure of non-compactness, the semigroup theory and monotone iterative technique. Finally, an example is provided to show the application of our main.

Existence and uniqueness of solutions for fractional q-difference Schrödinger equations

The aim of this paper is to investigate the existence and uniqueness of solutions for a class of fractional q-difference Schrodinger equations appeared in Li et al. (Appl Math Lett 46:100–105, 2015). By use of two completely different approaches—\(u_0\)-positive operator and monotone iterative technique, this paper not only establishes excellent conditions to ensure the existence and uniqueness of solutions for this problem, but also constructs iterative sequences to approximate the solution. As applications, examples are presented to illustrate our main results.

Fixed point results in set $P_{h,e}$ with applications to fractional differential equations

In this paper, without assuming operators to be continuous or compact, by employing monotone iterative technique on ordered Banach space, we at first establish new fixed point theorems for some kinds of nonlinear mixed monotone operators in set $P_{h,e}$. Then, we study a new form of fractional two point boundary value problem depending on a certain constant and give the existence and uniqueness of solutions. We also show that the unique solution exists in set $P_{h,e}$ or $P_{h}$ and can be uniformly approximated by constructing two iterative sequences for any initial values. At the end, a concrete example is given to illustrate our abstract results. The conclusions of this article enrich the fixed point theorems and provide new methods to deal with nonlinear differential equations.

Numerical Results for Linear Sequential Caputo Fractional Boundary Value Problems

The generalized monotone iterative technique for sequential 2 q order Caputo fractional boundary value problems, which is sequential of order q, with mixed boundary conditions have been developed in our earlier paper. We used Green’s function representation form to obtain the linear iterates as well as the existence of the solution of the nonlinear problem. In this work, the numerical simulations for a linear nonhomogeneous sequential Caputo fractional boundary value problem for a few specific nonhomogeneous terms with mixed boundary conditions have been developed. This in turn will be used as a tool to develop the accurate numerical code for the linear nonhomogeneous sequential Caputo fractional boundary value problem for any nonhomogeneous terms with mixed boundary conditions. This numerical result will be essential to solving a nonlinear sequential boundary value problem, which arises from applications of the generalized monotone method.

Existence and iteration of positive solution for fractional integral boundary value problems with p-Laplacian operator

This paper is concerned with an integral boundary value problem of fractional differential equations with p-Laplacian operator. Sufficient conditions ensuring the existence of extremal solutions for the given problem are obtained. Our results are based on the method of upper and lower solutions and monotone iterative technique.

Monotone Iterative Technique for a Class of Four Point BVPs with Reversed Ordered Upper and Lower Solutions

Consider the class of four point nonlinear BVPs [Formula: see text] [Formula: see text] where [Formula: see text] is continuous, [Formula: see text], [Formula: see text] such that [Formula: see text] and [Formula: see text]. In this paper, we demonstrate an iterative technique. The iterative scheme is deduced by using quasilinearization. Then we consider upper-lower solutions in well ordered and reverse ordered cases and prove existence of solutions under some sufficient conditions. We show that under certain conditions, generated sequences are monotone, uniformly convergent and converges to the solution of the above problem. We also provide examples which validate that all the conditions derived in this paper, are realistic and can be satisfied. We have also plotted upper and lower solutions for the test examples and have shown that under the conditions, the derived upper and lower solutions are monotonic in nature.

Iterative positive solutions to a coupled fractional differential system with the multistrip and multipoint mixed boundary conditions

Using the monotone iterative technique, we investigate the existence of iterative positive solutions to a coupled system of fractional differential equations supplemented with multistrip and multipoint mixed boundary conditions. It is worth mentioning that the nonlinear terms of the system depend on the lower fractional-order derivatives of the unknown functions and the boundary conditions involve the combination of the multistrip fractional integral and the multipoint value of the unknown functions in [0,1].

Monotone iterative technique for Sobolev type fractional integro-differential equations with fractional nonlocal conditions

By applying monotone iterative technique coupled with the method of lower and upper solutions, sufficient conditions are obtained for the existence of mild solution of Sobolev type fractional integro-differential equations with fractional nonlocal conditions. The results are obtained under monotonicity and measure of noncompactness conditions of the nonlinear functions. At last, an example is given to show the applicability of main results.

Discrete monotone method for space-fractional nonlinear reaction\u2013diffusion equations

A discrete monotone iterative method is reported here to solve a space-fractional nonlinear diffusion–reaction equation. More precisely, we propose a Crank–Nicolson discretization of a reaction–diffusion system with fractional spatial derivative of the Riesz type. The finite-difference scheme is based on the use of fractional-order centered differences, and it is solved using a monotone iterative technique. The existence and uniqueness of solutions of the numerical model are analyzed using this approach, along with the technique of upper and lower solutions. This methodology is employed also to prove the main numerical properties of the technique, namely, the consistency, stability, and convergence. As an application, the particular case of the space-fractional Fisher’s equation is theoretically analyzed in full detail. In that case, the monotone iterative method guarantees the preservation of the positivity and the boundedness of the numerical approximations. Various numerical examples are provided to illustrate the validity of the numerical approximations. More precisely, we provide an extensive series of comparisons against other numerical methods available in the literature, we show detailed numerical analyses of convergence in time and in space against fractional and integer-order models, and we provide studies on the robustness and the numerical performance of the discrete monotone method.

Monotone iterative techniques together with Hyers‐Ulam‐Rassias stability

In this article, the first purpose is treating a coupled system of nonlinear boundary value problems (BVPs) of fractional‐order differential equations (FODEs) for existence of solutions. The corresponding fractional‐order derivative is taken in Riemann‐Liouville sense. The require results for iterative solutions are obtained by using monotone iterative techniques combine with the method of upper and lower solutions. In this regard, two sequences are established for upper and lower solutions, respectively, in which one is monotonically increasing and converges to upper solution, while other one is monotonically decreasing converges to lower solution of the considered problem. The second purpose is discussing different kinds Ulam stability results for the proposed problem. Some applications of our results are also provided.