Existence and uniqueness of positive solutions of a system of nonlinear algebraic equations
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Existence and uniqueness of positive solutions of a system of nonlinear algebraic equations István Gy˝ori1 · Ferenc Hartung1 Nahed A. Mohamady1
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© Akadémiai Kiadó, Budapest, Hungary 2016
m Abstract In this paper we consider the nonlinear system γi (xi ) = j=1 gi j (x j ), 1 ≤ i ≤ m. We give sufficient conditions which imply the existence and uniqueness of positive solutions of the system. Our theorem extends earlier results known in the literature. Several examples illustrate the main result. Keywords Nonlinear algebraic system · Positive solution · Existence · Uniqueness Mathematics Subject Classification 47J05
1 Introduction Nonlinear or linear algebraic systems appear as steady-state equations in continuous and discrete dynamical models (e.g., reaction–diffusion equations [14,19], neural networks [5, 6,15,22] compartmental systems [2,4,11,12,16,17], population models [13,21]). Next we mention some typical models. Compartmental systems are used to model many processes in pharmacokinetics, metabolism, epidemiology and ecology. We refer to [16,17] as surveys of basic theory and applications of linear and nonlinear compartmental systems without and with delays. A standard form of a linear compartmental system with delays is
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Ferenc Hartung [email protected] István Gy˝ori [email protected] Nahed A. Mohamady [email protected]
1
Department of Mathematics, University of Pannonia, Veszprém, Hungary
123
I. Gy˝ori et al.
q˙i (t) = −kii qi (t) +
m
ki j q j (t − τi j ) + Ii ,
i = 1, . . . , m.
(1.1)
j=1 j =i
Here qi (t) is the mass of the ith compartment at time t, while ki j > 0 represent the transfer or rate coefficients and Ii ≥ 0 is the inflow to the ith compartment. A possible generalization of (1.1) used in several applications is a compartmental system, where it is assumed that the intercompartmental flows are functions of the state of the donor compartments only in the form ki j f j (q j ) with some positive nonlinear function f j . So we get the nonlinear donor-controlled compartmental system (see, e.g., [2,4]) q˙i (t) = −kii f i (qi (t)) +
m
ki j f j (q j (t − τi j )) + Ii ,
i = 1, . . . , m.
(1.2)
j=1 j =i
Next we consider an ecological system of m species which are living in a symbiotic relationship with the other species (see [10]): ⎛ ⎞ m ⎜ ⎟ ki j x j + bi ⎠ , x˙i = xi ⎝−kii xi + i = 1, . . . , m. (1.3) j=1 j =i
Here kii > 0 represents the measure of the mortality due to intraspecific competition, the terms bi ≥ 0 represent the per capita growth due to external (inexhaustible) sources of energy, and the coefficients ki j ( j = i) are nonnegative due to the symbiosis. Cellular neural networks were introduced by Chua and Yang [7] in 1988, and since then they have been applied in many scientific and engineering applications. Here we consider the Hopfield neural network studied in [5] Ci u˙ i =
m
Ti j g j (u j ) −
j=1
ui + Ii , Ri
i = 1, . . . , m,
(1.4)
where Ci > 0, Ri > 0 and Ii are capacity, resistance, bias, respectively, Ti j is
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