Vol. 18, No. 1, pp. 11–20 2013 for University of Łódź Pressc
STABILITY OF THE VOLTERRA INTEGRODIFFERENTIAL EQUATION
M. JANFADA AND GH. SADEGHI
Abstract. In this paper, the Hyers-Ulam stability of the Volterra integrod-ifferential equation
x0(t) = g(t, x(t)) + Zt
0
K(t, s, x(s))ds, and the Volterra equation
x(t) = g(t, x(t)) + Zt
0
K(t, s, x(s))ds,
on the finite interval [0, T ], T > 0, are studied, where the state x(t) take values in a Banach space X.
AMS Subject Classification. Primary 39B22; Secondary 39B82, 34G20. Key words and phrases. Hyers-Ulam stability, Volterra integrodifferential equation, Volterra equation, C0-semigroup.
1. Introduction and preliminaries
A classical question in the theory of functional equations is the following: ”When is it true that a function which approximately satisfies a functional equation E must be close to an exact solution of E ?” If there exists an affirmative answer we say that the equation E is stable [8]. During the last decades several stability problems for various functional equations have been investigated by numerous mathematicians. We refer the reader to the survey articles [8, 9, 23] and monographs [6, 10, 13, 24] and references therein.
Consider the Volterra integrodifferential equation
(1) x0(t) = g(t, x(t)) +
Z t
0
K(t, s, x(s))ds. If for given differentiable function x(t), satisfying
kx0(t) − g(t, x(t)) − Z t
0
K(t, s, x(s))dsk ≤ φ(t),
φ(t) > 0, t ∈ [0, T ], there exists a solution y(t) of (1) such that for some C > 0,
then we say that (1) has the Hyers-Ulam stability. A similar definition can be considered for the Voltera equation
(2) x(t) = g(t, x(t)) +
Z t
0
K(t, s, x(s))ds.
These equations and their special and general versions with different view points, have been studied by many authors. See [1, 4, 5, 14, 15, 17, 18, 19, 20] and the references given therein.
For a nonempty set X, a function d : X ×X → [0, ∞] is called a generalized metric on X if and only if d satisfies
(M1) d(x, y) = 0 if and only if x = y. (M2) d(x, y) = d(y, x), for all x, y ∈ X.
(M3) d(x, z) ≤ d(x, y) + d(y, z), for all x, y, z ∈ X.
Trivially the only one difference of the generalized metric from the usual metric is that the range of the former is permitted to include infinity. We now introduce one of fundamental results of fixed point theory. For the proof, we refer to [16]. This theorem will play an important role in proving our main results.
Theorem 1. Let (X, d) be a generalized complete metric space. Assume that Λ : X → X is a strictly contractive operator with the Lipschitz constant L < 1. If there exists a nonnegative integer k such that d(Λk+1x, Λkx) < 1, for some x ∈ X, then the following are true:
(a) The sequence {Λnx} converges to a fixed point x∗ of Λ; (b) x∗ is the unique fixed point of Λ in
X∗ = {y ∈ X : d(Λkx, y) < ∞}; (c) If y ∈ X∗, then d(y, x∗) ≤ 1−L1 d(Λy, y).
In this paper, using this theorem, we shall study the Hyers-Ulam stability of (1) and (2). Next some applicable examples of these equations and their Hyers-Ulam stability will be considered.
2. Hyers–Ulam stability
Cădariu and Radu [2] studied the stability of the Cauchy additive func-tional equation using the fixed point method. Applying such a clever idea, they could present another proof for the Hyers-Ulam stability of that equa-tion [3, 11, 22]. Also Soon-Mo Jung [12] used this idea for studying the stability of the following Voltrra integral equation
y(x) = Z x
0
As a recent work on the stability of integral equations, one can see [25]. In this section, by using the idea of Cădariu, Radu and Jung, we will study the Hyers-Ulam stability of the integrodifferential Volterra integral equation and the Volterra integral equation.
Theorem 2. Suppose X is a Banach space and L, L1, L2 and T are positive
constant for which 0 < L1+ (L1+ L2)L + L2T L < 1. Let g : [0, T ] × X → X ,
K : [0, T ] × [0, T ] × X → X and φ : [0, T ] → (0, ∞) be continuous and satisfy kg(t, x) − g(t, y)k ≤ L1kx − yk, kK(t, s, x) − K(t, s, y)k ≤ L2kx − yk, (3) and Z t 0 φ(s)ds ≤ Lφ(t),
for all s, t ∈ [0, T ] and x, y ∈ X . If f : [0, T ] → X is a differentiable function satisfies
(4) kf0(t) − g(t, f (t)) − Z t
0
K(t, s, f (s))dsk ≤ φ(t), t ∈ [0, T ],
then there exists a unique differentiable function f0 : [0, T ] → X such that
for each t ∈ [0, T ] (5) f00(t) = g(t, f0(t)) + Z t 0 K(t, s, f0(s))ds, and (6) kf0(t) − f00(t)k + kf (t) − f0(t)k ≤ 1 + L 1 − L1+ (L1+ L2)L + L2T L φ(t). Proof. Put M := {x : [0, T ] → X : x is differentiable} and define a mapping d : M × M → [0, ∞] by
d(x, y) = inf{C ∈ [0, ∞] : kx0(t) − y0(t)k + kx(t) − y(t)k ≤ Cφ(t), t ∈ [0, T ]}. We show that (M, d) is a complete generalized metric space. We just prove the triangle inequality and the completeness of this space. Assume that d(x, y) > d(x, z) + d(z, y), for some x, y, z ∈ M . Then there exists t0∈ [0, T ]
with kx0(t0) − y0(t0)k + kx(t0) − y(t0)k > d(x, z) + d(z, y)φ(t0). (7) Thus, by definition of d, kx0(t0) − y0(t0)k + kx(t0) − y(t0)k > kx0(t0) − z0(t0)k + kx(t0) − z(t0)k + kz0(t0) − y0(t0)k + kz(t0) − y(t0)k,
which is a contradiction. Now we show that (M, d) is complete. Let {xn} be a Cauchy sequence in (M, d). This, by definition of d, implies that (8) ∀ε>0∃Nε∈N∀m,n≥Nε∀t∈[0,T ] kx
0
n(t) − x 0
m(t)k + kxn(t) − xm(t)k < εφ(t).
By continuity of φ on compact interval [0, T ], (8) implies that {xn} and {x0 n}
are uniformly convergent on [0, T ]. So there exists a differentiable function x such that {xn} and {x0
n} are uniformly convergent to x and x0, respectively.
Hence x ∈ M and from (8), letting m → ∞, we have ∀ε>0 ∃Nε∈N∀n≥Nε∀t∈[0,T ] kx 0 n(t) − x 0 (t)k + kxn(t) − x(t)k ≤ εφ(t). Consequently ∀ε>0 ∃Nε∈N∀n≥Nε d(xn, x) ≤ ε and so (M, d) is complete. Now define Λ : M → M by (9) Λ(x(t)) = Z t 0 g(τ, x(τ ))dτ + Z t 0 Z t 0 K(τ, s, x(s))dsdτ.
First we show that Λ is strictly contractive. Suppose x, y ∈ M , Cxy ∈ [0, ∞]
and d(x, y) ≤ Cxy. Thus for all t ∈ [0, T ],
kx0(t) − y0(t)k + kx(t) − y(t)k ≤ Cxyφ(t). Hence by (3) d dt Λx(t) − Λy(t) + kΛx(t) − Λy(t)k = = g(t, x(t)) − g(t, y(t)) + Z t 0 K(t, s, x(s)) − K(t, s, y(s))ds + + Z t 0 g(τ, x(τ )) − g(τ, y(τ ))dτ + Z t 0 Z t 0 K(τ, s, x(s)) − K(τ, s, y(s))dsdτ ≤ L1kx(t) − y(t)k + L2 Z t 0 kx(τ ) − y(τ )k dτ + + L1 Z t 0 kx(τ ) − y(τ )k dτ + L2T Z t 0 kx(s) − y(s)k ds ≤L1+ (L1+ L2)L + L2T L Cxyφ(t).
This implies that
So Λ is strictly contractive, since 0 < L1+ (L1+ L2)L + L2T L < 1. On the
other hand, trivially f ∈ M and by (4) d dt Λf (t) − f (t) + kΛf (t) − f (t)k ≤ φ(t) + Z t 0 φ(s)ds = (1 + L)φ(t). Consequently (11) d(Λf, f ) ≤ 1 + L < ∞.
It follows from Theorem 1 (a) that there exists a unique element f0 ∈ M∗=
{y ∈ M : d(Λf, y) < ∞} such that Λf0 = f0, or equivalently
f0(t) = Z t 0 g(τ, f0(τ ))dτ + Z t 0 Z t 0 K(τ, s, f0(s))dsdτ.
Now the facts that f0 is differentiable and g, K are continuous, imply that
f00(t) = g(t, f0(t)) +
Z t
0
K(t, s, f0(s))ds.
Also from Theorem 1 (c) and (11), we have
d(f, f0) ≤ 1 1 − (L1+ (L1+ L2)L + L2T L) d(Λf, f ) ≤ ≤ (1 + L) 1 − (L1+ (L1+ L2)L + L2T L) .
In view of definition of d we can conclude that the inequality (6) holds, for all t ∈ [0, T ]. Put θ = 1−(L (1+L)
1+(L1+L2)L+L2T L).
Let h be another differentiable function satisfying (5), (6). Then f ∈ M , d(f, h) < θ and
(12) h0(t) = g(t, h(t)) +
Z t 0
K(t, s, h(s))ds.
For proving the uniqueness of f0, it is enough to show that h is a fixed point
d(Λf, h) < ∞. From (12) and the fact that d(f, h) < θ, we obtain d dt(Λf (t) − h(t)) + kΛf (t) − h(t)k = = kg(t, f (t)) − g(t, h(t)) − Z t 0 K(t, s, f (s))ds + Z t 0 K(t, s, h(s))dsk+ + Z t 0 g(t, f (τ )) − g(τ, h(τ ))dτ − Z t 0 Z t 0 K(τ, s, f (s))ds + K(τ, s, h(s))dsdτ ≤ L1kf (t) − h(t)k + L2 Z t 0 kf (s) − h(s)ds+ + TL1kf (t) − h(t)k + L2 Z t 0 kf (s) − h(s)dsk ≤ (L1+ L2)(1 + T )θφ(t),
which implies that d(Λf, h) ≤ (L1 + L2T )θ < ∞. This completes the
proof.
In the next theorem the Hyers-Ulam stability of the Volterra integral equation is studied. This theorem is an extension of Theorem 2.1 of [12]. Theorem 3. Suppose X is a Banach space and L, L1, L2 and T are positive
constants for which 0 < (L1 + L2)L < 1. Let g : [0, T ] × X → X , K :
[0, T ] × [0, T ] × X → X and φ : [0, T ] → (0, ∞) be continuous functions satisfying kg(t, x) − g(t, y)k ≤ L1kx − yk kK(t, s, x) − K(t, s, y)k ≤ L2kx − yk (13) Z t 0 φ(s)ds ≤ Lφ(t),
for all s, t ∈ [0, T ] and x, y ∈ X . If f : [0, T ] → X is a continuous function satisfies
(14) kf (t) − g(t, f (t)) − Z t
0
K(t, s, f (s))dsk ≤ φ(t), t ∈ [0, T ]. Then there exists a unique continuous function f0: [0, T ] → X such that (15) f0(t) = g(t, f0(t)) + Z t 0 K(t, s, f0(s))ds and kf (t) − f0(t)k ≤ 1 1 − (L1+ L2)L φ(t). (16)
Proof. With
M := {x : [0, T ] → X : x is continuous }, define d : M × M → [0, ∞] by
d(x, y) = inf{C ∈ [0, ∞] : kx(t) − y(t)k ≤ Cφ(t), t ∈ [0, T ]}.
With a similar argument to the proof of Theorem 2, one can see that (M, d) is a complete generalized metric space. Now define Λ on M as in the proof of Theorem 2. One can verify that for any x, y ∈ M ,
d(Λx, Λy) ≤ (L1+ L2)Ld(x, y).
The fact that 0 < (L1 + L2)L < 1 implies that Λ is strictly contractive.
Also from (14), we obtain d(Λf, f ) ≤ 1 < ∞ and so by Theorem 1, Λ has a unique fixed point f0 in the set M∗:= {y ∈ M : d(Λf, y) < ∞}.
Let h be another continuous function satisfying (15), (16). Thus f ∈ M , d(f, h) < 1−(L1 1+L2)L and (17) h(t) = g(t, h(t)) + Z t 0 K(t, s, h(s))ds.
For proving the uniqueness of h, it is enough to show that h is a fixed point of Λ and h ∈ M∗. Using (15), we have Λh = h. Also form (15) and the fact that d(f, h) < 1−(L1 1+L2)L, we obtain kΛf (t) − h(t)k = kg(t, f (t)) − g(t, h(t)) − Z t 0 K(t, s, f (s))ds+ + Z t 0 K(t, s, h(s))dsk ≤ ≤ (L1+ L2) 1 − (L1+ L2)L φ(t),
which implies that d(Λf, h) < ∞. This completes the proof. In the sequel some examples and applications of our discussion are pre-sented.
We recall that for a Banach space X , a one-parameter family {T(t)}t≥0 in B(X ), the space of all bounded linear operators, is called a C0-semigroup of
operators if for all s, t ≥ 0,
T(s + t) = T(s)T(t) and T(0) = I( the identity operator), and for any x ∈ X, limt→0+T(t)x = x. In this case the operator A : D(A) ⊂ X → X , where D(A) = x ∈ X : lim t→0+ T(t)x − x t exists ,
defined by A(x) = limt→0+ T(t)x−xt is called the infinitesimal generator {T(t)}t≥0. One can see [21] or [7] for a comprehensive reference of
semi-group of operators theory.
Example 1. Let X be a Banach space, A ∈ B(X ) with kAk ≤ 1, T ∈ (0, ∞) and a(.) ∈ W1,1(R+, C), where W1,1 is the Sobolev space. Put T(t) =
etA, t ≥ 0. For any f ∈ W1,1(R+, X ), from Corollary 7.27 [7], we know that
there exists a unique solution u ∈ C1(R, X ) ∩ C(R, X ) satisfying the Volterra integrodifferential equation
(18) u0(t) = Au(t) + f (t) + Z t
0
a(t − s)Au(s)ds. Now define g : [0, T ] × X → X and K : [0, T ] × [0, T ] × X → X by
g(t, x) = Ax + f (t), K(t, s, x) = a(t − s)Ax. Trivially
kg(t, x) − g(t, y)k ≤ kAkkx − yk, and continuity of a implies that
kK(t, s, x) − K(t, s, y)k = |a(t − s)|kAkkx − yk ≤ M kAkkx − yk, for some M > 0. Suppose
0 < L < 1 − kAk kAk(1 + M + M T ), α ≥ L1 and φ(t) = ρeαt, ρ > 0. If ku0(t) − Au(t) − f (t) − Z t 0 a(t − s)Au(s)dsk ≤ φ(t),
for appropriate f and a(.), then with L1= kAk, and L2 = M kAk, by
Theo-rem 2, there exists a unique solution u0(t) of (18) such that
ku(t) − u0(t)k + ku0(t) − u00(t)k ≤
1 + L
1 − 1 − (L1+ (L1+ L2)L + L2T L)
φ(t) Thus we obtain the Hyers–Ulam stability of the equation 18.
Example 2. Suppose X is a Banach space, T ∈ (0, ∞) and {T(t)}t≥0 is a C0-semigroup of bounded linear operators with the infinitesimal generator
(A, D(A)). Let B ∈ C([0, T ], B(X )), the space of all continuous function from [0, T ] into B(X ). For x0∈ D(A), consider the integral equation
(19) u(t) = T(t)x0+
Z t
0
This equation has a solution (see 9.21 of [7]). Define g : [0, T ] × X → X and K : [0, T ] × [0, T ] × X → X , by
g(t, x) = T(t)x0, K(t, s, x) = T(t − s)B(s)x.
Trivially for any x, y ∈ X , kg(t, x) − g(t, y)k = 0. Also from Theorem I.2.2[21], we know that there exist M, ω > 0, such that for all t ≥ 0, kT(t)k ≤ M etw. On the other hand continuity of B : [0, T ] → B(X ) implies that kB(s)k ≤ M0, for some M0 > 0 and all s ∈ [0, T ]. Thus
kK(t, s, x) − K(t, s, y)k = kT(t − s)B(s)(x − y)k ≤ M M0eT ωkx − yk.
Now suppose 0 < L ≤ 1
M M0eT ω. For a fixed α ≥ 1
L and ρ > 0, if φ(t) = ρe αt,
then the conditions (13) hold. Thus if ku(t) − T(t)x0−
Z t
0
T(t − s)B(s)u(s)dsk ≤ φ(t), t ∈ [0, T ], by Theorem 3, there exists a unique solution u0(t) of (19) such that
ku(t) − u0(t)k ≤
1 1 − M M0eT ωL
φ(t). Hence it conclude the Hyers-Ulam Stability of (19).
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Mohammad Janfada
Department of Pure Mathematics, Ferdowsi University of Mashhad Mashhad, P.O. Box 1159-91775, Iran
E-mail: mjanfada@gmail.com
Gh. Sadeghi
Department of Mathematics, Hakim Sabzevary University Sabzevar, P.O. Box 397, Iran
