Ulam stability for a delay differential equation

Diana Otrocol
(original), ISI/JCR, paper

Abstract

We study the Ulam–Hyers stability and generalized Ulam–Hyers–Rassias stability for a delay differential equation. Some examples are given.

Authors

D. Otrocol
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

V.A. Ilea
(Babes Bolyai Univ.)

Keywords

Ulam–Hyers stability, Ulam–Hyers–Rassias stability, delay differential equation

Cite this paper as:

D. Otrocol, V. Ilea, Ulam stability for a delay differential equation, Cent. Eur. J. Math., Vol. 11(7) (2013), pp. 1296-1303, doi: 10.2478/s11533-013-0233-9

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https://ictp.acad.ro/otrocol/papers/2013-Otrocol-Ilea-CEJM-Ulam%20stability%20for%20dde.pdf

About this paper

Journal

Central European Journal of Mathematics

Publisher Name

Versita, Warsaw, Poland

DOI

http://doi.org/10.2478/s11533-013-0233-9

Print ISSN

1895-1074

Online ISSN
MR

MR3047057

ZBL

Google Scholar

https://scholar.google.ro/scholar?oi=bibs&hl=en&cites=17142862393831458783

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[2] Castro L.P., Ramos A., Hyers–Ulam–Rassias stability for a class of nonlinear Volterra integral equations, Banach J. Math. Anal., 2009, 3(1), 36–43

[3] Guo D., Lakshmikantham V., Liu X., Nonlinear Integral Equations in Abstract Spaces, Math. Appl., 373, Kuwer, Dordrecht, 1996

[4] Hyers D.H., Isac G., Rassias Th.M., Stability of Functional Equations in Several Variables, Progr. Nonlinear Differential Equations Appl., 34, Birkhäuser, Boston, 1998

[5] Jung S.-M., A fixed point approach to the stability of a Volterra integral equation, Fixed Point Theory Appl., 2007, # 57064

[6] Kolmanovskiı V., Myshkis A., Applied Theory of Functional-Differential Equations, Math. Appl. (Soviet Ser.), 85, Kluwer, Dordrecht, 1992

[7] Otrocol D., Ulam stabilities of differential equation with abstract Volterra operator in a Banach space, Nonlinear Funct. Anal. Appl., 2010, 15(4), 613–619

[8] Petru T.P., Petrușel A., Yao J.-C., Ulam–Hyers stability for operatorial equations and inclusions via nonself operators, Taiwanese J. Math., 2011, 15(5), 2195–2212

[9] Radu V., The fixed point alternative and the stability of functional equations, Fixed Point Theory, 2003, 4(1), 91–96

[10] Rassias Th.M., On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 1978, 72(2), 297–300

[11] Rus I.A., Generalized Contractions and Applications, Cluj University Press, Cluj-Napoca, 2001

[12] Rus I.A., Gronwall lemmas: ten open problems, Sci. Math. Jpn., 2009, 70(2), 221–228

[13] Rus I.A., Ulam stability of ordinary differential equations, Stud. Univ. Babeș-Bolyai Math., 2009, 54(4), 125–133

[14] Rus I.A., Remarks on Ulam stability of the operatorial equations, Fixed Point Theory, 2009, 10(2), 305–320

[15] Ulam S.M., A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, 8, Interscience, New York–London, 1960

Ulam stability for a delay differential equation

Diana Otrocol∗ and Veronica Ilea∗∗
Abstract.

In this paper we study the Ulam-Hyers stability and generalized Ulam-Hyers-Rassias stability for a delay differential equation. Some examples are given.

MSC 2000: 34K20, 34L05, 47H10.
Keywords: Ulam-Hyers stability, Ulam-Hyers-Rassias stability, delay differential equation.

∗ “T. Popoviciu” Institute of Numerical Analysis, Romanian Academy, Fântânele 57, Cluj-Napoca, 400320, Romania.
∗∗ Department of Mathematics, Faculty of Mathematics and Computer Science, “Babeş-Bolyai” University, M. Kogălniceanu 1, Cluj-Napoca, RO-400084, Romania.

1. Introduction

In the last 3030 years, the stability theory of functional equations was strongly developed. Very important contributions to this subject were brought by Ulam [15], Rassias [10], Hyers et al. [4], Jung [5], Guo et al. [3], Kolmanovskii and Myshkis [6] and Radu [9]. Our results are connected to some recent papers of Castro and Ramos [2] and Jung [5] (where integral and differential equations are considered), Bota-Boriceanu and Petruşel [1] and Petru et al. [8] (where the Ulam-Hyers stability for operatorial equations and inclusions are discussed). Following [13] and [7], in our paper we will investigate Ulam-Hyers stability, generalized Ulam-Hyers-Rassias stability for the following differential equation with modification of the argument

x′​(t)=f​(t,x​(t),x​(g​(t))),t∈I⊂ℝ¯,x^{\prime}(t)=f(t,x(t),x(g(t))),\ t\in I\subset\overline{\mathbb{R}},

where

  • (i)

    I=[a,b]I=[a,b] or I=[a,∞[,a,b∈ℝI=[a,\infty[,\ a,b\in\mathbb{R};

  • (ii)

    f∈C​([a,b]×ℝ2,ℝ),g∈C​([a,b],[a−h,b]),g​(t)≤t,h>0f\in C([a,b]\times\mathbb{R}^{2},\mathbb{R}),\ g\in C([a,b],[a-h,b]),\ g(t)\leq t,\ h>0, respectively f∈C([a,∞[×ℝ2,ℝ),g∈C([a,∞[,[a−h,∞[,g(t)≤t,h>0f\in C([a,\infty[\times\mathbb{R}^{2},\mathbb{R}),\ g\in C([a,\infty[,[a-h,\infty[,\ g(t)\leq t,\ h>0.

By a solution of the above equation we understand a function x∈C​([a−h,b],ℝ)∩C1​([a,b],ℝ)x\in C([a-h,b],\mathbb{R})\cap C^{1}([a,b],\mathbb{R}), respectively x∈C([a−h,∞[,ℝ)∩C1([a,∞[,ℝ)x\in C([a-h,\infty[,\mathbb{R})\cap C^{1}([a,\infty[,\mathbb{R}), that verifies the equation.

2. Preliminaries

We begin our considerations with some notions and results from Ulam stability (see [13], [14]), for the case I=[a,b]I=[a,b].

For f∈C​(I×ℝ2,ℝ)f\in C(I\times\mathbb{R}^{2},\mathbb{R}), ε>0\varepsilon>0, φ∈C​([a−h,b],ℝ+)\varphi\in C([a-h,b],\mathbb{R}_{+}) and ψ∈C​([a−h,a],ℝ)\psi\in C([a-h,a],\mathbb{R}) we consider the following Cauchy problem

(2.1) x′​(t)=f​(t,x​(t),x​(g​(t))),t∈Ix^{\prime}(t)=f(t,x(t),x(g(t))),\ t\in I
(2.2) x​(t)=ψ​(t),t∈[a−h,a]x(t)=\psi(t),\ t\in[a-h,a]

and the following inequations

(2.3) |y′​(t)−f​(t,y​(t),y​(g​(t)))|≤ε,t∈I\left|y^{\prime}(t)-f(t,y(t),y(g(t)))\right|\leq\varepsilon,\ t\in I
(2.4) |y′​(t)−f​(t,y​(t),y​(g​(t)))|≤φ​(t),t∈I.\left|y^{\prime}(t)-f(t,y(t),y(g(t)))\right|\leq\varphi(t),\ t\in I.
Definition 2.1.

The equation (2.1) is Ulam-Hyers stable if there exists a real number c>0c>0 such that for each ε>0\varepsilon>0 and for each solution y∈C1​([a−h,b],ℝ)y\in C^{1}([a-h,b],\mathbb{R}) of (2.3) there exists a solution x∈C1​([a−h,b],ℝ)x\in C^{1}([a-h,b],\mathbb{R}) of (2.1) with

|y​(t)−x​(t)|≤c​ε,∀t∈[a−h,b].\left|y(t)-x(t)\right|\leq c\varepsilon,\ \ \forall t\in[a-h,b].
Definition 2.2.

The equation (2.1) is generalized Ulam-Hyers-Rassias stable with respect to φ\varphi, if there exists cφ>0c_{\varphi}>0, such that for each solution y∈C1​([a−h,b],ℝ)y\in C^{1}([a-h,b],\mathbb{R}) of the inequation (2.4) there exists a solution x∈C1​([a−h,b],ℝ)x\in C^{1}([a-h,b],\mathbb{R}) of (2.1) with

|y​(t)−x​(t)|≤cφ​φ​(t),∀t∈[a−h,b].\left|y(t)-x(t)\right|\leq c_{\varphi}\varphi(t),\ \forall t\in[a-h,b].
Remark 2.3.

A function y∈C1​(I,ℝ)y\in C^{1}(I,\mathbb{R}) is a solution of (2.3) if and only if there exists a function h∈C​(I,ℝ)h\in C(I,\mathbb{R}) (which depends on yy) such that

  • (i)

    |h​(t)|≤ε,∀t∈I;\left|h(t)\right|\leq\varepsilon,\ \forall t\in I;

  • (ii)

    y′​(t)=f​(t,y​(t),y​(g​(t)))+h​(t),∀t∈I.y^{\prime}(t)=f(t,y(t),y(g(t)))+h(t),\ \forall t\in I.

Remark 2.4.

A function y∈C1​(I,ℝ)y\in C^{1}(I,\mathbb{R}) is a solution of (2.4) if and only if there exists a function h~∈C​(I,ℝ)\widetilde{h}\in C(I,\mathbb{R}) (which depends on yy) such that

  • (i)

    |h~​(t)|≤φ​(t),∀t∈I;\left|\widetilde{h}(t)\right|\leq\varphi(t),\ \forall t\in I;

  • (ii)

    y′​(t)=f​(t,y​(t),y​(g​(t)))+h~​(t),∀t∈I.y^{\prime}(t)=f(t,y(t),y(g(t)))+\widetilde{h}(t),\ \forall t\in I.

Remark 2.5.

If y∈C1​(I,ℝ)y\in C^{1}(I,\mathbb{R}) is as solution of the inequation (2.3), then yy is a solution of the following integral inequation

|y​(t)−y​(a)−∫atf​(s,y​(s),y​(g​(s)))​𝑑s|≤(t−a)​ε,∀t∈I.\left|y(t)-y(a)-\int\nolimits_{a}^{t}f(s,y(s),y(g(s)))ds\right|\leq(t-a)\varepsilon,\ \forall t\in I.
Remark 2.6.

If y∈C1​(I,ℝ)y\in C^{1}(I,\mathbb{R}) is as solution of the inequation (2.4), then yy is a solution of the following integral inequation

|y​(t)−y​(a)−∫atf​(s,y​(s),y​(g​(s)))​𝑑s|≤∫atφ​(s)​𝑑s,∀t∈I.\left|y(t)-y(a)-\int\nolimits_{a}^{t}f(s,y(s),y(g(s)))ds\right|\leq\int\nolimits_{a}^{t}\varphi(s)ds,\ \forall t\in I.

Analogously, one may have the above definitions and remarks for the case I=[a,∞[I=[a,\infty[, the interval [a−h,b][a-h,b] would be replaced by [a−h,∞[[a-h,\infty[.

In the sequel we shall use the following Picard operator definition and the well-known Gronwall lemma and abstract Gronwall lemma (see, e.g. Rus [12]).

Definition 2.7.

(Rus [11]) Let (X,d)(X,d) be a metric space. An operator A:X→XA:X\rightarrow X is a Picard operator if there exists x∗∈Xx^{\ast}\in X such that:

  1. (i)

    FA={x∗}F_{A}=\{x^{\ast}\} where FA:={x∈X∣A​(x)=x}F_{A}:=\{x\in X\mid A(x)=x\} is the fixed point set of AA;

  2. (ii)

    the sequence (An​(x0))n∈ℕ(A^{n}(x_{0}))_{n\in\mathbb{N}} converges to x∗x^{\ast} for all x0∈Xx_{0}\in X.

Lemma 2.8.

(Gronwall Lemma) Let g,h∈C​([a,b],ℝ+)g,\ h\in C([a,b],\mathbb{R}_{+}) be two functions. We suppose that gg is increasing. If x∈C​([a,b],ℝ+)x\in C([a,b],\mathbb{R}_{+}) is a solution of the inequation

x​(t)≤g​(t)+∫abh​(s)​x​(s)​𝑑s,t∈[a,b],x(t)\leq g(t)+\int_{a}^{b}h(s)x(s)ds,\ t\in[a,b],

then

x​(t)≤g​(t)​exp⁡(∫abh​(s)​𝑑s),t∈[a,b].x(t)\leq g(t)\exp\left(\int_{a}^{b}h(s)ds\right)\!\!,\ t\in[a,b].
Lemma 2.9.

(Abstract Gronwall Lemma) Let (X,d,≤)(X,d,\leq) be an ordered metric space and A:X→XA:X\rightarrow X an operator. We suppose that:

(i) AA is a Picard operator (FA={xA∗}F_{A}=\{x_{A}^{*}\});

(ii) AA is an increasing operator.

Then we have: (a) x∈X,x≤A​(x)⟹x≤xA∗x\in X,\ x\leq A(x)\Longrightarrow x\leq x_{A}^{\ast};

(b) x∈X,x≥A​(x)⟹x≥xA∗x\in X,\ x\geq A(x)\Longrightarrow x\geq x_{A}^{\ast}.

3. Ulam-Hyers stability on a compact interval I=[a,b]I=[a,b]

In this section we present conditions for the equation (2.1) to admit the Ulam-Hyers stability on a compact interval I=[a,b]I=[a,b].

Theorem 3.1.

We suppose that

  • (a)

    f∈C​([a,b]×ℝ2,ℝ),g∈C​([a,b],[a−h,b]),g​(t)≤t,h>0;f\in C([a,b]\times\mathbb{R}^{2},\mathbb{R}),\ g\in C([a,b],[a-h,b]),\ g(t)\leq t,\ h>0;

  • (b)

    there exists Lf>0L_{f}>0 such that ∀t∈[a,b],ui,vi∈ℝ,i=1,2\forall t\in[a,b],u_{i},v_{i}\in\mathbb{R},i=1,2, we have

    |f​(t,u1,u2)−f​(t,v1,v2)|≤Lf​∑i=12|ui−vi|;\left|f(t,u_{1},u_{2})-f(t,v_{1},v_{2})\right|\leq L_{f}\sum_{i=1}^{2}\left|u_{i}-v_{i}\right|;
  • (c)

    2​(b−a)​Lf<1.2(b-a)L_{f}<1.

Then

  • (i)

    the problem (2.1)–(2.2) has a unique solution in C​([a−h,b],ℝ)∩C1​([a,b],ℝ);C([a-h,b],\mathbb{R})\cap C^{1}([a,b],\mathbb{R});

  • (ii)

    the equation (2.1) is Ulam-Hyers stable.

Proof.

(i) In the condition (a),(a), the problem (2.1)–(2.2) is equivalent to the integral equation

x​(t)={ψ​(t),t∈[a−h,a],ψ​(t)+∫atf​(s,x​(s),x​(g​(s)))​𝑑s,t∈[a,b].x(t)=\begin{cases}\psi(t),\ &t\in[a-h,a],\\ \psi(t)+{\displaystyle\int_{a}^{t}f(s,x(s),x(g(s)))ds},\ &t\in[a,b].\end{cases}

Let X:=C​([a−h,b],ℝ)X:=C([a-h,b],\mathbb{R}) and Bf:X→XB_{f}:X\rightarrow X be given by

Bf​(x)​(t):={ψ​(t),t∈[a−h,a],ψ​(t)+∫atf​(s,x​(s),x​(g​(s)))​𝑑s,t∈[a,b].B_{f}(x)(t):=\begin{cases}\psi(t),\ &t\in[a-h,a],\\ \psi(t)+{\displaystyle\int_{a}^{t}f(s,x(s),x(g(s)))ds},\ &t\in[a,b].\end{cases}

We show that BfB_{f} is a contraction on XX with respect to the Chebyshev norm.

|Bf​(x)​(t)−Bf​(y)​(t)|=0,∀x,y∈C​([a−h,b],ℝ),t∈[a−h,a].\left|B_{f}(x)(t)-B_{f}(y)(t)\right|=0,\ \ \forall x,y\in C([a-h,b],\mathbb{R}),\ t\in[a-h,a].
|Bf​(x)​(t)−Bf​(y)​(t)|\displaystyle\left|B_{f}(x)(t)-B_{f}(y)(t)\right|
≤|∫atf​(s,x​(s),x​(g​(s)))​𝑑s−∫atf​(s,y​(s),y​(g​(s)))​𝑑s|\displaystyle\leq\left|\int_{a}^{t}f(s,x(s),x(g(s)))ds-\int_{a}^{t}f(s,y(s),y(g(s)))ds\right|
≤Lf​(maxa−h≤t≤b​|x​(s)−y​(s)|+maxa−h≤t≤b​|x​(g​(s))−y​(g​(s))|)​(b−a)\displaystyle\leq L_{f}\left(\underset{a-h\leq t\leq b}{\max}\left|x(s)-y(s)\right|+\underset{a-h\leq t\leq b}{\max}\left|x(g(s))-y(g(s))\right|\right)(b-a)
≤2​(b−a)​Lf​‖x−y‖,∀x,y∈C​([a−h,b],ℝ),t∈[a,b].\displaystyle\leq 2(b-a)L_{f}\left\|x-y\right\|,\ \forall x,y\in C([a-h,b],\mathbb{R}),\ t\in[a,b].

So,

‖Bf​(x)−Bf​(y)‖≤2​(b−a)​Lf​‖x−y‖,∀x,y∈C​([a−h,b],ℝ),\left\|B_{f}(x)-B_{f}(y)\right\|\leq 2(b-a)L_{f}\left\|x-y\right\|,\ \forall x,y\in C([a-h,b],\mathbb{R}),

i.e., BfB_{f} is a contraction w.r.t. the Chebyshev norm on X.X. The proof follows from the Banach contraction principle.

(ii) Let y∈C​([a−h,b],ℝ)∩C1​([a,b],ℝ)y\in C([a-h,b],\mathbb{R})\cap C^{1}([a,b],\mathbb{R}) be a solution of the inequation (2.3). We denote by x∈C​([a−h,b],ℝ)∩C1​([a,b],ℝ)x\in C([a-h,b],\mathbb{R})\cap C^{1}([a,b],\mathbb{R}) the unique solution of the Cauchy problem

x′​(t)\displaystyle x^{\prime}(t) =f​(t,x​(t),x​(g​(t))),t∈[a,b],\displaystyle=f(t,x(t),x(g(t))),\ t\in[a,b],
x​(t)\displaystyle x(t) =y​(t),t∈[a−h,a].\displaystyle=y(t),\ t\in[a-h,a].

From condition (a) we have

x​(t)={y​(t),t∈[a−h,a],y​(a)+∫atf​(s,x​(s),x​(g​(s)))​𝑑s,t∈[a,b].x(t)=\begin{cases}y(t),\ &t\in[a-h,a],\\ y(a)+{\displaystyle\int_{a}^{t}f(s,x(s),x(g(s)))ds},\ &t\in[a,b].\end{cases}

Remark 2.5 gives

|y​(t)−y​(a)−∫atf​(s,y​(s),y​(g​(s)))​𝑑s|≤(t−a)​ε,t∈[a,b].\left|y(t)-y(a)-\int_{a}^{t}f(s,y(s),y(g(s)))ds\right|\leq(t-a)\varepsilon,\ t\in[a,b].

It follows that |y​(t)−x​(t)|=0,\left|y(t)-x(t)\right|=0, for t∈[a−h,a]t\in[a-h,a] and for t∈[a,b]t\in[a,b] we have

(3.1) |y​(t)−x​(t)|≤\displaystyle\left|y(t)-x(t)\right|\leq
≤|y​(t)−y​(a)−∫atf​(s,y​(s),y​(g​(s)))​𝑑s|+\displaystyle\leq\left|y(t)-y(a)-\int_{a}^{t}f(s,y(s),y(g(s)))ds\right|+
+∫at|f​(s,y​(s),y​(g​(s)))−f​(s,x​(s),x​(g​(s)))|​𝑑s\displaystyle\quad+\int_{a}^{t}\left|f(s,y(s),y(g(s)))-f(s,x(s),x(g(s)))\right|ds
≤(t−a)​ε+Lf​(∫at|y​(s)−x​(s)|​𝑑s+∫at|y​(g​(s))−x​(g​(s))|​𝑑s).\displaystyle\leq(t\!-\!a)\varepsilon+\!L_{f}\Big(\int_{a}^{t}\left|y(s)\!-\!x(s)\right|ds\!+\!\int_{a}^{t}\left|y(g(s))\!-\!x(g(s))\right|ds\Big).

According to the last inequality, for u∈C​([a−h,b],ℝ+)u\in C([a-h,b],\mathbb{R}_{+}) we consider the following operator A:C​([a−h,b],ℝ+)→C​([a−h,b],ℝ+)A:C([a-h,b],\mathbb{R}_{+})\rightarrow C([a-h,b],\mathbb{R}_{+}) defined by

A​(u)​(t):={0,t∈[a−h,a],(t−a)​ε+Lf​∫atu​(s)​𝑑s+Lf​∫atu​(g​(s))​𝑑s,t∈[a,b].A(u)(t):=\begin{cases}0,\ &t\in[a-h,a],\\ (t-a)\varepsilon+L_{f}{\displaystyle\int_{a}^{t}u(s)ds+L_{f}\int_{a}^{t}u(g(s))ds},\ &t\in[a,b].\end{cases}

In order to verify that AA is a Picard operator (Definition 2.7) we prove that AA is a contraction.

For t∈[a,b]t\in[a,b]:

|A​(u)​(t)−A​(v)​(t)|≤\displaystyle\left|A(u)(t)-A(v)(t)\right|\leq
≤Lf​(∫at|u​(s)−v​(s)|​𝑑s+∫at|u​(g​(s))−v​(g​(s))|​𝑑s)\displaystyle\leq L_{f}\left(\int_{a}^{t}\left|u(s)-v(s)\right|ds+\int_{a}^{t}\left|u(g(s))-v(g(s))\right|ds\right)
≤Lf​(maxa−h≤t≤b​|u​(s)−v​(s)|+maxa−h≤t≤b​|u​(g​(s))−v​(g​(s))|)​(b−a)\displaystyle\leq L_{f}\left(\underset{a-h\leq t\leq b}{\max}\left|u(s)-v(s)\right|+\underset{a-h\leq t\leq b}{\max}\left|u(g(s))-v(g(s))\right|\right)(b-a)
≤2​(b−a)​Lf​‖u−v‖,∀u,v∈C​([a−h,b],ℝ+).\displaystyle\leq 2(b-a)L_{f}\left\|u-v\right\|,\ \forall u,v\in C([a-h,b],\mathbb{R}_{+}).

So,‖A​(u)−A​(v)‖≤2​(b−a)​Lf​‖u−v‖,∀u,v∈C​([a−h,b],ℝ+),\ \left\|A(u)-A(v)\right\|\leq 2(b-a)L_{f}\left\|u-v\right\|,\ \forall u,v\in C([a-h,b],\mathbb{R}_{+}), i.e., AA is a contraction w.r.t. the Chebyshev norm on C​([a−h,b],ℝ+).C([a-h,b],\mathbb{R}_{+}). Applying the Banach contraction principle, we have that AA is Picard operator and FA={u∗}F_{A}=\{u^{\ast}\}. Then

u∗​(t)=(t−a)​ε+Lf​∫atu∗​(s)​𝑑s+Lf​∫atu∗​(g​(s))​𝑑s,t∈[a,b].u^{\ast}(t)=(t-a)\varepsilon+L_{f}\int_{a}^{t}u^{\ast}(s)ds+L_{f}\int_{a}^{t}u^{\ast}(g(s))ds,\ t\in[a,b].

The solution u∗u^{\ast} is increasing and (u∗)′≥0.(u^{\ast})^{\prime}\geq 0. So, u∗​(g​(t))≤u∗​(t)u^{\ast}(g(t))\leq u^{\ast}(t) and

u∗​(t)≤(t−a)​ε+2​Lf​∫atu∗​(s)​𝑑s.u^{\ast}(t)\leq(t-a)\varepsilon+2L_{f}\int_{a}^{t}u^{\ast}(s)ds.

From the Gronwall Lemma we obtain

u∗​(t)≤c​ε,t∈[a−h,b],where​c:=(b−a)​exp⁡(2​Lf​(b−a)).u^{\ast}(t)\leq c\varepsilon,\ t\in[a-h,b],\ \text{where}\ c:=(b-a)\exp(2L_{f}(b-a)).

In particular, if u:=|y−x|u:=\left|y-x\right|, from (3.1), u​(t)≤A​(u)​(t)u(t)\leq A(u)(t) and applying the abstract Gronwall lemma we obtain u​(t)≤u∗​(t)u(t)\leq u^{\ast}(t) (AA is a Picard and an increasing operator). It follows that

|y​(t)−x​(t)|≤c​ε,t∈[a−h,b],\left|y(t)-x(t)\right|\leq c\varepsilon,\ t\in[a-h,b],

i.e., the equation (2.1) is Ulam-Hyers stable. ∎

4. Generalized Ulam-Hyers-Rassias stability on I=[a,∞[I=[a,\infty[

In this section we present conditions for the equation (2.1) to admit the generalized Ulam-Hyers-Rassias stability on the interval I=[a,∞[I=[a,\infty[.

Theorem 4.1.

We suppose that

  • (a)

    f∈C([a,∞[×ℝ2,ℝ),g∈C([a,∞[,[a−h,∞[),g(t)≤t,h>0;f\in C([a,\infty[\times\mathbb{R}^{2},\mathbb{R}),\ g\in C([a,\infty[,[a-h,\infty[),\ g(t)\leq t,\ h>0;

  • (b)

    there exists lf∈L1([a,∞[,ℝ+)l_{f}\in L^{1}([a,\infty[,\mathbb{R}_{+}) such that ∀t∈[a,∞[,ui,vi∈ℝ,i=1,2\forall t\in[a,\infty[,u_{i},v_{i}\in\mathbb{R},i=1,2, we have

    |f​(t,u1,u2)−f​(t,v1,v2)|≤lf​(t)​(|u1−v1|+|u2−v2|);\displaystyle\left|f(t,u_{1},u_{2})-f(t,v_{1},v_{2})\right|\leq l_{f}(t)(\left|u_{1}-v_{1}\right|+\left|u_{2}-v_{2}\right|);
  • (c)

    the function φ∈C[a,∞[\varphi\in C[a,\infty[ is increasing;

  • (d)

    there exists λ>0\lambda>0 such that

    ∫atφ(s)ds≤λφ(t),t∈[a,∞[.\int_{a}^{t}\varphi(s)ds\leq\lambda\varphi(t),\ t\in[a,\infty[.

Then

  • (i)

    the problem (2.1)–(2.2) has a unique solution in C([a−h,∞[,ℝ)∩C1([a,∞[,ℝ);C([a-h,\infty[,\mathbb{R})\cap C^{1}([a,\infty[,\mathbb{R});

  • (ii)

    the equation (2.1) is generalized Ulam-Hyers-Rassias stable with respect to φ\varphi.

Proof.

The proof follows the same steps as in Theorem 3.1. Let y∈C([a−h,∞[,ℝ)∩C1([a,∞[,ℝ)y\in C([a-h,\infty[,\mathbb{R})\cap C^{1}([a,\infty[,\mathbb{R}) be a solution of the inequation (2.4). The equation (2.1) has a unique solution in C([a−h,∞[,ℝ)∩C1([a,∞[,ℝ).C([a-h,\infty[,\mathbb{R})\cap C^{1}([a,\infty[,\mathbb{R}). We denote by x∈C([a−h,∞[,ℝ)∩C1([a,∞[,ℝ)x\in C([a-h,\infty[,\mathbb{R})\cap C^{1}([a,\infty[,\mathbb{R}) the unique solution of the Cauchy problem

x′​(t)\displaystyle x^{\prime}(t) =f(t,x(t),x(g(t))),t∈[a,∞[,\displaystyle=f(t,x(t),x(g(t))),\ t\in[a,\infty[,
x​(a)\displaystyle x(a) =y​(t),t∈[a−h,a].\displaystyle=y(t),\ t\in[a-h,a].

So

x​(t)={y​(t),t∈[a−h,a]y​(a)+∫atf​(s,x​(s),x​(g​(s)))​𝑑s,t∈[a,∞[.x(t)=\begin{cases}y(t),\ &t\in[a-h,a]\\ y(a)+\int_{a}^{t}f(s,x(s),x(g(s)))ds,\ &t\in[a,\infty[.\end{cases}

Remark 2.6 gives

|y​(t)−y​(a)−∫atf​(s,y​(s),y​(g​(s)))​𝑑s|≤\displaystyle\left|y(t)-y(a)-\int_{a}^{t}f(s,y(s),y(g(s)))ds\right|\leq
≤∫atφ(s)ds≤λφ(t),t∈[a,∞[.\displaystyle\leq\int_{a}^{t}\varphi(s)ds\leq\lambda\varphi(t),\ t\in[a,\infty[.

From the above relations, for t∈[a−h,a]t\in[a-h,a] we have |y​(t)−x​(t)|=0\left|y(t)-x(t)\right|=0 and for t∈[a,∞[t\in[a,\infty[, we obtain

|y​(t)−x​(t)|\displaystyle\left|y(t)-x(t)\right| ≤|y​(t)−y​(a)−∫atf​(s,y​(s),y​(g​(s)))​𝑑s|+\displaystyle\leq\left|y(t)-y(a)-\int_{a}^{t}f(s,y(s),y(g(s)))ds\right|+
+∫at|f​(s,y​(s),y​(g​(s)))−f​(s,x​(s),x​(g​(s)))|​𝑑s\displaystyle\quad+\int_{a}^{t}\left|f(s,y(s),y(g(s)))-f(s,x(s),x(g(s)))\right|ds
≤λ​φ​(t)+∫atlf​(s)​|y​(s)−x​(s)|​𝑑s+∫atlf​(s)​|y​(g​(s))−x​(g​(s))|​𝑑s.\displaystyle\leq\lambda\varphi(t)\!+\!\int_{a}^{t}\!l_{f}(s)\!\left|y(s)\!-\!x(s)\right|ds\!+\!\int_{a}^{t}\!l_{f}(s)\!\left|y(g(s))\!-\!x(g(s))\right|ds.

As in the proof of Theorem 3.1 (ii), it follows that

|y(t)−x(t)|≤λφ(t)exp(∫at2lf(s)ds)=cφφ(t),t∈[a,∞[,\left|y(t)-x(t)\right|\leq\lambda\varphi(t)\exp\left(\int_{a}^{t}2l_{f}(s)ds\right)=c_{\varphi}\varphi(t),\ t\in[a,\infty[,

where cφ:=λ​exp⁡(∫at2​lf​(s)​𝑑s)c_{\varphi}:=\lambda\exp\left(\int_{a}^{t}2l_{f}(s)ds\right), i.e., the equation (2.1) is generalized Ulam-Hyers-Rassias stable. ∎

5. Applications

Here we present some consequences of the above theory.

Example 5.1.

We consider the following Cauchy problem

(5.1) x′​(t)=f​(t,x​(t),x​(t−h)),t∈[a,b)\displaystyle x^{\prime}(t)=f(t,x(t),x(t-h)),\ t\in[a,b)
(5.2) x​(a)=x0\displaystyle x(a)=x_{0}

and the following inequations

|y′​(t)−f​(t,y​(t),y​(t−h))|≤ε,t∈[a,b)\left|y^{\prime}(t)-f(t,y(t),y(t-h))\right|\leq\varepsilon,\ t\in[a,b)
|y′​(t)−f​(t,y​(t),y​(t−h))|≤φ​(t),t∈[a,b).\left|y^{\prime}(t)-f(t,y(t),y(t-h))\right|\leq\varphi(t),\ t\in[a,b).

In this case, from Theorem 3.1 we have:

Theorem 5.2.

We suppose that

  • (a)

    f∈C​([a,b]×ℝ2,ℝ);f\in C([a,b]\times\mathbb{R}^{2},\mathbb{R});

  • (b)

    there exists Lf>0L_{f}>0 such that ∀t∈[a,b],ui,vi∈ℝ,i=1,2\forall t\in[a,b],u_{i},v_{i}\in\mathbb{R},i=1,2 we have

    |f​(t,u1,u2)−f​(t,v1,v2)|≤Lf​∑i=12|ui−vi|;\left|f(t,u_{1},u_{2})-f(t,v_{1},v_{2})\right|\leq L_{f}\sum_{i=1}^{2}\left|u_{i}-v_{i}\right|;

Then

  • (i)

    the problem (5.1)–(5.2) has a unique solution in C​([a−h,b],ℝ)∩C1​([a,b],ℝ);C([a-h,b],\mathbb{R})\cap C^{1}([a,b],\mathbb{R});

  • (ii)

    the equation (5.1) is Ulam-Hyers stable.

Let b=+∞b=+\infty. The conditions (a)(a)-(d)(d) from Theorem 4.1 are the same, so the problem (5.1)–(5.2) has a unique solution in C([a−h,∞[,ℝ)∩C1([a,∞[,ℝ)C([a-h,\infty[,\mathbb{R})\cap C^{1}([a,\infty[,\mathbb{R}) and the equation (5.1) is generalized Ulam-Hyers-Rassias stable on [a,∞[[a,\infty[.

Theorem 5.3.

We suppose that

  • (a)

    f∈C([a,∞[×ℝ2,ℝ);f\in C([a,\infty[\times\mathbb{R}^{2},\mathbb{R});

  • (b)

    there exists lf∈L1([a,∞[,ℝ+)l_{f}\in L^{1}([a,\infty[,\mathbb{R}_{+}) such that ∀t∈[a,∞[,ui,vi∈ℝ,i=1,2\forall t\in[a,\infty[,u_{i},v_{i}\in\mathbb{R},i=1,2, we have

    |f​(t,u1,u2)−f​(t,v1,v2)|≤lf​(t)​(|u1−v1|+|u2−v2|);\left|f(t,u_{1},u_{2})-f(t,v_{1},v_{2})\right|\leq l_{f}(t)(\left|u_{1}-v_{1}\right|+\left|u_{2}-v_{2}\right|);
  • (c)

    the function φ∈C[a,∞[\varphi\in C[a,\infty[ is increasing;

  • (d)

    there exists λ>0\lambda>0 such that

    ∫0tφ(s)ds≤λφ(t),t∈[a,∞[.\int_{0}^{t}\varphi(s)ds\leq\lambda\varphi(t),\ t\in[a,\infty[.

Then

  • (i)

    the problem (5.1)–(5.2) has a unique solution in C([a−h,∞[,ℝ)∩C1([a,∞[,ℝ);C([a-h,\infty[,\mathbb{R})\cap C^{1}([a,\infty[,\mathbb{R});

  • (ii)

    the equation (5.1) is generalized Ulam-Hyers-Rassias stable with respect to φ\varphi.

Example 5.4.

We consider the following Cauchy problem

(5.3) x′​(t)=f​(t,x​(t),x​(t2)),t∈[0,1]\displaystyle x^{\prime}(t)=f(t,x(t),x(t^{2})),\ t\in[0,1]
(5.4) x​(0)=x0\displaystyle x(0)=x_{0}

and the following inequations

|y′​(t)−f​(t,y​(t),y​(t2))|≤ε,t∈[0,1)\left|y^{\prime}(t)-f(t,y(t),y(t^{2}))\right|\leq\varepsilon,\ t\in[0,1)
|y′​(t)−f​(t,y​(t),y​(t2))|≤φ​(t),t∈[0,1).\left|y^{\prime}(t)-f(t,y(t),y(t^{2}))\right|\leq\varphi(t),\ t\in[0,1).

For this example, Theorem 3.1 and Theorem 4.1 become

Theorem 5.5.

We suppose that

  • (a)

    f∈C​([0,1]×ℝ2,ℝ);f\in C([0,1]\times\mathbb{R}^{2},\mathbb{R});

  • (b)

    there exists Lf>0L_{f}>0 such that ∀t∈[0,1],ui,vi∈ℝ,i=1,2\forall t\in[0,1],u_{i},v_{i}\in\mathbb{R},i=1,2, we have

    |f​(t,u1,u2)−f​(t,v1,v2)|≤Lf​∑i=12|ui−vi|;\left|f(t,u_{1},u_{2})-f(t,v_{1},v_{2})\right|\leq L_{f}\sum_{i=1}^{2}\left|u_{i}-v_{i}\right|;

Then

  • (i)

    the problem (5.3)–(5.4) has a unique solution in C​([0,1],ℝ)∩C1​([0,1],ℝ);C([0,1],\mathbb{R})\cap C^{1}([0,1],\mathbb{R});

  • (ii)

    the equation (5.3) is Ulam-Hyers stable.

Theorem 5.6.

We suppose that

  • (a)

    f∈C([0,∞[×ℝ2,ℝ);f\in C([0,\infty[\times\mathbb{R}^{2},\mathbb{R});

  • (b)

    there exists lf∈L1([0,∞[,ℝ+)l_{f}\in L^{1}([0,\infty[,\mathbb{R}_{+}) such that ∀t∈[0,∞[,ui,vi∈ℝ,i=1,2\forall t\in[0,\infty[,u_{i},v_{i}\in\mathbb{R},i=1,2, we have

    |f​(t,u1,u2)−f​(t,v1,v2)|≤lf​(t)​(|u1−v1|+|u2−v2|);\left|f(t,u_{1},u_{2})-f(t,v_{1},v_{2})\right|\leq l_{f}(t)(\left|u_{1}-v_{1}\right|+\left|u_{2}-v_{2}\right|);
  • (c)

    the function φ∈C[0,∞[\varphi\in C[0,\infty[ is increasing;

  • (d)

    there exists λ>0\lambda>0 such that

    ∫0tφ(s)ds≤λφ(t),t∈[0,∞[.\int_{0}^{t}\varphi(s)ds\leq\lambda\varphi(t),\ t\in[0,\infty[.

Then

  • (i)

    the problem (5.3)–(5.4) has a unique solution in C([0,∞[,ℝ)∩C1([0,∞[,ℝ);C([0,\infty[,\mathbb{R})\cap C^{1}([0,\infty[,\mathbb{R});

  • (ii)

    the equation (5.3) is generalized Ulam-Hyers-Rassias stable with respect to φ\varphi.

Acknowledgement The authors would like to thank the referees for their useful and valuable suggestions.

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