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Abstract and Applied Analysis
VolumeΒ 2011Β (2011), Article IDΒ 571795, 7 pages
http://dx.doi.org/10.1155/2011/571795
Research Article

Bounds of Solutions of Integrodifferential Equations

Department of Mathematics, Faculty of Electrical Engineering and Communication, TechnickΓ‘ 8, Brno University of Technology, 61600 Brno, Czech Republic

Received 20 January 2011; Accepted 24 February 2011

Academic Editor: Miroslava RůžičkovÑ

Copyright Β© 2011 ZdenΔ›k Ε marda. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Some new integral inequalities are given, and bounds of solutions of the following integro-differential equation are determined: ∫π‘₯β€²(𝑑)βˆ’β„±(𝑑,π‘₯(𝑑),𝑑0π‘˜(𝑑,𝑠,π‘₯(𝑑),π‘₯(𝑠))𝑑𝑠)=β„Ž(𝑑), π‘₯(0)=π‘₯0, where β„ŽβˆΆπ‘…+→𝑅, π‘˜βˆΆπ‘…2+×𝑅2→𝑅, β„±βˆΆπ‘…+×𝑅2→𝑅 are continuous functions, 𝑅+=[0,∞).

1. Introduction

Ou Yang [1] established and applied the following useful nonlinear integral inequality.

Theorem 1.1. Let 𝑒 and β„Ž be nonnegative and continuous functions defined on 𝑅+ and let 𝑐β‰₯0 be a constant. Then, the nonlinear integral inequality 𝑒2(𝑑)≀𝑐2ξ€œ+2𝑑0β„Ž(𝑠)𝑒(𝑠)𝑑𝑠,π‘‘βˆˆπ‘…+(1.1) implies ξ€œπ‘’(𝑑)≀𝑐+𝑑0β„Ž(𝑠)𝑑𝑠,π‘‘βˆˆπ‘…+.(1.2)

This result has been frequently used by authors to obtain global existence, uniqueness, boundedness, and stability of solutions of various nonlinear integral, differential, and integrodifferential equations. On the other hand, Theorem 1.1 has also been extended and generalized by many authors; see, for example, [2–19]. Like Gronwall-type inequalities, Theorem 1.1 is also used to obtain a priori bounds to unknown functions. Therefore, integral inequalities of this type are usually known as Gronwall-Ou Yang type inequalities.

In the last few years there have been a number of papers written on the discrete inequalities of Gronwall inequality and its nonlinear version to the Bihari type, see [13, 16, 20]. Some applications discrete versions of integral inequalities are given in papers [21–23].

Pachpatte [11, 12, 14–16] and Salem [24] have given some new integral inequalities of the Gronwall-Ou Yang type involving functions and their derivatives. Lipovan [7] used the modified Gronwall-Ou Yang inequality with logarithmic factor in the integrand to the study of wave equation with logarithmic nonlinearity. Engler [5] used a slight variant of the Haraux's inequality for determination of global regular solutions of the dynamic antiplane shear problem in nonlinear viscoelasticity. Dragomir [3] applied his inequality to the stability, boundedness, and asymptotic behaviour of solutions of nonlinear Volterra integral equations.

In this paper, we present new integral inequalities which come out from above-mentioned inequalities and extend Pachpatte's results (see [11, 16]) especially. Obtained results are applied to certain classes of integrodifferential equations.

2. Integral Inequalities

Lemma 2.1. Let 𝑒, 𝑓, and 𝑔 be nonnegative continuous functions defined on 𝑅+. If the inequality 𝑒(𝑑)≀𝑒0+ξ€œπ‘‘0ξ‚΅ξ€œπ‘“(𝑠)𝑒(𝑠)+𝑠0𝑔(𝜏)(𝑒(𝑠)+𝑒(𝜏))π‘‘πœπ‘‘π‘ (2.1) holds where 𝑒0 is a nonnegative constant, π‘‘βˆˆπ‘…+, then 𝑒(𝑑)≀𝑒0ξ‚Έξ€œ1+𝑑0ξ‚΅ξ€œπ‘“(𝑠)exp𝑠0ξ‚΅ξ‚΅ξ€œ2𝑔(𝜏)+𝑓(𝜏)1+𝜏0𝑔(𝜎)π‘‘πœŽξ‚Άξ‚Άπ‘‘πœπ‘‘s(2.2) for π‘‘βˆˆπ‘…+.

Proof. Define a function 𝑣(𝑑) by the right-hand side of (2.1) 𝑣(𝑑)=𝑒0+ξ€œπ‘‘0ξ‚΅ξ€œπ‘“(𝑠)𝑒(𝑠)+𝑠0𝑔(𝜏)(𝑒(𝑠)+𝑒(𝜏))π‘‘πœπ‘‘π‘ .(2.3) Then, 𝑣(0)=𝑒0, 𝑒(𝑑)≀𝑣(𝑑) and π‘£ξ…ž(ξ€œπ‘‘)=𝑓(𝑑)𝑒(𝑑)+𝑓(𝑑)𝑑0ξ€œπ‘”(𝑠)(𝑒(𝑑)+𝑒(𝑠))𝑑𝑠≀𝑓(𝑑)𝑣(𝑑)+𝑓(𝑑)𝑑0𝑔(𝑠)(𝑣(𝑑)+𝑣(𝑠))𝑑𝑠.(2.4) Define a function π‘š(𝑑) by ξ€œπ‘š(𝑑)=𝑣(𝑑)+𝑑0ξ€œπ‘”(𝑠)𝑣(𝑠)𝑑𝑠+𝑣(𝑑)𝑑0𝑔(𝑠)𝑑𝑠,(2.5) then π‘š(0)=𝑣(0)=𝑒0, 𝑣(t)β‰€π‘š(𝑑), π‘£ξ…žπ‘š(𝑑)≀𝑓(𝑑)π‘š(𝑑),(2.6)ξ…ž(𝑑)=2𝑔(𝑑)𝑣(𝑑)+π‘£ξ…žξ‚΅ξ€œ(𝑑)1+𝑑0ξ‚Άξ‚Έξ‚΅ξ€œπ‘”(𝑠)π‘‘π‘ β‰€π‘š(𝑑)2𝑔(𝑑)+𝑓(𝑑)1+𝑑0.𝑔(𝑠)𝑑𝑠(2.7) Integrating (2.7) from 0 to 𝑑, we have π‘š(𝑑)≀𝑒0ξ‚΅ξ€œexp𝑑0ξ‚΅ξ‚΅ξ€œ2𝑔(𝑠)+𝑓(𝑠)1+𝑠0𝑔(𝜎)π‘‘πœŽξ‚Άξ‚Άπ‘‘π‘ .(2.8) Using (2.8) in (2.6), we obtain π‘£ξ…ž(𝑑)≀𝑒0ξ‚΅ξ€œπ‘“(𝑑)exp𝑑0ξ‚΅ξ‚΅ξ€œ2𝑔(𝑠)+𝑓(𝑠)1+𝑠0𝑔(𝜎)π‘‘πœŽξ‚Άξ‚Άπ‘‘π‘ .(2.9) Integrating from 0 to 𝑑 and using 𝑒(𝑑)≀𝑣(𝑑), we get inequality (2.2). The proof is complete.

Lemma 2.2. Let 𝑒, 𝑓, and 𝑔 be nonnegative continuous functions defined on 𝑅+, 𝑀(𝑑) be a positive nondecreasing continuous function defined on 𝑅+. If the inequality ξ€œπ‘’(𝑑)≀𝑀(𝑑)+𝑑0ξ‚΅ξ€œπ‘“(𝑠)𝑒(𝑠)+𝑠0𝑔(𝜏)(𝑒(𝑠)+𝑒(𝜏))π‘‘πœπ‘‘π‘ ,(2.10) holds, where 𝑒0 is a nonnegative constant, π‘‘βˆˆπ‘…+, then ξ‚Έξ€œπ‘’(𝑑)≀𝑀(𝑑)1+𝑑0ξ‚΅ξ€œπ‘“(𝑠)exp𝑠0ξ‚΅ξ‚΅ξ€œ2𝑔(𝜏)+𝑓(𝜏)1+𝜏0𝑔(𝜎)π‘‘πœŽξ‚Άξ‚Άπ‘‘πœπ‘‘π‘ ,(2.11) where π‘‘βˆˆπ‘…+.

Proof. Since the function 𝑀(𝑑) is positive and nondecreasing, we obtain from (2.10) 𝑒(𝑑)ξ€œπ‘€(𝑑)≀1+𝑑0𝑓(𝑠)𝑒(𝑠)+ξ€œπ‘€(𝑠)𝑠0𝑔(𝜏)𝑒(𝑠)+𝑀(𝑠)𝑒(𝜏)𝑀(𝜏)π‘‘πœπ‘‘π‘ .(2.12) Applying Lemma 2.1 to inequality (2.12), we obtain desired inequality (2.11).

Lemma 2.3. Let 𝑒, 𝑓, 𝑔, and β„Ž be nonnegative continuous functions defined on 𝑅+, and let 𝑐 be a nonnegative constant.
If the inequality 𝑒2(𝑑)≀𝑐2ξ‚Έξ€œ+2𝑑0ξ‚΅ξ€œπ‘“(𝑠)𝑒(𝑠)𝑒(𝑠)+𝑠0𝑔(𝜏)(𝑒(𝜏)+𝑒(𝑠))π‘‘πœ+β„Ž(𝑠)𝑒(𝑠)𝑑𝑠(2.13) holds for π‘‘βˆˆπ‘…+, then ξ‚Έξ€œπ‘’(𝑑)≀𝑝(𝑑)1+𝑑0ξ‚΅ξ€œπ‘“(𝑠)exp𝑠0ξ‚΅ξ‚΅ξ€œ2𝑔(𝜏)+𝑓(𝜏)1+𝜏0ξ‚Άξ‚Ή,𝑔(𝜎)π‘‘πœŽξ‚Άξ‚Άπ‘‘πœπ‘‘π‘ (2.14) where ξ€œπ‘(𝑑)=𝑐+𝑑0β„Ž(𝑠)𝑑𝑠.(2.15)

Proof. Define a function 𝑧(𝑑) by the right-hand side of (2.13) 𝑧(𝑑)=𝑐2ξ‚Έξ€œ+2𝑑0ξ‚΅ξ€œπ‘“(𝑠)𝑒(𝑠)𝑒(𝑠)+𝑠0𝑔(𝜏)(𝑒(𝜏)+𝑒(𝑠))π‘‘πœ+β„Ž(𝑠)𝑒(𝑠)𝑑𝑠.(2.16) Then 𝑧(0)=𝑐2, βˆšπ‘’(𝑑)≀𝑧(𝑑) and π‘§ξ…ž(ξ‚Έξ‚΅ξ€œπ‘‘)=2𝑓(𝑑)𝑒(𝑑)𝑒(𝑑)+𝑑0ξ‚Άξ‚Ήβˆšπ‘”(𝑠)(𝑒(𝑑)+𝑒(𝑠))𝑑𝑠+β„Ž(𝑑)𝑒(𝑑)≀2ξ‚Έξ‚΅βˆšπ‘§(𝑑)𝑓(𝑑)ξ€œπ‘§(𝑑)+𝑑0ξ‚€βˆšπ‘”(𝑠)βˆšπ‘§(𝑑)+.𝑧(𝑠)𝑑𝑠+β„Ž(𝑑)(2.17) Differentiating βˆšπ‘§(𝑑) and using (2.17), we get π‘‘ξ‚€βˆšπ‘‘π‘‘ξ‚=𝑧𝑧(𝑑)ξ…ž(𝑑)2βˆšξ‚΅βˆšπ‘§(𝑑)≀𝑓(𝑑)π‘§ξ€œ(𝑑)+𝑑0π‘”ξ‚€βˆš(𝑠)π‘§βˆš(𝑑)+𝑧(𝑠)𝑑𝑠+β„Ž(𝑑).(2.18) Integrating inequality (2.18) from 0 to 𝑑, we have βˆšξ€œπ‘§(𝑑)≀𝑝(𝑑)+𝑑0ξ‚΅βˆšπ‘“(𝑠)ξ€œπ‘§(𝑠)+𝑠0ξ‚€βˆšπ‘”(𝜏)βˆšπ‘§(𝑠)+𝑧(𝜏)π‘‘πœπ‘‘π‘ ,(2.19) where 𝑝(𝑑) is defined by (2.15), 𝑝(𝑑) is positive and nondecreasing for π‘‘βˆˆπ‘…+. Now, applying Lemma 2.2 to inequality (2.19), we get βˆšξ‚Έξ€œπ‘§(𝑑)≀𝑝(𝑑)1+𝑑0ξ‚΅ξ€œπ‘“(𝑠)exp𝑠0ξ‚΅ξ‚΅ξ€œ2𝑔(𝜏)+𝑓(𝜏)1+𝜏0𝑔(𝜎)π‘‘πœŽξ‚Άξ‚Άπ‘‘πœπ‘‘π‘ .(2.20) Using (2.20) and the fact that βˆšπ‘’(𝑑)≀𝑧(𝑑), we obtain desired inequality (2.14).

3. Application of Integral Inequalities

Consider the following initial value problemπ‘₯ξ…ž(ξ‚΅ξ€œπ‘‘)βˆ’β„±π‘‘,π‘₯(𝑑),𝑑0ξ‚Άπ‘˜(𝑑,𝑠,π‘₯(𝑑),π‘₯(𝑠))𝑑𝑠=β„Ž(𝑑),π‘₯(0)=π‘₯0,(3.1) where β„ŽβˆΆπ‘…+→𝑅, π‘˜βˆΆπ‘…2+×𝑅2→𝑅, β„±βˆΆπ‘…+×𝑅2→𝑅 are continuous functions. We assume that a solution π‘₯(𝑑) of (3.1) exists on 𝑅+.

Theorem 3.1. Suppose that ||π‘˜ξ€·π‘‘,𝑠,𝑒1,𝑒2ξ€Έ||ξ€·||𝑒≀𝑓(𝑑)𝑔(𝑠)1||+||𝑒2||ξ€Έξ€·for𝑑,𝑠,𝑒1,𝑒2ξ€Έβˆˆπ‘…2+×𝑅2,||ℱ𝑑,𝑒1,𝑣1ξ€Έ||||𝑒≀𝑓(𝑑)1||+||𝑣1||ξ€·for𝑑,𝑒1,𝑣1ξ€Έβˆˆπ‘…+×𝑅2,(3.2) where 𝑓, 𝑔 are nonnegative continuous functions defined on 𝑅+. Then, for the solution π‘₯(𝑑) of (3.1) the inequality ||||ξ‚Έξ€œπ‘₯(𝑑)β‰€π‘Ÿ(𝑑)1+𝑑0ξ‚΅ξ€œπ‘“(𝑠)exp𝑠0ξ‚΅ξ‚΅ξ€œ2𝑔(𝜏)+𝑓(𝜏)1+𝜏0ξ‚Άξ‚Ή,||π‘₯𝑔(𝜎)π‘‘πœŽξ‚Άξ‚Άπ‘‘πœπ‘‘π‘ π‘Ÿ(𝑑)=0||+ξ€œπ‘‘0||||β„Ž(𝑑)𝑑𝑑(3.3) holds on 𝑅+.

Proof. Multiplying both sides of (3.1) by π‘₯(𝑑) and integrating from 0 to 𝑑 we obtain π‘₯2(𝑑)=π‘₯20ξ€œ+2𝑑0ξ‚Έξ‚΅ξ€œπ‘₯(𝑠)ℱ𝑠,π‘₯(𝑠),𝑠0ξ‚Άξ‚Ήπ‘˜(𝑠,𝜏,π‘₯(𝑠),π‘₯(𝜏))π‘‘πœ+π‘₯(𝑠)β„Ž(𝑠)𝑑𝑠.(3.4) From (3.2) and (3.4), we get ||||π‘₯(𝑑)2≀||π‘₯0||2ξ€œ+2𝑑0ξ‚Έ||||Γ—ξ‚΅||||+ξ€œπ‘“(𝑠)π‘₯(𝑠)π‘₯(𝑠)𝑠0ξ€·||||+||||ξ€Έξ‚Ά+||||||||𝑔(𝜏)π‘₯(𝑠)π‘₯(𝜏)π‘‘πœβ„Ž(𝑠)π‘₯(𝑠)𝑑𝑠.(3.5) Using inequality (2.14) in Lemma 2.3, we have ||||ξ‚Έξ€œπ‘₯(𝑑)β‰€π‘Ÿ(𝑑)1+𝑑0ξ‚΅ξ€œπ‘“(𝑠)exp𝑠0ξ‚΅ξ‚΅ξ€œ2𝑔(𝜏)+𝑓(𝜏)1+𝜏0𝑔(𝜎)π‘‘πœŽξ‚Άξ‚Άπ‘‘πœπ‘‘π‘ ,(3.6) where ||π‘₯π‘Ÿ(𝑑)=0||+ξ€œπ‘‘0||||β„Ž(𝑑)𝑑𝑑,(3.7) which is the desired inequality (3.3).

Remark 3.2. It is obvious that inequality (3.3) gives the bound of the solution π‘₯(𝑑) of (3.1) in terms of the known functions.

Acknowledgment

This author was supported by the Council of Czech Government grant MSM 00216 30503 and MSM 00216 30529 and by the Grant FEKTS-11-2-921 of Faculty of Electrical Engineering and Communication.

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