Hindawi Advances in Mathematical Physics Volume 2017, Article ID 9692685, 6 pages https://doi.org/10.1155/2017/9692685
Research Article Hyers-Ulam Stability and Existence of Solutions for Nigmatullinβs Fractional Diffusion Equation Zhuoyan Gao1 and JinRong Wang2 1
College of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan, Shanxi 030031, China Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China
2
Correspondence should be addressed to JinRong Wang;
[email protected] Received 2 December 2016; Accepted 26 January 2017; Published 21 February 2017 Academic Editor: Ming Mei Copyright Β© 2017 Zhuoyan Gao and JinRong Wang. 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 discuss stability of time-fractional order heat conduction equations and prove the Hyers-Ulam and generalized Hyers-UlamRassias stability of time-fractional order heat conduction equations via fractional Green function involving Wright function. In addition, an interesting existence result for solution is given.
πΏ
1. Introduction Motivated by a famous question of Ulam concerning the stability of group homomorphisms, Hyers and Rassias introduced the concepts of Hyers-Ulam and Hyers-Ulam-Rassias stability, respectively, in the case of the Cauchy functional equation in Banach spaces, which received a great influence in the development of the generalized Hyers-Ulam-Rassias stability of all kinds of functional equations. There are many interesting results on this topic in the case of functional equations, ordinary differential equations, partial differential equations, and impulsive differential equations; see, for example, [1β11] and the recent survey [12, 13]. Recently, Hegyi and Jung [14] presented the generalized Hyers-Ulam-Rassias stability of the classical Laplaceβs equation Ξπ’ = 0 in the class of spherically symmetric functions via harmonic functions method. Meanwhile, the same topic of fractional evolution equations via functional analysis methods has attracted attention of researchers. However, to the best of our knowledge, stability of fractional partial differential equations via direct analysis methods has not been discussed yet. In this paper, we study the stability of Nigmatullinβs timefractional order diffusion equation (see [15, Chapter 6]) πΏ
πΌ
π·0+,π‘ π’ (π₯, π‘) = π
2π
2
π’ (π₯, π‘) , π₯ β R, π‘ > 0, π > 0, ππ₯2
πΌβ1
π·0+,π‘ π’ (π₯, 0) = π (π₯) ,
lim π’ (π₯, π‘) = 0,
π‘βΒ±β
(1) and existence of solution to nonlinear problem πΏ
πΌ
π·0+,π‘ π’ (π₯, π‘) = π
2π
2
π’ (π₯, π‘) + π (π₯, π‘, π’ (π₯, π‘)) , ππ₯2 π₯ β R, π‘ > 0, π > 0, (2)
πΏ
πΌβ1
π·0+,π‘ π’ (π₯, 0) = π (π₯) ,
lim π’ (π₯, π‘) = 0,
π‘βΒ±β
where π is a continuous function on R and π will be πΌ assumed to satisfy certain conditions and the symbol πΏ π·0+,π‘ denotes the Riemann-Liouville time-fractional derivatives of the order πΌ β (0, 1] (see [15, Chapter 6, p.349, (6.1.12)]) πΏ
πΌ
π·0+,π‘ π’ (π₯, π‘) = (
π‘ 1 π’ (π₯, π ) π [πΌ]+1 ππ , ) β« ππ‘ Ξ (1 β {πΌ}) 0 (π‘ β π ){πΌ}
(3)
π₯ β R, π‘ > 0, and [πΌ] and {πΌ} denote the integral and fractional parts of πΌ and Ξ(β
) is the Euler-Gamma function.
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2. Preliminaries
and its solution is given by
The two parameter Mittag-Leffler functions πΈπΌ,π½ (π§) are π defined πΈπΌ,π½ (π§) = ββ π=0 (π§ /Ξ(ππΌ + π½)), π§ β R, and πΌ, π½ are positive real numbers. Next, πΈπΌ (π§) = πΈπΌ,1 (π§). The solvability of (1) has been reported in [15, Chapter 6.2.1]. Here we collect the following result.
π’ (π₯, π‘) = β«
ββ
π’ (π₯, π‘) = β«
ββ
πΌ
πΊ (π₯ β π, π‘) π (π) ππ,
πΊ1 (π₯, π‘) =
=
(4)
|π’ (π₯, π‘)| β€ π,
1 β πΌβ1 β« π‘ πΈπΌ,πΌ (βπ2 π¦2 π‘πΌ ) cos (π¦π₯) ππ¦ π 0 (5)
β«
+β
ββ
+β
πΊ1 (π₯ β π, π‘) ππ = β«
ββ +β
βπΌ π§π 1 π (πΌ, π½; π§) = β = β« π βπ½ ππ +π§π ππ , 2ππ π!Ξ (πΌπ + π½) π»π π=0
=β«
ββ
provided that the integral in the right-hand side of (15) is convergent, where π»π denotes the Hankel path of integration in the complex π -plane. Note that [17, Lemma 2] for any π > 0 and π‘ > 0, πΈπΌ,πΌ (βπ‘πΌ π) β€ 1/Ξ(πΌ). Then, we have the following estimation. Lemma 2. For any π‘ > 0, π₯ β R, πΊπΌ (π₯, π‘) β€ π‘πΌ /Ξ(πΌ), and (6)
where +β
β«
ββ
σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨π (π)σ΅¨σ΅¨σ΅¨ ππ fl ππ .
π π2 π’ (π₯, π‘) , π₯ β R, π‘ > 0, π’ (π₯, π‘) = π2 ππ‘ ππ₯2 π’ (π₯, 0) = π (π₯) ,
β
(9)
(13) πβπ₯ ). 2πβπ‘
1/πΌ
β
1/πΌ
Lemma 5. Let π(πΌ, π) = β«0 π1/πΌ πβπ ππ/ sin(ππΌ)ππΌπ4 . Then π (πΌ, π) π‘ cos (π¦π₯) ππ¦. β« ππ‘1+πΌ 0 π¦4 +β
ββ
Remark 4. If πΌ = 1, then (1) becomes a classical heat conduction equation
1 βπ2 π ππ = 1, βπ
Μ (πΌ, π) = β«0 π1/πΌ πβπ ππ/ sin(ππΌ)ππΌπ§2 . Μ (πΌ, π§)/π‘πΌ+1 , where π π For more asymptotic expansions on Mittag-Leffler functions, one can refer to [19, Lemmas 2.2, 2.3, 2.4]. Next, we give asymptotic property of πΊπΌ (π₯, π‘).
π’ (π₯, π‘) = β«
(8)
1 β1/2 β|π₯βπ|2 /4π2 π‘ π‘ π ππ 2πβπ
Note that [18, Lemma 3], for all π‘ > 0, |π‘πΌβ1 EπΌ,πΌ (βπ§π‘πΌ )| β€
(7)
Remark 3. Obviously, (7) can be fulfilled. For example, π(π) = 2 πβ(πβπ) , π β (π, π). Moreover, we can obtain 1 +β ππ¦π₯ π‘πΌ πΏ (π₯) , πΏ (π₯) = β« π ππ¦. Ξ (πΌ) 2π ββ
(12)
(π =
πΊπΌ (π₯, π‘) β€
Obviously, we have the following remarks.
πΊπΌ (π₯, π‘) β€
(11)
where we use the fact that
β
σ΅¨σ΅¨ ππ π‘πΌ σ΅¨σ΅¨ +β σ΅¨ σ΅¨σ΅¨ πΌ , σ΅¨σ΅¨β« πΊ (π₯ β π, π‘) π (π) ππσ΅¨σ΅¨σ΅¨ β€ σ΅¨σ΅¨ πΞ (πΌ) σ΅¨σ΅¨ ββ
(10)
If |π(π₯)| β€ π, then (4) satisfies the following inequality:
1 (πΌ/2)β1 πΌ πΌ |π₯| π (β , ; β π‘βπΌ/2 ) π‘ 2π 2 2 π
1 +β ππ¦π₯ πΌβ1 = β« π π‘ πΈπΌ,πΌ (βπ2 π¦2 π‘πΌ ) ππ¦, 2π ββ
1 β1/2 β|π₯|2 /4π2 π‘ π‘ π , 2πβπ
2 2 1 1 |π₯| π (β , ; β π‘β1/2 ) = πβ|π₯| /4π π‘ . 2 2 π
where fractional Green function πΊπΌ (π₯, π‘) involving Wright function π(πΌ, π½; π§) is given by πΊπΌ (π₯, π‘) =
πΊ1 (π₯ β π, π‘) π (π) ππ,
where
Lemma 1 (see [15, Corollary 6.1] or [16, (4.19)]). Equation (1) is solvable, and its solution has the form +β
+β
(14)
πΊπΌ (π₯ β π, π‘) π (π) ππ
π (πΌ, π) +β π‘ cos (π¦ (π₯ β π)) ππ¦π (π) ππ. β€ β« β« ππ‘1+πΌ ββ 0 π¦4
(15)
3. Fractional Duhamelβs Principle, Stability Concepts, and Remarks The standard Duhamel principle adopts the idea form ODEs in studying Cauchy problem for inhomogeneous partial differential equations by linking Cauchy problem for corresponding homogeneous equation. In this section, we establish a fractional Duhamel principle which helps us to study Ulamβs stability of (1) and existence of solution to (2).
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Lemma 6. Let πΉ(π₯, π‘) be jointly continuous on R Γ (0, +β). The solution of Cauchy problem for inhomogeneous partial differential equations of the type πΏ
πΌ π·0,π‘ π¦ (π₯, π‘)
=π
2π
2
π¦ (π₯, π‘) + πΉ (π₯, π‘) , ππ₯2
πΌβ1
π·0,π‘ π¦ (π₯, 0) = π (π₯) ,
π¦ (π₯, π‘) = β«
ββ
+β
0
ββ
+β« β«
(17) πΊπΌ (π₯ β π, π‘ β π) πΉ (π, π) ππ ππ.
Proof. Using the superposition principle, the following Cauchy problem (16) can be decomposed into two Cauchy problems: πΏ
πΏ
πΌ
π·0,π‘ π¦1 (π₯, π‘) = π
2π
2
π¦1 (π₯, π‘) , π₯ β R, π‘ > 0, ππ₯2
πΌ
2π
2
Definition 7. Equation (1) is Hyers-Ulam stable if there exists a number π > 0 such that for each solution π¦ of inequality (23) there exists a solution π’ of (1) with σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨π¦ (π₯, π‘) β π’ (π₯, π‘)σ΅¨σ΅¨σ΅¨ β€ ππ,
(ii) πΏ π·0+,π‘ π¦(π₯, π‘) = π2 (π2 π¦(π₯, π‘)/ππ₯2 ) + πΉ(π₯, π‘). Remark 9. If π¦(π₯, π‘) is a solution of inequality (23), then π¦ is a solution of the following integral inequality:
π¦2 (π₯, π‘) + πΉ (π₯, π‘) , ππ₯2
σ΅¨σ΅¨ σ΅¨σ΅¨ +β σ΅¨ σ΅¨σ΅¨ πΌ σ΅¨σ΅¨π¦ (π₯, π‘) β β« πΊ (π₯ β π, π‘) π (π) ππσ΅¨σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨ ββ
π¦1 (π₯, π‘) = β«
ββ
πΌ
πΊ (π₯ β π, π‘) π (π) ππ.
(20)
By virtue of homogeneous theorem the solution of (19) can be written as π¦2 (π₯, π‘) = β« π (π₯, π‘; π) ππ, 0
(21)
πΏ
πΌβ1 π·0,π‘ π (π₯, π)
ββ
2π
2
π (π₯, π‘) , π₯ β R, π‘ > π, ππ₯2
πΊ (π₯ β π, π‘ β π) ππ ππ.
π‘
+β
0
ββ
(27)
(π‘ β π)πΌ σ΅¨σ΅¨ σ΅¨ σ΅¨πΉ (π, π)σ΅¨σ΅¨σ΅¨ ππ ππ. πΞ (πΌ) σ΅¨
Let π(π₯, π‘) : R Γ (0, +β) β [0, +β) be a nonnegative function.
where π = π(π₯, π‘; π) is the solution of πΌ
0
(26)
πΌ
σ΅¨σ΅¨ σ΅¨σ΅¨ +β σ΅¨ σ΅¨σ΅¨ πΌ σ΅¨σ΅¨π¦ (π₯, π‘) β β« πΊ (π₯ β π, π‘) π (π) ππσ΅¨σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨ ββ σ΅¨σ΅¨ π‘ +β σ΅¨σ΅¨ σ΅¨ σ΅¨ β€ σ΅¨σ΅¨σ΅¨β« β« πΊπΌ (π₯ β π, π‘ β π) πΉ (π, π) ππ ππσ΅¨σ΅¨σ΅¨ σ΅¨σ΅¨ 0 ββ σ΅¨σ΅¨ β€ πβ« β«
π‘
π·0,π‘ π (π₯, π‘) = π
+β
By Remark 8 and (17), we get
By Lemma 1, the solution of (18) is
πΏ
π‘
β€ πβ« β«
πΌβ1
π·0,π‘ π¦2 (π₯, 0) = 0.
+β
(25)
Remark 8. A function π¦(π₯, π‘) is a solution of inequality (23) if and only if there is πΉ(π₯, π‘) such that
π₯ β R, π‘ > 0, (19) πΏ
π₯ β R, π‘ > 0.
πΌ
(18)
πΌβ1
π·0,π‘ π¦2 (π₯, π‘) = π
(24)
(i) |πΉ(π₯, π‘)| β€ π, π₯ β R, π‘ > 0;
π·0,π‘ π¦1 (π₯, 0) = π (π₯) , πΏ
σ΅¨σ΅¨ σ΅¨ π2 π¦ (π₯, π‘) σ΅¨σ΅¨σ΅¨ σ΅¨σ΅¨πΏ πΌ σ΅¨σ΅¨ π·0+,π‘ π¦ (π₯, π‘) β π2 σ΅¨σ΅¨ β€ πΜ π (π‘) , 2 σ΅¨σ΅¨ σ΅¨σ΅¨ ππ₯ σ΅¨ σ΅¨
Now we are ready to introduce the Hyers-Ulam and generalized Hyers-Ulam-Rassias stability concepts for (1).
πΊπΌ (π₯ β π, π‘) π (π) ππ
π‘
(23)
π₯ β R, π‘ > 0, π > 0, 0 < πΌ β€ 1.
is +β
σ΅¨ σ΅¨σ΅¨ π2 π¦ (π₯, π‘) σ΅¨σ΅¨σ΅¨ σ΅¨σ΅¨πΏ πΌ σ΅¨σ΅¨ β€ π, σ΅¨σ΅¨ π·0+,π‘ π¦ (π₯, π‘) β π2 σ΅¨σ΅¨ ππ₯2 σ΅¨σ΅¨σ΅¨ σ΅¨ π₯ β R, π‘ > 0, π > 0, 0 < πΌ β€ 1,
π₯ β R, π‘ > 0, (16) πΏ
Consider (1) and the following two inequalities:
(22)
= πΉ (π₯, π) .
By virtue of homogeneous theorem, Lemma 1, and (21) we obtain π¦(π₯, π‘) = π¦1 (π₯, π‘) + π¦2 (π₯, π‘) which is the desired result. Μ (π‘) : R β [0, +β) be a nonnegative and Let π > 0 and π increasing function.
Definition 10. Equation (1) is generalized Hyers-UlamRassias stable if there exists a number π > 0 such that for each solution π¦ of inequality (24) there exists a solution π’ of (1) with σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨π¦ (π₯, π‘) β π’ (π₯, π‘)σ΅¨σ΅¨σ΅¨ β€ ππ (π₯, π‘) , π₯ β R, π‘ > 0.
(28)
Remark 11. A function π¦(π₯, π‘) is a solution of inequality (24) if and only if there is πΉ(π₯, π‘) such that Μ (π‘), π₯ β R, π‘ > 0; (i) |πΉ(π₯, π‘)| β€ π πΌ
(ii) πΏ π·0+,π‘ π¦(π₯, π‘) = π2 (π2 π¦(π₯, π‘)/ππ₯2 ) + πΉ(π₯, π‘).
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Remark 12. If π¦(π₯, π‘) is a solution of inequality (24), then π¦ is a solution of the following integral inequality: +β σ΅¨σ΅¨ σ΅¨σ΅¨σ΅¨ σ΅¨σ΅¨π¦ (π₯, π‘) β β« πΊπΌ (π₯ β π, π‘) π (π) ππσ΅¨σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨ ββ σ΅¨ σ΅¨ π‘
β€β« β« 0
+β
ββ
σ΅¨ Μ σ΅¨σ΅¨ πΊπΌ (π₯ β π, π‘ β π) σ΅¨σ΅¨σ΅¨π (π)σ΅¨σ΅¨ ππ ππ
π‘
+β
0
ββ
β€ πβ« β«
(29)
Theorem 15. Assume that there exists π > 0 such that β«
+β
ββ
π’ (π₯, π‘) = β«
(π₯,π‘)βRΓ(0,+β) 0
ββ
σ΅¨ σ΅¨σ΅¨ πΌ σ΅¨σ΅¨πΊ (π₯ β π, π‘ β π)σ΅¨σ΅¨σ΅¨ ππ ππ = π
σ΅¨σ΅¨ σ΅¨σ΅¨ +β σ΅¨σ΅¨ σ΅¨ πΌ σ΅¨σ΅¨π¦ (π₯, π‘) β β« πΊ (π₯ β π, π‘) π (π) ππσ΅¨σ΅¨σ΅¨ σ΅¨σ΅¨ ββ σ΅¨σ΅¨ (30)
Then (1) is Hyers-Ulam stable on finite time interval [0, π] with respect to π and π. Proof. Let π¦(π₯, π‘) be a solution of inequality (23) and π’(π₯, π‘) the solution of Cauchy problem (1), and its expression is π’ (π₯, π‘) = β«
ββ
πΌ
πΊ (π₯ β π, π‘) π (π) ππ.
Keeping in mind (26), we have σ΅¨σ΅¨ σ΅¨σ΅¨ +β σ΅¨ σ΅¨σ΅¨ πΌ σ΅¨σ΅¨π¦ (π₯, π‘) β β« πΊ (π₯ β π, π‘) π (π) ππσ΅¨σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨ ββ π‘
+β
0
ββ
β€ πβ« β«
σ΅¨ σ΅¨σ΅¨ πΌ σ΅¨σ΅¨πΊ (π₯ β π, π‘ β π)σ΅¨σ΅¨σ΅¨ ππ ππ.
By (30) we obtain σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨π¦ (π₯, π‘) β π’ (π₯, π‘)σ΅¨σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨ +β σ΅¨ σ΅¨ = σ΅¨σ΅¨σ΅¨π¦ (π₯, π‘) β β« πΊπΌ (π₯ β π, π‘) π (π) ππσ΅¨σ΅¨σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨ ββ π‘
+β
0
ββ
β€ πβ« β«
(32)
π‘
+β
(π₯,π‘)β(0,π)Γ(0,π] 0
ββ
β« β«
σ΅¨ σ΅¨σ΅¨ 1 σ΅¨σ΅¨πΊ (π₯ β π, π‘ β π)σ΅¨σ΅¨σ΅¨ ππ ππ σ΅¨ σ΅¨
= π, where we use (13) in Remark 4.
0
ββ
(38)
πΌ
Μ (π) ππ ππ. πΊ (π₯ β π, π‘ β π) π
Hence, we have π‘ +β (π‘ β π)πΌ σ΅¨σ΅¨ σ΅¨ σ΅¨σ΅¨ Μ (π)σ΅¨σ΅¨σ΅¨σ΅¨ ππ ππ σ΅¨σ΅¨π σ΅¨σ΅¨π¦ (π₯, π‘) β π’ (π₯, π‘)σ΅¨σ΅¨σ΅¨ β€ β« β« 0 ββ πΞ (πΌ)
ππ‘πΌ+1 β€ fl ππ (π₯, π‘) , π (πΌ + 1) Ξ (πΌ)
(39)
which implies that (1) is Hyers-Ulam-Rassias stable with respect to π(π₯, π‘) fl π‘πΌ+1 and π fl π/π(πΌ + 1)Ξ(πΌ). The proof is completed.
π‘
+β
0
ββ
β« β«
σ΅¨σ΅¨ πΌ Μ (π)σ΅¨σ΅¨σ΅¨σ΅¨ ππ ππ β€ ππ (π₯, π‘) , σ΅¨σ΅¨πΊ (π₯ β π, π‘ β π) π
(40)
for each π₯ β R, π‘ > 0. (33)
which is reasonable due to the fact that (see [15, (6.3.7)]) (Fπ₯ Lπ‘ πΊπΌ ) (π₯, π‘) =
Example 14. Set π > 0, πΌ = 1, π₯ β (0, π), and π‘ β (0, π]. Then the following heat conduction equation
sup
(37)
Remark 16. Condition (36) can be changed to
which implies that (1) is Hyers-Ulam stable with respect to π and π. The proof is completed.
ππ,1 =
+β
(31)
σ΅¨ σ΅¨σ΅¨ πΌ σ΅¨σ΅¨πΊ (π₯ β π, π‘ β π)σ΅¨σ΅¨σ΅¨ ππ ππ = ππ,
π π2 π’ (π₯, π‘) , π₯ β (0, π) , π‘ β (0, π] , π’ (π₯, π‘) = π2 ππ‘ ππ₯2 is Ulam-Hyers stable with respect to π and
π‘
β€β« β«
< +β.
+β
πΊπΌ (π₯ β π, π‘) π (π) ππ.
Keeping in mind (29), we have
Theorem 13. Assume that there exists ππ,πΌ > 0 such that β« β«
+β
ββ
In this section, we present the stability results.
sup
(36)
Then (1) is generalized Hyers-Ulam-Rassias stable with respect to π(π‘) and π.
4. Hyers-Ulam Stability
+β
for each π₯ β R, π‘ > 0.
Proof. Let π¦(π₯, π‘) be a solution of inequality (24) and π’(π₯, π‘) the solution of Cauchy problem (1), and its expression is
π (π‘ β π)πΌ σ΅¨σ΅¨ Μ (π)σ΅¨σ΅¨σ΅¨σ΅¨ ππ ππ. σ΅¨π πΞ (πΌ) σ΅¨
π‘
σ΅¨σ΅¨ Μ σ΅¨σ΅¨ σ΅¨σ΅¨π (π)σ΅¨σ΅¨ ππ ππ β€ π,
(34)
π‘πΌ
π‘πΌ Μ ) (π₯) , (Fπ₯ π + π2 |π₯|2 π₯ β R, π‘ > 0,
where the symbols Fπ₯ and Lπ‘ denote the Fourier transform and Laplace transform, respectively. Μ (π‘) = |π‘|V , V =ΜΈ 0, one has (see [15, (1.3.55)]) Note that, for π Μ ) (π₯) = 2V+1 βπ (Fπ₯ π
(35)
(41)
Ξ ((V + 1) /2) βVβ1 |π₯| Ξ (βV/2)
Vπ = β2 sin ( ) Ξ (V + 1) |π₯|βVβ1 . 2
(42)
Then we can choose π(π₯, π‘) = |π₯|βVβ1 π‘πΌ /(π‘πΌ + π2 |π₯|2 ) and π = 2|Ξ(V + 1)|.
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Μ (π‘) = π‘2 . Then the following heat Example 17. Set πΌ = 1 and π conduction equation π2 π’ (π₯, π‘) π , π’ (π₯, π‘) = π2 ππ‘ ππ₯2
π₯ β R, π‘ > 0,
(43)
5. Existence of Solution Let B be the Banach space of all bounded continuous function π’(π₯, π‘) on R with fixed π‘ equipped with the norm Μ = βπ’(β
, π‘)βB = supπ₯βR |π’(π₯, π‘)|. Consider the Banach space B πΆ([0, π‘0 ], B) of all bounded continuous function defined on [0, π] equipped with the norm βπ’(β
, π‘)βBΜ = ΜπΌ = supπ‘β[0,π] |π’(β
, π‘)|. Next, we introduce Banach space B 1βπΌ {π’ : π‘ π’(β
, π‘) β πΆ([0, π‘0 ], B)} equipped with the norm βπ’βBΜ πΌ = supπ‘β[0,π] π‘1βπΌ |π’(β
, π‘)|. We introduce the following assumptions: [H1]: π(π₯, π‘, π’(π₯, π‘)) is jointly continuous on R Γ [0, π] Γ R with π(π₯, π‘, 0) = 0. [H2]: For any π > 0, there exists a constant πΏ > 0 such that
β€ πΏ βπ’ (β
, π‘) β V (β
, π‘)βB
(44)
for all π‘ β [0, π] and all π’, V β B with βπ’(β
, π‘)βB β€ π and βV(β
, π‘)βB β€ π. Define σ΅©σ΅© σ΅©σ΅© +β ΜπΌ : σ΅©σ΅©σ΅©π’ (β
, β
) β β« πΊπΌ (β
β π, β
) π (π) ππσ΅©σ΅©σ΅© Ξ₯ = {π’ β B σ΅©σ΅© Μ σ΅©σ΅© ββ σ΅©BπΌ σ΅©
β€β«
(45)
σ΅© σ΅© β€ σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅©B , π‘ β [0, π‘π ]} ,
Observing the definition of Ξ₯ and triangular inequality, one has 1 σ΅© σ΅© ) σ΅©σ΅©πσ΅©σ΅© . (49) βπ’βBΜ πΌ β€ (1 + Ξ (πΌ) σ΅© σ΅©B 0,
Next, we show that Q : Ξ₯ β Ξ₯. By [H2] with π(π₯, π‘, 0) = σ΅©σ΅© σ΅©σ΅© +β σ΅© σ΅©σ΅© πΌ σ΅©σ΅©(Qπ’) (β
, π‘) β β« πΊ (β
β π, π‘) π (π) ππσ΅©σ΅©σ΅© σ΅©σ΅©B ββ σ΅©σ΅© β€β«
(π‘ β π)πΌβ1 σ΅©σ΅© σ΅© σ΅©π (β
, π, π’ (β
, π))σ΅©σ΅©σ΅©B ππ Ξ (πΌ) σ΅©
π‘ πΏ σ΅© σ΅© β€ β« (π‘ β π)πΌβ1 ππ σ΅©σ΅©σ΅©π’ (β
, π)σ΅©σ΅©σ΅©B Ξ (πΌ) 0
=
Multiplying π‘1βπΌ to the above inequality and taking the supremum, we have σ΅©σ΅© σ΅©σ΅© +β σ΅© σ΅©σ΅© πΌ σ΅©σ΅©Qπ’ β β« πΊ (β
β π, β
) π (π) ππσ΅©σ΅©σ΅© ββ σ΅©σ΅©BΜ πΌ σ΅©σ΅© (51) πΏπ‘ 1 β€ βπ’ (β
, β
)βBΜ β€ βπ’ (β
, β
)βBΜ < βπ’ (β
, β
)βBΜ . Ξ (πΌ + 1) 2 Now we check that Q is contraction. Observe that β(Qπ’) (β
, π‘) β (QV) (β
, π‘)βB π‘
0
πΏ (π‘ β π)πΌβ1 βπ’ (β
, π) β V (β
, π)βB ππ Ξ (πΌ)
Q (π’) (π₯, π‘)
πΏπ‘πΌ βπ’ (β
, π‘) β V (β
, π‘)βB . Ξ (πΌ + 1)
βQπ’ β QVβBΜ πΌ β€
π‘
+β
0
ββ
πΊπΌ (π₯ β π, π‘ β π) π (π, π, π’ (π, π)) ππ ππ.
(52)
Multiplying π‘1βπΌ to the above inequality and taking the supremum, we have
ΜπΌ given by Proof. Define Q : Ξ₯ β B
+β« β«
(50)
πΏπ‘πΌ σ΅©σ΅© σ΅© σ΅©π’ (β
, π)σ΅©σ΅©σ΅©B . Ξ (πΌ + 1) σ΅©
β€
ββ
π‘
0
Theorem 18. Assume that [H1] and [H2] hold. Then (2) has unique solution in Ξ₯ and βπ’βBΜ πΌ β€ (1 + 1/Ξ(πΌ))βπβB .
πΊπΌ (π₯ β π, π‘) π (π) ππ
(47)
Multiplying π‘1βπΌ to the above inequality and taking the supremum, we have σ΅©σ΅© σ΅©σ΅©σ΅© +β πΌ σ΅©σ΅©β« πΊ (β
β π, β
) π (π) ππσ΅©σ΅©σ΅© β€ 1 σ΅©σ΅©σ΅©πσ΅©σ΅©σ΅© Μ . (48) σ΅©σ΅© ββ σ΅©σ΅© Μ σ΅© σ΅©B σ΅© σ΅©BπΌ Ξ (πΌ)
β€β«
+β
π‘πΌ σ΅©σ΅© σ΅© σ΅©πΏ (β
β π) π (π)σ΅©σ΅©σ΅©B ππ Ξ (πΌ) σ΅©
π‘πΌ σ΅©σ΅© σ΅© σ΅©π (π)σ΅©σ΅©σ΅©B . Ξ (πΌ) σ΅©
=
where π‘π = min{Ξ(πΌ + 1)/2πΏ, (Ξ(πΌ + 1)/2πΏ)1/πΌ }.
=β«
+β
ββ
is Hyers-Ulam-Rassias stable with respect to π(π₯, π‘) = |π₯|β3 π‘2 /(π‘2 + π2 |π₯|2 ) and π = 2, where we use (13) in Remark 4 again.
σ΅© σ΅©σ΅© σ΅©σ΅©π (β
, π‘, π’ (β
, π‘)) β π (β
, π‘, V (β
, π‘))σ΅©σ΅©σ΅©B
Note that σ΅©σ΅© σ΅©σ΅© +β σ΅© σ΅©σ΅© πΌ σ΅©σ΅©β« πΊ (β
β π, π‘) π (π) ππσ΅©σ΅©σ΅© σ΅©σ΅©B σ΅©σ΅© ββ
(46)
πΏπ‘πΌ βπ’ (β
, β
) β V (β
, β
)βBΜ πΌ Ξ (πΌ + 1)
(53) 1 β€ βπ’ (β
, β
) β V (β
, β
)βBΜ πΌ . 2 By contraction mapping principle, Q has a unique fixed point in Ξ₯ which is the solution of (2).
6
Competing Interests The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments This work is supported by National Natural Science Foundation of China (11661016), Training Object of High Level and Innovative Talents of Guizhou Province ((2016)4006), Unite Foundation of Guizhou Province ([2015]7640), and Graduate Course of Guizhou University (ZDKC[2015]003).
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