Solvability for a third order discontinuous fully equation with nonlinear functional boundary conditions

We prove an existence and location result for the third order functional nonlinear boundary value problem u′′′(t) = f(t,u,u′(t),u′′(t)), for t∈[a,b], 0 = L₀(u,u′,u(t₀)), 0 = L₁(u,u′,u′(a),u′′(a)), 0 = L₂(u,u′,u′(b),u′′(b)), with t₀∈[a,b] given, f:I×C(I)×R²→R is a L¹- Carathéodory function allowing some discontinuities on t and L₀,L₁, L₂ are continuous functions depending functionally on u and u′. The arguments make use of an a priori estimate on u′′, lower and upper solutions method and degree theory. Applications to a multipoint problem and to a beam equation will be presented

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

On a elastic beam fully equation with nonlinear boundary conditions

We study the fourth-order nonlinear boundary value problem u^{iv}=f(t,u,u′,u′′,u′′′), 0<t<1, u(0)=A, u′(0)=B, g(u′′(0), u′′′(0))=0, h(u′′(1),u′′′(1))=0, with f:[0,1]×R⁴→R a continuous function veryfing a Nagumo-type condition, A,B∈R and g,h:R²→R are continuous functions with adequate monotonicities. For this model of the bending of an elastic beam, clamped at the left end-point, we obtained an existence and location result by lower and upper-solution method and degree theory. Similar results are presented for the same beam fully equation with different types of boundary conditions

A fourth order BVP of Sturm-Liouville with asymmetric unbounded nonlinearities

It is obtained an existence and location result for the fourth order boundary value problem of Sturm-Liouville type u^{(iv)}(t)=f(t,u(t),u′(t),u′′(t),u′′′(t)), for t∈[0,1], u(0)=u(1)=A, k₁u′′′(0)-k₂u′′(0)=0, k₃u′′′(1)+k₄u′′(1)=0, where f:[0,1]×R⁴→R is a continuous function and A,k_{i}∈R, for i=1,...,4, are such that k₁,k₃>0, k₂,k₄≥0. We assume that f verifies a one-sided Nagumo type growth condition which allows an asymmetric unbounded behaviour on the nonlinearity. The arguments make use of an a priori estimate on the third derivative of a class of solutions, the lower and upper solutions method and degree theory

Lower and upper solutions for a fully nonlinear beam equation

In this paper the two point fourth order boundary value problem is considered u^{(iv)}=f(t,u,u',u'',u'''), 0<t<1, u(0)=u'(1)=u''(0)=u'''(1)=0, where is a continuous function satisfying a Nagumo-type condition. We prove the existence of a solution lying between lower and upper solutions using an a priori estimation, lower and upper solutions method and degree theory. The same arguments can be used, with adequate modifications, for any type of two-point boundary value problem, including all derivatives until order three, with the second and the third derivatives given in different end-points. An application to the extended Fisher-Kolmogorov problem will be obtained

Existence and location result for a fourth order boundary value problems

In the present work we prove an existence and location result for the fourth order fully nonlinear equation u^{(iv)}=f(t,u,u′,u′′,u′′′), 0<t<1, with the Lidstone boundary conditions u(0)=u′′(0)=u(1)=u′′(1)=0, where f:[0,1]×R⁴→R is a continuous function satisfying a Nagumo type condition. The existence of at least a solution lying between a pair of well ordered lower and upper solutions is obtained using an a priori estimates, lower and upper solutions method and degree theory

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

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

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