In this article we consider nonlinear differential equations with $\psi$-exponential and $\psi$-ordinary dichotomous linear part in a Banach space. By the help of the fixed point principle of Banach sufficient conditions are found for the existence of $\psi$-bounded solutions of these equations on $\mathbb{R}$ and $\mathbb{R}_+$

Georgieva, Atanaska

Kiskinov, Hristo

Kostadinov, Stepan

Zahariev, Andrey

Electronic journal of qualitative theory of differential equations

Atanaska Georgieva

Hristo Kiskinov

Stepan Kostadinov

Andrey Zahariev

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Existence of solutions of nonlinear differential equations with $\psi$-exponential or $\psi$-ordinary dichotomous linear part in a Banach space

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Electronic Journal of Qualitative Theory of Differential Equations2014, No. 2, 1–10; http://www.math.u-szeged.hu/ejqtde/Existence of solutions of nonlinear differentialequations with ψ-exponential or ψ-ordinarydichotomous linear part in a Banach spaceAtanaska Georgieva, Hristo Kiskinov, Stepan KostadinovB andAndrey ZaharievFaculty of Mathematics and Informatics, Plovdiv University, 236 Bulgaria Blvd., Plovdiv, BG–4003,BulgariaReceived 12 July 2013, appeared 13 March 2014Communicated by László HatvaniAbstract. In this article we consider nonlinear differential equations with ψ-exponentialand ψ-ordinary dichotomous linear part in a Banach space. By the help of the fixedpoint principle of Banach sufficient conditions are found for the existence of ψ-boundedsolutions of these equations on R and R+.Keywords: ψ-dichotomy for ordinary differential equations, ψ-boundedness.2010 Mathematics Subject Classification: 34G20, 34D09, 34C11.1 IntroductionThe problem of ψ-boundedness and ψ-stability of the solutions of differential equations in fi-nite dimensional Euclidean spaces has been studied by many auhtors, as e.g. Akinyele [1],Constantin [6]. In these papers, the function ψ is a scalar continuous function (and increasing,differentiable and bounded in [1], nondecreasing and such that ψ(t) ≥ 1 on R+ in [6]). InDiamandescu [8–15] and Boi [2–4] ψ is a nonnegative continuous diagonal matrix function.Inspired by the famous monographs of Coppel [5], Daleckii and Krein [7] and Massera andSchaeffer [17], where the important notion of exponential and ordinary dichotomy is consid-ered in detail, Diamandescu [8–12] and Boi [2–4] introduced and studied the ψ-dichotomy forlinear differential equations in finite dimensional Euclidean space.In our paper [16] we introduced the concept of ψ-dichotomy for arbitrary Banach spaces,where ψ is an arbitrary bounded invertible linear operator.In this paper nonlinear perturbed differential equations with ψ-dichotomous linear part areconsidered in an arbitrary Banach space. We will show that some properties of these equationswill be influenced by the corresponding ψ-dichotomous homogeneous linear equation. Suffi-cient conditions for the existence of ψ-bounded solutions of this equations onR andR+ in caseof ψ-exponential or ψ-ordinary dichotomy are found.BCorresponding author. Email: stkostadinov@uni-plovdiv.bg2 A. Georgieva, H. Kiskinov, S. Kostadinov, A. Zahariev2 PreliminariesLet X be an arbitrary Banach space with norm | · | and identity I. Let LB(X) be the spaceof all linear bounded operators acting in X with the norm ‖ · ‖. By J we shall denote R orR+ = [0,∞).We consider the nonlinear differential equationdxdt= A(t)x + F(t, x), (2.1)the corresponding linear homogenous equationdxdt= A(t)x (2.2)and the appropriate inhomogeneous equationdxdt= A(t)x + f (t), (2.3)where A(·) : J → LB(X), f (·) : J → X are strongly measurable and Bochner integrable on thefinite subintervals of J and F(·, ·) : J × X → X is a continuous function with respect to t.By a solution of equation (2.1) (or (2.2) or (2.3)) we will understand a continuous functionx(t) that is differentiable (in the sence that it is representable in the form x(t) =∫ ta y(τ)dτ of aBochner integral of a strongly measurable function y) and satisfies (2.1) (or (2.2) or (2.3)) almosteverywhere.By V(t) we will denote the Cauchy operator of (2.2).Let RL(X) be the subspace of all invertible operators in LB(X) and ψ(·) : J → RL(X) becontinuous for any t ∈ J operator-function.Definition 2.1 ( [16]). A function u(·) : J → X is said to be ψ-bounded on J if ψ(t)u(t) isbounded on J.Let Cψ(X) denote the Banach space of all ψ-bounded and continuous functions with valuesin X with the norm||| f |||Cψ = supt∈J|ψ(t) f (t)|.Definition 2.2 ( [16]). The equation (2.2) is said to have a ψ-exponential dichotomy on J if thereexist a pair of mutually complementary projections P1 and P2 = I − P1 and positive constantsN1, N2, ν1, ν2 such that||ψ(t)V(t)P1V−1(s)ψ−1(s)|| ≤ N1e−ν1(t−s) (s ≤ t; s, t ∈ J) (2.4)||ψ(t)V(t)P2V−1(s)ψ−1(s)|| ≤ N2e−ν2(s−t) (t ≤ s; s, t ∈ J) (2.5)The equation (2.2) is said to have a ψ-ordinary dichotomy on J if (2.4) and (2.5) hold withν1 = ν2 = 0.Remark 2.3. For ψ(t) = I for all t ∈ J we obtain the notion of exponential and ordinarydichotomy in [5, 7, 17].Nonlinear equations with ψ-dichotomous linear part 3Let us introduce the principal Green function of (2.3) with the projections P1 and P2 fromthe definition for ψ-exponential dichotomyG(t, s) ={V(t)P1V−1(s) (t > s; t, s ∈ J)−V(t)P2V−1(s) (t < s; t, s ∈ J).(2.6)Clearly G is continuous except at t = s where it has a jump discontinuity.Definition 2.4. Let r > 0 be an arbitrary number. We say that the conditions (H) are fulfilled ifthere exist positive functions m(t), k(t) such thatH1. |ψ(t)F(t, x)| ≤ m(t) (|ψ(t)x| ≤ r, t ∈ J)H2. |ψ(t)(F(t, x1)− F(t, x2))| ≤ k(t)|ψ(t)(x1 − x2)| (|ψ(t)x1|, |ψ(t)x2| ≤ r, t ∈ J)Definition 2.5. The nonnegative function m(t) is said to be integrally bounded on J if the fol-lowing inequality holds:B(m(t)) = supt∈J∫ t+1tm(s)ds < ∞.Definition 2.6. We say that the function F(t, x) belongs to the class EDψ(a1, a2, r) if the con-ditions (H) are fulfilled, the functions m(t), k(t) are integrally bounded on J and B(m(t)) ≤a1, B(k(t)) ≤ a2.For each integrable on J function m(t) we introduce the notationL(m(t)) =∫Jm(s)ds.Definition 2.7. We say that the function F(t, x) belongs to the class Dψ(a1, a2, r) if the conditions(H) are fulfilled, the functions m(t), k(t) are integrable on J and L(m(t)) ≤ a1, L(k(t)) ≤ a2.3 Main resultsTheorem 3.1. Let the following conditions be fulfilled:1. The linear part of (2.1) has ψ-exponential dichotomy on R with projections P1 and P2.2. The function F(t, x) belongs to the class EDψ(a1, a2, r).Then for an arbitrary r > 0 for sufficient small values of a1, a2 the equation (2.1) has a uniquesolution x(t), which is defined for t ∈ R and for which |ψ(t)x(t)| ≤ r (t ∈ R).Proof. Let J = R. We consider in the space Cψ(X) the operator Q : Cψ(X)→ Cψ(X) defined bythe formulaQx(t) =∫JG(t, τ)F(τ, x(τ))dτ (3.1)where G is defined by (2.6).Let x(t) be a solution of equation (2.1) that remains for t ∈ J in the ballSψ,r = {x : |||x|||Cψ ≤ r}.4 A. Georgieva, H. Kiskinov, S. Kostadinov, A. ZaharievThen the function F(t, x(t)) is ψ-bounded on J and it follows (see [16, Theorem 3.6]) that suchsolution satisfies the integral equationx(t) = Qx(t). (3.2)The converse is also true: a solution of the integral equation (3.2) which remains for t ∈ J in theball Sψ,r satisfies the differential equation (2.1) for t ∈ J.Now we shall show that the ball Sψ,r is invariant with respect to Q and the operator Q iscontracting.First we shall prove that the operator Q maps the ball Sψ,r into itself. Indeed we have|ψ(t)Qx(t)| ≤∣∣∣∣ψ(t) ∫J G(t, τ)F(τ, x(τ))dτ∣∣∣∣ .We have|ψ(t)Qx(t)| ≤∣∣∣∣ψ(t) ∫J G(t, τ)F(τ, x(τ))dτ∣∣∣∣≤∫J‖ψ(t)G(t, τ)ψ−1(τ)‖ |ψ(τ)F(τ, x(τ))|dτ=∫t≤τ‖ψ(t)G(t, τ)ψ−1(τ)‖ |ψ(τ)F(τ, x(τ))|dτ+∫t≥τ‖ψ(t)G(t, τ)ψ−1(τ)‖ |ψ(τ)F(τ, x(τ))|dτ≤ N2∫t≤τe−ν2(τ−t)m(τ)dτ + N1∫t≥τe−ν1(t−τ)m(τ)dτ≤ N2∫s≥0e−ν2sm(t + s)ds + N1∫s≤0eν1sm(t + s)ds≤ N2a1∞∑k=0e−ν2k + N1a1∞∑k=0e−ν1k =N2a11− e−ν2 +N1a11− e−ν1 .Hence by a1 ≤ r(N21−e−ν2 +N11−e−ν1)−1we obtain∣∣∣∣ψ(t) ∫J G(t, τ)F(τ, x(τ))dτ∣∣∣∣ ≤ r.Thus the operator Q maps the ball Sψ,r into itself.Now we shall prove that the operator Q is a contraction in the ball Sψ,r. Let x1, x2 ∈ Sψ,r.We obtain|ψ(t)Qx1(t)− ψ(t)Qx2(t)| ≤∣∣∣∣ψ(t) ∫J G(t, τ)(F(τ, x1(τ))− F(τ, x2(τ))dτ∣∣∣∣≤∫J∥∥∥ψ(t)G(t, τ)ψ−1(τ)∥∥∥ |ψ(τ)(F(τ, x1(τ))− F(τ, x2(τ))| dτ≤∫J∥∥∥ψ(t)G(t, τ)ψ−1(τ)∥∥∥ k(τ)|ψ(τ)(x1(τ))− x2(τ))|dτ≤∫J∥∥∥ψ(t)G(t, τ)ψ−1(τ)∥∥∥ k(τ)dτ supτ∈J|ψ(τ)(x1(τ))− x2(τ))|≤(N2a21− e−ν2 +N1a21− e−ν1)supτ∈J|ψ(τ)(x1(τ))− x2(τ))|.Nonlinear equations with ψ-dichotomous linear part 5Hence|||Qx1 −Qx2|||Cψ ≤(N2a21− e−ν2 +N1a21− e−ν1)|||x1 − x2|||Cψ .Thus by a2 <(N21−e−ν2 +N11−e−ν1)−1the operator Q is a contraction in the ball Sψ,r.From Banach’s fixed point principle the existence of a unique fixed point of the operator Qfollows.Corollary 3.2. If the conditions of Theorem 3.1 are fulfilled and if, moreover, F(t, 0) = 0 (t ∈ R) thenx = 0 is a unique solution of (2.1) in Cψ(X).Proof. Let F(t, 0) = 0 (t ∈ R). Then from H2 it follows|ψ(t)F(t, x(t))| ≤ k(t)|ψ(t)x(t)| (t ∈ R).Thus every solution x(t) except x(t) ≡ 0 (t ∈ R) will leave any ball Sψ,r1 (r1 < r) by t → ∞ ort→ −∞.Theorem 3.3. Let the following conditions be fulfilled:1. The linear part of (2.1) has ψ-ordinary dichotomy on R with projections P1 and P2.2. The function F(t, x) belongs to the class Dψ(a1, a2, r).Then for each r > 0 for sufficient small values of a1, a2 the equation (2.1) has a unique solution x(t),which is defined for t ∈ R and for which |ψ(t)x(t)| ≤ r (t ∈ R).Proof. Let J = R. In the proof of Theorem 3.1 it was mentioned that each solution x(t) ofequation (2.1) that remains for t ∈ J in the ball Sψ,r satisfies the integral equationx(t) =∫JG(t, τ)F(τ, x(τ))dτand vice versa.We consider again in the space Cψ(X) the operator Q : Cψ(X)→ Cψ(X) defined in (3.1).For |ψ(t)Qx(t)| we obtain the following estimate:|ψ(t)Qx(t)| ≤∣∣∣∣ψ(t) ∫J G(t, τ)F(τ, x(τ))dτ∣∣∣∣ .With a1 ≤ r max{N1, N2} we have|ψ(t)Qx(t)| ≤∣∣∣∣ψ(t) ∫J G(t, τ)F(τ, x(τ))dτ∣∣∣∣≤∫J∥∥∥ψ(t)G(t, τ)ψ−1(τ)∥∥∥ |ψ(τ)F(τ, x(τ))|dτ=∫t≤τ∥∥∥ψ(t)G(t, τ)ψ−1(τ)∥∥∥ |ψ(τ)F(τ, x(τ))|dτ+∫t≥τ∥∥∥ψ(t)G(t, τ)ψ−1(τ)∥∥∥ |ψ(τ)F(τ, x(τ))|dτ≤ N2∫t≤τm(τ)dτ + N1∫t≥τm(τ)dτ≤ max{N1, N2}∫Jm(τ)dτ ≤ max{N1, N2}a1 ≤ r.Thus the operator Q maps the ball Sψ,r into itself.6 A. Georgieva, H. Kiskinov, S. Kostadinov, A. ZaharievNow we shall prove that the operator Q is a contraction in the ball Sψ,r. Let x1, x2 ∈ Sψ,r.We obtain|ψ(t)Qx1(t)− ψ(t)Qx2(t)| ≤∣∣∣∣ψ(t) ∫J G(t, τ)(F(τ, x1(τ))− F(τ, x2(τ))dτ∣∣∣∣≤∫J∥∥∥ψ(t)G(t, τ)ψ−1(τ)∥∥∥ |ψ(τ)(F(τ, x1(τ))− F(τ, x2(τ))|dτ≤∫J∥∥∥ψ(t)G(t, τ)ψ−1(τ)∥∥∥ k(τ)|ψ(τ)(x1(τ))− x2(τ))|dτ≤∫J∥∥∥ψ(t)G(t, τ)ψ−1(τ)∥∥∥ k(τ)dτ supτ∈J|ψ(τ)(x1(τ))− x2(τ))|≤ (max{N1, N2}a2) supτ∈J|ψ(τ)(x1(τ))− x2(τ))|.Hence|||Qx1 −Qx2|||Cψ ≤ (a2 max{N1, N2}) |||x1 − x2|||Cψ .Thus by a2 < (max{N1, N2})−1 the operator Q is a contraction in the ball Sψ,r.From Banach’s fixed point principle the existence of a unique fixed point of the operator Qfollows.Theorem 3.4. Let the following conditions be fulfilled:1. The linear part of (2.1) has ψ-exponential dichotomy on R+ with projections P1 and P2.2. The function F(t, x) belongs to the class EDψ(a1, a2, r).Then for any r > 0 by sufficient small a1, a2 there exists ρ < r such that the equation (2.1) hasfor each ξ ∈ X1 = P1X with |ψ(0)ξ| ≤ ρ a unique solution x(t) on R+ for which P1x(0) = ξ and|ψ(t)x(t)| ≤ r (t ∈ R+).Proof. Let J = R+ and x(t) be a solution of equation (2.1) that remains for t ∈ J in the ballSψ,r = {x : |||x|||Cψ ≤ r}. From the results obtained in [16, Theorem 3.6 and Remark 3.8] itfollows that such x(t) satisfies the integral equationx(t) = V(t)ξ +∫JG(t, τ)F(τ, x(τ))dτ (3.3)where ξ = P1x(0). The converse is also true: a solution of the integral equation (3.3) satisfiesthe differential equation (2.1) for t ∈ J.Let ξ ∈ X1 and |ψ(0)ξ| ≤ ρ < r. We consider in the space Cψ(X) the operator Q : Cψ(X)→Cψ(X) defined by the formulaQx(t) = V(t)ξ +∫JG(t, τ)F(τ, x(τ))dτ (3.4)First we shall prove, that the operator Q maps the ball Sψ,r into itself. Indeed we have|ψ(t)Qx(t)| ≤ |ψ(t)V(t)ξ|+∣∣∣∣ψ(t) ∫J G(t, τ)F(τ, x(τ))dτ∣∣∣∣ .For the first addend with ρ ≤ r2N1 we obtain|ψ(t)V(t)ξ| ≤ N1e−ν1t|ψ(0)ξ| ≤ N1e−ν1tρ ≤ r2.Nonlinear equations with ψ-dichotomous linear part 7Using the same technique and notations as in the proof of Theorem 3.1 we obtain for the secondaddend the estimate ∣∣∣∣ψ(t) ∫J G(t, τ)F(τ, x(τ))dτ∣∣∣∣ ≤ N2a11− e−ν2 + N1a11− e−ν1 .Hence by a1 ≤ r2(N21−e−ν2 +N11−e−ν1)−1we obtain∣∣∣∣ψ(t) ∫J G(t, τ)F(τ, x(τ))dτ∣∣∣∣ ≤ r2.Thus the operator Q maps the ball Sψ,r into itself.Now we shall prove that the operator Q is a contraction in the ball Sψ,r. Let x1, x2 ∈ Sψ,r.We obtain as in the proof of Theorem 3.1 the estimate|||Qx1 −Qx2|||Cψ ≤(N2a21− e−ν2 +N1a21− e−ν1)|||x1 − x2|||CψBy a2 <(N21−e−ν2 +N11−e−ν1)−1the operator Q is a contraction in the ball Sψ,r.From Banach’s fixed point principle the existence of a unique fixed point of the operator Qfollows.Theorem 3.5. Let the following conditions be fulfilled:1. The linear part of (2.1) has ψ-ordinary dichotomy on R+ with projections P1 and P2.2. The function F(t, x) belongs to the class Dψ(a1, a2, r).Then for any r > 0 by sufficiently small a1, a2 there exists ρ < r such that the equation (2.1) hasfor each ξ ∈ X1 = P1X with |ψ(0)ξ| ≤ ρ a unique solution x(t) on R+ for which P1x(0) = ξ and|ψ(t)x(t)| ≤ r (t ∈ R+).Proof. Let J = R+, ξ ∈ X1 and |ψ(0)ξ| ≤ ρ < r. We consider again in the space Cψ(X) theoperator Q : Cψ(X)→ Cψ(X) defined by the formula (3.4).First we shall prove, that the operator Q maps the ball Sψ,r into itself. We have|ψ(t)Qx(t)| ≤ |ψ(t)V(t)ξ|+∣∣∣∣ψ(t) ∫J G(t, τ)F(τ, x(τ))dτ∣∣∣∣ .For the first addend with ρ ≤ r2N1 we obtain|ψ(t)V(t)ξ| ≤ N1|ψ(0)ξ| ≤ N1ρ ≤ r2.For the second addend with a1 ≤ r2 max{N1,N2} as in the proof of Theorem 3.3 we have∣∣∣∣ψ(t) ∫J G(t, τ)F(τ, x(τ))dτ∣∣∣∣ ≤ max{N1, N2}a1 ≤ r2.Thus the operator Q maps the ball Sψ,r into itself.Let x1, x2 ∈ Sψ,r. As in the proof of Theorem 3.3 we obtain the estimate|||Qx1 −Qx2|||Cψ ≤ (a2 max{N1, N2}) |||x1 − x2|||Cψ .Hence by a2 < (max{N1, N2})−1 the operator Q is a contraction in the ball Sψ,r.8 A. Georgieva, H. Kiskinov, S. Kostadinov, A. ZaharievFrom the fixed point principle of Banach it follows the existence of a unique fixed point ofthe operator Q.In the proof of Theorem 3.4 it was already mentioned that every solution of the differentialequation (2.1) which lies in the ball Sψ,r fulfil the equalityx(t) = Qx(t)and vice versa.Corollary 3.6. Let the conditions of Theorem 3.5 hold and let x1(t) and x2(t) be two solutions whoseinitial values fulfil P1x1(0) = ξ and P1x2(0) = η. Let N = max{N1, N2}.Then for Na2 < 1 the following estimate holds|ψ(t)(x1(t)− x2(t))| ≤ N1− Na2 |ψ(0)(ξ − η)| (t ∈ R+).Proof. Applying the presentation (3.3) for the solutions x1 and x2 we obtainx1(t)− x2(t) = V(t)(ξ − η) +∫ ∞0G(t, τ)(F(τ, x1(τ))− F(τ, x2(τ)))dτ.From here and the conditions of Theorem 3.5 for u(t) = ψ(t)(x1(t)− x2(t)) we obtain|u(t)| ≤ N|ψ(0)(ξ − η)|+ N∫ ∞0k(τ)u(τ)dτ.Let us consider the equationu(t) = α+ N∫ ∞0k(τ)u(τ)dτ, (3.5)where α = N|ψ(0)(ξ − η)|. Let us introduce the functional Φ : C → R+, where C is the spaceof all bounded functions on R+ with values in R+ by the formula(Φu)(t) = N∫ ∞0k(τ)u(τ)dτ.For the norm of Φ we obtain the estimate‖Φ‖ ≤ N∫ ∞0k(τ)dτ ≤ Na2.For sufficiently small a2 we have ‖Φ‖ ≤ 1.Let IC be the identity of the space C. Then the equation (IC − Φ)u = α has a boundedsolution u(t), i.e. there exists a constant c = supt∈R+ |u(t)| < ∞. We shall estimate the constantc from equation (3.5):c ≤ α+ Nc∫ ∞0k(τ)dτ ≤ α+ Nca2,i.e.c ≤ α1− Na2 .Finally we obtain|ψ(t)(x1(t)− x2(t))| ≤ N|ψ(0)(ξ − η)|1− Na2 .Nonlinear equations with ψ-dichotomous linear part 9AcknowledgementsThis research has been partially supported by Plovdiv University NPD grant NI13 FMI-002.References[1] O. AKINYELE, On partial stability and boundedness of degree k, Atti. Accad. Naz. LinceiRend. Cl. Sci. Fis. Mat. Natur. 8 65(1978), 259–264. MR0586610[2] P. N. BOI, On the ψ-dichotomy for homogeneous linear differential equations, Electron. J.Differ. Equ. 2006, No. 40, 1–12. MR2213584[3] P. N. BOI, Existence of ψ-bounded solutions for nonhomogeneous linear differential equa-tions, Electron. J. Differ. Equ. 2007, No. 52, 1–10. MR2299606[4] P. N. BOI, Behavior at infinity of ψ-evanescent solutions to linear differential equations,Electron. J. Differ. Equ. 2010, No. 102, 1–9. MR2680305[5] W. A. 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Existence of solutions of nonlinear differential equations with $\psi$ -exponential or $\psi$ -ordinary dichotomous linear part in a Banach space

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Existence of solutions of nonlinear differential equations with ψ\psiψ-exponential or ψ\psiψ-ordinary dichotomous linear part in a Banach space

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Existence of solutions of nonlinear differential equations with $\psi$ -exponential or $\psi$ -ordinary dichotomous linear part in a Banach space