Let a and b be real numbers satisfying a < b and let f (x,y,z,λ,µ) be a continuous function on (a,b) x R4. In Chapter's 1 and 2 we consider the generalized boundary value problem
y” = f(x,y,y1(a),y’(b)),
y(a) = 0, and y(b) =0,
where a solution y(x) is a function in C’[a,b] ∩ C2(a,b). Let g(x,y,z) satisfy the Caratheodory condition on [a,b] x R2 . in Chapter 3 we consider the boundary value problem
y" = g(x,y,y’),
y(a) = 0, and y(b) = 0 ,
where a solution y(x) is a function with an absolutely continuous first derivative on [a,b].
Both problems have the same Green's function and we formulate them as fixed point problems. If f is Lipschitz continuous with sufficiently small Lipschitz constants we prove the generalized problem has a unique solution.
We extend the notions of lower and upper solution and Nagumo condition introduced for the boundary value problem to apply to the generalized problem and prove their existence is sufficient to ensure the generalized problem has a solution. We prove lower and upper solutions exist and f satisfies a Nagumo condition if f grows 'less than linearly' with respect to its variables; we show greater than linear growth may be tolerated if f is assumed to be the sum of terms with appropriate signs and growth rates. We extend a second set of conditions introduced by Nagumo to bound the derivative of solutions for the boundary value problem to the generalized problem; we show the existence of lower and upper solutions and associated bounding functions for the derivative is sufficient to ensure the generalized problem has a solution. We prove lower and upper solutions and associated bounding functions for the derivative exist if f satisfies appropriate growth conditions.
Let h (x,z,λ,µ) be a continuous function on (a,b) x R3 and g(x,y,z) be continuous on (a,b) x R2. We prove the generalized problem has at most one solution if f has the form
f(x,y,z,λ,µ) = g(x,y,z) + h(x,z,λ,µ)
where g is strictly increasing as a function of y and h satisfies appropriate monotonicity conditions.
If F is a set of solutions of the generalized problem, then a solution z(x) in F is said to be maximal if z(x) ≥ y(x) on [a,b] for any y(x) in F. We prove that if there exist lower and upper solutions and f satisfies a Nagumo condition, then a maximal solution in the non empty family of solutions associated with these auxiliary functions may not exist unless f is decreasing as a function of λ and increasing as a function of µ.
We introduce lower and upper solutions for the boundary value problem under the Caratheudcry condition; we show that if lower and upper solution exist and g satisfies a Nagumo condition, then the boundary value problem has a maxinal and a minimal solution in the family of solutions associated with these auxiliary functions.
Let A, B, y, and λ belong to Rn and f(x,y,λ) be a 'measurable' function from [a,b] x R2n to Rn. In Chapters 4 and 5 we consider the two point boundary value problem
y' = f(x,y,λ),
y(a) = A , and y(b) = B ,
where a solution is a parameter value λ and a function y(x, λ) whose components are absolutely continuous on [a,b].
Following Kibenko and Perov we use the method of shooting with the initial value problem varying the parameter λ, thus reducing the problem to an onto mapping problem. By approximating f by a family of Lipschitz continuous functions we show that the multi valued mapping is the limit of a family of continuous functions. We prove an onto mapping principle for multi valued mappings which are limits of families of continuous functions and use this result to prove the two point problem has a solution if f satisfies appropriate growth conditions.
Finally, we prove the generalized boundary value problem has a solution if |f| is bounded using an existence result for the first order system. Kibenko and Perov obtained a perturbation for the first order system, in which they perturbed the boundary conditions. We prove a complementary result in which we perturb the equation rather than the boundary conditions.