Hindawi Publishing Corporation e Scientific World Journal Volume 2014, Article ID 878395, 8 pages http://dx.doi.org/10.1155/2014/878395
Research Article Some Properties of Solutions of a Functional-Differential Equation of Second Order with Delay Veronica Ana Ilea1 and Diana Otrocol2 1 2
Babes¸-Bolyai University, Kog˘alniceanu no. 1, RO-400084 Cluj-Napoca, Romania “T. Popoviciu” Institute of Numerical Analysis, Romanian Academy, P.O. Box. 68-1, 400110 Cluj-Napoca, Romania
Correspondence should be addressed to Diana Otrocol; [email protected] Received 13 August 2013; Accepted 7 October 2013; Published 10 February 2014 Academic Editors: F. Khani and H. Xu Copyright © 2014 V. Ana Ilea and D. Otrocol. 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. Existence, uniqueness, data dependence (monotony, continuity, and differentiability with respect to parameter), and Ulam-Hyers stability results for the solutions of a system of functional-differential equations with delays are proved. The techniques used are Perov’s fixed point theorem and weakly Picard operator theory.
1. Introduction Functional-differential equations with delay arise when modeling biological, physical, engineering, and other processes whose rate of change of state at any moment of time 𝑡 is determined not only by the present state but also by past state. The description of certain phenomena in physics has to take into account that the rate of propagation is finite. For example, oscillation in a vacuum tube can be described by the following equation in dimensionless variables [1, 2]: 𝑥 (𝑡) + 2𝑟𝑥 (𝑡) + 𝜔2 𝑥 (𝑡) + 2𝑞𝑥 (𝑡 − 1) = 𝜖𝑥3 (𝑡 − 1) . (1) In this equation, time delay is due to the fact that the time necessary for electrons to pass from the cathode to the anode in the tube is finite. The same equation has been used in the theory of stabilization of ships [2]. The dynamics of an autogenerator with delay and second-order filter was described in [3] by the equation 𝑥 (𝑡) + 2𝛿𝑥 (𝑡) + 𝑥 (𝑡) = 𝑓 (𝑥 (𝑡 − ℎ)) .
(2)
The model of ship course stabilization under conditions of uncertainty may be described by the following equation [4]: 𝐼𝑥 (𝑡) + 𝐻𝑥 (𝑡) = −𝐾Ψ (𝑡) + 𝑏0 𝜉0 (𝑡) , 𝑥 (𝑡0 ) = 𝑥0 ,
𝑥 (𝑡0 ) = 0,
(3)
with 𝑥(𝑡) being the angle of the deviation from course, Ψ(𝑡) the turning angle of the rudder, and 𝜉0 (𝑡) the stochastic disturbance. In the process of mathematical modeling, often small delays are neglected; that is why sometimes false conclusions appear. As an example we can give the following equation [1]: 𝑥 (𝑡) + 𝑥 (𝑡) + 𝑥 (𝑡) = 𝑎 [𝑥 (𝑡 − ℎ) + 𝑥 (𝑡 − ℎ) + 𝑥 (𝑡 − ℎ)] ,
(4)
which is asymptotically stable for ℎ = 0 but unstable for arbitrary ℎ > 0. Here 𝑎 > 1. If ℎ = 0 the above system is asymptotically stable. The characteristic equation is Δ(𝑧) = (𝑧2 +𝑧+1)(1−𝑎𝑒−ℎ𝑧 ) and has the following zeros with positive real part if ℎ > 0 : 𝑧𝑘 = 1/ℎ(In 𝑎+2𝑘𝜋𝑖), 𝑖2 = −1, 𝑘 = 0, ±1, . . .. So the trivial solution is unstable for any ℎ > 0. In this paper, we continue the research in this field and develop the study of the following general functional differential equation with delay: 𝑥 (𝑡) = 𝑓 (𝑡, 𝑥 (𝑡) , 𝑥 (𝑡) , 𝑥 (𝑡 − ℎ) , 𝑥 (𝑡 − ℎ)) , 𝑡 ∈ [𝑎, 𝑏] , 𝑥 (𝑡) = 𝜑 (𝑡) ,
𝑡 ∈ [𝑎 − ℎ, 𝑎] .
(5)
2
The Scientific World Journal
Existence, uniqueness, data dependence (monotony, continuity, and differentiability with respect to parameter), and Ulam-Hyers stability results of solution for the Cauchy problem are obtained. Our results are essentially based on Perov’s fixed point theorem and weakly Picard operator technique, which will be presented in Section 2. More results about functional and integral differential equations using these techniques can be found in [5–8]. The problem (5) is equivalent to the following system: 𝑥 (𝑡) = 𝑧 (𝑡) ,
𝑡 ∈ [𝑎, 𝑏] ,
𝑧 (𝑡) = 𝑓 (𝑡, 𝑥 (𝑡) , 𝑧 (𝑡) , 𝑥 (𝑡 − ℎ) , 𝑧 (𝑡 − ℎ)) ,
(6)
𝑡 ∈ [𝑎, 𝑏] , with the initial conditions 𝑥 (𝑡) = 𝜑 (𝑡) , 𝑧 (𝑡) = 𝜑 (𝑡) ,
𝑡 ∈ [𝑎 − ℎ, 𝑎] , 𝑡 ∈ [𝑎 − ℎ, 𝑎] .
(7)
By a solution of the system (6) we understand a function ( 𝑥𝑧 ) ∈ 𝐶([𝑎−ℎ, 𝑏], R2 )∩𝐶1 ([𝑎, 𝑏], R2 ) that verifies the system. We suppose that (C1 ) 𝑎 < 𝑏, ℎ > 0; (C2 ) 𝑓 ∈ 𝐶([𝑎, 𝑏] × R4 , R), 𝜑 ∈ 𝐶1 ([𝑎 − ℎ, 𝑎], R); (C3 ) there exists 𝐿 1 , 𝐿 2 > 0 such that ∀𝑡 ∈ [𝑎, 𝑏], 𝑢𝑖 , V𝑖 , 𝑢̃𝑖 , ̃V𝑖 ∈ R, 𝑖 = 1, 2, we have 𝑓 (𝑡, 𝑢1 , V1 , 𝑢2 , V2 ) − 𝑓 (𝑡, 𝑢̃1 , ̃V1 , 𝑢̃2 , ̃V2 ) ≤ 𝐿 1 max (𝑢1 − 𝑢̃1 , 𝑢2 − 𝑢̃2 ) + 𝐿 2 max (V1 − ̃V1 , V2 − ̃V2 ) .
If ( 𝑥𝑧 ) ∈ 𝐶([𝑎 − ℎ, 𝑏], R2 ) ∩ 𝐶1 ([𝑎, 𝑏], R2 ) is a solution of the problem (6)-(7), then ( 𝑥𝑧 ) is a solution of the following integral system:
𝜑 { ( ) (𝑡) , { { { 𝜑 { { { 𝑡 𝑥 ( ) (𝑡) = { 𝜑 (𝑎) + ∫ 𝑧 (𝑠) 𝑑𝑠 𝑧 { { 𝑎 { ) ( { 𝑡 { { 𝜑 (𝑎) + ∫ 𝑓 (𝑠, 𝑥 (𝑠) , 𝑧 (𝑠) , 𝑥 (𝑠 − ℎ) , 𝑧 (𝑠 − ℎ)) 𝑑𝑠 { 𝑎 If ( 𝑥𝑧 ) ∈ 𝐶([𝑎−ℎ, 𝑏], R2 ) is a solution of (9), then ( 𝑥𝑧 ) ∈ 𝐶([𝑎− ℎ, 𝑏], R2 ) ∩ 𝐶1 ([𝑎, 𝑏], R2 ) and ( 𝑥𝑧 ) is a solution of (6)-(7).
for 𝑡 ∈ [𝑎 − ℎ, 𝑎] , (9) for 𝑡 ∈ [𝑎, 𝑏] .
Moreover, the system (6) is equivalent to the functional integral system
𝑥 { { ( ) (𝑡) , for 𝑡 ∈ [𝑎 − ℎ, 𝑎] , { { 𝑧 { { { { 𝑥 𝑡 ( ) (𝑡) = { 𝑧 (𝑠) 𝑑𝑠 𝑥 + ∫ (𝑎) 𝑧 { { { 𝑎 { ) for 𝑡 ∈ [𝑎, 𝑏] . ( 𝑡 { { { 𝑧 (𝑎) + ∫ 𝑓 (𝑠, 𝑥 (𝑠) , 𝑧 (𝑠) , 𝑥 (𝑠 − ℎ) , 𝑧 (𝑠 − ℎ)) 𝑑𝑠 { 𝑎 We consider the operators 𝐵, 𝐸 : 𝐶([𝑎 − ℎ, 𝑏], R2 ) → 𝐶([𝑎 − 𝐵 (𝑥) 𝐸 (𝑥) ℎ, 𝑏], R2 ), 𝐵 = ( 𝐵1 𝑥𝑧 ), 𝐸 = ( 𝐸1 𝑥𝑧 ) defined by 𝐵 ( 𝑥𝑧 ) (𝑡) := 2( 𝑧 ) 2( 𝑧 ) the right hand side of (9), for 𝑡 ∈ [𝑎 − ℎ, 𝑏] and 𝐸 ( 𝑥𝑧 ) (𝑡) := the right hand side of (10), for 𝑡 ∈ [𝑎 − ℎ, 𝑏].
2. Preliminaries In this section, we introduce notations, definitions, and preliminary results which are used throughout this paper; see [9–17]. Let (𝑋, 𝑑) be a metric space and 𝐴 : 𝑋 → 𝑋 an operator. We will use the following notations: 𝐹𝐴 := {𝑥 ∈ 𝑋 | 𝐴(𝑥) = 𝑥}—the fixed points set of 𝐴;
(8)
(10)
𝐼(𝐴) := {𝑌 ⊂ 𝑋 | 𝐴(𝑌) ⊂ 𝑌, 𝑌 ≠ 0}—the family of the nonempty invariant subset of 𝐴; 𝐴𝑛+1 := 𝐴 ∘ 𝐴𝑛 , 𝐴0 = 1𝑋 , 𝐴1 = 𝐴, 𝑛 ∈ N. Definition 1. Let (𝑋, 𝑑) be a metric space. An operator 𝐴 : 𝑋 → 𝑋 is a Picard operator (PO) if there exists 𝑥∗ ∈ 𝑋 such that (i) 𝐹𝐴 = {𝑥∗ }; (ii) the sequence (𝐴𝑛 (𝑥0 ))𝑛∈N converges to 𝑥∗ for all 𝑥0 ∈ 𝑋. Definition 2. Let (𝑋, 𝑑) be a metric space. An operator 𝐴 : 𝑋 → 𝑋 is a weakly Picard operator (WPO) if the sequence
The Scientific World Journal
3
(𝐴𝑛 (𝑥))𝑛∈N converges for all 𝑥 ∈ 𝑋 and its limit (which may depend on 𝑥) is a fixed point of 𝐴. Definition 3. If 𝐴 is a weakly Picard operator, then we consider the operator 𝐴∞ defined by 𝐴∞ : 𝑋 → 𝑋,
𝐴∞ (𝑥) := lim 𝐴𝑛 (𝑥) . 𝑛→∞
(11)
Remark 4. It is clear that 𝐴∞ (𝑋) = 𝐹𝐴 . Definition 5. Let 𝐴 be a weakly Picard operator and 𝑐 > 0. The operator 𝐴 is 𝑐-weakly Picard operator if 𝑑 (𝑥, 𝐴∞ (𝑥)) ≤ 𝑐𝑑 (𝑥, 𝐴 (𝑥)) ,
∀𝑥 ∈ 𝑋.
(12)
The following concept is important for our further considerations. Definition 6. Let (𝑋, 𝑑) be a metric space and 𝑓 : 𝑋 → 𝑋 an operator. The fixed point equation 𝑥 = 𝑓 (𝑥)
(13)
is Ulam-Hyers stable if there exists a real number 𝑐𝑓 > 0 such that for each 𝜀 > 0 and each solution 𝑦∗ of the inequation 𝑑 (𝑦, 𝑓 (𝑦)) ≤ 𝜀,
(14)
there exists a solution 𝑥∗ of (13) such that 𝑑 (𝑦∗ , 𝑥∗ ) ≤ 𝑐𝑓 𝜀.
(15)
Now we have the following. Theorem 7 (see [17]). If 𝑓 : 𝑋 → 𝑋 is 𝑐-WPO, then the equation 𝑥 = 𝑓 (𝑥)
(16)
is Ulam-Hyers stable. Another result from the WPO theory is the following (see, e.g., [11]). Theorem 8 (fibre contraction principle). Let (𝑋, 𝑑) and (𝑌, 𝜌) be two metric spaces and 𝐴 : 𝑋 × 𝑌 → 𝑋 × 𝑌, 𝐴 = (𝐵, 𝐶), (𝐵 : 𝑋 → 𝑋, 𝐶 : 𝑋 × 𝑌 → 𝑌) a triangular operator. One supposes that (i) (𝑌, 𝜌) is a complete metric space; (ii) the operator 𝐵 is Picard operator; (iii) there exists 𝑙 ∈ [0, 1) such that 𝐶(𝑥, ⋅) : 𝑌 → 𝑌 is a 𝑙-contraction, for all 𝑥 ∈ 𝑋; (iv) if (𝑥∗ , 𝑦∗ ) ∈ 𝐹𝐴, then 𝐶(⋅, 𝑦∗ ) is continuous in 𝑥∗ . Then the operator 𝐴 is Picard operator. Throughout this paper we denote by 𝑀𝑚𝑚 (R+ ) the set of all 𝑚×𝑚 matrices with positive elements and by 𝐼 the identity 𝑚 × 𝑚 matrix. A square matrix 𝑄 with nonnegative elements is said to be convergent to zero if 𝑄𝑘 → 0 as 𝑘 → ∞. It is known that the property of being convergent to zero is equivalent to each of the following three conditions (see [9, 10]):
(a) 𝐼 − 𝑄 is nonsingular and (𝐼 − 𝑄)−1 = 𝐼 + 𝑄 + 𝑄2 + ⋅ ⋅ ⋅ (where 𝐼 stands for the unit matrix of the same order as 𝑄); (b) the eigenvalues of 𝑄 are located inside the open unit disc of the complex plane; (c) 𝐼 − 𝑄 is nonsingular and (𝐼 − 𝑄)−1 has nonnegative elements. We finish this section by recalling the following fundamental result (see [9, 18]). Theorem 9 (Perov’s fixed point theorem). Let (𝑋, 𝑑) with 𝑑(𝑥, 𝑦) ∈ R𝑚 be a complete generalized metric space and 𝐴 : 𝑋 → 𝑋 an operator. One supposes that there exists a matrix 𝑄 ∈ 𝑀𝑚𝑚 (R+ ), such that (i) 𝑑(𝐴(𝑥), 𝐴(𝑦)) ≤ 𝑄𝑑(𝑥, 𝑦), for all 𝑥, 𝑦 ∈ 𝑋; (ii) 𝑄𝑛 → 0 as 𝑛 → ∞. Then (a) 𝐹𝐴 = {𝑥∗ };
(b) 𝐴𝑛 (𝑥) = 𝑥∗ as 𝑛 → ∞, ∀𝑥 ∈ 𝑋; (c) 𝑑(𝐴𝑛 (𝑥), 𝑥∗ ) ≤ (𝐼 − 𝑄)−1 𝑄𝑛 𝑑(𝑥0 , 𝐴(𝑥0 )).
3. Main Results In this section, we present existence, uniqueness, and data dependence (monotony, continuity, and differentiability with respect to parameter) results of solution for the Cauchy problem (6)-(7). 3.1. Existence and Uniqueness. Using Perov’s fixed point theorem, we obtain existence and uniqueness theorem for the solution of the problem (6)-(7). Theorem 10. One supposes that (i) the conditions (C1 )–(C3 ) are satisfied;
(ii) 𝑄𝑛 → 0 as 𝑛 → ∞, where 𝑄 = (𝑏 − 𝑎) ( 𝐿01
1 𝐿2
).
Then, ∗
(a) the problem (6)-(7) has a unique solution ( 𝑥𝑧∗ ) ∈ 𝐶1 ([𝑎, 𝑏], R2 ); 0
𝑛
(b) for all ( 𝑥𝑧0 ) ∈ 𝐶1 ([𝑎, 𝑏], R2 ), the sequence ( 𝑥𝑧𝑛 )𝑛∈N 𝑛+1
𝑛
defined by ( 𝑥𝑧𝑛+1 ) = 𝐵 ( 𝑥𝑧𝑛 ) converges uniformly to ∗
( 𝑥𝑧∗ ), for all 𝑡 ∈ [𝑎, 𝑏], and 𝑥𝑛 𝑥∗ ( 𝑛 ) − ( ∗ ) 𝑧 𝑧
0 𝑥 𝑥1 ≤ (𝐼 − 𝑄) 𝑄 ( 0 ) − ( 1 ) ; 𝑧 𝑧 −1
(17)
𝑛
(c) the operator 𝐵 is Picard operator in (𝐶([𝑎 − ℎ, 𝑏], 𝑢𝑛𝑖𝑓
R2 ), →);
4
The Scientific World Journal (d) the operator 𝐸 is weakly Picard operator in (𝐶([𝑎 −
Proof. We have that
𝑢𝑛𝑖𝑓
2
ℎ, 𝑏], R ), →). Proof. Consider on the space 𝑋 := 𝐶([𝑎 − ℎ, 𝑏], R2 ) the norm 𝑢1 ( ) − (𝑢2 ) := max (𝑢1 − 𝑢2) , (18) V1 V2 V1 − V2 which endows 𝑋 with the uniform convergence. 𝜒 𝜒 Let 𝑋 𝜒 = {( 𝑥𝑧 ) ∈ 𝑋 | ( 𝑥𝑧 ) |[𝑎−ℎ,𝑎] = ( 𝜒 ) , for ( 𝜒 ) ∈ (
𝜒
𝑥∗ 𝑥∗ ( ∗) = 𝐸 ( ∗) , 𝑧 𝑧
𝜒 𝑋 𝜒 ( )∈𝐶([𝑎−ℎ,𝑎],R2 ) ( ) 𝜒 𝜒
𝑦∗ 𝑦∗ 𝑦̃∗ ( ∗ ) ≤ 𝐸∞ ( ∗ ) = 𝐸∞ ( ∗ ) ̃ 𝑤 𝑤 𝑤
(
(2) 𝐵|𝑋
(
𝜒 ) 𝜒
𝜒 , ) 𝜒
= 𝐸|𝑋
(
𝐵(𝑋
(
𝜒 ) ) 𝜒
⊂𝑋
(
∞
∗
where ( 𝑥𝑧̃̃∗ ) ∈ 𝑋( 𝑥 )|
𝜒 ; ) 𝜒
𝑧
𝜒 . ) 𝜒
On the other hand, for 𝑡 ∈ [𝑎 − ℎ, 𝑎] ∪ [𝑎, 𝑏] 𝑥 𝐵 (𝑥1 ) 𝐵1 ( 2 ) 1 𝑧 𝑧2 1 ( )−( ) 𝑥 𝐵 (𝑥1 ) 𝐵2 ( 2 ) 2 𝑧 𝑧2 1
(19)
Theorem 11. One supposes that (i) the conditions (a), (b), and (c) in Theorem 10 are satisfied; ̃
(ii) 𝑢𝑖 , V𝑖 , 𝑢̃𝑖 , ̃V𝑖 ∈ R, ( 𝑢V𝑖𝑖 ) ≤ ( 𝑢̃V𝑖𝑖 ), 𝑖 = 1, 2 imply that (20)
Let ( 𝑥𝑧∗ ) be a solution of (6) and ( 𝑤𝑦 ∗ ) a solution of the system 𝑦 (𝑡) ≤ 𝑤 (𝑡) ,
(ii) 𝑓2 (𝑡, ⋅, ⋅, ⋅, ⋅) : R4 → R is increasing, ∀𝑡 ∈ [𝑎, 𝑏]. 𝑥∗
Let ( 𝑧∗𝑖 ) be a solution of the system 𝑖
𝑥 (𝑡) = 𝑧 (𝑡) ,
(21)
𝑡 ∈ [𝑎, 𝑏] . 𝑦∗ 𝑥∗ 𝑦∗ 𝑥∗ ( ∗ ) ≤ ( ∗ ) implies that ( ∗ ) ≤ ( ∗ ) . 𝑤 [𝑎−ℎ,𝑎] 𝑧 [𝑎−ℎ,𝑎] 𝑤 𝑧 (22)
𝑡 ∈ [𝑎, 𝑏] ,
𝑧 (𝑡) = 𝑓𝑖 (𝑡, 𝑥 (𝑡) , 𝑧 (𝑡) , 𝑥 (𝑡 − ℎ) , 𝑧 (𝑡 − ℎ)) ,
(25)
𝑡 ∈ [𝑎, 𝑏] . Then 𝑥1∗ 𝑥2∗ 𝑥3∗ ( ∗ ) ≤ ( ∗ ) ≤ ( ∗ ) 𝑧1 𝑧2 𝑧3 [𝑎−ℎ,𝑎] [𝑎−ℎ,𝑎] [𝑎−ℎ,𝑎] 𝑥∗
imply that ( 𝑧1∗1 ) |
𝑥∗
𝑥∗
≤ ( 𝑧∗2 ) | 2
[𝑎−ℎ,𝑎]
≤ ( 𝑧∗3 ) | 3
[𝑎−ℎ,𝑎]
(26) .
Proof. We consider the operators 𝐸𝑖 corresponding to each system (25). The operators 𝐸𝑖 , 𝑖 = 1, 2, 3 are weakly Picard operators. Taking into consideration the condition (ii), 𝐸2 is increasing. From (i) we have 𝐸1 ≤ 𝐸2 ≤ 𝐸3 . On the other hand, we have that 𝑥̃∗ 𝑥∗ ( ∗𝑖 ) = 𝐸𝑖∞ ( ∗𝑖 ) , 𝑧̃𝑖 𝑧𝑖
𝑡 ∈ [𝑎, 𝑏] ,
𝑤 (𝑡) ≤ 𝑓 (𝑡, 𝑦 (𝑡) , 𝑤 (𝑡) , 𝑦 (𝑡 − ℎ) , 𝑤 (𝑡 − ℎ)) ,
Then
(i) 𝑓1 ≤ 𝑓2 ≤ 𝑓3 ;
[𝑎−ℎ,𝑎]
∗
∗
.
Theorem 12 (comparison theorem). Let 𝑓𝑖 ∈ 𝐶([𝑎, 𝑏] × R4 , R), 𝑖 = 1, 2, 3, be as in Theorem 10. One supposes that
ˇ 3.2. Inequalities of Caplygin Type. Now we establish the ˇ Caplygin type inequalities.
∀𝑡 ∈ [𝑎, 𝑏] .
[𝑎−ℎ,𝑎]
3.3. Data Dependence: Monotony. In this subsection, we study the monotony of the solution of the problem (6)-(7) with respect to 𝜑 and 𝑓.
0 1 𝑥1 𝑥 ≤ (𝑏 − 𝑎) ( ) ( ) − ( 2 ) , 𝐿 1 𝐿 2 𝑧1 𝑧2 whence 𝐵 is a contraction in (𝑋, ‖ ⋅ ‖) with 𝑄 = (𝑏 − 𝑎) ( 𝐿01 𝐿12 ). Applying Perov’s theorem we obtain (a), (b), and (c). Moreover, the operator 𝐵 is 𝑐-PO and 𝐸 is 𝑐-WPO with −1 𝑐 = [1 − (𝑏 − 𝑎) ( 𝐿01 𝐿12 )] .
𝑓 (𝑡, 𝑢1 , V1 , 𝑢2 , V2 ) ≤ 𝑓 (𝑡, 𝑢̃1 , ̃V1 , 𝑢̃2 , ̃V2 ) ,
(24)
𝑥̃∗ 𝑥∗ ≤ 𝐸 ( ∗) = ( ∗) , ̃𝑧 𝑧
is a
partition of 𝑋, and from [12] we have (1) 𝐵(𝑋) ⊂ 𝑋
(23)
From Theorem 10, (c), 𝐸 is a weakly Picard operator. From condition (ii), we obtain that 𝐸∞ is increasing [11]. So
)
𝐶([𝑎 − ℎ, 𝑎], R2 )}. Then 𝑋 = ∪
𝑦∗ 𝑦∗ ( ∗) ≤ 𝐸 ( ∗) . 𝑤 𝑤
∗
where ( 𝑥𝑧̃̃∗ ) ∈ 𝑋( 𝑥 )| 𝑧
[𝑎−ℎ,𝑎]
𝑖 = 1, 2, 3,
(27)
. The proof follows from the abstract
comparison Lemma (see [11]). 3.4. Data Dependence: Continuity. Consider the problem (6)(7) with the dates 𝑓𝑖 ∈ 𝐶([𝑎, 𝑏]×R4 , R), 𝑖 = 1, 2 and suppose that 𝑓𝑖 satisfy the conditions from Theorem 10 with the same Lipshitz constants. We obtain the data dependence result.
The Scientific World Journal
5
Theorem 13. Let 𝑓𝑖 ∈ 𝐶([𝑎, 𝑏] × R4 , R), 𝜑𝑖 ∈ 𝐶([𝑎 − ℎ, 𝑎], R), 𝑖 = 1, 2, be as in Theorem 10. One supposes that (i) there exists 𝜂1 , 𝜂2 > 0 such that 𝜑 𝜑 𝜂 1 ( ) (𝑡) − ( 2 ) (𝑡) ≤ ( 1 ) , 𝜂2 𝜑2 𝜑1
∀𝑡 ∈ [𝑎 − ℎ, 𝑎] ;
(28)
(ii) there exists 𝜂3 > 0 such that 𝑓1 (𝑡, 𝑢1 , 𝑢2 , 𝑢3 , 𝑢4 ) − 𝑓2 (𝑡, 𝑢1 , 𝑢2 , 𝑢3 , 𝑢4 ) ≤ 𝜂3 , ∀𝑡 ∈ [𝑎, 𝑏] , 𝑢𝑖 ∈ R, 𝑖 = 1, 4.
(29)
and since 𝑄𝑛 → ∞, as 𝑛 → ∞ implies that (𝐼 − 𝑄)−1 ∈ 𝑀22 (R+ ), we finally obtain
Then 𝑥∗ 𝑥∗ ( ∗ ) (⋅, 𝜑1 , 𝜑1 , 𝑓1 ) − ( ∗ ) (⋅, 𝜑2 , 𝜑2 , 𝑓2 ) 𝑧 𝑧 𝜂1 ≤ (𝐼 − 𝑄) ( ), 𝜂2 + (𝑏 − 𝑎) 𝜂3
𝑥∗ ≤ 𝐵𝜑1 ,𝑓1 (( ∗ ) (⋅, 𝜑1 , 𝜑1 , 𝑓1 )) 𝑧 𝑥∗ −𝐵𝜑1 ,𝑓1 (( ∗ ) (⋅, 𝜑2 , 𝜑2 , 𝑓2 )) 𝑧 ∗ 𝑥 + 𝐵𝜑1 ,𝑓1 (( ∗ ) (⋅, 𝜑2 , 𝜑2 , 𝑓2 )) 𝑧 𝑥∗ −𝐵𝜑2 ,𝑓2 (( ∗ ) (⋅, 𝜑2 , 𝜑2 , 𝑓2 )) 𝑧 𝑥∗ ≤ 𝑄 ( ∗ ) (⋅, 𝜑1 , 𝜑1 , 𝑓1 ) 𝑧 𝜂1 𝑥∗ ), − ( ∗ ) (⋅, 𝜑2 , 𝜑2 , 𝑓2 ) + ( 𝑧 𝜂2 + (𝑏 − 𝑎) 𝜂3 (33)
(30)
𝑥∗ 𝑥∗ ( ∗ ) (⋅, 𝜑1 , 𝜑1 , 𝑓1 ) − ( ∗ ) (⋅, 𝜑2 , 𝜑2 , 𝑓2 ) 𝑧 𝑧 𝜂 1 ≤ (𝐼 − 𝑄)−1 ( ). 𝜂2 + (𝑏 − 𝑎) 𝜂3
(34)
−1
∗
where ( 𝑥𝑧∗ ) (⋅, 𝜑, 𝜑 , 𝑓) denote the unique solution of (6)-(7). Proof. Consider the operators 𝐵𝜑𝑖 ,𝑓𝑖 , 𝑖 = 1, 2. From Theorem 10, it follows that 𝑥 𝑥 𝐵𝜑1 ,𝑓1 ( 1 ) − 𝐵𝜑1 ,𝑓1 ( 2 ) 𝑧1 𝑧2 𝑥 𝑥 𝑥 ≤ 𝑄 ( 1 ) − ( 2 ) , ∀ ( 𝑖 ) ∈ 𝑋. 𝑧2 𝑧𝑖 𝑧1
3.5. Data Dependence: Differentiability. Consider the following differential system with parameter: 𝑥 (𝑡) = 𝑧 (𝑡) ,
𝑡 ∈ [𝑎, 𝑏] ,
𝑧 (𝑡) = 𝑓 (𝑡, 𝑥 (𝑡) , 𝑧 (𝑡) , 𝑥 (𝑡 − ℎ) , 𝑧 (𝑡 − ℎ) ; 𝜆) ,
(35)
𝑡 ∈ [𝑎, 𝑏] , 𝜆 ∈ 𝐽, with the initial conditions (31)
𝑥 (𝑡) = 𝜑 (𝑡) ,
𝑧 (𝑡) = 𝜑 (𝑡) ,
𝑡 ∈ [𝑎 − ℎ, 𝑎] , 𝑡 ∈ [𝑎 − ℎ, 𝑎] ,
(36)
where 𝐽 ⊂ R is a compact interval. Suppose that the following conditions are satisfied:
Additionally, 𝜂1 𝑥 𝑥 ). 𝐵𝜑1 ,𝑓1 ( ) − 𝐵𝜑2 ,𝑓2 ( ) ≤ ( 𝜂2 + (𝑏 − 𝑎) 𝜂3 𝑧 𝑧
(32)
(C1 ) 𝑎 < 𝑏, ℎ > 0, 𝐽 ⊂ R a compact interval; (C2 ) 𝑓 ∈ 𝐶([𝑎, 𝑏] × R4 × 𝐽, R); (C3 ) 𝜑 ∈ 𝐶1 ([𝑎 − ℎ, 𝑎], R); (C4 ) there exists 𝐿 𝑓 > 0, such that
Thus, 𝑥∗ ( ∗ ) (⋅, 𝜑1 , 𝜑1 , 𝑓1 ) 𝑧
𝑥∗ − ( ∗ ) (⋅, 𝜑2 , 𝜑2 , 𝑓2 ) 𝑧
𝑥∗ = 𝐵𝜑1 ,𝑓1 (( ∗ ) (⋅, 𝜑1 , 𝜑1 , 𝑓1 )) 𝑧
𝑥 −𝐵𝜑2 ,𝑓2 (( ∗ ) (⋅, 𝜑2 , 𝜑2 , 𝑓2 )) 𝑧
𝜕𝑓 (𝑡, 𝑢 , 𝑢 , 𝑢 , 𝑢 ; 𝜆) 1 2 3 4 ≤ 𝐿 1 , 𝑢1 , 𝑢3 ∈ R, 𝜕𝑢𝑖 𝑖 = 1, 3, 𝜆 ∈ 𝐽; 𝜕𝑓 (𝑡, 𝑢 , 𝑢 , 𝑢 , 𝑢 ; 𝜆) 1 2 3 4 ≤ 𝐿 2 , 𝑢2 , 𝑢4 ∈ R, 𝜕𝑢 𝑖
(37)
𝑖 = 2, 4, 𝜆 ∈ 𝐽;
∗
(C5 ) for 𝑄 = (𝑏 − 𝑎) ( 𝐿01
1 𝐿2
), we have 𝑄𝑛 → 0 as 𝑛 → ∞.
6
The Scientific World Journal
Then, from Theorem 10, we have that the problem (6)-(7) ∗ has a unique solution, ( 𝑥𝑧∗ ) ∈ 𝐶1 ([𝑎, 𝑏], R2 ). We prove that ∗ ( 𝑥𝑧∗ ) ∈ 𝐶1 (𝐽, R2 ), ∀𝑡 ∈ [𝑎 − ℎ, 𝑏]. For this we consider the system
𝑥 (𝑡; 𝜆) = 𝑧 (𝑡; 𝜆) ,
Theorem 14. Consider the problem (38)–(36) and suppose the conditions (C1 )–(C5 ) hold. Then, ∗
(i) Equations (38)–(36) have a unique solution ( 𝑥𝑧∗ ) (⋅, 𝜆), in 𝐶([𝑎 − ℎ, 𝑏] × 𝐽, R2 );
𝑡 ∈ [𝑎, 𝑏] , 𝜆 ∈ 𝐽,
𝑧 (𝑡; 𝜆) = 𝑓 (𝑡, 𝑥 (𝑡; 𝜆) , 𝑧 (𝑡; 𝜆) , 𝑥 (𝑡 − ℎ; 𝜆) , 𝑧 (𝑡 − ℎ; 𝜆) ; 𝜆) , 𝑡 ∈ [𝑎, 𝑏] , 𝜆 ∈ 𝐽,
(38)
∗
(ii) ( 𝑥𝑧∗ ) (⋅, 𝜆) ∈ 𝐶1 (𝐽, R2 ), ∀𝑡 ∈ [𝑎 − ℎ, 𝑏]. Proof. The problem (38)–(36) is equivalent with the following functional-integral system:
∗
with ( 𝑥𝑧∗ ) ∈ 𝐶([𝑎 − ℎ, 𝑏] × 𝐽, R2 ) ∩ 𝐶1 ([𝑎, 𝑏] × 𝐽, R2 ).
𝑡
𝑥 ( ) (𝑡; 𝜆) = ( 𝑧
𝜑 (𝑎) + ∫ 𝑧 (𝑠; 𝜆) 𝑑𝑠
𝑎
𝑡
𝜑 (𝑎) + ∫ 𝑓 ( 𝑠, 𝑥 (𝑠; 𝜆) , 𝑧 (𝑠; 𝜆) , 𝑥 (𝑠 − ℎ; 𝜆) , 𝑧 (𝑠 − ℎ; 𝜆) ; 𝜆 ) 𝑑𝑠
),
(39)
𝑎
∗
/𝜕𝜆 ), from (39)-(40), we Supposing that there exists ( 𝜕𝑥 𝜕𝑧∗ /𝜕𝜆 obtain that
for 𝑡 ∈ [𝑎, 𝑏] and
𝜑 𝑥 ( ) (𝑡; 𝜆) = ( ) (𝑡) , 𝑧 𝜑
for 𝑡 ∈ [𝑎 − ℎ, 𝑎] .
𝑡 𝜕𝑧 (𝑠; 𝜆) 𝜕𝑥∗ =∫ 𝑑𝑠, 𝜕𝜆 𝜕𝜆 𝑎
(40)
Now let us take the operator 𝐴 : 𝐶([𝑎 − ℎ, 𝑏] × 𝐽, R2 ) → 𝐶([𝑎 − ℎ, 𝑏] × 𝐽, R2 ), defined by
𝑡 𝜕𝑧∗ = ∫ ((𝜕𝑓 (𝑠, 𝑥 (𝑠; 𝜆) , 𝑧 (𝑠; 𝜆) , 𝑥 (𝑠 − ℎ; 𝜆) , 𝜕𝜆 𝑎 −1
𝑧 (𝑠 − ℎ; 𝜆) ; 𝜆) ) × (𝜕𝑢1 ) )
𝜕𝑥 (𝑠; 𝜆) 𝑑𝑠 𝜕𝜆
𝑡
+ ∫ ((𝜕𝑓 (𝑠, 𝑥 (𝑠; 𝜆) , 𝑧 (𝑠; 𝜆) , 𝑥 (𝑠 − ℎ; 𝜆) ,
𝑥 𝐴1 ( ) 𝑧 𝑥 𝐴 ( ) (𝑡; 𝜆) := ( ) 𝑧 𝑥 𝐴2 ( ) 𝑧
𝑎
−1
𝑧 (𝑠 − ℎ; 𝜆) ; 𝜆) ) × (𝜕𝑢2 ) ) 𝑡
+ ∫ ((𝜕𝑓 (𝑠, 𝑥 (𝑠; 𝜆) , 𝑧 (𝑠; 𝜆) , 𝑥 (𝑠 − ℎ; 𝜆) ,
:= the right hand side of (39) , for 𝑡 ∈ [𝑎, 𝑏] , 𝑥 𝐴 ( ) (𝑡; 𝜆) := ( 𝑧
𝜕𝑧 (𝑠; 𝜆) 𝑑𝑠 𝜕𝜆
𝑥 𝐴1 ( ) 𝑧 ) 𝑥 𝐴2 ( ) 𝑧
𝑎
−1
𝑧 (𝑠 − ℎ; 𝜆) ; 𝜆) ) × (𝜕𝑢3 ) )
(41) 𝑡
+ ∫ ((𝜕𝑓 (𝑠, 𝑥 (𝑠; 𝜆) , 𝑧 (𝑠; 𝜆) , 𝑥 (𝑠 − ℎ; 𝜆) , 𝑎
−1
𝑧 (𝑠 − ℎ; 𝜆) ; 𝜆) ) × (𝜕𝑢4 ) )
:= the right hand side of (40) , for 𝑡 ∈ [𝑎 − ℎ, 𝑎] .
𝜕𝑥 (𝑠 − ℎ; 𝜆) 𝑑𝑠 𝜕𝜆
𝜕𝑧 (𝑠 − ℎ; 𝜆) 𝑑𝑠 𝜕𝜆
𝑡
+ ∫ ((𝜕𝑓 (𝑠, 𝑥 (𝑠; 𝜆) , 𝑧 (𝑠; 𝜆) , 𝑥 (𝑠 − ℎ; 𝜆) , 𝑎
Let 𝑋 = 𝐶([𝑎 − ℎ, 𝑏] × 𝐽, R2 ). It is clear, from the proof of the Theorem 10, that in the condition (C1 )–(C5 ), the operator 𝐴 : (𝑋, ‖ ⋅ ‖) → (𝑋, ‖ ⋅ ‖) is Picard operator. ∗ Let ( 𝑥𝑧∗ ) be the unique fixed point of 𝐴.
𝑧 (𝑠 − ℎ; 𝜆) ; 𝜆) ) × (𝜕𝜆)−1 ) 𝑑𝑠, (42) for all 𝑡 ∈ [𝑎, 𝑏], 𝜆 ∈ 𝐽.
The Scientific World Journal
7
This relation suggests that we consider the following operator:
From Theorem 8 the operator 𝐷 is Picard operator; that is, the sequences
𝐶 : 𝑋 × 𝑋 → 𝑋,
𝑛+1 𝑛 𝑥12 𝑥12 ( 𝑛+1 ) := 𝐴 ( 𝑛 ) , 𝑧12 𝑧12
𝑢 𝑥 𝑢 𝑥 (( 12 ) , ( 13 )) → 𝐶 (( 12 ) , ( 13 )) , 𝑧12 𝑢24 𝑧12 𝑢24
(43)
𝑛+1 𝑢13 ( 𝑛+1 ) 𝑢24
where 𝐶 (( 𝑥𝑧1212 ) , ( 𝑢𝑢1324 )) (𝑡; 𝜆) = 0 for 𝑡 ∈ [𝑎 − ℎ, 𝑎], 𝜆 ∈ 𝐽, and 𝑥 𝑢 𝐶1 (( 12 ) , ( 13 )) 𝑧 𝑢24 𝑥 𝑢 12 𝐶 (( 12 ) , ( 13 )) (𝑡; 𝜆) := ( ), 𝑧12 𝑢24 𝑥12 𝑢13 𝐶2 (( ) , ( )) 𝑧12 𝑢24 (44)
:=
𝑛 ∈ N, converge uniformly, with respect to 𝑡 ∈ [𝑎 − ℎ, 𝑏], 𝜆 ∈ 𝑢𝑛
𝑥𝑛
𝑢0
𝑥0
𝐽, to (( 𝑧𝑛12 ) , ( 𝑢13𝑛 )) ∈ 𝐹𝐷, for all ( 𝑧012 ) , ( 𝑢130 ) ∈ 𝑋. 12 24 12 24 If we take (
where 𝑡 𝜕𝑧 (𝑠; 𝜆) 𝑥 𝑢 𝑑𝑠, 𝐶1 (( 12 ) , ( 13 )) (𝑡; 𝜆) := ∫ 𝑧12 𝑢24 𝜕𝜆 𝑎
(
𝑥 𝑢 𝐶2 (( 12 ) , ( 13 )) (𝑡; 𝜆) 𝑧12 𝑢24 𝑡
(46)
𝑛 𝑛 𝑥12 𝑢13 𝐶 (( 𝑛 ) , ( 𝑛 )) , 𝑧12 𝑢24
0 𝑢13 0 𝑢24
0 𝑥12 0 𝑧12
0 ) = ( ), 0
)=(
0 𝜕𝑥12 𝜕𝜆 0 𝜕𝑧12 𝜕𝜆
(47) 0 ) = ( ), 0
then
:= ∫ ((𝜕𝑓 (𝑠, 𝑥 (𝑠; 𝜆) , 𝑧 (𝑠; 𝜆) , 𝑥 (𝑠 − ℎ; 𝜆) , 𝑎
1 𝑢13 ( 1) 𝑢24
−1
𝑧 (𝑠 − ℎ; 𝜆) ; 𝜆) ) × (𝜕𝑢1 ) ) 𝑢1 (𝑠; 𝜆) 𝑑𝑠 𝑡
=(
+ ∫ ((𝜕𝑓 (𝑠, 𝑥 (𝑠; 𝜆) , 𝑧 (𝑠; 𝜆) , 𝑥 (𝑠 − ℎ; 𝜆) , 𝑎
1 𝜕𝑥12 𝜕𝜆 1 𝜕𝑧12 𝜕𝜆
).
(48)
By induction we prove that
−1
𝑧 (𝑠 − ℎ; 𝜆) ; 𝜆) ) × (𝜕𝑢2 ) ) 𝑢2 (𝑠; 𝜆) 𝑑𝑠
𝑛 𝜕𝑥12 𝜕𝜆 ( 𝑛 ) = ( 𝑛 ), 𝑢24 𝜕𝑧12 𝜕𝜆
𝑡
𝑛 𝑢13
+ ∫ ((𝜕𝑓 (𝑠, 𝑥 (𝑠; 𝜆) , 𝑧 (𝑠; 𝜆) , 𝑥 (𝑠 − ℎ; 𝜆) , 𝑎
−1
𝑧 (𝑠 − ℎ; 𝜆) ; 𝜆) ) × (𝜕𝑢3 ) ) 𝑢3 (𝑠 − ℎ; 𝜆) 𝑑𝑠 𝑡
𝑥𝑛
+ ∫ ((𝜕𝑓 (𝑠, 𝑥 (𝑠; 𝜆) , 𝑧 (𝑠; 𝜆) , 𝑥 (𝑠 − ℎ; 𝜆) , 𝑎
−1
𝑧 (𝑠 − ℎ; 𝜆) ; 𝜆) ) × (𝜕𝑢4 ) ) 𝑢4 (𝑠 − ℎ; 𝜆) 𝑑𝑠 𝑡
+ ∫ ((𝜕𝑓 (𝑠, 𝑥 (𝑠; 𝜆) , 𝑧 (𝑠; 𝜆) , 𝑥 (𝑠 − ℎ; 𝜆) , 𝑎
𝑧 (𝑠 − ℎ; 𝜆) ; 𝜆) ) × (𝜕𝜆)−1 ) 𝑑𝑠, (45) for 𝑡 ∈ [𝑎, 𝑏], 𝜆 ∈ 𝐽. Here we use the notations 𝑢1 (𝑠; 𝜆) := 𝜕𝑥(𝑠; 𝜆)/𝜕𝜆, 𝑢2 (𝑠; 𝜆) := 𝜕𝑧(𝑠; 𝜆)/𝜕𝜆, 𝑢3 (𝑠 − ℎ; 𝜆) := 𝜕𝑥(𝑠 − ℎ; 𝜆)/𝜕𝜆, and 𝑢4 (𝑠 − ℎ; 𝜆) := 𝜕𝑧(𝑠 − ℎ; 𝜆)/𝜕𝜆. In this way, we have the triangular operator 𝐷 : 𝑋 × 𝑋 → 𝑋 × 𝑋, (( 𝑥𝑧1212 ) , ( 𝑢𝑢1324 )) → (𝐴 ( 𝑥𝑧1212 ) , 𝐶(( 𝑥𝑧1212 ) , ( 𝑢𝑢1324 ))), where 𝐴 is Picard operator and 𝐶(( 𝑥𝑧1212 ) , ( ⋅⋅ )) : 𝑋 → 𝑋 is 𝑄𝐶contraction with 𝑄𝐶 = (𝑏 − 𝑎) ( 𝐿01 𝐿12 ).
unif
∀𝑛 ∈ N.
𝑥∗
𝜕𝑥𝑛 /𝜕𝜆
(49)
unif
𝑢∗
So, ( 𝑧𝑛12 ) → ( 𝑧∗12 ) , as 𝑛 → ∞, and ( 𝜕𝑧𝑛12/𝜕𝜆 ) → ( 𝑢13∗ ), 12 12 12 24 as 𝑛 → ∞. From a Weierstrass argument we get that there exists ∗ 𝜕𝑥12 𝜕𝜆 ( ∗ ), 𝜕𝑧12 𝜕𝜆
𝑖 = 1, 2,
∗ 𝜕𝑥12 𝑢∗ 𝜕𝜆 ( ∗ ) = ( 13 ∗ ). 𝑢24 𝜕𝑧12 𝜕𝜆
(50)
3.6. Ulam-Hyers Stability. We start this section by presenting the Ulam-Hyers stability concept (see [15, 16]).
8
The Scientific World Journal
For 𝑓 ∈ 𝐶([𝑎, 𝑏] × R4 , R), 𝜀 > 0, and 𝜓 ∈ 𝐶([𝑎 − ℎ, 𝑏], R+ ), ℎ > 0, we consider the system 𝑥 (𝑡) = 𝑧 (𝑡) ,
𝑡 ∈ [𝑎, 𝑏] ,
𝑧 (𝑡) = 𝑓 (𝑡, 𝑥 (𝑡) , 𝑧 (𝑡) , 𝑥 (𝑡 − ℎ) , 𝑧 (𝑡 − ℎ)) ,
(51)
𝑡 ∈ [𝑎, 𝑏] and the following inequations 𝑥 (𝑡) − 𝑧 (𝑡) ≤ 𝜀, 𝑡 ∈ [𝑎, 𝑏] , 𝑧 (𝑡) − 𝑓 (𝑡, 𝑥 (𝑡) , 𝑧 (𝑡) , 𝑥 (𝑡 − ℎ) , 𝑧 (𝑡 − ℎ)) ≤ 𝜀,
(52)
𝑡 ∈ [𝑎, 𝑏] , 𝑥 (𝑡) − 𝑧 (𝑡) ≤ 𝜓 (𝑡) ,
𝑡 ∈ [𝑎, 𝑏] ,
𝑧 (𝑡) − 𝑓 (𝑡, 𝑥 (𝑡) , 𝑧 (𝑡) , 𝑥 (𝑡 − ℎ) , 𝑧 (𝑡 − ℎ)) ≤ 𝜓 (𝑡) , 𝑡 ∈ [𝑎, 𝑏] . (53) Definition 15. The system (51) is Ulam-Hyers stable if there exists a real number 𝑐 > 0 such that for each 𝜀 > 0 and for each solution ( 𝑤𝑦 ) ∈ 𝐶2 ([𝑎 − ℎ, 𝑏], R2 ) of (52) there exists a solution ( 𝑥𝑧 ) ∈ 𝐶2 ([𝑎 − ℎ, 𝑏], R2 ) of (51) with 𝑦 𝑥 (54) ( ) (𝑡) − ( ) (𝑡) ≤ 𝑐𝜀, ∀𝑡 ∈ [𝑎 − ℎ, 𝑏] . 𝑧 𝑤 Theorem 16. One supposes that (i) the conditions (C1 )–(C3 ) are satisfied; (ii) 𝑄𝑛 → 0 as 𝑛 → ∞, where 𝑄 = (𝑏 − 𝑎) ( 𝐿01
1 𝐿2
).
Then the system (6) is Ulam-Hyers stable. Proof. The system (6) is equivalent with the functional integral system (10). We consider the operator 𝐸 : 𝐶([𝑎 − ℎ, 𝑏], R2 ) → 𝐶([𝑎 − ℎ, 𝑏], R2 ), defined by 𝐸 ( 𝑥𝑧 ) (𝑡) := the right hand side of (10), for 𝑡 ∈ [𝑎 − ℎ, 𝑏]. So 𝑥 𝑥 ( ) = 𝐸 ( ) (𝑡) , 𝑧 𝑧
𝑡 ∈ [𝑎 − ℎ, 𝑏] .
(55)
From Theorem 10, 𝐸 is 𝑐-WPO with 𝑐 = [1 − (𝑏− −1 𝑎) ( 𝐿01 𝐿12 )] . Applying Theorem 7 we obtain that (6) is UlamHyers stable. Remark 17. Another proof for the above theorem can be done using Gronwall lemma [8, 14–17].
Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment The work of the first author was partially supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, Project no. PN-II-ID-PCE-2011-3-0094.
References [1] V. Kolmanovski˘ı and A. Myshkis, Applied Theory of FunctionalDifferential Equations, Kluwer Academic Publishers Group, Dordrecht, Germany, 1992. [2] E. Pinney, Ordinary Difference-Differential Equations, University of California Press, Berkeley, Calif, USA, 1958. [3] V. V. Guljaev, A. S. Dmitriev, and V. E. Kislov, “Strange attractors in the circle: selfoscillating systems,” Doklady Akademii Nauk SSSR, vol. 282, no. 2, pp. 53–66, 1985. [4] V. B. Kolmanovski˘ı and V. R. Nosov, Stability of FunctionalDifferential Equations, Academic Press, London, UK, 1986. [5] V. A. Ilea and D. Otrocol, “On a D. V. Ionescu’s problem for functional-differential equations,” Fixed Point Theory, vol. 10, no. 1, pp. 125–140, 2009. [6] I. M. Olaru, “An integral equation via weakly Picard operators,” Fixed Point Theory, vol. 11, no. 1, pp. 97–106, 2010. [7] I. M. Olaru, “Data dependence for some integral equations,” Studia. Universitatis Babes¸-Bolyai. Mathematica, vol. 55, no. 2, pp. 159–165, 2010. [8] D. Otrocol and V. Ilea, “Ulam stability for a delay differential equation,” Central European Journal of Mathematics, vol. 11, no. 7, pp. 1296–1303, 2013. [9] R. Precup, “The role of matrices that are convergent to zero in the study of semilinear operator systems,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 703–708, 2009. [10] I. A. Rus, Principles ans Applications of the Fixed Point Theory, Dacia, Cluj-Napoca, Romania, 1979, Romanian. [11] I. A. Rus, Generalized contractions and applications, Cluj University Press, Cluj-Napoca, Romania, 2001. [12] I. A. Rus, “Functional-differential equations of mixed type, via weakly Picard operators,” Seminar on Fixed Point Theory ClujNapoca, vol. 3, pp. 335–345, 2002. [13] I. A. Rus, “Picard operators and applications,” Scientiae Mathematicae Japonicae, vol. 58, no. 1, pp. 191–219, 2003. [14] I. A. Rus, “Gronwall lemmas: ten open problems,” Scientiae Mathematicae Japonicae, vol. 70, no. 2, pp. 221–228, 2009. [15] I. A. Rus, “Ulam stability of ordinary differential equations,” Studia. Universitatis Babes¸-Bolyai. Mathematica, vol. 54, no. 4, pp. 125–133, 2009. [16] I. A. Rus, “Remarks on Ulam stability of the operatorial equations,” Fixed Point Theory, vol. 10, no. 2, pp. 305–320, 2009. [17] I. A. Rus, “Ulam stability of the operatorial equations,” in Functional Equations in Mathematical Analysis, T. M. Rassias and J. Brzdek, Eds., chapter 23, Springer, 2011. [18] A. I. Perov and A. V. Kibenko, “On a general method to study boundary value problems,” Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, vol. 30, pp. 249–264, 1966.
Research Article Some Properties of Solutions of a Functional-Differential Equation of Second Order with Delay Veronica Ana Ilea1 and Diana Otrocol2 1 2
Babes¸-Bolyai University, Kog˘alniceanu no. 1, RO-400084 Cluj-Napoca, Romania “T. Popoviciu” Institute of Numerical Analysis, Romanian Academy, P.O. Box. 68-1, 400110 Cluj-Napoca, Romania
Correspondence should be addressed to Diana Otrocol; [email protected] Received 13 August 2013; Accepted 7 October 2013; Published 10 February 2014 Academic Editors: F. Khani and H. Xu Copyright © 2014 V. Ana Ilea and D. Otrocol. 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. Existence, uniqueness, data dependence (monotony, continuity, and differentiability with respect to parameter), and Ulam-Hyers stability results for the solutions of a system of functional-differential equations with delays are proved. The techniques used are Perov’s fixed point theorem and weakly Picard operator theory.
1. Introduction Functional-differential equations with delay arise when modeling biological, physical, engineering, and other processes whose rate of change of state at any moment of time 𝑡 is determined not only by the present state but also by past state. The description of certain phenomena in physics has to take into account that the rate of propagation is finite. For example, oscillation in a vacuum tube can be described by the following equation in dimensionless variables [1, 2]: 𝑥 (𝑡) + 2𝑟𝑥 (𝑡) + 𝜔2 𝑥 (𝑡) + 2𝑞𝑥 (𝑡 − 1) = 𝜖𝑥3 (𝑡 − 1) . (1) In this equation, time delay is due to the fact that the time necessary for electrons to pass from the cathode to the anode in the tube is finite. The same equation has been used in the theory of stabilization of ships [2]. The dynamics of an autogenerator with delay and second-order filter was described in [3] by the equation 𝑥 (𝑡) + 2𝛿𝑥 (𝑡) + 𝑥 (𝑡) = 𝑓 (𝑥 (𝑡 − ℎ)) .
(2)
The model of ship course stabilization under conditions of uncertainty may be described by the following equation [4]: 𝐼𝑥 (𝑡) + 𝐻𝑥 (𝑡) = −𝐾Ψ (𝑡) + 𝑏0 𝜉0 (𝑡) , 𝑥 (𝑡0 ) = 𝑥0 ,
𝑥 (𝑡0 ) = 0,
(3)
with 𝑥(𝑡) being the angle of the deviation from course, Ψ(𝑡) the turning angle of the rudder, and 𝜉0 (𝑡) the stochastic disturbance. In the process of mathematical modeling, often small delays are neglected; that is why sometimes false conclusions appear. As an example we can give the following equation [1]: 𝑥 (𝑡) + 𝑥 (𝑡) + 𝑥 (𝑡) = 𝑎 [𝑥 (𝑡 − ℎ) + 𝑥 (𝑡 − ℎ) + 𝑥 (𝑡 − ℎ)] ,
(4)
which is asymptotically stable for ℎ = 0 but unstable for arbitrary ℎ > 0. Here 𝑎 > 1. If ℎ = 0 the above system is asymptotically stable. The characteristic equation is Δ(𝑧) = (𝑧2 +𝑧+1)(1−𝑎𝑒−ℎ𝑧 ) and has the following zeros with positive real part if ℎ > 0 : 𝑧𝑘 = 1/ℎ(In 𝑎+2𝑘𝜋𝑖), 𝑖2 = −1, 𝑘 = 0, ±1, . . .. So the trivial solution is unstable for any ℎ > 0. In this paper, we continue the research in this field and develop the study of the following general functional differential equation with delay: 𝑥 (𝑡) = 𝑓 (𝑡, 𝑥 (𝑡) , 𝑥 (𝑡) , 𝑥 (𝑡 − ℎ) , 𝑥 (𝑡 − ℎ)) , 𝑡 ∈ [𝑎, 𝑏] , 𝑥 (𝑡) = 𝜑 (𝑡) ,
𝑡 ∈ [𝑎 − ℎ, 𝑎] .
(5)
2
The Scientific World Journal
Existence, uniqueness, data dependence (monotony, continuity, and differentiability with respect to parameter), and Ulam-Hyers stability results of solution for the Cauchy problem are obtained. Our results are essentially based on Perov’s fixed point theorem and weakly Picard operator technique, which will be presented in Section 2. More results about functional and integral differential equations using these techniques can be found in [5–8]. The problem (5) is equivalent to the following system: 𝑥 (𝑡) = 𝑧 (𝑡) ,
𝑡 ∈ [𝑎, 𝑏] ,
𝑧 (𝑡) = 𝑓 (𝑡, 𝑥 (𝑡) , 𝑧 (𝑡) , 𝑥 (𝑡 − ℎ) , 𝑧 (𝑡 − ℎ)) ,
(6)
𝑡 ∈ [𝑎, 𝑏] , with the initial conditions 𝑥 (𝑡) = 𝜑 (𝑡) , 𝑧 (𝑡) = 𝜑 (𝑡) ,
𝑡 ∈ [𝑎 − ℎ, 𝑎] , 𝑡 ∈ [𝑎 − ℎ, 𝑎] .
(7)
By a solution of the system (6) we understand a function ( 𝑥𝑧 ) ∈ 𝐶([𝑎−ℎ, 𝑏], R2 )∩𝐶1 ([𝑎, 𝑏], R2 ) that verifies the system. We suppose that (C1 ) 𝑎 < 𝑏, ℎ > 0; (C2 ) 𝑓 ∈ 𝐶([𝑎, 𝑏] × R4 , R), 𝜑 ∈ 𝐶1 ([𝑎 − ℎ, 𝑎], R); (C3 ) there exists 𝐿 1 , 𝐿 2 > 0 such that ∀𝑡 ∈ [𝑎, 𝑏], 𝑢𝑖 , V𝑖 , 𝑢̃𝑖 , ̃V𝑖 ∈ R, 𝑖 = 1, 2, we have 𝑓 (𝑡, 𝑢1 , V1 , 𝑢2 , V2 ) − 𝑓 (𝑡, 𝑢̃1 , ̃V1 , 𝑢̃2 , ̃V2 ) ≤ 𝐿 1 max (𝑢1 − 𝑢̃1 , 𝑢2 − 𝑢̃2 ) + 𝐿 2 max (V1 − ̃V1 , V2 − ̃V2 ) .
If ( 𝑥𝑧 ) ∈ 𝐶([𝑎 − ℎ, 𝑏], R2 ) ∩ 𝐶1 ([𝑎, 𝑏], R2 ) is a solution of the problem (6)-(7), then ( 𝑥𝑧 ) is a solution of the following integral system:
𝜑 { ( ) (𝑡) , { { { 𝜑 { { { 𝑡 𝑥 ( ) (𝑡) = { 𝜑 (𝑎) + ∫ 𝑧 (𝑠) 𝑑𝑠 𝑧 { { 𝑎 { ) ( { 𝑡 { { 𝜑 (𝑎) + ∫ 𝑓 (𝑠, 𝑥 (𝑠) , 𝑧 (𝑠) , 𝑥 (𝑠 − ℎ) , 𝑧 (𝑠 − ℎ)) 𝑑𝑠 { 𝑎 If ( 𝑥𝑧 ) ∈ 𝐶([𝑎−ℎ, 𝑏], R2 ) is a solution of (9), then ( 𝑥𝑧 ) ∈ 𝐶([𝑎− ℎ, 𝑏], R2 ) ∩ 𝐶1 ([𝑎, 𝑏], R2 ) and ( 𝑥𝑧 ) is a solution of (6)-(7).
for 𝑡 ∈ [𝑎 − ℎ, 𝑎] , (9) for 𝑡 ∈ [𝑎, 𝑏] .
Moreover, the system (6) is equivalent to the functional integral system
𝑥 { { ( ) (𝑡) , for 𝑡 ∈ [𝑎 − ℎ, 𝑎] , { { 𝑧 { { { { 𝑥 𝑡 ( ) (𝑡) = { 𝑧 (𝑠) 𝑑𝑠 𝑥 + ∫ (𝑎) 𝑧 { { { 𝑎 { ) for 𝑡 ∈ [𝑎, 𝑏] . ( 𝑡 { { { 𝑧 (𝑎) + ∫ 𝑓 (𝑠, 𝑥 (𝑠) , 𝑧 (𝑠) , 𝑥 (𝑠 − ℎ) , 𝑧 (𝑠 − ℎ)) 𝑑𝑠 { 𝑎 We consider the operators 𝐵, 𝐸 : 𝐶([𝑎 − ℎ, 𝑏], R2 ) → 𝐶([𝑎 − 𝐵 (𝑥) 𝐸 (𝑥) ℎ, 𝑏], R2 ), 𝐵 = ( 𝐵1 𝑥𝑧 ), 𝐸 = ( 𝐸1 𝑥𝑧 ) defined by 𝐵 ( 𝑥𝑧 ) (𝑡) := 2( 𝑧 ) 2( 𝑧 ) the right hand side of (9), for 𝑡 ∈ [𝑎 − ℎ, 𝑏] and 𝐸 ( 𝑥𝑧 ) (𝑡) := the right hand side of (10), for 𝑡 ∈ [𝑎 − ℎ, 𝑏].
2. Preliminaries In this section, we introduce notations, definitions, and preliminary results which are used throughout this paper; see [9–17]. Let (𝑋, 𝑑) be a metric space and 𝐴 : 𝑋 → 𝑋 an operator. We will use the following notations: 𝐹𝐴 := {𝑥 ∈ 𝑋 | 𝐴(𝑥) = 𝑥}—the fixed points set of 𝐴;
(8)
(10)
𝐼(𝐴) := {𝑌 ⊂ 𝑋 | 𝐴(𝑌) ⊂ 𝑌, 𝑌 ≠ 0}—the family of the nonempty invariant subset of 𝐴; 𝐴𝑛+1 := 𝐴 ∘ 𝐴𝑛 , 𝐴0 = 1𝑋 , 𝐴1 = 𝐴, 𝑛 ∈ N. Definition 1. Let (𝑋, 𝑑) be a metric space. An operator 𝐴 : 𝑋 → 𝑋 is a Picard operator (PO) if there exists 𝑥∗ ∈ 𝑋 such that (i) 𝐹𝐴 = {𝑥∗ }; (ii) the sequence (𝐴𝑛 (𝑥0 ))𝑛∈N converges to 𝑥∗ for all 𝑥0 ∈ 𝑋. Definition 2. Let (𝑋, 𝑑) be a metric space. An operator 𝐴 : 𝑋 → 𝑋 is a weakly Picard operator (WPO) if the sequence
The Scientific World Journal
3
(𝐴𝑛 (𝑥))𝑛∈N converges for all 𝑥 ∈ 𝑋 and its limit (which may depend on 𝑥) is a fixed point of 𝐴. Definition 3. If 𝐴 is a weakly Picard operator, then we consider the operator 𝐴∞ defined by 𝐴∞ : 𝑋 → 𝑋,
𝐴∞ (𝑥) := lim 𝐴𝑛 (𝑥) . 𝑛→∞
(11)
Remark 4. It is clear that 𝐴∞ (𝑋) = 𝐹𝐴 . Definition 5. Let 𝐴 be a weakly Picard operator and 𝑐 > 0. The operator 𝐴 is 𝑐-weakly Picard operator if 𝑑 (𝑥, 𝐴∞ (𝑥)) ≤ 𝑐𝑑 (𝑥, 𝐴 (𝑥)) ,
∀𝑥 ∈ 𝑋.
(12)
The following concept is important for our further considerations. Definition 6. Let (𝑋, 𝑑) be a metric space and 𝑓 : 𝑋 → 𝑋 an operator. The fixed point equation 𝑥 = 𝑓 (𝑥)
(13)
is Ulam-Hyers stable if there exists a real number 𝑐𝑓 > 0 such that for each 𝜀 > 0 and each solution 𝑦∗ of the inequation 𝑑 (𝑦, 𝑓 (𝑦)) ≤ 𝜀,
(14)
there exists a solution 𝑥∗ of (13) such that 𝑑 (𝑦∗ , 𝑥∗ ) ≤ 𝑐𝑓 𝜀.
(15)
Now we have the following. Theorem 7 (see [17]). If 𝑓 : 𝑋 → 𝑋 is 𝑐-WPO, then the equation 𝑥 = 𝑓 (𝑥)
(16)
is Ulam-Hyers stable. Another result from the WPO theory is the following (see, e.g., [11]). Theorem 8 (fibre contraction principle). Let (𝑋, 𝑑) and (𝑌, 𝜌) be two metric spaces and 𝐴 : 𝑋 × 𝑌 → 𝑋 × 𝑌, 𝐴 = (𝐵, 𝐶), (𝐵 : 𝑋 → 𝑋, 𝐶 : 𝑋 × 𝑌 → 𝑌) a triangular operator. One supposes that (i) (𝑌, 𝜌) is a complete metric space; (ii) the operator 𝐵 is Picard operator; (iii) there exists 𝑙 ∈ [0, 1) such that 𝐶(𝑥, ⋅) : 𝑌 → 𝑌 is a 𝑙-contraction, for all 𝑥 ∈ 𝑋; (iv) if (𝑥∗ , 𝑦∗ ) ∈ 𝐹𝐴, then 𝐶(⋅, 𝑦∗ ) is continuous in 𝑥∗ . Then the operator 𝐴 is Picard operator. Throughout this paper we denote by 𝑀𝑚𝑚 (R+ ) the set of all 𝑚×𝑚 matrices with positive elements and by 𝐼 the identity 𝑚 × 𝑚 matrix. A square matrix 𝑄 with nonnegative elements is said to be convergent to zero if 𝑄𝑘 → 0 as 𝑘 → ∞. It is known that the property of being convergent to zero is equivalent to each of the following three conditions (see [9, 10]):
(a) 𝐼 − 𝑄 is nonsingular and (𝐼 − 𝑄)−1 = 𝐼 + 𝑄 + 𝑄2 + ⋅ ⋅ ⋅ (where 𝐼 stands for the unit matrix of the same order as 𝑄); (b) the eigenvalues of 𝑄 are located inside the open unit disc of the complex plane; (c) 𝐼 − 𝑄 is nonsingular and (𝐼 − 𝑄)−1 has nonnegative elements. We finish this section by recalling the following fundamental result (see [9, 18]). Theorem 9 (Perov’s fixed point theorem). Let (𝑋, 𝑑) with 𝑑(𝑥, 𝑦) ∈ R𝑚 be a complete generalized metric space and 𝐴 : 𝑋 → 𝑋 an operator. One supposes that there exists a matrix 𝑄 ∈ 𝑀𝑚𝑚 (R+ ), such that (i) 𝑑(𝐴(𝑥), 𝐴(𝑦)) ≤ 𝑄𝑑(𝑥, 𝑦), for all 𝑥, 𝑦 ∈ 𝑋; (ii) 𝑄𝑛 → 0 as 𝑛 → ∞. Then (a) 𝐹𝐴 = {𝑥∗ };
(b) 𝐴𝑛 (𝑥) = 𝑥∗ as 𝑛 → ∞, ∀𝑥 ∈ 𝑋; (c) 𝑑(𝐴𝑛 (𝑥), 𝑥∗ ) ≤ (𝐼 − 𝑄)−1 𝑄𝑛 𝑑(𝑥0 , 𝐴(𝑥0 )).
3. Main Results In this section, we present existence, uniqueness, and data dependence (monotony, continuity, and differentiability with respect to parameter) results of solution for the Cauchy problem (6)-(7). 3.1. Existence and Uniqueness. Using Perov’s fixed point theorem, we obtain existence and uniqueness theorem for the solution of the problem (6)-(7). Theorem 10. One supposes that (i) the conditions (C1 )–(C3 ) are satisfied;
(ii) 𝑄𝑛 → 0 as 𝑛 → ∞, where 𝑄 = (𝑏 − 𝑎) ( 𝐿01
1 𝐿2
).
Then, ∗
(a) the problem (6)-(7) has a unique solution ( 𝑥𝑧∗ ) ∈ 𝐶1 ([𝑎, 𝑏], R2 ); 0
𝑛
(b) for all ( 𝑥𝑧0 ) ∈ 𝐶1 ([𝑎, 𝑏], R2 ), the sequence ( 𝑥𝑧𝑛 )𝑛∈N 𝑛+1
𝑛
defined by ( 𝑥𝑧𝑛+1 ) = 𝐵 ( 𝑥𝑧𝑛 ) converges uniformly to ∗
( 𝑥𝑧∗ ), for all 𝑡 ∈ [𝑎, 𝑏], and 𝑥𝑛 𝑥∗ ( 𝑛 ) − ( ∗ ) 𝑧 𝑧
0 𝑥 𝑥1 ≤ (𝐼 − 𝑄) 𝑄 ( 0 ) − ( 1 ) ; 𝑧 𝑧 −1
(17)
𝑛
(c) the operator 𝐵 is Picard operator in (𝐶([𝑎 − ℎ, 𝑏], 𝑢𝑛𝑖𝑓
R2 ), →);
4
The Scientific World Journal (d) the operator 𝐸 is weakly Picard operator in (𝐶([𝑎 −
Proof. We have that
𝑢𝑛𝑖𝑓
2
ℎ, 𝑏], R ), →). Proof. Consider on the space 𝑋 := 𝐶([𝑎 − ℎ, 𝑏], R2 ) the norm 𝑢1 ( ) − (𝑢2 ) := max (𝑢1 − 𝑢2) , (18) V1 V2 V1 − V2 which endows 𝑋 with the uniform convergence. 𝜒 𝜒 Let 𝑋 𝜒 = {( 𝑥𝑧 ) ∈ 𝑋 | ( 𝑥𝑧 ) |[𝑎−ℎ,𝑎] = ( 𝜒 ) , for ( 𝜒 ) ∈ (
𝜒
𝑥∗ 𝑥∗ ( ∗) = 𝐸 ( ∗) , 𝑧 𝑧
𝜒 𝑋 𝜒 ( )∈𝐶([𝑎−ℎ,𝑎],R2 ) ( ) 𝜒 𝜒
𝑦∗ 𝑦∗ 𝑦̃∗ ( ∗ ) ≤ 𝐸∞ ( ∗ ) = 𝐸∞ ( ∗ ) ̃ 𝑤 𝑤 𝑤
(
(2) 𝐵|𝑋
(
𝜒 ) 𝜒
𝜒 , ) 𝜒
= 𝐸|𝑋
(
𝐵(𝑋
(
𝜒 ) ) 𝜒
⊂𝑋
(
∞
∗
where ( 𝑥𝑧̃̃∗ ) ∈ 𝑋( 𝑥 )|
𝜒 ; ) 𝜒
𝑧
𝜒 . ) 𝜒
On the other hand, for 𝑡 ∈ [𝑎 − ℎ, 𝑎] ∪ [𝑎, 𝑏] 𝑥 𝐵 (𝑥1 ) 𝐵1 ( 2 ) 1 𝑧 𝑧2 1 ( )−( ) 𝑥 𝐵 (𝑥1 ) 𝐵2 ( 2 ) 2 𝑧 𝑧2 1
(19)
Theorem 11. One supposes that (i) the conditions (a), (b), and (c) in Theorem 10 are satisfied; ̃
(ii) 𝑢𝑖 , V𝑖 , 𝑢̃𝑖 , ̃V𝑖 ∈ R, ( 𝑢V𝑖𝑖 ) ≤ ( 𝑢̃V𝑖𝑖 ), 𝑖 = 1, 2 imply that (20)
Let ( 𝑥𝑧∗ ) be a solution of (6) and ( 𝑤𝑦 ∗ ) a solution of the system 𝑦 (𝑡) ≤ 𝑤 (𝑡) ,
(ii) 𝑓2 (𝑡, ⋅, ⋅, ⋅, ⋅) : R4 → R is increasing, ∀𝑡 ∈ [𝑎, 𝑏]. 𝑥∗
Let ( 𝑧∗𝑖 ) be a solution of the system 𝑖
𝑥 (𝑡) = 𝑧 (𝑡) ,
(21)
𝑡 ∈ [𝑎, 𝑏] . 𝑦∗ 𝑥∗ 𝑦∗ 𝑥∗ ( ∗ ) ≤ ( ∗ ) implies that ( ∗ ) ≤ ( ∗ ) . 𝑤 [𝑎−ℎ,𝑎] 𝑧 [𝑎−ℎ,𝑎] 𝑤 𝑧 (22)
𝑡 ∈ [𝑎, 𝑏] ,
𝑧 (𝑡) = 𝑓𝑖 (𝑡, 𝑥 (𝑡) , 𝑧 (𝑡) , 𝑥 (𝑡 − ℎ) , 𝑧 (𝑡 − ℎ)) ,
(25)
𝑡 ∈ [𝑎, 𝑏] . Then 𝑥1∗ 𝑥2∗ 𝑥3∗ ( ∗ ) ≤ ( ∗ ) ≤ ( ∗ ) 𝑧1 𝑧2 𝑧3 [𝑎−ℎ,𝑎] [𝑎−ℎ,𝑎] [𝑎−ℎ,𝑎] 𝑥∗
imply that ( 𝑧1∗1 ) |
𝑥∗
𝑥∗
≤ ( 𝑧∗2 ) | 2
[𝑎−ℎ,𝑎]
≤ ( 𝑧∗3 ) | 3
[𝑎−ℎ,𝑎]
(26) .
Proof. We consider the operators 𝐸𝑖 corresponding to each system (25). The operators 𝐸𝑖 , 𝑖 = 1, 2, 3 are weakly Picard operators. Taking into consideration the condition (ii), 𝐸2 is increasing. From (i) we have 𝐸1 ≤ 𝐸2 ≤ 𝐸3 . On the other hand, we have that 𝑥̃∗ 𝑥∗ ( ∗𝑖 ) = 𝐸𝑖∞ ( ∗𝑖 ) , 𝑧̃𝑖 𝑧𝑖
𝑡 ∈ [𝑎, 𝑏] ,
𝑤 (𝑡) ≤ 𝑓 (𝑡, 𝑦 (𝑡) , 𝑤 (𝑡) , 𝑦 (𝑡 − ℎ) , 𝑤 (𝑡 − ℎ)) ,
Then
(i) 𝑓1 ≤ 𝑓2 ≤ 𝑓3 ;
[𝑎−ℎ,𝑎]
∗
∗
.
Theorem 12 (comparison theorem). Let 𝑓𝑖 ∈ 𝐶([𝑎, 𝑏] × R4 , R), 𝑖 = 1, 2, 3, be as in Theorem 10. One supposes that
ˇ 3.2. Inequalities of Caplygin Type. Now we establish the ˇ Caplygin type inequalities.
∀𝑡 ∈ [𝑎, 𝑏] .
[𝑎−ℎ,𝑎]
3.3. Data Dependence: Monotony. In this subsection, we study the monotony of the solution of the problem (6)-(7) with respect to 𝜑 and 𝑓.
0 1 𝑥1 𝑥 ≤ (𝑏 − 𝑎) ( ) ( ) − ( 2 ) , 𝐿 1 𝐿 2 𝑧1 𝑧2 whence 𝐵 is a contraction in (𝑋, ‖ ⋅ ‖) with 𝑄 = (𝑏 − 𝑎) ( 𝐿01 𝐿12 ). Applying Perov’s theorem we obtain (a), (b), and (c). Moreover, the operator 𝐵 is 𝑐-PO and 𝐸 is 𝑐-WPO with −1 𝑐 = [1 − (𝑏 − 𝑎) ( 𝐿01 𝐿12 )] .
𝑓 (𝑡, 𝑢1 , V1 , 𝑢2 , V2 ) ≤ 𝑓 (𝑡, 𝑢̃1 , ̃V1 , 𝑢̃2 , ̃V2 ) ,
(24)
𝑥̃∗ 𝑥∗ ≤ 𝐸 ( ∗) = ( ∗) , ̃𝑧 𝑧
is a
partition of 𝑋, and from [12] we have (1) 𝐵(𝑋) ⊂ 𝑋
(23)
From Theorem 10, (c), 𝐸 is a weakly Picard operator. From condition (ii), we obtain that 𝐸∞ is increasing [11]. So
)
𝐶([𝑎 − ℎ, 𝑎], R2 )}. Then 𝑋 = ∪
𝑦∗ 𝑦∗ ( ∗) ≤ 𝐸 ( ∗) . 𝑤 𝑤
∗
where ( 𝑥𝑧̃̃∗ ) ∈ 𝑋( 𝑥 )| 𝑧
[𝑎−ℎ,𝑎]
𝑖 = 1, 2, 3,
(27)
. The proof follows from the abstract
comparison Lemma (see [11]). 3.4. Data Dependence: Continuity. Consider the problem (6)(7) with the dates 𝑓𝑖 ∈ 𝐶([𝑎, 𝑏]×R4 , R), 𝑖 = 1, 2 and suppose that 𝑓𝑖 satisfy the conditions from Theorem 10 with the same Lipshitz constants. We obtain the data dependence result.
The Scientific World Journal
5
Theorem 13. Let 𝑓𝑖 ∈ 𝐶([𝑎, 𝑏] × R4 , R), 𝜑𝑖 ∈ 𝐶([𝑎 − ℎ, 𝑎], R), 𝑖 = 1, 2, be as in Theorem 10. One supposes that (i) there exists 𝜂1 , 𝜂2 > 0 such that 𝜑 𝜑 𝜂 1 ( ) (𝑡) − ( 2 ) (𝑡) ≤ ( 1 ) , 𝜂2 𝜑2 𝜑1
∀𝑡 ∈ [𝑎 − ℎ, 𝑎] ;
(28)
(ii) there exists 𝜂3 > 0 such that 𝑓1 (𝑡, 𝑢1 , 𝑢2 , 𝑢3 , 𝑢4 ) − 𝑓2 (𝑡, 𝑢1 , 𝑢2 , 𝑢3 , 𝑢4 ) ≤ 𝜂3 , ∀𝑡 ∈ [𝑎, 𝑏] , 𝑢𝑖 ∈ R, 𝑖 = 1, 4.
(29)
and since 𝑄𝑛 → ∞, as 𝑛 → ∞ implies that (𝐼 − 𝑄)−1 ∈ 𝑀22 (R+ ), we finally obtain
Then 𝑥∗ 𝑥∗ ( ∗ ) (⋅, 𝜑1 , 𝜑1 , 𝑓1 ) − ( ∗ ) (⋅, 𝜑2 , 𝜑2 , 𝑓2 ) 𝑧 𝑧 𝜂1 ≤ (𝐼 − 𝑄) ( ), 𝜂2 + (𝑏 − 𝑎) 𝜂3
𝑥∗ ≤ 𝐵𝜑1 ,𝑓1 (( ∗ ) (⋅, 𝜑1 , 𝜑1 , 𝑓1 )) 𝑧 𝑥∗ −𝐵𝜑1 ,𝑓1 (( ∗ ) (⋅, 𝜑2 , 𝜑2 , 𝑓2 )) 𝑧 ∗ 𝑥 + 𝐵𝜑1 ,𝑓1 (( ∗ ) (⋅, 𝜑2 , 𝜑2 , 𝑓2 )) 𝑧 𝑥∗ −𝐵𝜑2 ,𝑓2 (( ∗ ) (⋅, 𝜑2 , 𝜑2 , 𝑓2 )) 𝑧 𝑥∗ ≤ 𝑄 ( ∗ ) (⋅, 𝜑1 , 𝜑1 , 𝑓1 ) 𝑧 𝜂1 𝑥∗ ), − ( ∗ ) (⋅, 𝜑2 , 𝜑2 , 𝑓2 ) + ( 𝑧 𝜂2 + (𝑏 − 𝑎) 𝜂3 (33)
(30)
𝑥∗ 𝑥∗ ( ∗ ) (⋅, 𝜑1 , 𝜑1 , 𝑓1 ) − ( ∗ ) (⋅, 𝜑2 , 𝜑2 , 𝑓2 ) 𝑧 𝑧 𝜂 1 ≤ (𝐼 − 𝑄)−1 ( ). 𝜂2 + (𝑏 − 𝑎) 𝜂3
(34)
−1
∗
where ( 𝑥𝑧∗ ) (⋅, 𝜑, 𝜑 , 𝑓) denote the unique solution of (6)-(7). Proof. Consider the operators 𝐵𝜑𝑖 ,𝑓𝑖 , 𝑖 = 1, 2. From Theorem 10, it follows that 𝑥 𝑥 𝐵𝜑1 ,𝑓1 ( 1 ) − 𝐵𝜑1 ,𝑓1 ( 2 ) 𝑧1 𝑧2 𝑥 𝑥 𝑥 ≤ 𝑄 ( 1 ) − ( 2 ) , ∀ ( 𝑖 ) ∈ 𝑋. 𝑧2 𝑧𝑖 𝑧1
3.5. Data Dependence: Differentiability. Consider the following differential system with parameter: 𝑥 (𝑡) = 𝑧 (𝑡) ,
𝑡 ∈ [𝑎, 𝑏] ,
𝑧 (𝑡) = 𝑓 (𝑡, 𝑥 (𝑡) , 𝑧 (𝑡) , 𝑥 (𝑡 − ℎ) , 𝑧 (𝑡 − ℎ) ; 𝜆) ,
(35)
𝑡 ∈ [𝑎, 𝑏] , 𝜆 ∈ 𝐽, with the initial conditions (31)
𝑥 (𝑡) = 𝜑 (𝑡) ,
𝑧 (𝑡) = 𝜑 (𝑡) ,
𝑡 ∈ [𝑎 − ℎ, 𝑎] , 𝑡 ∈ [𝑎 − ℎ, 𝑎] ,
(36)
where 𝐽 ⊂ R is a compact interval. Suppose that the following conditions are satisfied:
Additionally, 𝜂1 𝑥 𝑥 ). 𝐵𝜑1 ,𝑓1 ( ) − 𝐵𝜑2 ,𝑓2 ( ) ≤ ( 𝜂2 + (𝑏 − 𝑎) 𝜂3 𝑧 𝑧
(32)
(C1 ) 𝑎 < 𝑏, ℎ > 0, 𝐽 ⊂ R a compact interval; (C2 ) 𝑓 ∈ 𝐶([𝑎, 𝑏] × R4 × 𝐽, R); (C3 ) 𝜑 ∈ 𝐶1 ([𝑎 − ℎ, 𝑎], R); (C4 ) there exists 𝐿 𝑓 > 0, such that
Thus, 𝑥∗ ( ∗ ) (⋅, 𝜑1 , 𝜑1 , 𝑓1 ) 𝑧
𝑥∗ − ( ∗ ) (⋅, 𝜑2 , 𝜑2 , 𝑓2 ) 𝑧
𝑥∗ = 𝐵𝜑1 ,𝑓1 (( ∗ ) (⋅, 𝜑1 , 𝜑1 , 𝑓1 )) 𝑧
𝑥 −𝐵𝜑2 ,𝑓2 (( ∗ ) (⋅, 𝜑2 , 𝜑2 , 𝑓2 )) 𝑧
𝜕𝑓 (𝑡, 𝑢 , 𝑢 , 𝑢 , 𝑢 ; 𝜆) 1 2 3 4 ≤ 𝐿 1 , 𝑢1 , 𝑢3 ∈ R, 𝜕𝑢𝑖 𝑖 = 1, 3, 𝜆 ∈ 𝐽; 𝜕𝑓 (𝑡, 𝑢 , 𝑢 , 𝑢 , 𝑢 ; 𝜆) 1 2 3 4 ≤ 𝐿 2 , 𝑢2 , 𝑢4 ∈ R, 𝜕𝑢 𝑖
(37)
𝑖 = 2, 4, 𝜆 ∈ 𝐽;
∗
(C5 ) for 𝑄 = (𝑏 − 𝑎) ( 𝐿01
1 𝐿2
), we have 𝑄𝑛 → 0 as 𝑛 → ∞.
6
The Scientific World Journal
Then, from Theorem 10, we have that the problem (6)-(7) ∗ has a unique solution, ( 𝑥𝑧∗ ) ∈ 𝐶1 ([𝑎, 𝑏], R2 ). We prove that ∗ ( 𝑥𝑧∗ ) ∈ 𝐶1 (𝐽, R2 ), ∀𝑡 ∈ [𝑎 − ℎ, 𝑏]. For this we consider the system
𝑥 (𝑡; 𝜆) = 𝑧 (𝑡; 𝜆) ,
Theorem 14. Consider the problem (38)–(36) and suppose the conditions (C1 )–(C5 ) hold. Then, ∗
(i) Equations (38)–(36) have a unique solution ( 𝑥𝑧∗ ) (⋅, 𝜆), in 𝐶([𝑎 − ℎ, 𝑏] × 𝐽, R2 );
𝑡 ∈ [𝑎, 𝑏] , 𝜆 ∈ 𝐽,
𝑧 (𝑡; 𝜆) = 𝑓 (𝑡, 𝑥 (𝑡; 𝜆) , 𝑧 (𝑡; 𝜆) , 𝑥 (𝑡 − ℎ; 𝜆) , 𝑧 (𝑡 − ℎ; 𝜆) ; 𝜆) , 𝑡 ∈ [𝑎, 𝑏] , 𝜆 ∈ 𝐽,
(38)
∗
(ii) ( 𝑥𝑧∗ ) (⋅, 𝜆) ∈ 𝐶1 (𝐽, R2 ), ∀𝑡 ∈ [𝑎 − ℎ, 𝑏]. Proof. The problem (38)–(36) is equivalent with the following functional-integral system:
∗
with ( 𝑥𝑧∗ ) ∈ 𝐶([𝑎 − ℎ, 𝑏] × 𝐽, R2 ) ∩ 𝐶1 ([𝑎, 𝑏] × 𝐽, R2 ).
𝑡
𝑥 ( ) (𝑡; 𝜆) = ( 𝑧
𝜑 (𝑎) + ∫ 𝑧 (𝑠; 𝜆) 𝑑𝑠
𝑎
𝑡
𝜑 (𝑎) + ∫ 𝑓 ( 𝑠, 𝑥 (𝑠; 𝜆) , 𝑧 (𝑠; 𝜆) , 𝑥 (𝑠 − ℎ; 𝜆) , 𝑧 (𝑠 − ℎ; 𝜆) ; 𝜆 ) 𝑑𝑠
),
(39)
𝑎
∗
/𝜕𝜆 ), from (39)-(40), we Supposing that there exists ( 𝜕𝑥 𝜕𝑧∗ /𝜕𝜆 obtain that
for 𝑡 ∈ [𝑎, 𝑏] and
𝜑 𝑥 ( ) (𝑡; 𝜆) = ( ) (𝑡) , 𝑧 𝜑
for 𝑡 ∈ [𝑎 − ℎ, 𝑎] .
𝑡 𝜕𝑧 (𝑠; 𝜆) 𝜕𝑥∗ =∫ 𝑑𝑠, 𝜕𝜆 𝜕𝜆 𝑎
(40)
Now let us take the operator 𝐴 : 𝐶([𝑎 − ℎ, 𝑏] × 𝐽, R2 ) → 𝐶([𝑎 − ℎ, 𝑏] × 𝐽, R2 ), defined by
𝑡 𝜕𝑧∗ = ∫ ((𝜕𝑓 (𝑠, 𝑥 (𝑠; 𝜆) , 𝑧 (𝑠; 𝜆) , 𝑥 (𝑠 − ℎ; 𝜆) , 𝜕𝜆 𝑎 −1
𝑧 (𝑠 − ℎ; 𝜆) ; 𝜆) ) × (𝜕𝑢1 ) )
𝜕𝑥 (𝑠; 𝜆) 𝑑𝑠 𝜕𝜆
𝑡
+ ∫ ((𝜕𝑓 (𝑠, 𝑥 (𝑠; 𝜆) , 𝑧 (𝑠; 𝜆) , 𝑥 (𝑠 − ℎ; 𝜆) ,
𝑥 𝐴1 ( ) 𝑧 𝑥 𝐴 ( ) (𝑡; 𝜆) := ( ) 𝑧 𝑥 𝐴2 ( ) 𝑧
𝑎
−1
𝑧 (𝑠 − ℎ; 𝜆) ; 𝜆) ) × (𝜕𝑢2 ) ) 𝑡
+ ∫ ((𝜕𝑓 (𝑠, 𝑥 (𝑠; 𝜆) , 𝑧 (𝑠; 𝜆) , 𝑥 (𝑠 − ℎ; 𝜆) ,
:= the right hand side of (39) , for 𝑡 ∈ [𝑎, 𝑏] , 𝑥 𝐴 ( ) (𝑡; 𝜆) := ( 𝑧
𝜕𝑧 (𝑠; 𝜆) 𝑑𝑠 𝜕𝜆
𝑥 𝐴1 ( ) 𝑧 ) 𝑥 𝐴2 ( ) 𝑧
𝑎
−1
𝑧 (𝑠 − ℎ; 𝜆) ; 𝜆) ) × (𝜕𝑢3 ) )
(41) 𝑡
+ ∫ ((𝜕𝑓 (𝑠, 𝑥 (𝑠; 𝜆) , 𝑧 (𝑠; 𝜆) , 𝑥 (𝑠 − ℎ; 𝜆) , 𝑎
−1
𝑧 (𝑠 − ℎ; 𝜆) ; 𝜆) ) × (𝜕𝑢4 ) )
:= the right hand side of (40) , for 𝑡 ∈ [𝑎 − ℎ, 𝑎] .
𝜕𝑥 (𝑠 − ℎ; 𝜆) 𝑑𝑠 𝜕𝜆
𝜕𝑧 (𝑠 − ℎ; 𝜆) 𝑑𝑠 𝜕𝜆
𝑡
+ ∫ ((𝜕𝑓 (𝑠, 𝑥 (𝑠; 𝜆) , 𝑧 (𝑠; 𝜆) , 𝑥 (𝑠 − ℎ; 𝜆) , 𝑎
Let 𝑋 = 𝐶([𝑎 − ℎ, 𝑏] × 𝐽, R2 ). It is clear, from the proof of the Theorem 10, that in the condition (C1 )–(C5 ), the operator 𝐴 : (𝑋, ‖ ⋅ ‖) → (𝑋, ‖ ⋅ ‖) is Picard operator. ∗ Let ( 𝑥𝑧∗ ) be the unique fixed point of 𝐴.
𝑧 (𝑠 − ℎ; 𝜆) ; 𝜆) ) × (𝜕𝜆)−1 ) 𝑑𝑠, (42) for all 𝑡 ∈ [𝑎, 𝑏], 𝜆 ∈ 𝐽.
The Scientific World Journal
7
This relation suggests that we consider the following operator:
From Theorem 8 the operator 𝐷 is Picard operator; that is, the sequences
𝐶 : 𝑋 × 𝑋 → 𝑋,
𝑛+1 𝑛 𝑥12 𝑥12 ( 𝑛+1 ) := 𝐴 ( 𝑛 ) , 𝑧12 𝑧12
𝑢 𝑥 𝑢 𝑥 (( 12 ) , ( 13 )) → 𝐶 (( 12 ) , ( 13 )) , 𝑧12 𝑢24 𝑧12 𝑢24
(43)
𝑛+1 𝑢13 ( 𝑛+1 ) 𝑢24
where 𝐶 (( 𝑥𝑧1212 ) , ( 𝑢𝑢1324 )) (𝑡; 𝜆) = 0 for 𝑡 ∈ [𝑎 − ℎ, 𝑎], 𝜆 ∈ 𝐽, and 𝑥 𝑢 𝐶1 (( 12 ) , ( 13 )) 𝑧 𝑢24 𝑥 𝑢 12 𝐶 (( 12 ) , ( 13 )) (𝑡; 𝜆) := ( ), 𝑧12 𝑢24 𝑥12 𝑢13 𝐶2 (( ) , ( )) 𝑧12 𝑢24 (44)
:=
𝑛 ∈ N, converge uniformly, with respect to 𝑡 ∈ [𝑎 − ℎ, 𝑏], 𝜆 ∈ 𝑢𝑛
𝑥𝑛
𝑢0
𝑥0
𝐽, to (( 𝑧𝑛12 ) , ( 𝑢13𝑛 )) ∈ 𝐹𝐷, for all ( 𝑧012 ) , ( 𝑢130 ) ∈ 𝑋. 12 24 12 24 If we take (
where 𝑡 𝜕𝑧 (𝑠; 𝜆) 𝑥 𝑢 𝑑𝑠, 𝐶1 (( 12 ) , ( 13 )) (𝑡; 𝜆) := ∫ 𝑧12 𝑢24 𝜕𝜆 𝑎
(
𝑥 𝑢 𝐶2 (( 12 ) , ( 13 )) (𝑡; 𝜆) 𝑧12 𝑢24 𝑡
(46)
𝑛 𝑛 𝑥12 𝑢13 𝐶 (( 𝑛 ) , ( 𝑛 )) , 𝑧12 𝑢24
0 𝑢13 0 𝑢24
0 𝑥12 0 𝑧12
0 ) = ( ), 0
)=(
0 𝜕𝑥12 𝜕𝜆 0 𝜕𝑧12 𝜕𝜆
(47) 0 ) = ( ), 0
then
:= ∫ ((𝜕𝑓 (𝑠, 𝑥 (𝑠; 𝜆) , 𝑧 (𝑠; 𝜆) , 𝑥 (𝑠 − ℎ; 𝜆) , 𝑎
1 𝑢13 ( 1) 𝑢24
−1
𝑧 (𝑠 − ℎ; 𝜆) ; 𝜆) ) × (𝜕𝑢1 ) ) 𝑢1 (𝑠; 𝜆) 𝑑𝑠 𝑡
=(
+ ∫ ((𝜕𝑓 (𝑠, 𝑥 (𝑠; 𝜆) , 𝑧 (𝑠; 𝜆) , 𝑥 (𝑠 − ℎ; 𝜆) , 𝑎
1 𝜕𝑥12 𝜕𝜆 1 𝜕𝑧12 𝜕𝜆
).
(48)
By induction we prove that
−1
𝑧 (𝑠 − ℎ; 𝜆) ; 𝜆) ) × (𝜕𝑢2 ) ) 𝑢2 (𝑠; 𝜆) 𝑑𝑠
𝑛 𝜕𝑥12 𝜕𝜆 ( 𝑛 ) = ( 𝑛 ), 𝑢24 𝜕𝑧12 𝜕𝜆
𝑡
𝑛 𝑢13
+ ∫ ((𝜕𝑓 (𝑠, 𝑥 (𝑠; 𝜆) , 𝑧 (𝑠; 𝜆) , 𝑥 (𝑠 − ℎ; 𝜆) , 𝑎
−1
𝑧 (𝑠 − ℎ; 𝜆) ; 𝜆) ) × (𝜕𝑢3 ) ) 𝑢3 (𝑠 − ℎ; 𝜆) 𝑑𝑠 𝑡
𝑥𝑛
+ ∫ ((𝜕𝑓 (𝑠, 𝑥 (𝑠; 𝜆) , 𝑧 (𝑠; 𝜆) , 𝑥 (𝑠 − ℎ; 𝜆) , 𝑎
−1
𝑧 (𝑠 − ℎ; 𝜆) ; 𝜆) ) × (𝜕𝑢4 ) ) 𝑢4 (𝑠 − ℎ; 𝜆) 𝑑𝑠 𝑡
+ ∫ ((𝜕𝑓 (𝑠, 𝑥 (𝑠; 𝜆) , 𝑧 (𝑠; 𝜆) , 𝑥 (𝑠 − ℎ; 𝜆) , 𝑎
𝑧 (𝑠 − ℎ; 𝜆) ; 𝜆) ) × (𝜕𝜆)−1 ) 𝑑𝑠, (45) for 𝑡 ∈ [𝑎, 𝑏], 𝜆 ∈ 𝐽. Here we use the notations 𝑢1 (𝑠; 𝜆) := 𝜕𝑥(𝑠; 𝜆)/𝜕𝜆, 𝑢2 (𝑠; 𝜆) := 𝜕𝑧(𝑠; 𝜆)/𝜕𝜆, 𝑢3 (𝑠 − ℎ; 𝜆) := 𝜕𝑥(𝑠 − ℎ; 𝜆)/𝜕𝜆, and 𝑢4 (𝑠 − ℎ; 𝜆) := 𝜕𝑧(𝑠 − ℎ; 𝜆)/𝜕𝜆. In this way, we have the triangular operator 𝐷 : 𝑋 × 𝑋 → 𝑋 × 𝑋, (( 𝑥𝑧1212 ) , ( 𝑢𝑢1324 )) → (𝐴 ( 𝑥𝑧1212 ) , 𝐶(( 𝑥𝑧1212 ) , ( 𝑢𝑢1324 ))), where 𝐴 is Picard operator and 𝐶(( 𝑥𝑧1212 ) , ( ⋅⋅ )) : 𝑋 → 𝑋 is 𝑄𝐶contraction with 𝑄𝐶 = (𝑏 − 𝑎) ( 𝐿01 𝐿12 ).
unif
∀𝑛 ∈ N.
𝑥∗
𝜕𝑥𝑛 /𝜕𝜆
(49)
unif
𝑢∗
So, ( 𝑧𝑛12 ) → ( 𝑧∗12 ) , as 𝑛 → ∞, and ( 𝜕𝑧𝑛12/𝜕𝜆 ) → ( 𝑢13∗ ), 12 12 12 24 as 𝑛 → ∞. From a Weierstrass argument we get that there exists ∗ 𝜕𝑥12 𝜕𝜆 ( ∗ ), 𝜕𝑧12 𝜕𝜆
𝑖 = 1, 2,
∗ 𝜕𝑥12 𝑢∗ 𝜕𝜆 ( ∗ ) = ( 13 ∗ ). 𝑢24 𝜕𝑧12 𝜕𝜆
(50)
3.6. Ulam-Hyers Stability. We start this section by presenting the Ulam-Hyers stability concept (see [15, 16]).
8
The Scientific World Journal
For 𝑓 ∈ 𝐶([𝑎, 𝑏] × R4 , R), 𝜀 > 0, and 𝜓 ∈ 𝐶([𝑎 − ℎ, 𝑏], R+ ), ℎ > 0, we consider the system 𝑥 (𝑡) = 𝑧 (𝑡) ,
𝑡 ∈ [𝑎, 𝑏] ,
𝑧 (𝑡) = 𝑓 (𝑡, 𝑥 (𝑡) , 𝑧 (𝑡) , 𝑥 (𝑡 − ℎ) , 𝑧 (𝑡 − ℎ)) ,
(51)
𝑡 ∈ [𝑎, 𝑏] and the following inequations 𝑥 (𝑡) − 𝑧 (𝑡) ≤ 𝜀, 𝑡 ∈ [𝑎, 𝑏] , 𝑧 (𝑡) − 𝑓 (𝑡, 𝑥 (𝑡) , 𝑧 (𝑡) , 𝑥 (𝑡 − ℎ) , 𝑧 (𝑡 − ℎ)) ≤ 𝜀,
(52)
𝑡 ∈ [𝑎, 𝑏] , 𝑥 (𝑡) − 𝑧 (𝑡) ≤ 𝜓 (𝑡) ,
𝑡 ∈ [𝑎, 𝑏] ,
𝑧 (𝑡) − 𝑓 (𝑡, 𝑥 (𝑡) , 𝑧 (𝑡) , 𝑥 (𝑡 − ℎ) , 𝑧 (𝑡 − ℎ)) ≤ 𝜓 (𝑡) , 𝑡 ∈ [𝑎, 𝑏] . (53) Definition 15. The system (51) is Ulam-Hyers stable if there exists a real number 𝑐 > 0 such that for each 𝜀 > 0 and for each solution ( 𝑤𝑦 ) ∈ 𝐶2 ([𝑎 − ℎ, 𝑏], R2 ) of (52) there exists a solution ( 𝑥𝑧 ) ∈ 𝐶2 ([𝑎 − ℎ, 𝑏], R2 ) of (51) with 𝑦 𝑥 (54) ( ) (𝑡) − ( ) (𝑡) ≤ 𝑐𝜀, ∀𝑡 ∈ [𝑎 − ℎ, 𝑏] . 𝑧 𝑤 Theorem 16. One supposes that (i) the conditions (C1 )–(C3 ) are satisfied; (ii) 𝑄𝑛 → 0 as 𝑛 → ∞, where 𝑄 = (𝑏 − 𝑎) ( 𝐿01
1 𝐿2
).
Then the system (6) is Ulam-Hyers stable. Proof. The system (6) is equivalent with the functional integral system (10). We consider the operator 𝐸 : 𝐶([𝑎 − ℎ, 𝑏], R2 ) → 𝐶([𝑎 − ℎ, 𝑏], R2 ), defined by 𝐸 ( 𝑥𝑧 ) (𝑡) := the right hand side of (10), for 𝑡 ∈ [𝑎 − ℎ, 𝑏]. So 𝑥 𝑥 ( ) = 𝐸 ( ) (𝑡) , 𝑧 𝑧
𝑡 ∈ [𝑎 − ℎ, 𝑏] .
(55)
From Theorem 10, 𝐸 is 𝑐-WPO with 𝑐 = [1 − (𝑏− −1 𝑎) ( 𝐿01 𝐿12 )] . Applying Theorem 7 we obtain that (6) is UlamHyers stable. Remark 17. Another proof for the above theorem can be done using Gronwall lemma [8, 14–17].
Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment The work of the first author was partially supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, Project no. PN-II-ID-PCE-2011-3-0094.
References [1] V. Kolmanovski˘ı and A. Myshkis, Applied Theory of FunctionalDifferential Equations, Kluwer Academic Publishers Group, Dordrecht, Germany, 1992. [2] E. Pinney, Ordinary Difference-Differential Equations, University of California Press, Berkeley, Calif, USA, 1958. [3] V. V. Guljaev, A. S. Dmitriev, and V. E. Kislov, “Strange attractors in the circle: selfoscillating systems,” Doklady Akademii Nauk SSSR, vol. 282, no. 2, pp. 53–66, 1985. [4] V. B. Kolmanovski˘ı and V. R. Nosov, Stability of FunctionalDifferential Equations, Academic Press, London, UK, 1986. [5] V. A. Ilea and D. Otrocol, “On a D. V. Ionescu’s problem for functional-differential equations,” Fixed Point Theory, vol. 10, no. 1, pp. 125–140, 2009. [6] I. M. Olaru, “An integral equation via weakly Picard operators,” Fixed Point Theory, vol. 11, no. 1, pp. 97–106, 2010. [7] I. M. Olaru, “Data dependence for some integral equations,” Studia. Universitatis Babes¸-Bolyai. Mathematica, vol. 55, no. 2, pp. 159–165, 2010. [8] D. Otrocol and V. Ilea, “Ulam stability for a delay differential equation,” Central European Journal of Mathematics, vol. 11, no. 7, pp. 1296–1303, 2013. [9] R. Precup, “The role of matrices that are convergent to zero in the study of semilinear operator systems,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 703–708, 2009. [10] I. A. Rus, Principles ans Applications of the Fixed Point Theory, Dacia, Cluj-Napoca, Romania, 1979, Romanian. [11] I. A. Rus, Generalized contractions and applications, Cluj University Press, Cluj-Napoca, Romania, 2001. [12] I. A. Rus, “Functional-differential equations of mixed type, via weakly Picard operators,” Seminar on Fixed Point Theory ClujNapoca, vol. 3, pp. 335–345, 2002. [13] I. A. Rus, “Picard operators and applications,” Scientiae Mathematicae Japonicae, vol. 58, no. 1, pp. 191–219, 2003. [14] I. A. Rus, “Gronwall lemmas: ten open problems,” Scientiae Mathematicae Japonicae, vol. 70, no. 2, pp. 221–228, 2009. [15] I. A. Rus, “Ulam stability of ordinary differential equations,” Studia. Universitatis Babes¸-Bolyai. Mathematica, vol. 54, no. 4, pp. 125–133, 2009. [16] I. A. Rus, “Remarks on Ulam stability of the operatorial equations,” Fixed Point Theory, vol. 10, no. 2, pp. 305–320, 2009. [17] I. A. Rus, “Ulam stability of the operatorial equations,” in Functional Equations in Mathematical Analysis, T. M. Rassias and J. Brzdek, Eds., chapter 23, Springer, 2011. [18] A. I. Perov and A. V. Kibenko, “On a general method to study boundary value problems,” Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, vol. 30, pp. 249–264, 1966.
