Copyright MMIX ARACNE editrice S.r.l. via Raffaele Garofalo, 133 A/B Roma (06)

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1 Sergiy Borysenko Gerardo Iovane Differential Models: Dynamical Systems with Perturbations

2 Copyright MMIX ARACNE editrice S.r.l. via Raffaele Garofalo, 133 A/B Roma (06) ISBN 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

3 Contents 1 About Some New Integral Inequalities Wendroff Type for Discontinuous Functions Introduction Integral inequalities for discontinuous functions with discontinuities nonlipschitztype Inequalitieswithretardation Some New Integral Inequalities of Bellman-Bihari Type With Delay for Discontinuous Functions Introduction Integro-functional inequalities of Bellman-Bihari type Inequalities Containing Multiple Integrals About A Generalization of Bellman-Bihari-Type Inequalities for Discontinuous Functions and Their Applications Introduction Integro-suminequalities Previouslyresults Some new integro-functional inequalities Bellman-Bihari type Wendroff s Type Inequalities for Discontinuous Functions Lipschitztypediscontinuity Integral inequalities for discontinuous functions with discontinuitiesofnonlipschitztype Inequalitieswithretardation GeneraltheoremsofBiharitype Someapplications Investigations of the Properties Motion for Essential Nonlinear Systems Perturbed by Impulses on Some Hypersurfaces Introduction StagingProblems EstimatesSolutionSystemofVariation

4 2 4.4 Stability(Lipschitzcase) CaseA) CaseB) Attraction CaseA) CaseB) Integro-Sum Inequalities and Motion Stability of Systems With Impulse Perturbations Introduction Objectforinvestigations Bellman-Bihari-Rakhmatullina method for discontinuous functions satisfyingintegralinequalities Boundedness and stability conditions on nonlinear approximation Integro-sum representations and stability of motion About Estimates of Solutions for Nonlinear Hyperbolic Equations With Impulsive Perturbations on Some Hypersurfaces Introduction Objectforinvestigation Mainresults Global Attractor for Non-Autonomous Wave Equation Without Uniqueness of Solution Introduction Settingoftheproblem Elements of abstract theory of global attractors for multivalued nonautonomousdynamicalsystems Propertiesofsolutionsoftheproblem(2.1),(2.2) Themainresults Global Attractor for Impulsive Reaction-Diffusion Equation Introduction SettingoftheProblem Elements of abstract theory of global attractors of nonautonomous multi-valueddynamicalsystems The family of MSP, generated by the problem (2.1) (2.4) Themainresult Asymptotic Behavior of Impulsive Non-Autonomous Wave Equation Introduction Settingoftheproblem

5 3 9.3 The family of multi-valued semi-processes, generated by the problem(2.1),(3.1) Themainresult Analysis of the Properties of the Solutions of Impulsive Systems of Differential Equations Introduction BasicnotationsandPreviouslyresults Analysis of the properties of the solution system (A) Conclusions Analysis of the Properties of the Solutions of Impulsive Systems of Integro-Differential Equations Introduction Preliminaries and Fundamental Results MainResults Conclusions About New Integro - Sum Inequality Gronwall - Bellman Type And Its Application For the Problem of Boundedness Introduction A new analogy of Gronwall - Bellman type inequality Conditions of boundedness solutions On Some Integro - Sum Gronwall - Bellman - Bihari - Wendroff s Type Inequalities Introduction One-dimensional integro - sum functional inequalities Wendroff stypeinequalities Remark About the New Conditions of the Solvability of Chaplygin s Problem for Integro-Sum Inequalities Introduction Chaplygin s Problem for Integral Inequalities About Conditions of Solvability of the Chaplygin s Problem for PiecewiseContinuousFunctions The Problem of Reduction Functional Integral Inequalities for Function of Several Variables to One-dimensional Inequalities Conclusions

6 4 15 The Bihari s Result Generalization for the Discontinuous Functions of Two Variables Introduction Bihari s lemma for the function of two variables Mainresult Conclusions Investigations of the Properties Solutions Impulsive Differential Systems on Linear Approximation Introduction Preliminaryconsiderations Mainresults To the Question of Mixed Type System Simulation in the Tasks of Analysis and Control Introduction Tasksetting Conclusions Necessary Conditions for Control of Objects With Distributed Constants Introduction Tasksetting Task of optimal control for the objects with distributed constants, which are described by non-linear differential equations with particle derivatives of elliptic type Task of optimal control for the objects with distributed constants, which are described by non-linear differential equations with particlederivativesofparabolictype Task of optimal control for the objects with distributed constants, which are described by non-linear differential equations with particlederivativesofhyperbolictype Conclusion

7 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

8 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.

9 Integral inequalities Integral inequalities for discontinuous functions with discontinuities non Lipschitz type Theorem 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

10 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 If γ i =0,a(t, x) =c = const > 0, from the result of the theorem 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

11 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;

12 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 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 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)

13 Integral inequalities 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 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)