Many mathematical models arising from physics, chemistry, biology, economics etc. are given by first-order differential systems whose time-dependent variables stay for specific quantities, most often subject by their nature to the condition of positivity. In this connection, there is a huge literature on positive solutions to various classes of problems. However, there are cases where the positivity is only required on a subinterval of time. For instance, in case of the Lotka-Volterra predator-prey model, we may be interested that the size of the prey population $x\left(t\right)$ on a given time interval $J$ - for example the reproduction season - be larger than a given threshold $x_{0},$ and the size of predator population $y\left(t\right)$ on that time interval be less than $y_{0}.$ Under the substitutions $u=x-x_{0}$ and $v=y_{0}-y,$ the desired threshold conditions become the positivity conditions $u\geq 0$ and $v\geq 0$ on the prescribed time interval $J.$ Analogous problems appear in economic models when the size of production of some goods in some periods of the year needs to be sufficiently larger to satisfy the market demand, or on the contrary, to be sufficiently smaller to avoid excessive stocks. Having in mind these examples as a main motivation, in this paper we seek solutions to differential systems on a given interval, which are necessarily positive only on a prescribed subinterval.

Additionally in this paper, we treat differential systems with nonlocal conditions. In particular, they include the initial value condition, boundary value conditions, multi-point and integral constraints. As a motivating example, we have the integral condition $\int_{\Omega}u=1$ which appears in many physical models where $u$ is a probability distribution. The research on differential equations with nonlocal conditions began with the pioneer works of Cioranescu [5], Whyburn [17] and Conti [6], and have gained a special attention in the last decades motivated by concrete applications in different domains (see, e.g., [1]-[4], [8]-[16] and [18]).

More exactly in this paper we study the existence, localization and multiplicity of solutions to the following problem:

where $\left[a,b\right]$ is a fixed subinterval of $\left[0,1\right],$ $f:[0,1]\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is continuous, and $\ g:C([0,1],\mathbb{R}^{n})\rightarrow\mathbb{R}^{n}$ a continuous linear mapping. We shall seek solutions $u\in C^{1}([0,1],\mathbb{R}^{n}).$

Taking $\alpha(u):=u(a)+g(u),$ the nonlocal condition $g\left(u\right)=0$ can be replaced by

Our approach to problem (1.1) is based on fixed point principles and fixed point index theory, and takes into consideration the support $\left[a,c\right]$ of $\alpha,$ namely the smallest subinterval $\left[a,c\right]\subseteq\left[a,b\right]$ such that $\alpha(u)=\alpha(v)$ for every $u,v\in C([0,1],\mathbb{R}^{n})$ having the same restriction to $\left[a,c\right].$

Recall that for a closed convex subset $D$ of a Banach $X,$ a subset $U\subseteq D$ open in $D,$ and a completely continuous operator $N:\overline{U}\rightarrow D$ such that $N\left(u\right)\neq u$ on the $\partial_{D}U=\overline{U}\setminus U,$ one can consider the fixed point index of $N$ over $U$ with respect to $D,$ which is denoted by $i\left(N,U,D\right)$ (see, e.g., [7, p. 238]) and has the following properties: $i\left(N,U,D\right)\neq 0$ implies that $N$ has at least one fixed point in $U$ (existence);$\ i\left(u_{0},U,D\right)=1\$for the any constant operator $N\equiv u_{0}\in U$ (normalization); the homotopy invariance; and additivity with respect to $U.$ We have the following lemma that is used in the proof of our main result.

We shall use the notation $\alpha\left[1\right],$ for the matrix whose columns are

Thus, in virtue of the linearity of $\alpha,$ for any constant vector-valued function $c=(c_{1},\cdot\cdot,c_{n}),$ using the decomposition

one has that $\alpha\left(c\right)$ is the product of matrix $\alpha[1]$ with the column vector $c,\$i.e.,

Also the notation $I$ will stay for the identity matrix of size $n.$

Finally, for a vector $x=\left(x_{1},\cdot\cdot,x_{n}\right)\in\mathbb{R}^{n},$ by $\left|x\right|$ we shall mean the column vector whose elements are $\left|x_{1}\right|,\cdot\cdot,\left|x_{n}\right|;$ for a scalar function $v\in C[0,1],$ we shall denote by $\left|v\right|_{\infty}$ its supremum norm, and for a vector-valued function $u\in C([0,1],\mathbb{R}^{n})$ of scalar components $u_{1},\cdot\cdot,u_{n},$ we shall denote by $\left|u\right|_{\infty}$ the column vector of elements $\left|u_{1}\right|_{\infty},\cdot\cdot,\left|u_{n}\right|_{\infty}.$ Also, for any subinterval $\left[a,b\right]$ of $\left[0,1\right],$ by $\int_{a}^{b}u\left(t\right)dt$ we shall mean the column vector of the elements $\int_{a}^{b}u_{1}\left(t\right)dt,\cdot\cdot,\int_{a}^{b}u_{n}\left(t\right)dt.$ Thus, when referred to vectors or vector-valued functions, the equalities and inequalities from below have to be seen as equalities and inequalities between column vectors.

First, if the matrix $I-\alpha\left[1\right]$ is non-singular, then our differential system subject to the condition $u\left(a\right)=\alpha\left(u\right)$ is equivalent to the integral type equation in $C\left(\left[0,1\right],\mathbb{R}^{n}\right),$ namely

To prove this it suffices to integrate the differential equation from $a$ to an arbitrary $t,$ to obtain

then to apply the linear mapping $\alpha$ and use the condition $u\left(a\right)=\alpha\left(u\right),$ to obtain

whence

Now replacing in (2.2) gives (2.1).

The solutions $u\in C\left(\left[0,1\right],\mathbb{R}^{n}\right)$ of (2.1) are the fixed points of the operator $N:C\left(\left[0,1\right],\mathbb{R}^{n}\right)\rightarrow C\left(\left[0,1\right],\mathbb{R}^{n}\right)$ given by

which is completely continuous due to the continuity of $f$ and $\alpha.$

Looking for solutions that are positive on the prescribed subinterval $\left[a,b\right]$ of $\left[0,1\right],$ we are let to consider the set

Obviously, $K$ is a wedge (i.e., closed, convex and with $\lambda K\subset K$ for every $\lambda\in\mathbb{R}_{+}$). In order to use fixed point principles and index, we need that the condition $N\left(K\right)\subset K$ is satisfied. It is easily seen that this happens if additionally the following positivity properties hold:

where $\left[a,c\right]$ is the support of $\alpha.$ Thus $f\left(t,\cdot\right)$ is not necessarily positive on $\mathbb{R}_{+}^{n}$ for $t\in\left[0,1\right]\setminus\left[a,b\right],$ but $\alpha\left(u\right)$ is positive even for functions $u$ which are not positive on the whole interval $\left[0,1\right].$ Also note that under condition (2.4), one has $\alpha\left[1\right]\in\mathcal{M}_{n\times n}(\mathbb{R}_{+})$ and then the hypothesis (2.5) is equivalent to the fact that the eigenvalues of the matrix $\alpha\left[1\right]$ are located inside the unit circle.

Our main result is the following theorem on the existence, localization and multiplicity of solutions to problem (1.1).

For two vectors $x,y\in\mathbb{R}_{+}^{n}$ with $y\leq x,$ and $i=1,\cdot\cdot,n,$ we denote:

Also $\underline{f}^{x}\left(s\right),\ \ \overline{f}^{x}\left(s\right),\ \mu^{x}$ and $\underline{f}^{y,x}\left(s\right)$ are the vector-valued functions of the corresponding scalar functions from the above table. For instance,

which is looked as a column vector.