Bounds of Solutions of Integrodifferential Equations

Abstract

Some new integral inequalities are given, and bounds of solutions of the following integro-differential equation are determined: 𝑥′(𝑡)−ℱ(𝑡,𝑥(𝑡),∫^𝑡0𝑘(𝑡,𝑠,𝑥(𝑡),𝑥(𝑠))𝑑𝑠)=ℎ(𝑡), 𝑥(0)=𝑥₀, where ℎ∶𝑅₊→𝑅, 𝑘∶𝑅²⁺×𝑅²→𝑅, ℱ∶𝑅₊×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 𝑢 and ℎ be nonnegative and continuous functions defined on 𝑅₊ and let 𝑐≥0 be a constant. Then, the nonlinear integral inequality 𝑢²(𝑡)≤𝑐²+2^𝑡0ℎ(𝑠)𝑢(𝑠)𝑑𝑠,𝑡∈𝑅₊(1.1) implies 𝑢(𝑡)≤𝑐+^𝑡0ℎ(𝑠)𝑑𝑠,𝑡∈𝑅₊.(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 𝑢, 𝑓, and 𝑔 be nonnegative continuous functions defined on 𝑅₊. If the inequality 𝑢(𝑡)≤𝑢₀+^𝑡0𝑓(𝑠)𝑢(𝑠)+^𝑠0𝑔(𝜏)(𝑢(𝑠)+𝑢(𝜏))𝑑𝜏𝑑𝑠(2.1) holds where 𝑢₀ is a nonnegative constant, 𝑡∈𝑅₊, then 𝑢(𝑡)≤𝑢₀1+^𝑡0𝑓(𝑠)exp^𝑠02𝑔(𝜏)+𝑓(𝜏)1+^𝜏0𝑔(𝜎)𝑑𝜎𝑑𝜏𝑑s(2.2) for 𝑡∈𝑅₊.

Proof. Define a function 𝑣(𝑡) by the right-hand side of (2.1) 𝑣(𝑡)=𝑢₀+^𝑡0𝑓(𝑠)𝑢(𝑠)+^𝑠0𝑔(𝜏)(𝑢(𝑠)+𝑢(𝜏))𝑑𝜏𝑑𝑠.(2.3) Then, 𝑣(0)=𝑢₀, 𝑢(𝑡)≤𝑣(𝑡) and 𝑣^(𝑡)=≤𝑓𝑓((𝑡𝑡))𝑣𝑢((𝑡𝑡))++𝑓𝑓((𝑡𝑡))^𝑡0_𝑡0𝑔(𝑠)(𝑢(𝑡)+𝑢(𝑠))𝑑𝑠𝑔(𝑠)(𝑣(𝑡)+𝑣(𝑠))𝑑𝑠.(2.4) Define a function 𝑚(𝑡) by 𝑚(𝑡)=𝑣(𝑡)+^𝑡0𝑔(𝑠)𝑣(𝑠)𝑑𝑠+𝑣(𝑡)^𝑡0𝑔(𝑠)𝑑𝑠,(2.5) then 𝑚(0)=𝑣(0)=𝑢₀, 𝑣(t)≤𝑚(𝑡), 𝑚𝑣^(𝑡)≤𝑓(𝑡)𝑚(𝑡),(2.6)_(𝑡)=≤2𝑚𝑔((𝑡)𝑡)𝑣2(𝑔𝑡)(𝑡+)+𝑣^𝑓(𝑡()𝑡)11++^𝑡0𝑔_𝑡0(𝑠)𝑑𝑠𝑔(𝑠)𝑑𝑠.(2.7) Integrating (2.7) from 0 to 𝑡, we have 𝑚(𝑡)≤𝑢₀exp^𝑡02𝑔(𝑠)+𝑓(𝑠)1+^𝑠0𝑔(𝜎)𝑑𝜎𝑑𝑠.(2.8) Using (2.8) in (2.6), we obtain 𝑣^(𝑡)≤𝑢₀𝑓(𝑡)exp^𝑡02𝑔(𝑠)+𝑓(𝑠)1+^𝑠0𝑔(𝜎)𝑑𝜎𝑑𝑠.(2.9) Integrating from 0 to 𝑡 and using 𝑢(𝑡)≤𝑣(𝑡), we get inequality (2.2). The proof is complete.

Lemma 2.2. Let 𝑢, 𝑓, and 𝑔 be nonnegative continuous functions defined on 𝑅₊, 𝑤(𝑡) be a positive nondecreasing continuous function defined on 𝑅₊. If the inequality 𝑢(𝑡)≤𝑤(𝑡)+^𝑡0𝑓(𝑠)𝑢(𝑠)+^𝑠0𝑔(𝜏)(𝑢(𝑠)+𝑢(𝜏))𝑑𝜏𝑑𝑠,(2.10) holds, where 𝑢₀ is a nonnegative constant, 𝑡∈𝑅₊, then 𝑢(𝑡)≤𝑤(𝑡)1+^𝑡0𝑓(𝑠)exp^𝑠02𝑔(𝜏)+𝑓(𝜏)1+^𝜏0𝑔(𝜎)𝑑𝜎𝑑𝜏𝑑𝑠,(2.11) where 𝑡∈𝑅₊.

Proof. Since the function 𝑤(𝑡) is positive and nondecreasing, we obtain from (2.10) 𝑢(𝑡)𝑤(𝑡)≤1+^𝑡0𝑓(𝑠)𝑢(𝑠)𝑤(𝑠)+^𝑠0𝑔(𝜏)𝑢(𝑠)𝑤(𝑠)+𝑤𝑢((𝜏𝜏))𝑑𝜏𝑑𝑠.(2.12) Applying Lemma 2.1 to inequality (2.12), we obtain desired inequality (2.11).

Lemma 2.3. Let 𝑢, 𝑓, 𝑔, and ℎ be nonnegative continuous functions defined on 𝑅₊, and let 𝑐 be a nonnegative constant.
If the inequality 𝑢²(𝑡)≤𝑐²+2^𝑡0𝑓(𝑠)𝑢(𝑠)𝑢(𝑠)+^𝑠0𝑔(𝜏)(𝑢(𝜏)+𝑢(𝑠))𝑑𝜏+ℎ(𝑠)𝑢(𝑠)𝑑𝑠(2.13) holds for 𝑡∈𝑅₊, then 𝑢(𝑡)≤𝑝(𝑡)1+^𝑡0𝑓(𝑠)exp^𝑠02𝑔(𝜏)+𝑓(𝜏)1+^𝜏0𝑔(𝜎)𝑑𝜎𝑑𝜏𝑑𝑠,(2.14) where 𝑝(𝑡)=𝑐+^𝑡0ℎ(𝑠)𝑑𝑠.(2.15)

Proof. Define a function 𝑧(𝑡) by the right-hand side of (2.13) 𝑧(𝑡)=𝑐²+2^𝑡0𝑓(𝑠)𝑢(𝑠)𝑢(𝑠)+^𝑠0𝑔(𝜏)(𝑢(𝜏)+𝑢(𝑠))𝑑𝜏+ℎ(𝑠)𝑢(𝑠)𝑑𝑠.(2.16) Then 𝑧(0)=𝑐², 𝑢(𝑡)≤√𝑧(𝑡) and 𝑧^(𝑡)=≤22√𝑓(𝑡)𝑢(𝑡)𝑢(𝑡)+^𝑡0𝑔(𝑠)(𝑢(𝑡)+𝑢(𝑠))𝑑𝑠+ℎ(𝑡)𝑢(𝑡)𝑧(𝑡)𝑓(𝑡)√𝑧(𝑡)+^𝑡0𝑔(𝑠)√𝑧(𝑡)+√𝑧(𝑠)𝑑𝑠+ℎ(𝑡).(2.17) Differentiating √𝑧(𝑡) and using (2.17), we get 𝑑𝑑𝑡√𝑧(𝑡)=𝑧^(𝑡)≤𝑓2√(𝑡)𝑧(𝑡√)𝑧(𝑡)+^𝑡0𝑔(𝑠)√𝑧(𝑡)+√𝑧(𝑠)𝑑𝑠+ℎ(𝑡).(2.18) Integrating inequality (2.18) from 0 to 𝑡, we have √𝑧(𝑡)≤𝑝(𝑡)+^𝑡0𝑓(𝑠)√𝑧(𝑠)+^𝑠0𝑔(𝜏)√𝑧(𝑠)+√𝑧(𝜏)𝑑𝜏𝑑𝑠,(2.19) where 𝑝(𝑡) is defined by (2.15), 𝑝(𝑡) is positive and nondecreasing for 𝑡∈𝑅₊. Now, applying Lemma 2.2 to inequality (2.19), we get √𝑧(𝑡)≤𝑝(𝑡)1+^𝑡0𝑓(𝑠)exp^𝑠02𝑔(𝜏)+𝑓(𝜏)1+^𝜏0𝑔(𝜎)𝑑𝜎𝑑𝜏𝑑𝑠.(2.20) Using (2.20) and the fact that 𝑢(𝑡)≤√𝑧(𝑡), we obtain desired inequality (2.14).

3. Application of Integral Inequalities

Consider the following initial value problem𝑥^(𝑡)−ℱ𝑡,𝑥(𝑡),^𝑡0𝑘(𝑡,𝑠,𝑥(𝑡),𝑥(𝑠))𝑑𝑠=ℎ(𝑡),𝑥(0)=𝑥₀,(3.1) where ℎ∶𝑅₊→𝑅, 𝑘∶𝑅²⁺×𝑅²→𝑅, ℱ∶𝑅₊×𝑅²→𝑅 are continuous functions. We assume that a solution 𝑥(𝑡) of (3.1) exists on 𝑅₊.

Theorem 3.1. Suppose that ||𝑘𝑡,||𝑠ℱ,𝑢₁,𝑡,𝑢𝑢₂₁||≤𝑓(𝑡)𝑔(𝑠)||𝑢₁||+||𝑢₂||for𝑡,𝑠,𝑢₁,𝑢₂∈𝑅²⁺×𝑅²,,𝑣₁||≤𝑓(𝑡)||𝑢₁||+||𝑣₁||for𝑡,𝑢₁,𝑣₁∈𝑅₊×𝑅²,(3.2) where 𝑓, 𝑔 are nonnegative continuous functions defined on 𝑅₊. Then, for the solution 𝑥(𝑡) of (3.1) the inequality ||𝑥(𝑡)||≤𝑟(𝑡)1+^𝑡0𝑓(𝑠)exp^𝑠02𝑔(𝜏)+𝑓(𝜏)1+^𝜏0𝑟(𝑡)=||𝑥₀𝑔(𝜎)𝑑𝜎𝑑𝜏𝑑𝑠,||+^𝑡0||ℎ(𝑡)||𝑑𝑡(3.3) holds on 𝑅₊.

Proof. Multiplying both sides of (3.1) by 𝑥(𝑡) and integrating from 0 to 𝑡 we obtain 𝑥²(𝑡)=𝑥²⁰+2^𝑡0𝑥(𝑠)ℱ𝑠,𝑥(𝑠),^𝑠0𝑘(𝑠,𝜏,𝑥(𝑠),𝑥(𝜏))𝑑𝜏+𝑥(𝑠)ℎ(𝑠)𝑑𝑠.(3.4) From (3.2) and (3.4), we get ||𝑥(𝑡)||²≤||𝑥₀||²+2^𝑡0𝑓(𝑠)||𝑥(𝑠)||×||𝑥(𝑠)||+^𝑠0𝑔(𝜏)||𝑥(𝑠)||+||𝑥(𝜏)||𝑑𝜏+||ℎ(𝑠)||||𝑥(𝑠)||𝑑𝑠.(3.5) Using inequality (2.14) in Lemma 2.3, we have ||𝑥(𝑡)||≤𝑟(𝑡)1+^𝑡0𝑓(𝑠)exp^𝑠02𝑔(𝜏)+𝑓(𝜏)1+^𝜏0𝑔(𝜎)𝑑𝜎𝑑𝜏𝑑𝑠,(3.6) where 𝑟(𝑡)=||𝑥₀||+^𝑡0||ℎ(𝑡)||𝑑𝑡,(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 𝑥(𝑡) 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.