Some properties of non-locally bounded solutions for Abel integral equations are given. The case in which there exists two non-trivial solutions for such equations is also studied. Besides, some known results about existence, uniqueness and attractiveness of solutions for some Volterra equations are improved

Arias, M. R.

Benítez Suárez, Rafael

Repositori d'Objectes Digitals per a l'Ensenyament la Recerca i la Cultura

PROCEEDINGS OF THE FOURTH INTERNATIONAL CONFERENCE ONDYNAMICAL SYSTEMS AND DIFFERENTIAL EQUATIONSMay 24 – 27, 2002, Wilmington, NC, USA pp. 42–47PROPERTIES OF SOLUTIONS FOR NONLINEAR VOLTERRAINTEGRAL EQUATIONSM. R. AriasDpto. Matema´ticas. Universidad de ExtremaduraAv. Elvas s/n06071, Badajoz. SpainR. Ben´ıtezDpto. Matema´ticas. Universidad de ExtremaduraAv. Virgen del Puerto, 210600, Plasencia (Ca´ceres). SpainAbstract. Some properties of non-locally bounded solutions for Abel integral equa-tions are given. The case in which there exists two non-trivial solutions for suchequations is also studied. Besides, some known results about existence, uniquenessand attractiveness of solutions for some Volterra equations are improved.1. Introduction. In this paper we study the behaviour of solutions for nonlinearVolterra integral equationsu(x) =∫ x0k(x− s)g(u(s))ds, (1)where k and g are known functions, called kernel and nonlinearity respectively, andu is the unknown. This equation shall be referred to as equation (k, g) and theintegral operatorTkgf(x) =∫ x0k(x− s)g(f(s))dsas the associated operator to the equation (k, g). From now on we suppose that gis a continuous strictly increasing function such that g(0) = 0, g′(x) > 0 almosteverywhere and g transforms null sets into null sets. These assumptions will bedenoted by (B).Abel integral equations are particular cases of Volterra equations with powerkernels k(x) = xα with α > −1. In this case the associated operator to equation(xα, g) will be denoted by Tαg.Note that, when g(0) = 0, the function 0 is a solution of (k, g) known as trivialsolution. We are interested only in non-trivial solutions, i.e. positive solutions usuch that u 6≡ 0 “near zero”. We shall say that some property holds near zero whenthe property holds on an interval (0, δ) for some positive δ. Therefore, from now onsolution means non-trivial solutions.1991 Mathematics Subject Classification. Primary: 45Gxx; Secondary: 45D05, 45E10, 47G10.Key words and phrases. Abel equations, Volterra equations, nonlinear integral equations, re-pelling solutions, multiple solutions.The first author has been partially supported by Junta de Extremadura, grant MOV02B082.The second author has been partially supported by Junta de Extremadura, grant MOV02B015.Both authors have been partially supported by MCyT, project BFM2001-0849.42SOLUTIONS FOR NONLINEAR VOLTERRA INTEGRAL EQUATIONS 43We shall say that f is globally attracted by u when (Tnkgf(x))n∈N converges tou(x) for any x in the domain of u, where Tnkg denotes the composition of Tkg withitself n-times; and we shall say that f is locally attracted by u when (Tnkgf)n∈Nconverges to u near zero.In this paper, first, we are going to improve some results about existence, unique-ness and attractiveness of solutions for Volterra integral equations with locallybounded kernels. Next, we will introduce some properties about the behaviourof solutions for nonlinear Abel integral equations.2. Locally bounded kernels. Throughout this section we consider equations(k, g) where the nonlinearity g satisfies (B) and k is a positive measurable andlocally bounded kernel defined on R+ such thatK(x) =∫ x0k(s)dsis a finite strictly increasing function; these assumptions for kernels together withassumptions (B) will be denoted by (GC).Let (k, g) be an equation satisfying (GC), where kernel k is also a continuousfunction. It is known that (k, g) has, at most, a unique solution u; and any positivemeasurable function f , such that f 6≡ 0 near zero, is globally attracted by u. Theseassertions are followed from Proposition 2.1 and Lemma 3.1 in [2]. The aim ofthis section is to prove the truthfulness of above assertions when the continuityof kernel is removed and the equation (k, g) just satisfies (GC). Note that [2] isstrongly based on [1]. Therefore, to attain our aim we need to analyse the use ofkernel’s continuity in [1, 2].In [1] are considered equations (k, g) with continuous kernels and satisfying(GC). In that paper, the continuity of kernels was just considered at the beginningof the third section in order to guarantee the continuity of any solution. Note that,even if we remove the continuity of kernel k, any solution of equation (k, g) satis-fying (GC) is still a continuous function; this is followed from next lemma, whereM(R) and L1(R) denote sets constituted by Lebesgue measurable and Lebesgueintegrable functions, respectively, defined on R.Lemma 1. Let f ∈ L1(R), h ∈ M(R), and suppose that h is bounded. Then theconvolutionf ∗ h(x) =∫Rf(x− s)h(s)dsdefines a function that is bounded and uniformly continuous on R.This lemma appears in [6, pag 357] as exercise 6. To prove the continuity of locallybounded solutions when (k, g) just satisfies (GC), we apply this lemma locally.That is, we fix an interval [0, b], for any positive b, and we define the auxiliarykernel k˜ and the nonlinearity g˜ in the following wayk˜(x) ={k(x) x ∈ [0, b],0 x ∈ R/[0, b] and g˜(x) ={g(x) x ∈ [0, d],0 x ∈ R/[0, d],where d = maxx∈[0,b]{u(x)}. Note that k˜ ∈ L1(R), g˜ ◦ u is a bounded function ofM(R) and u(x) = (k ∗ g ◦ u)(x) = (k˜ ∗ g˜ ◦ u)(x), x ∈ [0, b]. Therefore u is uniformlycontinuous on [0, b].The results presented in [2] were obtained considering equations (k, g) with con-tinuous kernels and satisfying (GC). In that paper, there are two reasons to consider44 M. R. ARIAS AND R. BENI´TEZthe continuity of the kernel. The first one is that we were trying to extend the re-sults presented in [1]. As we have seen in previous paragraph, the continuity of thekernel is not necessary to guarantee the truthfulness of those results. The other oneis that we needed, at the beginning of the second section, to define a continuousincreasing kernel k¯ such that k ≤ k¯. To do it we defined the following k¯,k¯(x) ={0 x < 0,sup{k(s) : s ∈ [0, x]} x ≥ 0.Note that the locally boundedness of k is sufficient to find a positive continuousincreasing kernel such that k is upper bounded by it.To sum up, there is no problem to replace continuous kernels in papers [1, 2] bylocally bounded ones. Therefore, for equations (k, g) verifying (GC), we have thatTheorem 1. There exists at most a unique positive solution. Moreover such so-lution is a uniformly continuous, increasing function and a global attractor of allpositive functions f such that f 6≡ 0 near zero.3. Abel kernels. A particular case of equations considered in previous section, isthe nonlinear Abel integral equationu(x) =∫ x0(x− s)αg(u(s)) ds, α > 0. (2)Therefore, equation (2) has at most one solution which is a continuous function anda global attractor.For α ∈ (−1, 0), Abel kernels are non-locally bounded functions. In this caseAbel equations may have locally bounded solutions, non-locally bounded solutionsor both (see [3]). In this section, our aim is to study the attracting behaviour ofnon-locally bounded solutions.First, let us consider Abel integral equationsu(x) =∫ x0(x− s)α(u(s))β ds,where α ∈ (−1, 0) and β > −1/α > 1. This kind of equation has a non-locallybounded solution (see [4]) u(x) = dxγ , beingγ =1 + α1− β and d = B(α+ 1, γβ + 1)1/(1−β), (3)where B(x, y) denotes the Euler beta function. Note that γ < 0 and d > 0. We willshow in next theorem that u does not attract the functions of the familyU = {cxγ : c > 0, c 6= d}.Theorem 2. If f ∈ U then we have that:(a) For c > d, limn→+∞ Tnαβf(x) = +∞, with x ∈ (0,+∞).(b) For c < d, limn→+∞ Tnαβf(x) = 0, with x ∈ (0,+∞).Proof. For f ∈ U we have thatTαβf(x) =∫ x0(x− s)α(csγ)β ds = cβB(α+ 1, γβ + 1)xγ = ϕ(c)xγ ,beingϕ(c) = cβB(α+ 1, γβ + 1), (4)SOLUTIONS FOR NONLINEAR VOLTERRA INTEGRAL EQUATIONS 45so Tnαβf(x) = ϕn(c)xγ , where ϕn denotes the composition of ϕ with itself n-times.We have to study the sequence (ϕn(c))n∈N in order to calculate the limit of thesequence(Tnαβf(x))n∈N.It is easy to show that 0 and d are the only fixed points of the ϕ. We have thatϕ′(c) = βcβ−1B(α+ 1, γβ + 1) ≥ 0,andϕ′′(c) = β(β − 1)cβ−2B(α+ 1, γβ + 1) ≥ 0,so both, ϕ and ϕ′ are strictly increasing functions. Note that ϕ′(d) = β > 1. Then(a) For c > d, ϕ(c) > ϕ(d) = d. Moreover, by the Mean Value Theorem we havethat there exists a ξ ∈ (d, c) such thatϕ(c)− d = ϕ(c)− ϕ(d) = ϕ′(ξ)(c− d) > ϕ′(d)(c− d) > c− d.Then we can assure that ϕ(c) > c, and therefore, by the monotony of ϕ,(ϕn(c))n∈N is an increasing sequence. This sequence is not upper boundedsince any limit of such sequence, by the continuity of ϕ, would be a fixedpoint of ϕ greater than d, which is absurd, so limn→∞ ϕn(c) = +∞.(b) In a similar way we obtain, for c < d, that the sequence (ϕn(c))n∈N is strictlydecreasing and converges to 0.Then we can conclude thatlimn→+∞Tnαβf(x) = limn→+∞ϕn(c)xγ ={0, if c ∈ [0, d),+∞, if c > d,for x ∈ (0,+∞); which completes the proof.For positive β, in particular for β > −α−1, operators Tαβ are monotone increas-ing. Therefore, it is immediate to see that last Theorem remains true onUˆ = {f ∈ L1(R) : f ≥ cxγ , c > d or f ≤ cxγ , c < d}.In the rest of this section, we are going to see that the behaviour described inTheorem 2 remains true for a wider class of Abel integral equations than thoseconsidered in last Theorem. To do it, first, we need to recall briefly the examples,presented in [3], about nonlinear Abel equations (xα, g) with two solutions, onelocally bounded and the other one non-locally bounded. Let us consider an Abelintegral equation (xα, g˜) with a locally bounded solution u˜, where g˜ is a nonlinearityverifying (B) and an Abel integral equation (xα, xβ) with a non-locally boundedsolution uβ(x) = dxγ , where α ∈ (−1, 0), β > −α−1, and d and γ are defined in(3). Then, for any fixed positive ε, we define the following nonlinearity:g(x) ={Cg˜(x) if 0 ≤ x < εxβ if x ≥ ε, (5)where C is the unique constant that makes g continuous. The equation (xα, g) hastwo solutions, one, u1, is locally bounded and another one, u2 is not. Solution u1is equal to u˜ near zero and u2 ≡ uβ near zero.In order to study the attracting behaviour of the solutions when both kind ofsolutions exist, we need the following lemma. In the proof of next lemma we needthe proposition 2.3 of [3]; this proposition requires that the nonlinearity has an ω-lower estimation, with ω ∈ (0, 1). A function g has an ω-lower estimation if g ≥ xωnear zero.46 M. R. ARIAS AND R. BENI´TEZLemma 2. Let u1 and u2 denote the locally bounded solution and the non-locallybounded solution of equation (xα, g), respectively. Then u2 ≥ u1.Proof. Let us suppose that there exists an x1 such that u2(x1) < u1(x1), and let usconsider the functionu3 = min{u1, u2}.Since u3 ≤ u1, proposition 2.3 in [3] assures the convergence of (Tmαgu3)m∈N to u1.On the other hand, since u3 ≤ u2 and the operator Tαg is monotone, we have thatTmαgu3 ≤ u2 for every natural m. Therefore,Tmαgu3(x1) ≤ u2(x1) < u1(x1), ∀m ∈ N,which is impossible since limm→∞ Tmαgu3(x1) = u1(x1).For next theorem we will consider Abel equations (xα, g), where α ∈ (−1, 0) andg is defined as in (5), assuming the existence of an ω–lower estimation for g˜, withω ∈ (0, 1), and β > −α−1; and d and γ are defined in (3).Theorem 3. Let f be a function such that f(x) = cxγ near zero. Then we havethat:(a) For c > d, limm→∞ Tmαgf(x) = +∞ on (0,+∞).(b) For c < d, limm→∞ Tmαgf(x) = u1(x) on the domain of definition of u1.Proof.(a) From the definition of g, this case is followed from Theorem 2 near zero, i.e.,there exists an interval, denoted by (0, a], where limm→∞ Tmαgf(x) = +∞.For x > a, let f(x) = lim infm→∞ Tmαgf(x); it is just necessary to note thatf ≡ +∞ on (0, a], so Tαgf(x) = +∞ for all positive x. Therefore, the limitof (Tmαgf)m∈N exists and is +∞ on (a,+∞).(b) Let (0, µ] be the biggest interval where f(x) = cxγ , and let M be any positiveconstant greater than ε and min{f(x) : x ∈ (0, µ]}. Taking into account thedefinition of M and f on (0, µ], there exists a positive δ0 such that f(x) ≥Mon (0, δ0]. In particular, taking δ0 = (M/c)1/γ , we get the biggest interval onwhich the above assertion is true . Then,Tαgf(x) =∫ x0(x− s)αg(f(s)) ds =∫ x0(x− s)α(csγ)β ds = ϕ(c)xγ ,for all x ∈ (0, δ0], where ϕ is defined in (4). Since c < d, as was seen inpart (b) of the proof of Theorem 2, we have that Tαgf(x) < f(x) on (0, δ0].So,(Tmαgf)m∈N is a monotone decreasing sequence; note that Tαg is a mono-tone operator. And from the non-locally boundedness of f it follows thatthis sequence is lower bounded by the locally bounded solution u1 on (0, δ0].Therefore,(Tmαgf)m∈N has a pointwise limit which will be denoted by v.Now we have to prove that v is locally bounded near zero. Let us definethe sequence δm = (M/ϕm(c))1/γ , being ϕm = ϕ ◦ ϕ ◦ . . . ◦ ϕ, m-times. Thissequence is decreasing and converges to zero, as it is followed immediatelyfrom part (b) of the proof of Theorem 2.Now, from the construction of (δm)m∈N and the definition of M and f , wehave, for any natural m, that Tmαgf ≥M on (0, δm] and Tmαgf(δm−1) < M .For any t ∈ (0, δ0] let us consider any natural number p such that δp < t.Let ct be a positive constant such that (M/ϕp(ct))1/γ = t. Since δp < t wehave that c < ct. Therefore, f < fct holds on (0, δ0], where fct(x) = ctxγSOLUTIONS FOR NONLINEAR VOLTERRA INTEGRAL EQUATIONS 47and T pαgf(t) < Tpαgfct(t) = M . In conclusion, we have that for any t ∈ (0, δ0]there exists a natural p such that T pαgf(t) < M . Since v(t) is the infimum of(T pαgf(t))p∈N, then v < M on (0, δ0].From the locally boundedness of v, by virtue of the Monotone ConvergenceTheorem, we have that v is a locally bounded solution for equation (xα, g)and then, since u1 is the unique locally bounded solution of such equation,v ≡ u1.Remark 1. Note that, since β > 1, then for any δ > 0, the nonlinearity g definedin (5) verifies the following condition,∫ +∞δ(sg(s))1/(α+1)dss< +∞. (6)This condition, given by W. Mydlarczyk [5, Theorem 2.4], is equivalent to theexistence of a blow-up for the locally bounded solution. Therefore, from Lemma2, since the locally bounded solution is upper bounded by the non-locally boundedone, we can conclude that both solutions have a blow-up.Acknowledgments. We are very grateful to the referees for their useful and suit-able comments used to improve this paper.REFERENCES[1] M.R. Arias. Existence and Uniqueness of Solutions for Nonlinear Volterra equations, Math.Proc. Camb. Phil. Soc. 129 (2000), 361-370.[2] M.R. Arias, R. Ben´ıtez. A note of the uniqueness and the attrative behaviour of solutions fornonlinear Volterra equations, J. Integral Equations App. 13 (4) (Winter 2001), 305-310.[3] M.R. Arias, R. Ben´ıtez. Aspects of the behaviour of solutions for nonlinear Abel equations,Nonlinear Anal. T.M.A. (accepted).[4] R. Gorenflo, S. Vesella, “Abel integral equations, analysis and applications”, Lecture Notesin Mathematics, 1461. Ed. Springer-Verlag, Berlin Eidelberg, 1991.[5] W. Mydlarczyk. The Blow-Up Solutions of Integral Equations, Colloquium Mathematicum79 (1) (1999), 147-156.[6] K. R. Stromberg. “An introduction to classical real analysis”, Cole Advanced Books & Soft-ware (Wadsworth & Brooks), 1981.Received September 2002; in revised March 2003.E-mail address: arias@unex.esE-mail address: rbenitez@unex.es

Properties of solutions for nonlinear Volterra integral equations

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