Research Article Ulam-Hyers-Rassias Stability of a Hyperbolic Partial Differential Equation
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1 International Scholarly Research Network ISRN Mathematical Analysis Volume 212, Article ID 69754, 1 pages doi:1.542/212/69754 Research Article Ulam-Hyers-Rassias Stability of a Hyperbolic Partial Differential Equation Nicolaie Lungu 1 and Cecilia Crăciun 2 1 Department of Mathematics, Technical University of Cluj-Napoca, Street C. Daicoviciu 15, 42, Cluj-Napoca, Romania 2 Department of Mathematics, Colfe s School, Horn Park Lane, London SE12 8AW, UK Correspondence should be addressed to Cecilia Crăciun, ceciliacraciun@yahoo.com Received 21 October 211; Accepted 3 November 211 Academic Editors: K. H. Kwon and J. V. Stokman Copyright q 212 N. Lungu and C. Crăciun. 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. We consider a nonlinear hyperbolic partial differential equation in a general form. Using a Gronwall-type lemma we prove results on the Ulam-Hyers stability and the generalised Ulam-Hyers- Rassias stability of this equation. 1. Introduction Results on Ulam stability for the functional equations are well known see, e.g., 1 4. In this paper we consider different types of Ulam stability of a nonlinear hyperbolic partial differential equation. Following results from Rus 5 and Rus and Lungu 6, we present some Ulam-Hyers and Ulam-Hyers-Rassias stabilities for the following equation: 2 u ) f (x, y, u ), u ( ) u ( ) ) x, y, x, y, x<a, y<b, 1.1 where f C,a,b B 3, B and B, is a real or complex Banach space. 2. Preliminaries In this section we present some definitions and results on different types of Ulam stability for a hyperbolic partial differential equation. We also present a Gronwall type lemma, which will be useful in the proof of the main results.
2 2 ISRN Mathematical Analysis Let a, b,, ε>, ϕ C,a,b, R and B, be a real or complex Banach space. We consider the following hyperbolic partial differential equation: 2 u ) f (x, y, u ), u ( ) u ( ) ) x, y, x, y, x<a, y<b, 2.1 where f C,a,b B 3, B. We also consider the following inequalities: ) f (x, y, v ), v ) f (x, y, v ), v ) f (x, y, v ), v ( ) v ( ) ) x, y, x, y ε, x,a, y,b, 2.2 ( ) v ( ) ) x, y, x, y ϕ ), x,a, y,b, ( ) v ( ) ) x, y, x, y εϕ ), x,a, y,b Definition 2.1. A function u is a solution to 2.1 if u C 1,a,b, B, 2 u C,a,b, B 2.5 and u satisfies 2.1. From 5, 6 we have the following definitions and results. Definition 2.2. Equation 2.1 is Ulam-Hyers stable if there exist the real numbers C 1 f, C2 f,and C 3 > such that for any ε>and for any solution v to the inequality 2.2 there exists a f solution u to 2.1 with v ) u ) C 1 f ε, v v x,a, y,b, ( ) u( ) x, y x, y C 2 fε, x,a, y,b, ) u ) C 3 f ε, x,a, y,b. 2.6
3 ISRN Mathematical Analysis 3 Definition 2.3. Equation 2.1 is generalised Ulam-Hyers-Rassias stable if there exist the real numbers C 1 f,ϕ, C2 f,ϕ,andc3 > such that for any ε>and for any solution v to the inequality f,ϕ 2.3 there exists a solution u to 2.1 with v ) u ) C 1 f,ϕ ϕ ), v v x,a, y,b, ( ) u( ) x, y x, y C 2 f,ϕ ϕ ), x,a, y,b, ( ) u ( ) x, y x, y C 3 f,ϕ ϕ ), x,a, y,b. 2.7 Remark 2.4. A function v is a solution to the inequality 2.2 if and only if, there exists a function g C,a,b, B, which depends on v, such that i For all ε>, g x, y ε, for all x,a, for all y,b ; ii For all x,a, for all y,b : ) f (x, y, v ), v ( ) v ( ) ) x, y, x, y g ). 2.8 Remark 2.5. A function v is a solution to the inequality 2.3 if and only if, there exists a function g C,a,b, B, which depends on v, such that i g x, y ϕ x, y, for all x,a, for all y,b ; ii For all x,a, for all y,b : ) f (x, y, v ), v ( ) v ( ) ) x, y, x, y g ). 2.9 Remark 2.6. A function v is a solution to the inequality 2.4 if and only if, there exists a function g C,a,b, B, which depends on v, such that i For all ε>, g x, y εϕ x, y, for all x,a, for all y,b ; ii For all x,a, for all y,b : ) f (x, y, v ), v ( ) v ( ) ) x, y, x, y g ). 2.1 Throughout this paper we denote ( ) u( ) ( ) u ( ) u 1 x, y x, y, u2 x, y x, y, 2.11 ( ) v ( ) ( ) v ( ) v 1 x, y x, y, v2 x, y x, y.
4 4 ISRN Mathematical Analysis Theorem 2.7. If v is a solution to the inequality 2.2, then v, v 1,v 2 satisfies the following system of integral inequalities: v ) v x, v (,y ) v, v ) 1 v1 x, f s, t, v s, t,v 1 s, t,v 2 s, t ds dt εxy, f x, t, v x, t,v 1 x, t,v 2 x, t dt εy, v ( ) ( ) x 2 x, y v2,y f ( s, y, v ( s, y ) ( ) ( )),v 1 s, y,v2 s, y ds εx, 2.12 for all x,a and y,b. Proof. By Remark 2.4 we have that v ) v x, v (,y ) v, v 1 ) v1 x, f s, t, v s, t,v 1 s, t,v 2 s, t ds dt f x, t, v x, t,v 1 x, t,v 2 x, t dt g s, t ds dt g x, t dt 2.13 ( ) ( ) x v 2 x, y v2,y f ( s, y, v ( s, y ) ( ) ( )) x,v 1 s, y,v2 s, y ds g ( s, y ) ds, and we have the following inequalities: v ) v x, v (,y ) v, f s, t, v s, t,v 1 s, t,v 2 s, t ds dt g s, t ds dt g s, t ds dt εxy, v ) y 1 v1 x, f x, t, v x, t,v 1 x, t,v 2 x, t dt g x, t dt εy, v ( ) ( ) x 2 x, y v2,y f ( s, y, v ( s, y ) ( ) ( )) x ( ),v 1 s, y,v2 s, y ds g s, y ds εx The following two theorems are obtained in a similar fashion.
5 ISRN Mathematical Analysis 5 Theorem 2.8. If v is a solution to the inequality 2.3, then v, v 1,v 2 satisfies the following system of integral inequalities: v ) v x, v (,y ) v, f s, t, v s, t,v 1 s, t,v 2 s, t ds dt ϕ s, t ds dt, v ) y 1 v1 x, f x, t, v x, t,v 1 x, t,v 2 x, t dt ϕ x, t dt, v ( ) ( ) x 2 x, y v2,y f ( s, y, v ( s, y ) ( ) ( )) x,v 1 s, y,v2 s, y ds ϕ ( s, y ) ds, for all x,a and y,b Theorem 2.9. If v is a solution to the inequality 2.4, then v, v 1,v 2 satisfies the following system of integral inequalities: v ) v x, v (,y ) v, f s, t, v s, t,v 1 s, t,v 2 s, t ds dt ε ϕ s, t ds dt, v ) y 1 v1 x, f x, t, v x, t,v 1 x, t,v 2 x, t dt ε ϕ x, t dt, v ( ) ( ) x 2 x, y v2,y f ( s, y, v ( s, y ) ( ) ( )) x,v 1 s, y,v2 s, y ds ε ϕ ( s, y ) ds, for all x,a and y,b The following Gronwall type lemma is an important tool in proving the main results of this paper. Lemma 2.1 see 7, also 8. One assumes that i u, v, h C R n, R ; ii for any t t one has t u t h t v s u s ds; t 2.17 iii h t is positive and nondecreasing. Then, u t h t exp t s v r dr, for any t t. 2.18
6 6 ISRN Mathematical Analysis 3. Ulam-Hyers Stability In this section we present a result on the existence and uniqueness of the solution to 2.1 and derive a result on Ulam-Hyers stability for the same equation in the case of a< and b<. Theorem 3.1. One assumes that i a<, b< ; ii f C,a,b B 3, B ; iii there exists L f > such that f, z1,z 2,z 3 ) f, t1,t 2,t 3 ) Lf max{ z i t i,i 1, 2, 3}, 3.1 for all x,a, y,b and z 1,z 2,z 3,t 1,t 2,t 3 B. Then, a for φ C 1,a, B and ψ C 1,b, B 2.1 has a unique solution, which satisfies u x, φ x, u (,y ) ψ ( y ), x,a, y,b ; 3.2 b 2.1 is Ulam-Hyers stable. Proof. a This is a known result see, e.g, 9, 1. b Let v be a solution to the inequality 2.2 and let u be the unique solution to 2.1, which satisfies the following conditions: u x, v x,, u (,y ) v (,y ), x,a, y,b. 3.3 From Theorem 2.7, the hypothesis iii, and Gronwall Lemma 2.1, it follows that ( ) ( ) v x, y u x, y v ) v x, v (,y ) v, f s, t, v s, t,v 1 s, t,v 2 s, t ds dt f s, t, v s, t,v1 s, t,v 2 s, t f s, t, u s, t,u 1 s, t,u 2 s, t ds dt εxy L f max v i s, t u i s, t ds dt i {1,2,3} εab exp ( L f ab ) C 1 f ε, where C1 f : ab exp( L f ab ). 3.4
7 ISRN Mathematical Analysis 7 Similarly we have ( ) ( ) v 1 x, y u1 x, y v ) 1 v1 x, f x, t, v x, t,v 1 x, t,v 2 x, t dt f x, t, v x, t,v1 x, t,v 2 x, t f x, t, u x, t,u 1 x, t,u 2 x, t dt εy L f max v i x, t u i x, t dt i {1,2,3} εb exp ( L f b ) C 2 f ε, where C2 f : b exp( L f b ), ( ) ( ) v2 x, y u2 x, y v ( ) ( ) x 2 x, y v2,y f ( s, y, v ( s, y ) ( ) ( )),v 1 s, y,v2 s, y ds εx L f ( ( ) ( ) ( )) ( ( ) ( ) ( )) f s, y, v s, y,v1 s, y,v2 s, y f s, y, u s, y,u1 s, y,u2 s, y ds ( ) ( ) max vi s, y ui s, y ds i {1,2,3} εa exp ( L f a ) C 3 f ε, where C3 f : a exp( L f a ). 3.5 Remark 3.2. In general, if a or b, then 2.1 is not Ulam-Hyers stable. 4. Generalised Ulam-Hyers-Rassias Stability In this section we prove the generalised Ulam-Hyers-Rassias stability of the hyperbolic partial differential equation 2.1. We consider 2.1 and the inequality 2.3 in the case a and b. Theorem 4.1. One assumes that i C,, B 3, B ; ii there exists l f C 1,,, R such that f, z 1,z 2,z 3 ) f, t1,t 2,t 3 ) l f ) max{ zi t i, i 1, 2, 3}, 4.1 for all x, y, ;
8 8 ISRN Mathematical Analysis iii there exist λ 1 ϕ,λ 2 ϕ,λ 3 ϕ > such that ϕ s, t ds dt λ 1 ϕϕ ), ϕ x, t dt λ 2 ϕϕ ), ϕ ( s, y ) ds λ 3 ϕϕ ), x, y,, x, y,, x, y, ; 4.2 iv ϕ : R R R is increasing. Then 2.1 (a and b ) is generalised Ulam-Hyers-Rassias stable. Proof. Let v be a solution to the inequality 2.3. Denote by u the unique solution to the Darboux problem: 2 u ( ) ( ( ) ( ) ( )) x, y f x, y, u x, y,u1 x, y,u2 x, y, x, y,, u x, v x,, x,, u (,y ) v (,y ), y,. 4.3 If u is a solution to the Darboux problem 4.3, then u, u 1,u 2 is a solution to the following system: u ) v x, v (,y ) v, u 1 ) v1 x, f x, t, u x, t,u 1 x, t,u 2 x, t dt, f s, t, u s, t,u 1 s, t,u 2 s, t ds dt, 4.4 ( ) ( ) x u 2 x, y v2,y f ( s, y, u ( s, y ) ( ) ( )),u 1 s, y,u2 s, y ds. From Theorem 2.8 and the hypothesis iii, it follows that v ) v x, v (,y ) v, ϕ s, t ds dt λ 1 ϕϕ ), x, y,, v ) y 1 v1 x, f x, t, v x, t,v 1 x, t,v 2 x, t dt f s, t, v s, t,v 1 s, t,v 2 s, t ds dt ϕ x, t dt λ 2 ϕϕ ), x, y,, v ( ) ( ) x 2 x, y v2,y f ( s, y, v ( s, y ) ( ) ( )),v 1 s, y,v2 s, y ds ϕ ( s, y ) ds λ 3 ϕϕ ), x, y,. 4.5
9 ISRN Mathematical Analysis 9 Using 4.5 gives us ( ) ( ) v x, y u x, y v ) v x, v (,y ) v, f s, t, v s, t,v 1 s, t,v 2 s, t ds dt f s, t, v s, t,v 1 s, t,v 2 s, t f s, t, u s, t,u 1 s, t,u 2 s, t ds dt λ 1 ϕϕ ) l f s, t max i {1,2,3} v i s, t u i s, t ds dt. 4.6 From Lemma 2.1 it follows that v ) u ) ( λ 1 ϕ exp ( ) ( ) v x, y u x, y C 1 f,ϕ ϕ ), ) l f s, t ds dt ϕ ), x, y,, x, y, or 4.7 where C 1 f,ϕ : λ1 ϕ exp Similarly, we have l f s, t ds dt. v1 ) u1 ) λ 2 ϕ ϕ ) l f x, t max i {1,2,3} v i x, t u i x, t dt, 4.8 and from Lemma 2.1 we get ( ) ( ) v1 x, y u1 x, y λ 2 ϕ exp ( v 1 ) u1 ) C 2 f,ϕ ϕ ), ) l f x, t dt ϕ ), x, y,, x, y, or 4.9 where C 2 f,ϕ : λ2 ϕ exp l f x, t dt. Also, v2 ) u2 ) λ 3 ϕ ϕ ) By using Lemma 2.1 we obtain ( ) ( ( ) ( )) l f s, y max vi s, y ui s, y ds. 4.1 i {1,2,3} ( ( ) ( ) v 2 x, y u2 x, y λ 3 ϕ exp v2 ) u2 ) C 3 f,ϕ ϕ ), l f ( s, y ) ds )ϕ ), x, y, or x, y,, 4.11 where C 3 f,ϕ : λ3 ϕ exp l f s, y ds. So, 2.1 is generalised Ulam-Hyers-Rassias stable.
10 1 ISRN Mathematical Analysis References 1 L. Cădariu and V. Radu, The fixed points method for the stability of some functional equations, Carpathian Mathematics, vol. 23, no. 1-2, pp , D. H. Hyers, G. Isac, and T. M. Rassias, Stability of Functional Equations in Several Variables,Birkhäuser, Basel, Switzerland, D. Popa, Functional Equations. Set-Valued Solutions Stability, Technical University Press, Cluj-Napoca, Romania, D. Popa, On the stability of the general linear equation, Results in Mathematics, vol.53,no.3-4,pp , I. A. Rus, Ulam stability of ordinary differential equations, Studia. Universitatis Babeş-Bolyai. Mathematica, vol. 54, no. 4, pp , I. A. Rus and N. Lungu, Ulam stability of a nonlinear hyperbolic partial differential equation, Carpathian Mathematics, vol. 24, no. 3, pp , V. Lakshmikantham, S. Leela, and A. A. Martynyuk, Stability Analysis of Nonlinear Systems, vol. 125, Marcel Dekker Inc., New York, NY, USA, B. G. Pachpatte, Inequalities for Differential and Integral Equations, Academic Press, San Diego, Calif, USA, N. Lungu, Qualitative Problems in the Theory of Hyperbolic Differential Equations, Digital Data, Cluj- Napoca, Romania, N. Lungu and I. A. Rus, Hyperbolic differential inequalities, Libertas Mathematica, vol. 21, pp. 35 4, 21.
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