Nagumo and Landesman-Lazer type conditions for nonlinear second order systems

We study the existence of solutions for a nonlinear second order system of ordinary differential equations under various boundary conditions. Assuming suitable Nagumo type conditions we prove the existence of at least one solution applying the method of upper and lower solutions. Moreover, using topological degree methods we prove the existence of solutions under Landesman-Lazer type conditions.     

Boundary value problems for first order functional differential equations with impulsive integral conditions

In this paper, by using the method of lower and upper solutions coupled with the monotone iterative technique, we give conditions for existence of extreme solutions for first order functional differential equations with a new impulsive integral condition. Some comparison results are also formulated.     

Inequalities for solutions of singular initial problems for Carathéodory systems via Ważewski’s principle

In this paper we give sufficient conditions for solvability of a singular initial problem formulated for Carathéodory systems of ordinary differential equations. The existence of solutions is proved by the supposition that corresponding auxiliary lower and upper singular problems have solutions. The proof technique uses a notion of a regular polyfacial subset which is developed for Carathéodory systems of ordinary differential equations and a modification of the topological method for such systems given by Palamides, Sficas and Staikos. An application concerning the existence of positive solutions for a special class of singular problems is given as well.     

Existence and uniqueness of solutions of second-order three-point boundary value problems with upper and lower solutions in the reversed order

This paper studies the existence and uniqueness of solutions of second-order three-point boundary value problems with lower and upper solutions in the reversed order, obtains the sufficient conditions for the existence and uniqueness of solutions by use of the monotone iterative method, and gives the iterative sequence for solving a solution and its error estimate formula under the condition of unique solution.     

The monotone method for Neumann functional differential equations with upper and lower solutions in the reverse order

In this paper, we show that the monotone technique produces two monotone sequences that converge uniformly to extremal solutions of second order functional differential equations and ?$?$-Laplacian equations with Neumann boundary value conditions. Moreover, we obtain existence results assuming upper and lower solutions in the reverse order.     

Nonlinear second-order equations with functional implicit impulses and nonlinear functional boundary conditions

In this paper we present existence results for solutions of nonlinear second-order boundary value problems with impulses. Our impulses are applied at p points in the interval and given implicitly by nonlinear functions of the solution. Moreover we allow functional dependence on the solution. Our existence results follow from the existence of a pair of well ordered lower and upper solutions. We generalize earlier results of Cabada and Tomec?ek, allowing more general compatible boundary conditions, impulses and φ-Laplacian equations.     

Multiplicity of sign-changing solutions for some four-point boundary value problem

In this paper we obtain using Leray–Schauder degree theory some multiplicity results for sign-changing solutions of a four-point boundary value problem. We assume the existence of a pair of strict lower and upper solutions and some additional conditions on the nonlinear term f$f$.     

Impulsive boundary value problems with nonlinear boundary conditions

In this paper, the upper and lower solution method and Schauder’s fixed point theorem are employed in the study of boundary value problems for a class of second-order impulsive ordinary differential equations with nonlinear boundary conditions. We prove the existence of solutions to the problem under the assumption that there exist lower and upper solutions associated with the problem.     

Fixed point theorems for a class of nonlinear operators in Banach spaces and applications

In this paper, we study a class of nonlinear operator equations with more extensive conditions in ordered Banach spaces. By using the cone theory and Banach contraction mapping principle, the existence and uniqueness of solutions for such equations are investigated without demanding the existence of upper and lower solutions and compactness and continuity conditions. The results in this paper are applied to a class of abstract semilinear evolution equations with noncompact semigroup in Banach spaces and the initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces. The results obtained here improve and generalize many known results.     

Unbounded upper and lower solutions method for Sturm–Liouville boundary value problem on infinite intervals

Unbounded upper and lower solutions theories are established for the Sturm–Liouville boundary value problem of a second order ordinary differential equation on infinite intervals. By using such techniques and the Schäuder fixed point theorem, the existence of solutions as well as the positive ones is obtained. Nagumo conditions play an important role in the nonlinear term involved in the first-order derivatives.     

Differential equations with bounded positive Green’s functions and generalized Aizerman’s hypothesis

We derive conditions for the positivity and boundedness of the Green functions of the higher order linear nonautonomous ODE. By virtue of these conditions, the existence of positive solutions for a class of nonlinear equations is proved. In addition, upper and lower estimates for the Green functions are established. Moreover, it is shown that nonlinear equations, having separated nonautonomous linear parts, satisfy the generalized Aizerman hypothesis on absolute stability, if they have the positive Green functions.     

Quasilinearization for functional differential equations with retardation and anticipation

The technique of generalised quasilinearization is developed, using the method of lower and upper solutions and the monotone iterative technique (MIT), to prove the existence of unique solutions for functional differential equations (FDE) with retardation and anticipation.     

Systems of first order differential inclusions with maximal monotone terms

In this paper, we establish the existence of solutions to systems of first order differential inclusions with maximal monotone terms satisfying the periodic boundary condition. Our proofs rely on the theory of maximal monotone operators, and the Schauder and the Kakutani fixed point theorems. A notion of solution-tube to these problems is introduced. This notion generalizes the notion of upper and lower solutions of first order differential equations.     

Impulsive mild solutions for semilinear differential inclusions with nonlocal conditions in Banach spaces

In this paper we deal with the existence of impulsive mild solutions for semilinear differential inclusions with nonlocal conditions, where the linear part generates an evolution system and the nonlinearity satisfies the lower Scorza-Dragoni property. Our theorems extend the existence propositions proved by Fan in 2010. An example is presented.     

The generalized quasilinearization method for second-order three-point boundary value problems

In this paper, we obtain the existence and uniqueness results for general second-order three-point boundary value problems by applying the method of upper and lower solutions together with Leray–Schauder degree, as well as obtaining monotone iteration schemes which converge quadratically to the unique solution of some specific second-order three-point boundary value problem by using the method of quasilinearization.     

Monotone iterative technique for functional differential equations with retardation and anticipation

The method of the monotone iterative technique together with coupled lower and upper solutions is employed to prove existence and uniqueness results for functional differential equations involving retardation and anticipation.     

On the existence of traveling waves for delayed reaction-diffusion equations

We study the existence of traveling wave solutions for reaction-diffusion equations with nonlocal delay, where reaction terms are not necessarily monotone. The existence of traveling wave solutions for reaction-diffusion equations with nonlocal delays is obtained by combining upper and lower solutions for associated integral equations and the Schauder fixed point theorem. The smoothness of upper and lower solutions is not required in this paper.     

Monotone method for first order functional differential equations with retardation and anticipation

This paper discusses first order functional differential equations with retardation and anticipation. By using the method of upper and lower solutions coupled with monotone iterative technique, new existence results are obtained which improve known results in the literature.     

Doubly Nonlinear Degenerate Parabolic Systems with Coupled Nonlinear Boundary Conditions

In this paper, we study the global existence and the global nonexistence of doubly nonlinear degenerate parabolic systems with nonlinear boundary conditions. We first prove a local existence result by the regularization method. Next, we construct a weak comparison principle. Then we discuss the large time behavior of solutions by using a modified upper and lower solution methods and constructing various upper and lower solutions. Necessary and sufficient conditions on the global existence of all positive (weak) solutions are obtained.     

Nagumo type existence results of Sturm-Liouville BVP for impulsive differential equations

In this paper, the method of upper and lower solutions and the Schauder degree theory are employed in the study of Sturm-Liouville boundary value problems for second order impulsive differential equations. We obtain the existence of at least three solutions to the problem under the assumption that the nonlinear term f satisfies a Nagumo condition with respect to the first order derivative.