A note on a class of problems for a higher-order fully nonlinear equation under one-sided Nagumo-type condition

The purpose of this work is to establish existence and location results for the higher order fully nonlinear differential equation u⁽ⁿ⁾(t)=f(t,u(t),u′(t),…,u⁽ⁿ⁻¹⁾(t)), n≥2, with the boundary conditions u^{(i)}(a) = A, for i=0,⋯,n-3, u⁽ⁿ⁻¹⁾(a) = B, u⁽ⁿ⁻¹⁾(b)=C or u^{(i)}(a)=A, for i=0,⋯,n-3, c₁u⁽ⁿ⁻²⁾(a)-c₂u⁽ⁿ⁻¹⁾(a)=B, c₃u⁽ⁿ⁻²⁾(b)+c₄u⁽ⁿ⁻¹⁾(b)=C, with A_{i},B,C∈R, for i=0,⋯,n-3, and c₁, c₂, c₃, c₄ real positive constants. It is assumed that f:[a,b]×Rⁿ⁻¹→R is a continuous function satisfying one-sided Nagumo-type conditions which allows an asymmetric unbounded behavior on the nonlinearity. The arguments are based on Leray-Schauder topological degree and lower and upper solutions method

Solvability of some third order boundary value problem with asymmetric unbounded nonlinearities

The purpose of this work is to establish existence and location results for the higher order fully nonlinear differential equation u⁽ⁿ⁾(t)=f(t,u(t),u′(t),…,u⁽ⁿ⁻¹⁾(t)), n≥2, with the boundary conditions u^{(i)}(a) = A_{i}, for i=0,⋯,n-3, u⁽ⁿ⁻¹⁾(a) = B, u⁽ⁿ⁻¹⁾(b)=C or u^{(i)}(a)=A_{i}, for i=0,⋯,n-3, c₁u⁽ⁿ⁻²⁾(a)-c₂u⁽ⁿ⁻¹⁾(a)=B, c₃u⁽ⁿ⁻²⁾(b)+c₄u⁽ⁿ⁻¹⁾(b)=C, with A_{i},B,C ∈ R, for i=0,⋯,n-3, and c₁, c₂, c₃, c₄ real positive constants. It is assumed that f:[a,b]×Rⁿ⁻¹→R is a continuous function satisfying one-sided Nagumo-type conditions which allows an asymmetric unbounded behaviour on the nonlinearity. The arguments are based on Leray-Schauder topological degree and lower and upper solutions method

A third order boundary value problem with one-sided Nagumo condition

In this paper we present an existence and location result for the third order separated boundary value problem composed by the differential equation u′′′(t)=f(t,u(t),u′(t),u′′(t)) with the boundary conditons u(a)=A, u′′(a)=0 and u′′(b)=0, where f:[a,b]×R³→R is a continuous funtion and A∈R. One-sided Nagumo condition, lower and upper solutions, a priori estimates and Leray-Schauder degree play an important role in the arguments

Existence result for a third-order ODE with nonlinear boundary conditions in presence of a sign-type Nagumo control

In this work we provide an existence and location result for the third order nonlinear differential equation u′′′(t)=f(t,u(t),u′(t),u′′(t)) where f:[a,b]×R³→R is a continuous function, and two types of boundary conditions u(a)=A, φ(u′(b),u′′(b))=0, u′′(a)=B, or u(a)=A, ψ(u′(a),u′′(a))=0, u′′(b)=C, with φ, ψ:R²→R continuous functions and monotonous in the second variable and A,B,C∈R. We assume that f satisfy a one-sided Nagumo-type condition which allows an asymmetric unbounded behavior on the nonlinearity. The arguments used concern Leray-Schauder degree theory and lower and upper solutions technique