Existence of Solutions for a Class of Nonlinear Neumann BVPs in the Presence of Upper and Lower Solutions

In this article, we explore the monotone iterative technique (MI-technique) to study the existence of solutions for a class of nonlinear Neumann 4-point, boundary value problems (BVPs) defined as, \begin{eqnarray*} \begin{split} -\z^{(2)}(\y)=\x(\y,\z,\z^{(1)}),\quad 0<\y<1,\\ \z^{(1)}(0)=\lambda \z^{(1)}(\beta_1 ),\quad \z^{(1)}(1)=\delta \z^{(1)}(\beta_2), \end{split} \end{eqnarray*} where $ 0<\beta_1 \leq \beta_2 <1$ and $\lambda$, $\delta\in (0,1)$. The nonlinear term $ \x(\y,\z,\z^{(1)}): \Omega\rightarrow \mathbb{R} $, where $\Omega =[0,1]\times \mathbb{R}^2 $, is Lipschitz in $ \z^{(1)}(\y)$ and one sided Lipschitz in $ \z(\y)$. Using lower solution $l(\y)$ and upper solutions $u(\y)$, we develop MI-technique, which is based on quasilinearization. To construct the sequences of upper and lower solutions which are monotone, we prove maximum principle as well as anti maximum principle. Then under certain assumptions, we prove that these sequence converges uniformly to the solution $ \z(\y)$ in the specific region, where $ \frac{\partial\x}{\partial\z}<0 $ or $ \frac{\partial\x}{\partial\z}>0 $. To demonstrate that the proposed technique is effective, we compute the solution of the nonlinear multi-point BVPs. We don’t require sign restriction which is very common and very strict condition.

Convergence of approximate solutions for singular difference systems with maxima

In this paper, by introducing a new singular fractional difference comparison theorem, the existence of maximal and minimal quasi-solutions are proved for the singular fractional difference system with maxima combined with the method of upper and lower solutions and the monotone iterative technique. Finally, we give an example to show the validity of the established results.

Existence Theory for Positive Iterative Solutions to a Type of Boundary Value Problem

We introduce some research results on a type of third-order boundary value problem for positive iterative solutions. The existence of solutions to these problems was proved using the monotone iterative technique. As an examination of the proposed method, an example to illustrate the effectiveness of our results was presented.

Monotone Iterative Method for ψ-Caputo Fractional Differential Equation with Nonlinear Boundary Conditions

The main contribution of this paper is to prove the existence of extremal solutions for a novel class of ψ-Caputo fractional differential equation with nonlinear boundary conditions. For this purpose, we utilize the well-known monotone iterative technique together with the method of upper and lower solutions. Finally, we provide an example along with graphical representations to confirm the validity of our main results.

Iterative Solutions for the Differential Equation with p -Laplacian on Infinite Interval

This paper systematically investigates a class of fourth-order differential equation with p -Laplacian on infinite interval in Banach space. By means of the monotone iterative technique, we establish not only the existence of positive solutions but also iterative schemes under the suitable conditions. At last, we give an example to demonstrate the application of the main result.

Lower and upper solutions method to the fully elastic cantilever beam equation with support

The aim of this paper is to consider a fully cantilever beam equation with one end fixed and the other connected to a resilient supporting device, that is, $$ \textstyle\begin{cases} u^{(4)}(t)=f(t,u(t),u'(t),u''(t),u'''(t)), \quad t\in [0,1], \\ u(0)=u'(0)=0, \\ u''(1)=0,\qquad u'''(1)=g(u(1)), \end{cases} $$ { u ( 4 ) ( t ) = f ( t , u ( t ) , u ′ ( t ) , u ″ ( t ) , u ‴ ( t ) ) , t ∈ [ 0 , 1 ] , u ( 0 ) = u ′ ( 0 ) = 0 , u ″ ( 1 ) = 0 , u ‴ ( 1 ) = g ( u ( 1 ) ) , where $f:[0,1]\times \mathbb{R}^{4}\rightarrow \mathbb{R}$ f : [ 0 , 1 ] × R 4 → R , $g: \mathbb{R}\rightarrow \mathbb{R}$ g : R → R are continuous functions. Under the assumption of monotonicity, two existence results for solutions are acquired with the monotone iterative technique and the auxiliary truncated function method.

Monotone iterative technique for nonlinear differential equation of fractional order

In this paper, we mainly study the existence of solution of fractional differential equations. Firstly, the existence of the maxmum solution and minmum solution of the differential equation are proved by using the fixed point theorem and the monotone iteration method. Secondly, the existence of the solution of the original equation is proved by using the newly constructed differential equation. Finally, the application of the monotone iteration method is given through an example.

Analysis and Computation of Solutions for a Class of Nonlinear SBVPs Arising in Epitaxial Growth

In this work, the existence and nonexistence of stationary radial solutions to the elliptic partial differential equation arising in the molecular beam epitaxy are studied. Since we are interested in radial solutions, we focus on the fourth-order singular ordinary differential equation. It is non-self adjoint, it does not have exact solutions, and it admits multiple solutions. Here, λ∈R measures the intensity of the flux and G is stationary flux. The solution depends on the size of the parameter λ. We use a monotone iterative technique and integral equations along with upper and lower solutions to prove that solutions exist. We establish the qualitative properties of the solutions and provide bounds for the values of the parameter λ, which help us to separate existence from nonexistence. These results complement some existing results in the literature. To verify the analytical results, we also propose a new computational iterative technique and use it to verify the bounds on λ and the dependence of solutions for these computed bounds on λ.