On some problems with nonlocal integral condition

Abstract We study the second order nonlinear boundary value problems with non‐local integral conditions and construct the Fucik type spectrum for these problems.


Introduction
The Fučík equation is a simple formally nonlinear equation with piece-wise linear right hand side. It was intensively investigated together with various boundary conditions ( [6,7,10]), for instance, with the Dirichlet boundary conditions A set of (µ, λ) such that the problem has nontrivial solutions is called the Fučík spectrum. The problem (1.1), (1.2) generalizes classical spectral problems with one parameter and reduces to the classical problem if µ = λ. The Fučík spectrum is a two-dimensional set which may have interesting properties. The Fučík spectrum for the problem (1.1), (1.2) is well known and consists of infinite set of curves (branches) which can be obtained analytically and graphically. The Fučík spectra for the Dirichlet and Neumann problems are similar, but the spectrum in the case of one (or two) of boundary conditions being in integral form differs essentially. It was studied in the work of the author ( [11]) and both analytical and graphical description was given in the case of boundary conditions x(0) = 0, In this work we study more complicated case x(0) = 0, x(1) = γ 1 0 x(s)ds, γ ∈ R. (1.4) which includes both the conditions (1.2) and (1.3). We give description of the spectrum for various γ and discover new properties of the spectrum. The paper is organized as follows. In Section 2 we give results on the spectrum for the well known Fučík problem (1.1), (1.2). We give the basic notations also. In Section 3 we present the results on the spectrum for the Fučík type problem (1.1), (1.4). Our goal is to study the spectrum of the problem (1.1), (1.4). The properties of the spectrum for different values of γ are given in Section 4. In Section 5 we consider more general problem and describe some properties of the spectrum. This study is based on some previous works. The first one is the author's work [11], where the spectrum of the equation (1.1) together with the integral condition (1.3) was considered and some properties of the spectrum were presented. Some branches of the spectrum for the problem (1.1), (1.4) consist of several components similarly to the spectrum for the problem (1.1), (1.3). We also use papers [3,8,9], where the eigenvalue problem for one-dimentional differential operator together with integral conditions (1.4) was considered. Notice that the problem (1.1), (1.4) generalizes classical spectral problems with one parameter and reduces to the problem (1.5) if µ = λ. We note that nonlocal boundary conditions (including integral conditions) are formulated for many applied problems, see e.g. [1,2], where also numerical algorithms for solution of such problems are proposed and investigated.

The Basic Statements
Consider the classical Fučík problem (1.1), (1.2). First, we describe the decomposition of the spectrum into branches F + i and F − i (i = 0, 1, 2, . . .). Proposition 1. The Fučík spectrum consists of the set of the curves

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The first five pairs of branches of the spectrum to the problem (1.1), (1.2) are depicted in Figure 1.

The Spectrum of the Problem
Now consider the problem (1.1), (1.4). The expressions for the branches of the spectrum for this problem are given in the next theorem. The meaning of the notation of the spectrum branches is the same as earlier.
Proof. The proof of this theorem is similar to the proof given in the work [11] for the case of (µ, λ) being in the first quadrant. We will restrict to some comments. First of all, we obtain the expressions for F + 0 . Let's suppose that the solution without zeroes in the interval (0, 1) exists and x ′ (0) > 0. In this case we obtain that problem (1.1), (1.4) reduces to the eigenvalue problem. The solution of the problem must satisfy the condition (1.4), thus the expression for F + 0 for positive µ values is obtained. Analogously we do analysis for nonpositive µ values.
The idea of the proof for other branches is similar. We consider the eigenvalue problems in the intervals between two consecutive zeroes of the solution and use the conditions of the solutions for these problems. For example we will prove this theorem for µ > π 2 , λ < 0 (this means that (µ, λ) are in the fourth quadrant). Suppose that (µ, λ) ∈ F + 1− and let x(t) be the corresponding nontrivial solution of the problem (1.1), (1.4). The solution has only one zero in (0, 1) and x ′ (0) > 0. Let this zero be denoted by τ . Consider a solution of the problem (1.1), (1.4) first in the interval (0, τ ) and then in the interval (τ, 1). We obtain that the problem (1.1), (1.4) in these intervals reduces to the linear eigenvalue problems. So in the interval (0, τ ) we have the problem x ′′ = −µx with boundary conditions x(0) = x(τ ) = 0. In the interval (τ, 1) we have the problem x ′′ = −λx (λ < 0) with boundary condition x(τ ) = 0, notice that x ′ (τ ) < 0. Let the value of −λ be denoted by δ.
In view of (1.4) a solution of the problem (1.1), (1.4) x(t) must satisfy the condition Now we consider a solution of the equation we obtain The integral value is the following Multiplying both sides by 1/(A √ µ) and replacing δ by −λ, we obtain the expression for Rewrite the equation (4.1) in the form The studies of the equation (4.3) show that it is solvable only for γ > 2. For γ = 2 we obtain that µ = 0 is the branch F + 0 of the spectrum. Therefore the branch Proof. It follows from Lemma 1 that for 0 < γ < 2 the branches F ± 0 can be continued in the second and the fourth quadrants analogously as for the spectrum of the problem (1.1), (1.2) (see [5]), for γ > 2 the branches F ± 0 are in the second, the third and the fourth quadrants, but for γ = 2 the branches F ± 0 are the axes µ = 0 and λ = 0. For γ < 0 and γ > 2 the branches F ± 1 consist of three parts and form the continuous curves. These branches are located in the first, in the second and in the fourth quadrants. Let us consider the expression for F + Consider the right side of equation (4.4). Let λ tends to 0. Using L'Hospital's rule, we obtain Now consider the last part of the equation (4.4). Let λ tends to 0.

Proof.
It is clear that the odd-numbered branches and the even-numbered ones (and vice versa also) intersect at the points in which the problem (1.1), (1.4) reduces to the problem (1.1), (1.3). It follows that these points are the same as for the problem (1.1), (1.3) (see [11]). ⊓ ⊔ Remark 2. It was observed in [3,9] that some points of the spectrum for problems    Next two theorems are direct consequences of Lemmas.     Several first branches of the spectrum to the problem (1.1), (1.4) for 0 ≤ γ < 2 are depicted in Figure 2, the dashed curves is the spectrum for the problem (1.1), (1.2), the red ones -F + i branches, the blue curves -F − i branches of the spectrum for the problem (1.1), (1.4).  Remark 3. The spectrum of the problem (1.1), (1.4) for γ = 2 is in the first quadrant and the axes also belong the spectrum. For γ = 20 and γ = 100 the branches F ± 0 exist also, but they are far away from the axes (see Figure 4).

Remark 4.
In the work [11] the Fučik equation was considered together with the conditions and some properties of the spectrum were described. The problem (1.1), (4.6) is a particular case of the problem (1.1), (1.4). Indeed, rewrite the second condition of the (4.6) (for α = 1) in the form

More General Problem
Now we consider the Fučík equation together with nonlocal boundary conditions The analog of this problem for one-dimensional differential operator was considered in [3,8,9].
The solution of the problem (5.1), . It follows from the boundary condi- The proof of this fact is clear. Let us change the variable in the problem (5.7), (5.8) as follows X(τ ) = x(1 − t). Then we obtain Acknowledgement. The author wishes to thank the anonymous referee for the constructive suggestions.