Sergiy Borysenko Gerardo Iovane Differential Models: Dynamical Systems with Perturbations
Copyright MMIX ARACNE editrice S.r.l. www.aracneeditrice.it info@aracneeditrice.it via Raffaele Garofalo, 133 A/B 00173 Roma (06) 93781065 ISBN 978 88 548 2562 8 No part of this book may be reproduced in any form, by print, photoprint, microfilm, microfiche, or any other means, without written permission from the publisher. 1 st edition: June 2008
Contents 1 About Some New Integral Inequalities Wendroff Type for Discontinuous Functions 5 1.1 Introduction... 5 1.2 Integral inequalities for discontinuous functions with discontinuities nonlipschitztype... 7 1.3 Inequalitieswithretardation... 15 2 Some New Integral Inequalities of Bellman-Bihari Type With Delay for Discontinuous Functions 21 2.1 Introduction... 21 2.2 Integro-functional inequalities of Bellman-Bihari type....... 23 2.3 Inequalities Containing Multiple Integrals.............. 27 3 About A Generalization of Bellman-Bihari-Type Inequalities for Discontinuous Functions and Their Applications 33 3.1 Introduction... 33 3.2 Integro-suminequalities... 35 3.2.1 Previouslyresults... 35 3.2.2 Some new integro-functional inequalities Bellman-Bihari type 48 3.3 Wendroff s Type Inequalities for Discontinuous Functions..... 51 3.3.1 Lipschitztypediscontinuity... 51 3.3.2 Integral inequalities for discontinuous functions with discontinuitiesofnonlipschitztype... 53 3.3.3 Inequalitieswithretardation... 57 3.3.4 GeneraltheoremsofBiharitype... 59 3.3.5 Someapplications... 60 4 Investigations of the Properties Motion for Essential Nonlinear Systems Perturbed by Impulses on Some Hypersurfaces 65 4.1 Introduction... 65 4.2 StagingProblems... 66 4.3 EstimatesSolutionSystemofVariation... 67 1
2 4.4 Stability(Lipschitzcase)... 70 4.4.1 CaseA)... 72 4.4.2 CaseB)... 73 4.5 Attraction... 76 4.5.1 CaseA)... 78 4.5.2 CaseB)... 80 5 Integro-Sum Inequalities and Motion Stability of Systems With Impulse Perturbations 85 5.1 Introduction... 85 5.2 Objectforinvestigations... 86 5.3 Bellman-Bihari-Rakhmatullina method for discontinuous functions satisfyingintegralinequalities... 88 5.4 Boundedness and stability conditions on nonlinear approximation. 90 5.5 Integro-sum representations and stability of motion......... 93 6 About Estimates of Solutions for Nonlinear Hyperbolic Equations With Impulsive Perturbations on Some Hypersurfaces 97 6.1 Introduction... 97 6.2 Objectforinvestigation... 98 6.3 Mainresults... 99 7 Global Attractor for Non-Autonomous Wave Equation Without Uniqueness of Solution 107 7.1 Introduction... 107 7.2 Settingoftheproblem... 108 7.3 Elements of abstract theory of global attractors for multivalued nonautonomousdynamicalsystems... 111 7.4 Propertiesofsolutionsoftheproblem(2.1),(2.2)... 113 7.5 Themainresults... 117 8 Global Attractor for Impulsive Reaction-Diffusion Equation 123 8.1 Introduction... 123 8.2 SettingoftheProblem... 124 8.3 Elements of abstract theory of global attractors of nonautonomous multi-valueddynamicalsystems... 125 8.4 The family of MSP, generated by the problem (2.1) (2.4)..... 128 8.5 Themainresult... 131 9 Asymptotic Behavior of Impulsive Non-Autonomous Wave Equation 135 9.1 Introduction... 135 9.2 Settingoftheproblem... 136
3 9.3 The family of multi-valued semi-processes, generated by the problem(2.1),(3.1).... 137 9.4 Themainresult... 141 10 Analysis of the Properties of the Solutions of Impulsive Systems of Differential Equations 145 10.1Introduction... 145 10.2BasicnotationsandPreviouslyresults... 147 10.3 Analysis of the properties of the solution system (A)........ 149 10.4Conclusions... 159 11 Analysis of the Properties of the Solutions of Impulsive Systems of Integro-Differential Equations 161 11.1Introduction... 161 11.2 Preliminaries and Fundamental Results............... 163 11.3MainResults... 168 11.4Conclusions... 174 12 About New Integro - Sum Inequality Gronwall - Bellman Type And Its Application For the Problem of Boundedness 175 12.1Introduction... 175 12.2 A new analogy of Gronwall - Bellman type inequality........ 176 12.3 Conditions of boundedness solutions................. 178 13 On Some Integro - Sum Gronwall - Bellman - Bihari - Wendroff s Type Inequalities 185 13.1Introduction... 185 13.2 One-dimensional integro - sum functional inequalities........ 186 13.3 Wendroff stypeinequalities... 188 13.4Remark... 191 14 About the New Conditions of the Solvability of Chaplygin s Problem for Integro-Sum Inequalities 193 14.1Introduction... 193 14.2 Chaplygin s Problem for Integral Inequalities... 194 14.3 About Conditions of Solvability of the Chaplygin s Problem for PiecewiseContinuousFunctions... 195 14.4 The Problem of Reduction Functional Integral Inequalities for Function of Several Variables to One-dimensional Inequalities...... 198 14.5Conclusions... 204
4 15 The Bihari s Result Generalization for the Discontinuous Functions of Two Variables 207 15.1Introduction... 207 15.2 Bihari s lemma for the function of two variables........... 208 15.3Mainresult... 210 15.4Conclusions... 214 16 Investigations of the Properties Solutions Impulsive Differential Systems on Linear Approximation 217 16.1Introduction... 217 16.2Preliminaryconsiderations... 219 16.3Mainresults... 222 17 To the Question of Mixed Type System Simulation in the Tasks of Analysis and Control 229 17.1Introduction... 229 17.2Tasksetting... 230 17.3Conclusions... 244 18 Necessary Conditions for Control of Objects With Distributed Constants 245 18.1Introduction... 245 18.2Tasksetting... 246 18.3 Task of optimal control for the objects with distributed constants, which are described by non-linear differential equations with particle derivatives of elliptic type..................... 249 18.4 Task of optimal control for the objects with distributed constants, which are described by non-linear differential equations with particlederivativesofparabolictype... 252 18.5 Task of optimal control for the objects with distributed constants, which are described by non-linear differential equations with particlederivativesofhyperbolictype... 257 18.6Conclusion... 260
Chapter 1 About Some New Integral Inequalities Wendroff Type for Discontinuous Functions 1.1 Introduction Our Chapter is devoted to the development of the integral inequalities method as tool for investigating the qualitative characteristic solutions of different kinds of equations: differential equations, difference equations, partial differential equations, impulsive differential equations [3], [32], [56], [51], [52], [101], [103], [152], [153]. By using the results by R.P. Agarwal [3], R.P. Agarwal, S. Leela [7],O. Akinyele [4], D.Bainov, I. Simeonov [12], R. Bellman [20], I. Bihari [21], B.K. Bondge, B.G. Pachpatte [[23], [25], [26]], T.H. Gronwall [78], V. Lakshamikantham [102], V. Lakshamikantham, S. Leela [112], V. Lakshamikantham, S. Leela, M. Mohana Rao Rama [116], V. Lakshamikantham, D. Bainov, P.S. Simeonov [106], B.G. Pachpatte [[144]-[146]], M.H. Shih, C.C.Yeh [169], V.M. Singare, B.G. Pachpatte [170], D.R. Snow [171], [172], E. Thandapani, R.P. Agarwal [176], W. Walter [180], D. Willet, J.S.W. Wong [183], C.C. Yeh [184], [185], [186], C.C. Yeh, M.H. Shih [187], E.C. Young [188], A.I. Zahariev, D.D. Bainov [189] for continuous and discrete generalization of the Gronwall-Bellman-Bihari type inequalities and the results by N.V.Azbelev, Z.B.Tsalyuk [8], L.F.Rakhmatullina [155] for investigating the conditions of the solvability Chaplygin s problem for the integral equation (inequality) of Volterra type (one-dimensional, continuous case), we con- 5
6 About Some New Integral Inequalities Wendroff... sider new integral inequalities for discontinuous functions of Wendroff type. From our results we also deduce new generalizations of previous results by S.D.Borysenko [33], [39], [43], by D.Borysenko [44], by A.M. Samoilenko, S.D. Borysenko [159], [160] for integro-sum inequalities. It is useful to note that in the earlier articles [33], [36], [159], integro-sum inequalities for the piecewise-continuous functions of such type: Z t ϕ(t) ψ(t)+ K(t, s, ϕ(s))ds + X μ(t, t i )σ k (ϕ(t i 0)), <t i<t were investigated; here ϕ(t),ψ(t),μ(t, t i ),σ i (u) are continuous nonnegative functions (i =1, 2,...) fort 0, excepted for ϕ(t), which has the first kind of discontinuities in the points {t k },k= 1, :0 <t 1 <, lim i t i =. The kernel K(t, s, u), which is nonnegative at t s, is determined in domain t s, u k = const > 0, and at fixed t and s it is nondecreasing with respect to u; the functions σ k (u) are continuous nonnegative and non-decreasing with respect to u. In the work [39] the first author obtained such estimate: ϕ(t) ξ ψ (t), t [0, + ], where ξ ψ (t) is some solution of the integro-sum equation: Z t ξ(t) =ψ(t)+ K(t, s, ξ(s))ds + X μ(t, t i )σ i (ξ(t i 0)) <t i<t continuous in each of the intervals [t i,t i+1 [,i=0, 1,... and has some first kind of discontinuities in the points {t i }. From the result [39], the results in [33], [165] follows for integro-sum inequalities. In all the previously described results for integro-sum inequalities before the results in [44] (see [33], [36], [40],[41], [51], [52], [56], [83], [77], [161], [163], [165]) were considered Lipschitz type of nonlinearities for σ k (u). In the works [39], [43], [164] were obtained generalization integral inequalities Bellman-Bihari type for functions of two independent variables with jumps (finite) in some fixed points from an open domain Ω R 2 +. In this Chapter, similarly to the results in [44] for one-dimensional case, we investigate the integral inequalities for the functions of two independent variables (Wendroff type) with non-lipschitz type of functions, which characterize the values of discontinuity (the results [39], [43], [164] follow automatically as particular cases of the results of this article). In the Section 2, we obtain new analogy of the results by Bellman and Bihari for discontinuous functions (2-dimensional case). In Section 3, we generalize the results of section 2 for integro-sum functional inequalities with delay.
Integral inequalities... 7 1.2 Integral inequalities for discontinuous functions with discontinuities non Lipschitz type Theorem 1.2.1 Let a nonnegative function ϕ(t, x), determined in the domain Ω =[ k,j 1 Ω kj = {(t, x) :t [t k 1,t k [,x [x k 1,x k [}, k =1, 2,..., j =1, 2,...] be continuous in Ω, withtheexceptionofthepoints{t i,x i } where there are finite jumps: ϕ(t i 0,x i 0) 6= ϕ(t i +0,x i +0), i =1, 2,... and satisfying in Ω such an integro-sum inequality: ϕ(t, x) a(t, x)+ b(ξ,η)ϕ(ξ,η)dξdη + X + γ i ϕ m (t i 0,x i 0), (2.1) (,)<(t i,x i)<(t,x) with m>0, 0, 0, where a(t, x) > 0, (t, x) Ω and nondecreasing with respect to (t, x): p P, q Q = a(p, q) a(p, Q), (p, q) Ω, (P, Q) Ω; γ i =const 0, i N, b 0 and satisfying such a condition: b(ξ,η) =0, if (ξ,η) Ω ij,i6= j for arbitraries i, j =1, 2,... Here (t k,x k ) < (t k+1,t k+1 ), if t k < t k+1,x k < x k+1, k =0, 1, 2,... and lim t i =, i lim x i =. i
8 About Some new Integral Inequalities Wendroff... Then the function ϕ(t, x) satisfies such estimates: ϕ(t, x) a(t, x) Q (t [1 + γ 0,)<(t i,x i)<(t,x) ia m 1 (t i,x i )] exp[ R t R x b(ξ,η)dξdη], if 0 <m 1; (2.2) ϕ(t, x) a(t, x) Q (t [1 + γ 0,)<(t i,x i)<(t,x) ia m 1 (t i,x i )] exp[m R t R x b(ξ,η)dξdη], if m 1. Remark 1.2.1 If γ i =0,a(t, x) =c = const > 0, from the result of the theorem 1.2.1 the classical result by Wendroff follows (see [12]). If m =1, in a particular case we obtain the result [39]. For one-dimensional case (ϕ(t, x) = ϕ(t)) the result of the theorem coincides with the result which was presented in [44]; if also a(t, x) =a(t), b(u, v) =0, the result of the theorem coincides with the discrete analogous case (for m =1, Corollary 4.12, p.183 [3], similar theorem 4.21, p.190, if m 6= 1). Proof. Denote W (t, x) = ϕ(t,x) a(t,x),w(, )=1.From the inequality (2.1) it follows W (t, x) 1+ + b(ξ,η)w (ξ,η)dξdη + X γ i a m 1 (t i,x i )[W (t i 0,x i 0)] m. (2.3) (,)<(t i,x i)<(t,x) Consider the domain Ω 11 = {(t, x) :t [,t 1 [,x [,x 1 [}. The inequality (2.3) is reduced in such a form W (t, x) 1+ W (t, x) exp[ b(ξ,η)w (ξ,η)dξdη = b(ξ,η)dξdη], (t, x) Ω 11. (2.4) Let us assume that (t, x) Ω 22 = {(t, x) :t [t 1,t 2 [,x [x 1,x 2 [}, then
Integral inequalities... 9 Z x1 Z t1 W (t, x) 1+γ 1 a m 1 (t 1,x 1 )exp[m b(ξ,η)dξdη]+ Z t1 Z x1 Z ξ Z η + b(ξ,η)exp[ b(s, t)dsdt]dξdη + + t1 x 1 b(ξ,η)w(ξ,η)dξdη Z t1 Z x1 = γ 1 a m 1 (t 1,x 1 )exp[m Z t1 Z x1 +exp[ b(s, t)dsdt]+ b(ξ,η)dξdη]+ t 1 x 1 b(ξ,η)w(ξ,η)dξdη; W (t, x) (1 + γ 1 a m 1 (t 1,x 1 )) exp[ R t 1 R x1 b(t, s)dtds]+ + R t R x t 1 x 1 b(ξ,η)w(ξ,η)dξdη, if 0 <m 1 R x1 (1 + γ 1 a m 1 (t i,x i )) exp[m R t 1 b(t, s)dtds]+ + R t R x t 1 x 1 b(ξ,η)w(ξ,η)dξdη, if m 1 W (t, x) ( (1 + γ1 a m 1 (t 1,x 1 )) exp[ R t R x b(τ,s)dτds], if 0 <m 1; (1 + γ 1 a m 1 (t 1,x 1 )) exp[m R t R x b(τ,s)dτds], if m 1; (t, x) Ω 22. Supposing, that (2.2) is justified in the domain Ω kk. Then for (t, x) Ω k+1,k+1 the following inequality takes place W (t, x) kx i=1 exp[ i 1 Y γ i a m 1 (t i,x i ) (1 + γ j a m 1 (t i,x i )) Z ti Z xi Z ξ Z η j=1 b(ξ,η)dξdη]+ kx exp[ b(σ, v)dσdv]dξdη + ky Z tk (1 + γ j a m 1 (t i,x i )) exp[ j=1 Z ti t i 1 Z xi i=1 t k Z xk x i 1 i 1 Y (1 + γ j a m 1 (t i,x i )) j=1 x k b(ξ,η)dξdη b(ξ,η)dξdη]+ t k x k b(ξ,η)dξdη, if 0 <m 1;
10 About Some new Integral Inequalities Wendroff... W (t, x) ky (1 + γ j a m 1 (t i,x i )) exp[m j=1 Z tk Z xk b(ξ,η)dξdη]+ + b(ξ,η)dξdη, t k x k if m 1. Then ( Q k j=1 W (t, x) (1 + γ ja m 1 (t i,x i )) exp[ R t R x b(ξ,η)dξdη], if 0 <m 1; Q k j=1 (1 + γ ja m 1 (t i,x i )) exp[m R t R x b(ξ,η)dξdη], if m 1 (t, x) Ω k+1,k+1. From the last inequalities by taking into account that W = ϕ a the estimates (2.2) follow in all the domain Ω. Theorem 1.2.2 Let a nonnegative function ϕ(t, z), determined in domain Ω, be continuous in Ω except in the points {t i,x i } points of finite jumps and satisfy the following integro-sum inequality ϕ(t, x) a(t, x)+ + X (,)<(t i,x i)<(t,x) b(ξ,η)ϕ m (ξ,η) dξdη + γ i ϕ m (t i 0,x i 0), (2.5) m>0, m6= 1, where a, b, γ i satisfy the conditions of the theorem 1.2.1. Then for (t, x) Ω, the function ϕ(t, x) satisfies the inequalities: ϕ(t, x) a(t, x) if 0 <m<1; ϕ(t, x) a(t, x) Y (, )<(t i,x i )<(t,x) (1 + γ i a m 1 (t i,x i )) [1 + (1 m) a m 1 (ξ,η)b(ξ,η)dξdη] 1/1 m, (2.6) Y (, )<(t i,x i )<(t,x) 1 (m 1) (1 + mγ i a m 1 (t i,x i )) Y (, )<(t i,x i )<(t,x) a m 1 (ξ,η)b(ξ,η)dξdη (1 + mγ i a m 1 (t i,x i )) ¾ 1 m 1 m 1, (2.7)
Integral inequalities... 11 if m>1, for an arbitrary (t, x) Ω : Y (,)<(t i,x i)<(t,x) a m 1 (ξ,η)b(ξ,η)dξdη 1 m, (1 + mγ i a m 1 (t i,x i )) < (1 + 1 m 1 )1/m 1. (2.8) Proof. As in the theorem 1.2.1 the following inequality can be written W (t, x) 1+ b(ξ,η)a m 1 (ξ,η)w m (ξ,η)dξdη + x X 0 γ i a m 1 (t i,x i )W m (t i 0,x i 0). (2.9) (,)<(t i,x i)<(t,x) By using the generalization of the result by Bihari for a continuous function of two independent variables, in domain Ω 11, such an inequality for the function W (t, x) takes place: W (t, x) [1 + (1 m) if only 0 <m<1, (t, x) Ω 11 ; b(ξ,η)a m 1 (ξ,η)dξdη] 1/1 m, W(t, x) [1 (m 1) b(ξ,η)a m 1 (ξ,η)dξdη] 1 m 1, if only m>1, (t, x) Ω 1 satisfying the condition: b(ξ,η)a m 1 (ξ,η)dξdη < 1 m 1. (2.10) Taking into account (2.8) the inequality (2.10) takes place (t, x) Ω 11. By considering (t, x) Ω 22 and using the relation (2.5), (2.8), (2.9) it is possible to write such an inequality: Z x1 Z t1 W (t, x) γ 1 a m 1 (t 1,x 1 )[1 + (1 m) b(ξ,η)a m 1 (ξ,η)dξdη] m/1 m + Z t1 Z x1 [1 + (1 m) b(ξ,η)a m 1 (ξ,η)dξdη] 1/1 m + t 1 x 1 b(ξ,η)a m 1 (ξ,η)w m (ξ,η)dξdη. (2.11)