In this paper, the Hyers-Ulam stability of the Volterra integrodifferential equation and the Volterra equation on the finite interval [0, T], T > 0, are studied, where the state x(t) take values in a Banach space X

Janfada, Mohammad

Sadeghi, Gh.

English

In this paper, the Hyers-Ulam stability of the Volterra integrodifferential equation 
$$ x^{'} (t) = g(t,x(t)) + \int_0^t K(t,s,x(s))ds $$
and the Volterra equation 
$$ x(t)=g(t,x(t)) + \int_0^t K(t,s,x(s))ds $$
on the finite interval $ [0, T], T &gt; 0 $, are studied, where the state $ x(t) $ take values in a Banach space $X$

Biblioteka Nauki - repozytorium artykuÅÃ³w

Stability of the Volterra Integrodifferential Equation

Repozytorium Uniwersytetu Łódzkiego (University of Lodz Repository)

Folia Mathematica Acta Universitatis LodziensisVol. 18, No. 1, pp. 11–20 c© 2013 for University of Łódź PressSTABILITY OF THE VOLTERRAINTEGRODIFFERENTIAL EQUATIONM. JANFADA AND GH. SADEGHIAbstract. In this paper, the Hyers-Ulam stability of the Volterra integrod-ifferential equationx′(t) = g(t, x(t)) +∫ t0K(t, s, x(s))ds,and the Volterra equationx(t) = g(t, x(t)) +∫ t0K(t, s, x(s))ds,on the finite interval [0, T ], T > 0, are studied, where the state x(t) take valuesin a Banach space X.AMS Subject Classification. Primary 39B22; Secondary 39B82, 34G20.Key words and phrases. Hyers-Ulam stability, Volterra integrodifferentialequation, Volterra equation, C0-semigroup.1. Introduction and preliminariesA classical question in the theory of functional equations is the following:”When is it true that a function which approximately satisfies a functionalequation E must be close to an exact solution of E?” If there exists anaffirmative answer we say that the equation E is stable [8]. During the lastdecades several stability problems for various functional equations have beeninvestigated by numerous mathematicians. We refer the reader to the surveyarticles [8, 9, 23] and monographs [6, 10, 13, 24] and references therein.Consider the Volterra integrodifferential equation(1) x′(t) = g(t, x(t)) +∫ t0K(t, s, x(s))ds.If for given differentiable function x(t), satisfying‖x′(t)− g(t, x(t))−∫ t0K(t, s, x(s))ds‖ ≤ φ(t),φ(t) > 0, t ∈ [0, T ], there exists a solution y(t) of (1) such that for someC > 0,‖x(t)− y(t)‖ ≤ Cφ(t),12 M. JANFADA AND GH. SADEGHIthen we say that (1) has the Hyers-Ulam stability. A similar definition canbe considered for the Voltera equation(2) x(t) = g(t, x(t)) +∫ t0K(t, s, x(s))ds.These equations and their special and general versions with different viewpoints, 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 generalizedmetric 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 usualmetric is that the range of the former is permitted to include infinity.We now introduce one of fundamental results of fixed point theory. For theproof, we refer to [16]. This theorem will play an important role in provingour main results.Theorem 1. Let (X, d) be a generalized complete metric space. Assumethat Λ : X → X is a strictly contractive operator with the Lipschitz constantL < 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 inX∗ = {y ∈ X : d(Λkx, y) <∞};(c) If y ∈ X∗, then d(y, x∗) ≤ 11−Ld(Λy, y).In this paper, using this theorem, we shall study the Hyers-Ulam stabilityof (1) and (2). Next some applicable examples of these equations and theirHyers-Ulam stability will be considered.2. Hyers–Ulam stabilityCădariu 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 thestability of the following Voltrra integral equationy(x) =∫ x0f(τ, y(τ))dτ.STABILITY OF THE VOLTERRA... 13As 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 studythe Hyers-Ulam stability of the integrodifferential Volterra integral equationand the Volterra integral equation.Theorem 2. Suppose X is a Banach space and L,L1, L2 and T are positiveconstant for which 0 < L1+(L1+L2)L+L2TL < 1. Let g : [0, T ]×X → X ,K : [0, T ]× [0, T ]×X → X and φ : [0, T ]→ (0,∞) be continuous and satisfy‖g(t, x)− g(t, y)‖ ≤ L1‖x− y‖,‖K(t, s, x)−K(t, s, y)‖ ≤ L2‖x− y‖,(3)and∫ t0φ(s)ds ≤ Lφ(t),for all s, t ∈ [0, T ] and x, y ∈ X . If f : [0, T ]→ X is a differentiable functionsatisfies(4) ‖f ′(t)− g(t, f(t))−∫ t0K(t, s, f(s))ds‖ ≤ φ(t), t ∈ [0, T ],then there exists a unique differentiable function f0 : [0, T ] → X such thatfor each t ∈ [0, T ](5) f ′0(t) = g(t, f0(t)) +∫ t0K(t, s, f0(s))ds,and(6) ‖f ′(t)− f ′0(t)‖+ ‖f(t)− f0(t)‖ ≤1 + L1− L1 + (L1 + L2)L+ L2TLφ(t).Proof. PutM := {x : [0, T ]→ X : x is differentiable}and define a mapping d : M ×M → [0,∞] byd(x, y) = inf{C ∈ [0,∞] : ‖x′(t)− y′(t)‖+ ‖x(t)− y(t)‖ ≤ Cφ(t), t ∈ [0, T ]}.We show that (M,d) is a complete generalized metric space. We just provethe triangle inequality and the completeness of this space. Assume thatd(x, y) > d(x, z)+d(z, y), for some x, y, z ∈M . Then there exists t0 ∈ [0, T ]with‖x′(t0)− y′(t0)‖+ ‖x(t0)− y(t0)‖ >(d(x, z) + d(z, y))φ(t0).(7)Thus, by definition of d,‖x′(t0)− y′(t0)‖+ ‖x(t0)− y(t0)‖ >‖x′(t0)− z′(t0)‖+ ‖x(t0)− z(t0)‖+ ‖z′(t0)− y′(t0)‖+ ‖z(t0)− y(t0)‖,14 M. JANFADA AND GH. SADEGHIwhich 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 ] ‖x′n(t)−x′m(t)‖+ ‖xn(t)−xm(t)‖ < εφ(t).By continuity of φ on compact interval [0, T ], (8) implies that {xn} and {x′n}are uniformly convergent on [0, T ]. So there exists a differentiable function xsuch that {xn} and {x′n} are uniformly convergent to x and x′, respectively.Hence x ∈M and from (8), letting m→∞, we have∀ε>0 ∃Nε∈N∀n≥Nε∀t∈[0,T ] ‖x′n(t)− x′(t)‖+ ‖xn(t)− x(t)‖ ≤ εφ(t).Consequently∀ε>0 ∃Nε∈N∀n≥Nε d(xn, x) ≤ εand so (M,d) is complete.Now define Λ : M →M by(9) Λ(x(t)) =∫ t0g(τ, x(τ))dτ +∫ t0∫ t0K(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 ],‖x′(t)− y′(t)‖+ ‖x(t)− y(t)‖ ≤ Cxyφ(t).Hence by (3)∥∥∥∥ ddt(Λx(t)− Λy(t))∥∥∥∥+ ‖Λx(t)− Λy(t)‖ ==∥∥∥∥g(t, x(t))− g(t, y(t)) + ∫ t0(K(t, s, x(s))−K(t, s, y(s)))ds∥∥∥∥++∥∥∥∥∫ t0g(τ, x(τ))− g(τ, y(τ))dτ +∫ t0∫ t0K(τ, s, x(s))−K(τ, s, y(s))dsdτ∥∥∥∥≤ L1 ‖x(t)− y(t)‖+ L2∫ t0‖x(τ)− y(τ)‖ dτ++ L1∫ t0‖x(τ)− y(τ)‖ dτ + L2T∫ t0‖x(s)− y(s)‖ ds≤(L1 + (L1 + L2)L+ L2TL)Cxyφ(t).This implies that(10) d(Λx,Λy) ≤ (L1 + (L1 + L2)L+ L2TL)d(x, y).STABILITY OF THE VOLTERRA... 15So Λ is strictly contractive, since 0 < L1 + (L1 +L2)L+L2TL < 1. On theother hand, trivially f ∈M and by (4)∥∥∥∥ ddt(Λf(t)− f(t))∥∥∥∥+ ‖Λf(t)− f(t)‖ ≤ φ(t) + ∫ t0φ(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 equivalentlyf0(t) =∫ t0g(τ, f0(τ))dτ +∫ t0∫ t0K(τ, s, f0(s))dsdτ.Now the facts that f0 is differentiable and g,K are continuous, imply thatf ′0(t) = g(t, f0(t)) +∫ t0K(t, s, f0(s))ds.Also from Theorem 1 (c) and (11), we haved(f, f0) ≤ 11− (L1 + (L1 + L2)L+ L2TL)d(Λf, f) ≤≤ (1 + L)1− (L1 + (L1 + L2)L+ L2TL) .In view of definition of d we can conclude that the inequality (6) holds, forall t ∈ [0, T ]. Put θ = (1+L)1−(L1+(L1+L2)L+L2TL) .Let h be another differentiable function satisfying (5), (6). Then f ∈M ,d(f, h) < θ and(12) h′(t) = g(t, h(t)) +∫ t0K(t, s, h(s))ds.For proving the uniqueness of f0, it is enough to show that h is a fixed pointof Λ and h ∈ M∗. Using (12), one can see that Λh = h. We show that16 M. JANFADA AND GH. SADEGHId(Λf, h) <∞. From (12) and the fact that d(f, h) < θ, we obtain∥∥∥∥ ddt (Λf(t)− h(t))∥∥∥∥+ ‖Λf(t)− h(t)‖ == ‖g(t, f(t))− g(t, h(t))−∫ t0K(t, s, f(s))ds+∫ t0K(t, s, h(s))ds‖++∥∥∥∥∫ t0(g(t, f(τ))− g(τ, h(τ)))dτ−∫ t0∫ t0(K(τ, s, f(s))ds+K(τ, s, h(s))ds)dτ∥∥∥∥≤ L1‖f(t)− h(t)‖+ L2∫ t0‖f(s)− h(s)ds++ T(L1‖f(t)− h(t)‖+ L2∫ t0‖f(s)− h(s)ds)‖≤ (L1 + L2)(1 + T )θφ(t),which implies that d(Λf, h) ≤ (L1 + L2T )θ < ∞. This completes theproof. In the next theorem the Hyers-Ulam stability of the Volterra integralequation 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 positiveconstants 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 functionssatisfying‖g(t, x)− g(t, y)‖ ≤ L1‖x− y‖‖K(t, s, x)−K(t, s, y)‖ ≤ L2‖x− y‖(13) ∫ t0φ(s)ds ≤ Lφ(t),for all s, t ∈ [0, T ] and x, y ∈ X . If f : [0, T ] → X is a continuous functionsatisfies(14) ‖f(t)− g(t, f(t))−∫ t0K(t, s, f(s))ds‖ ≤ 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)) +∫ t0K(t, s, f0(s))dsand‖f(t)− f0(t)‖ ≤ 11− (L1 + L2)Lφ(t).(16)STABILITY OF THE VOLTERRA... 17Proof. WithM := {x : [0, T ]→ X : x is continuous },define d : M ×M → [0,∞] byd(x, y) = inf{C ∈ [0,∞] : ‖x(t)− y(t)‖ ≤ 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 proofof 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 aunique 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) < 11−(L1+L2)L and(17) h(t) = g(t, h(t)) +∫ t0K(t, s, h(s))ds.For proving the uniqueness of h, it is enough to show that h is a fixed pointof Λ and h ∈M∗. Using (15), we have Λh = h. Also form (15) and the factthat d(f, h) < 11−(L1+L2)L , we obtain‖Λf(t)− h(t)‖ = ‖g(t, f(t))− g(t, h(t))−∫ t0K(t, s, f(s))ds++∫ t0K(t, s, h(s))ds‖ ≤≤ (L1 + L2)1− (L1 + L2)Lφ(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 inB(X ), the space of all bounded linear operators, is called a C0-semigroup ofoperators 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 , whereD(A) ={x ∈ X : limt→0+T(t)x− xtexists},18 M. JANFADA AND GH. SADEGHIdefined 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 ‖A‖ ≤ 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 thatthere exists a unique solution u ∈ C1(R,X )∩C(R,X ) satisfying the Volterraintegrodifferential equation(18) u′(t) = Au(t) + f(t) +∫ t0a(t− s)Au(s)ds.Now define g : [0, T ]×X → X and K : [0, T ]× [0, T ]×X → X byg(t, x) = Ax+ f(t), K(t, s, x) = a(t− s)Ax.Trivially‖g(t, x)− g(t, y)‖ ≤ ‖A‖‖x− y‖,and continuity of a implies that‖K(t, s, x)−K(t, s, y)‖ = |a(t− s)|‖A‖‖x− y‖ ≤M‖A‖‖x− y‖,for some M > 0. Suppose0 < L <1− ‖A‖‖A‖(1 +M +MT ) ,α ≥ 1L and φ(t) = ρeαt, ρ > 0. If‖u′(t)− Au(t)− f(t)−∫ t0a(t− s)Au(s)ds‖ ≤ φ(t),for appropriate f and a(.), then with L1 = ‖A‖, and L2 = M‖A‖, by Theo-rem 2, there exists a unique solution u0(t) of (18) such that‖u(t)− u0(t)‖+ ‖u′(t)− u′0(t)‖ ≤1 + L1− 1− (L1 + (L1 + L2)L+ L2TL)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 isa 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 functionfrom [0, T ] into B(X ). For x0 ∈ D(A), consider the integral equation(19) u(t) = T(t)x0 +∫ t0T(t− s)B(s)u(s)ds, t ∈ [0, T ].STABILITY OF THE VOLTERRA... 19This equation has a solution (see 9.21 of [7]). Define g : [0, T ]×X → X andK : [0, T ]× [0, T ]×X → X , byg(t, x) = T(t)x0, K(t, s, x) = T(t− s)B(s)x.Trivially for any x, y ∈ X , ‖g(t, x) − g(t, y)‖ = 0. Also from TheoremI.2.2[21], we know that there existM,ω > 0, such that for all t ≥ 0, ‖T(t)‖ ≤Metw. On the other hand continuity of B : [0, T ] → B(X ) implies that‖B(s)‖ ≤M0, for some M0 > 0 and all s ∈ [0, T ]. Thus‖K(t, s, x)−K(t, s, y)‖ = ‖T(t− s)B(s)(x− y)‖ ≤MM0eTω‖x− y‖.Now suppose 0 < L ≤ 1MM0eTω. For a fixed α ≥ 1L and ρ > 0, if φ(t) = ρeαt,then the conditions (13) hold. Thus if‖u(t)− T(t)x0 −∫ t0T(t− s)B(s)u(s)ds‖ ≤ φ(t), t ∈ [0, T ],by Theorem 3, there exists a unique solution u0(t) of (19) such that‖u(t)− u0(t)‖ ≤ 11−MM0eTωLφ(t).Hence it conclude the Hyers-Ulam Stability of (19).References[1] A. F. Bachurskaya, Uniqueness and convergence of successive approximationsfor a class of Volterra equations, Differentsial’nye Uraveniya 10(9) (1974),pp. 1722- 1724.[2] L. Cădariu and V. Radu, On the stability of the Cauchy functional equation: a fixedpoint approach, in Iteration Theory (ECIT Š02), vol. 346 of Grazer Math. Ber., pp.43-52, Karl-Franzens-Univ. Graz, Graz, Austria, 2004.[3] L. Cădariu and V. Radu, Fixed points and the stability of Jensen’s functional equa-tion, J. Inequal. Pure Appl. Math., 4, no. 1, Art. 4 (2003).[4] A. Constantin, Topological transversality: Application to an integrodifferential equa-tion, J. Math. Anal. Appl. 197 (1996), pp. 855-863.[5] C. Corduneanu, Integral Equations and Stability of Feedback Systems, AcademicPress, New York, 1993.[6] S. Czerwik, Stability of Functional Equations of Ulam–Hyers–Rassias Type, HadronicPress, Palm Harbor, Florida, 2003.[7] K.J. Engle and R. Nagle, One-parameter Semigroups for Linear Evaluation Equa-tions, Springer-Verlag, New York 2000.[8] G.L. Forti, Hyers-Ulam stability of functional equations in several variables, Aequa-tiones Math. 50 , no. 1-2 (1995), pp. 143-190.[9] D.H. Hyers and Th.M. Rassias, Approximate homomorphisms, Aequationes Math.44 (1992), pp. 125-153.[10] D.H. Hyers, G. Isac and Th.M. Rassias, Stability of Functional Equations in SeveralVariables, Birkhäuser, Basel, 1998.20 M. JANFADA AND GH. SADEGHI[11] S.-M. Jung, A fixed point approach to the stability of isometries, J. Math. Anal. Appl.,329, no. 2(2007), pp. 879-890.[12] S.-M. Jung, A fixed point approach to the stability of a Volterra integral equation,Fixed Point Theory Appl., 2007, Art. ID 57064, 9 pp.[13] S.-M. Jung, Hyers–Ulam–Rassias Stability of Functional Equations in MathematicalAnalysis, Hadronic Press lnc., Palm Harbor, Florida, 2001.[14] M. A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equa-tions, Pergamon Press, Oxford, 1964.[15] M. Kwapisz, On the existence and uniqueness of solutions of a certain integral- func-tional equation, Ann. Polon. Math. 31 (1975), pp. 23-41.[16] B. Margolits and J. Diaz, A fixed point theorem of the alternative for contractions ona generalized complete metric space, Bull. Amer. Math. Soc., 74 (1968), pp. 305-309.[17] R. K. Miller, Nonlinear Volterra Integral Equations, W. A. Benjamin, Menlo Park,CA, 1971.[18] B. G. Pachpatte, Inequalities for Differential and Integral Equations, Academic Press,New York, 1998.[19] B. G. Pachpatte, On a certain iterated Volterra integrodifferential equation, An. Sti.Univ. Al. I. Cuza Iasi, Tomul LIV (2008), pp. 175-186.[20] B. G. Pachpatte, On certain Volterra integral and Volterra integrodifferential equa-tions, Facta Univ., Ser. Math. Inform. 23, (2008), pp. 1-12.[21] A. Pazy, Semigroups of Operators and Applications to Partial Differential Equations,Springer-Verlag, New York 1983.[22] V. Radu, The fixed point alternative and the stability of functional equations, FixedPoint Theory, 4, no. 1 (2003), pp. 91-96.[23] Th. M. Rassias, On the stability of functional equations and a problem of Ulam, ActaAppl. Math. 62, no. 1(2000), pp. 23-130.[24] Th. M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Aca-demic Publishers, Dordrecht, Boston and London, 2003.[25] R. Saadati, S. M. Vaezpour, and B. E. Rhoades , T-Stability Approach to VariationalIteration Method for Solving Integral Equations, Fixed Point Theory Appl. 2009, Art.ID 393245, 9pp.Mohammad JanfadaDepartment of Pure Mathematics, Ferdowsi University of MashhadMashhad, P.O. Box 1159-91775, IranE-mail: mjanfada@gmail.comGh. SadeghiDepartment of Mathematics, Hakim Sabzevary UniversitySabzevar, P.O. Box 397, IranE-mail: ghadir54@yahoo.com

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Stability of the Volterra Integrodifferential Equation

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Stability of the Volterra Integrodifferential Equation

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