Stability of the second order partial differential equations

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:

$$\left\{\begin{array}{c}{y}_x\left(x,t\right)=f\left(x,t,y\left(x,t\right)\right),\\ a{y}_x\left(x,t\right)+b{y}_t\left(x,t\right)=f\left(x,t,y\left(x,t\right)\right),\\ p\left(x,t\right)y_{xt}\left(x,t\right)+q\left(x,t\right)y_t\left(x,t\right)+p_t\left(x,t\right)y_x\left(x,t\right)-p_x\left(x,t\right)y_t\left(x,t\right)=f\left(x,t,y\left(x,t\right)\right),\\ p\left(x,t\right)y_{xx}\left(x,t\right)+q\left(x,t\right)y_x\left(x,t\right)=f\left(x,t,y\left(x,t\right)\right).\end{array}\right.$$

2000 Mathematics Subject Classification. 26D10; 34K20; 39B52; 39B82; 46B99.

1. Introduction

Let X be a normed space over a scalar field $K$, and let I be an open interval. Assume that, for any function f : I → X (y = f (x)) satisfying the differential inequality

$$\left|\right|a_n\left(t\right)y^{\left(n\right)}\left(t\right)+a_{n-1}\left(t\right)y^{\left(n-1\right)}\left(t\right)+\cdots +a_1\left(t\right)y^{\prime }\left(t\right)+a_0\left(t\right)y\left(t\right)+h\left(t\right)\left|\right|\le \epsilon$$

for all t ∈ I, where ε ≥ 0, there exists a function f₀ : I → X satisfying

$$\left\{\begin{array}{c}{y}_0=f_0\left(x\right),\\ a_n\left(t\right)y_0^{\left(n\right)}\left(t\right)+a_{n-1}\left(t\right)y_0^{\left(n-1\right)}\left(t\right)+\cdots +a_1\left(t\right){y^{\prime }}_0\left(t\right)+a_0\left(t\right)y_0\left(t\right)+h\left(t\right)=0\end{array}\right.$$

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:

  $$y_x\left(x,t\right)=f\left(x,t,y\left(x,t\right)\right);$$

• (2) The first order nonlinear partial differential equation:

  $$a{y}_x\left(x,t\right)+b{y}_t\left(x,t\right)=f\left(x,t,y\left(x,t\right)\right)$$

for all a, b ∈ ℝ;

• (3) The second order nonlinear partial differential equation:

  $$p\left(x,t\right)y_{xx}\left(x,t\right)+q\left(x,t\right)y_x\left(x,t\right)=f\left(x,t,y\left(x,t\right)\right)$$

  (1.1)

under the following condition:

$$p_{xx}\left(x,t\right)=q_x\left(x,t\right).$$

(1.2)

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:

  $$p\left(x,t\right)y_{xt}\left(x,t\right)+q\left(x,t\right)y_t\left(x,t\right)+p_t\left(x,t\right)y_x\left(x,t\right)-p_x\left(x,t\right)y_t\left(x,t\right)=f\left(x,t,y\left(x,t\right)\right)$$

under the following condition:

$$p_{xt}\left(x,t\right)=q_t\left(x,t\right).$$

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

$$\mathsf{\text{d}}\left(Tx,Ty\right)\le \alpha \mathsf{\text{d}}\left(x,y\right)$$

for all x, y ∈ X. Then, there exists a unique a ∈ X such that Ta = a. Moreover, a = lim _{n →∞} Tⁿ x and

$$\mathsf{\text{d}}\left(a,x\right)\le \frac{1}{1-\alpha }\mathsf{\text{d}}\left(x,Tx\right)$$

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

$$\underset{c}{\overset{x}{\int }}L\left(\tau ,t\right)\phi \left(\tau ,t\right)\mathsf{\text{d}}\tau <\beta \phi \left(x,t\right);$$

(2.1)

$$|K\left(x,t,u\left(x,t\right)\right)-K\left(x,t,v\left(x,t\right)\right)|\le L\left(x,t\right)|u\left(x,t\right)-v\left(x,t\right)|$$

(2.2)

for all x, t ∈ I and u, v ∈ C (I × I). Let y : I × I → ℝ be such that

$$|y_x\left(x,t\right)-K\left(x,t,y\left(x,t\right)\right)|\le \phi \left(x,t\right)$$

(2.3)

for all x, t ∈ I. Then, there exists a unique continuously differentiable function y₀ : I × I → ℝ such that

$$y_0\left(x,t\right)=y\left(c,t\right)+\underset{c}{\overset{x}{\int }}K\left(\tau ,t,y_0\left(\tau ,t\right)\right)\mathsf{\text{d}}\tau$$

(consequently, y₀ is a solution to y _x (x, t) = K(x, t, y(x,t))) and

$$|y\left(x,t\right)-y_0\left(x,t\right)|\le \frac{\beta }{1-\beta }\phi \left(x,t\right)$$

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:

$$\mathsf{\text{d}}\left(u,v\right)=\mathsf{\text{su}}{\mathsf{\text{p}}}_{x,t\in I}\frac{|u\left(x,t\right)-v\left(x,t\right)|}{\phi \left(x,t\right)}$$

and the operator

$$\left(Tu\right)\left(x,t\right)=y\left(c,t\right)+\underset{c}{\overset{x}{\int }}K\left(\tau ,t,u\left(\tau ,t\right)\right)\mathsf{\text{d}}\tau$$

for all u ∈ X. Using (2.1) and (2.2), we have

$$\begin{array}{ll}\hfill \mathsf{\text{d}}\left(Tu,Tv\right)& =\mathsf{\text{su}}{\mathsf{\text{p}}}_{x,t\in I}\frac{\left|{\int }_c^x\left[K\left(\tau ,t,u\left(\tau ,t\right)\right)-K\left(\tau ,t,v\left(\tau ,t\right)\right)\right]\mathsf{\text{d}}\tau \right|}{\phi \left(x,t\right)}\\ \le \mathsf{\text{su}}{\mathsf{\text{p}}}_{x,t\in I}\frac{{\int }_c^{x}L\left(\tau ,t\right)|u\left(\tau ,t\right)-v\left(\tau ,t\right)|\mathsf{\text{d}}\tau }{\phi \left(x,t\right)}\\ =\mathsf{\text{su}}{\mathsf{\text{p}}}_{x,t\in I}\left[\frac{{\int }_c^{x}L\left(\tau ,t\right)\phi \left(\tau ,t\right)\frac{|u\left(\tau ,t\right)-v\left(\tau ,t\right)|}{\phi \left(\tau ,t\right)}\mathsf{\text{d}}\tau }{\phi \left(x,t\right)}\right]\\ \le \mathsf{\text{su}}{\mathsf{\text{p}}}_{x,t\in I}\frac{{\int }_c^{x}L\left(\tau ,t\right)\phi \left(\tau ,t\right)\mathsf{\text{su}}{\mathsf{\text{p}}}_{\tau ,t\in I}\frac{|u\left(\tau ,t\right)-v\left(\tau ,t\right)|}{\phi \left(\tau ,t\right)}\mathsf{\text{d}}\tau }{\phi \left(x,t\right)}\\ =\mathsf{\text{d}}\left(u,v\right)\mathsf{\text{su}}{\mathsf{\text{p}}}_{x,t\in I}\frac{{\int }_c^{x}L\left(\tau ,t\right)\phi \left(\tau ,t\right)\mathsf{\text{d}}\tau }{\phi \left(\tau ,t\right)}\\ \le \beta \mathsf{\text{d}}\left(u,v\right).\end{array}$$

Now, by Theorem 1.1, there exists a unique y₀ ∈ X such that $T{y}_0=y_0$, that is,

$$y_0\left(x,t\right)=y\left(c,t\right)+\underset{c}{\overset{x}{\int }}K\left(\tau ,t,y_0\left(\tau ,t\right)\right)\mathsf{\text{d}}\tau .$$

Moreover, by Theorem 1.1, we have

$$\mathsf{\text{d}}\left(y_0,y\right)\le \frac{1}{1-\beta }\mathsf{\text{d}}\left(y,Ty\right)$$

(2.4)

for all y ∈ X. It follows from (2.3) that

$$-\phi \left(x,t\right)\le y_x\left(x,t\right)-K\left(x,t,y\left(x,t\right)\right)\le \phi \left(x,t\right)$$

for all x, t ∈ I. If we integrate each term in the above inequality from c to x, then we get

$$\begin{array}{c}\left|y(x,t)-\left(y(c,t)-{\displaystyle \underset{c}{\overset{x}{\int }}K}(\tau ,t,y(\tau ,t)\right)\text{d}\tau \right|\le {\displaystyle \underset{c}{\overset{x}{\int }}\phi }(\tau ,t)\text{d}\tau \\ \le {\displaystyle \underset{c}{\overset{x}{\int }}L}(\tau ,t)\phi (\tau ,t)\text{d}\tau \\ \le \beta \phi (x,t).\end{array}$$

Now, we have

$$\frac{|y\left(x,t\right)-\left(Ty\right)\left(x,t\right)|}{\phi \left(x,t\right)}\le \beta \Rightarrow \mathsf{\text{su}}{\mathsf{\text{p}}}_{x,t\in I}\frac{|y\left(x,t\right)-\left(Ty\right)\left(x,t\right)|}{\phi \left(x,t\right)}\le \beta .$$

Thus, we get

$$\mathsf{\text{d}}\left(y,Ty\right)\le \beta .$$

(2.5)

Therefore, by (2.4) and (2.5), we see that

$$|y\left(x,t\right)-y_0\left(x,t\right)|\le \frac{\beta }{1-\beta }\phi \left(x,t\right)$$

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

$$\begin{array}{c}\underset{c}{\overset{x}{\int }}L\left(\tau ,t\right)\phi \left(\tau ,t\right)\mathsf{\text{d}}\tau \le \beta \phi \left(x,t\right);\\ h\left(c,t\right)=-\left[p\left(c,t\right)y_x\left(c,t\right)-p_x\left(c,t\right)y\left(c,t\right)+q\left(c,t\right)y\left(c,t\right)\right];\\ K\left(x,t,y\left(x,t\right)\right)\\ =-{\left(p\left(x,t\right)\right)}^{-1}\left[\left(p_x\left(x,t\right)-q\left(x,t\right)\right)y\left(x,t\right)+h\left(c,t\right)-\underset{c}{\overset{x}{\int }}f(\tau ,t,y\left(\tau ,t\right)\mathsf{\text{d}}\tau \right]\end{array}$$

and

$$|K\left(x,t,u\left(x,t\right)\right)-K\left(x,t,v\left(x,t\right)\right)|\le L\left(x,t\right)|u\left(x,t\right)-v\left(x,t\right)|$$

for all c, x, t ∈ I and h, u, v, y ∈ C (I × I). Let y : I × I → ℝ be a function such that:

$$|p\left(x,t\right)y_{xx}\left(x,t\right)+q\left(x,t\right)y_x\left(x,t\right)-f\left(x,t,y\left(x,t\right)\right)|\le \phi \left(x,t\right)$$

(2.7)

for all x, t ∈ I and (1.2) holds. Then, there exists a unique solution y₀ : I × I → ℝ of (1.1) such that

$$|y\left(x,t\right)-y_0\left(x,t\right)|\le \frac{\beta }{1-\beta }\phi \left(x,t\right).$$

Proof. It follows from (1.2) and (2.7) that

$$\begin{array}{l}|p(x,t)y_{xx}(x,t)+q(x,t)y_x(x,t)-f(x,t,y(x,t))|\\ =|(p(x,t)y_x(x,t)-p_x(x,t)y(x,t){)}_x+(q(x,t)y(x,t){)}_x\\ +[p_{xx}(x,t)-q_x(x,t)]y(x,t)-f(x,t,y(x,t))|\\ =|(p(x,t)y_{xx}(x,t)-p_x(x,t)y(x,t){)}_x+(q(x,t)y(x,t){)}_x-f(x,t,y(x,t))|\\ \le \phi (x,t).\end{array}$$

Thus, we have

$$\begin{array}{c}-\phi \left(x,t\right)\\ \le {\left(p\left(x,t\right)y_{xx}\left(x,t\right)-p_x\left(x,t\right)y\left(x,t\right)\right)}_x+{\left(q\left(x,t\right)y\left(x,t\right)\right)}_x-f\left(x,t,y\left(x,t\right)\right)\\ \le \phi \left(x,t\right).\end{array}$$

(2.8)

By using (2.8), we get

$$\begin{array}{c}|p\left(x,t\right)y_x\left(x,t\right)-p_x\left(x,t\right)y\left(x,t\right)+q\left(x,t\right)y\left(x,t\right)+h\left(c,t\right)-\underset{c}{\overset{x}{\int }}f\left(\tau ,t,y\left(\tau ,t\right)\right)\mathsf{\text{d}}\tau |\\ =\left|p\left(x,t\right)\right|\left|y_x\left(x,t\right)+{\left(p\left(x,t\right)\right)}^{-1}\left(\left(q\left(x,t\right)-p_x\left(x,t\right)\right)y\left(x,t\right)+h\left(c,t\right)\right.\right.\\ -\left.\left.\underset{c}{\overset{x}{\int }}f\left(\tau ,t,y\left(\tau ,t\right)\right)\mathsf{\text{d}}\tau \right)\right|\\ \le \underset{c}{\overset{x}{\int }}\phi \left(\tau ,t\right)\mathsf{\text{d}}\tau ,\end{array}$$

(2.9)

where

$$h\left(c,t\right)=-\left[p\left(c,t\right)y_x\left(c,t\right)-p_x\left(c,t\right)y\left(c,t\right)+q\left(c,t\right)y\left(c,t\right)\right].$$

From (2.9), it follows that

$$\begin{array}{c}\left|y_x\left(x,t\right)+{\left(p\left(x,t\right)\right)}^{-1}\left(\left(q\left(x,t\right)-p_x\left(x,t\right)\right)y\left(x,t\right)+h\left(c,t\right)-\underset{c}{\overset{x}{\int }}f\left(\tau ,t,y\left(\tau ,t\right)\right)\mathsf{\text{d}}\tau \right)\right|\\ \le |p\left(x,t\right){|}^{-1}\underset{c}{\overset{x}{\int }}\phi \left(\tau ,t\right)\mathsf{\text{d}}\tau .\end{array}$$

From $p\left(x,t\right)=\frac{p\left(x,t\right)\left[1+{\left(p\left(x,t\right)\right)}^2\right]}{1+{\left(p\left(x,t\right)\right)}^2}$, without less of generality, we can assume that |p(x, t)| ≥ 1.

Now, By putting

$$\begin{array}{c}K\left(x,t,y\left(x,t\right)\right)\\ =-{\left(p\left(x,t\right)\right)}^{-1}\left[\left(p_x\left(x,t\right)-q\left(x,t\right)\right)y\left(x,t\right)+h\left(c,t\right)-\underset{c}{\overset{x}{\int }}f(\tau ,t,y(\tau ,t)\mathsf{\text{d}}\tau \right]\end{array}$$

in the above inequality, we get

$$\begin{array}{ll}\hfill |y_x\left(x,t\right)-K\left(x,t,y\left(x,t\right)\right)|& \le |p\left(x,t\right){|}^{-1}\underset{c}{\overset{x}{\int }}\phi \left(\tau ,t\right)\mathsf{\text{d}}\tau \\ \le \underset{c}{\overset{x}{\int }}\phi \left(\tau ,t\right)\mathsf{\text{d}}\tau \\ \le \underset{c}{\overset{x}{\int }}L\left(\tau ,t\right)\phi \left(\tau ,t\right)\mathsf{\text{d}}\tau \\ \le \beta \phi \left(x,t\right)\\ \le \phi \left(x,t\right).\end{array}$$

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,

$$\mu \left(x,t\right)\left[p\left(x,t\right)y_{xx}\left(x,t\right)+q\left(x,t\right)y_x-f\left(x,t,y\left(x,t\right)\right)\right]=0$$

(2.10)

and

$${\left(\mu \left(x,t\right)p\left(x,t\right)\right)}_{xx}-{\left(q\left(x,t\right)\mu \left(x,t\right)\right)}_x=0,$$

(2.11)

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

$$\begin{array}{c}\underset{c}{\overset{x}{\int }}L\left(x,t\right)\phi \left(\tau ,t\right)\mathsf{\text{d}}\tau \le \beta \phi \left(x,t\right);\\ h\left(c,t\right)=-\left[\mu \left(c,t\right)p\left(x,t\right)y_x\left(c,t\right)-{\left(\mu p\right)}_x\left(c,t\right)y\left(c,t\right)+\mu \left(c,t\right)q\left(c,t\right)y\left(c,t\right)\right];\\ K\left(x,t,y\left(x,t\right)\right)=-{\left(\mu \left(x,t\right)p\left(x,t\right)\right)}^{-1}\left[\left(\mu \left(x,t\right)q\left(x,t\right)-{\left(\mu q\right)}_x\left(x,t\right)\right)y\left(x,t\right)\right.\\ \left.+\left(h\left(c,t\right)-\underset{c}{\overset{x}{\int }}\mu \left(\tau ,t\right)f\left(\tau ,t,y\left(\tau ,t\right)\right)\mathsf{\text{d}}\tau \right)\right]\end{array}$$

and

$$|K\left(x,t,u\left(x,t\right)\right)-K\left(x,t,v\left(x,t\right)\right)|\le L\left(x,t\right)|u\left(x,t\right)-V\left(x,t\right)|.$$

for all c, x, t ∈ I and h, u, v ∈ C (I × I). Let y : I × I → ℝ be a function such that

$$\left|\mu \left(x,t\right)\right||p\left(x,t\right)y_{xx}\left(x,t\right)+q\left(x,t\right)y_x\left(x,t\right)-f\left(x,t,y\left(x,t\right)\right)|\le \phi \left(x,t\right)$$

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

$$|y\left(x,t\right)-y_0\left(x,t\right)|\le \frac{\beta }{1-\beta }\phi \left(x,t\right).$$

Proof. It follows from Theorem 2.2 that

$$y_0\left(x,t\right)=y\left(c,t\right)+\underset{c}{\overset{x}{\int }}K\left(\tau ,t,y\left(\tau ,t\right)\right)\mathsf{\text{d}}\tau$$

with

$$\begin{array}{c}K\left(x,t,y\left(x,t\right)\right)=-{\left(\mu \left(x,t\right)p\left(x,t\right)\right)}^{-1}\left[\left(\mu \left(x,t\right)q\left(x,t\right)-{\left(\mu q\right)}_x\left(x,t\right)\right)y\left(x,t\right)\right.\\ \left.+\left(h\left(c,t\right)-\underset{c}{\overset{x}{\int }}\mu \left(\tau ,t\right)f\left(\tau ,t,y\left(\tau ,t\right)\right)\mathsf{\text{d}}\tau \right)\right]\end{array}$$

and

$$h\left(c,t\right)=-\left[\mu \left(c,t\right)p\left(x,t\right)y_x\left(c,t\right)-{\left(\mu p\right)}_x\left(c,t\right)y\left(c,t\right)+\mu \left(c,t\right)q\left(c,t\right)y\left(c,t\right)\right]$$

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:

$$a{y}_x\left(x,t\right)+b{y}_t\left(x,t\right)+g\left(x\right)y\left(x,t\right)+h\left(x\right)=0$$

for all a ≥ 0 and b > 0.

Now, we consider the generalization of this equation as follows:

$$a{y}_x\left(x,t\right)+b{y}_t\left(x,t\right)=f\left(x,t,y\left(x,t\right)\right)$$

(2.12)

for all a, b ∈ ℝ with a ≠ 0 and b ≠ 0. Let ζ and η be defined by

$$\zeta =x-\frac{a}{b}t,\eta =\frac{1}{b}t.$$

(2.13)

If we define $\tilde{y}\left(\zeta ,\eta \right)=y\left(\zeta +a\eta ,b\eta \right)=y\left(x,t\right)$, then, by (2.13), we have

$$\begin{array}{c}{y}_x\left(x,t\right)={\tilde{y}}_{\zeta }\left(\zeta ,\eta \right)\frac{\partial \zeta }{\partial x}+\tilde{y}\left(\zeta ,\eta \right)\frac{\partial \eta }{\partial x},\\ y_t\left(x,t\right)={\tilde{y}}_{\zeta }\left(\zeta ,\eta \right)\frac{\partial \zeta }{\partial t}+{\tilde{y}}_{\eta }\left(\zeta ,\eta \right)\frac{\partial \eta }{\partial t}=-\frac{a}{b}{\tilde{y}}_{\zeta }\left(\zeta ,\eta \right)+\frac{1}{b}{\tilde{y}}_{\eta }\left(\zeta ,\eta \right).\end{array}$$

Thus, we see that $a{y}_x\left(x,t\right)+b{y}_t\left(x,t\right)={\tilde{y}}_{\eta }\left(\zeta ,\eta \right)$, and so we can rewrite the equation (2.12) as follows:

$${\tilde{y}}_{\eta }\left(\zeta ,\eta \right)=\tilde{f}\left(\zeta ,\eta ,{\tilde{y}}_{\eta }\left(\zeta ,\eta \right)\right).$$

(2.14)

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:

$$\begin{array}{c}p\left(x,t\right)y_{xt}\left(x,t\right)+q\left(x,t\right)y_t\left(x,t\right)+p_t\left(x,t\right)y_x\left(x,t\right)-p_x\left(x,t\right)y_t\left(x,t\right)\\ =f\left(x,t,y\left(x,t\right)\right).\end{array}$$

(2.15)

Now, we prove the Hyers-Ulam-Rassias stability of (2.15) under the condition:

$$p_{xt}\left(x,t\right)=q_t\left(x,t\right)$$

(2.16)

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

$$\begin{array}{l}{\displaystyle \underset{c}{\overset{x}{\int }}L}(\tau ,t)\phi (\tau ,t)\text{d}\tau \le \beta \phi (x,t);\\ h(x,c)=-[p(x,c)y_x(x,c)-p_x(x,c)y(x,c)+q(x,c)y(x,c)];\\ K(x,t,y(x,t))\\ =-(p(x,t){)}^{-1}\left[(p_x(x,t)-q(x,t))y(x,t)+h(x,c)-{\displaystyle \underset{c}{\overset{t}{\int }}f}(x,\tau ,y(x,\tau )\text{d}\tau \right]\end{array}$$

and

$$|K\left(x,t,u\left(x,t\right)\right)-K\left(x,t,v\left(x,t\right)\right)|\le L\left(x,t\right)|u\left(x,t\right)-v\left(x,t\right)|$$

for all c, x, t ∈ I and h, y, u, v ∈ C (I × I). Let y : I × I → ℝ be a function such that

$$\begin{array}{c}|p\left(x,t\right)y_{xt}\left(x,t\right)+q\left(x,t\right)y_t\left(x,t\right)+p_t\left(x,t\right)y_x\left(x,t\right)-p_x\left(x,t\right)y_t\left(x,t\right)-f\left(x,t,y\left(x,t\right)\right)|\\ \le \phi \left(x,t\right)\end{array}$$

(2.17)

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

$$|y\left(x,t\right)-y_0\left(x,t\right)|\le \frac{\beta }{1-\beta }\phi \left(x,t\right).$$

Proof. By (2.17) and (2.16), we see that

$$\begin{array}{l}|p(x,t)y_{xt}(x,t)+q(x,t)y_t(x,t)+p_t(x,t)y_x(x,t)-p_x(x,t)y_t(x,t)-f(x,t,y(x,t))|\\ =|(p(x,t)y_x(x,t)-p_x(x,t)y(x,t)+q(x,t)y(x,t){)}_t\\ +[p_{xt}(x,t)-q_t(x,t)]y(x,t)-f(x,t,y(x,t))|\\ =|(p(x,t)y_x(x,t)-p_x(x,t)y(x,t)+q(x,t)y(x,t){)}_t-f(x,t,y(x,t))|\end{array}$$

Thus, we have

$$\begin{array}{c}-\phi \left(x,t\right)\\ \le {\left(p\left(x,t\right)y_x\left(x,t\right)-p_x\left(x,t\right)y\left(x,t\right)+q\left(x,t\right)y\left(x,t\right)\right)}_t-f\left(x,t,y\left(x,t\right)\right)\\ \le \phi \left(x,t\right).\end{array}$$

(2.18)

It follows from (2.18) that

$$\begin{array}{c}\left|p\left(x,t\right)y_x\left(x,t\right)-p_x\left(x,t\right)y\left(x,t\right)+q\left(x,t\right)y\left(x,t\right)+h\left(c,t\right)-\underset{c}{\overset{t}{\int }}f\left(x,\tau ,y\left(x,\tau \right)\right)\mathsf{\text{d}}\tau \right|\\ =|p\left(x,t\right){|}^{-1}\left|y_x\left(x,t\right)+{\left(p\left(x,t\right)\right)}^{-1}\left(\left(q\left(x,t\right)-p_x\left(x,t\right)\right)y\left(x,t\right)+h\left(c,t\right)\right.\right.\\ \left.-\left.\underset{c}{\overset{t}{\int }}f\left(x,\tau ,y\left(x,\tau \right)\right)\mathsf{\text{d}}\tau \right)\right|\\ \le \underset{c}{\overset{t}{\int }}\phi \left(x,\tau \right)\mathsf{\text{d}}\tau ,\end{array}$$

(2.19)

where

$$h\left(x,c\right)=-\left[p\left(x,c\right)y_x\left(x,c\right)-p_x\left(x,c\right)y\left(x,c\right)+q\left(x,c\right)y\left(x,c\right)\right].$$

From (2.19), we obtain

$$\begin{array}{c}\left|y_x\left(x,t\right)+{\left(p\left(x,t\right)\right)}^{-1}\left(\left(q\left(x,t\right)-p_x\left(x,t\right)\right)y\left(x,t\right)+h\left(c,t\right)-\underset{c}{\overset{t}{\int }}f\left(x,\tau ,y\left(x,\tau \right)\right)\mathsf{\text{d}}\tau \right)\right|\\ \le |p\left(x,t\right){|}^{-1}\underset{c}{\overset{t}{\int }}\phi \left(x,\tau \right)\mathsf{\text{d}}\tau .\end{array}$$

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.