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Lyapunov type inequalities for mixed nonlinear Riemann–Liouville fractional differential equations with a forcing term
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Lyapunov type inequalities for mixed nonlinear Riemann–Liouville fractional differential equations with a forcing term Ravi P. Agarwal a , Abdullah Özbekler b,a,∗ a
Department of Mathematics, Texas A&M University-Kingsville, 700 University Blvd., Kingsville, TX 78363-8202, USA
b
Department of Mathematics, Atilim University 06836, Incek, Ankara, Turkey
article
info
abstract
Article history: Received 4 February 2016 Received in revised form 26 September 2016
In this paper, we present some new Lyapunov and Hartman type inequalities for Riemann–Liouville fractional differential equations of the form
MSC: 34A08 34A40 34C10
where p, q, f are real-valued functions and 0 < γ < 1 < µ < 2. No sign restrictions are imposed on the potential functions p, q and the forcing term f . The inequalities obtained generalize and compliment the existing results for the special cases of this equation in the literature. © 2016 Elsevier B.V. All rights reserved.
Keywords: Lyapunov type inequality Sub-linear Super-linear Forced Fractional Riemann–Liouville
(a Dα x)(t ) + p(t ) | x(t ) |µ−1 x(t ) + q(t ) | x(t ) |γ −1 x(t ) = f (t ),
1. Introduction We recall Lyapunov inequalities for Hill’s equation x′′ (t ) + ν(t )x(t ) = 0,
(1.1)
where ν(t ) ∈ L1 [a, b] is a real-valued function. If x(t ) is a nontrivial solution of Eq. (1.1) having two consecutive zeros at a and b, where a, b ∈ R with a < b, then the inequality b
|ν(t )|dt > a
4
(1.2)
b−a
holds. This striking inequality was first proved by Lyapunov [1] and is known as ‘‘Lyapunov inequality’’. Later Wintner [2] and thereafter some authors achieved to replace the function ‘‘|ν(t )|’’ in Ineq. (1.2) by the function ‘‘ν + (t )’’ i.e. they obtained the following inequality: b
ν + (t )dt > a
∗
4 b−a
,
Corresponding author at: Department of Mathematics, Atilim University 06836, Incek, Ankara, Turkey. E-mail addresses: [email protected] (R.P. Agarwal), [email protected], [email protected] (A. Özbekler).
http://dx.doi.org/10.1016/j.cam.2016.10.009 0377-0427/© 2016 Elsevier B.V. All rights reserved.
(1.3)
2
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where ν + (t ) = max{ν(t ), 0}, and the constant 4 on the right hand side of Ineq. (1.3) (and Ineq. (1.2)) is the best possible largest number (see [1] and [3, Thm. 5.1]). In [3], Hartman obtained a more general inequality than both (1.2) and (1.3): b
(b − t )(t − a)ν + (t )dt > b − a.
(1.4)
a
Since (b − t )(t − a) ≤ (b − a)2 /4 for all t ∈ (a, b), Ineq. (1.4) implies Ineq. (1.3). The Lyapunov inequality and its generalizations have been used successfully in connection with oscillation and Sturmian theory, asymptotic theory, disconjugacy, eigenvalue problems and various properties of the solutions of (1.1) and related equations, see for instance [2–20] and the references cited therein. For some extensions to Hamiltonian systems, higher order differential equations, nonlinear and half-linear differential equations, difference and dynamic equations, functional and impulsive differential equations, we refer in particular to [21–23,7,8,24–41]. In this work, we obtain Lyapunov type inequalities for the Riemann–Liouville fractional forced nonlinear differential equations of order α ∈ (0, 2]
(a Dα x)(t ) + p(t )|x(t )|µ−1 x(t ) + q(t )|x(t )|γ −1 x(t ) = f (t ),
(1.5)
subject to Dirichlet (2-point) boundary conditions x(a) = x(b) = 0,
(1.6)
where p, q, f ∈ C[t0 , ∞) and the nonlinearities satisfy 0 < γ < 1 < µ < 2. Moreover, no sign restrictions are imposed on the potentials p and q, and the forcing term f . It is clear that there are two special cases of Eq. (1.5); one is the forced sub-linear (p(t ) = 0) fractional equation
(a Dα x)(t ) + q(t )|x(t )|γ −1 x(t ) = f (t );
0 < γ < 1,
(1.7)
and the other is the forced super-linear (q(t ) = 0) fractional equation
(a Dα x)(t ) + p(t )|x(t )|µ−1 x(t ) = f (t );
1 < µ < 2,
(1.8)
If α = 2 and f (t ) = 0, letting γ → 1− (or µ → 1+ ) in Eq. (1.7) (or Eq. (1.8)) results in Hill’s equation (1.1) with ν(t ) = q(t ) (or ν(t ) = p(t )), and hence, our results extend the classical Hartman [3] and Lyapunov [1] results. Classical Lyapunov Ineq. (1.2) for Hill’s equation (1.1) proved to be very useful in various problems related with differential equations and, since then, many improvements and generalizations of Ineq. (1.2) have appeared in the literature. To the best of our knowledge, among these generalizations, the first result in which a fractional derivative is used instead of the (classical) ordinary derivative in Hill’s equation (1.1) is the publication of Ferreira [42] in 2013. Since in recent years fractional calculus has become an important branch of mathematics (for an excellent introduction, see Kilbas et al. [43]), and finds wide variety of real world applications, our work fills the gap in the literature. As far as the Lyapunov type inequalities for fractional differential equations are considered, there are some new works, see for example [44–49]. In 2015, O’Regan and Samet [44] were concerned with the problem of finding new Lyapunov type inequalities for fractional boundary value problem
(a Dα x)(t ) + q(t )x(t ) = 0;
a < t < b, 3 < α ≤ 4 x(a) = x′ (a) = x(b) = x′ (b) = 0. In the same year, Jleli et al. [48] considered the Caputo fractional differential equation under Robin boundary conditions
(Ca Dα x)(t ) + q(t )x(t ) = 0;
a < t < b, 1 < α ≤ 2
x(a) − x′ (a) = x(b) + x′ (b) = 0, and they got a corresponding Lyapunov type inequality. Later Rong and Bai [45] concerned with the Caputo fractional differential equation under boundary condition involving the Caputo fractional derivative
(Ca Dα x)(t ) + q(t )x(t ) = 0; x ( a) = 0 ,
C β aD x
(
a < t < b, 1 < α ≤ 2
)(b) = 0,
where 0 < β ≤ 1. As far as the nonlinear equations are concerned, it seems there is only a single work [50] in the literature. Chidouh and Torres [46] focused on the Riemann–Liouville nonlinear fractional boundary value problem
(a Dα x)(t ) + q(t )g (x(t )) = 0; x(a) = x(b) = 0,
a < t < b, 1 < α ≤ 2
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where g ∈ C(R, R) is a concave and nondecreasing function. However, there is hardly any result for forced nonlinear fractional equations especially for forced sub/super-linear fractional differential equations. The first and probably the only work in this direction will be our paper. In particular, our work compliments and improves the work of Ferreira [42]. We begin with the concept of fractional integral and derivative of order α ≥ 0. Definition 1.1. Let α ≥ 0 and f be a real function defined on [a, b]. The Riemann–Liouville fractional integral of order α is defined by (a I 0 f )(t ) = f (t ) and
(a I α f )(t ) =
1
Γ (α)
t
(t − s)α−1 f (s)ds
(1.9)
a
for t ∈ [a, b], where α is a positive constant. Definition 1.2. The Riemann–Liouville fractional derivative of order α ≥ 0 is defined by
(a D0 f )(t ) = f (t ) and m−α (a Dα f )(t ) = (Dm f )(t ) a I
for α > 0, where m is the smallest integer greater or equal than α . These differential operators of arbitrary order are nonlocal and this feature has been used to model several real world phenomena, typically by substituting an ordinary derivative by a fractional one in a differential equation (see e.g. [51]). Ferreira considered the following Riemann–Liouville linear fractional differential equation of order α ∈ (0, 2]
(a Dα x)(t ) + ν(t )x(t ) = 0
(1.10)
satisfying the Dirichlet boundary conditions (1.6) where a < b and ν ∈ C[a, b]. To obtain an inequality similar to Ineq. (1.2), he wrote Prb. (1.10)–(1.6) as an equivalent integral equation and then used some properties of its Green’s function (which was constructed in [52] for a = 0 and b = 1). We state Ferreira’s results in the following: Lemma 1.1. x(t ) is a solution of the boundary value Prb. (1.10)–(1.6) if and only if x(t ) satisfies the integral equation x(t ) =
b
G(t , s)ν(s)x(s)ds,
(1.11)
a
where
(t − a)(b − s) α−1 − (t − s)α−1 1 b−a G(t , s) = (t − a)(b − s) α−1 Γ (α) b−a
if
a ≤ s ≤ t ≤ b; (1.12)
if
a≤t ≤s≤b
is the Green’s function of Prb. (1.10)–(1.6). Lemma 1.2. The Green’s function G(t , s) given in (1.12) of Prb. (1.10)–(1.6) satisfies the following properties: (i) G(t , s) ≥ 0 for all a ≤ t, s ≤ b. (ii) maxt ∈[a,b] G(t , s) = G(s, s), s ∈ [a, b]. (iii) G(s, s) has a unique maximum, given by max G(s, s) = G
s∈[a,b]
a+b a+b 2
,
2
=
1
Γ (α)
b−a 4
α−1
.
(1.13)
Theorem 1.3 (Lyapunov Type Inequality). Let x(t ) be a nontrivial solution of Eq. (1.10) satisfying the Dirichlet boundary conditions (1.6) where a < b. If x(t ) ̸= 0 in (a, b), then the inequality b
|ν(t )|dt > Γ (α) a
holds.
4 b−a
α−1 (1.14)
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Note that for α = 2 Ineq. (1.14) reduces to Lyapunov’s classical Ineq. (1.2) for Hill’s equation (1.1). When α = 2, Eq. (1.5) reduces to the forced nonlinear differential equation of the form x′′ (t ) + p(t )|x(t )|µ−1 x(t ) + q(t )|x(t )|γ −1 x(t ) = f (t ).
(1.15)
Very recently, present authors [53] obtained the following Hartman and Lyapunov type inequalities for Prb. (1.15)–(1.6). Theorem 1.4 (Hartman Type Inequality). Let x(t ) be a nontrivial solution of Eq. (1.15) satisfying the Dirichlet boundary conditions (1.6). If x(t ) ̸= 0 in (a, b), then the inequality b
(b − t )(t − a)(p + q )(t )dt +
+
b
(b − t )(t − a) µ0 p+ (t ) + γ0 q+ (t ) + |f (t )| dt
a
a
>
(b − a)2
(1.16)
4
holds, where
µ0 = (2 − µ)µµ/(2−µ) 22/(µ−2) > 0
(1.17)
γ0 = (2 − γ )γ γ /(2−γ ) 22/(γ −2) > 0.
(1.18)
and
Theorem 1.5 (Lyapunov Type Inequality). Let x(t ) be a nontrivial solution of Eq. (1.15) satisfying the Dirichlet boundary conditions (1.6). If x(t ) ̸= 0 in (a, b), then the inequality b
(p+ + q+ )(t )dt
b
a
µ0 p+ (t ) + γ0 q+ (t ) + |f (t )| dt
>
a
4
(1.19)
(b − a)2
holds, where the constants µ0 and γ0 are the same as in (1.17) and (1.18). We remark that the Lyapunov type inequalities have been studied by many authors, see for instance the survey paper [54] and the references therein, but to the best of our knowledge there is no result in the literature for the nonlinear fractional differential equations of the form (1.5); in fact, even for its particular cases, i.e. for Eqs. (1.7) and (1.8). 2. Main results In this section we state and prove our results. Throughout this section we shall denote by u± = max{±u, 0}. In what follows we will need the following lemma, see [50, Lemma 2.1] (see also [53]). We include its proof here for completeness. Lemma 2.1. If A is positive, and B, z are nonnegative, then Az 2 − Bz α + (2 − α)α α/(2−α) 22/(α−2) A−α/(2−α) B2/(2−α) ≥ 0
(2.1)
for any α ∈ (0, 2) with equality holding if and only if B = z = 0. Proof. Let
F (z ) = Az 2 − Bz α ,
z≥0
(2.2)
where A > 0 and B ≥ 0. Clearly, when z = 0 or B = 0, (2.1) is obvious. On the other hand, if B > 0, then it is easy to see that F attains its minimum at z0 = (α A−1 B/2)1/(2−α) and
Fmin = −(2 − α)α α/(2−α) 22/(α−2) A−α/(2−α) B2/(2−α) . Thus, (2.1) holds. Note that if B > 0, then Ineq. (2.1) is strict.
We are now in the position to state and prove our first main result. Theorem 2.1 (Hartman Type Inequality). Let x(t ) be a nontrivial solution of Eq. (1.5) satisfying the Dirichlet boundary conditions (1.6). If x(t ) > 0 in (a, b), then the inequality b
[(b − t )(t − a)]α−1 p+ (t ) + q+ (t ) dt
a
>
b
[(b − t )(t − a)]α−1 µ0 p+ (t ) + γ0 q+ (t ) + f − (t ) dt
a
Γ 2 (α) 4
(b − a)2α−2
holds, where the constants µ0 and γ0 are the same as in (1.17) and (1.18).
(2.3)
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Proof. First, we note that the solution of the Riemann–Liouville linear fractional differential equation of order α ∈ (0, 2]
− (a Dα z )(t ) = H (t )
(2.4)
satisfying the Dirichlet boundary conditions (1.6) can be represented by z (t ) =
b
G(t , s)H (s)ds,
(2.5)
a
where G(t , s) is the Green’s function of Prb. (2.4)–(1.6) given as in (1.12). It is clear from Lemma 1.2(ii) that
(b − s)(s − a) 0 ≤ G(t , s) ≤ G(s, s) = b−a
α−1 (2.6)
for all (t , s) ∈ [a, b] × [a, b]. Now let x(t ) be a positive solution of Eq. (1.5) with x(a) = x(b) = 0, where a < b are consecutive zeros. Then using (1.5), (2.4) and (2.5), x(t ) can be expressed as x(t ) =
b
G(t , s) p(s)xµ (s) + q(s)xγ (s) − f (s) ds.
(2.7)
a
Let x(c ) = maxt ∈(a,b) x(t ). Then by (2.6) and (2.7), we have x(c ) =
b
G(c , s) p(s)xµ (s) + q(s)xγ (s) − f (s) ds
a b
G(s, s) p+ (s)xµ (s) + q+ (s)xγ (s) + f − (s) ds
≤
a
=
b
1
Γ (α)(b − a)α−1
[(b − s)(s − a)]α−1 p+ (s)xµ (s) + q+ (s)xγ (s) + f − (s) ds
a
= P0 xµ (c ) + Q0 xγ (c ) + F0 ,
(2.8)
where P0 =
Γ (α)(b − a)α−1
Q0 =
b
1
b
1
[(b − s)(s − a)]α−1 p+ (s)ds,
(2.9)
a
Γ (α)(b − a)α−1
[(b − s)(s − a)]α−1 q+ (s)ds
(2.10)
a
and F0 =
b
1
Γ (α)(b − a)α−1
[(b − s)(s − a)]α−1 f − (s)ds.
a
On the other hand, Ineq. (2.1) in Lemma 2.1 with A = B = 1, implies that xµ (c ) < x2 (c ) + µ0
and xγ (c ) < x2 (c ) + γ0 .
Using these inequalities and Ineq. (2.8) we find the following quadratic inequality:
(P0 + Q0 )x2 (c ) − x(c ) + µ0 P0 + γ0 Q0 + F0 > 0.
(2.11)
But this is only possible when
(P0 + Q0 )(µ0 P0 + γ0 Q0 + F0 ) > 1/4, which is the same as (2.3). This completes the proof of Theorem 2.1.
Theorem 2.2 (Hartman Type Inequality). Let x(t ) be a nontrivial solution of Eq. (1.5) satisfying the Dirichlet boundary conditions (1.6). If x(t ) < 0 in (a, b), then the inequality b
[(b − t )(t − a)]
α−1
a
>
p (t ) + q (t ) dt
+
+
b
[(b − t )(t − a)]
α−1
µ0 p (t ) + γ0 q (t ) + f (t ) dt +
+
+
a
Γ 2 (α) 4
(b − a)2α−2
holds, where the constants µ0 and γ0 are the same as in (1.17) and (1.18).
(2.12)
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Proof. Let x(t ) be a negative solution of Eq. (1.5) with x(a) = x(b) = 0, where a < b are consecutive zeros. In fact, if x(t ) < 0 for t ∈ (a, b), then we can consider −x(t ) as a positive solution of
(a Dα x)(t ) + p(t )|x(t )|β−1 x(t ) + q(t )|x(t )|γ −1 x(t ) = −f (t ).
(2.13)
Then using (1.5), (2.4) and (2.5), x(t ) can be expressed as x(t ) =
b
G(t , s) p(s)xβ (s) + q(s)xγ (s) + f (s) ds.
(2.14)
a
Let x(c ) = maxt ∈(a,b) x(t ). Then by (2.6) and (2.14), we have x(c ) =
b
G(c , s) p(s)xµ (s) + q(s)xγ (s) + f (s) ds
a b
G(s, s) p+ (s)xµ (s) + q+ (s)xγ (s) + f + (s) ds
≤ a
=
(b −
b
1 a)α−1
[(b − s)(s − a)]α−1 p+ (s)xµ (s) + q+ (s)xγ (s) + f + (s) ds
a
µ
= P0 x (c ) + Q0 xγ (c ) + F0 ,
(2.15)
where P0 and Q0 are defined in (2.9) and (2.10), and
1
F0 =
(b − a)α−1
b
[(b − s)(s − a)]α−1 f + (s)ds.
a
Now repeating the same steps as in Theorem 2.1, we obtain (2.12) which completes the proof of Theorem 2.2.
From Theorems 2.1 and 2.2, the next result immediately follows. Theorem 2.3 (Hartman Type Inequality). Let x(t ) be a nontrivial solution of Eq. (1.5) satisfying the Dirichlet boundary conditions (1.6). If x(t ) ̸= 0 in (a, b), then the inequality b
[(b − t )(t − a)]
α−1
p (t ) + q (t ) dt
+
+
b
a
[(b − t )(t − a)]
α−1
µ0 p+ (t ) + γ0 q+ (t ) + |f (t )| dt
a
Γ 2 (α)
>
4
(b − a)2α−2
(2.16)
holds, where the constants µ0 and γ0 are the same as in (1.17) and (1.18). Proof. Let x(t ) be a nontrivial solution of Eq. (1.5) with x(a) = x(b) = 0, where a < b are consecutive zeros. Since either x(t ) > 0 or x(t ) < 0 for t ∈ (a, b) and f ± (t ) ≤ |f (t )|, by (2.3) and (2.12) we obtain (2.16). This completes the proof of Theorem 2.3. Remark 1. When α = 2, Theorem 2.3 coincides with [53, Theorem. 2.3], i.e. Theorem 1.4. Corollary 2.4 (Disconjugacy). If b
[(b − t )(t − a)]
α−1
p (t ) + q (t ) dt
+
+
a b
[(b − t )(t − a)]
×
α−1
µ0 p+ (t ) + γ0 q+ (t ) + |f (t )| dt
≤
Γ 2 (α)
a
4
(b − a)2α−2
then Eq. (1.5) is disconjugate in [a, b], where the constants µ0 and γ0 are the same as in (1.17) and (1.18). Ineq. (1.13) yields that Ineq.’s (2.3), (2.12) and (2.16) in Theorems 2.1–2.3, immediately imply the following Lyapunov type inequalities for Eq. (1.5). Theorem 2.5 (Lyapunov Type Inequality). Let x(t ) be a nontrivial solution of Eq. (1.5) satisfying the Dirichlet boundary conditions (1.6). If x(t ) > 0 in (a, b), then the inequality b
p+ (t ) + q+ (t ) dt
a
b
µ0 p+ (t ) + γ0 q+ (t ) + f − (t ) dt
a
holds, where the constants µ0 and γ0 are the same as in (1.17) and (1.18).
>
42α−3 Γ 2 (α)
(b − a)2α−2
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Theorem 2.6 (Lyapunov Type Inequality). Let x(t ) be a nontrivial solution of Eq. (1.5) satisfying the Dirichlet boundary conditions (1.6). If x(t ) < 0 in (a, b), then the inequality b
p+ (t ) + q+ (t ) dt
b
µ0 p+ (t ) + γ0 q+ (t ) + f + (t ) dt
>
a
a
42α−3 Γ 2 (α)
(b − a)2α−2
holds, where the constants µ0 and γ0 are the same as in (1.17) and (1.18). Theorem 2.7 (Lyapunov Type Inequality). Let x(t ) be a nontrivial solution of Eq. (1.5) satisfying the Dirichlet boundary conditions (1.6). If x(t ) ̸= 0 in (a, b), then the inequality b
p+ (t ) + q+ (t ) dt
b
µ0 p+ (t ) + γ0 q+ (t ) + |f (t )| dt
>
a
a
42α−3 Γ 2 (α)
(b − a)2α−2
holds, where the constants µ0 and γ0 are the same as in (1.17) and (1.18). Remark 2. When α = 2, then Theorem 2.7 coincides with [53, Theorem 2.4], i.e. Theorem 1.5. Corollary 2.8 (Disconjugacy). If b
p+ (t ) + q+ (t ) dt
b
a
µ0 p+ (t ) + γ0 q+ (t ) + |f (t )| dt
≤
a
42α−3 Γ 2 (α)
(b − a)2α−2
then Eq. (1.5) is disconjugate in [a, b], where the constants µ0 and γ0 are the same as in (1.17) and (1.18). Next we present two new results where Eq. (1.5) has two special type of nonlinearities, namely forced sub-linear or forced super-linear, i.e., p(t ) ≡ 0 or q(t ) ≡ 0, respectively. Theorem 2.9. Let x(t ) be a nontrivial solution of Eq. (1.7) satisfying the Dirichlet boundary conditions (1.6). If x(t ) ̸= 0 in (a, b), then the following hold: (i) Hartman type inequality; b
[(b − t )(t − a)]α−1 q+ (t )dt
a
b
[(b − t )(t − a)]α−1 γ0 q+ (t ) + |f (t )| dt
>
a
Γ 2 (α) 4
(b − a)2α−2 .
(ii) Lyapunov type inequality; b
q+ (t )dt
b
a
+ γ0 q (t ) + |f (t )| dt
>
a
42α−3 Γ 2 (α)
(b − a)2α−2
,
where γ ∈ (0, 1) and the constant γ0 is the same as in (1.18). Theorem 2.10. Let x(t ) be a nontrivial solution of Eq. (1.8) satisfying the Dirichlet boundary conditions (1.6). If x(t ) ̸= 0 in (a, b), then the following hold: (i) Hartman type inequality; b
[(b − t )(t − a)]α−1 p+ (t )dt
a
b
[(b − t )(t − a)]α−1 µ0 p+ (t ) + |f (t )| dt
a
>
Γ 2 (α) 4
(b − a)2α−2 .
(ii) Lyapunov type inequality; b
p+ (t )dt a
b
µ0 p+ (t ) + |f (t )| dt
a
>
42α−3 Γ 2 (α)
(b − a)2α−2
,
where µ ∈ (1, 2) and the constant µ0 is the same as in (1.17). When γ → 1− (or µ → 1+ ), Eq. (1.7) (or Eq. (1.8)) reduces to forced Riemann–Liouville linear fractional differential equation of order α ∈ (0, 2]
(a Dα x)(t ) + ν(t )x(t ) = f (t ), where ν(t ) = q(t ) (or ν(t ) = p(t )). Since lim µ0 = lim γ0 = 1/4,
µ→1+
γ →1−
we have the following result from Theorems 2.9 and 2.10.
(2.17)
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Corollary 2.11. Let x(t ) be a nontrivial solution of Eq. (2.17) satisfying the Dirichlet boundary conditions (1.6). If x(t ) ̸= 0 in (a, b), then the following hold: (i) Hartman type inequality; b
[(b − t )(t − a)]
α−1 +
ν (t )dt
b
α−1
[(b − t )(t − a)]
ν (t ) + 4|f (t )| dt +
> Γ 2 (α)(b − a)2α−2 .
(2.18)
a
a
(ii) Lyapunov type inequality; b
ν (t )dt +
b
+ ν (t ) + 4|f (t )| dt
a
a
>
42α−2 Γ 2 (α)
(b − a)2α−2
.
(2.19)
Remark 3. In the case when x(t ) > 0 or x(t ) < 0 for t ∈ (a, b), Hartman and Lyapunov type inequalities for forced sub-linear equations (1.7), forced super-linear equation (1.8) and forced linear equation (2.17) can be formulated by replacing the term ‘‘|f (t )|’’ by ‘‘f − (t )’’ (‘‘f + (t )’’) in Theorems 2.9 and 2.10 and Corollary 2.11. Remark 4. When f (t ) ≡ 0, Ineq. (2.19) in Corollary 2.11 reduces to b
ν + (t )dt > Γ (α)
4
α−1 (2.20)
b−a
a
which is sharper than Ineq. (1.14) given by Ferreira [42] in Theorem 1.3 since |ν| ≥ ν + . Moreover, when f (t ) ≡ 0 and α = 2, Ineq.’s (2.18) and (2.19) coincide with the classical Lyapunov and Hartman inequalities, i.e. Ineq.’s (1.4) and (1.3). Example 2.12. For the fractional equation
(0 D3/2 x)(t ) + k1 |x(t )|1/2 x(t ) + k2 |x(t )|−1/2 x(t ) = k3 ;
t ≥0
(2.21)
where kj , j = 1, 2, 3, are real constants with k1 , k2 > 0, if the solution x(t ) has consecutive zeros at 0 and b > 0, then in view of (2.16) the following inequality must be satisfied −8/3
(k1 + k2 )(2 3 k1 + 2 −8 3
b
[(b − t )t ]
3k2 + |k3 |)
1/2
0
2 dt
>
1 4
Γ 2 (3/2)b.
(2.22)
Since the integral b
1 (b − t )tdt = π b2 8
0
√ and Γ (3/2) = π /2, Ineq. (2.22) turns out b>
1
−8/3
π (k1 + k2 )(2 3 k1 + 2 7
4
−8 3
−1/3 3k2 + |k3 |) .
(2.23)
Thus Eq. (2.21) is disconjugate on [0, b] if
b≤
1 4
−1/3 π 7 (k1 + k2 )(2−8 33 k1 + 2−8/3 3k2 + |k3 |) .
On the other hand, if we use the Lyapunov type inequality given in Theorem 2.7, then we obtain the inequality b>
4
π
−1/3 (k1 + k2 )(2−8 33 k1 + 2−8/3 3k2 + |k3 |) .
(2.24)
We note that Ineq. (2.24) is better than Ineq. (2.23) and hence we may conclude that Eq. (2.21) is disconjugate on [0, b] if
b≤
4
π
(k1 + k2 )(2−8 33 k1 + 2−8/3 3k2 + |k3 |)
−1/3
.
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3. Concluding remarks We conclude this paper with the following remark. The results obtained in this paper for Eq. (1.5) can be easily extended to the Riemann–Liouville fractional forced nonlinear differential equations of order α ∈ (0, 2] with positive and negative coefficients
(a Dα x)(t ) ± p(t )|x(t )|µ−1 x(t ) ∓ q(t )|x(t )|γ −1 x(t ) = f (t ) or more generally; equations of the form
(a Dα x)(t ) +
n
qk (t )|x(t )|σk −1 x(t ) = f (t )
k =1
where 0 < σ1 < · · · < σm < 1 < σm+1 < · · · < σn < 2 with no sign restrictions on the potential functions qk , k = 1, . . . , n, and the forcing term f . The formulation of these results is left to the reader. It will be of interest to find similar results for the mixed nonlinear equations of the form Eq. (1.5) or Riemann–Liouville fractional forced super-linear equation of order α ∈ (0, 2]
(a Dα x)(t ) + p(t )|x(t )|µ−1 x(t ) = f (t ) for µ ∈ [2, ∞). Acknowledgments This work was carried out when the second author was on academic leave, visiting TAMUK (Texas A&M UniversityKingsville) and he wishes to thank TAMUK. This work is partially supported by TUBITAK (2219) (The Scientific and Technological Research Council of Turkey). References [1] A.M. Liapunov, Probleme général de la stabilité du mouvement, (French Translation of a Russian paper dated 1893), Ann. Fac. Sci. Univ. Toulouse 2 (1907) 27–247. Reprinted as Ann. Math. Studies, No. 17, Princeton, 1947. [2] A. Wintner, On the nonexistence of conjugate points, Amer. J. Math. 73 (1951) 368–380. [3] P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964, and Birkhuser, Boston 1982. [4] A. Beurling, Un théoréme sur les fonctions bornées et uniformément continues sur l’axe réel, Acta Math. 77 (1945) 127–136. [5] G. Borg, On a Liapunoff criterion of stability, Amer. J. Math. 71 (1949) 67–70. [6] R.C. Brown, D.B. Hinton, Opials inequality and oscillation of 2nd order equations, Proc. Amer. Math. Soc. 125 (1997) 1123–1129. [7] S.S. Cheng, Lyapunov inequalities for differential and difference equations, Fasc. Math. 23 (1991) 25–41. [8] R.S. Dahiya, B. Singh, A Liapunov inequality and nonoscillation theorem for a second order nonlinear differential-difference equations, J. Math. Phys. Sci. 7 (1973) 163–170. [9] S.B. Eliason, Lyapunov inequalities and bounds on solutions of certain second order equations, Canad. Math. Bull. 17 (4) (1974) 499–504. [10] H. Hochstadt, A new proof of stability estimate of Lyapunov, Proc. Amer. Math. Soc. 14 (1963) 525–526. [11] M.K. Kwong, On Lyapunovs inequality for disfocality, J. Math. Anal. Appl. 83 (1981) 486–494. [12] C. Lee, C. Yeh, C. Hong, R.P. Agarwal, Lyapunov and Wirtinger inequalities, Appl. Math. Lett. 17 (2004) 847–853. [13] D.S. Mitrinović, J.E. Pečarić, A.M. Fink, Inequalities Involving Functions and their Integrals and Derivatives, Mathematics and its Applications, in: (East European Series), vol. 53, Kluwer Academic Publishers Group, Dordrecht, 1991. [14] P.L. Napoli, J.P. Pinasco, Estimates for eigenvalues of quasilinear elliptic systems, J. Differential Equations 227 (2006) 102–115. [15] Z. Nehari, Some eigenvalue estimates, J. Anal. Math. 7 (1959) 79–88. [16] Z. Nehari, On an inequality of Lyapunov, in: Studies in Mathematical Analysis and Related Topics, Stanford University Press, Stanford, CA, 1962. [17] B.G. Pachpatte, Inequalities related to the zeros of solutions of certain second order differential equations, Facta Univ. Ser. Math. Inform. 16 (2001) 35–44. [18] T.W. Reid, A matrix equation related to an non-oscillation criterion and Lyapunov stability, Quart. Appl. Math. Soc. 23 (1965) 83–87. [19] T.W. Reid, A matrix Lyapunov inequality, J. Math. Anal. Appl. 32 (1970) 424–434. [20] B. Singh, Forced oscillation in general ordinary differential equations, Tamkang J. Math. 6 (1975) 5–11. [21] G.Sh. Guseinov, A. Zafer, Stability criteria for linear periodic impulsive Hamiltonian systems, J. Math. Anal. Appl. 35 (2007) 1195–1206. [22] G.Sh. Guseinov, A. Zafer, Stability criterion for second order linear impulsive differential equations with periodic coefficients, Math. Nachr. 281 (2008) 1273–1282. [23] Z. Kayar, A. Zafer, Stability criteria for linear Hamiltonian systems under impulsive perturbations, Appl. Math. Comput. 230 (2014) 680–686. [24] D. Cakmak, Lyapunov-type integral inequalities for certain higher order differential equations, Appl. Math. Comput. 216 (2010) 368–373. [25] S.S. Cheng, A discrete analogue of the inequality of Lyapunov, Hokkaido Math. 12 (1983) 105–112. [26] O. Došlý, P. Řehák, Half-Linear Differential Equations, Elsevier Ltd., Heidelberg, 2005. [27] A. Elbert, A half-linear second order differential equation, Colloq. Math. Soc. János Bolyai 30 (1979) 158–180. [28] S.B. Eliason, A Lyapunov inequality for a certain non-linear differential equation, J. Lond. Math. Soc. 2 (1970) 461–466. [29] S.B. Eliason, Lyapunov type inequalities for certain second order functional differential equations, SIAM J. Appl. Math. 27 (1) (1974) 180–199. [30] G.Sh. Guseinov, B. Kaymakcalan, Lyapunov inequalities for discrete linear Hamiltonian systems, Comput. Math. Appl. 45 (2003) 1399–1416. [31] L. Jiang, Z. Zhou, Lyapunov inequality for linear Hamiltonian systems on time scales, J. Math. Anal. Appl. 310 (2005) 579–593. [32] B.G. Pachpatte, On Lyapunov-type inequalities for certain higher order differential equations, J. Math. Anal. Appl. 195 (1995) 527–536. [33] B.G. Pachpatte, Lyapunov type integral inequalities for certain differential equations, Georgian Math. J. 4 (2) (1997) 139–148. [34] S. Panigrahi, Lyapunov-type integral inequalities for certain higher order differential equations, Electron. J. Differential Equations 2009 (28) (2009) 1–14.
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N. Parhi, S. Panigrahi, On Liapunov-type inequality for third-order differential equations, J. Math. Anal. Appl. 233 (2) (1999) 445–460. N. Parhi, S. Panigrahi, Liapunov-type inequality for higher order differential equations, Math. Slovaca 52 (1) (2002) 31–46. A. Tiryaki, M. Unal, D. Cakmak, Lyapunov-type inequalities for nonlinear systems, J. Math. Anal. Appl. 332 (2007) 497–511. M. Unal, D. Cakmak, A. Tiryaki, A discrete analogue of Lyapunov-type inequalities for nonlinear systems, Comput. Math. Appl. 55 (2008) 2631–2642. M. Unal, D. Cakmak, Lyapunov-type inequalities for certain nonlinear systems on time scales, Turkish J. Math. 32 (2008) 255–275. X. Yang, On Liapunov-type inequality for certain higher-order differential equations, Appl. Math. Comput. 134 (2003) 307–317. X. Yang, Lyapunov-type inequality for a class of even-order differential equations, Appl. Math. Comput. 215 (2010) 3884–3890. R.A.C. Ferreira, A Lyapunov type inequality for a fractional boundary value problem, Fract. Calc. Appl. Anal. 16 (4) (2013) 978–984. A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and Applications of Fractional Differential Equations, in: North-Holland Mathematics Studies, vol. 204, Elsevier, Amsterdam, 2006. D. O’Regan, B. Samet, Lyapunov-type inequalities for a class of fractional differential equations, J. Inequal. Appl. 2015 (247) (2015) 1–10. J. Rong, C. Bai, Lyapunov-type inequality for a fractional differential equation with fractional boundary conditions, Adv. Differential Equations 2015 (82) (2015) 1–10. A. Chidouh, D.F.M. Torres, A generalized Lyapunovs inequality for a fractional boundary value problem, J. Comput. Appl. Math. 312 (2017) 192–197. M. Jleli, B. Samet, Lyapunov-type inequalities for fractional boundary-value problems, Electron. J. Differential Equations (88) (2015) 11. M. Jleli, L. Ragoub, B. Samet, A Lyapunov-type inequality for a fractional differential equation under a Robin boundary condition, J. Funct. Spaces (2015) Art. ID 468536, 5 pp. M. Jleli, B. Samet, Lyapunov-type inequalities for a fractional differential equation with mixed boundary conditions, Math. Inequal. Appl. 18 (2) (2015) 443–451. R.P. Agarwal, A. Özbekler, Lyapunov type inequalities for even order differential equations with mixed nonlinearities, J. Inequal. Appl. 2015 (46) (2015) 1–10. M. Rahimy, Applications of fractional differential equations, Appl. Math. Sci. (Ruse) 4 (49–52) (2010) 2453–2461. Z. Bai, H. Lü, Positive solutions for boundary value problem of nonlinear fractional differential equation, J. Math. Anal. Appl. 311 (2) (2005) 495–505. R.P. Agarwal, A. Özbekler, Disconjugacy via Lyapunov and Vallée-Poussin type inequalities for forced differential equations, Appl. Math. Comput. 265 (2015) 456–468. A. Tiryaki, Recent developments of Lyapunov-type inequalities, Adv. Dynam. Sys. Appl. 5 (2) (2010) 231–248.
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Contents lists available at ScienceDirect
Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam
Lyapunov type inequalities for mixed nonlinear Riemann–Liouville fractional differential equations with a forcing term Ravi P. Agarwal a , Abdullah Özbekler b,a,∗ a
Department of Mathematics, Texas A&M University-Kingsville, 700 University Blvd., Kingsville, TX 78363-8202, USA
b
Department of Mathematics, Atilim University 06836, Incek, Ankara, Turkey
article
info
abstract
Article history: Received 4 February 2016 Received in revised form 26 September 2016
In this paper, we present some new Lyapunov and Hartman type inequalities for Riemann–Liouville fractional differential equations of the form
MSC: 34A08 34A40 34C10
where p, q, f are real-valued functions and 0 < γ < 1 < µ < 2. No sign restrictions are imposed on the potential functions p, q and the forcing term f . The inequalities obtained generalize and compliment the existing results for the special cases of this equation in the literature. © 2016 Elsevier B.V. All rights reserved.
Keywords: Lyapunov type inequality Sub-linear Super-linear Forced Fractional Riemann–Liouville
(a Dα x)(t ) + p(t ) | x(t ) |µ−1 x(t ) + q(t ) | x(t ) |γ −1 x(t ) = f (t ),
1. Introduction We recall Lyapunov inequalities for Hill’s equation x′′ (t ) + ν(t )x(t ) = 0,
(1.1)
where ν(t ) ∈ L1 [a, b] is a real-valued function. If x(t ) is a nontrivial solution of Eq. (1.1) having two consecutive zeros at a and b, where a, b ∈ R with a < b, then the inequality b
|ν(t )|dt > a
4
(1.2)
b−a
holds. This striking inequality was first proved by Lyapunov [1] and is known as ‘‘Lyapunov inequality’’. Later Wintner [2] and thereafter some authors achieved to replace the function ‘‘|ν(t )|’’ in Ineq. (1.2) by the function ‘‘ν + (t )’’ i.e. they obtained the following inequality: b
ν + (t )dt > a
∗
4 b−a
,
Corresponding author at: Department of Mathematics, Atilim University 06836, Incek, Ankara, Turkey. E-mail addresses: [email protected] (R.P. Agarwal), [email protected], [email protected] (A. Özbekler).
http://dx.doi.org/10.1016/j.cam.2016.10.009 0377-0427/© 2016 Elsevier B.V. All rights reserved.
(1.3)
2
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where ν + (t ) = max{ν(t ), 0}, and the constant 4 on the right hand side of Ineq. (1.3) (and Ineq. (1.2)) is the best possible largest number (see [1] and [3, Thm. 5.1]). In [3], Hartman obtained a more general inequality than both (1.2) and (1.3): b
(b − t )(t − a)ν + (t )dt > b − a.
(1.4)
a
Since (b − t )(t − a) ≤ (b − a)2 /4 for all t ∈ (a, b), Ineq. (1.4) implies Ineq. (1.3). The Lyapunov inequality and its generalizations have been used successfully in connection with oscillation and Sturmian theory, asymptotic theory, disconjugacy, eigenvalue problems and various properties of the solutions of (1.1) and related equations, see for instance [2–20] and the references cited therein. For some extensions to Hamiltonian systems, higher order differential equations, nonlinear and half-linear differential equations, difference and dynamic equations, functional and impulsive differential equations, we refer in particular to [21–23,7,8,24–41]. In this work, we obtain Lyapunov type inequalities for the Riemann–Liouville fractional forced nonlinear differential equations of order α ∈ (0, 2]
(a Dα x)(t ) + p(t )|x(t )|µ−1 x(t ) + q(t )|x(t )|γ −1 x(t ) = f (t ),
(1.5)
subject to Dirichlet (2-point) boundary conditions x(a) = x(b) = 0,
(1.6)
where p, q, f ∈ C[t0 , ∞) and the nonlinearities satisfy 0 < γ < 1 < µ < 2. Moreover, no sign restrictions are imposed on the potentials p and q, and the forcing term f . It is clear that there are two special cases of Eq. (1.5); one is the forced sub-linear (p(t ) = 0) fractional equation
(a Dα x)(t ) + q(t )|x(t )|γ −1 x(t ) = f (t );
0 < γ < 1,
(1.7)
and the other is the forced super-linear (q(t ) = 0) fractional equation
(a Dα x)(t ) + p(t )|x(t )|µ−1 x(t ) = f (t );
1 < µ < 2,
(1.8)
If α = 2 and f (t ) = 0, letting γ → 1− (or µ → 1+ ) in Eq. (1.7) (or Eq. (1.8)) results in Hill’s equation (1.1) with ν(t ) = q(t ) (or ν(t ) = p(t )), and hence, our results extend the classical Hartman [3] and Lyapunov [1] results. Classical Lyapunov Ineq. (1.2) for Hill’s equation (1.1) proved to be very useful in various problems related with differential equations and, since then, many improvements and generalizations of Ineq. (1.2) have appeared in the literature. To the best of our knowledge, among these generalizations, the first result in which a fractional derivative is used instead of the (classical) ordinary derivative in Hill’s equation (1.1) is the publication of Ferreira [42] in 2013. Since in recent years fractional calculus has become an important branch of mathematics (for an excellent introduction, see Kilbas et al. [43]), and finds wide variety of real world applications, our work fills the gap in the literature. As far as the Lyapunov type inequalities for fractional differential equations are considered, there are some new works, see for example [44–49]. In 2015, O’Regan and Samet [44] were concerned with the problem of finding new Lyapunov type inequalities for fractional boundary value problem
(a Dα x)(t ) + q(t )x(t ) = 0;
a < t < b, 3 < α ≤ 4 x(a) = x′ (a) = x(b) = x′ (b) = 0. In the same year, Jleli et al. [48] considered the Caputo fractional differential equation under Robin boundary conditions
(Ca Dα x)(t ) + q(t )x(t ) = 0;
a < t < b, 1 < α ≤ 2
x(a) − x′ (a) = x(b) + x′ (b) = 0, and they got a corresponding Lyapunov type inequality. Later Rong and Bai [45] concerned with the Caputo fractional differential equation under boundary condition involving the Caputo fractional derivative
(Ca Dα x)(t ) + q(t )x(t ) = 0; x ( a) = 0 ,
C β aD x
(
a < t < b, 1 < α ≤ 2
)(b) = 0,
where 0 < β ≤ 1. As far as the nonlinear equations are concerned, it seems there is only a single work [50] in the literature. Chidouh and Torres [46] focused on the Riemann–Liouville nonlinear fractional boundary value problem
(a Dα x)(t ) + q(t )g (x(t )) = 0; x(a) = x(b) = 0,
a < t < b, 1 < α ≤ 2
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where g ∈ C(R, R) is a concave and nondecreasing function. However, there is hardly any result for forced nonlinear fractional equations especially for forced sub/super-linear fractional differential equations. The first and probably the only work in this direction will be our paper. In particular, our work compliments and improves the work of Ferreira [42]. We begin with the concept of fractional integral and derivative of order α ≥ 0. Definition 1.1. Let α ≥ 0 and f be a real function defined on [a, b]. The Riemann–Liouville fractional integral of order α is defined by (a I 0 f )(t ) = f (t ) and
(a I α f )(t ) =
1
Γ (α)
t
(t − s)α−1 f (s)ds
(1.9)
a
for t ∈ [a, b], where α is a positive constant. Definition 1.2. The Riemann–Liouville fractional derivative of order α ≥ 0 is defined by
(a D0 f )(t ) = f (t ) and m−α (a Dα f )(t ) = (Dm f )(t ) a I
for α > 0, where m is the smallest integer greater or equal than α . These differential operators of arbitrary order are nonlocal and this feature has been used to model several real world phenomena, typically by substituting an ordinary derivative by a fractional one in a differential equation (see e.g. [51]). Ferreira considered the following Riemann–Liouville linear fractional differential equation of order α ∈ (0, 2]
(a Dα x)(t ) + ν(t )x(t ) = 0
(1.10)
satisfying the Dirichlet boundary conditions (1.6) where a < b and ν ∈ C[a, b]. To obtain an inequality similar to Ineq. (1.2), he wrote Prb. (1.10)–(1.6) as an equivalent integral equation and then used some properties of its Green’s function (which was constructed in [52] for a = 0 and b = 1). We state Ferreira’s results in the following: Lemma 1.1. x(t ) is a solution of the boundary value Prb. (1.10)–(1.6) if and only if x(t ) satisfies the integral equation x(t ) =
b
G(t , s)ν(s)x(s)ds,
(1.11)
a
where
(t − a)(b − s) α−1 − (t − s)α−1 1 b−a G(t , s) = (t − a)(b − s) α−1 Γ (α) b−a
if
a ≤ s ≤ t ≤ b; (1.12)
if
a≤t ≤s≤b
is the Green’s function of Prb. (1.10)–(1.6). Lemma 1.2. The Green’s function G(t , s) given in (1.12) of Prb. (1.10)–(1.6) satisfies the following properties: (i) G(t , s) ≥ 0 for all a ≤ t, s ≤ b. (ii) maxt ∈[a,b] G(t , s) = G(s, s), s ∈ [a, b]. (iii) G(s, s) has a unique maximum, given by max G(s, s) = G
s∈[a,b]
a+b a+b 2
,
2
=
1
Γ (α)
b−a 4
α−1
.
(1.13)
Theorem 1.3 (Lyapunov Type Inequality). Let x(t ) be a nontrivial solution of Eq. (1.10) satisfying the Dirichlet boundary conditions (1.6) where a < b. If x(t ) ̸= 0 in (a, b), then the inequality b
|ν(t )|dt > Γ (α) a
holds.
4 b−a
α−1 (1.14)
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Note that for α = 2 Ineq. (1.14) reduces to Lyapunov’s classical Ineq. (1.2) for Hill’s equation (1.1). When α = 2, Eq. (1.5) reduces to the forced nonlinear differential equation of the form x′′ (t ) + p(t )|x(t )|µ−1 x(t ) + q(t )|x(t )|γ −1 x(t ) = f (t ).
(1.15)
Very recently, present authors [53] obtained the following Hartman and Lyapunov type inequalities for Prb. (1.15)–(1.6). Theorem 1.4 (Hartman Type Inequality). Let x(t ) be a nontrivial solution of Eq. (1.15) satisfying the Dirichlet boundary conditions (1.6). If x(t ) ̸= 0 in (a, b), then the inequality b
(b − t )(t − a)(p + q )(t )dt +
+
b
(b − t )(t − a) µ0 p+ (t ) + γ0 q+ (t ) + |f (t )| dt
a
a
>
(b − a)2
(1.16)
4
holds, where
µ0 = (2 − µ)µµ/(2−µ) 22/(µ−2) > 0
(1.17)
γ0 = (2 − γ )γ γ /(2−γ ) 22/(γ −2) > 0.
(1.18)
and
Theorem 1.5 (Lyapunov Type Inequality). Let x(t ) be a nontrivial solution of Eq. (1.15) satisfying the Dirichlet boundary conditions (1.6). If x(t ) ̸= 0 in (a, b), then the inequality b
(p+ + q+ )(t )dt
b
a
µ0 p+ (t ) + γ0 q+ (t ) + |f (t )| dt
>
a
4
(1.19)
(b − a)2
holds, where the constants µ0 and γ0 are the same as in (1.17) and (1.18). We remark that the Lyapunov type inequalities have been studied by many authors, see for instance the survey paper [54] and the references therein, but to the best of our knowledge there is no result in the literature for the nonlinear fractional differential equations of the form (1.5); in fact, even for its particular cases, i.e. for Eqs. (1.7) and (1.8). 2. Main results In this section we state and prove our results. Throughout this section we shall denote by u± = max{±u, 0}. In what follows we will need the following lemma, see [50, Lemma 2.1] (see also [53]). We include its proof here for completeness. Lemma 2.1. If A is positive, and B, z are nonnegative, then Az 2 − Bz α + (2 − α)α α/(2−α) 22/(α−2) A−α/(2−α) B2/(2−α) ≥ 0
(2.1)
for any α ∈ (0, 2) with equality holding if and only if B = z = 0. Proof. Let
F (z ) = Az 2 − Bz α ,
z≥0
(2.2)
where A > 0 and B ≥ 0. Clearly, when z = 0 or B = 0, (2.1) is obvious. On the other hand, if B > 0, then it is easy to see that F attains its minimum at z0 = (α A−1 B/2)1/(2−α) and
Fmin = −(2 − α)α α/(2−α) 22/(α−2) A−α/(2−α) B2/(2−α) . Thus, (2.1) holds. Note that if B > 0, then Ineq. (2.1) is strict.
We are now in the position to state and prove our first main result. Theorem 2.1 (Hartman Type Inequality). Let x(t ) be a nontrivial solution of Eq. (1.5) satisfying the Dirichlet boundary conditions (1.6). If x(t ) > 0 in (a, b), then the inequality b
[(b − t )(t − a)]α−1 p+ (t ) + q+ (t ) dt
a
>
b
[(b − t )(t − a)]α−1 µ0 p+ (t ) + γ0 q+ (t ) + f − (t ) dt
a
Γ 2 (α) 4
(b − a)2α−2
holds, where the constants µ0 and γ0 are the same as in (1.17) and (1.18).
(2.3)
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Proof. First, we note that the solution of the Riemann–Liouville linear fractional differential equation of order α ∈ (0, 2]
− (a Dα z )(t ) = H (t )
(2.4)
satisfying the Dirichlet boundary conditions (1.6) can be represented by z (t ) =
b
G(t , s)H (s)ds,
(2.5)
a
where G(t , s) is the Green’s function of Prb. (2.4)–(1.6) given as in (1.12). It is clear from Lemma 1.2(ii) that
(b − s)(s − a) 0 ≤ G(t , s) ≤ G(s, s) = b−a
α−1 (2.6)
for all (t , s) ∈ [a, b] × [a, b]. Now let x(t ) be a positive solution of Eq. (1.5) with x(a) = x(b) = 0, where a < b are consecutive zeros. Then using (1.5), (2.4) and (2.5), x(t ) can be expressed as x(t ) =
b
G(t , s) p(s)xµ (s) + q(s)xγ (s) − f (s) ds.
(2.7)
a
Let x(c ) = maxt ∈(a,b) x(t ). Then by (2.6) and (2.7), we have x(c ) =
b
G(c , s) p(s)xµ (s) + q(s)xγ (s) − f (s) ds
a b
G(s, s) p+ (s)xµ (s) + q+ (s)xγ (s) + f − (s) ds
≤
a
=
b
1
Γ (α)(b − a)α−1
[(b − s)(s − a)]α−1 p+ (s)xµ (s) + q+ (s)xγ (s) + f − (s) ds
a
= P0 xµ (c ) + Q0 xγ (c ) + F0 ,
(2.8)
where P0 =
Γ (α)(b − a)α−1
Q0 =
b
1
b
1
[(b − s)(s − a)]α−1 p+ (s)ds,
(2.9)
a
Γ (α)(b − a)α−1
[(b − s)(s − a)]α−1 q+ (s)ds
(2.10)
a
and F0 =
b
1
Γ (α)(b − a)α−1
[(b − s)(s − a)]α−1 f − (s)ds.
a
On the other hand, Ineq. (2.1) in Lemma 2.1 with A = B = 1, implies that xµ (c ) < x2 (c ) + µ0
and xγ (c ) < x2 (c ) + γ0 .
Using these inequalities and Ineq. (2.8) we find the following quadratic inequality:
(P0 + Q0 )x2 (c ) − x(c ) + µ0 P0 + γ0 Q0 + F0 > 0.
(2.11)
But this is only possible when
(P0 + Q0 )(µ0 P0 + γ0 Q0 + F0 ) > 1/4, which is the same as (2.3). This completes the proof of Theorem 2.1.
Theorem 2.2 (Hartman Type Inequality). Let x(t ) be a nontrivial solution of Eq. (1.5) satisfying the Dirichlet boundary conditions (1.6). If x(t ) < 0 in (a, b), then the inequality b
[(b − t )(t − a)]
α−1
a
>
p (t ) + q (t ) dt
+
+
b
[(b − t )(t − a)]
α−1
µ0 p (t ) + γ0 q (t ) + f (t ) dt +
+
+
a
Γ 2 (α) 4
(b − a)2α−2
holds, where the constants µ0 and γ0 are the same as in (1.17) and (1.18).
(2.12)
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Proof. Let x(t ) be a negative solution of Eq. (1.5) with x(a) = x(b) = 0, where a < b are consecutive zeros. In fact, if x(t ) < 0 for t ∈ (a, b), then we can consider −x(t ) as a positive solution of
(a Dα x)(t ) + p(t )|x(t )|β−1 x(t ) + q(t )|x(t )|γ −1 x(t ) = −f (t ).
(2.13)
Then using (1.5), (2.4) and (2.5), x(t ) can be expressed as x(t ) =
b
G(t , s) p(s)xβ (s) + q(s)xγ (s) + f (s) ds.
(2.14)
a
Let x(c ) = maxt ∈(a,b) x(t ). Then by (2.6) and (2.14), we have x(c ) =
b
G(c , s) p(s)xµ (s) + q(s)xγ (s) + f (s) ds
a b
G(s, s) p+ (s)xµ (s) + q+ (s)xγ (s) + f + (s) ds
≤ a
=
(b −
b
1 a)α−1
[(b − s)(s − a)]α−1 p+ (s)xµ (s) + q+ (s)xγ (s) + f + (s) ds
a
µ
= P0 x (c ) + Q0 xγ (c ) + F0 ,
(2.15)
where P0 and Q0 are defined in (2.9) and (2.10), and
1
F0 =
(b − a)α−1
b
[(b − s)(s − a)]α−1 f + (s)ds.
a
Now repeating the same steps as in Theorem 2.1, we obtain (2.12) which completes the proof of Theorem 2.2.
From Theorems 2.1 and 2.2, the next result immediately follows. Theorem 2.3 (Hartman Type Inequality). Let x(t ) be a nontrivial solution of Eq. (1.5) satisfying the Dirichlet boundary conditions (1.6). If x(t ) ̸= 0 in (a, b), then the inequality b
[(b − t )(t − a)]
α−1
p (t ) + q (t ) dt
+
+
b
a
[(b − t )(t − a)]
α−1
µ0 p+ (t ) + γ0 q+ (t ) + |f (t )| dt
a
Γ 2 (α)
>
4
(b − a)2α−2
(2.16)
holds, where the constants µ0 and γ0 are the same as in (1.17) and (1.18). Proof. Let x(t ) be a nontrivial solution of Eq. (1.5) with x(a) = x(b) = 0, where a < b are consecutive zeros. Since either x(t ) > 0 or x(t ) < 0 for t ∈ (a, b) and f ± (t ) ≤ |f (t )|, by (2.3) and (2.12) we obtain (2.16). This completes the proof of Theorem 2.3. Remark 1. When α = 2, Theorem 2.3 coincides with [53, Theorem. 2.3], i.e. Theorem 1.4. Corollary 2.4 (Disconjugacy). If b
[(b − t )(t − a)]
α−1
p (t ) + q (t ) dt
+
+
a b
[(b − t )(t − a)]
×
α−1
µ0 p+ (t ) + γ0 q+ (t ) + |f (t )| dt
≤
Γ 2 (α)
a
4
(b − a)2α−2
then Eq. (1.5) is disconjugate in [a, b], where the constants µ0 and γ0 are the same as in (1.17) and (1.18). Ineq. (1.13) yields that Ineq.’s (2.3), (2.12) and (2.16) in Theorems 2.1–2.3, immediately imply the following Lyapunov type inequalities for Eq. (1.5). Theorem 2.5 (Lyapunov Type Inequality). Let x(t ) be a nontrivial solution of Eq. (1.5) satisfying the Dirichlet boundary conditions (1.6). If x(t ) > 0 in (a, b), then the inequality b
p+ (t ) + q+ (t ) dt
a
b
µ0 p+ (t ) + γ0 q+ (t ) + f − (t ) dt
a
holds, where the constants µ0 and γ0 are the same as in (1.17) and (1.18).
>
42α−3 Γ 2 (α)
(b − a)2α−2
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Theorem 2.6 (Lyapunov Type Inequality). Let x(t ) be a nontrivial solution of Eq. (1.5) satisfying the Dirichlet boundary conditions (1.6). If x(t ) < 0 in (a, b), then the inequality b
p+ (t ) + q+ (t ) dt
b
µ0 p+ (t ) + γ0 q+ (t ) + f + (t ) dt
>
a
a
42α−3 Γ 2 (α)
(b − a)2α−2
holds, where the constants µ0 and γ0 are the same as in (1.17) and (1.18). Theorem 2.7 (Lyapunov Type Inequality). Let x(t ) be a nontrivial solution of Eq. (1.5) satisfying the Dirichlet boundary conditions (1.6). If x(t ) ̸= 0 in (a, b), then the inequality b
p+ (t ) + q+ (t ) dt
b
µ0 p+ (t ) + γ0 q+ (t ) + |f (t )| dt
>
a
a
42α−3 Γ 2 (α)
(b − a)2α−2
holds, where the constants µ0 and γ0 are the same as in (1.17) and (1.18). Remark 2. When α = 2, then Theorem 2.7 coincides with [53, Theorem 2.4], i.e. Theorem 1.5. Corollary 2.8 (Disconjugacy). If b
p+ (t ) + q+ (t ) dt
b
a
µ0 p+ (t ) + γ0 q+ (t ) + |f (t )| dt
≤
a
42α−3 Γ 2 (α)
(b − a)2α−2
then Eq. (1.5) is disconjugate in [a, b], where the constants µ0 and γ0 are the same as in (1.17) and (1.18). Next we present two new results where Eq. (1.5) has two special type of nonlinearities, namely forced sub-linear or forced super-linear, i.e., p(t ) ≡ 0 or q(t ) ≡ 0, respectively. Theorem 2.9. Let x(t ) be a nontrivial solution of Eq. (1.7) satisfying the Dirichlet boundary conditions (1.6). If x(t ) ̸= 0 in (a, b), then the following hold: (i) Hartman type inequality; b
[(b − t )(t − a)]α−1 q+ (t )dt
a
b
[(b − t )(t − a)]α−1 γ0 q+ (t ) + |f (t )| dt
>
a
Γ 2 (α) 4
(b − a)2α−2 .
(ii) Lyapunov type inequality; b
q+ (t )dt
b
a
+ γ0 q (t ) + |f (t )| dt
>
a
42α−3 Γ 2 (α)
(b − a)2α−2
,
where γ ∈ (0, 1) and the constant γ0 is the same as in (1.18). Theorem 2.10. Let x(t ) be a nontrivial solution of Eq. (1.8) satisfying the Dirichlet boundary conditions (1.6). If x(t ) ̸= 0 in (a, b), then the following hold: (i) Hartman type inequality; b
[(b − t )(t − a)]α−1 p+ (t )dt
a
b
[(b − t )(t − a)]α−1 µ0 p+ (t ) + |f (t )| dt
a
>
Γ 2 (α) 4
(b − a)2α−2 .
(ii) Lyapunov type inequality; b
p+ (t )dt a
b
µ0 p+ (t ) + |f (t )| dt
a
>
42α−3 Γ 2 (α)
(b − a)2α−2
,
where µ ∈ (1, 2) and the constant µ0 is the same as in (1.17). When γ → 1− (or µ → 1+ ), Eq. (1.7) (or Eq. (1.8)) reduces to forced Riemann–Liouville linear fractional differential equation of order α ∈ (0, 2]
(a Dα x)(t ) + ν(t )x(t ) = f (t ), where ν(t ) = q(t ) (or ν(t ) = p(t )). Since lim µ0 = lim γ0 = 1/4,
µ→1+
γ →1−
we have the following result from Theorems 2.9 and 2.10.
(2.17)
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Corollary 2.11. Let x(t ) be a nontrivial solution of Eq. (2.17) satisfying the Dirichlet boundary conditions (1.6). If x(t ) ̸= 0 in (a, b), then the following hold: (i) Hartman type inequality; b
[(b − t )(t − a)]
α−1 +
ν (t )dt
b
α−1
[(b − t )(t − a)]
ν (t ) + 4|f (t )| dt +
> Γ 2 (α)(b − a)2α−2 .
(2.18)
a
a
(ii) Lyapunov type inequality; b
ν (t )dt +
b
+ ν (t ) + 4|f (t )| dt
a
a
>
42α−2 Γ 2 (α)
(b − a)2α−2
.
(2.19)
Remark 3. In the case when x(t ) > 0 or x(t ) < 0 for t ∈ (a, b), Hartman and Lyapunov type inequalities for forced sub-linear equations (1.7), forced super-linear equation (1.8) and forced linear equation (2.17) can be formulated by replacing the term ‘‘|f (t )|’’ by ‘‘f − (t )’’ (‘‘f + (t )’’) in Theorems 2.9 and 2.10 and Corollary 2.11. Remark 4. When f (t ) ≡ 0, Ineq. (2.19) in Corollary 2.11 reduces to b
ν + (t )dt > Γ (α)
4
α−1 (2.20)
b−a
a
which is sharper than Ineq. (1.14) given by Ferreira [42] in Theorem 1.3 since |ν| ≥ ν + . Moreover, when f (t ) ≡ 0 and α = 2, Ineq.’s (2.18) and (2.19) coincide with the classical Lyapunov and Hartman inequalities, i.e. Ineq.’s (1.4) and (1.3). Example 2.12. For the fractional equation
(0 D3/2 x)(t ) + k1 |x(t )|1/2 x(t ) + k2 |x(t )|−1/2 x(t ) = k3 ;
t ≥0
(2.21)
where kj , j = 1, 2, 3, are real constants with k1 , k2 > 0, if the solution x(t ) has consecutive zeros at 0 and b > 0, then in view of (2.16) the following inequality must be satisfied −8/3
(k1 + k2 )(2 3 k1 + 2 −8 3
b
[(b − t )t ]
3k2 + |k3 |)
1/2
0
2 dt
>
1 4
Γ 2 (3/2)b.
(2.22)
Since the integral b
1 (b − t )tdt = π b2 8
0
√ and Γ (3/2) = π /2, Ineq. (2.22) turns out b>
1
−8/3
π (k1 + k2 )(2 3 k1 + 2 7
4
−8 3
−1/3 3k2 + |k3 |) .
(2.23)
Thus Eq. (2.21) is disconjugate on [0, b] if
b≤
1 4
−1/3 π 7 (k1 + k2 )(2−8 33 k1 + 2−8/3 3k2 + |k3 |) .
On the other hand, if we use the Lyapunov type inequality given in Theorem 2.7, then we obtain the inequality b>
4
π
−1/3 (k1 + k2 )(2−8 33 k1 + 2−8/3 3k2 + |k3 |) .
(2.24)
We note that Ineq. (2.24) is better than Ineq. (2.23) and hence we may conclude that Eq. (2.21) is disconjugate on [0, b] if
b≤
4
π
(k1 + k2 )(2−8 33 k1 + 2−8/3 3k2 + |k3 |)
−1/3
.
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9
3. Concluding remarks We conclude this paper with the following remark. The results obtained in this paper for Eq. (1.5) can be easily extended to the Riemann–Liouville fractional forced nonlinear differential equations of order α ∈ (0, 2] with positive and negative coefficients
(a Dα x)(t ) ± p(t )|x(t )|µ−1 x(t ) ∓ q(t )|x(t )|γ −1 x(t ) = f (t ) or more generally; equations of the form
(a Dα x)(t ) +
n
qk (t )|x(t )|σk −1 x(t ) = f (t )
k =1
where 0 < σ1 < · · · < σm < 1 < σm+1 < · · · < σn < 2 with no sign restrictions on the potential functions qk , k = 1, . . . , n, and the forcing term f . The formulation of these results is left to the reader. It will be of interest to find similar results for the mixed nonlinear equations of the form Eq. (1.5) or Riemann–Liouville fractional forced super-linear equation of order α ∈ (0, 2]
(a Dα x)(t ) + p(t )|x(t )|µ−1 x(t ) = f (t ) for µ ∈ [2, ∞). Acknowledgments This work was carried out when the second author was on academic leave, visiting TAMUK (Texas A&M UniversityKingsville) and he wishes to thank TAMUK. This work is partially supported by TUBITAK (2219) (The Scientific and Technological Research Council of Turkey). References [1] A.M. Liapunov, Probleme général de la stabilité du mouvement, (French Translation of a Russian paper dated 1893), Ann. Fac. Sci. Univ. Toulouse 2 (1907) 27–247. Reprinted as Ann. Math. Studies, No. 17, Princeton, 1947. [2] A. Wintner, On the nonexistence of conjugate points, Amer. J. Math. 73 (1951) 368–380. [3] P. Hartman, Ordinary Differential Equations, Wiley, New York, 1964, and Birkhuser, Boston 1982. [4] A. Beurling, Un théoréme sur les fonctions bornées et uniformément continues sur l’axe réel, Acta Math. 77 (1945) 127–136. [5] G. Borg, On a Liapunoff criterion of stability, Amer. J. Math. 71 (1949) 67–70. [6] R.C. Brown, D.B. Hinton, Opials inequality and oscillation of 2nd order equations, Proc. Amer. Math. Soc. 125 (1997) 1123–1129. [7] S.S. Cheng, Lyapunov inequalities for differential and difference equations, Fasc. Math. 23 (1991) 25–41. [8] R.S. Dahiya, B. Singh, A Liapunov inequality and nonoscillation theorem for a second order nonlinear differential-difference equations, J. Math. Phys. Sci. 7 (1973) 163–170. [9] S.B. Eliason, Lyapunov inequalities and bounds on solutions of certain second order equations, Canad. Math. Bull. 17 (4) (1974) 499–504. [10] H. Hochstadt, A new proof of stability estimate of Lyapunov, Proc. Amer. Math. Soc. 14 (1963) 525–526. [11] M.K. Kwong, On Lyapunovs inequality for disfocality, J. Math. Anal. Appl. 83 (1981) 486–494. [12] C. Lee, C. Yeh, C. Hong, R.P. Agarwal, Lyapunov and Wirtinger inequalities, Appl. Math. Lett. 17 (2004) 847–853. [13] D.S. Mitrinović, J.E. Pečarić, A.M. Fink, Inequalities Involving Functions and their Integrals and Derivatives, Mathematics and its Applications, in: (East European Series), vol. 53, Kluwer Academic Publishers Group, Dordrecht, 1991. [14] P.L. Napoli, J.P. Pinasco, Estimates for eigenvalues of quasilinear elliptic systems, J. Differential Equations 227 (2006) 102–115. [15] Z. Nehari, Some eigenvalue estimates, J. Anal. Math. 7 (1959) 79–88. [16] Z. Nehari, On an inequality of Lyapunov, in: Studies in Mathematical Analysis and Related Topics, Stanford University Press, Stanford, CA, 1962. [17] B.G. Pachpatte, Inequalities related to the zeros of solutions of certain second order differential equations, Facta Univ. Ser. Math. Inform. 16 (2001) 35–44. [18] T.W. Reid, A matrix equation related to an non-oscillation criterion and Lyapunov stability, Quart. Appl. Math. Soc. 23 (1965) 83–87. [19] T.W. Reid, A matrix Lyapunov inequality, J. Math. Anal. Appl. 32 (1970) 424–434. [20] B. Singh, Forced oscillation in general ordinary differential equations, Tamkang J. Math. 6 (1975) 5–11. [21] G.Sh. Guseinov, A. Zafer, Stability criteria for linear periodic impulsive Hamiltonian systems, J. Math. Anal. Appl. 35 (2007) 1195–1206. [22] G.Sh. Guseinov, A. Zafer, Stability criterion for second order linear impulsive differential equations with periodic coefficients, Math. Nachr. 281 (2008) 1273–1282. [23] Z. Kayar, A. Zafer, Stability criteria for linear Hamiltonian systems under impulsive perturbations, Appl. Math. Comput. 230 (2014) 680–686. [24] D. Cakmak, Lyapunov-type integral inequalities for certain higher order differential equations, Appl. Math. Comput. 216 (2010) 368–373. [25] S.S. Cheng, A discrete analogue of the inequality of Lyapunov, Hokkaido Math. 12 (1983) 105–112. [26] O. Došlý, P. Řehák, Half-Linear Differential Equations, Elsevier Ltd., Heidelberg, 2005. [27] A. Elbert, A half-linear second order differential equation, Colloq. Math. Soc. János Bolyai 30 (1979) 158–180. [28] S.B. Eliason, A Lyapunov inequality for a certain non-linear differential equation, J. Lond. Math. Soc. 2 (1970) 461–466. [29] S.B. Eliason, Lyapunov type inequalities for certain second order functional differential equations, SIAM J. Appl. Math. 27 (1) (1974) 180–199. [30] G.Sh. Guseinov, B. Kaymakcalan, Lyapunov inequalities for discrete linear Hamiltonian systems, Comput. Math. Appl. 45 (2003) 1399–1416. [31] L. Jiang, Z. Zhou, Lyapunov inequality for linear Hamiltonian systems on time scales, J. Math. Anal. Appl. 310 (2005) 579–593. [32] B.G. Pachpatte, On Lyapunov-type inequalities for certain higher order differential equations, J. Math. Anal. Appl. 195 (1995) 527–536. [33] B.G. Pachpatte, Lyapunov type integral inequalities for certain differential equations, Georgian Math. J. 4 (2) (1997) 139–148. [34] S. Panigrahi, Lyapunov-type integral inequalities for certain higher order differential equations, Electron. J. Differential Equations 2009 (28) (2009) 1–14.
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