Electronic Journal of Qualitative Theory of Differential Equations 2008, No. 17, 1-7; http://www.math.u-szeged.hu/ejqtde/

THE FREEZING METHOD FOR VOLTERRA INTEGRAL EQUATIONS IN A BANACH SPACE ^{∗}

M. I. Gil’ Department of Mathematics Ben Gurion University of the Negev P.0. Box 653, Beer-Sheva 84105, Israel E-mail: gilmi@cs.bgu.ac.il

Abstract

The ”freezing” method for ordinary differential equations is extended to the Volterra integral equations in a Banach space of the type

x(t) − ∫_{0}^{t} K(t, t − s)x(s)ds = f(t) (t ≥ 0),

where K(t, s) is an operator valued function ”slowly” varying in the first argument. Besides, sharp explicit stability conditions are derived. Subject Classification: 45M10, 45N05 Key words: Volterra integral equations, Banach space, stability

1 Introduction and statement of the basic lemma

Stability and boundedness of Volterra integral and integrodifferential equations have been extensively considered for a long time (see the well-known books [1, 4], recent papers [5, 8, 15, 16] and papers listed below). The basic method in the theory of stability and boundedness of Volterra integral equations is the direct Liapunov method. But finding the Liapunov functionals is a difficult mathematical problem. The other approach is connected with an interpretation of the Volterra equations as operator equations in appropriate spaces. Such an approach was used in many papers, cf. [3, 6, 7, 12, 14, 16] and references therein. In this paper, for a class of Volterra equations in a Banach space we establish explicit sufficient stability conditions which are also necessary stability conditions when the integral operator is a convolution. Our results improve the well known ones in the case of the considered equations.

———————————————————–

∗ This research was supported by the Kameah fund.

The approach suggested below is based on the extension of the ”freezing” method which was introduced by V.M. Alekseev for linear ordinary differential equations cf. [2] (see also [9, Section 3.2]). That method was already extended to difference equations [11].

Let X be a Banach space with a norm ∥.∥ and the unit operator I, R_{+} := [0, ∞), and C(ω, X) is the space of continuous functions defined on a set ω ⊂ ℝ with values in X and equipped with the sup-norm |.|_{C(ω)} = |.|_{C(ω,X)}. L^{p}(ω, X) (1 ≤ p < ∞) is the space of functions defined on ω with values in X and equipped with the

|f|_{L^{p}(ω)} = [∫_{ω} ∥f(t)∥^{p}dt]^{1/p}.

Consider in X the equation

(1.1)  x(t) − ∫_{0}^{t} K(t, t − s)x(s)ds = f(t) (f ∈ C(R_{+}, X), t ≥ 0),

where K(t, s) is a functions defined on [0 ≤ s ≤ t < ∞], whose values are bounded operators in X, and for any fixed τ ≥ 0, K(τ, .) is integrable and bounded on R_{+}. In addition,

(1.2)  ∫_{0}^{t} ∥K(t, s) − K(τ, s)∥ds ≤ q|t − τ| (q = const; t, τ ≥ 0).

A solution of Equation (1.1) is a continuous function defined on R_{+} and satisfying (1.1) for all finite t > 0. The existence of solutions under consideration is checked below.

Note that the approach suggested below enables us to consider also the equation

x(t) − ∫_{0}^{t} K(t − s, s)x(s)ds = f(t) (t ≥ 0)

under condition (1.2). It is clear that under (1.2) the function K(τ, s), for a fixed τ, admits the Laplace transform

K̃_{τ}(z) := ∫_{0}^{∞} e^{−zs}K(τ, s)ds (Rez ≥ c_{0} = const).

Besides, it is assumed that the operator W_{τ}(z) := I − K̃_{τ}(z) is invertible for all z ∈ C_{+} := {z ∈ ℂ : Re z ≥ 0} and W_{τ}^{−1}(iy) ∈ L^{1}(ℝ). Introduce the ”local Green function”

G_{τ}(t) := \frac{1}{2π} ∫_{−∞}^{∞} e^{iyt}W_{τ}^{−1}(iy)dy.

We will say that Equation (1.1) is stable, if for any f ∈ C(R_{+}, X) a solution x of (1.1) satisfies the inequality

(1.3)  |x|_{C(R_{+})} ≤ a_{0}|f|_{C(R_{+})},

where the constant a_{0} does not depend on f.

Lemma 1.1 Under condition (1.2), let

(1.4)  q ∫_{0}^{∞} s sup_{τ≥0} ∥G_{τ}(s)∥ds < 1.

Then Equation (1.1) is stable. Moreover, constant a_{0} in (1.3) is explicitly pointed below.

This lemma is proved in the next section.

2 Proof of Lemma 1.1

Consider the equation

(2.1)  x(t) − ∫_{0}^{t} K(τ, t − s)x(s)ds = f(t) (t ≥ 0)

with a fixed τ ≥ 0. Applying to (2.1) the Laplace transform, we have

x̃(z) − K̃_{τ}(z)x̃(z) = f̃(z),

where x̃(z) and f̃(z) are the Laplace transforms to x(t) and f(t), respectively, z is the dual variable. Hence,

x̃(z) = W_{τ}^{−1}(z)f̃(z).

So

(2.2)  x(t) = ∫_{0}^{t} G_{τ}(t − s)f(s)ds.

Now rewrite (1.1) in the form

(2.3)  x(t) − ∫_{0}^{t} K(τ, t − s)x(s)ds = f_{0}(t, τ) + f(t) (t ≥ 0).

with

f_{0}(t, τ) = ∫_{0}^{t}(K(t, t − s) − K(τ, t − s))x(s)ds.

So according to (2.2),

(2.4)  x(t) = ∫_{0}^{t} G_{τ}(t − s)(f(s) + f_{0}(s, τ))ds = F (t) + ∫_{0}^{t} G_{τ}(t − s)f_{0}(s, τ)ds,

where

F (t) = ∫_{0}^{t} G_{τ}(t − s)f(s)ds.

With the notation

w(t) := sup_{τ≥0} ∥G_{τ}(t)∥

we thus get

|F |_{C(R_{+})} ≤ |f|_{C(R_{+})} sup_{t} ∫_{0}^{t} w(t − s)ds = |w|_{L^{1}(R_{+})} |f|_{C(R_{+})}.

Due to (1.3)

∥f_{0}(t, τ)∥ ≤ ∫_{0}^{t} ∥(K(τ, t − s) − K(t, t − s))x(s)∥ds ≤ |x|_{C(0,t)}q|t − τ|.

Now (2.4) implies

∥x(t)∥ ≤ |w|_{L^{1}(R_{+})} |f|_{C(R_{+})} + q ∫_{0}^{t} w(t − s)|x|_{C(0,s)}|s − τ|ds.

EJQTDE, 2008 No. 17, p. 3 Take t = τ. Then

∥x(τ)∥ ≤ |w|_{L^{1}(R_{+})} |f|_{C(R_{+})} + q ∫_{0}^{τ} w(τ − s)|_{X}|x|_{C(0,s)}(τ − s)ds.

Hence,

∥x(τ)∥ ≤ |w|_{L^{1}(R_{+})} |f|_{C(R_{+})} + |x|_{C(0,τ)} ∫_{0}^{τ}(τ − s)w(τ − s)ds_{1} ≤
|w|_{L^{1}(R_{+})} |f|_{C(R_{+})} + |x|_{C(0,τ)}Θ,

where

Θ = q ∫_{0}^{∞} sw(s)ds.

Therefore, for any t_{0} > 0,

sup_{τ≤t_{0}} ∥x(τ)∥ ≤ |w|_{L^{1}(R_{+})} |f|_{C(R_{+})} + sup_{τ≤t_{0}} |x|_{C(0,τ)}Θ.

Now condition (1.4) implies

|x|_{C(0,t_{0})} ≤ \frac{|w|_{L^{1}(R_{+})} |f|_{C(R_{+})}}{1 − Θ} .

Since the right hand part does not depend on t_{0}, inequality (1.3) follows. Besides,

a_{0} = \frac{|w|_{L^{1}(R_{+})}}{1 − Θ} .

The existence of solutions is due to the Neumann series

x = ∑_{k=0}^{∞} V^{k}f,

where V is the Volterra integral operator defined in (1.1). The lemma is proved. □

3 The main result

First, note that

tG_{τ}(t) = t\frac{1}{2πi} ∫_{−i∞}^{i∞} e^{zt}W_{τ}^{−1}(z)dz = \frac{1}{2πi} ∫_{−i∞}^{+i∞} e^{zt}T (z)dz,

where

T_{τ}(z) := −\frac{dW_{τ}^{−1}(z)}{dz} = W_{τ}^{−1}(z)\frac{dW_{τ}(z)}{dz}W_{τ}^{−1}(z).

For a number b > 0 and Re z > −b, let T_{τ}(z) be regular and

(3.1)  ψ_{b} := sup_{τ≥0} \frac{1}{2π} ∫_{−∞}^{∞} ∥T_{τ}(iy − b)∥dy < ∞.

EJQTDE, 2008 No. 17, p. 4 Then

∥tG_{τ}(t)∥ ≤ e^{−bt} \frac{1}{2π} ∫_{−∞}^{∞} ∥T (iy − b)∥dy = e^{−bt}ψ_{b}.

So

∫_{0}^{∞} t sup_{τ} ∥G_{τ}(t)∥dt ≤ ψ_{b} ∫_{0}^{∞} e^{−bt}dt = \frac{ψ_{b}}{b} .

Now Lemma 1.1 implies our main result. Theorem 3.1 Under condition (1.2), for a positive b and all z with Re z > −b, let T_{τ}(z) be regular, and the conditions (3.1) and qψ_{b} < b hold. Then Equation (1.1) is stable.

To illustrate this result, consider in X the equation

(3.2)  x(t) − A(t) ∫_{0}^{t} e^{−(t−s)h}x(s)ds = f(t) (h = const > 0, t ≥ 0),

where A(t) is a variable bounded operator in X satisfying

(3.3)  ∥A(t) − A(τ)∥ ≤ q_{1}|t − τ| (t, τ ≥ 0).

Take K(t, s) = A(t)e^{−sh}. Then

(3.4)  ∫_{0}^{t} ∥K(t, s) − K(τ, s)∥ds ≤ q_{1}∥A(t) − A(τ)∥ ∫_{0}^{t} e^{−sh}ds ≤ \frac{q_{1}}{h} |t − τ| (t, τ ≥ 0).

So (1.2) holds with q = q_{1}/h. We also have

K̃_{τ}(z) := A(τ) ∫_{0}^{∞} e^{−zs}e^{−hs}ds = \frac{A(τ)}{z + h}

and

W_{τ}(z) := I − \frac{A(τ)}{z + h}.

Hence,

T_{τ}(z) = (I − \frac{A(τ)}{z + h})^{−2} \frac{A(τ)}{(z + h)^{2}} = A(τ)((z + h)I − A(τ))^{−2}.

So

(3.5)  ∥T_{τ}(z)∥ ≤ ∥A(τ)∥ ∥((z + h)I − A(τ))^{−1}∥^{2} (τ ≥ 0).

Note that some estimates for resolvents of nonselfadjoint operators can be found in [10]. For instance, take X = L^{2}(0, 1) and

A(t)w(y) = a(t, y) ∫_{0}^{1} m(y, y_{1})w(y_{1})dy_{1} (y ∈ [0, 1]),

where a(t, .) for all t ≥ 0 is a scalar measurable function satisfying the conditions

sup_{t≥0,y∈[0,1]} |a(t, y)| < ∞

and

(3.6)  |a(t, y) − a(τ, y)| ≤ q_{0}|t − τ| (y ∈ [0, 1]; t, τ ≥ 0).

In addition, the scalar function m(., .) satisfies the condition

N_{m} := [∫_{0}^{1} ∫_{0}^{1} |m(y, y_{1})|^{2}dy dy_{1}]^{1/2} < ∞.

That is, we consider the equation

(3.7) u(t, y) = f(t, y) + a(t, y) ∫_{0}^{t} e^{−h(t−s)} ∫_{0}^{1} m(y, y_{1})u(s, y_{1})dy_{1}ds (0 ≤ y ≤ 1; t ≥ 0),

where f(t, .) ∈ L^{2}(0, 1). By the Schwarz inequaliy, for any w ∈ L^{2}(0, 1) we get

∥(A(t) − A(τ))w∥^{2} = ∫_{0}^{1} |(a(t, y) − a(τ, y)) ∫_{0}^{1} m(y, y_{1})w(y_{1})dy_{1}|^{2}dy ≤ (q_{0}|t − τ|N_{m})^{2}∥w∥^{2}.

That is, (3.3) holds with q_{1} = q_{0}N_{m}. So according to (3.4), condition (1.2) is valid with q = q_{0}N_{m}/h. Furthermore, clearly,

∥A(τ)∥ ≤ c(a, m) := sup_{τ,y} |a(τ, y)|N_{m} (τ ≥ 0).

Assume that

(3.8)  2c(a, m) < h

and take b = h/2. Then by (3.5),

∥T_{τ}(−b + iy)∥ ≤ \frac{c(a, m)}{(\sqrt{y^{2} + h^{2}/4} − c(a, m))^{2}} (τ ≥ 0).

So we have the inequality ψ_{b} ≤ ψ̃_{h}, where

ψ̃_{h} := \frac{c(a, m)}{2π} ∫_{−∞}^{∞} \frac{dy}{(\sqrt{y^{2} + h^{2}/4} − c(a, m))^{2}} < ∞.

Thus under conditions (3.6) and (3.8), thanks to Theorem 3.1, Equation (3.7) is stable provided 2q_{0}N_{m}ψ̃_{h} < h^{2}.

References

[1] Burton, T. A., Volterra Integral and Differential Equations, Acad. Press, New York, 1983.

[2] Bylov, B. F., Grobman, B. M., Nemyckii V. V. and Vinograd R.E. The Theory of Lyapunov Exponents, Nauka, Moscow, 1966. (In Russian).

[3] Becker, L. C. Function bounds for solutions of Volterra equations and exponential asymptotic stability. Nonlinear Anal., Theory Methods Appl. 67, No. 2 (A), 382-397 (2007).

[4] Corduneanu, C., Integral Equations and Applications. Cambridge Univ. Press, Cambridge, 1990.

[5] Crisci, M.R., V.B. Kolmanovskii, E. Russo, A. Vecchio, Stability of continuous and discrete Volterra integro-differential equations by Lyapunov approach, J. Integral Equations Appl., 7 (4), 393-411 (1995).

[6] Diamandescu, A., On the Ψ-stability of a nonlinear Volterra integro-differential system. Electron. J. Differ. Equ. 2005, Paper No. 56, 14 p., electronic only (2005).

[7] Diamandescu, A., On the Ψ-conditional asymptotic stability of the solutions of a nonlinear Volterra integro-differential system. Electron. J. Differ. Equ. 2007, Paper No. 29, 13 p., electronic only (2007)

[8] Funakubo, M, Hara, T. and Sakata, S., On the uniform asymptotic stability for a linear integro-differential equation of Volterra type. J. Math. Anal. Appl. 324, No. 2, 1036-1049 (2006).

[9] Gil’, M. I., Stability of Finite and Infinite Dimensional Systems, Kluwer Academic Publishers, Boston-Dordrecht-London, 1998.

[10] Gil’, M. I., Operator Functions and Localization of Spectra, Lecture Notes In Mathematics, Vol. 1830, Springer Verlag, 2003.

[11] Gil’, M. I. Difference Equations in Normed Spaces. Stability and Oscillations, North-Holland, Mathematics Studies 206, Elsevier, Amsterdam, 2007.

[12] Gil’, M.I. and Kloeden P. E., Solution estimates of nonlinear vector Volterra-Stieltjes equations, Analysis and Appl., 1, No 2, 165-175 (2003).

[13] Islam, M. and Raffoul, Y., Stability properties of linear Volterra integrodifferential equations with nonlinear perturbation, Commun. Appl. Anal. 7, No. 3, 405-416 (2003).

[14] Islam, M., Raffoul, Y., Stability in linear Volterra integrodifferential equations with nonlinear perturbation, J. Integral Equations Appl. 17, No. 3, 259-276 (2005).

[15] Weng, P., Existence and global stability of positive periodic solution in a logistic integrodifferential equation with feedback control. Ann. Differ. Equations 16, No.3, 281-290 (2000).

[16] Zhang, Bo, Necessary and sufficient conditions for stability in Volterra equations of nonconvolution type. Dyn. Syst. Appl. 14, No. 3-4, 525-550 (2005).

(Received February 14, 2008)