Journal of Inequalities in Pure and Applied

A NUMERICAL METHOD IN TERMS OF THE THIRD FOR A DELAY EQUATION FROM BIOMATHEMATICS volume 6, issue 2, article 42, 2005. Received 02 March, 2003; ALEXANDRU BICA AND CRACIUN˘ IANCU accepted 01 March, 2005. Communicated by: S.S. Dragomir Department of Mathematics University of Oradea Str. Armatei Romane 5 Oradea, 3700, Romania. EMail: [email protected] Abstract Faculty of Mathematics and Informatics Babe¸s-Bolyai University Contents Cluj-Napoca, Str. N. Kogalniceanu no.1 Cluj-Napoca, 3400, Romania. EMail: [email protected] JJ II J I Home Page Go Back Close c 2000 Victoria University ISSN (electronic): 1443-5756 Quit 024-03 Abstract

This paper presents a numerical method for approximating the positive, bounded and smooth solution of a delay integral equation which occurs in the study of the spread of epidemics. We use the cubic spline interpolation and obtain an algorithm based on a perturbed trapezoidal quadrature rule.

2000 Mathematics Subject Classification: Primary 45D05; Secondary 65R32, A Numerical Method in Terms of 92C60. the Third Derivative for a Delay Key words: Delay Integral Equation, Successive approximations, Perturbed trape- Integral Equation from zoidal quadrature rule. Biomathematics Contents Alexandru Bica and Craciun˘ Iancu

1 Introduction ...... 3 Title Page 2 Existence and Uniqueness of the Positive Smooth Solution .. 4 3 Approximation of the Solution on the Initial Interval ...... 7 Contents 4 Main Results ...... 11 JJ II References J I Go Back Close Quit Page 2 of 33

J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au 1. Introduction Consider the delay integral equation:

Z t (1.1) x(t) = f(s, x(s))ds. t−τ

This equation is a mathematical model for the spread of certain infectious diseases with a contact rate that varies seasonally. Here x(t) is the proportion of infectives in the population at time t, τ > 0, is the lenght of time in which an A Numerical Method in Terms of the Third Derivative for a Delay individual remains infectious and f(t, x(t)) is the proportion of new infectives Integral Equation from per unit time. Biomathematics There are known results about the existence of a positive bounded solution Alexandru Bica and Craciun˘ Iancu (see [3], [8]), which is periodic in certain conditions ([9]), or about the exis- tence and uniqueness of the positive periodic solution (in [10], [11]). In [8] the Title Page author obtains the existence and uniqueness of the continuous positive bounded solution, and in [6] a numerical method for the approximation of this solution Contents is provided using the trapezoidal quadrature rule. JJ II Here, we obtain the existence and uniqueness of the positive, bounded and smooth solution and use the cubic spline of interpolation from [5] to approx- J I imate this solution on the initial interval [−τ, 0]. We suppose that on [−τ, 0], Go Back the solution Φ is known only in the discrete moments t , i = 0, n, and use the i Close values Φ(ti) = yi, i = 0, n for the spline interpolation. Afterward, we outline a numerical method and an algorithm to approximate the solution on [0,T ], with Quit T > 0 fixed, using the quadrature rule from [1] and [2]. Page 3 of 33

J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au 2. Existence and Uniqueness of the Positive Smooth Solution Impose the initial condition x(t) = Φ(t), t ∈ [−τ, 0] to equation (1.1) and consider T > 0, be fixed. We obtain the initial value problem:

 R t f(s, x(s))ds, ∀t ∈ [0,T ]  t−τ (2.1) x(t) =  Φ(t), ∀t ∈ [−τ, 0]. A Numerical Method in Terms of the Third Derivative for a Delay Suppose that the following conditions are fulfilled: Integral Equation from Biomathematics 1 (i) Φ ∈ C [−τ, 0] and we have Alexandru Bica and Craciun˘ Iancu

0 Z 0 b = Φ(0) = f(s, x(s))ds with Φ (0) = f(0, b) − f(−τ, Φ(−τ)); Title Page −τ Contents (ii) b > 0 and ∃a, M, β ∈ R, M > 0, 0 < a ≤ β such that a ≤ Φ(t) ≤ β, ∀t ∈ [−τ, 0]; JJ II (iii) f ∈ C([−τ, T ] × [a, β]), f(t, x) ≥ 0, f(t, y) ≤ M, ∀t ∈ [−τ, T ], ∀x ≥ J I 0, ∀y ∈ [a, β]; Go Back (iv) Mτ ≤ β and there is an integrable g such that f(t, x) ≥ g(t), ∀t ∈ Close [−τ, T ], ∀x ≥ a and Quit Z t Page 4 of 33 g(s)ds ≥ a, ∀t ∈ [0,T ]; t−τ J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au (v) ∃L > 0 such that |f(t, x) − f(t, y)| ≤ L |x − y| , ∀t ∈ [−τ, T ], ∀x, y ∈ [a, ∞).

Then, we obtain the following result: Theorem 2.1. Suppose that assumptions (i)-(v) are satisfied. Then the equation (1.1) has a unique continuous solution x(t) on [−τ, T ], with a ≤ x(t) ≤ β, ∀t ∈ [−τ, T ] such that x(t) = Φ(t) for t ∈ [−τ, 0]. Also,

max {|xn(t) − x(t)| : t ∈ [0,T ]} −→ 0 A Numerical Method in Terms of the Third Derivative for a Delay as n → ∞ where x (t) = Φ(t) for t ∈ [−τ, 0] , n ∈ , x (t) = b and Integral Equation from n N 0 Biomathematics t Z Alexandru Bica and Craciun˘ Iancu xn(t) = f(s, xn−1(s))ds t−τ Title Page for t ∈ [0,T ] , n ∈ N∗. Moreover, the solution x belongs to C1[−τ, T ]. Contents Proof. From [9] under the conditions (i), (ii), (iv), (v), it follows that the exis- tence of an unique positive continuous on [−τ, T ] solution for (1.1) such that JJ II x(t) ≥ a, ∀t ∈ [−τ, T ] and x(t) = Φ(t) for t ∈ [−τ, 0]. Using Theorem 2 from J I [7] we conclude that max {|xn(t) − x(t)| : t ∈ [0,T ]} −→ 0 as n → ∞ . From (iv) we see that Go Back Z t Z t Close x(t) = f(s, x(s))ds ≤ Mds = Mτ ≤ β, ∀t ∈ [0,T ]. Quit t−τ t−τ Page 5 of 33 Because a ≤ Φ(t) ≤ β for t ∈ [−τ, 0] and x(t) = Φ(t) for t ∈ [−τ, 0] we deduce that a ≤ x(t) ≤ β, ∀t ∈ [−τ, T ], and the solution is bounded. Since J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au R t x is a solution for (1.1) we have x(t) = t−τ f(s, x(s))ds, for all t ∈ [0,T ], and because f ∈ C([−τ, T ] × [a, β]) we can state that x is differentiable on [0,T ] , and x0 is continuous on [0,T ]. From condition (iii) it follows that x is differentiable with x0 continuous on [−τ, 0] (including the continuity in the point t = 0). Then x ∈ C1[−τ, T ] and the proof is complete. Corollary 2.2. In the conditions of the Theorem2.1, if f ∈ C1([−τ, T ] × [a, β]), Φ ∈ C2[−τ, 0] and

∂f ∂f A Numerical Method in Terms of Φ00(0) = (0, b) + (0, b)[f(0, b) − f(−τ, , Φ(−τ))] the Third Derivative for a Delay ∂t ∂x Integral Equation from ∂f ∂f Biomathematics − (−τ, Φ(−τ)) − (−τ, Φ(−τ)) Φ0(−τ), ∂t ∂x Alexandru Bica and Craciun˘ Iancu then x ∈ C2[−τ, T ]. Title Page Proof. Follows directly from the above theorem. Contents JJ II J I Go Back Close Quit Page 6 of 33

J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au 3. Approximation of the Solution on the Initial Interval Suppose that on the interval [−τ, 0] the function Φ is known only in the points, ti, i = 0, n, which form the uniform partition

(3.1) ∆n : −τ = t0 < t1 < ··· < tn−1 < tn = 0,

τ where we have the values Φ(ti) = yi, i = 0, n and ti+1 − ti = h = n , A Numerical Method in Terms of ∀i = 0, n − 1. the Third Derivative for a Delay Integral Equation from Let Biomathematics

1  ∆2y ∆3y ∆4y  Alexandru Bica and Craciun˘ Iancu m = ∆y − 0 + 0 − 0 , and 0 h 0 2 3 4  4  1 2 3 11∆ y0 Title Page M0 = 2 ∆ y0 − ∆ y0 + , h 12 Contents where, JJ II 2 ∆y0 = y1 − y0, ∆ y0 = y2 − 2y1 + y0, J I 3 4 ∆ y0 = y3 − 3y2 + 3y1 − y0, ∆ y0 = y4 − 4y3 + 6y2 − 4y1 + y0. Go Back Close We build a cubic spline of interpolation which corresponds to the following conditions: Quit

0 00 Page 7 of 33 (3.2) Φ(ti) = yi, i = 0, n, Φ (t0) = m0, Φ (t0) = M0.

J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au This spline function is s ∈ C2[−τ, 0] which, according to [5], have on the each subinterval [ti−1, ti], i = 1, n, the expression:

Mi − Mi−1 3 Mi−1 2 (3.3) s(t) = · (t − ti−1) + · (t − ti−1) 6hi 2 + mi−1 · (t − ti−1) + yi−1, 0 00 for t ∈ [ti−1, ti] where yi = s(ti), mi = s (ti),Mi = s (ti), i = 0, n. For these values, according to [5], there exists the recurence relations:  A Numerical Method in Terms of yi − yi−1 6mi−1  Mi = 6 · − − 2Mi−1 the Third Derivative for a Delay  h2 h Integral Equation from (3.4) , i = 1, n. Biomathematics  yi − yi−1 1  m = 3 · − 2m − M · h Alexandru Bica and Craciun˘ Iancu  i h i−1 2 i−1 Then, from [5] Lemma 2.1, there exists a unique cubic spline function of interpolation, s, which satisfy the conditions: Title Page  s(t ) = y , ∀i = 0, n Contents  i i  0 (3.5) s (t0) = m0 JJ II  00  s (t0) = M0, J I and on the each subinterval of ∆n is defined by the relation (3.3). Go Back 0 00 Between the values of s and s on the knots there exists the relations: Close  yi − yi−1 − mi−1 · h  Mi + 2Mi−1 = 6 · Quit  h2 (3.6) , i = 1, n. Page 8 of 33  2 (mi − mi−1)  Mi + Mi−1 = h J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au This spline function s approximates the solution Φ on the interval [−τ, 0] and we have s(ti) = Φ(ti) = yi, ∀i = 0, n. Remark 1. Since Φ ∈ C2[−τ, 0], we can estimate the error of this approxima- tion, kΦ − sk, where k·k is the Cebyševˇ norm on the set of continuous functions on an compact interval of the real axis: kuk = max{|u(t)| : t lies in an com- pact interval}, for any u on this interval. If we know the 1 00 R 0 00 2  2 value kΦ k2 = −τ [Φ (t)] dt , then for each t ∈ [−τ, 0] we have A Numerical Method in Terms of 3 √ 3 the Third Derivative for a Delay 00 2 2 00 2 |Φ(t) − s(t)| ≤ kΦ − sk ≤ kΦ k2 · h ≤ τ kΦ k · h , Integral Equation from Biomathematics according to [4] page 127. Else, if we know only the values yi = Φ(ti), i = 0, n Alexandru Bica and Craciun˘ Iancu then |Φ(t) − s(t)| ≤ kΦ − sk , ∀t ∈ [−τ, 0], and for kΦ − sk we use the error estimation from [7], since Φ ∈ C2[−τ, 0] and then Φ is a Lipschitzian function on [−τ, 0]. Title Page In [7], it has been shown that Contents

kΦ − sk ≤ max{ks − F1k , ks − F2k} JJ II J I where Go Back

ks − F1k = max{ai : i = 1, n} Close ks − F k = max{b : i = 1, n} 2 i Quit Page 9 of 33

J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au and

ai = (s − F1) |[ti−1,ti] ,

bi = (s − F2) |[ti−1,ti] ,

F1(t) = sup{Φ(tk) − kΦkL · |t − tk| : k = 0, n},

F2(t) = inf{Φ(tk) + kΦkL · |t − tk| : k = 0, n}, with kΦ | k = max{| [t , t ; Φ] |: i = 1, n} A Numerical Method in Terms of ∆n L i−1 i the Third Derivative for a Delay Integral Equation from if [ti−1, ti; Φ] = [Φ(ti) − Φ(ti−1)]/(ti − ti−1) is the divided difference of the Biomathematics function Φ on the knots ti−1, ti. Alexandru Bica and Craciun˘ Iancu

Title Page Contents JJ II J I Go Back Close Quit Page 10 of 33

J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au 4. Main Results From Theorem 2.1 follows that the equation (1.1) has a unique positive, bounded and smooth solution on [−τ, T ]. Let ϕ be this solution, which, by virtue of The- orem 2.1, can be obtained by successive approximations method on [0,T ]. So, we have :  ϕm(t) = Φ(t) , for t ∈ [−τ, 0], ∀m ∈ N and   R 0  ϕ0(t) = Φ(0) = b = −τ f(s, Φ(s))ds, A Numerical Method in Terms of  the Third Derivative for a Delay  R t R t  ϕ1(t) = f(s, ϕ0(s))ds = f(s, b)ds, Integral Equation from  t−τ t−τ Biomathematics (4.1) , for t ∈ [0,T ]. ···  Alexandru Bica and Craciun˘ Iancu   R t  ϕm(t) = f(s, ϕm−1(s))ds,  t−τ  Title Page  ··· Contents To obtain the sequence of successive approximations (4.1 ) we compute the JJ II using a quadrature rule. ∗ J I We assume that there is l ∈ N such that T = lτ. On each interval [iτ, (i + 1)τ], i = 0, l − 1 we establish an equidistant partition. Then on the interval Go Back [−τ, T ] q = l · n + n + 1 we have knots which realize the division: Close

(4.2) −τ = t0 < t1 < ··· < tn−1 < 0 = tn < tn+1 < ··· < tq−1 < tq = T, Quit Page 11 of 33 having ti+1 − ti = h, ∀i = n, q − 1. We can see that tj − τ = tj−n, ∀j = n, q.

J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au In the aim to compute the integrals from (4.1) we use the following quadra- ture rule of N.S. Barnett and S.S. Dragomir in [1]:

n−1 Z b (b − a) X (4.3) F (t)dt = [F (t ) + F (t )] 2n i i+1 a i=0 (b − a)2 − [F 0(a) − F 0(b)] + R (F ), 12n2 n where b − a A Numerical Method in Terms of t = a + i · , i = 0, n, the Third Derivative for a Delay i n Integral Equation from Biomathematics and 4 (b − a) Alexandru Bica and Craciun˘ Iancu |R (F )| ≤ · kF 000k , if F ∈ C3[a, b]. n 160n3 Here, we consider the function F defined by F (t) = f(t, x(t)), for t ∈ [−τ, T ]. Title Page Since the solution of (2.1) verify the relation, Contents x0(t) = f(t, x(t)) − f(t − τ, x(t − τ)), ∀t ∈ [0,T ], JJ II we point out the following connection between the smoothness of x and f. J I Remark 2. In the conditions of Corollary 2.2, if Φ ∈ C3[−τ, 0] , f ∈ C2([−τ, T ]× Go Back [a, β]), and  Close 000 d ∂f ∂f 0 Φ (0) = lim (t, ϕ(t)) + (t, ϕ(t))ϕ (t) Quit t>0,t→0 dt ∂t ∂x ∂f ∂f  Page 12 of 33 − (t − τ, ϕ(t − τ)) − (t − τ, ϕ(t − τ)) ϕ0(t − τ) , ∂t ∂x J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au then x ∈ C3[−τ, T ]. If x ∈ C3[−τ, T ] and f ∈ C3([−τ, T ] × [a, β]) then F ∈ C3[−τ, T ] and 000 000 F (t) = [f(t, x(t))]t , ∀t ∈ [−τ, T ]. For this function F we apply the quadra- ture rule (4.3) and obtain the approximative values of the solution ϕ at the points tk, k = n + 1, q, as in the following formula:

Z tk (4.4) ϕm(tk) = f(s, ϕm−1(s))ds tk−τ A Numerical Method in Terms of n−1 τ X the Third Derivative for a Delay = [f(tk+i − τ, ϕm−1(tk+i − τ)) Integral Equation from 2n Biomathematics i=0

+ f(tk+i+1 − τ, ϕm−1(tk+i+1 − τ))] Alexandru Bica and Craciun˘ Iancu τ 2 ∂f − (tk, ϕm−1(tk)) 12n2 ∂t Title Page ∂f + (t , ϕ (t )) · ϕ0 (t ) Contents ∂x k m−1 k m−1 k ∂f JJ II − (tk − τ, ϕm−1(tk − τ)) ∂t J I ∂f  − (t − τ, ϕ (t − τ)) · ϕ0 (t − τ) + r(n) (f), ∂x k m−1 k m−1 k m,k Go Back Close ∀ m ∈ ∗ and ∀k = n + 1, q, where, N Quit 0 ϕm−1(t) = f(t, ϕm−2(t))−f(t−τ, ϕm−2(t−τ)), ∀t ∈ [0,T ], ∀m ∈ N, m ≥ 2, Page 13 of 33 and ϕ0 (t) = 0, ϕ0 (t) = f(t, b) − f(t − τ, b), ∀t ∈ [0,T ]. 0 1 J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au To estimate the remainder we need to obtain an upper bound for the third 000 derivative [f(t, x(t))]t . After elementary we have for t ∈ [−τ, T ]:

∂3f ∂3f [f(t, x(t))]000 = (t, x(t)) + 3 (t, x(t)) · x0(t) t ∂t3 ∂t2∂x ∂3f ∂3f + 3 (t, x(t)) · [x0(t)]2 + (t, x(t)) [x0(t)]3 ∂t∂x2 ∂x3 ∂2f ∂2f + 3 (t, x(t))x00(t) + 3 (t, x(t))x0(t)x00(t) ∂t∂x ∂x2 A Numerical Method in Terms of ∂f the Third Derivative for a Delay + (t, x(t))x000(t). Integral Equation from ∂x Biomathematics We denote Alexandru Bica and Craciun˘ Iancu

M0 = M = max {|f(t, x)| : t ∈ [−τ, T ], x ∈ [a, β]} Title Page α  |α|  ∂ f ∂ f(t, x) Contents = max : t ∈ [−τ, T ], x ∈ [a, β], α1 + α2 = |α| ∂tα1 ∂xα2 ∂tα1 ∂xα2   JJ II ∂f ∂f M1 = max , , ∂t ∂x J I  2 2 2  ∂ f ∂ f ∂ f Go Back M2 = max 2 , , 2 , ∂t ∂t∂x ∂x Close  3 3 3 3  ∂ f ∂ f ∂ f ∂ f M3 = max , , , . Quit ∂t3 ∂t2∂x ∂t∂x2 ∂x3 Page 14 of 33

J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au Consequently, we obtain the estimations:

000 2 3 [f(t, x(t)]t ≤ M3(1 + 6M0 + 12M0 + 8M0 ) 2 + M2(8M1 + 32M0M1 + 32M0 M1) 3 000 + 4M1 + 8M0M3 = M , ∀t ∈ [−τ, T ], and τ 4M 000 (n) ∗ rm,k(f) ≤ , ∀m ∈ N , ∀k = n + 1, q. 160n3 A Numerical Method in Terms of the Third Derivative for a Delay Then, to compute the integrals (4.1), we can use the following algorithm: Integral Equation from Biomathematics

Alexandru Bica and Craciun˘ Iancu ϕ1(tk) n−1 τ X Title Page = [f(t − τ, ϕ (t − τ)) 2n k+i 0 k+i i=0 Contents +f(t − τ, ϕ (t − τ))] k+i+1 0 k+i+1 JJ II τ 2 ∂f ∂f − (t , ϕ (t )) + (t , ϕ (t )) · ϕ0 (t ) 12n2 ∂t k 0 k ∂x k 0 k 0 k J I  Go Back ∂f ∂f 0 − (tk − τ, ϕ0(tk − τ)) − (tk − τ, ϕ0(tk − τ)) · ϕ0(tk − τ) ∂t ∂x Close (n) + r1,k (f) Quit (n) =: ϕf1(tk) + r1,k (f), Page 15 of 33

J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au ϕ2(tk) n−1 τ X = [f(t − τ, ϕ (t − τ)) 2n k+i 1 k+i i=0

(4.5) +f(tk+i+1 − τ, ϕ1(tk+i+1 − τ))] τ 2 ∂f ∂f − (t , ϕ (t )) + (t , ϕ (t )) · ϕ0 (t ) 12n2 ∂t k 1 k ∂x k 1 k 1 k  ∂f ∂f 0 − (tk − τ, ϕ1(tk − τ)) − (tk − τ, ϕ1(tk − τ)) · ϕ1(tk − τ) A Numerical Method in Terms of ∂t ∂x the Third Derivative for a Delay (n) Integral Equation from + r1,k (f) Biomathematics n−1 Alexandru Bica and Craciun˘ Iancu τ X h (n) = f(t − τ, ϕ (t − τ) + r (f)) 2n k+i f1 k+i 1,k+i−n i=0 (n) i Title Page + f(tk+i+1 − τ, ϕf1(tk+i+1 − τ) + r1,k+i+1−n(f)) Contents τ 2 ∂f − · (t , ϕ (t ) + r(n)(f)) 12n2 ∂t k f1 k 1,k JJ II ∂f (n) J I + (tk, ϕf1(tk) + r1,k (f)) · (f(tk, b) − f(tk − τ, b)) ∂x Go Back ∂f − (t − τ, ϕ (t − τ) + r(n) (f)) ∂t k f1 k 1,k−n Close ∂f  − (t − τ, ϕ (t − τ) + r(n) (f))(f(t − τ, b) − f(t − 2τ, b)) Quit ∂x k f1 k 1,k−n k k Page 16 of 33 (n) + r2,k (f) J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au n−1 τ X = [f(t − −τ, ϕ (t − τ)) + f(t − τ, ϕ (t − τ))] 2n k+i f1 k+i k+i+1 f1 k+i+1 i=0 τ 2 ∂f ∂f − · (t , ϕ (t )) + (t , ϕ (t )) · (f(t , b) − f(t − τ, b)) 12n2 ∂t k f1 k ∂x k f1 k k k ∂f − (t − τ, ϕ (t − τ)) ∂t k f1 k  ∂f ^(n) − (tk − τ, ϕ1(tk − τ)) · (f(tk − τ, b) − f(tk − 2τ, b)) + r2,k (f) ∂x f A Numerical Method in Terms of the Third Derivative for a Delay ^(n) Integral Equation from =: ϕ^2(tk) + r2,k (f), ∀k = n + 1, q. Biomathematics

We have the remainder estimation: Alexandru Bica and Craciun˘ Iancu 4 000  2  (n) τ M τ M2 (1 + 2M) rg(f) ≤ 1 + τL + , ∀k = n + 1, q. 2,k 160n3 6n2 Title Page By induction, for m ≥ 3 we obtain: Contents

(4.6) ϕm(tk) JJ II n−1  τ X (n) J I = f(t − τ, ϕ (t − τ) + r ^ (f)) 2n k+i ]m−1 k+i m−1,k+i−n i=0 Go Back + f(tk+i+1 − τ, ϕ]m−1(tk+i+1 − τ, Close i Quit ϕ]m−1(tk+i+1 − τ) + rm−1^,k+i+1−n(f)) τ 2 ∂f ∂f Page 17 of 33 − · (t , ϕ (t ) + r^(n) (f)) + (t , ϕ (t ) 12n2 ∂t k ]m−1 k m−1,k ∂x k ]m−1 k J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au ∂f + r^(n) (f)) · ϕ0 (t ) − (t − τ, ϕ (t − τ) + r(^n) (f)) m−1,k m−1 k ∂t k ]m−1 k m−1,k−n ∂f  − (t − τ, ϕ (t − τ) + r(^n) (f)) · ϕ0 (t − τ) ∂x k ]m−1 k m−1,k−n m−1 k n−1  τ X (n) = f(t − τ, ϕ (t − τ) + r ^ (f)) 2n k+i ]m−1 k+i m−1,k+i−n i=0 i ^ A Numerical Method in Terms of +f(tk+i+1 − τ, ϕ]m−1(tk+i+1 − τ) + rm−1,k+i+1−n(f)) the Third Derivative for a Delay Integral Equation from 2  Biomathematics τ ∂f ^(n) ∂f − · (tk, ϕ]m−1(tk) + rm−1,k(f)) + (tk, ϕ]m−1(tk) 12n2 ∂t ∂x Alexandru Bica and Craciun˘ Iancu

^(n) ^(n) + rm−1,k(f)) · (f(tk, ϕ]m−2(tk) + rm−2,k(f)) Title Page ∂f − f(t − τ, ϕ (t − τ) + r(^n) (f)) − (t − τ, ϕ (t − τ) Contents k ]m−2 k m−2,k−n ∂t k ]m−1 k JJ II ∂f + r(^n) (f)) − (t − τ, ϕ (t − τ) m−1,k−n ∂x k ]m−1 k J I Go Back (^n) (^n) + rm−1,k−n(f)) · (f(tk − τ, ϕ]m−2(tk − τ) + rm−2,k−n(f)) Close  (n^) (n) Quit − f(tk − 2τ, ϕ]m−2(tk − 2τ) + rm−2,k−2n(f))) + rm,k(f) Page 18 of 33

J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au n−1 τ X = f t − τ, ϕ (t − τ) + f t − τ, ϕ (t − τ) 2n k+i ]m−1 k+i k+i+1 ]m−1 k+i+1 i=0 τ 2 ∂f ∂f − · t , ϕ (t ) + t , ϕ (t ) · (f t , ϕ (t ) 12n2 ∂t k ]m−1 k ∂x k ]m−1 k k ]m−2 k ∂f − f(t − τ, ϕ (t − τ)) − t − τ, ϕ (t − τ) k ]m−2 k ∂t k ]m−1 k ∂f  A Numerical Method in Terms of − tk − τ, ϕ]m−1(tk − τ) · (f(tk − τ), ϕ]m−2(tk − τ) the Third Derivative for a Delay ∂x Integral Equation from  Biomathematics  ^(n) − f tk − 2τ, ϕm−2(tk − 2τ) ) +r (f) ] m,k Alexandru Bica and Craciun˘ Iancu

^(n) =: ϕfm(tk) + rm,k(f), Title Page Contents ∀m ∈ N , m ≥ 2 , ∀k = n + 1, q. JJ II Remark 3. We can see that, for k = n, 2n we have tk − τ ∈ [−τ, 0] and then J I 0 0 0 Go Back ϕm−1(tk − τ) = Φ (tk − τ) = s (tk − τ) = mk−n Close for m ∈ N, m ≥ 2 in (4.5) and (4.6). Quit Page 19 of 33

J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au For the remainders we have the estimations:

(n) (4.7) rg(f) m,k 4 000  2  τ M (n) τ M2(1 + 2M) ≤ + r^ (f) · τL + 160n3 m−1,k 6n2 2 τ LM2 (n) (n) + r^ (f) · r^ (f) 3n2 m−1,k m−2,k 4 000 τ M m−1 m−1 A Numerical Method in Terms of ≤ 3 1 + τL + ... + τ L the Third Derivative for a Delay 160n Integral Equation from m m−2 4 000   τ L M2 · (2M + 1) τ M 1 Biomathematics + 2 · 3 + O 5 6n 160n n Alexandru Bica and Craciun˘ Iancu 4 000 m m 6 000 m−2 m−2 τ M (1 − τ L ) τ M τ L M2 · (2M + 1) = 3 + 5 160n (1 − τL) 960n Title Page  1  + O , Contents n5 JJ II ∀m ∈ N, m ≥ 3, ∀k = n + 1, q. For instance, if m = 3 we have the estimation: J I 4 000 Go Back ^(n) τ M 2 2 (4.8) r3,k (f) ≤ 1 + τL + τ L 160n3 Close τ 6M 000M · (1 + 2τL) (2M + 1) τ 8M 000M 2 · (2M + 1)2 + 2 + 2 Quit 960n5 5760n7 τ 10(M 000)2M L · (1 + τL) τ 12(M 000)2M 2L · (2M + 1) Page 20 of 33 + 2 + 2 , 76800n8 460800n10 J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au ∀k = n + 1, q. We obtain the following result: Theorem 4.1. Considering the initial value problem (2.1) under the conditions from Corollary 2.2 and Remark 2, if f ∈ C3 ([−τ, T ] × [a, β]) , τL < 1 and the exact solution ϕ is approximated by the sequence (ϕm(tk)) ∗ , k = 1, q , on f k∈N the equidistant points (3.1), through the successive approximation method (4.1), combined with the quadrature rule (4.3), then the following error estimation holds : A Numerical Method in Terms of τ m · Lm τ 4M 000 the Third Derivative for a Delay (4.9) |ϕ(t ) − ϕ (t )| ≤ · kϕ − ϕ k + Integral Equation from k fm k 0 1 C[0,T ] 3 1 − τL 160n (1 − τL) Biomathematics τ 6M 000τ m−2Lm−2M · (2M + 1)  1  + 2 + O , Alexandru Bica and Craciun˘ Iancu 960n5 n5 ∗ ∀m ∈ N , m ≥ 2, ∀k = n + 1, q. Title Page Proof. We have Contents |ϕ(t ) − ϕ (t )| ≤ |ϕ(t ) − ϕ (t )|+|ϕ (t ) − ϕ (t )| , ∀m ∈ ∗, ∀k = n, q. k fm k k m k m k fm k N JJ II From Banach’s fixed point principle we have J I τ m · Lm |ϕ(t ) − ϕ (t )| ≤ kϕ − ϕ k ≤ · kϕ − ϕ k , Go Back k m k m 1 − τL 0 1 C[0,T ] Close ∀m ∈ N∗, ∀k = n + 1, q. Also, Quit g(n) ∗ |ϕm(tk) − ϕm(tk)| ≤ rm,k(f) , ∀m ∈ N , m ≥ 2, ∀k = n + 1, q. f Page 21 of 33 Using the remainder estimation (4.7), the proof is completed. J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au If F 000 is Lipschitzian we can use a recent formula of N.S. Barnett and S.S. Dragomir from Corollary 1 in [2], Z b (b − a) (b − a)2 F (t)dt = [F (a) + F (b)] − [F 0(b) − F 0(a)] + R(F ), a 2 12 L(b−a)5 with |R(F )| ≤ 720 , where L is the Lipschitz constant. A composite quadra- ture formula can be easy obtained, considering an uniform partition of [a, b] b−a b−a with the knots ti = a + i · n , i = 0, n and the step h = n , A Numerical Method in Terms of n−1 Z b 2 the Third Derivative for a Delay (b − a) X (b − a) 0 0 Integral Equation from F (t)dt = [F (ti) + F (ti+1)]− 2 [F (a) − F (b)]+Rn(F ), 2n 12n Biomathematics a i=0 having the remainder estimation, Alexandru Bica and Craciun˘ Iancu L (b − a)5 |R (F )| ≤ . n 720n4 Title Page Here, we use these formulas for F (t) = f(t, x(t)) and obtain, Contents

Z tk (4.10) f(t, x(t))dt JJ II tk−τ J I Z tk = F (t)dt Go Back tk−τ n−1 Close τ X = [F (t − τ) + F (t − τ)] 2n k+i k+i+1 Quit i=0 τ 2 Page 22 of 33 − [F 0(t ) − F 0(t − τ)] + R (F ), ∀k = n + 1, q. 12n2 k k n J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au with τ 5L |R (F )| ≤ 3 , n 720n4 000 000 if ∃L3 > 0 such that |F (u) − F (v)| ≤ L3 |u − v| , ∀u, v ∈ [−τ, T ]. From (4.7) and (4.3) we see that the relations (4.5) and (4.6) remains unchanged in this case, and for the remainders the estimations (4.7) becomes:

τ 5L r(n)(f) ≤ 3 , ∀k = n + 1, q 1,k 4 720n A Numerical Method in Terms of the Third Derivative for a Delay 5  2  (n) τ L3 τ M2 (1 + 2M) Integral Equation from rg(f) ≤ 1 + τL + , ∀k = n + 1, q Biomathematics 2,k 720n4 6n2 Alexandru Bica and Craciun˘ Iancu

(n) (4.11) rg(f) m,k Title Page 5  2  τ L3 (n) τ M2(1 + 2M) Contents ≤ + r^ (f) · τL + 720n4 m−1,k 6n2 2 JJ II τ LM2 (n) (n) + r^ (f) · r^ (f) 3n2 m−1,k m−2,k J I 5 m m 7 m−2 m−2   τ L3(1 − τ L ) τ L3τ L M2 · (2M + 1) 1 Go Back ≤ 4 + 6 + O 6 , 720n (1 − τL) 4320n n Close

∀m ∈ N∗, ∀k = n + 1, q. Quit Page 23 of 33

J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au For instance, the estimation (4.8) becomes:

5  2  (n) τ L3 (n) τ M2(1 + 2M) rg(f) ≤ + rg(f) · τL + 3,k 720n4 2,k 6n2 2 τ LM2 (n) (n) + rg(f) · r (f) 3n2 2,k 1,k τ 5L ≤ 3 1 + τL + τ 2L2 720n4 τ 7L M · (1 + 2τL) (2M + 1) τ 9L M 2 · (2M + 1)2 A Numerical Method in Terms of + 3 2 + 3 2 the Third Derivative for a Delay 4320n6 25920n8 Integral Equation from τ 12(L )2M L · (1 + τL) τ 14(L )2M 2L · (2M + 1) Biomathematics + 3 2 + 3 2 , 1825200n10 10951200n12 Alexandru Bica and Craciun˘ Iancu ∀k = n + 1, q. Title Page In this way we obtain the following result: Contents Theorem 4.2. If f ∈ C3([−τ, T ] × [α, β]), Φ ∈ C3[−τ, 0], having, JJ II d ∂f ∂f J I Φ000(0) = lim (t, ϕ(t)) + (t, ϕ(t))ϕ0(t) t>0,t→0 dt ∂t ∂x Go Back ∂f ∂f  − (t − τ, ϕ(t − τ)) − (t − τ, ϕ(t − τ)) ϕ0(t − τ) , Close ∂t ∂x Quit ∂3f ∂3f ∂3f and the functions ∂t3 , ∂t∂x2 and ∂x3 are Lipschitzian in t and x, then ϕ ∈ Page 24 of 33 3 000 C [−τ, T ],F is Lipschitzian with a Lipschitz constant L3 > 0 and the fol-

J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au lowing estimation holds: τ m · Lm τ 5L |ϕ(t ) − ϕ (t )| ≤ · kϕ − ϕ k + 3 k fm k 1 − τL 0 1 C[0,T ] 160n3 (1 − τL) τ 7L τ m−2Lm−2M · (2M + 1)  1  + 3 2 + O , 4320n6 n6 ∗ ∀m ∈ N , m ≥ 2, ∀k = n + 1, q. Proof. From Remark 2, we infer that ϕ ∈ C3[−τ, T ] and because f ∈ C3([−τ, T ] 3 ∂f A Numerical Method in Terms of ×[α, β]) we see that F ∈ C [−τ, T ] and ∂x is Lipschitzian in t and x. For this the Third Derivative for a Delay 0 reason there exist L01,L01 > 0 such that Integral Equation from Biomathematics ∂f ∂f (u, x) − (v, x) ≤ L01 |u − v| Alexandru Bica and Craciun˘ Iancu ∂x ∂x

∂f ∂f 0 (u, x) − (u, y) ≤ L |x − y| , ∂x ∂x 01 Title Page ∂3f Contents ∀u, v ∈ [−τ, T ], ∀x, y ∈ [α, β]. Since ∂t3 is Lipschitzian in t and x there exist 0 L30,L30 > 0 such that JJ II 3 3 ∂ f ∂ f (u, x(u)) − (v, x(v)) J I ∂t3 ∂t3 Go Back 3 3 3 3 ∂ f ∂ f ∂ f ∂ f ≤ (u, x(u)) − (v, x(u)) + (v, x(u)) − (v, x(v)) Close ∂t3 ∂t3 ∂t3 ∂t3 0 Quit ≤ L30 |u − v| + L30 |x(u) − x(v)| 0 0 ≤ L30 |u − v| + L30 · kx k |u − v| Page 25 of 33 0 ≤ (L30 + 2L30M) |u − v| , J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au ∀u, v ∈ [−τ, T ]. We have x0(t) = f(t, x(t)) − f(t − τ, x(t − τ)), ∀t ∈ [0,T ] 0 ∂3f ∂3f and kx k ≤ 2M. Similar, since ∂t∂x2 and ∂x3 are Lipschitzian in t and x there 0 0 exist L12,L12 > 0 and L03,L03 > 0 such that we have:

3 3 ∂ f ∂ f 0 0 (u, x(u)) − (v, x(v)) ≤ (L12 + L · kx k) |u − v| , ∂t∂x2 ∂t∂x2 12 3 3 ∂ f ∂ f 0 0 (u, x(u)) − (v, x(v)) ≤ (L03 + L · kx k) |u − v| , ∂x3 ∂x3 03 A Numerical Method in Terms of ∀u, v ∈ [−τ, T ]. Now, we can state that the Third Derivative for a Delay Integral Equation from Biomathematics ∂f ∂f 0 0 (u, x(u)) − (v, x(v)) ≤ (L01 + L · kx k) |u − v| , Alexandru Bica and Craciun˘ Iancu ∂x ∂x 01

3 0 00 ∀u, v ∈ [−τ, T ], and since x ∈ C [−τ, T ] we have that x and x are Lips- Title Page chitzian. Also, we have F 000(t) = [f(t, x(t))]000 and t Contents

3 3 000 ∂ f ∂ f 0 JJ II [f(t, x(t))]t = 3 (t, x(t)) + 3 2 (t, x(t))x (t) ∂t ∂t ∂x J I ∂3f ∂3f + 3 (t, x(t)) [x0(t)]2 + (t, x(t)) [x0(t)]3 ∂t∂x2 ∂x3 Go Back ∂2f ∂2f + 3 (t, x(t))x00(t) + 3 (t, x(t))x0(t)x00(t) Close ∂t∂x ∂x2 ∂f Quit + (t, x(t))x000(t). ∂x Page 26 of 33

J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au From all the above we can see that:

|F 000(u) − F 000(v)| 3 0 ∂ f 0 ≤ (L30 + 2L M) |u − v| + 3 · kx k |u − v| 30 ∂t2∂x 0 0 0 2 0 0 0 3 + 3(L12 + L12 · kx k) · kx k |u − v| + (L03 + L03 · kx k) · kx k · |u − v| 2 2 ∂ f 000 ∂ f 00 2 + 3 · kx k |u − v| + 3 2 · kx k |u − v| ∂t∂x ∂x A Numerical Method in Terms of 0 0 000 the Third Derivative for a Delay + (L01 + L · kx k) · kx k |u − v| 01 Integral Equation from ≤ L3 |u − v| , ∀u, v ∈ [−τ, T ]. Biomathematics Alexandru Bica and Craciun˘ Iancu Here we have

0 2 0 Title Page L3 = L30 + 2L30M + 6M1M3(2M + 1) + 12M (L12 + 2L12M) 3 0  2 + 8M (L03 + 2L03M) + 6M2(2M + 1) M2(2M + 1) + 2M1 Contents + 12M M 2(2M + 1)2 + 2(L + 2L0 M) 2 1 01 01 JJ II × (2M + 1)[M (2M + 1) + 2M 2] > 0, 2 1 J I 00 00 2 since kx k ≤ 2M1(2M + 1) and kx k ≤ 2(2M + 1)[M2(2M + 1) + 2M1 ], Go Back having for t ∈ [0,T ] Close ∂f ∂f x00(t) = (t, x(t)) + (t, x(t))x0(t) Quit ∂t ∂x ∂f ∂f Page 27 of 33 − (t − τ, x(t − τ)) − (t − τ, x(t − τ)) x0(t − τ), ∂t ∂x J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au and ∂2f ∂2f x000(t) = (t, x(t)) + (t, x(t))x0(t) ∂t2 ∂t∂x  ∂2f ∂2f  + x0(t) · (t, x(t)) + (t, x(t))x0(t) ∂t∂x ∂x2 ∂f ∂2f + x00(t) (t, x(t)) − (t − τ, x(t − τ)) ∂x ∂t2 ∂2f − (t − τ, x(t − τ)) · x0(t − τ) A Numerical Method in Terms of ∂t∂x the Third Derivative for a Delay  ∂2f Integral Equation from + x0(t − τ) · (t − τ, x(t − τ)) Biomathematics ∂t∂x Alexandru Bica and Craciun˘ Iancu ∂2f  + (t − τ, x(t − τ)) · x0(t − τ) ∂x2 ∂f Title Page + x00(t − τ) (t − τ, x(t − τ)). ∂x Contents 000 Then, F is Lipschitzian, and so we can apply the quadrature rule (4.9) from JJ II [2] and we have the inequality (4.11). Using the estimation from Banach’s fixed point principle we obtain the desired estimation. J I Remark 4. To approximate the solution ϕ on [0,T ] we can use the cubic spline Go Back of interpolation s, defined by the interpolatory conditions: Close   s(tk) = ϕfm(tk), ∀k = n + 1, q Quit  s(tn) = yn Page 28 of 33  00 00  s (tn) = Mn = 0, s (tq) = Mq = 0 J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au ∗ for a fixed m ∈ N , on the knots tk, k = n, q, as in [4, p. 119–128]:

s : [0,T ] → R, s(t) = si(t), ∀t ∈ [ti−1, ti], ∀i = n + 1, q where,

3 3 2 Mi(t − ti−1) + Mi−1(ti − t) (6ϕi−1 − Mi−1hi ) · (ti − t) si(t) = + 6hi 6hi (6ϕ − M h2) · (t − t ) i i i i−1 A Numerical Method in Terms of + , ∀t ∈ [ti−1, ti], i = n + 1, q, 6hi the Third Derivative for a Delay Integral Equation from 00 Biomathematics and ϕk = s(tk),Mk = s (tk), ∀k = n, q. The values Mk, k = n, q are obtained from the relations Alexandru Bica and Craciun˘ Iancu

h M M (h + h ) h M i i−1 + i i i+1 + i+1 i+1 Title Page 6 3 6 ϕ − ϕ ϕ − ϕ Contents = i+1 i − i i−1 , i = n + 1, q − 1, hi+1 hi JJ II 3 where Mn = Mq = 0. Since f ∈ C ([−τ, T ] × [α, β]) we infer that ϕ ∈ J I C4[0,T ]. If Φ ∈ C4[−τ, 0] and Φ(0) = lim ϕ(t) then ϕ ∈ C4[−τ, T ]. In t>0,t→0 Go Back these hypothesis, for t ∈ [0,T ]\{ t , k = n, q}, we have the following error k Close estimation: Quit m m τ · L (n) 5 IV 4 |ϕ(t) − s(t)| ≤ · kϕ0 − ϕ1k + rg(f) + · ϕ · h , Page 29 of 33 1 − τL m,k 384

J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au T τ where h = q−n−1 = n . We have xIV (t) ∂3f ∂3f ∂3f = (t, x(t)) + 3 (t, x(t)) · x0(t) + 3 (t, x(t)) [x0(t)]2 ∂t3 ∂t2∂x ∂t∂x2 ∂3f ∂2f + (t, x(t)) [x0(t)]3 + 3 (t, x(t))x00(t) ∂x3 ∂t∂x 2 ∂ f 0 00 ∂f 000 + 3 2 (t, x(t)) · x (t) · x (t) + (t, x(t)) · x (t) A Numerical Method in Terms of ∂x ∂x the Third Derivative for a Delay 3 3 ∂ f ∂ f 0 Integral Equation from − (t − τ, x(t − τ)) − 3 (t − τ, x(t − τ)) · x (t − τ) Biomathematics ∂t3 ∂t2∂x ∂3f Alexandru Bica and Craciun˘ Iancu − 3 (t − τ, x(t − τ)) · [x0(t − τ)]2 ∂t∂x2 ∂3f − (t − τ, x(t − τ)) · [x0(t − τ)]3 Title Page ∂x3 Contents ∂2f ∂2f − 3 (t − τ))x00(t − τ) − 3 (t − τ, x(t − τ)) · x0(t − τ)x00(t − τ) ∂t∂x ∂x2 JJ II ∂f − (t − τ, x(t − τ))x000(t − τ), J I ∂x Go Back ∀t ∈ [0,T ]. Since Close 0 00 k f k≤ M, kϕ k ≤ 2M, kϕ k ≤ 2M1(1 + 2M), Quit kϕ000k ≤ 2M (1 + 2M)2 + 4M 2(1 + 2M), 2 1 Page 30 of 33

J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au we have the following estimation:

IV 0 0 2 0 3 ϕ ≤ 2M3 + 6M3 kϕ k + 6M3 kϕ k + 2M3 kϕ k 00 0 00 000 + 6M2 kϕ k + 6M2 kϕ k · kϕ k + 2M1 kϕ k 3 2 3 ≤ 2M3(1 + 2M) + 16M1M2(1 + 2M) + 8M1 (1 + 2M).

A Numerical Method in Terms of the Third Derivative for a Delay Integral Equation from Biomathematics

Alexandru Bica and Craciun˘ Iancu

Title Page Contents JJ II J I Go Back Close Quit Page 31 of 33

J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au References

[1] N.S. BARNETT AND S.S. DRAGOMIR, Perturbed trapezoidal inequality in terms of the third derivative and applications, RGMIA Res. Rep. Coll., 4(2) (2001), 221–233.

[2] N.S. BARNETT AND S.S. DRAGOMIR, Perturbed trapezoidal inequality in terms of the fourth derivative and applications, Korean J. Comput. Appl. Math., 9(1) (2002), 45–60. A Numerical Method in Terms of [3] D. GUO AND V. LAKSMIKANTHAM, Positive solutions of nonlinear the Third Derivative for a Delay Integral Equation from integral equations arising in infectious diseases, J. Math. Anal. Appl., 134 Biomathematics (1988), 1–8. Alexandru Bica and Craciun˘ Iancu [4] C. IACOB (Ed.), Classic and Modern Mathematics, Ed. Tehnica,˘ Bu- cure¸sti,1983 (in Romanian). Title Page [5] C. IANCU, On the cubic spline of interpolation, Seminar of Functional Contents Analysis and Numerical Methods, Cluj-Napoca, Preprint 4, 1981, 52–71. JJ II [6] C. IANCU, A numerical method for approximating the solution of an in- tegral equation from biomathematics, Studia Univ. "Babe¸s-Bolyai",Math- J I ematica, XLIII(4) (1998). Go Back

[7] C. IANCU AND C. MUSTA¸TA,˘ Error estimation in the approximation of Close functions by interpolation cubic splines, Mathematica- Rev. Anal. Numér. Quit Théor. Approx., 29(52)(1) (1987), 33–39. Page 32 of 33

J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au [8] R. PRECUP, Positive solutions of the initial value problem for an integral equation modelling infectious diseases, Seminar of Fixed Point Theory, Cluj-Napoca, Preprint 3, 1991, 25–30. [9] R. PRECUP, Periodic solutions for an integral equation from biomathe- matics via Leray-Schauder principle, Studia Univ. ”Babes-Bolyai”, Math- ematica, XXXIX(1) (1994). [10] I.A. RUS, Principles and Applications of the Fixed Point Theory, Ed. Da- cia, Cluj-Napoca, 1979 (in Romanian). A Numerical Method in Terms of the Third Derivative for a Delay [11] I.A. RUS, A delay integral equation from biomathematics, Seminar of Dif- Integral Equation from Biomathematics ferential Equations, Cluj-Napoca, Preprint 3, 1989, 87–90. Alexandru Bica and Craciun˘ Iancu

Title Page Contents JJ II J I Go Back Close Quit Page 33 of 33

J. Ineq. Pure and Appl. Math. 6(2) Art. 42, 2005 http://jipam.vu.edu.au