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Stability of the second order partial differential equations
Journal of Inequalities and Applications volume 2011, Article number: 81 (2011)
Abstract
We say that a functional equation (ξ) is stable if any function g satisfying the functional equation (ξ) approximately is near to a true solution of (ξ).
In this paper, by using Banach's contraction principle, we prove the stability of nonlinear partial differential equations of the following forms:
2000 Mathematics Subject Classification. 26D10; 34K20; 39B52; 39B82; 46B99.
1. Introduction
Let X be a normed space over a scalar field , and let I be an open interval. Assume that, for any function f : I → X (y = f (x)) satisfying the differential inequality
for all t ∈ I, where ε ≥ 0, there exists a function f0 : I → X satisfying
and ||f (t) - f0 (t)|| ≤ K (ε) for any t ∈ I.
Then we say that the above differential equation has the Hyers-Ulam stability. If the above statement is also true, then we replace ε and K(ε) by φ(t) and ϕ(t), where φ, ϕ : I → [0, ∞) are functions not depending on f and f0 explicitly, then we say that the corresponding differential equation has the Hyers-Ulam-Rassias stability or the generalized Hyers-Ulam stability.
In 1998, the Hyers-Ulam stability of differential equation y' = y was first investigated by Alsina and Ger [1]. In 2002, this result has been generalized by Takahasi et al. [2] for the Banach space-valued differential equation y' = λy. In 2005, Jung [3] proved the generalized Hyers-Ulam stability of a linear differential equation of the first order. For more results on stability of differential equations, see also [4–7] and [8] and, for more details on the Hyers-Ulam stability and related topics, the readers refer to [9–17] and [18–20].
In this paper, we prove the Hyers-Ulam-Rassias stability of the following partial differential equations:
-
(1)
The first order nonlinear partial differential equation:
-
(2)
The first order nonlinear partial differential equation:
for all a, b ∈ ℝ;
-
(3)
The second order nonlinear partial differential equation:
(1.1)
under the following condition:
The differential equation (1.1) is the second order nonlinear partial differential equation, and we call it exact if the condition (1.2) holds.
-
(4)
The mixed type second order nonlinear partial differential equation:
under the following condition:
Theorem 1.1. (Banach's Contraction Principle) Let (X, d) be a complete matric space and T : X → X be a contraction, that is, there exists α ∈ [0,1) such that
for all x, y ∈ X. Then, there exists a unique a ∈ X such that Ta = a. Moreover, a = lim n →∞ Tn x and
for all x ∈ X.
2. Main results
In this section, let I = [a, b] be a closed interval with a < b and C (I × I) = {f : I × I → ℝ: f is continuous}. For the sake of convenience, assume that all the integrals and all the derivatives exist.
Theorem 2.1. Let c ∈ I φ : I × I → (0, ∞) be a continuous function, L : I × I → [1,∞) be an integrable function and K : I × I × ℝ → ℝ be a continuous function. Assume that there exists 0 < β < 1 such that
for all x, t ∈ I and u, v ∈ C (I × I). Let y : I × I → ℝ be such that
for all x, t ∈ I. Then, there exists a unique continuously differentiable function y0 : I × I → ℝ such that
(consequently, y0 is a solution to y x (x, t) = K(x, t, y(x,t))) and
for all x, t ∈ I.
Proof. Let X be the set of all continuously differentiable functions u : I × I → ℝ. We define a metric d and an operator T on X as follows, respectively:
and the operator
for all u ∈ X. Using (2.1) and (2.2), we have
Now, by Theorem 1.1, there exists a unique y0 ∈ X such that , that is,
Moreover, by Theorem 1.1, we have
for all y ∈ X. It follows from (2.3) that
for all x, t ∈ I. If we integrate each term in the above inequality from c to x, then we get
Now, we have
Thus, we get
Therefore, by (2.4) and (2.5), we see that
for all x, t ∈ I. This completes the proof. □
Theorem 2.2. Let c ∈ I, p, q : I × I → ℝ be continuous functions with p(x, t) ≠ 0 for all x, t ∈ I, φ : I × I → (0, ∞) be a continuous function, L : I × I → [1, ∞) be an integrable function, and f : I × I × ℝ → ℝ be a continuous function. Assume that there exists 0 < β < 1 such that
and
for all c, x, t ∈ I and h, u, v, y ∈ C (I × I). Let y : I × I → ℝ be a function such that:
for all x, t ∈ I and (1.2) holds. Then, there exists a unique solution y0 : I × I → ℝ of (1.1) such that
Proof. It follows from (1.2) and (2.7) that
Thus, we have
By using (2.8), we get
where
From (2.9), it follows that
From , without less of generality, we can assume that |p(x, t)| ≥ 1.
Now, By putting
in the above inequality, we get
Thus, the conclusions of the Theorem follows from Theorem 2.1. This completes the proof. □
If (1.1) is multiplied by a function μ(x, t) such that the resulting equation is exact, that is,
and
then we say that μ(x, t) is an integrating factor of the partial differential equation (1.1).
Corollary 2.3. Let p, q, μ : I × I → ℝ be continuous functions such that p(x, t) ≠ 0 and μ(x, t) ≠ 0 for all x, t ∈ I, and (2.10) holds. Assume that c ∈ I, L : I × I → [1, ∞) is an integrable function and f : I × I × ℝ → ℝ is a continuous function. Suppose that there exists 0 < β < 1 such that
and
for all c, x, t ∈ I and h, u, v ∈ C (I × I). Let y : I × I → ℝ be a function such that
for all x, t ∈ I and the condition (2.11) holds. Then, There exists a unique solution y0 : I × I → ℝ of (2.10) such that
Proof. It follows from Theorem 2.2 that
with
and
has the required properties. This completes the proof. □
Remark 2.4. In 2009, Jung [7] proved the Hyers-Ulam stability of linear partial differential equation of the first order of the following form:
for all a ≥ 0 and b > 0.
Now, we consider the generalization of this equation as follows:
for all a, b ∈ ℝ with a ≠ 0 and b ≠ 0. Let ζ and η be defined by
If we define , then, by (2.13), we have
Thus, we see that , and so we can rewrite the equation (2.12) as follows:
Now, we can use Theorem 2.1 for the generalized Hyers-Ulam stability of (2.14).
We consider the mixed type second order nonlinear partial differential equation:
Now, we prove the Hyers-Ulam-Rassias stability of (2.15) under the condition:
Theorem 2.5. Let c ∈ Ip, q : I × I → ℝ be continuous functions with p(x, t) ≠ 0 for all x, t ∈ I, φ : I × I → (0, ∞) be a continuous function, L : I × I → [1, ∞) be an integrable function, and f : I × I × ℝ → ℝ be a continuous function. Assume that there exists 0 < β < 1 such that
and
for all c, x, t ∈ I and h, y, u, v ∈ C (I × I). Let y : I × I → ℝ be a function such that
for all x, t ∈ I and the condition (2.16) holds. Then, there exists a unique solution y 0 : I × I → ℝ of (2.15) such that
Proof. By (2.17) and (2.16), we see that
Thus, we have
It follows from (2.18) that
where
From (2.19), we obtain
The rest of the proof is similar to that of Theorem 2.2. This completes the proof. □
Remark 2.6. We can define the integrating factor for the equation (2.15) and prove a corollary similar to Corollary 2.3 for Theorem 2.6.
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Acknowledgements
This work was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).
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AB carried out the molecular genetic studies, participated in the sequence alignment and drafted the manuscript. JY carried out the immunoassays. MT participated in the sequence alignment. ES participated in the design of the study and performed the statistical analysis. FG conceived of the study and participated in its design and coordination. All authors read and approved the final manuscript.
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Gordji, M.E., Cho, Y., Ghaemi, M. et al. Stability of the second order partial differential equations. J Inequal Appl 2011, 81 (2011). https://doi.org/10.1186/1029-242X-2011-81
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DOI: https://doi.org/10.1186/1029-242X-2011-81