ASYMPTOTICS OF INTEGRODIFFERENTIAL MODELS WITH INTEGRABLE KERNELS II

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1 IJMMS 23:5, PII. S Hindawi Publishing Corp. ASYMPTOTICS OF INTEGRODIFFERENTIAL MODELS WITH INTEGRABLE KERNELS II ANGELINA M. BIJURA Received 28 January 23 Nonlinear singularly perturbed Volterra integrodifferential equations with weakly singular kernels are investigated using singular perturbation methods, the Mellin transform technique, and the theory of fractional integration. 2 Mathematics Subject Classification: 4A6, 45J5, 45E, 26A33.. Introduction. The following singularly perturbed nonlinear Volterra integrodifferential equation is considered: εy t) = gt)+ Γ β) kt,s) t s) h s,ys) ) ds, <t T, y) = y β..) Here, ε is a positive parameter satisfying <ε and<β<. The functions g and h and the Volterra kernel k are continuously differentiable. Moreover, kt,t) and 2 ht,ψt)) are assumed to be nonzero for all t T,fora continuously differentiable function ψ. These functions may as well depend regularly on ε but it is assumed here that they are independent. As for the case when ht,y) = y investigated in [6],.) exhibits an initial layer or a layer of rapid transition at t =. It will be shown that the order of magnitude of the initial layer thickness is Oε /2 β) ), ε. In this region, referred as the inner layer, the solution yt;ε) of.) changes rapidly while in the rest of the domain, called the outer region, the solution varies slowly. To guarantee the existence of decaying solutions in the initial layer, appropriate layer stability assumption will be imposed. It is known from the standard theory of Volterra integral equations that this problem has a continuous solution yt;ε), t T, for all ε>. However, if one is interested in the solution for small values of ε, perturbation methods have to be applied. The structure of the integrodifferential equation suggests a regular approximation of the form y app t;ε) ε n y n t), ε..2) n=

2 354 ANGELINA M. BIJURA Assuming that the regular approximation has indeed the structure given in.2), the leading order term y t) satisfies = gt)+ Γ β) kt,s) t s) β h s,y s) ) ds, t T..3) This is a Volterra integral equation of the first kind. For this equation to have a continuous solution, gt) cannot be merely continuous; the forcing function must be smoother than the desired solution. Even if.3) has a solution y t) in C[,T], it may not approximate yt;ε) uniformly for all t T as ε, especially if y ) lim ε y,ε). This situation confirms that problem.) is singularly perturbed, and therefore, a second term in.2) is needed to correct the nonuniformity. Thus, one introduces a new scaled variable with a different magnitude in order to obtain a uniformly valid approximation. The idea is that if the initial layer region is described in terms of the new time scale, the layer region is stretched in such a way that its neighborhood gets magnified unproportionally to the rest of the domain under consideration, and therefore, no rapid variation in the solution should be exhibited. This new variable is, therefore, called the inner variable or the stretched variable. The two widely used techniques that employ the inner and outer expansions are the method of matched asymptotic expansion and the additive decomposition method. In the investigation presented here, the additive decomposition method is applied. Assuming that the initial layer stability condition holds, it is shown here that yt;ε) converges uniformly to y t) as ε, for all t T. The additive decomposition method was first applied to study.) in[]. A variety of interesting examples have been solved in []. However, the analysis fails to reveal the general structure of the formal approximation as the case β = is considered simultaneously with the case < β <. Also, the validity of the given formal approximationin [] is not demonstrated. The linear singularly perturbed Volterra integral equations with weakly singular kernels have been studied in [4]. It is shown in [4] that the initial layer correction term can be written in terms of the Mittag-Leffler function. The Mittag-Leffler function decays algebraically at infinity. The results in [4] reveal that singularly perturbed Volterra integral equations with weakly singular kernels exhibit narrower initial layer regions compared with similar equations with continuous kernels and integrodifferential equations with weakly singular kernels. The linear version of.) has been investigated in [6] using the theory of fractional integration. It is demonstrated in [6] that the linear scalar singularly perturbed Volterra integrodifferential equation has a wider initial layer width, of order Oε /2 β) ), ε, and that the formal approximate solution is an asymptotic solution up to the order of magnitude of Oε /2 β) ), ε. The rescaling of the initial layer width and magnitude of the solution) in [2, 3, 4, 5, 6] reveals differences in the order of magnitudes of the initial layer thickness and magnitudes of the solution in the layer region for the cases β = and β. This, in fact, suggests

3 ASYMPTOTICS OF INTEGRODIFFERENTIAL MODELS different starting asymptotic approximations. This paper aims at extending the results in [6] for the nonlinear problem. The main, new, and interesting result is the proof of asymptotic correctness. The method of additive decomposition will be used to construct an approximate solution of.). This will be proceeded by the construction of approximations valid in the inner and outer regions. The inner layer approximation is assumed to be negligible in the outer region, and therefore, a decay in the initial layer is crucial. In the application of the additive decomposition method, exponential decay in the initial layer simplifies the analysis as transcendentally small terms can be omitted from the asymptotic expansions. However, in the case considered here, the inner layer solution decays algebraically and this challenges the technique. Also, there are abnormalities when balancing terms of similar orders of ε. This shows the difficulties in analyzing the cases β = and < β < simultaneously. The abnormality occurs especially when one tries to derive the first- and higher-) order terms in the formal approximate solution. Therefore, this paper will restrict attention to the leading order solution. In Section 2, a review of some known results which are applied later in the analysis is presented. In Section 3, the application of the additive decomposition technique to integrodifferential equations of type.) is described and the leading order formal solution is derived. In Section 4, assumptions imposed on the data are stated and the formal approximate solution is proved to have the required properties. It is also shown in Section 4 that if y t;ε) satisfies.) approximately with a residual ρt;ε), thenρt;ε) = Oε), ε. Finally, in Section 5, the theorem on asymptotic correctness is presented which says: under given conditions, if y t;ε) is a formal approximate solution of.) and yt;ε) is the exact solution, then yt;ε) y t;ε) =Oε /2 β) ), ε, uniformly for all t T and all sufficiently small values of ε. 2. Mathematical preliminaries. The following results will be applied in the presentation, and therefore, for convenience, are stated below. The first result is that of applying the Mellin transformation to the asymptotic evaluation of integrals. One will find a detailed discussion in [7, 7]. The Parseval formula for Mellin transforms will be applied in Section 4, and therefore, is particularly stated. The Mellin transform of a locally integrable function yt) on, ) is defined by M[y;z] = t z yt)dt 2.) when the integral converges.

4 356 ANGELINA M. BIJURA Theorem 2.. Suppose that M[f; z] and M[h;z] are defined and holomorphic, each in some vertical strip, whose boundary is determined by the analytical structure of the corresponding function as t and as t. Suppose further that these strips overlap. Then the integral It) = f s)hts)ds 2.2) can be described in terms of the Mellin-Barnes integral: It) = c+i M [ fs); z ] M [ hts);z ] dz, 2.3) 2πi c i where Rez) = c lies in the overlapping strip. The second result involves the application of fractional calculus to obtain solutions of integrodifferential equations with weakly singular kernels. Theorem 2.2. The Volterra integrodifferential equation φ t) = κ + κ t s) β φs)ds, <t, φ) = φ, 2.4) Γ β) where >β>, κ and κ are constants, has the solution φt) = φ E 2 β κ t 2 β) +κ E 2 β κ s 2 β) ds. 2.5) Here, E γ is the Mittag-Leffler function of order γ, defined by E γ λt γ ) λ n t nγ =, t >, γ>, λ C. 2.6) Γ nγ +) n= The third and last result is the nonlinear generalization of the Gronwall s inequality. This has been proved in [8]. Let φ,ψ : [,ζ) [, ), ϕ : [,ζ) [, ) [, ) be con- Theorem 2.3. tinuous such that ϕt,u) ϕt,v) Mt,u)u v), t [,ζ), v u, 2.7) where M is nonnegative and continuous on [,ζ) [, ). Then for every nonnegative continuous solution of the inequality yt) φt)+ψt) ϕ s,ys) ) ds, t [,ζ), 2.8)

5 ASYMPTOTICS OF INTEGRODIFFERENTIAL MODELS the following estimate holds: yt) φt)+ψt) ϕ s,φs) ) exp M σ,φσ) ) ) ψσ)dσ ds 2.9) s for all t [,ζ). 3. Heuristic analysis and formal solution. To start, one seeks an approximate solution y app t;ε) in the form ) t y app t;ε) = ut;ε)+v ε ;ε, α τ = t,α>, εα 3.) and requires that lim vτ;ε) =. 3.2) τ The inner layer function vt/ ;ε), which is the second term needed in.2), corrects the nonuniformity in the initial layer. It is assumed that ut; ε) and vτ; ε) have asymptotic expansions of the form ut;ε) ε n u n t), n= vτ;ε) ε nα v n τ), 3.3) n= as ε, so that y app t;ε) ε n u n t)+ n= Moreover, one requires that for all n, n= ) t ε nα v n, ε. 3.4) lim τ v nτ) =. 3.5) The substitution of 3.4) in.), the expression of all terms in terms of the inner variable τ, and the examination of the dominant balance in the relation yield Hence, one chooses Ord ε α) = Ord β)), ε. 3.6) α = 2 β. 3.7) This implies that the singularly perturbed equation.) possesses an initial layer width of order ε /2 β) ), ε, meaning that the solution yt;ε) of.) is slowly varying for Oε /2 β) ) t T as ε, but changes rapidly on a small interval t Oε /2 β) ). Therefore, the initial layer region for problem.) is thicker compared with similar equations with continuous kernels see [2, 3, 5]) and integral equations with weakly singular kernels see [4]).

6 358 ANGELINA M. BIJURA 3.. Derivation of the formal approximate solution. From this point, the attention is restricted to ) t y t;ε) = u t)+v 3.8) since y app t;ε) y t;ε), ε. During the derivation of the formal solution, the following will be assumed that v τ) c τ β, τ, 3.9) where c and β are constants, β >. Now, let ) t yt;ε) := y t;ε) = u t)+v. 3.) Suppose that y t;ε) satisfies.) approximately, with a residual ρt;ε), then ) t εu t)+ε α v = gt)+ Γ β) )) kt,s) s t s) h s,u β s)+v ds ρt;ε), 3.) or equivalently, ) t ρt;ε) = εu t) ε α v +gt)+ Γ β) + Γ β) kt,s) t s) β h s,u s) ) ds { )) kt,s) s h s,u t s) β s)+v h s,u s) )} ds. 3.2) To obtain the leading order outer equation, one considers 3.2) in the limit, as ε and t> fixed. However, the last integral in 3.2) can be written as t β I ε t) = Γ β) { kt,ts) h s) β For a fixed t>, one can rewrite I ε t) as ts,u ts)+v ts )) h ts,u ts) )} ds. 3.3) I ε θ;t) = β) θ β φs)ψsθ;t)ds, 3.4) where θ = t/, φs) = Γ β) s) β, s<,, s<, ψsθ;t) = kt,ts) { h ts,u ts)+v θs) ) h ts,u ts) )}. 3.5)

7 ASYMPTOTICS OF INTEGRODIFFERENTIAL MODELS By suppressing the dependence of ψ on t, the following lemma establishes that when t>isfixed,i ε t) = O ), ε. Lemma 3.. Consider I β θ) = φs)ψsθ)ds. 3.6) Now, suppose that M[φ; z] and M[ψ;z] are defined and holomorphic, each in some vertical strip determined by the asymptotic behaviors of φ and ψ. Then it follows that, as θ, I β θ) = O θ ). 3.7) Proof. The form of the integral in 3.6) suggests the application of Mellin transform technique in the determination of an asymptotic expansion of I β θ), θ. This is a well-known technique which involves asymptotic behaviors of functions in the integrand. One should note that Taylor theorem implies that v θ) and ψθ) are O-equivalent, ψτ) = Ord v τ) ), τ, τ. 3.8) It is known from 3.8) thatv τ) = O), τ, and it has been assumed in 3.9) thatv τ) = Oτ β ), τ.sincei β θ) is absolutely convergent, the analytical strips of M[φ; z] and M[ψ;z] overlap. Let Re{z} =c lay in the overlapping strip, then the Parseval formula 2.3) implies that I β θ) = c+i θ z M[φ; z]m[ψ;z]dz, 3.9) 2πi c i where the following identity has been used: M [ ψsθ);z ] = θ z M[ψ;z]. 3.2) One observes from 2.) that Γ z) M[φ;z] = Γ z + β), 3.2) and therefore, I β θ) = c+i θ z Γ z) M[ψ;z] dz. 3.22) 2πi c i Γ 2 β z) The asymptotic evaluation of 3.22), as θ, involves the asymptotic behavior of ψτ), τ, which in turn involves the asymptotic behavior of v τ), τ, the result assumed in 3.9). The asymptotic relation in 3.8) implies that ψθ) c θ β, θ,β >. 3.23)

8 36 ANGELINA M. BIJURA It can be shown that M[ψ;z] has a simple pole at z = β and the singular part of the Laurent expansion of M[ψ;z] about this point is given by c z β. 3.24) To compute the asymptotic behavior of I β, the vertical path is displaced to the right. In doing so, the pole implied by Γ z) at z = is encountered before that of M[ψ;z] since β >, and hence, it provides the leading order contribution. Computing the relevant residues, one finds that where I β θ) Γ β) M[ψ;]θ, θ, 3.25) M[ψ;] = ψs)ds. 3.26) Substituting this result into I ε θ;t) yields, for a fixed t>, I ε t) = O ), ε. 3.27) Then, if ρt;ε) = o) as ε, the leading order outer equation is obtained from 3.2) byfixingt> and letting ε tend to zero. This gives = gt)+ Γ β) kt,s) t s) β h s,u s) ) ds, t. 3.28) The equation governing the leading order inner layer solution follows by substituting 3.28) in3.2) and expressing all terms in terms of τ: ρ τ;ε ) +εu τ ) +ε α v τ) τ = εα β) k τ, σ ) { h,u )+v Γ β) τ σ) β σ ) ) h,u ) )} dσ τ + εα β) k τ, σ ) {h σ,u Γ β) τ σ) β σ ) +v σ ) ) h σ,u σ ))} This equation is equivalent to ρ τ;ε ) = εu τ ) ε α v τ) + εα β) Γ β) τ +Oε), ε. { h,u )+v σ ) ) h,u ) )}) dσ. 3.29) k,) τ σ) β { h,u )+v σ ) ) h,u ) )} dσ 3.3)

9 ASYMPTOTICS OF INTEGRODIFFERENTIAL MODELS Multiplying throughout by,fixingτ>, and letting ε tend to zero, one has by assuming ρ ε γ τ;ε) = o), asε, v k,) τ τ) = τ σ) β{ h,u )+v σ ) ) h,u ) )} dσ, Γ β) τ>, v ) = y u ). 3.3) If u t) satisfies 3.28) andv τ) obeys 3.3), it follows from 3.2) that ρt;ε) = εu t) Γ β) + Γ β) kt,s) t s) β k,) t s) β { h { )) s h,u )+v h,u ) )} ds s,u s)+v s )) h s,u s) )} ds. 3.32) 4. Properties of the formal solution. From this section on, it will be assumed that H g ) the function gt) C 2 [,T] and that g) = ; H h ) the nonlinear term ht,y) is at least twice continuously differentiable with respect to both t and y; H k ) kt,s) is a C 2 -function on s t T and that there exists a positive constant η such that kt,t) 2 h t,u t) ) η, t T, k,) 2 h,υ) η 4.) for all υ between u ) and y. Note that H k ) is the initial layer stability condition. It forces initial layer solutions to decay and also restricts generally the size of the initial layer jump, see Proposition 4.. This assumption may look restrictive, but it is satisfied by most physical models since it is the criterion for general stability. Examples of such models include problems in reactor dynamics see [2, 3, 4]) and reaction diffusion models, see, for example, [6]. It is shown in this section that there are unique solutions u t) and v τ) that solve 3.28) and3.3), respectively, and that they have the important properties assumed in their derivation. Equation 3.28) is a Volterra integral equation of the first kind for u t). If one puts the resulting equation = gt)+ Γ β) h t,u t) ) = pt), 4.2) kt,s) ps)ds, t s) β t, 4.3)

10 362 ANGELINA M. BIJURA is a linear Volterra equation of the first kind. A unique solution pt) C [,T] exists under given conditions H g ) and kt,t), see []. Then the existence and uniqueness of a continuous solution u t) of 4.2) follows from the implicit function theorem and the fact that 2 ht,u t)) H k ). The cases kt,s) = k a constant and kt,s) = kt s) would give the function pt) exactly. Numerical approximation of u t) from 4.2) should then be easy. Proposition 4.. Suppose that H g ), H h ), and H k )hold.then3.3) has a C -solution v satisfying v τ) y u ) Γ β ) τβ 2, τ. 4.4) Proof. Consider 3.3); the standard theory of Volterra integrodifferential equations see, e.g., [3, 5]) ensures the existence of v τ) C [, ). Moreover, for v ) sufficiently small, v τ), τ. Using the identity with ϕ) ϕ) = ϕ ξ)dξ 4.5) ϕξ) = h,u )+ξv τ) ), 4.6) one writes h,u )+v τ) ) h,u ) ) = 2 h,u )+ξv τ) ) dξv τ). 4.7) Thus, 3.3) becomes v τ) = τ τ σ) β k,) 2 h,u )+ξv σ ) ) dξv σ )dσ, Γ β) v ) = y u ). 4.8) Assumption H k ) implies that v τ) is nonincreasing if v ) > and nondecreasing if v )< and that u )+ξv τ) lies between u ) and u )+ v ). Integrating both sides of 4.8), one obtains v τ) v ) = Γ 2 β) τ τ σ) β k,) 2 h,u )+ξv σ ) ) dξv σ )dσ. 4.9)

11 ASYMPTOTICS OF INTEGRODIFFERENTIAL MODELS It follows that, when v )>, and when v )<, Thus, η v τ) v ) Γ 2 β) η v τ) v ) Γ 2 β) v τ) v ) η Γ 2 β) τ τ τ τ σ) β v σ )dσ 4.) τ σ) β v σ )dσ. 4.) τ σ) β v σ ) dσ, τ >. 4.2) In particular, v τ) v ), τ. It follows from comparison theorems and the application of the Laplace transform method on that φτ) = v ) η Γ 2 β) τ τ σ) β φσ )dσ, τ >, 4.3) v τ) y u ) E2 β ητ 2 β ), τ. 4.4) Here, E 2 β is the Mittag-Leffler function defined in 2.6). The asymptotic expansions of the Mittag-Leffler function established in [4, 6] imply that v τ) y u ) ) ) i+ τ β 2)i Γ ), τ. 4.5) 2 β)i This completes the proof. i= Proposition 4.2. Suppose that ρt; ε) satisfies 3.32), then there exists a positive constant c which does not depend on ε and ε, such that ρt;ε) c ε, ε, 4.6) for all t T and all <ε ε. Proof. Equation 3.32) can also be written as ρt;ε) = εu t) + t β Γ β) s) [ψ β )) ts t,ts,v ψ,,v ts ))] ds, t, 4.7)

12 364 ANGELINA M. BIJURA where ψ t,ts,v ts )) [ = kt,ts) h ts,u ts)+v ts )) h ts,u ts) )]. 4.8) Consider the integral on the right-hand side of It;ε) = s) [ψ β t,ts,v ts )) ψ,,v ts ))] ds. 4.9) At t =, I;ε) =. For any fixed t> and ε sufficiently small, Lemma 3. establishes that ψt,ts,v s/ )) and ψ,,v s/ )) assume the same asymptotic expansion. Therefore, for all t T and all sufficiently small values of ε, It;ε) =. It then follows that ρt;ε) c ε 4.2) uniformly for all t T and all <ε ε, where c is a constant which depends on u t) and T. 5. Proof of asymptotic correctness. In this section, it is proved that the formal approximation defined in 3.8) is, indeed, an asymptotic approximation to the solution yt;ε) of.). The method is to adopt the theory of [9] on developing a rigorous theory of singular perturbation. Define the remainder χt;ε) = yt;ε) y t;ε), t T. 5.) Substituting this in.) and using 3.) gives εχ t;ε) = ρt;ε)+ Γ β) kt,s) t s) β { h s,y s;ε)+χs;ε) ) h s,y s;ε) )} ds, <t T, χ;ε) =. 5.2) Applying Taylor s theorem, one has an equivalent perturbed linear equation εχ kt,s) t;ε) = Gχ)t;ε)+ Γ β) t s) 2h s,y β s;ε) ) χs;ε)ds, <t T, χ;ε) =, 5.3)

13 ASYMPTOTICS OF INTEGRODIFFERENTIAL MODELS where Gχ)t;ε) is a sum of a small term and a term independent of χ. Moreover, where Gχ)t;ε) = ρt;ε)+ Γ β) kt,s) t s) β ϕ s,χ,y ;ε ) ds, 5.4) ϕ t,χ,y ;ε ) = χt;ε) 2 µ) 22 h t,y t;ε)+µχt;ε) ) dµ. 5.5) Under given conditions on the data, 5.3) has a continuous solution χt;ε) on [,T], T>, for all ε>. The interest here is in the boundedness of χt;ε) for sufficiently small values of ε. Let γt,s; ε) denote the differential resolvent kernel for the kernel in 5.3). Then γt,s;ε) satisfies γt,t;ε) =, γt,s;ε) =, s>t,and ε γt,s;ε) = t Γ β) ε 2 γt,s;ε) = Γ β) kt,σ ) t σ) 2h σ,y β σ ;ε) ) γσ,s;ε)dσ, 5.6a) kσ,s) γt,σ;ε) σ s) 2h s,y β s;ε) ) dσ. 5.6b) s s Again, under given conditions, the existence of a continuous solution γt,s; ε) of 5.6) is guaranteed, see [, Chapter ]. To this point, one may use the variation-of-constants formula to write 5.3) as χt;ε) = γt,s;ε)gχ)s;ε)ds, t T. 5.7) ε To verify 5.7), left multiply 5.3) byγt,s; ε) and integrate with respect to s, γt,s;ε)χ s;ε)ds = ε + εγ β) γt,s;ε)gχ)s;ε)ds γt,s;ε) s ks,σ ) s σ) β 2h σ,y σ ;ε) ) χσ;ε)dσ ds. 5.8) Applying integration by parts on the left-hand side and changing the order of integration in the second term on the right-hand side, one obtains χ t;ε) = ε + γt,s;ε)gχ)s;ε)ds { 2 γt,s;ε) + εγ β) s kσ,s) γt,σ;ε) σ s) 2h s,y β σ ;ε) ) } dσ χs;ε)ds. 5.9)

14 366 ANGELINA M. BIJURA But according to 5.6b), the second integral on the right-hand side is zero, so 5.7) is verified. The following lemma proves that the differential resolvent γt,s; ε) is uniformly bounded for all s t T and all sufficiently small ε. Lemma 5.. Suppose that H g ), H h ), and H k )hold.letγt,s;ε) satisfy 5.6). Then there exist positive constants c 2 and ε such that γt,s;ε) ds c2, t T, <ε ε. 5.) Proof. Consider 5.6a) and integrate both sides with respect to t to obtain γt,s;ε) = + εγ 2 β) s at,σ ;ε)t σ) β γσ,s;ε)dσ, 5.) where at,s;ε) = β) 2 h s,y s;ε) ) k s +t s)σ,s ) σ β dσ. 5.2) If t is replaced by t +s in 5.) and change of variables is performed, one has γt+s,s;ε) = + εγ 2 β) One will note that at,t;ε) = β) 2 h t,y t;ε) ) at +s,σ +s;ε)t σ) β γσ +s,s;ε)dσ. 5.3) kt,t) σ β dσ, t T. 5.4) It follows from H k ) that there is a positive number <δ<ηsuch that at,t;ε) δ< 5.5) for all t T and all <ε ε. It can be shown that at,s;ε) does not change the sign on t T. Thus, there exists a positive constant δ such that at,s;ε) δ, s t T, <ε ε. 5.6) Therefore, the differential resolvent γ satisfies γt+s,s;ε), s t T, <ε ε, δ γt+s,s;ε) εγ 2 β) t σ) β γσ +s,s;ε)dσ. 5.7)

15 ASYMPTOTICS OF INTEGRODIFFERENTIAL MODELS In this form, s is simply a parameter and one may write the above inequality as δ γt;ε) εγ 2 β) t σ) β γσ;ε)dσ, 5.8) and the proof of Lemma 5. may be directly applied to 5.8). Thus, to prove the lemma, it suffices to show that there exist a positive constant c 3 such that γs;ε)ds c 3, t T, <ε ε. 5.9) It has been proved in [6] that γt;ε) 2αe ɛ ϱ/ε) αt cos ɛ 2 t ), γs;ε)ds 4αεα ɛ ϱ, 5.2a) 5.2b) for all t T and all <ε ε, where ɛ, ɛ 2,andϱ are positive constants. This completes the proof of Lemma 5.. The main result in this paper is presented in the following theorem. Theorem 5.2. Suppose that H g ), H h ), and H k ) are satisfied and that Lemma 5. holds. Then.) has a continuous solution yt;ε) with the property that, if y t;ε) is the formal approximate solution, then there are positive constants c and ε, where c does not depend on ε such that yt;ε) y t;ε) c 5.2) for all t T and <ε ε. Proof. Consider 5.7) which is equivalent to χt;ε) = ε + εγ β) γt,s;ε)ρs;ε)ds ϕ χ,y,σ;ε ) σ γt,s;ε)ks,σ) s σ) β ds dσ, 5.22) where upon applying 5.2a), σ γt,s;ε)ks,σ) ds s σ) β ɛ 3 t σ) β ξ β e ɛξ/εα dξ, σ T. 5.23)

16 368 ANGELINA M. BIJURA Here, ɛ and ɛ 3 are positive constants, where ɛ 3 depends on α, kt,s), and T.A change of variables implies that χt;ε) ε γt,s;ε) ρs;ε) ds + ε α ɛ 4 Γ β) ϕ χ,y,σ;ε ) ξ β e ɛξ dξ dσ, 5.24) where ɛ 4 is a positive constant. Applying Proposition 4.2, 5.2b), and the definition of a Gamma function, one obtains t χt;ε) c2 +c 3 ε α ϕ χ,y,σ;ε ) dσ, t T, 5.25) where c 2 and c 3 are positive constants depending on T. Then, Theorem 2.3 yields the required result that χt;ε) c2 + c 2 2 e2c 2c 3 T 5.26) uniformly on t T and <ε ε. References [] J. S. Angell and W. E. Olmstead, Singularly perturbed Volterra integral equations. II, SIAMJ. Appl. Math ), no. 6, [2] A. M. Bijura, Rigorous results on the asymptotic solutions of singularly perturbed nonlinear Volterra integral equations, J. Integral Equations Appl. 4 22), no. 2, [3], Singularly perturbed systems of Volterra equations, J. Appl. Anal.822), no. 2, [4], Singularly perturbed Volterra integral equations with weakly singular kernels, Int. J. Math. Math. Sci. 3 22), no. 3, [5], Singularly perturbed Volterra integro-differential equations, Quaest. Math ), no. 2, [6], Asymptotics of integrodifferential models with integrable kernels, Int. J. Math. Math. Sci ), no. 25, [7] N. Bleistein and R. A. Handelsman, Asymptotic Expansions of Integrals, Holt, Rinehart and Winston, New York, 975. [8] S.S.Dragomir,On some nonlinear generalizations of Gronwall s inequality,makedon. Akad. Nauk. Umet. Oddel. Mat.-Tehn. Nauk. Prilozi 3 992), no. 2, [9] W. Eckhaus, Asymptotic Analysis of Singular Perturbations, Studies in Mathematics and Its Applications, vol. 9, North-Holland Publishing, New York, 979. [] R. Gorenflo and S. Vessella, Abel Integral Equations. Analysis and applications, Lecture Notes in Mathematics, vol. 46, Springer-Verlag, Berlin, 99. [] G. Gripenberg, S.-O. Londen, and O. Staffans, Volterra Integral and Functional Equations, Encyclopedia of Mathematics and Its Applications, vol. 34, Cambridge University Press, Cambridge, 99. [2] K. B. Hannsgen, A Volterra equation with completely monotonic convolution kernel, J. Math. Anal. Appl. 3 97),

17 ASYMPTOTICS OF INTEGRODIFFERENTIAL MODELS [3] J. J. Levin, The asymptotic behavior of the solution of a Volterra equation, Proc. Amer. Math. Soc ), [4] J. J. Levin and J. A. Nohel, On a system of integro-differential equations occurring in reactor dynamics, J. Math. Mech. 9 96), [5] R. K. Miller, Nonlinear Volterra Integral Equations, Mathematics Lecture Note Series, W. A. Benjamin, California, 97. [6] W. E. Olmstead and R. A. Handelsman, Diffusion in a semi-infinite region with nonlinear surface dissipation, SIAM Rev ), no. 2, [7] R. B. Paris and D. Kaminski, Asymptotics and Mellin-Barnes Integrals, Encyclopedia of Mathematics and Its Applications, vol. 85, Cambridge University Press, Cambridge, 2. Angelina M. Bijura: Department of Mathematics, University of Dar Es Salaam, P.O. Box 3562, Dar Es Salaam, Tanzania Current address: Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 77, South Africa address: abijura@maths.uct.ac.za

18 Mathematical Problems in Engineering Special Issue on Modeling Experimental Nonlinear Dynamics and Chaotic Scenarios Call for Papers Thinking about nonlinearity in engineering areas, up to the 7s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system. Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision. In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points. Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from Qualitative Theory of Differential Equations, allowing more precise analysis and synthesis, in order to produce new vital products and services. Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers. This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems. Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment. Authors should follow the Mathematical Problems in Engineering manuscript format described at Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at mts.hindawi.com/ according to the following timetable: Guest Editors José Roberto Castilho Piqueira, Telecommunication and Control Engineering Department, Polytechnic School, The University of São Paulo, São Paulo, Brazil; piqueira@lac.usp.br Elbert E. Neher Macau, Laboratório Associado de Matemática Aplicada e Computação LAC), Instituto Nacional de Pesquisas Espaciais INPE), São Josè dos Campos, São Paulo, Brazil ; elbert@lac.inpe.br Celso Grebogi, Department of Physics, King s College, University of Aberdeen, Aberdeen AB24 3UE, UK; grebogi@abdn.ac.uk Manuscript Due February, 29 First Round of Reviews May, 29 Publication Date August, 29 Hindawi Publishing Corporation