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A variety of dynamic α-conformable Steffensen-type inequality on a time scale measure space

  • The main objective of this work is to establish several new alpha-conformable of Steffensen-type inequalities on time scales. Our results will be proved by using time scales calculus technique. We get several well-known inequalities due to Steffensen, if we take α=1. Some cases we get continuous inequalities when T=R and discrete inequalities when T=Z.

    Citation: Ahmed A. El-Deeb, Osama Moaaz, Dumitru Baleanu, Sameh S. Askar. A variety of dynamic α-conformable Steffensen-type inequality on a time scale measure space[J]. AIMS Mathematics, 2022, 7(6): 11382-11398. doi: 10.3934/math.2022635

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  • The main objective of this work is to establish several new alpha-conformable of Steffensen-type inequalities on time scales. Our results will be proved by using time scales calculus technique. We get several well-known inequalities due to Steffensen, if we take α=1. Some cases we get continuous inequalities when T=R and discrete inequalities when T=Z.



    Riemann-Liouville fractional integral given by

    Iαa+ξ()=1Γ(α)χa(χ)α1ξ()dt.

    Many different concepts of fractional derivative maybe found in [9,10,11]. In [12] studied a conformable derivative:

    αf()=limϵ0f(+ϵ1α)f()ϵ.

    The time scale conformable derivatives was introduced by Benkhettou et al. [17].

    Further, in recent years, numerous mathematicians claimed that non-integer order derivatives and integrals are well suited to describing the properties of many actual materials, such as polymers. Fractional derivatives are a wonderful tool for describing memory and learning. a variety of materials and procedures inherited properties is one of the most significant benefits of fractional ownership. For more concepts and definition on time scales see [13,14,15,16,17,18,19,33,34,35].

    Continuous version of Steffensen's inequality [7] is written as: For 0g()1 on [a,b]. Then

    bbλf()dtbaf()g()dta+λaf()dt, (1.1)

    where λ=bag()dt.

    Supposing f is nondecreasing gets the reverse of (1.1).

    Also, the discrete inequality of Steffensen [6] is: For λ2n=1g()λ1. Then

    n=nλ2+1f()n=1f()g()λ1=1f(). (1.2)

    Recently, a large number of dynamic inequalities on time scales have been studied by a small number of writers who were inspired by a few applications (see [1,2,3,4,8,28,29,30,31,32,36,37,40,41,42,44,48,49,50,51,52,53]).

    In [5] Jakšetić et al. proved that, if ˆμ([c,d])=[a,b]g()dˆμ(), where [c,d][a,b]. Then

    [a,b]f()g()dˆμ()[c,d]f()g()dˆμ()+[a,c](f()f(d))g()dˆμ(),

    and

    [c,d]f()dˆμ()[d,b](f(c)f())g()dˆμ()[a,b]f()g()dˆμ().

    Anderson, in [3], studied the inequality:

    bbλϕ()baϕ()ψ()a+λaϕ(), (1.3)

    In [47] the authors have proved, for

    m+λ1mζ()d=kmζ()g()d,

    and

    nnλ2ζ()d=nkζ()g()d.

    If there exists a constant A such that r()/ζ()At is monotonic on the intervals [m,k], [k,n], and

    nmtq()g()d=m+λ1mtq()d+nnλ2tq()d,

    then

    nmr()g()dm+λ1mr()d+nnλ2r()d.

    In particularly, Anderson [3] proved

    nnλr()nmr()g()m+λmr().

    where m,nTκ with m<n, r, g:[m,n]TR are -integrable functions such that r is of one sign and nonincreasing and 0g()1 on [m,n]T and λ=nmg(), nλ,m+λT.

    We prove the next two needed results:

    Theorem 1.1. Assume q>0 with 0g()ζ() [m,n]T and λ is given from nmg()Δα=m+λmζ()Δα, then

    nmr()g()Δαm+λmr()ζ()Δα. (1.4)

    Also, provided with 0g()ζ() and nnλζ()Δα=nmg()Δα, we have

    nnλr()ζ()Δαnmr()g()Δα. (1.5)

    We get the reverse inequalities of (1.4) and (1.5) when assuming r/ζ is nondecreasing.

    Theorem 1.2. Assume ψ is integrable on time scales interval [m,n], with ζ()ψ()g()ψ()0[m,n]T and m+λmζ()Δα=nmg()Δα=nnλζ()Δα and g, r and ζ are Δα-integrable functions, ζ()g()0, we have

    nnλr()ζ()Δα+nm|(r()r(nλ))ψ()|Δαnmr()g()Δαm+λmr()ζ()Δαnm|(r()r(m+λ))ψ()|Δα, (1.6)

    and

    nnλr()ζ()Δαnnλ[r()ζ()(r()r(nλ))][ζ()g()]Δαnmr()g()Δαm+λm[r()ζ()(r()r(m+λ))][ζ()g()]Δαm+λmr()ζ()Δα. (1.7)

    Proof. The proof techniques of Theorems 1.6 and 1.7 are like to that in [4] and is removed.

    Several authors proved conformable Hardy's inequality [20,21], conformable Hermite-Hadamard's inequality [22,23,24], conformable inequality of Opial's [26,27] and conformable inequality of Steffensen's [25]. In [45] Anderson proved the followong results:

    Theorem 1.3. [45] Suppose α(0,1] and r1, r2R such that 0r1r2. Suppose :[r1,r2][0,) and Γ:[r1,r2][0,1] are α-fractional integrable functions on [r1,r2] with Π is decreasing, we get

    r2r2Π(ζ)dαζr2r1Π(ζ)Γ(ζ)dαζr1+r1Π(ζ)dαζ,

    where =α(r2r1)rα2rα1r2r1Γ(ζ)dαζ[0,r2r1].

    In [46] the authors gave an extension for Theorem 1.8:

    Theorem 1.4. Assume α(0,1] and r1, r2R such that 0r1r2. Suppose ,Γ,Σ:[r1,r2][0,) are integrable on [r1,r2] with the decreasing function Π and 0ΓΣ, we get

    r2r2Σ(ζ)Π(ζ)dαζr2r1Π(ζ)Γ(ζ)dαζr1+r1Σ(ζ)Π(ζ)dαζ,

    where =(r2r1)r2r1Σ(ζ)dαζr2r1Γ(ζ)dαζ[0,r2r1].

    In this paper, we prove and explore several novel speculations of the Steffensen inequality obtained in [47] through the conformable integral containing time scale concept. We furthermore recover certain known results as special cases of our results.

    Lemma 2.1. Assume ζ>0 is rd-continuous function on [m,n]T, g, r be rd-continuous on [m,n]T such that r/ζ nonincreasing function and 0g()1 [m,n]T. Then

    (Λ1)

    nmr()g()Δαm+λmr()Δα, (2.1)

    where λ is given by

    nmζ()g()Δα=m+λmζ()Δα.

    (Λ2)

    nnλr()Δαnmr()g()Δα, (2.2)

    such that

    nnλζ()Δα=nmζ()g()Δα.

    (2.1) and (2.2) are reversed when r/ζ is nondecreasing.

    Proof. Putting g()ζ()g() and r()r()/ζ() in (1.4), (1.5) to get (Λ1) and (Λ2) simultaneously.

    Lemma 2.2. Under the same hypotheses of Lemma 2.1. with ψ be integrable functions on [m,n]T and 0ψ()g()1ψ() for all [m,n]T. Then

    nnλr()Δα+nm|(r()ζ()r(nλ)ζ(nλ))ζ()ψ()|Δαnmr()g()Δαm+λmr()Δαnm|(r()ζ()r(m+λ)ζ(m+λ))ζ()ψ()|Δα,

    where λ is obtained from

    m+λmh()Δα=nmζ()g()Δα=nnλζ()Δα.

    Proof. Putting g()ζ()g(), r()r()/h() and ψ()ζ()ψ() in (1.6).

    Lemma 2.3. Under the same conditions of Lemma 2.1. Then

    nnλr()Δαnnλ(r()[r()ζ()r(nλ)ζ(nλ)]ζ()[1g()])Δαnmr()g()Δαm+λm(r()[r()ζ()r(a+λ)ζ(m+λ)]ζ()[1g()])Δαm+λmr()Δα,

    where λ is obtained from

    m+λmζ()Δα=nmg()Δα=nnλζ()Δα.

    Proof. Taking g()ζ()g() and r()r()/ζ() in (1.7).

    Theorem 2.1. Under the same conditions of Lemma 2.3 such that k(m,n) and λ1, λ2 are given from

    (Λ3)

    m+λ1mζ()Δα=kmζ()g()Δα,
    nnλ2ζ()Δα=nkζ()g()Δα.

    If rσ/ζAHk1[m,n] and

    nmϕ()ζ()g()Δα=m+λ1mϕ()ζ()Δα+nnλ2ϕ()ζ()Δα, (2.3)

    then

    nmrσ()g()Δαm+λ1mrσ()Δα+nnλ2rσ()Δα. (2.4)

    (2.4) is reversed if rσ/ζAHk2[m,n] and (2.3).

    (Λ4)

    kkλ1ζ()Δα=kmζ()g()Δα,
    k+λ2kζ()Δα=nkζ()g()Δα.

    If rσ/ζAHk1[m,n] and

    nmϕ()ζ()g()Δα=k+λ2kλ1ϕ()ζ()Δα, (2.5)

    then

    nmrσ()g()Δαk+λ2kλ1rσ()Δα. (2.6)

    If rσ/ζAHk2[m,n] and (2.5) satisfied, then we reverse (2.6).

    (Λ5) If λ1, λ2 be the same as in (Λ3) and rσ/ζAHk1[m,n] so that

    nmϕ()ζ()g()Δα=m+λ1m(ϕ()ζ()[ϕ()mλ1]ζ()[1g()])Δα+nnλ2(ϕ()ζ()[ϕ()n+λ2]ζ()[1g()])Δα, (2.7)

    then

    nmrσ()g()Δαm+λ1m(rσ()|rσ()ζ()rσ(m+λ1)ζ(m+λ1)|ζ()[1g()])Δα+nnλ2(rσ()|rσ()ζ()rσ(nλ2)ζ(nλ2)|ζ()[1g()])Δα. (2.8)

    If rσ/ζAHk2[m,n] and (2.7) satisfied, the inequality in (2.8) is reversed.

    (Λ6) If λ1, λ2 be defined as in (Λ4) and rσ/ζAHk1[m,n] and

    nmϕ()ζ()g()Δα=kkλ1(ϕ()ζ()[ϕ()k+λ1]ζ()[1g()])Δα=m+λ1m(ϕ()ζ()[ϕ()k+λ2]ζ()[1g()])Δα, (2.9)

    then

    nmrσ()g()Δαkkλ1(rσ()[rσ()ζ()rσ(kλ1)ζ(kλ1)]ζ()[1g()])Δα+k+λ2k(rσ()[rσ()ζ()rσ(k+λ2)ζ(k+λ2)]ζ()[1g()])Δα. (2.10)

    If rσ/ζAHk2[m,n] and (2.9) satisfied, we reverse (2.10).

    Proof. (Λ3) Consider rσ/ζAHk1[m,n], and R1()=rσ()Aϕ()ζ(), since A is given in Definition 2.1. Since R1/ζ:[m,k]TR, using Lemma 2.1(Λ1), we deduce

    0m+λ1mR1()ΔαkmR1()g()Δα=m+λ1mrσ()Δαkmrσ()g()ΔαA(m+λ1mϕ()ζ()Δαkmϕ()ζ()g()Δα). (2.11)

    As R1/ζ:[k,n]TR is nondecreasing, using Lemma 2.1(Λ2), we obtain

    0nkR1()g()Δαnnλ2R1()Δα=nkrσ()g()Δαnnλ2rσ()ΔαA(nkϕ()ζ()g()Δαnnλ2ϕ()ζ()Δα). (2.12)

    (2.11) and (2.12) imply that

    m+λ1mrσ()Δα+nnλ2rσ()Δαnmrσ()g()ΔαA(m+λ1mϕ()ζ()Δα+nnλ2ϕ()ζ()Δαnmϕ()ζ()g()Δα)

    Hence, if (2.3) is hold, then (2.4) holds. For rσ/ζAHk2[m,n], we get the some steps.

    (Λ4) Let rσ/ζAHk1[m,n], also R1(x)=rσ(x)Aϕ(x)ζ(x), where A as in Definition 2.1. R1/ζ:[m,k]TR is nonincreasing, so from Lemma 2.1(Λ1) we obtain

    0kmrσ()g()Δαkkλ1rσ()ΔαA(kmϕ()h()g()Δαkcλ1ϕ()ζ()Δα). (2.13)

    Using Lemma 2.1(Λ1) we have

    0k+λ2krσ()Δαnkrσ()g()ΔαA(k+λ2kϕ()ζ()Δαnkϕ()ζ()g()Δα). (2.14)

    Thus, from (2.13), (2.14), we get

    nmrσ()g()Δαk+λ2kλ1rσ()ΔαA(nmϕ()ζ()g()Δαk+λ2kλ1ϕ()ζ()Δα)

    Therefore, if nmϕ()ζ()g()Δα=k+λ2kλ1ϕ()ζ()Δα is satisfied, then (2.8) holds. Follow the same steps for rσ/ζAHk2[m,n].

    Using Lemma 2.3 and repeat the steps of Theorem 2.1(Λ3) and Theorem 2.1(Λ4) in the proof of (Λ5) and (Λ6) respectively.

    Corollary 2.1. The inequalities (2.4), (2.6), (2.8) and (2.10) of Theorem 2.1 letting T=R takes

    (i)nmfσ()g()dαm+λ1mrσ()dα+nnλ2rσ()dα. (2.15)
    (ii)nmrσ()g()dαk+λ2kλ1rσ()dα. (2.16)
    (iii)nmrσ()g()dαm+λ1m(rσ()[rσ()ζ()rσ(m+λ1)ζ(m+λ1)]ζ()[1g()])dα+nnλ2(rσ()[rσ()ζ()rσ(nλ2)ζ(nλ2)]ζ()[1g()])dα. (2.17)
    (iv)nmrσ()g()dαkkλ1(rσ()[rσ()ζ()rσ(kλ1)ζ(kλ1)]ζ()[1g()])dα+k+λ2k(rσ()[rσ()ζ()rσ(k+λ2)ζ(k+λ2)]ζ()[1g()])dα. (2.18)

    Corollary 2.2. We get [47,Theorems 8,10,21 and 22], if we put α=1 and ϕ()= in Corollary 2.1 [(i),(ii),(iii),(iv)] simultaneously.

    Corollary 2.3. In Corollary 2.1 taking T=Z, the results (2.15)–(2.18) will be equivalent to

    (i)n1=mr(+1)g()α1m+λ11=mr(+1)+n1=nλ2r(+1)α1.
    (ii)n1=mr(+1)g()α1k+λ21=kλ1r(+1)α1.
    (iii)n1=mr(+1)g()α1m+λ11=m(r(+1)[r(+1)ζ()r(a+λ1+1)ζ(m+λ1)]ζ()[1g()])α1+n1=nλ2(r(+1)[r(+1)ζ()r(nλ2+1)ζ(nλ2)]ζ()[1g()])α1.
    (iv)n1=mr(+1)g())α1k1=kλ1(r(+1)[r(+1)ζ()r(kλ1+1)ζ(kλ1)]ζ()[1g()]))α1+k+λ21=k(r(+1)[r(+1)ζ()r(k+λ2+1)ζ(k+λ2)]ζ()[1g()]))α1.

    Theorem 2.2. Under the assumptions in Lemma 2.1 with 0g()ζ() and λ1, λ2 be defined as

    (Λ7)

    m+λ1mζ()Δα=kmg()Δα,
    nnλ2ζ()Δα=nkg()Δα.

    If rσ/ζAHk1[m,n] and

    nmϕ()g()Δα=m+λ1mϕ()ζ()Δα+nnλ2ϕ()ζ()Δα, (2.19)

    then

    nmrσ()g()Δαm+λ1mrσ()ζ()Δα+nnλ2rσ()ζ()Δα. (2.20)

    (Λ8)

    kkλ1ζ()Δα=kmg()Δα,
    k+λ2kζ()Δα=nkg()Δα.

    If rσ/ζAHk1[m,n] and

    nmϕ()g()Δα=k+λ2kλ1ϕ()ζ()Δα, (2.21)

    then

    nmrσ()g()Δαk+λ2kλ1rσ()ζ()Δα. (2.22)

    If rσ/ζAHk2[m,n] and (2.19), (2.21) satisfied, we get the reverse of (2.20) and (2.22).

    Proof. By using Theorem 2.1 [(Λ3),(Λ4)] and by putting gg/h and ffh, we get the proof of (Λ7) and (Λ8).

    Corollary 2.4. In Theorem 2.2 [(Λ7),(Λ8)], assuming T=R, the following results obtains:

    (i)nmrσ()g()dαm+λ1mrσ()ζ()dα+nnλ2rσ()ζ()dα. (2.23)
    (ii)nmrσ()g()dαk+λ2kλ1rσ()ζ()dα. (2.24)

    Corollary 2.5. In Corollary 2.4 [(i),(ii)], when we put α=1 and ϕ()= then [47,Theorems 16 and 17] gotten.

    Corollary 2.6. In (2.23) and (2.24) letting T=Z, gets

    (i)n1=mr(+1)g()α1m+λ11=mr(+1)h()+n1=nλ2r(+1)h()α1.
    (ii)n1=mr(+1)g()α1k+λ21=kλ1r(+1)ζ()α1.

    Theorem 2.3. Using the same conditions in Lemma 2.3. Letting w:[m,n]TR be integrable with 0g()w() [m,n]T and

    (Λ9)m+λ1mw()ζ()Δα=kmζ()g()Δα,
    nnλ2w()ζ()Δα=nkζ()g()Δα.

    If rσ/ζAHk1[m,n] and

    nmϕ()ζ()g()Δα=m+λ1mϕ()w()ζ()Δα+nnλ2ϕ()w()ζ()Δα, (2.25)

    then

    nmrσ()g()Δαm+λ1mrσ()w()Δα+nnλ2rσ()w()Δα. (2.26)
    (Λ10)kkλ1w()ζ()Δα=kmζ()g()Δα,
    k+λ2kw()ζ()Δα=nkζ()g()Δα.

    If rσ/ζAHk1[m,n] and

    nmϕ()ζ()g()Δα=k+λ2kλ1ϕ()w()ζ()Δα, (2.27)
    nmrσ()g()Δαk+λ2kλ1rσ()w()Δα. (2.28)

    The inequalities in (2.26) and (2.28) are reversible if rσ/ζAHc2[a,b] and (2.25), (2.27) hold.

    Proof. In Theorem 2.1 [(Λ3),(Λ4)], ζ changes wq, g changes g/w and r changes rw.

    Corollary 2.7. In (2.26) and (2.28). Letting T=R, we have

    (i)nmrσ()g()dαm+λ1mrσ()w()dα+nnλ2rσ()w()dα. (2.29)
    (ii)nmrσ()g()dαk+λ2kλ1rσ()w()dα. (2.30)

    Corollary 2.8. In Corollary 2.7 [(i),(ii)], letting α=1 and ϕ()= we get [47,Theorems 18 and 19].

    Corollary 2.9. In (2.29) and (2.30), crossing T=Z, gets

    (i)n1=mr(+1)g()α1m+λ11=mr(+1)w()+n1=nλ2r(+1)w()α1.
    (ii)n1=mr(+1)g()α1k+λ21=kλ1r(+1)w()α1.

    Theorem 2.4. Using the same conditions in Lemma 2.1, and Theorem 2.1 [(Λ3),(Λ4)] with ψ:[m,n]TR be a integrable: 0ψ()g()1ψ().

    (Λ11) If rσ/ζAHk1[m,n] and

    nmϕ()ζ()g()Δα=m+λ1mϕ()ζ()Δαkm|ϕ()mλ1|ζ()ψ()Δα+nnλ2ϕ()ζ()Δα+nk|ϕ()n+λ2|ζ()ψ()Δα, (2.31)

    then

    nmrσ()g()Δαm+λ1mrσ()Δαkm|rσ()ζ()rσ(m+λ1)ζ(m+λ1)|ζ()ψ()Δα+nnλ2rσ()Δα+nk|rσ()ζ()rσ(nλ2)ζ(nλ2)|ζ()ψ()Δα. (2.32)

    (Λ12) If rσ/ζAHk1[m,n] and

    nmϕ()ζ()g()Δα=kkλ1ϕ()ζ()Δαkm|ϕ()k+λ1|ζ()ψ()Δα+nk|ϕ()kλ1|ζ()ψ()Δα, (2.33)

    then

    nmrσ()g()Δαk+λ2kλ1rσ()Δα+km|rσ()ζ()rσ(kλ1)ζ(kλ1)|ζ()ψ()Δαnk|rσ()ζ()rσ(k+λ2)ζ(k+λ2)|ζ()ψ()Δα. (2.34)

    If rσ/ζAHk2[m,n] and (2.31) and (2.33) satisfied, we get the reverse of (2.32) and (2.34).

    Proof. The same steps of Theorem 2.1 [(Λ3),(Λ4)] with Lemma 2.1, R1/ζ:[m,k]TR nonincreasing, R1/ζ:[k,n]TR nondecreasing.

    Corollary 2.10. In Theorem 2.4 [(Λ11),(Λ12)], letting T=R we get:

    (i)nmrσ()g()dαm+λ1mrσ()dαkm|rσ()ζ()rσ(m+λ1)ζ(m+λ1)|ζ()ψ()dα+nnλ2rσ()dα+nk|rσ()ζ()rσ(nλ2)ζ(nλ2)|ζ()ψ()dα. (2.35)
    (ii)nmrσ()g()dαk+λ2kλ1rσ()dα+km|rσ()ζ()rσ(kλ1)ζ(kλ1)|ζ()ψ()dαnk|rσ()ζ()rσ(k+λ2)ζ(k+λ2)|ζ()ψ()dα. (2.36)

    Corollary 2.11. In (2.35) and (2.36), we put α=1, with ϕ()= we get [47,Theorems 23 and 24].

    Corollary 2.12. Our results (2.35) and (2.36), by using T=Z gets

    (i)n1=mr(+1)g()α1m+λ11=mr(+1)α1k1=m|r(+1)ζ()r(m+λ1+1)ζ(m+λ1)|ζ()ψ()ˆ+n1=nλ2r(+1)α1+n1=k|r(+1)ζ()r(nλ2+1)ζ(nλ2)|ζ()ψ()α1.
    (ii)n1=mr(+1)g()α1k+λ21=kλ1r(+1)α1+k1=m|r(+1)ζ()r(kλ1+1)ζ(kλ1)|ζ()ψ()α1n1=k|r(+1)ζ()r(k+λ2+1)ζ(k+λ2)|h()ψ()α1.

    In this work, we explore new generalizations of the integral Steffensen inequality given in [38,39,43] by the utilization of the α-conformable derivatives and integrals, A few of these results are generalised to time scales. We also obtained the discrete and continuous case of our main results, in order to gain some fresh inequalities as specific cases.

    The authors extend their appreciation to the Research Supporting Project number (RSP-2022/167), King Saud University, Riyadh, Saudi Arabia.

    The authors declare no conflict of interest.



    [1] Y. Tian, W. Zhanshan, Composite slack-matrix-based integral inequality and its application to stability analysis of time-delay systems, Appl. Math. Lett., 120 (2021), 107252. https://doi.org/10.1016/j.aml.2021.107252 doi: 10.1016/j.aml.2021.107252
    [2] Y. Tian, W. Zhanshan, A new multiple integral inequality and its application to stability analysis of time-delay systems, Appl. Math. Lett., 105 (2020), 106325. https://doi.org/10.1016/j.aml.2020.106325 doi: 10.1016/j.aml.2020.106325
    [3] D. R. Anderson, Time-scale integral inequalities, J. Inequal. Pure Appl. Math., 6 (2005).
    [4] U. M. Ozkan, H. Yildirim, Steffensen's integral inequality on time scales, J. Inequal. Appl., 2007 (2007). https: //doi.org/10.1155/2007/46524
    [5] J. Jakšetić, J. Pečarić, K. S. Kalamir, Extension of Cerone's generalizations of Steffensen's inequality, Jordan J. Math. Stat., 8 (2015), 179–194.
    [6] J. C. Evard, H. Gauchman, Steffensen type inequalities over general measure spaces, Analysis, 17 (1997), 301–322. https://doi.org/10.1524/anly.1997.17.23.301 doi: 10.1524/anly.1997.17.23.301
    [7] J. F. Steffensen, On certain inequalities between mean values, and their application to actuarial problems, Scandinavian Actuar. J., 1918 (1918), 82–97. https://doi.org/10.1080/03461238.1918.10405302 doi: 10.1080/03461238.1918.10405302
    [8] K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, 1993.
    [9] V. Daftardar-Gejji, H. Jafari, Analysis of a system of nonautonomous fractional differential equations involving caputo derivatives, J. Math. Anal. Appl., 328 (2007), 1026–1033. https://doi.org/10.1016/j.jmaa.2006.06.007 doi: 10.1016/j.jmaa.2006.06.007
    [10] A. Kilbas, H. Srivastava, J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006.
    [11] I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Elsevier, 1998.
    [12] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65–70. https://doi.org/10.1016/j.cam.2014.01.002 doi: 10.1016/j.cam.2014.01.002
    [13] O. S. Iyiola, E. R. Nwaeze, Some new results on the new conformable fractional calculus with application using d'alambert approach, Progr. Fract. Differ. Appl., 2 (2016), 115–122. https://doi.org/10.18576/pfda/020204 doi: 10.18576/pfda/020204
    [14] O. S. Iyiola, G. O. Ojo, On the analytical solution of fornberg-whitham equation with the new fractional derivative, Pramana, 85 (2015), 567–575. https://doi.org/10.1007/s12043-014-0915-2 doi: 10.1007/s12043-014-0915-2
    [15] O. S. Iyiola, O. Tasbozan, A. Kurt, Y. Çenesiz, On the analytical solutions of the system of conformable time-fractional robertson equations with 1-d diffusion, Chaos Soliton. Fract., 94 (2017), 1–7.
    [16] T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57–66. https://doi.org/10.1016/j.cam.2014.10.016 doi: 10.1016/j.cam.2014.10.016
    [17] N. Benkhettou, S. Hassani, D. F. Torres, A conformable fractional calculus on arbitrary time scales, J. King Saud Univ. Sci., 28 (2016), 93–98. https://doi.org/10.1016/j.jksus.2015.05.003 doi: 10.1016/j.jksus.2015.05.003
    [18] E. R. Nwaeze, A mean value theorem for the conformable fractional calculus on arbitrary time scales, Progr. Fract. Differ. Appl., 2 (2016), 287–291. https://doi.org/10.18576/pfda/020406 doi: 10.18576/pfda/020406
    [19] E. R. Nwaeze, D. F. M. Torres, Chain rules and inequalities for the bht fractional calculus on arbitrary timescales, Arab. J. Math., 6 (2017), 13–20. https://doi.org/10.1007/s40065-016-0160-2 doi: 10.1007/s40065-016-0160-2
    [20] S. H. Sakerr, M. Kenawy, G. H. AlNemer, M. Zakarya, Some fractional dynamic inequalities of hardy's type via conformable calculus, Mathematics, 8 (2020), 434. https://doi.org/10.3390/math8030434 doi: 10.3390/math8030434
    [21] M. Zakaryaed, M. Altanji, G. H. AlNemer, A. El-Hamid, A. Hoda, C. Cesarano, et al., Fractional reverse coposn's inequalities via conformable calculus on time scales, Symmetry, 13 (2021), 542. https://doi.org/10.3390/sym13040542 doi: 10.3390/sym13040542
    [22] Y. M. Chu, M. A. Khan, T. Ali, S. S. Dragomir, Inequalities for α-fractional differentiable functions, J. Inequal. Appl., 2017 (2017), 1–12. https://doi.org/10.1186/s13660-017-1371-6 doi: 10.1186/s13660-017-1371-6
    [23] M. A. Khan, T. Ali, S. S. Dragomir, M. Z. Sarikaya, Hermite-Hadamard type inequalities for conformable fractional integrals, RACSAM Rev. R. Acad. A, 112 (2018), 1033–1048. https://doi.org/10.1007/s13398-017-0408-5 doi: 10.1007/s13398-017-0408-5
    [24] E. Set, A. Gözpnar, A. Ekinci, Hermite-Hadamard type inequalities via confortable fractional integrals, Acta Math. Univ. Comen., 86 (2017), 309–320.
    [25] M. Sarikaya, H. Yaldiz, H. Budak, Steffensen's integral inequality for conformable fractional integrals, Int. J. Anal. Appl., 15 (2017), 23–30.
    [26] M. Z. ASarikaya, C. C. Billisik, Opial type inequalities for conformable fractional integrals via convexity, Chaos Soliton. Fract., 2018.
    [27] M. Sarikaya, H. Budak, New inequalities of opial type for conformable fractional integrals, Turkish J. Math., 41 (2017), 1164–1173. https://doi.org/10.3906/mat-1606-91 doi: 10.3906/mat-1606-91
    [28] R. Agarwal, M. Bohner, A. Peterson, Inequalities on time scales: A survey, Math. Inequal. Appl., 4 (2001), 535–557.
    [29] R. Agarwal, D. O'Regan, S. Saker, Dynamic inequalities on time scales, Springer, Cham, 2014.
    [30] G. A. Anastassiou, Foundations of nabla fractional calculus on time scales and inequalities, Comput. Math. Appl., 59 (2010), 3750–3762. https://doi.org/10.1016/j.camwa.2010.03.072 doi: 10.1016/j.camwa.2010.03.072
    [31] G. A. Anastassiou, Principles of delta fractional calculus on time scales and inequalities, Math. Comput. Model., 52 (2010), 556–566. https://doi.org/10.1016/j.mcm.2010.03.055 doi: 10.1016/j.mcm.2010.03.055
    [32] G. A. Anastassiou, Integral operator inequalities on time scales, Int. J. Differ. Equ., 7 (2012), 111–137.
    [33] M. Bohner, A. Peterson, Dynamic equations on time scales: An introduction with applications, Birkhäuser Boston, Inc., Boston, MA, 2001.
    [34] M. Bohner, A. Peterson, Advances in dynamic equations on time scales, Birkhäuser Boston, Inc., Boston, MA, 2003.
    [35] S. Hilger, Analysis on measure chains—a unified approach to continuous and discrete calculus, Results Math., 18 (1990), 18–56. https://doi.org/10.1007/BF03323153 doi: 10.1007/BF03323153
    [36] A. A. El-Deeb, A. Abdeldaim, Z. A. Khan, On some generalizations of dynamic Opial-type inequalities on time scales, Adv. Differ.Equ., 2019. https://doi.org/10.1186/s13662-019-2268-0
    [37] W. N. Li, Some new dynamic inequalities on time scales, J. Math. Anal. Appl., 319 (2016), 802–814. https://doi.org/10.1016/j.jmaa.2005.06.065 doi: 10.1016/j.jmaa.2005.06.065
    [38] J. Pečarić, A. Josip, K. Perušić, Mercer and Wu- Srivastava generalisations of Steffensen's inequality, Appl. Math. Comput., 219 (2013), 10548–10558. https://doi.org/10.1016/j.amc.2013.04.028 doi: 10.1016/j.amc.2013.04.028
    [39] J. Pečarić, Notes on some general inequalities, Pub. Inst. Math., 32 (1982), 131–135.
    [40] M. Sahir, Dynamic inequalities for convex functions harmonized on time scales, J. Appl. Math. Phys., 5 (2017), 2360–2370. https://doi.org/10.4236/jamp.2017.512193 doi: 10.4236/jamp.2017.512193
    [41] S. H. Saker, A. A. El-Deeb, H. M. Rezk, R. P. Agarwal, On Hilbert's inequality on time scales, Appl. Anal. Discrete Math., 11 (2017), 399–423. https://doi.org/10.2298/AADM170428001S doi: 10.2298/AADM170428001S
    [42] Y. Tian, A. A. El-Deeb, F. Meng, Some nonlinear delay Volterra-Fredholm type dynamic integral inequalities on time scales, Discrete Dyn. Nat. Soc., 2018. https://doi.org/10.1155/2018/5841985
    [43] S. H. Wu, H. M. Srivastava, Some improvements and generalizations of Steffensen's integral inequality, Appl. Math. Comput., 192 (2007), 422–428. https://doi.org/10.1016/j.amc.2007.03.020 doi: 10.1016/j.amc.2007.03.020
    [44] Q. Sheng, M. Fadag, J. Henderson, J. M. Davis, An exploration of combined dynamic derivatives on time scales and their applications, Nonlinear Anal. Real World Appl., 7 (2006), 395–413. https://doi.org/10.1016/j.nonrwa.2005.03.008 doi: 10.1016/j.nonrwa.2005.03.008
    [45] D. R. Anderson, Taylor's formula and integral inequalities for conformable fractional derivatives, Contrib. Math. Eng., 2016. https://doi.org/10.1007/978-3-319-31317-7_2
    [46] M. Sarikaya, H. Yaldiz, H. Budak, Steffensen's integral inequality for conformable fractional integrals, Int. J. Anal. Appl., 15 (2017), 23–30.
    [47] J. Pečarić, K. S. Kalamir, Generalized Steffensen type inequalities involving convex functions, J. Funct. Spaces, 2014 (2014). https://doi.org/10.1155/2014/428030
    [48] S. O. Shah, A. Zada, M. Muzammil, M. Tayyab, R. Rizwan, On the Bielecki-Ulam's type stability results of first order non-linear impulsive delay dynamic systems on time scales, Qual. Theory Dyn. Syst., 2 (2020). https://doi.org/10.1007/s12346-020-00436-8
    [49] S. O. Shah, A. Zada, A. E. Hamza, Stability analysis of the first order non-linear impulsive time varying delay dynamic system on time scales, Qual. Theory Dyn. Syst., 18 (2019). https://doi.org/10.1007/s12346-019-00315-x
    [50] S. O. Shah, A. Zada, Existence, uniqueness and stability of solution to mixed integral dynamic systems with instantaneous and noninstantaneous impulses on time scales, Appl. Math. Comput., 359 (2019), 202–213. https://doi.org/10.1016/j.amc.2019.04.044 doi: 10.1016/j.amc.2019.04.044
    [51] S. O. Shah, A. Zada, C. Tunc, A. Asad, Bielecki- Ulam-Hyers stability of nonlinear Volterra impulsive integro-delay dynamic systems on time scales, Punjab Univ. J. Math., 53 (2021), 339–349.
    [52] S. O. Shah, A. Zada, On the stability analysis of non-linear Hammerstein impulsive integro-dynamic system on time scales with delay, Punjab Univ. J. Math., 51 (2019), 89–98.
    [53] A. Zada, S. O. Shah, Hyers-Ulam stability of first-order nonlinear delay differential equations with fractional integrable impulses, Hacet. J. Math. Stat., 47 (2018), 1196–1205. https://doi.org/10.15672/HJMS.2017.496 doi: 10.15672/HJMS.2017.496
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