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RIGHT FOCAL POINT BOUNDARY VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS
Abstract
In this work we consider solutions of the nth order differential equation y('(n)) = f(x,y,y'...,y('(n-1))) satisfying boundary conditions of the form y('(i))(x(,k)) = y(,ik), s(,k-1) = m(,1) + ... + m(,k-1) (LESSTHEQ) i (LESSTHEQ) m(,1) + ... + m(,k) - 1 = s(,k) - 1, 1 (LESSTHEQ) k (LESSTHEQ) r, 2 (LESSTHEQ) r (LESSTHEQ) n, where a < x(,1) < ... < x(,r) < b. Such a boundary value problem is called a right (m(,1),...,m(,r))-focal point boundary value problem. Assuming that f is continuous on (a,b) x (//R)('n), that solutions of initial value problems are unique, and that each right (1,1,...,1)-focal point boundary value problem has at most one solution, we first show that solutions of all right (m(,1),...,m(,r))-focal point boundary value problems, 2 (LESSTHEQ) r (LESSTHEQ) n - 1, are unique, when they exist. In addition to the above assumptions, we next assume that solutions of initial value problems extend to (a,b) and that a certain compactness condition on solutions is satisfied. Then we prove that all right (m(,1),...,m(,r))-focal point boundary value problems, 2 < r < n, have solutions on (a,b). Next, a Lipschitz condition is imposed on f. Assuming in addition only that f is continuous on (a,b) x (//R)('n), we obtain best interval lengths, in terms of the Lipschitz coefficients, of subintervals of (a,b) on which unique solutions of all right (m(,1),...,m(,r))-focal point boundary value problems exist. In the case n = 3, we prove that if f is continuous, if solutions of initial value problems are unique and extend, and if each right (m(,1),m(,2))-focal point boundary value problem for y''' = f(x,y,y',y'') has at most one solution, then each right (1,1,1)-focal point problem for y''' = f(x,y,y',y'') has at most one solution. Finally, a topological and convergence result is obtained based on the continuous dependence of solutions of right focal point problems on boundary conditions. We show that if y(,0)(x) is a solution of some right (m(,1),...,m(,r))-focal point boundary value problem, then there is a sequence {y(,j)(x)} of distinct solutions of right (1,1,...,1)-focal point boundary value problems such that {y(,j)('(i))(x)} converges to y(,0)('(i))(x) uniformly on compact subintervals of (a,b), for 0 (LESSTHEQ) i (LESSTHEQ) n - 1.
Subject Area
Mathematics
Recommended Citation
HENDERSON, JOHNNY LEE, "RIGHT FOCAL POINT BOUNDARY VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS" (1981). ETD collection for University of Nebraska-Lincoln. AAI8122594.
https://digitalcommons.unl.edu/dissertations/AAI8122594