Solvability of Integral Equations with Endogenous Delays
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Solvability of Integral Equations with Endogenous Delays Natali Hritonenko · Yuri Yatsenko
Received: 30 September 2011 / Accepted: 6 March 2013 © Springer Science+Business Media Dordrecht 2013
Abstract This paper performs a systematic study of some nonlinear integral equations with unknown lower limits of integration, which arise in economics, operations research, population biology and environmental sciences. A general investigation technique is developed for systems of such equations and several theorems about the existence and uniqueness of a solution are proven. The links to other functional equations with state-dependent delays are discussed. Applied interpretation of the obtained results is provided. Keywords Nonlinear Volterra integral equations · Endogenous delay · Uniqueness and existence of a solution Mathematics Subject Classification (2010) 45G15 · 37N40 · 91B76
1 Introduction The paper studies the nonlinear integral equations m t Kij (τ, t)xj (τ )dτ, xi (t) = j =1
zi (t) < t,
(1)
zj (t)
i = 1, . . . , n, m ≤ n, t ∈ [t0 , T ), T ≤ ∞,
(2)
when the kernel matrix K(τ, t) = {Kij (τ, t) ≥ 0, i = 1, . . . , m, j = 1, . . . , n}, τ ∈ [τ0 , T ), t ∈ [t0 , T ), is known and some or all components of x(t) = {xi (t), i = 1, . . . , n} and z(t) = {zi (t), i = 1, . . . , n}, t ∈ [t0 , T ), are unknown. The independent variable t stands for N. Hritonenko () Department of Mathematics, Prairie View A&M University, Prairie View, TX 77446, USA e-mail: [email protected] Y. Yatsenko School of Business, Houston Baptist University, Houston, TX 77074, USA e-mail: [email protected]
N. Hritonenko, Y. Yatsenko
time. Because of the inequality (2), the functions zi (t), i = 1, . . . , n, in the lower integration limits of (1) represent delays with respect to the current time t , so, we refer to the unknown components zi as the endogenous delays. Equations of type (1) are always nonlinear if, at least, one of the delays zi is endogenous (unknown). The endogenous delays bring essential mathematical challenges to the investigation, namely, the unknowns xi in the problem (1)–(2) depend on the unknown delays zi and Eq. (1) after differentiation involve the terms xi (zi (t)). Equations with such terms are known as the functional equations with state-dependent delays [16] and receive a growing attention in applications. 1.1 Equations with State-Dependent Delays The theory of functional equations with state-dependent delays has emerged from various applied fields such as physics, engineering, automatic control, neural networks, population ecology, cell biology, epidemiology, modeling of immune response and respiration systems, and others [2, 9, 11, 16]. This theory is a real crossroad of several modern areas of mathematical modeling and new theoretic results obtained in one application can be often used by other areas. As mentioned in the detailed survey [16], such equations are not simple, they differ considerably, and many challenging problems in this theory are still remained unsolved. Any advancement of this theo
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