q Non uniform difference calculus and classical integral inequalities

R E S E A R C H

Open Access

q

-Non uniform difference calculus and

classical integral inequalities

Gaspard Bangerezako

_{, Jean Paul Nuwacu}

_{and Enas M Shehata}

*_{Correspondence:}

[email protected] 1_{Department of Mathematics,} Faculty of Science, University of Burundi, Bujumbura, Burundi Full list of author information is available at the end of the article

Abstract

We first establishq-non uniform difference versions of the integral inequalities of Hölder, Cauchy–Schwarz, and Minkowski of classical mathematical analysis and then integral inequalities of Grönwall and Bernoulli based on the Lagrange method of linearq-non uniform difference equations of first order. Finally, we prove the

Lyapunov inequality for the solutions of theq-non uniform Sturm–Liouville equation.

Keywords: q-Non uniform difference calculus; Hölder; Cauchy–Schwarz; Minkowski; Grönwall; Bernoulli; Lyapunov inequalities; Lagrange method; Sturm–Liouville equation

1 Introduction

Considering the most general divided difference derivative [1,2],

Dft(s)=f(t(s+ 1

2)) –f(t(s– 1 2))

t(s+1₂) –t(s–1₂) , (1)

admitting the property that iff(t) =Pn(t(s)) is a polynomial of degree n in t(s), then Df(t(s)) =P˜n–1(t(s)) is a polynomial int(s) of degreen– 1, one is led to the following most important canonical forms fort(s) in order of increasing complexity:

t(s) =t(0); (2)

t(s) =s; (3)

t(s) =qs; (4)

t(s) =q

s_+q–s

2 ; q∈C,s∈Z. (5)

When the functiont(s) is given by (2)–(4), the divided difference derivative (1) leads to the ordinary differential derivativeDf(t) =_dtdf(t), finite difference derivative

f(s) =f(s+ 1) –f(s) =edsd _{– 1f(s), (6)}

andq-difference derivative (or Jackson derivative [3])

Dqf(t) =

f(qt) –f(t)

qt–t = qdtd _{– 1}

qt–t f(t), (7)

respectively (see also [4–6]). Whent(s) =x(qs_{) is given by (5), the corresponding derivative}

is usually referred to as theAskey–Wilsonfirst order divided difference operator [7] that one can write as follows:

Dfx(z)=f(x(q 1

2_{z)) –f(x(q}–12_z))

x(q12_{z) –x(q}–12_z)

, (8)

wherex(z) = z+₂z–1 is the well-knownJoukowskitransformation andz=qs.

The calculus related to the differential derivative, the continuous or differential calcu-lus, is clearly the classical one. The one related to derivatives (6),(7), (8) (difference,q -difference, andq-non uniform difference, respectively) is referred to as thediscrete calcu-lus. Its interest is twofold: On the one hand, it generalizes the continuous calculus; on the other hand, it uses a discrete variable.

This work is concerned with theq-non uniform difference calculus. We particularly aim to establishq-non uniform difference versions of the well-known in differential calculus integral inequalities of Hölder, Cauchy–Schwarz, Minkowski, Grönwall, Bernoulli, and Lyapunov. Another captivating work on the raised inequalities can be found in [8] for a calculus based on a derivative also generalizing (6) and (7), but one will remark that even if that work greatly inspired us, there is not any hierarchic relationship between the calculus considered there (see [9] for a general theory) and the one considered here (see also [10–12]) based on (8). We will note also that (8) is at our best knowledge the most general known divided difference derivative having the property of sending a polynomial of degreenin a polynomial of degreen– 1.

In the following lines, we first introduce basic concepts ofq-non uniform difference calculus necessary for the sequel, and then study the mentioned integral inequalities. The functions that are considered in theq-non uniform difference calculus are clearly defined on the set

T=

xqk,k∈Z

2 =

n

2

n∈Z

. (9)

2 q-Non uniform difference calculus

In this section, we discus the essential elements ofq-non uniform integral calculus and

q-non uniform linear difference equations of first order.

2.1 q-Non uniform integral calculus

2.1.1 Integration

We consider theq-non uniform divided difference derivative defined by (from now on

q∈R+_,s∈Z+_{, as indicated below)}

Dfx(z)def= f(x(q 1

2_{z)) –f(x(q}–12_z))

x(q12_{z) –x(q}–12_z)

x(z) =z+z –1

2 =x

qs=q

s_+q–s

2 , q∈R

+_,s∈Z+_{. (11)}

A functionf(x(z)) is said to beq-non uniform differentiableonx(z) iff the ratio on the r.h.s. (right-hand side) of (10) exists and is finite. Clearly, every continuous function onT (in the topology ofR) isq-non uniform differentiable on that set.

Let us suppose thatDF(x(z)) =f(x(z));z=qs_{or equivalently}

F(x(qs+12_{)) –F(x(q}s–12₎₎

x(qs+12_{) –x(q}s–12₎

=fxqs. (12)

We have

Fxqs–12_–Fxqs+12_=xqs–12_–xqs+12_fxqs_, then we have

Fxqs–Fxqs+1=xqs–xqs+1fxqs+12

Fxqs+1–Fxqs+2=xqs+1–xqs+2fxqs+32 ..

.

FxqN–1–FxqN=xqN–1–xqNfxqN–12_. By adding member by member, we get

Fxqs–FxqN=

N–1

r=s

xqr–xqr+1fxqr+12_.

Hence

x(z)

x(qN₎

fx(z)dqx(z)

def =

x(qN–1)

x(t)=x(z)

x(t) –x(qt)fxtq12

=

qN–1

t=z

x(t) –x(qt)fxtq12_{. (13)}

This integral sends a polynomial (inx(z)) of degreenin a polynomial of degreen+ 1 [12].

Replacingx(z) = z+z₂–1in this last equation, we have

x(z)

x(qN₎f

x(z)dqx(z) =

1 2(1 –q)

qN–1

t=z

t

1 – 1

qt2

fxtq12_{. (14)}

Assume first that 0 <q< 1. In this case, according to the constructionN≥s≥0, one will haveqN_≤z=qs_{andx(z)≤x(q}N_{), and the convenient writing of (14) is}

x(qN₎

x(z)

fx(z)dqx(z) =

1 2(1 –q)

qN–1

t=z

t

1

qt2 – 1

fxtq12_{. (15)}

On the other hand, if 1 <q<∞, we will havez≤qNandx(z)≤x(qN) and (14) becomes (15) again. On the other side, in (15), the factor

h(t) = (1 –q)t

1

qt2 – 1

; t=qs,s∈Z+

is always positive regardless of whether 0 <q< 1 or 1 <q<∞(let us agree from now on that 0 <q< 1 and therefore 0≤z≤1). This leads us to the following fundamentalpositivity

property of the integral in (15).

Property 2.1 If f(x(z))≥0and a=x(qα_)≤b=x(qβ_),then b

a

fx(z)dqx(z)≥0.

Corollary 2.1 If f(x(z))≥g(x(z))and a=x(qα_)≤b=x(qβ_),then b

a

fx(z)dqx(z)≥ b

a

gx(z)dqx(z).

2.1.2 Connection between the q-integral and q-non uniform integral

We have

x(qN)

x(z)

fx(z)dqx(z) =

1 2

z

qN

1

qz2– 1

fxq12_{zdqz. (16)}

ForN→ ∞, ∞

x(z)

fx(z)dqx(z) =

1 2

z

0

1

qz2 – 1

fxq12_zdqz

=1 2(1 –q)z

∞

i=0

1

q2i+1_z2– 1

fxqi+12_{z, (17)}

and the integral with lower and upper finite bound can be written as follows:

b

a

fx(z)dqx(z) =

∞

a

fx(z)dqx(z) –

∞

b

fx(z)dqx(z) (18)

a=x(qα_)≤b=x(qβ_{) (if the integrals on the r.h.s. of (18) exist).}

A functionf(x(z)) defined onTis said to beq-non uniform integrableon a finite interval [x(z),x(qN_{)] iff the sum on the r.h.s. of (15) exists and is finite. If the upper bound is infinite,}

Remark2.1 It is not difficult to notice that to deal with the case wheres≥N≥0 it would suffice to replace in the preceding formulasz=qs_byqN _{and vice versa.}

In this case, from (13), we obtain

x(qN₎

x(z)

fx(z)dqx(z)def= x(zq–1)

x(t)=x(qN₎

x(t) –x(qt)fxq12_t

=

zq–1

t=qN

x(t) –x(qt)fxq12_{t (19)}

or by replacingx(z) = z+z₂–1in (19)

x(qN)

x(z)

fx(z)dqx(z) =

1 2(1 –q)

zq–1

t=qN

t

1 – 1

qt2

fxq12_{t. (20)}

Passing to theq-integral, we will have

x(qN)

x(z)

fx(z)dqx(z) =

1 2

z

qN

1

qt2– 1

fxq12_td

qt. (21)

ForN→0,

x(z)

1

fx(z)dqx(z) =

1 2(q– 1)

1

t=q–1_z

t

1 – 1

qt2

fxq12_t

=1 2(q– 1)

s–1

i=0

qi

1 – 1

q2i+1

fxqi+12

=1 2

z

1

1 – 1

qt2

fxq12_td

qt. (22)

Finally, ifa=x(qα_)≤b=x(qβ_), b

a

fx(z)dqx(z)

def =

b

1

fx(z)dqx(z) – a

1

fx(z)dqx(z)

= 1

a

fx(z)dqx(z) –

1

b

fx(z)dqx(z). (23)

2.1.3 Fundamental principles of analysis

(i) We can formulate the statement of the fundamental principle of analysis as follows: “Theq-non uniform derivative of the integral of a function is this function itself ”. This corresponds to the formula

D x(z) x(qN₎

fx(z)dqx(z)

(ii) This is theq-non uniform version of the Newton–Leibnitz formula

x(z)

x(qN₎[Df]

x(z)dqx(z) =f

x(z)–fxqN. (25)

2.1.4 Integration by parts

By integrating the formula

fxq12_zD_gx(z)=D_{[fg]x(z)–gxq}–21_zD_{fx(z) (26)} and using the second fundamental principle of the analysis, one obtains

x(z)

x(qN₎

fxq12_zD_gx(z)dqx(z)

= [fg]x_x(₍z_q)N₎–

x(z)

x(qN₎g

xq–12_zD_{fx(z)dqx(z). (27)}

2.2 Linearq-non uniform difference equations of first order

The general linearq-non uniform difference equation of first order is given by

Dyx(z)=ax(z)yxq–12_{z+bx(z) (28)}

or

Dy˜x(z)=a˜x(z)y˜xq12_z+b˜_{x(z), (29)}

wherea(x(z)),b(x(z)),a˜(x(z)), andb˜(x(z)) are known, whiley(x(z)) andy˜(x(z)) are unknown functions to be determined.

Consider first the homogeneous equation corresponding to (28):

Dy0

x(z)=ax(z)y0

xq–12_{z. (30)}

Detailing, we get

y0(x(q 1

2z)) –y₀(x(q–12z))

x(q12_{z) –x(q}–12_z)

=ax(z)y0

xq–12_z,

which gives

y0

xq12_z=px(z)y0xq–12_z, where

px(z)= 1 +xzq12_–xzq–12_ax(z) = 1 +q

1 2 _–q–12

2

Using the recursion

y0

x(z)=pxq21_z–1_y0x(qz), we obtain

y0

x(z)=y0

x(z0) z

t=q–1_z0

pxtq12–1 ₍₃₂₎

or

y0

x(z)=

N–1

i=0

pxqi+12_z–1_y0xqN_{z. (33)}

ForN−→ ∞(orz0−→0), we obtain

y0

x(z)= _∞

i=0

pxzqi+12–1

y0

x(0). (34)

Let us define the exponential function

E

a,q12

x(z0);x(z) def

=

z

t=q–1_z 0

pxtq12–1_{. (35)}

It is clear that

E

a,q– 12

x(0);x(z)= ∞

i=0

pxzqi+12–1 ₍₃₆₎

and

E

a,q– 12

x(z0);∞

= ∞

i=0

pxq–iz0q– 1

2–1_{. (37)}

Similarly, for the homogeneous equation corresponding to (29)

Dy˜0

x(z)=a˜x(z)y˜0

xq12_{z, (38)} developing, we have

˜

y0

xq12_zp˜_x(z)=y˜_0xq–12_{z, (39)} where

˜

px(z)= 1 –xq12_z–xq–12_za˜_x(z) = 1 –q

1 2_–q–12

2

Using the recursion

˜

y0

x(z)=p˜xq12_zy˜ 0

x(qz), we obtain

˜

y0

x(z)=y˜0

x(z0) z

t=q–1_z 0

˜

pxtq12 ₍₄₁₎

or

˜

y0

x(z)=

N–1

i=0

˜

pxzqi+12˜_y 0

qNx. (42)

ForN−→ ∞(orz0−→0), we have

˜

y0

x(z)= ∞

i=0

˜

pxzqi+12_y˜_{0x(0). (43)}

Let us now define the second exponential function

E

˜

a;q12

x(z0),x(z) def

=

z

t=q–1_z 0

˜

pxtq12_{. (44)}

We have

E

˜

a;q12

x(0),x(z)= ∞

i=0

˜

pxtqi+12 ₍₄₅₎

and

E

˜

a,q12

x(z0);∞

= ∞

i=0

˜

pxqiz0q 1

2–1_{. (46)}

From (35) and (44) we obtain that:

Ifa˜(x(z)) = –a(x(z)), thenp˜(x(z)) =p(x(z)) andy0(x(z))y˜0(x(z)) =y0(x(z0))y˜0(x(z0)). Hence we have the following.

Theorem 2.1 If y(x(z))and˜y(x(z))are solutions of

Dyx(z)=ax(z)yxq–12_z

and

Dy˜x(z)= –ax(z)y˜xq12_z,

Direct proof

D(yy˜ ) =yxq–12_zD_y˜_x(z)+y˜_xzq21D_yx(z)

= –ax(z)y˜xzq12_yxzq–12_+ax(z)y˜_xzq12_yxzq–12_{= 0}

⇒D(yy˜ ) = 0; yy˜ = const.

As welly(x(z0))y˜(x(z0)) = 1, theny(x(z))y˜(x(z)) = 1. Corollary 2.2

E

a,q– 12

x(z0);x(z)

.E

–a,q12

x(z0);x(z)

= 1 =E

–a,q– 12

x(z0);x(z)

.E

a,q12

x(z0);x(z)

.

Let us consider now the non-homogeneous equations (28) and (29). To find the solution of the non-homogeneous equation

Dyx(z)=ax(z)yxzq–12_{+bx(z), (47)}

we assume thaty0(x(z)) is the solution of the corresponding homogeneous equation

Dyx(z)=ax(z)yxzq–12_{, (48)}

and use the Lagrange method (variation of constants method) by taking

yx(z)=y0

x(z)cx(z) (49) as the solution of (47), wherec(x(z)) is unknown. By placing (49) in (47), we have

Dy0

x(z)cx(z)=ax(z)cxzq–12_y0xzq–12_+bx(z) or

Dcx(z)y0

xzq12_+cxzq–12D_y 0

x(z) =ax(z)cxzq–12_y

0

xzq–12_+bx(z). Then, since

Dy0

x(z)=ax(z)y0

xzq–12_, we get

Dcx(z)y0

xzq12_+cxzq–12_ax(z)y 0

xzq–12 =ax(z)cxzq–12_y0xzq–12_+bx(z)

or

Dcx(z)y0

then

Dcx(z)=y–1₀ xzq12_{bx(z). (51)}

This gives us the following relation:

cx(z)=cx(z0)

+

x(z)

x(z0)

y–1₀ xtq12_bx(t)d

qx(t). (52)

Placing (52) in (49), we obtain

yx(z)=y0

x(z)cx(z0)

+

x(z)

x(z0)

y0

x(z)y–1₀ xtq12_{bx(t)dqx(t), (53)}

hence

cx(z0)

=y–1₀ x(z0)

yx(z0)

. (54)

The relations (53)–(54) give

yx(z)=Φx(z),x(z0)

yx(z0)

+

x(z)

x(z0)

Φx(z0),x

tq12_bx(t)d

qx(t)

, (55)

where

Φ(a,b) =y0(a)y–10 (b). (56)

We do the same with the equation associated to (28), equation (29), using the Lagrange method with

˜

yx(z)=c˜x(z)y˜0

x(z), (57)

wherec˜(x(z)) is an unknown function. Placing (57) in (29), we have

˜

cx(z)=˜cx(z0)

+

x(z)

x(z0)

˜

y–1₀ xtq–12_b˜_{x(t)dqx(t). (58)}

Placing (58) in (57), we obtain

˜

yx(z)=y˜0

x(z)˜cx(z0)

+

x(z)

x(z0)

˜

y0

x(z)y˜–1₀ xtq–12_b˜_{x(t)dqx(t), (59)}

hence

˜

cx(z0)

=˜y–1₀ x(z0)

˜

yx(z0)

. (60)

Now we can write the general solution of (29) like

˜

yx(z)=Φ˜x(z),x(z0)

˜

yx(z0)

+

x(z)

x(z0)

˜

Φx(z0),x

tq–12_b˜_x(t)dqx(t)

where

˜

Φ(a,b) =y˜0(a)y˜–10 (b). (62)

3 q-Non uniform difference integral inequalities

In this section, we will first establishq-non uniform versions of some integral inequali-ties of classical mathematical analysis such as the integral inequaliinequali-ties of Hölder, Cauchy– Schwarz, and Minkowski. It can be seen that the techniques used in classical analysis re-main valid here. Then we establishq-non uniform versions of some other integral inequal-ities based on the linearq-non uniform difference equations of first order and the corre-sponding Lagrange resolution method: the inequalities of Grönwall, Bernoulli; and finally we prove the Lyapunov inequality for the solutions of theq-non uniform Sturm–Liouville equation.

3.1 q-Non uniform Hölder and Cauchy–Schwarz inequalities

Theorem 3.1(q-non uniform Hölder inequality) Let a,b∈[1,∞[∩T.For all real-valued functions f,g,defined and q-non uniform integrable on[a,b],we have

b

a

fx(z)gx(z)dqx(z)

≤ b

a

fx(z)αdqx(z)

1

α b

a

gx(z)βdqx(z)

1

β

(63)

withα> 1and_α1+_β1 = 1.

Proof Let us first show that forA,B∈[1,∞[ we have

A1α_B

1

β ≤A

α +

B

β. (64)

Indeed, since 1_α+_β1 = 1,A,B∈[1,∞[ , A_αα +B_ββ runs through the segment [Aα_,Bβ_{], while}

logAα

α +

logBβ

β runs through the segment linking the points (A

α_,logAα_{) and (A}β_,logAβ_).

By concavity of the logarithm function, we conclude thatlog(A_αα +B_ββ)≥log_αAα +log_βBβ =

log(AB). By applying the exponential to the two members of this inequality, we obtain (64). Let us now take

Ax(z)= |f(x(z))|

α

b

a |f(x(z))|αdqx(z)

and Bx(z)= |g(x(z))|

β

b

a |g(x(z))|βdqx(z)

,

considering that

b

a

fx(z)αdqx(z) b

a

gx(z)βdqx(z)

= 0 (65)

integrating on [a,b], we have

b

a

|f(x(z))| (_ab|f(x(z))|α_d

qx(z))

1

α

|g(x(z))| (_ab|g(x(z))|β_d

qx(z))

1

β dqx(z)

=

b

a

A1α_B

1

β _d

qx(z)

≤ b a A α + B β

dqx(z)

= b a 1 α

|f(x(z))|α

b

a |f(x(z))|αdqx(z)

+ 1

β

|g(x(z))|β

b

a |g(x(z))|βdqx(z)

dqx(z)

=1

α b

a

|f(x(z))|α

b

a |f(x(z))|αdqx(z)

dqx(z) +

1

β b

a

|g(x(z))|β

b

a |g(x(z))|βdqx(z)

dqx(z)

=1

α +

1

β = 1,

which gives us directly theq-non uniform Hölder inequality

b

a

fx(z)gx(z)dqx(z)≤ b

a

fx(z)αdqx(z)

1

α b

a

gx(z)βdqx(z)

1

β

. (66)

If we takeα=β= 2 in theq-non uniform Hölder inequality (63), we have theq-non uniform Cauchy–Schwarz inequality.

Corollary 3.1 (q-non uniform Cauchy–Schwarz inequality) Let a,b∈[1,∞[∩T. For all real-valued functions f,g,defined and q-non uniform integrable on[a,b],we have

b

a

fx(z)gx(z)dqx(z)≤ b

a

fx(z)2dqx(z) b

a

gx(z)2dqx(z)

. (67)

3.2 q-Non uniform Minkowski inequality

We can now use the q-non uniform Hölder inequality to deduce the q-non uniform Minkowski inequality.

Theorem 3.2(q-non uniform Minkowski inequality) Let a,b∈[1,∞[∩T.For all real-valued functions f,g,defined and q-non uniform integrable on[a,b],we have

b

a

(f +g)x(z)αdqx(z)

1

α

≤ b

a

fx(z)αdqx(z)

1

α

+

b

a

gx(z)αdqx(z)

1

α

(68)

Proof We have

b

a

(f +g)x(z)αdqx(z) = b

a

(f +g)x(z)α–1(f+g)x(z)dqx(z)

≤ b

a

(f +g)x(z)α–1fx(z)dqx(z)

+

b

a

(f+g)x(z)α–1gx(z)dqx(z). (69)

Using theq-non uniform Hölder inequality withβ=_αα_–1, we will obtain

b

a

(f +g)x(z)αdqx(z)

≤ b

a

fx(z)αdqx(z)

1

α b

a

(f +g)x(z)(α–1)β|dqx(z)

1

β

+

b

a

gx(z)αdqx(z)

1

α b

a

(f+g)x(z)(α–1)β|dqx(z)

1

β

=

b

a

fx(z)αdqx(z)

1

α

+

b

a

gx(z)αdqx(z)

1

α

× b

a

(f+g)x(z)α|dqx(z)

1

β

. (70)

Dividing both sides of this inequality by{_ab|(f +g)(x(z))|α_|d qx(z)}

1

β_{, we have}

b

a

(f +g)x(z)αdqx(z)

1–_β1

≤ b

a

fx(z)αdqx(z)

1

α

+

b

a

gx(z)αdqx(z)

1

α

. (71)

As 1 –_β1=1_α, this gives us theq-non uniform Minkowski inequality.

3.3 q-Non uniform Grönwall inequality

Let us first introduce the following inequalities based on the Lagrange method for the linearq-non uniform difference non-homogeneous equations.

Lemma 3.1 Let y, f be real-valued functions defined and q-non uniform integrable on

[c,d],∀c,d∈[1,∞[∩T.Let a(x(z))such that p(x(z)) = 1 + (q12 –q–12)(z–z–1)a(x(z)) > 0.

That means a(x(z))≥0.

Suppose that y0(x(z))is a solution of

Dy0

x(z)=ax(z)y0

q–12_{x(z) (72)}

If

Dyx(z)≤ax(z)yxzq–12_+fx(z) ∀_x(z)∈_[1,∞_{[, (73)}

then

yx(z)≤y0

x(z)yx(z0)

+y0

x(z)

x(z)

x(z0)

y–1₀ xtq12_{fx(t)dqx(t). (74)}

Proof Lety0(x(z)) be the solution of the homogeneous equation

Dy0

x(z)=ax(z)y0

xzq–12 ₍₇₅₎

such thaty0(x(z0)) = 1. By looking for the functiony(x(z)) satisfying (73) by the Lagrange method

yx(z)=cx(z)y0

x(z), (76)

wherec(x(z)) is indeterminate, we replace (76) in (73) and we have

Dcx(z)y0

x(z)≤ax(z)cxzq12_y0xzq–12_{+fx(z) (77)}

or

y0

xzq12D_cx(z)+cxzq–12D_y0x(z)

≤ax(z)cx(z)y0

xzq12_{+fx(z). (78)}

Using (75), we have

y0

xzq12D_cx(z)+cxzq–12_ax(z)y0xzq–12

≤ax(z)cxzq–12_y 0

xzq–12_{+fx(z). (79)}

Simplifying this gives

y0

xzq12D_cx(z)≤_{fx(z) (80)}

or

Dcx(z)≤y–1₀ xzq12_{fx(z), (81)}

sincea(x(z))∈R+ _impliesy

0(x(z)) > 0. By integrating the two members of the equality fromx(z0) tox(z), we will have

cx(z)–cx(z0)

≤ x(z)

x(z0)

However, from (76) followsc(x(z0)) =y(x(z0)), given thaty0(x(z0)) = 1. Hence

cx(z)≤yx(z0)

+

x(z)

x(z0)

y–1₀ xtq12_{fx(t)dqx(t). (83)}

Placing (83) in (76), we get

yx(z)=cx(z)y0

x(z)

≤y0

x(z)yx(z0)

+

x(z)

x(z0)

y–1₀ xtq12_fx(t)dqx(t)

, (84)

which gives the desired result

yx(z)≤y0

x(z)yx(z0)

+y0

x(z)

x(z)

x(z0)

y–1₀ xtq12_{fx(t)dqx(t). (85)}

Taking account of Corollary2.2and the fact that by the definitionE

a,q– 12(x(z0);x(z0)) =

E

a,q12(x(z0);x(z0)) = 1, we obtain the following.

Corollary 3.2 If y,f,and a are functions satisfying the conditions of Lemma3.1,then

yx(z)≤E

a;q– 12

x(z0),x(z)

yx(z0)

+E

a;q– 12

x(z0),x(z) x(z)

x(z0)

E

–a;q12

x(z0),x

tq12_{fx(t)dqx(t). (86)}

Lemma 3.2 Let y, f be real-valued functions defined and q-non uniform integrable on

[c,d],∀c,d∈[1,∞[∩T.Let a(x(z))such that

px(z)= 1 –q12_–q–12_z–z–1_{ax(z)> 0.}

This means that a(x(z))≤0.

Suppose that y0(x(z))is a solution of

Dy0

x(z)=ax(z)y0

xzq12

such that y0(x(z0)) = 1.

In that case,if

Dyx(z)≤ax(z)y(xzq12_+fx(z), ∀_x(z)∈_[1,∞_{[ , (87)}

then,for all x(z)∈[1,∞[,we have

yx(z)≤y0

x(z)yx(z0)

+y0

x(z)

x(z)

x(z0)

Proof Lety0(x(z)) be a solution of the homogeneous equation

Dy0

x(z)=ax(z)y0

xzq12_{, y0x(z0)= 1. (89)} Looking for the functiony(x(z)) satisfying (87) by the Lagrange method

yx(z)=cx(z)y0

x(z), (90)

wherec(x(z)) is indeterminate, we replace the functiony(x(z)) in (87) and we have

Dcx(z)y0

x(z)≤ax(z)cxzq21_y0xzq21_+fx(z) or

cxzq12D_y0x(z)+y0xzq–12D_cx(z)

≤ax(z)cxzq12_y0xzq12_+fx(z).

Using (89), we have

cxzq12_ax(z)y0xzq12_+y0xzq–12D_cx(z)

≤ax(z)cxzq12_y 0

xzq12_{+fx(z). (91)} Simplifying this gives

y0

xzq–12D_cx(z)≤_{fx(z) (92)} or

Dcx(z)≤y–1₀ xzq–12_{fx(z) (93)}

sincea(x(z))≤0 impliesy0(x(z)) > 0. By integrating the two members of the equality above fromx(z0) tox(z), we will have

cx(z)–cx(z0)

≤ x(z)

x(z0)

y–1₀ xtq–12_{fx(t)dqx(t). (94)}

However, from (90) followsc(x(z0)) =y(x(z0)), given thaty0(x(z0)) = 1. This gives

cx(z)≤yx(z0)

+

x(z)

x(z0)

y–1₀ xtq–12_{fx(t)dqx(t). (95)}

Then we obtain

yx(z)=cx(z)y0

x(z)

≤y0

x(z)yx(z0)

+

x(z)

x(z0)

y–1₀ xtq–12_fx(t)dqx(t)

which gives the desired result

yx(z)≤y0

x(z)yx(z0)

+y0

x(z)

x(z)

x(z0)

y–1₀ xtq–12_{fx(t)dqx(t). (97)}

As well as for Corollary3.2, we deduce the following.

Corollary 3.3 If y,f,and a satisfy the conditions of Lemma3.2,then

yx(z)≤E

a;q12

x(z0),x(z)

yx(z0)

+E

a;q12

x(z0),x(z)

× x(z)

x(z0)

E

–a;q– 12

x(z0),x

tq–12_fx(t)d

qx(t). (98)

Theorem 3.3(q-non uniform Grönwall inequality) Let y,f be real-valued and q-non uni-form integrable functions on[c,d],∀c,d∈[1,∞[∩T,and a≥0.If

yx(z)≤fx(z)+

x(z)

x(z0)

yxtq–12_{ax(t)dqx(t), (99)}

then

yx(z)≤fx(z) +E

a,q– 12

x(z0),x(z)

× x(z)

x(z0)

ax(t)fxtq–12_E –a,q12

x(z0);x

tq12_{dqx(t). (100)}

Proof Let us define

νx(z)=

x(z)

x(z0)

yxtq–12_ax(t)d

qx(t). (101)

Thenν(x(z0)) = 0 andDν=y(x(zq– 1

2_))a(x(z)). Hence hypothesis (99) gives

yx(z)≤fx(z)+νx(z) (102)

and

Dνx(z)=yxzq–12_ax(z)≤_fxzq–12_+νxzq–12_{ax(z), (103)}

or

From Lemma3.1, inequality (104) gives

νx(z)≤νx(z0)

E

a,q– 12

x(z0);x(z)

+E

a,q– 12

x(z0);x(z)

× x(z)

x(z0)

ax(t)fxtq–12_E –a,q12

x(z0);x

tq12_{dqx(t), (105)}

and inequality (102) implies that (withν(x(z0)) = 0)

yx(z)≤fx(z) +E

a,q– 12

x(z0);x(z)

× x(z)

x(z0)

ax(z)fxtq–12_E –a,q12

x(z0);x

tq12_d

qx(t), (106)

which is theq-non uniform Grönwall inequality. As a direct consequence, we get the following results.

Corollary 3.4 Let y,f be real-valued functions defined and q-non uniform integrable on

[c,d],∀c,d∈[1,∞[∩Tand a(x(z))such that1 + (q12–q–12)(z–z–1)a(x(z)) > 0.That means

a(x(z))≥0.Then

yx(z)≤

x(z)

x(z0)

yxtq–12_ax(t)d

qx(t) (107)

for all x(z)implies that y(x(z))≤0.

Proof This is due to Theorem3.3withf(x(z))≡0 Corollary 3.5 Let a≥0andα∈R.If

yx(z)≤α+

x(z)

x0

yxtq–12_ax(t)d

qx(t) (108)

for all x0=x(z0) > 0,then y(x(z))≤αE

a,q– 12(x0,x(z)).

Proof By theq-non uniform Grönwall integral inequality (100), if we takef(x(z)) =α, then

yx(z)≤α+E

a,q– 12

x0,x(z) x(z)

x0

αax(t)E

–a,q12

x0,x

tq12_d

qx(t)

=α

1 –E

a,q– 12

x0,x(z) x(z)

x0

DE

–a,q12

x0,x(z)

dqx(t)

=α1 –E

a,q– 12

x0,x(z)

E

–a,q12

x0,x(z)

–E

–a,q12(x0,x0)

=α–αE

a,q– 12

x0,x(z)

E

–a;q12

x0;x(z)

+αE

a,q– 12

x0,x(z)

=αE

a,q– 12

x0,x(z)

3.4 q-Non uniform Bernoulli inequality

Theorem 3.4(q-non uniform Bernoulli inequality) Letα∈R.Then∀x(z), x0=x(z0)∈ [1,∞[with x(zq–12_{) >x0,we have}

E

a,q– 12

x0,x(z)

≥1 +αx(z) –x0

. (109)

Proof Let us takey(x(z)) =α(x(z) –x0),x(zq– 1

2) >x₀,∀z. ThenDy(x(z)) =_αand we have

αyxzq–12_+α=α2_xzq–12_–x 0

+α≥α=Dyx(z),

which implies thatDy(x(z))≤αy(x(zq–12_{)) +α.}

On the other hand, by Lemma3.1, we get (withy(x0) = 0)

yx(z)≤y(x0)E

a,q– 12

x0,x(z)

+E

a,q– 12

x0,x(z) x(z)

x0

aE

–a,q12

x0,x

tq12_dqx(t)

=E

a,q– 12

x0,x(z) x(z)

x0

(–)DE

–a,q12

x0,x(t)

dqx(t)

= –E

a,q– 12

x0,x(z)

E

–a,q12

x0,x(z)

– 1

= –1 +E

a,q– 12

x0,x(z)

.

That is whyE

a,q– 12(x0,x(z))≥1 +y(x(z)) = 1 +α(x(z) –x0).

3.5 q-Non uniform Lyapunov inequality

Let us now turn toq-non uniform Lyapunov inequality regarding the derivative and inte-gral introduced. For that, consider the followingq-non uniform Sturm–Liouville equation:

DD+_{ux(z)+fx(z)uxzq}1_{2= 0, (110)}

whereD+_{is defined byD}+_{f(x(z)) =}f(x(zq))–f(x(z))

x(zq)–x(z) . Let us define the functionFby

F(y) =

b

a

Dyx(z)2–fx(z)yxzq122_d

qx(z). (111)

Lemma 3.3 Let u be a non-trivial solution of the q-non uniform Sturm–Liouville equation

(110).Then,for all y belonging to the domain of F,the following equality remains verified:

F(y) –F(u) –F(y–u) = 2(y–u)(b)D+u(b) – 2(y–u)(a)D+u(a). (112)

Proof We have

F(y) –F(u) –F(y–u) =

b

a

–Dux(z)2–fx(z)yxq12_z–D_(y–u)x(z)2 –fx(z)(y–u)xq12_z2_d

qx(z)

= 2

b

a

–Dux(z)2+fx(z)uxzq122₊D_yx(z)D_ux(z) –fx(z)yxzq12_uxzq12_dqx(z)

= 2

b

a

–Dux(z)2–uxzq12DD+_ux(z)+D_yx(z)D_ux(z) +yxzq12DD+_ux(z)d

qx(z)

= 2

b

a

Dyx(z)Dux(z)+yxzq12DD+_ux(z)

–Dux(z)2+uxzq12DD+_ux(z)dqx(z),

(usingf(x(z))u(x(zq12_{)) = –}DD+_{u(x(z))), or}

F(y) –F(u) –F(y–u) = 2

b

a

Dyx(z)D+ux(z)–Dux(z)D+ux(z)dqx(z)

= 2

b

a

Dyx(z)–ux(z)D+ux(z)dqx(z)

= 2y(b) –u(b)D+u(b) – 2y(a) –u(a)D+u(a). Lemma 3.4 Let y be in the domain of F,then∀a,b∈[1,∞[∩Tand c,d∈[a,b]∩Tsuch that a≤c≤d≤b,we have

d

c

Dyx(z)2dqx(z)≥

(y(d) –y(c))2

d–c . (113)

Proof Let us take

ux(z)=y(d) –y(c)

d–c x(z) +

dy(c) –cy(d)

d–c .

ThenD+_{u(x(z)) =}y(d)–y(c)

d–c andDD+u(x(z)) = 0, which proves thatu(x(z)) is a solution of

(110) withf(x(z)) = 0,∀x(z)∈[1,∞[∩Tand

F(y) =

b

a

Dyx(z)2dqx(z),

∀y(x(z)) from the domain ofF. From Lemma3.3, we get

F(y) –F(u) –F(y–u) = 2(y–u)(b)D+u(b) – 2(y–u)(a)D+u(a) = 0. Consequently,

This leads us to the following result:

d

c

Dyx(z)2dqx(z)≥ d

c

Dux(z)2dqx(z)

=

d

c

y(d) –y(c)

d–c

2

dqx(z)

=(y(d) –y(c)) 2

d–c .

Theorem 3.5(q-non uniform Lyapunov inequality) Let f be a real-valued function de-fined and q-non uniform differentiable on[a,b],∀a,b∈[1,∞[∩T,a<b,and let u(x(z))be a non-trivial solution of equation(110)with u(a) =u(b) = 0,then

b

a

fx(z)dqx(z) = – b

a

DD+_u(x(z))

u(x(zq12))

dqx(z)≥

4

b–a. (114) Proof From Lemma3.3withy= 0 andu(a) =u(b) = 0, we have

F(u) =

b

a

Dux(z)2–fx(z)uxzq122_d

qx(z) = 0. (115)

LetM=max{u2(x(zq21));_x(_q12_z)∈[_a,_b]∩T}and_c∈[_a,_b]∩Tsuch that_u2(_c) =_M. Then

M=u2(c)≥u2xzq12_{> 0,} and

M

b

a

fx(z)dqx(z)≥ b

a

fx(z)u2xzq12_dqx(z)

=

b

a

Dux(z)2dqx(z)

=

c

a

Dux(z)2dqx(z) + b

c

Dux(z)2dqx(z)

≥(u(c) –u(a))2

c–a +

(u(b) –u(c))2

b–c

=M

1

c–a+

1

b–c

u(a) =u(b) = 0 andu2(c) =M

=M

(a+b– 2c)2 (c–a)(b–c)(b–a)+

4

b–a

≥M 4

b–a,

which implies that_abf(x(z))dqx(z)≥M_b–4_a.

Theq-non uniform Lyapunov inequality is thus proved. 4 Conclusion

Bernoulli based on the Lagrange method of linearq-non uniform difference equations of first order were established. Finally, the Lyapunov inequality for the solutions of the

q-non uniform Sturm–Liouville equation was proved.

Acknowledgements

The authors acknowledge the referees for noting some misprints and communicating some recent references on q-calculus. JPN acknowledges the Doctoral School of the University of Burundi for support.

Funding

The funding for this article comes from the Doctoral School of the University of Burundi in which the doctoral research of Jean Paul Nuwacu (one of the authors of this work) is conducted and supported.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Author details

1_{Department of Mathematics, Faculty of Science, University of Burundi, Bujumbura, Burundi.}2_{Department of}

Mathematics, Faculty of Science, Menoufia University, Menoufia, Egypt.

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Received: 9 January 2019 Accepted: 22 January 2019

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