It is proved that if a linear operator $\ell:C([a,b];\mathbb{R})\rightarrow L([a,b];\mathbb{R})$ is nonpositive and for the initial value problem $$u''(t)=\ell(u)(t)+q(t),\quad u(a)=c_1,\quad u'(a)=c_2 $$ the theorem on differential inequalities is valid, then $\ell$ is an $a$-Volterra operator

Stepankova, H.

Electronic journal of qualitative theory of differential equations

H. Stepankova

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A note on the theorem on differential inequalities

Hana Stepankova

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Electronic Journal of Qualitative Theory of Differential Equations2005, No. 7, 1-8; http://www.math.u-szeged.hu/ejqtde/A NOTE ON THE THEOREM ON DIFFERENTIALINEQUALITIESH. SˇTEˇPA´NKOVA´Abstract. It is proved that if a linear operator ` : C([a, b]; R) →L([a, b]; R) is nonpositive and for the initial value problemu′′(t) = `(u)(t) + q(t), u(a) = c1, u′(a) = c2the theorem on differential inequalities is valid, then ` is an a−Volterra op-erator.2000 Mathematics Subject Classification. 34K06, 34K12.Key words and phrases: Second order linear functional differentialequation, initial value problem, Volterra operator.The following notation is used throughout the paper.N is the set of natural numbers.R is the set of real numbers, R+ = [0, +∞[.C([a, b]; R) is the Banach space of continuous functions u : [a, b] → Rwith the norm ‖u‖C = max{|u(t)| : t ∈ [a, b]}.C([a, b]; R+) = {u ∈ C([a, b]; R) : u(t) ≥ 0 for t ∈ [a, b]}.C˜([a, b]; R) is the set of absolutely continuous functions u : [a, b] → R.C˜ ′([a, b]; R) is the set of functions u ∈ C˜([a, b]; R) such that u′ ∈ C˜([a, b]; R).C˜ ′loc([a, b[ ; R) is the set of functions u ∈ C˜([a, b]; R) such that u′ ∈C˜([a, β]; R) for every β ∈ ]a, b[ .C˜ ′loc(]a, b[ ; R) is the set of functions u ∈ C˜([a, b]; R) such that u′ ∈C˜([α, β]; R) for every [α, β] ⊂ ]a, b[ .L([a, b]; R) is the Banach space of Lebesgue integrable functions p : [a, b] →R with the norm ‖p‖L =b∫a|p(s)|ds.L([a, b]; R+) = {p ∈ L([a, b]; R) : p(t) ≥ 0 for almost all t ∈ [a, b]}.Lab is the set of linear bounded operators ` : C([a, b]; R) → L([a, b]; R).EJQTDE, 2005 No. 7, p. 1Pab is the set of operators ` ∈ Lab transforming the set C([a, b]; R+) intothe set L([a, b]; R+).We will say that ` ∈ Lab is an a−Volterra operator if for arbitrary b0 ∈]a, b] and v ∈ C([a, b]; R) satisfying the conditionv(t) = 0 for t ∈ [a, b0]we have`(v)(t) = 0 for almost all t ∈ [a, b0].We will say that an operator Ω : L([a, b]; R) → C([a, b]; R) is an a−Volterraoperator, if for arbitrary b0 ∈ ]a, b] and q ∈ L([a, b]; R) satisfying the conditionq(t) = 0 for almost all t ∈ [a, b0]we haveΩ(q)(t) = 0 for t ∈ [a, b0].In what follows, the equalities and inequalities with integrable functionsare understood to hold almost everywhere.Consider the problem on the existence and uniqueness of a solution of theequationu′′(t) = `(u)(t) + q(t) (1)satisfying the initial conditionsu(a) = c0, u′(a) = c1, (2)where ` ∈ Lab, q ∈ L([a, b]; R) and c0, c1 ∈ R . By a solution of the equation(1) we understand a function u ∈ C˜ ′([a, b]; R) satisfying this equation (almosteverywhere) in [a, b].Along with the problem (1), (2) consider the corresponding homogeneousproblemu′′(t) = `(u)(t), (10)u(a) = 0, u′(a) = 0. (20)The following result is well-known from the general theory of boundaryvalue problems for functional differential equations (see, e.g., [1, 2, 4, 5, 8]).EJQTDE, 2005 No. 7, p. 2Theorem 1. The problem (1), (2) is uniquely solvable iff the correspondinghomogeneous problem (10), (20) has only the trivial solution.Definition 1. We will say that an operator ` ∈ Lab belongs to the set H˜ab(a)if for every function u ∈ C˜ ′([a, b]; R) satisfyingu′′(t) ≥ `(u)(t) for t ∈ [a, b], (3)and (20), the inequalityu(t) ≥ 0 for t ∈ [a, b] (4)holds.Remark 1. It follows from Definition 1 that if ` ∈ H˜ab(a), then the homoge-neous problem (10), (20) has only the trivial solution. Therefore, accordingto Theorem 1 the problem (1), (2) is uniquely solvable. Moreover, the inclu-sion ` ∈ H˜ab(a) guarantees that if q ∈ L([a, b]; R+), then the unique solutionof the problem (1), (20) is nonnegative.Note also that ` ∈ H˜ab(a) iff a certain theorem on differential inequalitieshold. More precisely, whenever u, v ∈ C˜ ′([a, b]; R) satisfy the inequalitiesu′′(t) ≤ `(u)(t) + q(t), v′′(t) ≥ `(v)(t) + q(t) for t ∈ [a, b],u(a) = v(a), u′(a) = v′(a),thenu(t) ≤ v(t) for t ∈ [a, b].In the paper [7], sufficient conditions are established guaranteeing theinclusion ` ∈ H˜ab(a). In particular, in [7, Theorem 1.3] the following propo-sition is proved.Proposition 1. Let −` ∈ Pab be an a−Volterra operator and let there exista function γ ∈ C˜ ′loc(]a, b[ ; R) satisfyingγ′′(t) ≤ `(γ)(t) for t ∈ [a, b], (5)γ(t) > 0 for t ∈ ]a, b[ , (6)γ(a) + limt→a+γ′(t) 6= 0. (7)Then ` ∈ H˜ab(a).EJQTDE, 2005 No. 7, p. 3Below we will prove (see Theorem 3) that in Proposition 1 the conditionon ` to be a−Volterra operator is necessary. Analogous result for first orderfunctional differential equations is proved in [3].Before we formulate the main results, let us introduce the following defi-nition.Definition 2. Let the problem (10), (20) have only the trivial solution. De-note by Ω the operator, which assigns to every function q ∈ L([a, b]; R) thesolution of the problem (1), (20).Remark 2. From Theorem 1 it follows that the operator Ω is well defined.It is also clear that Ω is a linear operator which maps the set L([a, b]; R) intothe set C([a, b]; R).Remark 3. It follows from [5, Theorem 1.4.1] that the operator Ω is contin-uous (bounded) (see also [1, 4, 6]).Remark 4. It immediately follows from Definitions 1 and 2 that if ` ∈H˜ab(a), then the operator Ω is nonnegative, i.e., it transforms the set L([a, b]; R+)into the set C([a, b]; R+).Theorem 2. Let −` ∈ Pab and ` ∈ H˜ab(a). Then Ω is an a−Volterraoperator.Proof. Let t0 ∈ ]a, b[ and let the function q ∈ L([a, b]; R) be such thatq(t) = 0 for t ∈ [a, t0]. (8)We will show thatΩ(q)(t) = 0 for t ∈ [a, t0]. (9)Denote by u the solution of the problem (1), (20) and by v the solution ofthe problemv′′(t) = `(v)(t) + |q(t)|, (10)v(a) = 0, v′(a) = 0. (11)According to Remark 1 (see also Remark 4) and the assumption −` ∈ H˜ab(a),we havev(t) ≥ 0 for t ∈ [a, b], (12)u(t) ≤ v(t) for t ∈ [a, b]. (13)EJQTDE, 2005 No. 7, p. 4Since −` ∈ Pab, it follows from (10) and (12) thatv′′(t) ≤ |q(t)| for t ∈ [a, t0]Hence, on account of (8), (11) and (12), we obtainv(t) = 0 for t ∈ [a, t0]. (14)On the other hand, by virtue of (1), (10), (13), and the assumption −` ∈ Pab,we get(u(t)− v(t))′′ = `(u− v)(t) + q(t)− |q(t)| ≥ q(t)− |q(t)| for t ∈ [a, b].Hence in view of (8) and (14) we getu′′(t) ≥ v′′(t) = 0 for t ∈ [a, t0].The latter inequality, together with (13), (14) and (20), impliesu(t) = 0 for t ∈ [a, t0].Consequently (since u(t) = Ω(q)(t) for t ∈ [a, b]), the equality (9) is fulfilled.Theorem 3. Let −` ∈ Pab and ` ∈ H˜ab(a). Then ` is an a−Volterra opera-tor.Proof. Assume the contrary, let ` be not an a−Volterra operator. Thenthere exist v0 ∈ C([a, b]; R) and b0 ∈ ]a, b[ such thatv0(t) = 0 for t ∈ [a, b0]andmes{t ∈ [a, b0] : `(v0)(t) 6= 0} > 0.Without loss of generality we can assume thatmes{t ∈ [a, b0] : `(v0)(t) < 0} > 0. (15)First we will show thatΩ(`(|v0|))(t) = 0 for t ∈ [a, b0], (16)EJQTDE, 2005 No. 7, p. 5where Ω is the operator introduced in Definition 2.Choose a sequence of functions vk ∈ C˜′([a, b]; R), k ∈ N, such thatlimk→+∞‖vk − |v0|‖C = 0 (17)andvk(t) = 0 for t ∈ [a, b0], k ∈ N. (18)According to Remark 3 and (17), we getlimk→+∞‖Ω(`(vk))− Ω(`(|v0|))‖C = 0. (19)It is clear thatv′′k(t) = `(vk)(t) + qk(t) for t ∈ [a, b], k ∈ N, (20)whereqk(t)def= v′′k(t)− `(vk)(t) for t ∈ [a, b], k ∈ N. (21)Consequently,vk(t) = Ω(qk)(t) for t ∈ [a, b], k ∈ N. (22)It follows from (20)–(22) thatvk(t) = Ω(v′′k)(t)− Ω(`(vk))(t) for t ∈ [a, b], k ∈ N. (23)Hence, taking into account the fact that Ω is an a−Volterra operator (seeTheorem 2) and the condition (18), we obtainΩ(`(vk))(t) = −vk(t) = 0 for t ∈ [a, b0], k ∈ N.Thus, in view of (19), we get the equality (16).Let u be a solution of the problem (1), (20), whereq(t) ={−`(|v0|)(t) for t ∈ [a, b0[0 for t ∈ [b0, b]. (24)EJQTDE, 2005 No. 7, p. 6It is evident thatu(t) = Ω(q)(t) for t ∈ [a, b] (25)andq(t) ≥ 0 for t ∈ [a, b]. (26)Moreover, on account of the assumption −` ∈ Pab, the inequality`(|v0|)(t) ≤ `(v0)(t) for t ∈ [a, b]holds. Consequently, due to (15) and (24)mes{t ∈ [a, b0] : q(t) > 0} > 0. (27)According to Theorem 2, Ω is an a−Volterra operator. Hence by virtue of(16) and (24) we get from (25) thatu(t) = 0 for t ∈ [a, b0]. (28)On the other hand, the inequality (26) and the assumption ` ∈ H˜ab(a) implyu(t) ≥ 0 for t ∈ [a, b]. (29)In view of (29) and the assumption −` ∈ Pab, it follows from (1) thatu′′(t) ≤ q(t) for t ∈ [a, b].Hence, on account of (24), we obtainu′′(t) ≤ 0 for t ∈ [b0, b].The latter inequality, together with (28) and (29), yieldsu(t) = 0 for t ∈ [a, b].Therefore, it follows from (1) that q ≡ 0, which contradicts (27).EJQTDE, 2005 No. 7, p. 7References[1] N. V. Azbelev, V. P. Maksimov, L. F. Rakhmatullina, Introductionto the theory of functional differential equations (in Russian), Nauka,Moscow, 1991.[2] E. Bravyi, A note on the Fredholm property of boundary value problemsfor linear functional differential equations, Mem. Differential EquationsMath. Phys. 20 (2000), 133–135.[3] E. Bravyi, A. Lomtatidze, B. Pu˚zˇa, A note on the theorem on differentialinequalities, Georgian Math. J. 7 (2000), No. 4, 627–631.[4] R. Hakl, A. Lomtatidze, I. P. Stavroulakis, On a boundary value problemfor scalar linear functional differential equations, Abstract and Appl.Anal. (2004), No. 1, 45–67.[5] I. Kiguradze, B. Pu˚zˇa, Boundary value problems for systems of lin-ear functional differential equations, Folia, Facult. Scient. Natur. Univ.Masarykianae Brunensis, Mathematica 12, 2003.[6] I. Kiguradze, B. Pu˚zˇa, On boundary value problems for systems of linearfunctional differential equations, Czechoslovak Math. J. 47 (1997), No. 2,341–373.[7] A. Lomtatidze, H. Sˇteˇpa´nkova´, On sign constant and monotone solutionsof second order linear functional differential equations, Mem. DifferentialEquations Math. Phys. 33 (2004), 10–45.[8] Sˇ. Schwabik, M. Tvrdy´, O. Vejvoda, Differential and integral equations:boundary value problems and adjoints, Academia, Praha, 1979.(Received February 7, 2005)Author’s address:Hana Sˇteˇpa´nkova´Faculty of Education, University of South Bohemia,Jerony´mova 10, 371 15 Cˇeske´ Budeˇjovice,Czech RepublicE-mail: stepanh@pf.jcu.czEJQTDE, 2005 No. 7, p. 8

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