The minimizer for the second order differential problem with the integral condition

In this paper, we investigate the best fit solution for the second order differential problem with one initial and other integral conditions. We obtain the representation of that minimizer and present an example.


Introduction
In the paper [2], there was studied the best fit solution to the second order differential problem with one initial condition and other nonlocal two point condition.
If γξ 2 = 2 [3], it also has the unique solution, which is of the form For γξ 2 = 2, the problem (1)-(2) does not have the unique solution. This is the case that we are going to study below and obtain the so called best fit solution.

The vectorial problem
As in [2], we also consider the equivalent vectorial form Here f = (f, g 1 , g 2 ) ⊤ ∈ L 2 [0, 1] × R 2 denotes the right hand side of the problem. Let us note that the vectorial operator L has the following properties. Proof. The proof is obtained almost the same as Theorem 1 is proved in [2]. ⊓ ⊔ If the problem has the unique solution (γξ 2 = 2), its range is coincident with the entire space L 2 [0, 1] × R 2 . For the problem without the unique solution (γξ 2 = 2), we obtain the following range representation. Lemma 1. If γξ 2 = 2, the range of the operator L is of the form Proof. The general solution of the equation −u ′′ = f is given by Substituting it into nonlocal conditions, we get c 1 = g 1 and Since γξ 2 = 2, we solve Thus, vectorial functions of the form f = (f, g 1 , g 2 ) ⊤ , with the obtained g 2 expression via arbitrary g 1 and f , represent the range. ⊓ ⊔ According to [1], properties of L implies the closeness of N (L * ), where L * : is the adjoint operator of L. Then the nullspace and range theorem gives N (L * ) = R(L) ⊥ , which representation can also be derived in the following form.
with arbitrary g 1 ∈ R and f ∈ L 2 [0, 1]. Since g 1 and f obtain values independently, we take g 1 = 0, afterwards f = 0 and get two conditionsf (y)−g 2 γ ξ 0 G cl (x, y) dx = 0 and From the Fredholm alternative theorem, we obtain the solvability condition for the problem without the uniqueness.

Existence and representation of the minimizer
According to [1] and Theorem 1, the operator L has the Moore-Penrose inverse It is the unique solution to the four operator equations Moreover, the Moore-Penrose inverse describes the desired best fit solution which is also known as the minimum norm least squares solution or the best approximate solution. Let us note that the function u o has the minimum H 2 [0, 1] norm among all minimizers u g of the norm of the residual The minimizer u o to the problem Lu = f (which may be consistent or inconsistent) is always the minimizer to the consistent problem Lu = P R(L) f [1] (here and further P S denotes the orthogonal projector onto a subspace S). Let us solve it! First, we shortly denote the right hand side and calculate it as below The problem Lu =f is of the form (1)-(2) with the above calculated right hand sidef = (f ,g 1 ,g 2 ) ⊤ instead of f = (f, g 1 , g 2 ) ⊤ . Thus, we take the general solution of the differential equation −u ′′ =f , that is, Substituting it into the initial condition u(0) =g 1 , we find c 1 =g 1 . Let us note that the integral condition u(1) = γ ξ 0 u(x) dx +g 2 is satisfied trivially because we have the solvable problem without the uniqueness (γξ 2 = 2). Thus, we obtain the general solution of the form According to [1], we can find the minimizer from the formula u o = P N (L) ⊥ u g , that is also u o = u g − P N (L) u g . Let us note that N (L) = {cx, c ∈ R} and, thus, P N (L) (cx) = cx. Now we calculate and obtain P N (L)g 1 =g 1 · P N (L) 1 =g 1 · 3x/8. Similarly, we calculate 1] x ·f (y) dy with the kernel P N (L) G cl (x, y) = 1 8 · xy(1 − y 2 ). Here t denotes the integration variable. Then the formula u o = u g − P N (L) u g gives

Generalized Green's function
Substituting expressions ofg 1 andf , we rewrite this minimizer in the form Let us note that the unique solution (case γξ 2 = 2), which is given in Introduction, can also be represented in the relative form These functions v 1 , v 2 are called the biorthogonal fundamental system and G(x, y) is called the Green's function for the problem (1)-(2) (see [3]). Thus, because of the similarity, we call the functions v g,1 , v g,2 -the generalized biorthogonal fundamental system and G g (x, y) -the generalized Green's function for the problem (1)-(2) without the uniqueness (case γξ 2 = 2).