First order systems of odes with nonlinear nonlocal boundary conditions

In this article, we prove an existence of solutions for a non-local boundary value problem with nonlinearity in a nonlocal condition. Our method is based upon the Mawhin's coincidence theory.


Introduction
In this paper we consider the following ordinary differential equation (1) x with the non-local condition (2)  x k (s) dg k (s) .
The subject of nonlocal boundary conditions for ordinary differential equations has been a topic of various studies in mathematical articles for many years. The multi-point conditions such as F (x(t 1 ), x(t 2 ), ..., x(t n )) = 0 were studied at first ( [11]), then also the significantly nonlocal conditions with the values of the unknown function occurring over the entire domain (integral) became the subject of interest. It is easy to see that the conditions which there is the Stieltjes integral with respect to any function with the total variation in contain also multi-point problems.
Usually the matter of consideration are the second-order differential equations because of their supposed applications but sometimes also the firstorder differential equations are being considered as in the present paper ( [3] and [17]). And with our level of generality the second order differential equations can be treated as the first-order systems. The methods are typical: searching for the fixed point of integral operator using Contraction Principle, Schauder fixed-point theorem, topological-order methods, e.g. basing on Cone Expansion and Compression Theorem, or finally the Leray-Schauder degree of compact mapping or the Mawhin degree of coincidence.
In this paper both differential equations and boundary conditions are nonlinear what somehow forces to the use of the degree of coincidence -the linear part x ′ has the nontrivial kernel. Using this method and with such a generality of assumptions the theorems that can be obtained are the ones in which the Brouwer degree of the nonlinear part being not zero on the kernel of the linear part is the main assumption. In this paper there is only the degree of "the half" of the nonlinear operator, i.e. h and the assumptions regarding the other half of the nonlinear part are different.
Nonlinear boundary conditions have occurred before in works [5], [6], [15], [16] but they were of different nature than here: under Stieltjes integral there was the assumption of the unknown function with the nonlinear function. Therefore, the obtained results are not comparable with the previous works; those results present a new direction of research. It is possible only to notice the compatibility with the conventional results regarding the existence of the periodic solutions ( [10]). This problem will be explained further in paragraph 4.

Some preliminaries
In this section we recall some facts about a Fredholm operator and Mawhin's coincidence theory. This section is based on [4] (page .
Let X and Y be a Banach space. An linear operator L : X ⊃ dom L → Y is said to be a Fredholm operator if dim ker L < ∞, im L is closed in Y and codim im L < ∞. The index of the Fredholm operator is defined as follows ind L := dim ker L − codim im L.
If L is the Fredholm operator, then continuous projections P : X → X, Q : Y → Y such that im P = ker L, ker Q = im L exist. Thus X = ker L ⊕ ker P and Y = im L ⊕ im Q. It is apparent that ker L ∩ ker P = {0}, therefore we can consider the restriction L P := L| ker P : dom L ∩ ker P → Y which is invertible. A nonlinear operator N : X → Y is called L-compact if N maps bounded sets into bounded ones and K P, To obtain the results of the existence we use the following Mawhin's theorem. .
Now we return to main problem and present our notations. We set X : It is clear that Consequently, L is a Fredholm operator of index zero and we can use Mawhin's theory.

The existence of solutions
We know that our operator L is a Fredholm operator with index zero. Our purpose is to use the Mawhin's theory. In first step we define projections P : X → X by The description of P makes it evident that ker P = {x ∈ X : Then inverse operator is defined as and we have Since the first term is a composition of Nemytskii operator and Volterra integral operator and the second term is a finite rank we get the following Our main result is given in the following theorem there exists R > 0 such that: h(x) = 0 for r − < |x| ≤ r + and the Brouwer degree deg(h, B R k (0, r), 0) is defined and does not vanish for some r ∈ (r − , r + ]. Before we proceed to the proof we recall some notions regarding the Riemann-Stieltjes integral ( [12], page 9-11 and 105-123). Let g : [a, b] → R k and consider the sum  Recall that both sides expressions are vectors, which means that the first summand has coordinates The proof follows from the form of Riemmann-Stieltjes sums which converge to the integrals: Proof. The proof is carried out in two steps. In step 1, we prove that BVP (1, 2) has the solution under stronger assumptions: lim ε→0 + var(g, [ε, 1]) < |g(0 + ) − g(0)| and f (t, x), x < 0 for t ∈ (0, 1], |x| = R. Step 1. We know that the BVP (1), (2) is equivalent to (3). A linear operator L is a Fredholm operator with index zero and nonlinear N is Lcompact. If we prove other assumptions of Mawhin theorem we get the assertion.
Similar problem was considered in [13], with the difference that in second condition of (8) we have x ′ (1) = 1 0 x(s) dg(s).