PARAMETERS AND SOLUTIONS OF LINEAR AND NONLINEAR OSCILLATORS

Relationship between existence of solutions for certain classes of nonlinear boundary value problems and the smallest or the largest eigenvalue of the corresponding linear problem is obtained. Behavior of the solutions, as the parameter increases, is also studied.


R. LING
[i, 2, 3, 4], and in nuclear energy distribution [5,6].In these problems, the parameter has physical significance, such as the energy level or the stiffness factor of the system under consideration.
In this work, relationship between existence of solutions for classes of nonlinear boundary value problems with equations of the form (1.1) and the smallest or the largest eigenvalue of the corresponding linear problem is obtained.The case of the coefficient q(x) being a negative constant has been investigated in [7].Conditions on the coefficients of the equation, under which the solution remains bounded as the parameter increases, are obtained.

EXISTENCE OF SOLUTIONS FOR NONLINEAR BOUNDRY VALUE PROBLEMS AND EIGENVALUE
OF CORRESPONDING LINEAR PROBLEMS.
In this section, relationship between existence of solutions for equations of the form (i. i) with zero boundary conditions and the smallest or largest eigenvalue of the corresponding linear problem is obtained.The analysis used here is similar to that in [7].It would be assumed that the functions p(x) and q(x) are in the class C[0,1].
Suppose now that the largest elgenvalue of (2.3) and (2.4) is positive.Note first that if y is positive and M denotes its maximum, then To apply an existence theorem for nonlinear elgenvalue problems in [9], equa- tion (2.1) is written in the form Ly F(x, y) where Ly -y" + a(x)y, a(x) > 0, n F(x, y) [p(x) +a(x)]y-q(x)y To show that a positive solution of (2.1) and (2.2) exists, we must find curves u(x), v(x) such that 0 < u(x) < v(x), for all x e (0, I), and a(x) must be chosen so that F(x, y) is a monotonic increasing function of y for all (x, y) in the set s--{(x,y) o <_ x-< , u(x) _< y_< v(x)}.

Let u(x)
z l(x) ,,here z, (x From the fact that 8F n-I p(x) + a(x) q(x)ny 8Y F(x, y) is increasing in y in S if a(x) > q(x)ny n-I p (x) x e [0, I] x e [0, I] By [9], the nonlinear problem (2.1) and (2.2) has at least one solution in S.
PROOF.Suppose (2.1) and (2.2) has a negative solution.Then as in the proof of Theorem 2.1, it can be shown that if %1 is the largest eigenvalue of (2.3) and (2.4) and Zl is a corresponding eigenfunction, then I S q(x) yn Zl 0 dE 1 q(x) y Zl 0 dx and since n is odd, %1 is positive.
Conversely, suppose that the largest eigenvalue of (2.3) and (2.4) is positive.
Note first that if y is negative and m denotes its minimum at say x0, then y" -P(X0) m + q(x0)m To apply the existence theorem in [ 9], equation (2.1) is written in a form as in the proof of Theorem 2.1.To show that a negative solution of (2.1) and (2.2)   exists, this time we must find curves u(x), v(x) such that u(x) _< v(x) < 0, for x (0, I), v(0) --0, v(1) --0, Lv >_ F(x, v), u(O) < O, u(1) _< O, Lu-< F(x, u) and a(x) must be chosen so that F(x, y) is a monotonic increasing function of y for all (x, y) in the set -I < z l(x) < 0 and R <z l(x), for x s (0, i) then n -1 q(x) z 1 >-q(x) z 1 and so

Lv
From the fact that 3__[ n-I 3y p(x) + a(x) q(x)n y F(x, y) is increasing in y in S if n-I a(x) _> q(x)ny p(x) <QnR-Po,, so let a(x) >-Q n R-P0 It follows from [9] that the nonlinear problem (2.1) and (2.2) has at least one solution in S.
In the next Theorem, the nonlinear problem y"(x) + p(x)y + q(x) yn 0 and the corresponding linear eigenvalue problem z"(x) + p(x) z + Xq(x) z 0 z(0) z are considered.

BOUNDEDNESS OF THE SOLUTION AS THE PARAMETER INCREASES.
In this section, boundedness of the solution of y"(x) + p(x)y + Xq(x)y n 0 as the parameter I increases, is studied.It would be assumed that the functions p(x) and q(x) are in the class CI[0, i].