A. Ya. Lepin

Extremal solutions of boundary value problems

To prove the existence of a solution of a two-point boundary value problem for an nth-order operator equation by the a priori estimate method, we study extremal solutions of auxiliary boundary value problems for an nth-order differential equation with simplest right-hand side, which have a unique solution under certain restrictions on the boundary conditions.

On a positive solution of a three-point boundary value problem

where g ∈ C(I, [0,+∞)), f ∈ C([0,+∞) × (−∞, 0], [0,+∞)), α, η ∈ (0, 1), and f satisfies the following conditions. 1. The inequality f (x1, x1) ≤ f (x2, x2) is valid for all x1, x2 ∈ [0,+∞) and x1, x2 ∈ (−∞, 0] such that x1 ≤ x2 and x1 ≤ x2. 2a1. There exists an H1 > 0 such that |f (x, x′)| ≤ C (|x|+ |x′|) for all x ∈ [0,+∞) and x′ ∈ (−∞, 0] such that |x|+ |x′| ≤ H1. 2a2. There exists an H2 > 0 such that |f (x, x′)| ≥ D (|x|+ |x′|), D > 0, for all x ∈ [0,+∞) and x′ ∈ (−∞, 0] such that |x|+ |x′| ≥ H2. 2b1. There exists an H1 > 0 such that |f (x, x′)| ≥ C (|x|+ |x′|), C > 0, for all x ∈ [0,+∞) and x′ ∈ (−∞, 0] such that |x|+ |x′| ≤ H1. 2b2. There exists an H2 > 0 such that |f (x, x′)| ≤ D (|x|+ |x′|) for all x ∈ [0,+∞) and x′ ∈ (−∞, 0] such that |x|+ |x′| ≥ H2. A similar boundary value problem was considered in [2]. Remark 1. Conditions 1, 2a1, and 2a2 are contradictory. Indeed, it follows from the inequality |f(0, 0)| ≤ C(|0|+ |0|) = 0 that f(0, 0) = 0, and the inequality 0 ≤ f (0,−H2) ≤ f(0, 0) = 0 implies that f (0,−H2) = 0, but 0 = |f (0,−H2)| ≥ D (|0|+ |−H2|) = DH2 > 0. Remark 2. If Condition 1 is satisfied, then Condition 2b1 is equivalent to the condition f(0, 0) > 0. Indeed, 0 0) (∀x ∈ [0,+∞)) (∀x′ ∈ (−∞, 0]) (x ≥ H2 ⇒ f (x, x′) ≤ Dx) . Indeed, f (x, x′) ≤ f(x, 0) ≤ Dx. The converse is obvious. Consider the boundary value problem

Generalized lower and upper functions for the φ-Laplacian

We give definitions of generalized lower and upper functions and generalized solution, which differs from a solution in the conventional sense in that the derivative of a generalized solution can be equal to +∞ and −∞. We show how to use generalized solutions for obtaining classical results.

Boundary value problem for an nth-order operator equation

We consider a boundary value problem and prove that it has a solution.

Lower and upper functions and generalized solutions of a boundary value problem for a differential-operator equation

We suggest a new approach to lower and upper functions for higher-order boundary value problems and a weakening of the Schrader condition.

Necessary and sufficient conditions for the solvability of a class of boundary value problems

We obtain necessary and sufficient conditions for the solvability of boundary value problems with one- and two-point boundary conditions.

On a boundary value problem with integral boundary conditions

We study the existence of positive solutions of second-order ordinary differential equations with integral boundary conditions. The result generalizes the conditions obtained in [1] for the existence of positive solutions.

Extremal solutions of sixth-order boundary value problems

For sixth-order boundary value problems, we find extremal solutions that provide the best estimates in the proof of the existence of a solution by the method of a priori estimates.

Extremal solutions of boundary value problems of the fifth order

For boundary value problems of the fifth order, we find extremal solutions that provide the best estimates in the proof of the existence of a solution of the boundary value problem by the method of a priori estimates.

Boundary value problems with operator right-hand side

For a second-order boundary value problem with operator right-hand side and with functional boundary conditions, we prove solvability theorems with mixed and Dirichlet boundary conditions assuming the existence of a lower and an upper function. These theorems are analogs of theorems for the corresponding boundary value problems for an ordinary second-order differential equation with right-hand side satisfying the Carathéodory conditions.