Upper and Lower Solutions of Boundary Value Problems for Functional Differential Equations and Theorems on Functional Differential Inequalities

2000 ◽  
Vol 7 (3) ◽  
pp. 489-512 ◽  
Author(s):  
R. Hakl ◽  
I. Kiguradze ◽  
B. Půža
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Differential Inequalities ◽  
Bounded Operators ◽  
Nonlinear Operators ◽  
Functional Differential

Abstract Sufficient conditions are found for the existence of an upper and a lower solutions of the boundary value problem where and are linear bounded operators, and and are continuous, generally speaking nonlinear, operators. Kamke type theorems are proved on functional differential inequalities.

A Periodic Boundary Value Problem for Functional Differential Equations of Higher Order

2009 ◽  
Vol 16 (4) ◽  
pp. 651-665
Author(s):  
Robert Hakl ◽  
Sulkhan Mukhigulashvili
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Unique Solvability ◽  
Periodic Boundary ◽  
Boundary Value ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Periodic Boundary Value Problem ◽  
Bounded Operators ◽  
Functional Differential

Abstract On the interval [0,ω], consider the periodic boundary value problem where 𝑛 ≥ 2, 𝑙𝑖 : 𝐶([0,ω];𝑅) → 𝐿([0,ω];𝑅) (𝑖 = 0,…,𝑛 – 1) are linear bounded operators, 𝑞 ∈ 𝐿([0,ω];𝑅), 𝑐𝑗 ∈ 𝑅 (𝑗 = 0,…,𝑛 – 1). The effective sufficient conditions guaranteeing the unique solvability of the considered problem are established.

On a boundary value problem on an infinite interval for nonlinear functional differential equations

2017 ◽  
Vol 24 (2) ◽  
pp. 217-225 ◽  
Author(s):  
Ivan Kiguradze ◽  
Zaza Sokhadze
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Volterra Operator ◽  
Boundary Value ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Continuous Functions ◽  
Infinite Interval ◽  
Nonlinear Functional ◽  
Functional Differential

AbstractSufficient conditions are found for the solvability of the following boundary value problem:u^{(n)}(t)=f(u)(t),\qquad u^{(i-1)}(0)=\varphi_{i}(u^{(n-1)}(0))\quad(i=1,% \dots,n-1),\qquad\liminf_{t\to+\infty}\lvert u^{(n-2)}(t)|<+\infty,where {f\colon C^{n-1}(\mathbb{R}_{+})\to L_{\mathrm{loc}}(\mathbb{R}_{+})} is a continuous Volterra operator, and {\varphi_{i}\colon\mathbb{R}\to\mathbb{R}} ({i=1,\dots,n}) are continuous functions.

On the Solvability of Nonlinear Boundary Value Problems for Functional Differential Equations

1998 ◽  
Vol 5 (3) ◽  
pp. 251-262
Author(s):  
I. Kiguradze ◽  
B. Půža
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Linear Operators ◽  
Nonlinear Boundary Value Problems ◽  
Functional Differential ◽  
Continuous Operators

Abstract Sufficient conditions are established for the solvability of the boundary value problem where p : C(I; Rn ) × C(I; Rn ) → L(I; Rn ), q : C(I; Rn ) → L(I; Rn ), l : C(I; Rn ) × C(I; Rn ) → Rn , and cn : C(I; Rn ) → Rn are continuous operators, and p(x, ·) and l(x, ·) are linear operators for any fixed x ∈ C(I; Rn ).

scholarly journals Solvability of a linear non-local boundary value problem for nonlinear functional differential equations

2009 ◽  
Vol 43 (1) ◽  
pp. 189-201
Author(s):  
Zdeněk Opluštil
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Unique Solvability ◽  
Boundary Value ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Nonlinear Functional ◽  
Local Boundary ◽  
Non Local ◽  
Functional Differential

Abstract New sufficient conditions are established for the solvability as well as unique solvability of a linear non-local boundary value problem for nonlinear functional differential equations.

Nonlinear Three-Point Boundary Value Problems for a Class of Impulsive Functional Differential Equations

2009 ◽  
Vol 16 (4) ◽  
pp. 617-628
Author(s):  
Guoping Chen ◽  
Jianhua Shen
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Upper And Lower Solutions ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Lower Solution ◽  
Point Boundary ◽  
Functional Differential ◽  
Extreme Solutions

Abstract This paper is concerned with the existence of extreme solutions of nonlinear three-point boundary value problems for a class of first order impulsive functional differential equations. In the presence of a lower solution α and an upper solution β with the classical condition α ≤ β or the reversed ordering condition β ≤ α, some sufficient conditions for the existence of extreme solutions are obtained by using the method of upper and lower solutions coupled with the monotone iterative technique.

On Singular Boundary Value Problems for Functional Differential Equations of Higher Order

2001 ◽  
Vol 8 (4) ◽  
pp. 791-814
Author(s):  
I. Kiguradze ◽  
B. Půža ◽  
I. P. Stavroulakis
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Singular Boundary ◽  
Dimensional Vector ◽  
Singular Boundary Value Problems ◽  
Functional Differential

Abstract Sufficient conditions are established for the solvability of the boundary value problem 𝑥(𝑛) (𝑡) = 𝑓(𝑥)(𝑡), ℎ𝑖(𝑥) = 0 (𝑖 = 1, . . . , 𝑛), where 𝑓 is an operator (ℎ𝑖 (𝑖 = 1, . . . , 𝑛) are operators) acting from some subspace of the space of (𝑛 – 1)-times differentiable on the interval ]𝑎, 𝑏[ 𝑚-dimensional vector functions into the space of locally integrable on ]𝑎, 𝑏[ 𝑚-dimensional vector functions (into the space ).

scholarly journals On a boundary value problem for scalar linear functional differential equations

2004 ◽  
Vol 2004 (1) ◽  
pp. 45-67 ◽  
Author(s):  
R. Hakl ◽  
A. Lomtatidze ◽  
I. P. Stavroulakis
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value ◽  
Homogeneous Equation ◽  
Functional Differential Equations ◽  
Solution Space ◽  
Fredholm Alternative ◽  
Linear Boundary ◽  
Bounded Operators ◽  
Functional Differential ◽  
Linear Boundary Value Problem

Theorems on the Fredholm alternative and well-posedness of the linear boundary value problemu′(t)=ℓ(u)(t)+q(t),h(u)=c, whereℓ:C([a,b];ℝ)→L([a,b];ℝ)andh:C([a,b];ℝ)→ℝare linear bounded operators,q∈L([a,b];ℝ), andc∈ℝ, are established even in the case whenℓis not astrongly boundedoperator. The question on the dimension of the solution space of the homogeneous equationu′(t)=ℓ(u)(t)is discussed as well.

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