AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2014/483707 483707 Research Article On the Hyers-Ulam Stability of Differential Equations of Second Order Alqifiary Qusuay H. 1, 2 Jung Soon-Mo 3 Rogovchenko Yuriy 1 Computer Science and Mathematics College, University of Al-Qadisiyah Al-Diwaniyah Iraq qu.edu.iq 2 College of Mathematics, University of Belgrade, Belgrade Serbia bg.ac.rs 3 Mathematics Section, College of Science and Technology Hongik University Sejong 339–701 Republic of Korea 2014 772014 2014 08 01 2014 16 06 2014 25 06 2014 7 7 2014 2014 Copyright © 2014 Qusuay H. Alqifiary and Soon-Mo Jung. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

By using of the Gronwall inequality, we prove the Hyers-Ulam stability of differential equations of second order with initial conditions.

1. Introduction

In 1940, Ulam  posed a problem concerning the stability of functional equations: “Give conditions in order for a linear function near an approximately linear function to exist.”

A year later, Hyers  gave an answer to the problem of Ulam for additive functions defined on Banach spaces: let X and Y be real Banach spaces and ε>0. Then for every function f:XY satisfying (1)f(x+y)-f(x)-f(y)ε for all x,yX, there exists a unique additive function A:XY with the property (2)f(x)-A(x)ε for all xX.

After Hyers’s result, many mathematicians have extended Ulam’s problem to other functional equations and generalized Hyers’s result in various directions (see ). A generalization of Ulam’s problem was recently proposed by replacing functional equations with differential equations: the differential equation (3)φ(f(t),y(t),y(t),,y(n)(t))=0 has the Hyers-Ulam stability if for a given ε>0 and a function y such that (4)|φ(f(t),y(t),y(t),,y(n)(t))|ε, there exists a solution y0 of the differential equation such that |y(t)-y0(t)|K(ε) and limε0K(ε)=0.

Obloza seems to be the first author who has investigated the Hyers-Ulam stability of linear differential equations (see [7, 8]). Thereafter, Alsina and Ger published their paper , which handles the Hyers-Ulam stability of the linear differential equation y(t)=y(t): if a differentiable function y(t) is a solution of the inequality |y(t)-y(t)|ε for any t(a,), then there exists a constant c such that |y(t)-cet|3ε for all t(a,).

Those previous results were extended to the Hyers-Ulam stability of linear differential equations of first order and higher order with constant coefficients in , respectively. Furthermore, Jung has also proved the Hyers-Ulam stability of linear differential equations (see ). Rus investigated the Hyers-Ulam stability of differential and integral equations using the Gronwall lemma and the technique of weakly Picard operators (see [18, 19]). Recently, the Hyers-Ulam stability problems of linear differential equations of first order and second order with constant coefficients were studied by using the method of integral factors (see [20, 21]). The results given in [10, 15, 20] have been generalized by Cimpean and Popa  and by Popa and Raşa [23, 24] for the linear differential equations of nth order with constant coefficients. Furthermore, the Laplace transform method was recently applied to the proof of the Hyers-Ulam stability of linear differential equations (see ).

This paper consists of two main sections. In Section 2, we introduce some sufficient conditions under which each solution of the linear differential equation (11) is bounded. In Section 3, we prove the Hyers-Ulam stability of the linear differential equations of the form (11) as well as the nonlinear differential equations of the form (55) by using the Gronwall lemma that was recently introduced by Rus [18, 19] in studying the Hyers-Ulam stability of differential equations.

One of the advantages of this paper is that the authors have applied the Gronwall lemma, which is now recognized as a powerful method, for proving the Hyers-Ulam stability of various differential equations of second order.

2. Preliminaries

In this section, we first introduce and prove a lemma which is a kind of the Gronwall inequality.

Lemma 1.

Let u,υ:[0,)[0,) be integrable functions, let c>0 be a constant, and let t00 be given. If u satisfies the inequality (5)u(t)c+t0tu(τ)υ(τ)dτ for all tt0, then (6)u(t)cexp(t0tυ(τ)dτ) for all tt0.

Proof.

It follows from (5) that (7)u(t)υ(t)c+t0tu(τ)υ(τ)dτυ(t) for all tt0. Integrating both sides of the last inequality from t0 to t, we obtain (8)ln(c+t0tu(τ)υ(τ)dτ)-lnct0tυ(τ)dτ or (9)c+t0tu(τ)υ(τ)dτcexp(t0tυ(τ)dτ) for each tt0, which together with (5) implies that (10)u(t)cexp(t0tυ(τ)dτ) for all tt0.

In the following theorem, using Lemma 1, we investigate sufficient conditions under which every solution of the differential equation (11)u′′(t)+(1+ψ(t))u(t)=0 is bounded.

Theorem 2.

Let ψ:[0,)R be a differentiable function. Every solution u:[0,)R of the linear differential equation (11) is bounded provided that 0|ψ(t)|dt< and ψ(t)0 as t.

Proof.

First, we choose t0 to be large enough so that 1+ψ(t)1/2 for all tt0. Multiplying (11) by u(t) and integrating it from t0 to t, we obtain (12)12u(t)2+12u(t)2+t0tψ(τ)u(τ)u(τ)dτ=c1 for all tt0. Integrating by parts, this yields (13)12u(t)2+12u(t)2+12ψ(t)u(t)2-12t0tψ(τ)u(τ)2dτ=c2 for any tt0. Then it follows from (13) that (14)14u(t)212u(t)2+12·12u(t)212u(t)2+12(1+ψ(t))u(t)2=c2+12t0tψ(τ)u(τ)2dτ for all tt0. Thus, it holds that (15)u(t)24c2+2t0tψ(τ)u(τ)2dτ4|c2|+2t0t|ψ(τ)|u(τ)2dτ for any tt0.

In view of Lemma 1, (15), and our hypothesis, there exists a constant M1>0 such that (16)u(t)24|c2|exp(t0t2|ψ(τ)|dτ)<M12 for all tt0. On the other hand, since u is continuous, there exists a constant M2>0 such that |u(t)|M2 for all 0tt0, which completes the proof.

Corollary 3.

Let ϕ:[0,)R be a differentiable function satisfying ϕ(t)1 as t. Every solution u:[0,)R of the linear differential equation (17)u′′(t)+ϕ(t)u(t)=0 is bounded provided that 0|ϕ(t)|dt<.

3. Main Results on Hyers-Ulam Stability

Given constants L>0 and t00, let U(L;t0) denote the set of all functions u:[t0,)R with the following properties:

u  is twice continuously differentiable;

u(t0)=u(t0)=0;

t0|u(τ)|dτL.

We now prove the Hyers-Ulam stability of the linear differential equation (11) by using the Gronwall inequality.

Theorem 4.

Given constants L>0 and t00, assume that ψ:[t0,)R is a differentiable function with C=t0|ψ(τ)|dτ< and λ=inftt0ψ(t)>-1. If a function uU(L;t0) satisfies the inequality (18)|u′′(t)+(1+ψ(t))u(t)|ε for all tt0 and for some ε0, then there exist a solution u0U(L;t0) of the differential equation (11) and a constant K>0 such that (19)|u(t)-u0(t)|Kε for any tt0, where (20)K=2L1+λexp(C2(1+λ)).

Proof.

We multiply (18) with |u(t)| to get (21)-ε|u(t)|u(t)u′′(t)+u(t)u(t)+ψ(t)u(t)u(t)ε|u(t)| for all tt0. If we integrate each term of the last inequalities from t0 to t, then it follows from (ii) that (22)-εt0t|u(τ)|dτ12u(t)2+12u(t)2+t0tψ(τ)u(τ)u(τ)dτεt0t|u(τ)|dτ for any tt0.

Integrating by parts and using (iii), we have (23)-εL12u(t)2+12u(t)2+12ψ(t)u(t)2hhhhhhh-12t0tψ(τ)u(τ)2dτεL for all tt0.

Since 1+λ>0 holds for all tt0, it follows from (23) that (24)1+λ2u(t)212u(t)2+1+λ2u(t)212u(t)2+12(1+ψ(t))u(t)2εL+12t0tψ(τ)u(τ)2dτεL+12t0t|ψ(τ)|u(τ)2dτ or (25)u(t)22Lε1+λ+11+λt0t|ψ(τ)|u(τ)2dτ for any tt0.

Applying Lemma 1, we obtain (26)u(t)22Lε1+λexp(11+λt0t|ψ(τ)|dτ)2Lε1+λexp(C1+λ) for all tt0. Hence, it holds that (27)|u(t)|exp(C2(1+λ))2Lε1+λ for any tt0. Obviously, u0(t)0 satisfies (11) and the conditions (i), (ii), and (iii) such that (28)|u(t)-u0(t)|Kε for all tt0, where K=2L/(1+λ)exp(C/2(1+λ)).

If we set ϕ(t):=1+ψ(t), then the following corollary is an immediate consequence of Theorem 4.

Corollary 5.

Given constants L>0 and t00, assume that ϕ:[t0,)R is a differentiable function with C=t0|ϕ(τ)|dτ< and λ=inftt0ϕ(t)>0. If a function uU(L;t0) satisfies the inequality (29)|u′′(t)+ϕ(t)u(t)|ε for all tt0 and for some ε0, then there exist a solution u0U(L;t0) of the differential equation (17) and a constant K>0 such that (30)|u(t)-u0(t)|Kε for any tt0, where K=exp(C/2λ)2L/λ.

Example 6.

Let ϕ:[0,)R be a constant function defined by ϕ(t)=a for all t0 and for a constant a>0. Then, we have C=0|ϕ(τ)|dτ=0 and λ=inft0ϕ(t)=a. Assume that a twice continuously differentiable function u:[0,)R satisfies u(0)=u(0)=0, 0|u(τ)|dτL, and (31)|u′′(t)+ϕ(t)u(t)|=|u′′(t)+au(t)|ε for all t0 and for some ε0 and L>0. According to Corollary 5, there exists a solution u0:[0,)R of the differential equation, y′′(t)+ay(t)=0, such that (32)|u(t)-u0(t)|2Laε for any t0.

Indeed, if we define a function u:[0,)R by (33)u(t)=α(t+1)2cosat+2αa(t+1)2sinat-α, where we set α=(a/(a+a+2))L, then u satisfies the conditions stated in the first part of this example, as we see in the following. It follows from the definition of u that (34)u(t)=(2α(t+1)2-2α(t+1)3)cosat-(aα(t+1)2+4αa(t+1)3)sinat and, hence, we get u(0)=u(0)=0. Moreover, we obtain (35)|u(t)|2+a(t+1)2α+(4a-2)α(t+1)3,0|u(τ)|dτ=02+a(τ+1)2αdτ+0(4a-2)α(τ+1)3dτ=(2+a  )α+(2a-1)α=L. For any given ε>0, if we choose the constant α such that 0<αaε/(aa+4a+2a+12), then we can easily see that (36)|u′′(t)+au(t)||(-8(t+1)3+6(t+1)4)αcosatddddddd+(4a(t+1)3+1a12(t+1)4)αsinat-aα|(8(t+1)3-6(t+1)4)αgggggggi+(4a(t+1)3+1a12(t+1)4)α+aα=aa+4a+2a+12aαε for any t0.

Theorem 7.

Given constants L>0 and t00, assume that ψ:[t0,)(0,) is a monotone increasing and differentiable function. If a function uU(L;t0) satisfies the inequality (18) for all tt0 and for some ε>0, then there exists a solution u0U(L;t0) of the differential equation (11) such that (37)|u(t)-u0(t)|2Lεψ(t0) for any tt0.

Proof.

We multiply (18) with |u(t)| to get (38)-ε|u(t)|u(t)u′′(t)+u(t)u(t)+ψ(t)u(t)u(t)ε|u(t)| for all tt0. If we integrate each term of the last inequalities from t0 to t, then it follows from (ii) that (39)-εt0t|u(τ)|dτ12u(t)2+12u(t)2+t0tψ(τ)u(τ)u(τ)dτεt0t|u(τ)|dτ for any tt0.

Integrating by parts, the last inequalities together with (iii) yield (40)-εL12u(t)2+12u(t)2+12ψ(t)u(t)2ggggggg-12t0tψ(τ)u(τ)2dτεL for all tt0. Then we have (41)12ψ(t)u(t)212t0tψ(τ)u(τ)2dτ+εLεL+t0tψ(τ)ψ(τ)u(τ)2ψ(τ)2dτ for any tt0.

Applying Lemma 1, we obtain (42)12ψ(t)u(t)2εLexp(t0tψ(τ)ψ(τ)dτ)=εLψ(t)ψ(t0) for all tt0, since ψ:[t0,)(0,) is a monotone increasing function. Hence, it holds that (43)|u(t)|2Lεψ(t0) for any tt0. Obviously, u0(t)0 satisfies (11), u0U(L;t0), and the inequality (37) for all tt0.

Corollary 8.

Given constants L>0 and t00, assume that ϕ:[t0,)(1,) is a monotone increasing and differentiable function with ϕ(t0)=2. If a function uU(L;t0) satisfies the inequality (44)|u′′(t)+ϕ(t)u(t)|ε for all tt0 and for some ε>0, then there exists a solution u0U(L;t0) of the differential equation (17) such that (45)|u(t)-u0(t)|2Lε for any tt0.

If we set ϕ(t):=-ψ(t), then the following corollary is an immediate consequence of Theorem 7.

Corollary 9.

Given constants L>0 and t00, assume that ϕ:[t0,)(-,0) is a monotone decreasing and differentiable function with ϕ(t0)=-1. If a function uU(L;t0) satisfies the inequality (46)|u′′(t)+(1-ϕ(t))u(t)|ε for all tt0 and for some ε>0, then there exists a solution u0U(L;t0) of the differential equation (47)u′′(t)+(1-ϕ(t))u(t)=0 such that (48)|u(t)-u0(t)|2Lε for any tt0.

Example 10.

Let ϕ:[0,)(-,0) be a monotone decreasing function defined by ϕ(t)=e-t-2 for all t0. Then, we have ϕ(0)=-1. Assume that a twice continuously differentiable function u:[0,)R satisfies u(0)=u(0)=0, 0|u(τ)|dτL, and (49)|u′′(t)+(1-ϕ(t))u(t)|=|u′′(t)+(3-e-t)u(t)|ε for all t0 and for some ε>0 and L>0. According to Corollary 9, there exists a solution u0:[0,)R of the differential equation, y′′(t)+(3-e-t)y(t)=0, such that (50)|u(t)-u0(t)|2Lε for any t0.

Indeed, if we define a function u:[0,)R by (51)u(t)=α(t+1)3sint+12α(t+1)2cost-α2, where α is a real number with |α|(2/43)ε, then u satisfies the conditions stated in the first part of this example, as we see in the following. It follows from the definition of u that (52)u(t)=-3α(t+1)4sint-12α(t+1)2sint and, hence, we get u(0)=u(0)=0. Moreover, we obtain (53)|u(t)|3|α|(t+1)4+12|α|(t+1)2,0|u(τ)|dτ03|α|(τ+1)4dτ+012|α|(τ+1)2dτ=:L<. We can see that (54)|u′′(t)+(3-e-t)u(t)||12α(t+1)5sint-3α(t+1)4costgggggggg+(4-e-t)α(t+1)3sintgggggggg+2-e-t2α(t+1)2cost-3-e-t2α|12|α|(t+1)5+3|α|(t+1)4+4|α|(t+1)3+|α|(t+1)2+32|α|432|α|ε for any t0.

Now, we investigate the Hyers-Ulam stability of the nonlinear differential equation (55)u′′(t)+F(t,u(t))=0.

Theorem 11.

Given constants L>0 and t00, assume that F:[t0,)×R(0,) is a function satisfying F(t,u(t))/F(t,u(t))>0 and F(t,0)=1 for all tt0 and uU(L;t0). If a function u:[t0,)[0,) satisfies uU(L;t0) and the inequality (56)|u′′(t)+F(t,u(t))|ε for all tt0 and for some ε>0, then there exists a solution u0:[t0,)[0,) of the differential equation (55) such that (57)|u(t)-u0(t)|Lε for any tt0.

Proof.

We multiply (56) with |u(t)| to get (58)-ε|u(t)|u(t)u′′(t)+F(t,u(t))u(t)ε|u(t)| for all tt0. If we integrate each term of the last inequalities from t0 to t, then it follows from (ii) that (59)-εt0t|u(τ)|dτ12u(t)2+t0tF(τ,u(τ))u(τ)dτεt0t|u(τ)|dτ for any tt0.

Integrating by parts and using (iii), the last inequalities yield (60)-εL12u(t)2+F(t,u(t))u(t)-t0tF(τ,u(τ))u(τ)dτεL for all tt0. Then we have (61)F(t,u(t))u(t)εL+t0tF(τ,u(τ))u(τ)dτεL+t0tF(τ,u(τ))F(τ,u(τ))F(τ,u(τ))u(τ)dτ for any tt0.

Applying Lemma 1, we obtain (62)F(t,u(t))u(t)εLexp(t0tF(τ,u(τ))F(τ,u(τ))dτ)=εLF(t,u(t)) for all tt0. Hence, it holds that |u(t)|Lε for any tt0. Obviously, u0(t)0 satisfies (55) and u0U(L;t0) such that (63)|u(t)-u0(t)|Lε for all tt0.

In the following theorem, we investigate the Hyers-Ulam stability of the Emden-Fowler nonlinear differential equation of second order (64)u′′(t)+h(t)u(t)α=0 for the case where α is a positive odd integer.

Theorem 12.

Given constants L>0 and t00, assume that h:[t0,)(0,) is a differentiable function. Let α be an odd integer larger than 0. If a function u:[t0,)[0,) satisfies uU(L;t0) and the inequality (65)|u′′(t)+h(t)u(t)α|ε for all tt0 and for some ε>0, then there exists a solution u0:[t0,)[0,) of the differential equation (64) such that (66)|u(t)-u0(t)|(βLεh(t0))1/β for any tt0, where β=α+1.

Proof.

We multiply (65) with |u(t)| to get (67)-ε|u(t)|u(t)u′′(t)+h(t)u(t)αu(t)ε|u(t)| for all tt0. If we integrate each term of the last inequalities from t0 to t, then it follows from (ii) that (68)-εt0t|u(τ)|dτ12u(t)2+t0th(τ)u(τ)αu(τ)dτεt0t|u(τ)|dτ for any tt0.

Integrating by parts and using (iii), the last inequalities yield (69)-εL12u(t)2+h(t)u(t)α+1α+1-t0th(τ)u(τ)α+1α+1dτεL for all tt0. Then we have (70)h(t)u(t)α+1α+1εL+t0th(τ)u(τ)α+1α+1dτεL+t0th(τ)h(τ)h(τ)u(τ)α+1α+1dτ for any tt0.

Applying Lemma 1, we obtain (71)h(t)u(t)α+1α+1εLexp(t0th(τ)h(τ)dτ)εLh(t)h(t0) for all tt0, from which we have (72)u(t)α+1(α+1)Lεh(t0) for all tt0. Hence, it holds that (73)|u(t)|(βLεh(t0))1/β for any tt0, where we set β=α+1. Obviously, u0(t)0 satisfies (64) and u0U(L;t0). Moreover, we get (74)|u(t)-u0(t)|(βLεh(t0))1/β for all tt0.

Given constants L0, M>0, and t00, let U(L;M;t0) denote the set of all functions u:[t0,)R with the following properties:

u is twice continuously differentiable;

u(t0)=u(t0)=0;

|u(t)|L for all tt0;

t0|u(τ)|dτM for all tt0.

We now investigate the Hyers-Ulam stability of the differential equation of the form (75)u′′(t)+u(t)+h(t)u(t)β=0, where β is a positive odd integer.

Theorem 13.

Given constants L0, M>0, and t00, assume that h:[t0,)[0,) is a function satisfying C:=t0|h(τ)|dτ<. Let β be an odd integer larger than 0. If a function uU(L;M;t0) satisfies the inequality (76)|u′′(t)+u(t)+h(t)u(t)β|ε for all tt0 and for some ε>0, then there exists a solution u0:[t0,)R of the differential equation (75) such that (77)|u(t)-u0(t)|2Mεexp(CLβ-1β+1) for any tt0.

Proof.

We multiply (76) with |u(t)| to get (78)-ε|u(t)|u(t)u′′(t)+u(t)u(t)+h(t)u(t)βu(t)ε|u(t)| for all tt0. If we integrate each term of the last inequalities from t0 to t, then it follows from (ii) that (79)-εt0t|u(τ)|dτ12u(t)2+12u(t)2+t0th(τ)u(τ)βu(τ)dτεt0t|u(τ)|dτ for any tt0.

Integrating by parts and using (ii) and (iv), the last inequalities yield (80)-εM12u(t)2+12u(t)2+h(t)1β+1u(t)β+1ggggggggg-1β+1t0th(τ)u(τ)β+1dτεM for all tt0. Then it follows from (iii) that (81)12u(t)2εM+1β+1t0th(τ)u(τ)β+1dτεM+2β+1t0t12u(τ)2h(τ)u(τ)β-1dτεM+2β+1t0t12u(τ)2|h(τ)||u(τ)|β-1dτεM+2Lβ-1β+1t0t12u(τ)2|h(τ)|dτ for any tt0.

Applying Lemma 1, we obtain (82)12u(t)2εMexp(t0t2Lβ-1β+1|h(τ)|dτ)εMexp(2CLβ-1β+1) for all tt0. Hence, it holds that (83)|u(t)|2Mεexp(CLβ-1β+1) for any tt0. Obviously, u0(t)0 satisfies (75) and u0U(L;M;t0). Furthermore, we get (84)|u(t)-u0(t)|2Mεexp(CLβ-1β+1) for all tt0.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. 2013R1 A1A2005557).

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