Nagumo Condition Research Articles

Existence of multiple unbounded solutions for a three-point boundary value problems on an infinite time scales

Abstract In this paper, we consider the second-order three point boundary value problem on time scales with integral boundary conditions on a half-line. We will use the upper and lower solution method along with the Schauder’s fixed point theorem to establish the existence of at least one solution which lies between pairs of unbounded upper and lower solutions. Further, by assuming two pairs of unbounded upper and lower solutions, the Nagumo condition on the nonlinear term involved in the first-order derivative, we will establish the existence of multiple unbounded solutions on an infinite interval by using the topological degree theory. The results of this paper extend the results of Akcan and Çetin (2018), Akcan and Hamal (2014), Eloe, Kaufmann and Tisdell (2006), and generalize the results of Lian and Geng (2011). Examples are included to illustrate the validation of the results.

Periodic second-order systems and coupled forced Van der Pol oscillators

We present an existence and localization result for periodic solutions of second-order non-linear coupled planar systems, without requiring periodicity for the non-linearities. The arguments for the existence tool are based on a variation of the Nagumo condition and the Topological Degree Theory. The localization tool is based on a technique of orderless upper and lower solutions, that involves functions with translations. We apply our result to a system of two coupled Van der Pol oscillators with a forcing component.

Periodic solutions in reversible systems in second order systems with distributed delays

<abstract><p>In this paper, we study the existence and multiplicity of periodic solutions to a class of second-order nonlinear differential equations with distributed delay. Under assumptions that the nonlinearity is odd, differentiable at zero and satisfies the Nagumo condition, by applying the equivariant degree method, we prove that the delay equation admits multiple periodic solutions. The results are then illustrated by an example.</p></abstract>

Existence of solutions for higher order $\phi-$Laplacian BVPs on the half-line using a one-sided Nagumo condition with nonordered upper and lower solutions

In this paper, we consider the following \((n+1)\)st order bvp on the half line with a \(\phi-\)Laplacian operator \[ \begin{cases} (\phi(u^{(n)}))'(t) = f(t,u(t),\ldots,u^{(n)}(t)), & \text{a.e.},\, t\in [0,+\infty), \\ n \in \mathbb{N}\setminus\{0\}, \\ \\ u^{(i)}(0) = A_{i}, \, i=0,\ldots,n-2, \\ u^{(n-1)}(0) + au^{(n)}(0) = B, \\ u^{(n)}(+\infty) = C. \end{cases} \] The existence of solutions is obtained by applying Schaefer's fixed point theorem under a one-sided Nagumo condition with nonordered lower and upper solutions method where \(f\) is a \(L^{1}\)-Carathéodory function.

On periodic Ambrosetti-Prodi-type problems

<abstract><p>This work presents a discussion of Ambrosetti-Prodi-type second-order periodic problems. In short, the existence, non-existence, and multiplicity of solutions will be discussed on the parameter $ \lambda $. The arguments rely on a Nagumo condition, to guarantee an apriori bound on the first derivative, lower and upper-solutions method, and the Leray-Schauder's topological degree theory. There are two types of new results based on the parameter's variation: An existence and non-existence theorem and a multiplicity theorem, proving the existence of a bifurcation point. An application to a damped and forced pendulum is studied, suggesting a method to estimate the critical values of the parameter.</p></abstract>

New results and applications on the existence results for nonlinear coupled systems

In this manuscript, we study a certain classical second-order fully nonlinear coupled system with generalized nonlinear coupled boundary conditions satisfying the monotone assumptions. Our new results unify the existence criteria of certain linear and nonlinear boundary value problems (BVPs) that have been previously studied on a case-by-case basis; for example, Dirichlet and Neumann are special cases. The common feature is that the solution of each BVPs lies in a sector defined by well-ordered coupled lower and upper solutions. The tools we use are the coupled lower and upper solutions approach along with some results of fixed point theory. By means of the coupled lower and upper solutions approach, the considered BVPs are logically modified to new problems, known as modified BVPs. The solution of the modified BVPs leads to the solution of the original BVPs. In our case, we only require the Nagumo condition to get a priori bound on the derivatives of the solution function. Further, we extend the results presented in (Franco et al. in Extr. Math. 18(2):153–160, 2003; Franco et al. in Appl. Math. Comput. 153:793–802, 2004; Franco and O’Regan in Arch. Inequal. Appl. 1:423–430, 2003; Asif et al. in Bound. Value Probl. 2015:134, 2015). Finally, as an application, we consider the fully nonlinear coupled mass-spring model.

Successive approximations for a differential equation in a Banach space via Constantin condition

Abstract We give some sufficient conditions for the convergence of the sequence of successive approximations to the unique solution of first order Cauchy problem in a Banach space. Our approach is based on a generalized Nagumo condition due to A. Constantin and the properties of the Kuratowski measure of noncompactness.

Existence and location of solutions to fourth-order Lidstone coupled systems with dependence on odd derivatives

This paper addresses the existence and location results for coupled system with two fourth-order differential equations with dependence on all derivatives in nonlinearities and subject to Lidstone-type boundary conditions. To guarantee the existence and location of the solutions, we applied lower and upper solutions technique and degree theory. In this context, we highlight a new type of Nagumo condition to control the growth of the third derivatives and increases the number of applications, as well as a new type of definitions of upper and lower solutions for such coupled systems. Last section contains an application to a coupled system composed by two fourth order equations, which models the estimated bending of simply-supported beam with torsional solitons.

Existence of Solutions for $${\mathrm {m}}$$-Point Boundary Value Problems on an Infinite Time Scale

In this study, we present the existence result for the the second order $${\mathrm {m}}$$-point boundary value problems on infinite time scales. Nagumo condition, lower and upper solutions play an important role in the arguments.

Solutions of fixed period in the nonlinear wave equation on networks

The wave equation on network is defined by \(\partial _{tt}u=\Delta _{\mathfrak {G}}u+g(u)\), where \(u\in {\mathbb {R}}^{n}\) and the graph Laplacian \(\Delta _{\mathfrak {G}}\) is an operator on functions on n vertices. We suppose that \(g:\mathbb {R}^{n}\rightarrow \mathbb {R}^{n}\) is an odd continuous function that satisfies \(g(0)=0\), \(Dg(0)=0\) and the Nagumo condition. Assuming that the graph is invariant by a subgroup of permutations \(\Gamma \), using a \(\Gamma \)-equivariant topological invariant we prove the existence of multiple non-constant p-periodic solutions characterized by their symmetries.

Radially Symmetric Solutions of the p-Laplace Equation with Gradient Terms

We consider the Dirichlet problem for the p-Laplace equation with nonlinear gradient terms. In particular, these gradient terms cannot satisfy the Bernstein—Nagumo conditions. We obtain some sufficient conditions that guarantee the existence of a global bounded radially symmetric solution without any restrictions on the growth of the gradient term. Also we present some conditions on the function simulating the mass forces, which allow us to obtain a bounded radially symmetric solution under presence of an arbitrary nonlinear source.

Solvability of a boundary value problem with a Nagumo Condition

ABSTRACTThe aim of this paper is to discuss the existence and localization of solutions for a generalized Emden-Fowler equation involving a conformable derivative and with a Dirichlet boundary condition. Our approach is based on the lower and upper solutions method and Schauder fixed-point theorem under a Nagumo condition.

Sufficient conditions for the solvability of some third order functional boundary value problems on the half-line

This paper is concerned with the existence of bounded or unbounded solutions to third-order boundary value problem on the half-line with functional boundary conditions. The arguments are based on the Green functions, a Nagumo condition, Schauder fixed point theorem and lower and upper solutions method. An application to a Falkner-Skan equation with functional boundary conditions is given to illustrate our results.

Existence of Coupled Solutions of BVP for ϕ-Laplacian Impulsive Differential Equations

In this paper, we study the existence of coupled solutions of anti-periodic boundary value problems for impulsive differential equations with ϕ-Laplacian operator. Based on a pair of coupled lower and upper solutions and appropriate Nagumo condition, we prove the existence of coupled solutions for anti-periodic impulsive differential equations boundary value problems with ϕ-Laplacian operator.

Anti-periodic Boundary Value Problems of <i>φ</i>-Laplacian Impulsive Differential Equations

This paper concerned with the existence of solutions of anti-periodic boundary value problems for impulsive differential equations with φ-Laplacian operator. Firstly, the definition a pair of coupled lower and upper solutions of the problem is introduced. Then, under the approach of coupled upper and lower solutions together with Nagumo condition, we prove that there exists at least one solution of anti-periodic boundary value problems for impulsive differential equations with φ-Laplacian operator.

Unbounded solutions of third order three-point boundary value problems on a half-line

Abstract We consider the following third order three-point boundary value problem on a half-line: x'''(t)+q(t)f(t,x(t),x'(t),x''(t)) = 0, t ∈ (0,+∞), x'(0) = A, x(η) = B, x''(+∞) = C, where η ∈ (0,+∞), but fixed, and f : [0,+∞) × ℝ3 → ℝ satisfies Nagumo's condition. We apply Schauder's fixed point theorem, the upper and lower solution method, and topological degree theory, to establish existence theory for at least one unbounded solution, and at least three unbounded solutions. To demonstrate the usefulness of our results, we illustrate two examples.

Existence of solutions for fourth order three-point boundary value problems on a half-line

In this paper, we apply Schauder's fixed point theorem, the upper and lower solution method, and topological degree theory to establish the existence of unbounded solutions for the following fourth order three-point boundary value problem on a half- line x 0000 (t) + q(t) f(t, x(t), x 0 (t), x 00 (t), x 000 (t)) = 0, t2 (0,+¥), x 00 (0) = A, x(h) = B1, x 0 (h) = B2, x 000 (+¥) = C, where h2 (0,+¥), but fixed, and f : (0,+¥) R 4 ! R satisfies Nagumo's condition. We present easily verifiable sufficient conditions for the existence of at least one solu- tion, and at least three solutions of this problem. We also give two examples to illustrate the importance of our results.

Existence of Solutions to a Second Order Coupled System with Nonlinear Coupled Boundary Conditions

We study existence of solution in the presence of upper and lower solutions of some second-order nonlinear coupled ordinary differential system (ODS for short) depending on first order derivatives with nonlinear coupled boundary conditions (CBCs for short). Our method for nonlinear coupled system with nonlinear CBCs is new and it unifies the treatment of many different second order problems. Nagumo condition is used to define bound for the derivative of the solution. Coupled lower and upper solutions, Arzela-Ascoli theorem and Schauder's fixed point theorem play an important role in establishing the arguments.

BVPs for higher-order integro-differential equations with ϕ-Laplacian and functional boundary conditions

Abstract In this paper, we study the existence of solutions of a class of higher-order integro-differential boundary value problems with ϕ-Laplacian like operator and functional boundary conditions. By giving the definition of a pair of coupled lower and upper solutions and some new hypotheses, we obtain some new existence results for boundary value problems with ϕ-Laplacian like operator by employing the Schauder fixed point theorem and an appropriate Nagumo condition. Finally, an example is given to illustrate the results. MSC:39K10, 34B15.

Existence of Solutions for Riemann-Liouville Fractional Boundary Value Problem

By using the method of upper and lower solutions and fixed point theorems, the existence of solutions for a Riemann-Liouville fractional boundary value problem with the nonlinear term depending on fractional derivative of lower order is obtained under the classical Nagumo conditions. Also, some results concerning Riemann-Liouville fractional derivative at extreme points are established with weaker hypotheses, which improve some works in Al-Refai (2012). As applications, an example is presented to illustrate our main results.