KERNEL-RESOLVENT RELATIONS FOR AN INTEGRAL EQUATION

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1 Ø Ñ Å Ø Ñ Ø Ð ÈÙ Ð Ø ÓÒ DOI: 1.478/v Tatra Mt. Math. Publ. 48 (11), 5 4 KERNEL-RESOLVENT RELATIONS FOR AN INTEGRAL EQUATION Theodore A. Burton ABSTRACT. Weconideracalarintegralequationz(t) = a(t) C(t,)[ z()+ G(,z()) ] d where G(t,z) φ(t) z, C i convex, and a (L L )[, ). Related to thi i the linear equation x(t) = a(t) C(t,)x()d and the reolvent equation R(t,) = C(t,) C(t,u)R(u,)du. A Liapunov functional i contructed which give qualitative reult about all three equation. We have two goal. Firt, we are intereted in condition under which propertie of C are tranferred into propertie of the reolvent R which i ued in the variation- -of-parameter formula. We etablih condition on C and function b o that t C(t,)b()d a t and i in L [, ) impliethat R(t,)b()d a t and i in L [, ). Such reult are fundamental in proving that the olution z atifie z(t) a(t) a t and that ( ) dt z(t) a(t) <. Thi i in final form and no other verion will be ubmitted. 1. Introduction There are many important relation between the kernel, C(t, ), of the integral equation x(t) = a(t) C(t, )x() d, (1) where a i continuou for t and C i continuou for t <, and the reolvent, R(t, ), olving R(t,) = C(t,) C(t, u)r(u, ) du. () The function R i ued in the variation-of-parameter formula c 11 Mathematical Intitute, Slovak Academy of Science. 1 Mathematic Subject Claification: 47G5, 34D. Keyword: integral equation, Liapunov functional. 5

2 THEODORE A. BURTON x(t) = a(t) R(t, )a() d. (3) However, three uch relation tand out for their implicity and utility. Firt, if there i an α < 1 with then up t< up t< C(t,) d α, (4) R(t,) d α (1 α). (5) It i an old reult and it origin eem to have been lot. Among it many applicationithe immediateconcluionthata L [, )impliex L [, ) [, p. 54], where x olve (1). Next, if there i a β < 1 with then up t< up t< C(u,) du β, (6) R(u,) du β/(1 β) (7) yielding immediate L p [, ) reult for x. It wa firt proved in [, p. 54]. Notice that the econd coordinate of C may be thought of a the L coordinate, while the firt i the L p coordinate, propertie lot in the convolution cae. Finally, we come to a relation which will be modified for thi project. It wa obtained by Strau [9]. If (4) hold and if for each T > we have then lim T t lim T t C(t,) d =, (8) R(t,) d = (9) with immediate application to the nonlinear perturbed equation 6 z(t) = a(t) [ C(t, ) z()+g (,z() )] d (1)

3 KERNEL-RESOLVENT RELATIONS FOR AN INTEGRAL EQUATION with a and C a in (1) and G continuou for t < and z R. Now (1) i decompoed into (1) and z(t) = x(t) R(t,)G (,z() ) d (11) with x the olution of (1) and R the olution of (3). We call the pair (4) and (5), (6) and (7), (8) and (9) (with (4)) tranfer principle. In thi paper we focu on new one related to real-world problem and which avoid the mallne condition. In joint work with Dwiggin ([4], [5]) we removed (4) in the (8) (9) et, replacing it with a global Lipchitz condition on R with repect to and applying the reult to (11). The general problem of (1) and (11) i dicued in Strau [9], Miller [6], Miller-Nohel- -Wong [7], Ilam and Neugebauer [8], for example. In all of that work there are difficultie which eem to tem from a tep in the work, where we take the abolute value inide the integral and conider t R(t,)G (,z() ) d. The preent tudy differ in that we tudy that term without ever taking an abolute value. Thi reult in reduced aumption and, epecially, we never require either (4) or (6). It will turn out that C, itelf, can be arbitrarily large and o can variou integral of C. Thi i a ignificant contrat to the three aforementioned relation. The reult will enable u to ay that z(t) a(t) a t and that ( ) dt z(t) a(t) <. In thi paper we will extend thi lit of relation between C and R for the cae in which (8) hold and C i convex, C(t,), C (t,), C t (t,), C t (t,) for t <, a well a preent a detailed analyi of (1) and (11). The work i centered on (1) with two different aumption on G which illutrate the range of application of the idea. Firt, we aume that G(t,z) φ(t) z, φ L 1 [, ). In 198 Volterra [1] noted that many real-world problem were being modelled by integral and integro-differential equation with convex kernel. In ome cae, uch a vicoelaticity, convexity ha been deduced from firt principle, in other problem the model i decriptive. In the heat equation there i convexity in the kernel for t and it ha been hown in Burton [3] that uch ingularitie do not materially affect the behavior of olution. Subjectively, invetigator have choen convexity becaue it yield behavior coinciding with our intuitive idea of fading memory. The overwhelming reaon for our preference for a convex kernel come from the fact that Liapunov functional allow u to ue arbitrarily large kernel and avoid the Draconian condition found in (4) and (6). 7

4 THEODORE A. BURTON. Strategy In(1),G(t,z)iconideredtobeamallperturbationofx.Thetakitofind ignificant propertie of the olution, x, of (1), determine reaoned aumption ong(t,z),andprovethatz icloelyrelatedtox.inthiectionwewillillutrate one way in which that can be done very well under the aumption that C i convex. The work proceed in three tep. Step 1. For C convex, a uggetion of Volterra in 198 led to a Liapunov functional for a nonlinear form of (1) which lead to fine detail about the olution of (1). The linear form of that Liapunov function i (ee [, p. 131] or [1] for the contruction) V(t) = C (t,) x(u) du d+c(t,) having derivative (a hown in the appendix) atifying V (t) a (t) x (t) ( x(t) a(t) ). In addition, if there i a B > with C(t,t) B, then ( x(t) a(t) ) BV(t). x(u) du Thu, a L [, ) = x L [, ), x a L [, ) and if, in addition, C(t,t) B and a L, then x L. If we alo aume (8), then we can conclude that x(t) a(t) a t. The three aumption of a L, C convex, and (8) leave u with very extenive information about x. Step. A G i a mall perturbation, we expect to prove that z L L, z x L [, ), z(t) x(t). In the next ection we will find condition under which z (L L )[, ). Thoe are, o to peak, the coare propertie of z. Here, we will aume thoe propertie and ue that information to find the finer property of z(t) x(t) a t and z x L [, ). Next, a G i a mall perturbation for a linear equation we would ak that G(t, z) φ(t) z (1) and ponder the aumption for φ. 8

5 KERNEL-RESOLVENT RELATIONS FOR AN INTEGRAL EQUATION We note that when z L, then (11) can be conidered a z(t) = x(t) R(t, )b() d, where b() Kφ(), K i a bound on z. Thi i preciely what wa done in the correponding invetigation of ordinary differential equation. From our conjecture about the propertie of z, our goal i to prove f(t) := R(t,)b()d L [, ) and Thi will lead u directly to the aumption on φ. R(t,)b()d. (13) Step 3. Since we know how to find the mot intricate propertie of the olution of (1), we will contruct a linear equation containing f a it olution. Uing that we will how that (13) hold, concluding then from (1) that z x L [, ) and z(t) x(t) a(t) a t. More importantly, we will have dicovered how to parlay the propertie of t C(t,)b()d into the propertie of R lited in (13) whenever C i convex and (8) hold, extending the pair of propertie lited in the introduction without aking mallne condition on C. Here are the detail for contructing (13). Let b : [, ) R be continuou, write () a and then form R(t,)b() = C(t,)b() R(t,)b()d = which we will write a with f(t) = = C(t,)b()d C(t,)b()d f(t) = S(t) C(t,u)R(u,)dub(), R(t, )b() d, S(t) = C(t,u)R(u,)dub()d C(t, u) u R(u,)b()ddu C(t, u)f(u) du (14) C(t, )b() d. (15) 9

6 We define V(t) = C (t,) THEODORE A. BURTON f(u)du d +C(t,) and obtain, under the convexity aumption on C, that f(u)du (16) V (t) ( S(t) f(t) ) f (t)+s (t), (17) a it i een in the appendix. In addition, in the appendix we ee that when then C(t,t) B, (18) ( S(t) f(t) ) BV(t). (19) Thu, (17) and (19) yield the following reult with exact counterpart for (1). Ì ÓÖ Ñ.1º If C i convex, if C(t,t) B, and if S L [, ), then t t 1 ( ) ( ) du S(t) f(t) + S(u) f(u) + f (u)du B S (u)du. () Thi theorem give u, in a imple way, a relation between C and R. The relation i alo found in Burton-Dwiggin [5], but wa not ued in the way we ue it here; intead, we worked with the abolute value of R, a technique which we tudiouly avoid here. Ì ÓÖ Ñ.º Let C be convex anduppoethat C(t,)h()d a t for every continuou function h: [, ) R with h L [, ). If b: [, ) R i continuou and if S(t) L [, ) and S(t) a t, then the ame i true for f; here, b, S, and f atify (15). Proof. A C i convex, (17) hold. A S L [, ), the ame i true for f from (17). A f L [, ) it qualifie a the h of the theorem and o C(t,)f()d a t. Then from (14) we ee that f(t) S(t) a t. A S(t) o doe f. Thi complete the proof. Sequence of tep. In all of thi, everything tart with the given C(t,) being convex. Next, with our Liapunov functional in mind, b(t) mut be tentatively elected from the vector pace of function for which S L [, ). Automatically, f L [, ). Now, the vector pace from which b i drawn mut be refined o that S(t) a t ; here, (8) will play a central role. Finally, we mut how that f(t) S(t) a t and, again, that it will be a imple conequence of (8). In olving (1) we will have z(t) a(t) L [, ) and z(t) a(t) a t. That will be very trong convergence. In the context of thi problem, b(t) = G ( t,z(t) ) and b(t) φ(t) z(t) ; then our aumption on φ are baed 3

7 KERNEL-RESOLVENT RELATIONS FOR AN INTEGRAL EQUATION on reaon, not technical neceity. There i a continuum of uch function, b, and a ubcla i given in Burton-Dwiggin [5]. Two ubclae are imple and of coniderable interet; thee are b L 1 [, ) and b L [, ). We will focu on them after we have aured ourelve that z really doe atify the conjectured propertie of z (L L ). Thu, we mut paue here to obtain thoe propertie of z. 3. The Liapunov functional We will ue our Liapunov functional directly on (1), examine the derivative and deduce from that a et of condition needed to have z L [, ) and z L. Thu, the condition which appear in the theorem below are elected from the variou tep of the proof. Ì ÓÖ Ñ 3.1º Let C be convex, C(t,t) B for ome B >, and let (1) hold with a (L L )[, ), φ (L 1 L )[, ), (1) where a i from (1) and φ i introduced in (1). If x olve (1), if z olve (1), and if R olve (3), then x, z, and while G(t,z) (L 1 L )[, ). R(t,)G (,z() ) d (L L )[, ), () Proof. Define a Liapunov functional for (1) by [ V(t) = C (t,) z(u)+g ( u,z(u) )] du d +C(t,) [ z(u)+g ( u,z(u) )] du. (3) In the appendix we how that if C(t,t) B for ome B >, then ( z(t) a(t) ) BV(t) and that the derivative of V along a olution of (1) atifie V (t) [ z +G(t,z) ][ a(t) z(t) ] = [ za(t) z +a(t)g(t,z) zg(t,z) ]. (4) 31

8 THEODORE A. BURTON Ue (1) and note that for any ǫ (, ) 1 we can find M > with [ V (t) ǫz +Ma (t) z +ǫφ (t)z +Ma (t)+φz ]. Thu, [ V (t) ( 1+ǫ)z +Ma (t)+ ( φ+ǫφ (t) ) z ]. (5) We proceed in two tep, firt howing z L and then z L [, ). From (5) we have ( ) V (t) 4Ma (t)+ φ(t)+ǫφ (t) z (t) (6) and we note the algebraic relation (z a) ( 1 ) z a o that from (z a) BV we obtain ( ) 1 z a BV or z 4BV +a o ( ) ( ) V (t) 4Ma (t)+ φ(t)+ǫφ (t) 4BV(t)+4 φ(t)+ǫφ (t) a (t). By (1) we can find γ,µ L 1 [, ) with V (t) γ(t)+µ(t)v(t), (7) yielding V L [, ) and, becaue a L, o i z. Going back to (5) with z L [, ), we have (φ + ǫφ )z L 1 [, ) o now an integration of (5) yield z L [, ). We howed the ame for x in Section. Thee in (11) yield a well a G(t,z) (L 1 L )[, ). R(t,)G (,z() ) d (L L )[, ), (8) 4. An explicit kernel-reolvent relation Everything center around Theorem.. We mut have S(t) = C(t,)b()d L [, ) where b(t) majorizeg(t,z) with G(t,z) φ(t) z when we havez (L L ). In hort, given C(t,), what are our choice for φ? There i a whole continuum 3

9 KERNEL-RESOLVENT RELATIONS FOR AN INTEGRAL EQUATION of choice which lie between two prominent branche. Firt, we can tart with b L 1 [, ) when C i, roughly, in L. Another branch i found by taking b L [, )and,againroughly,c L 1.Wecanthenfillinbetweenthebranche uing Hölder inequality. Here i the firt branch. Ä ÑÑ 4.1º Let b(t) be continuou, b() d =: M <, and let up t< Then S(t) = C(t,)b()d L [, ). Proof. Let o that a required. L(t) := L(t) u u u M = M MJ C (u,)du =: J <. (9) C(u, )b() d du b() d u C (u,) b() ddu C (u,) b() ddu C (u,)du b() d b() d JM, Given C atifying the above condition, we chooe b a tated and we can be ure that S and f are in L [, ). Next, we want to be ure that f(t) = R(t,)b()d a t. We enure that in three tep. Notation. The ymbol C = up C(t,) and b [,T] = up b(). t< T Ò Ø ÓÒ 4.º A function C(t,) i aid to tend to zero on-average a t if it i bounded for t < and if for each T >, then (8) hold. 33

10 THEODORE A. BURTON Ä ÑÑ 4.3º Let up t C (t,)d =: M < and let C(t,) tend to zero on-average a t. If S L [, ) and f L [, ), then for (14) we have f(t) S(t) a t. (3) Proof. Let f (u)du =: H, C =: N, and S (t)dt =: P. For < T < t we have f(t) S(t) = T C(t,)f() d C(t,)f() d + T HN f ()d T T T C(t,)f() d C (t,)d+ C(t,) d +M T T C (t,)d f ()d. T f ()d For a given ǫ >, take T o large that the lat term i le than ǫ. Then take t o large that the next-to-lat term i le than ǫ. Ä ÑÑ 4.4º If C(t,) tend to zero on-average a t, if b L 1 [, ), and if b(t) i bounded on compact interval, then S(t) a t. Proof. For < T < t we have S(t) = C(t, )b() d T from which the reult follow. C(t,)b() d + C(t, )b() d T T b [,T] C(t,) d+ C b() d, T 34

11 KERNEL-RESOLVENT RELATIONS FOR AN INTEGRAL EQUATION Ì ÓÖ Ñ 4.5º Let b be continuou, let b L 1 [, ), let C be convex, let C(t,) tend to zero on-average a t, let (9) hold, and let C (t,)d L [, ). Then and C(t,)b()d a t R(t,)b()d a t. Proof. A b L 1 [, ) and continuou, while (9) hold, by Lemma 4.1 we have S L [, ) and o by convexity f L [, ). A C(t,) on-average and C (t,)d L, we have by Lemma 4.3 that f(t) S(t). Since S(t) by Lemma 4.4, f(t), a required. We can now offer a olution to our problem with (1). Ì ÓÖ Ñ 4.6º Let (1) and (1) hold, let C be convex, let (9) hold, let C (t,)d L [, ), and let C(t,) tend to zero on-average a t. If z olve (1), then z(t) a(t) a t and ( ) dt z(t) a(t) <. Proof. A (1) and (14) have the ame form, () tranlate into a reult for (1) with a (L L )[, ). Applying the technique of Lemma 4.3 to (1) yield x (L L )[, ), x a L [, ), and x(t) a(t) a t. By Theorem 3.1 we have G(t,z) (L 1 L )[, ), o we define b(t) = G ( t,z(t) ), f(t) = C(t,)b()d forming (14). R(t,)b()d, and S(t) = By Lemma 4.1 we have S L, o the ame i true for f. Then by Lemma 4.3 f(t) S(t) ; but b(t) i continuou o by Lemma 4.4, S(t), yielding f(t). Hence, z(t) x(t) a(t) a t. A f L, then z x L. It follow that z a L. Thi complete the proof. 5. A parallel reult We turn now to a reult at the other end of the continuum and work with a pair of L 1 condition on C intead of the L condition. 35

12 THEODORE A. BURTON Ä ÑÑ 5.1º If b L [, ) and i continuou, if there are poitive contant M and P with C(t,) d M and C(u,) du P, then S L [, ). Proof. We have S (u)du = u u = M MP C(u, )b() d du C(u,) d u C(u,) b ()ddu M C(u,) b ()dud = M b ()d MP b ()d. u C(u,) dub ()d C(u,) b ()ddu Ä ÑÑ 5.º If b L [, ), if for each T >, then b [,T] L, if C(t,) d M, and if C(t,) tend to zero on-average a t, then S(t) a t. Proof. Note that C <. We have S (t) = C(t, )b() d C(t,) d C(t,) b ()d M C(t,) b ()d (and for < T < t we have) = M T C(t,) b ()d + C(t,) b ()d T T M b [,T] C(t,) d + C and thi will tend to zero a t. T b ()d 36

13 KERNEL-RESOLVENT RELATIONS FOR AN INTEGRAL EQUATION Ä ÑÑ 5.3º If S L [, ), if C(t,) d M, if C i convex, and if C(t,) tend to zero on-average (o C i finite), then f L [, ), and f(t) S(t) a t. Proof. A S L and C i convex, we have f L. Alo, C and f i continuou. We ee that ( ) t f(t) S(t) = C(t,)f()d C(t,)d C(t,)f ()d and the ret i identical to that of Lemma 5.. Ì ÓÖ Ñ 5.4º Let b be continuouand in L [, ) and let C be convex and tend to zero on-average a t. Suppoe alo that there are poitive number M,P with C(t,)d M and C(u,)du P. Then C(t,)b()d L [, ), C(t,)b()d a t, and R(t,)b()d a t a well a being in L [, ). Proof. By Lemma 5.1 we have S L [, ). A C i convex, we now have f L [, ). A C(t,) on-average, f(t) S(t) a t by Lemma 5.3. By Lemma 5. we ee that S(t) a t and that complete the proof. WeofferacomparionbetweenTheorem4.5andTheorem5.4.InTheorem4.5 we had C (t,)d bounded and b(t) = G ( t,z(t) ) L 1. Thi allowed u to how that R(t,)G(,z() ) d L and tend to zero o z(t) x(t). We had to work with (11) becaue if we had worked directly with (1), then we would have needed z +G(t,z) L 1 ; it i very difficult to how z L 1 becaue that i known to imply a very trong kind of uniform aymptotic tability. By contrat, Theorem 5.4 will allow u to work directly with (1). Our Theorem 3.1 yield z + G(t,z) L o we take b(t) = z + G(t,z) and aume C(t,)d M and C(u,)du P obtaining C(t,)b()d L and tending to zero. That yield z a L and z a. Of coure, there are other benefit of the concluion that R(t,)b()d and i in L, a we will ee below. A example, firt let C(t,) = [1+t ] 3 o that C (t,)d i bounded. ThenTheorem 5.4failand weare forcedtotudy (11)uing Theorem4.5. Next, let C(t,) = [1 + t ] 4 3 o that C(t,)d and C(u,)du are bounded. We can ue Theorem 5.4 on (1) directly. Now we conider (1) with a different aumption on φ, namely G(t,z) φ(t) z, φ L [, ), φ(t) β < 1. (31) Thi i a harh condition a it ha z to dominate G(t,z) for every (t,z). 37

14 THEODORE A. BURTON Define V a in (3), but at (5) we now make ǫ o mall (and M o large) that β +ǫβ +ǫ µ < 1. Then from (5) we have [ ] V (t) ( 1+µ)z +Ma (t). (3) Accordingly, (1) i replaced by (31) and We now have a reult parallel to Theorem 3.1. a (L L )[, ). (33) Ì ÓÖ Ñ 5.5º Let C be convex, C(t,t) B for ome B >, and let (31) and (33) hold. If x olve (1), if z olve (1), and if R olve (3), then x,z,g(t,z) (L L )[, ). (34) Proof. From (3) we ee that V i bounded, that z i bounded, and that z L [, ). By (31), G(t,z) L [, ) and it i bounded. Clearly, x L [, ) and i bounded. Recall that in the proof of Theorem 4.6 we argued that x(t) a(t) a t and x a L. Taking b(t) = G(t,z), we form f(t) = S(t) C(t,)f()d. Note that b i continuou. We ummarize a follow. Ì ÓÖ Ñ 5.6º Let the condition of Theorem 5.4 and 5.5 hold. Then R(t,)G (,z() ) d a t and z a L [, ). A an example, we can let C(t,) = [1 + t ] (1+δ) for δ > and b(t) = φ(t) = [ + t] 3 4. All condition of Lemma 5.1, 5., and 5.3 hold. However, Lemma 4.5 i not atified. Theorem 5.6 hold, but Theorem 4.6 doe not. They are eentially different reult and there i a continuum of reult between them and they are obtained uing Hölder inequality in the proce of forming S(t), etablihing that S L. Each give different condition on b and C to enure that R(t,)b()d L and tend to zero. 38

15 KERNEL-RESOLVENT RELATIONS FOR AN INTEGRAL EQUATION 6. Appendix for The general nonlinear Liapunov functional (ee [1]) defined by V(t) = C (t,) and C convex atifie V (t) g(t,z) = g(t,z) [ g ( u,z(u) ) du d+c(t,) z(t) = a(t) C (t,) C(t, ) +g(t,z)c(t,) = g(t,z) C(t,)g (,z() ) d g ( u,z(u) ) dud +g(t,z)c(t,) g ( u,z(u) ) du When g(t,z) = z, thi readily yield g ( u,z(u) ) du t + g ( u,z(u) ) du C(t,)g (,z() ) d C(t,)g (,z() ) d = g(t,z) [ a(t) z(t) ]. V (t) a (t) z (t) ( z(t) a(t) ), a ued in Section. For the lower bound, ( ) t z(t) a(t) = C(t,)g (,z() ) d = C(t, ) C(t, ) g ( u,z(u) ) du t + g ( u,z(u) ) du + C (t,) C (t,) g ( u,z(u) ) du ] g ( u,z(u) ) du g ( u,z(u) ) du 39

16 C (t,) + THEODORE A. BURTON C (t,)d C(t,)+ g ( u,z(u) ) du C (t,) C (t,)d V(t) g ( u,z(u) ) du d [ ] = C(t,)+C(t,) t V(t) = C(t,t)V(t). REFERENCES [1] BURTON, T. A.: Boundedne and periodicity in integral and integro-differential equation, Differential Equation Dynam. Sytem 1 (1993), [] BURTON, T. A.: Liapunov Functional for Integral Equation, Trafford, Victoria, B. C., Canada, 8, ( [3] BURTON, T. A.: A Liapunov functional for a ingular integral equation, Nonlinear Anal. 73 (1), [4] BURTON, T. A. DWIGGINS, D. P.: Reolvent, integral equation, and limit et, Math. Bohem. 135 (1), [5] BURTON, T. A. DWIGGINS, D. P.: Smoothed integral equation, J. Math. Anal. Appl. 377 (11), [6] MILLER, R. K.: Nonlinear Volterra Integral Equation, W. A. Benjamin, Menlo Park, CA, [7] MILLER, R. K. NOHEL, J. A. WONG, J. S. W.: Perturbation of Volterra integral equation, J. Math. Anal. Appl. 5 (1969), [8] ISLAM, M. N. NEUGEBAUER, J. T.: Qualitative propertie of nonlinear Volterra integral equation, Electron. J. Qual. Theory Differ. Equ. 1 (8), [9] STRAUSS, A.: On a perturbed Volterra integral equation, J. Math. Anal. Appl. 3 (197), [1] VOLTERRA, V.: Sur la théorie mathématique de phénomè héréditaire, J. Math. Pur. Appl. 7 (198), Received Augut 7, 1 Northwet Reearch Intitute 73 Caroline St. Port Angele Wahington 9836 U.S.A. taburton@olypen.com 4