
Consider the second order nonlinear partial differential equation:
∂2tu=F(u,∂xu),(t,x)∈C×R.
Given small analytic data, Yamane was able to obtain the order of the lifespan of the solution with respect to the smallness parameter ε. On the other hand, Gourdin and Mechab studied the lifespan of the solution given small Gevrey data, but under the assumption that F is independent of u. In this paper, we considered non-vanishing Gevrey data and used the method of successive approximations to obtain a solution and constructively estimate its lifespan.
Citation: John Paolo O. Soto, Jose Ernie C. Lope, Mark Philip F. Ona. Lifespan of solutions to second order Cauchy problems with small Gevrey data[J]. AIMS Mathematics, 2024, 9(1): 1434-1442. doi: 10.3934/math.2024070
[1] | Saad Ihsan Butt, Muhammad Nasim Aftab, Hossam A. Nabwey, Sina Etemad . Some Hermite-Hadamard and midpoint type inequalities in symmetric quantum calculus. AIMS Mathematics, 2024, 9(3): 5523-5549. doi: 10.3934/math.2024268 |
[2] | Nattapong Kamsrisuk, Donny Passary, Sotiris K. Ntouyas, Jessada Tariboon . Quantum calculus with respect to another function. AIMS Mathematics, 2024, 9(4): 10446-10461. doi: 10.3934/math.2024510 |
[3] | Humaira Kalsoom, Muhammad Amer Latif, Muhammad Idrees, Muhammad Arif, Zabidin Salleh . Quantum Hermite-Hadamard type inequalities for generalized strongly preinvex functions. AIMS Mathematics, 2021, 6(12): 13291-13310. doi: 10.3934/math.2021769 |
[4] | Iqra Nayab, Shahid Mubeen, Rana Safdar Ali, Faisal Zahoor, Muath Awadalla, Abd Elmotaleb A. M. A. Elamin . Novel fractional inequalities measured by Prabhakar fuzzy fractional operators pertaining to fuzzy convexities and preinvexities. AIMS Mathematics, 2024, 9(7): 17696-17715. doi: 10.3934/math.2024860 |
[5] | Muhammad Bilal Khan, Muhammad Aslam Noor, Thabet Abdeljawad, Bahaaeldin Abdalla, Ali Althobaiti . Some fuzzy-interval integral inequalities for harmonically convex fuzzy-interval-valued functions. AIMS Mathematics, 2022, 7(1): 349-370. doi: 10.3934/math.2022024 |
[6] | Jorge E. Macías-Díaz, Muhammad Bilal Khan, Muhammad Aslam Noor, Abd Allah A. Mousa, Safar M Alghamdi . Hermite-Hadamard inequalities for generalized convex functions in interval-valued calculus. AIMS Mathematics, 2022, 7(3): 4266-4292. doi: 10.3934/math.2022236 |
[7] | Muhammad Uzair Awan, Muhammad Aslam Noor, Tingsong Du, Khalida Inayat Noor . On M-convex functions. AIMS Mathematics, 2020, 5(3): 2376-2387. doi: 10.3934/math.2020157 |
[8] | Muhammad Bilal Khan, Hari Mohan Srivastava, Pshtiwan Othman Mohammed, Dumitru Baleanu, Taghreed M. Jawa . Fuzzy-interval inequalities for generalized convex fuzzy-interval-valued functions via fuzzy Riemann integrals. AIMS Mathematics, 2022, 7(1): 1507-1535. doi: 10.3934/math.2022089 |
[9] | Yanping Yang, Muhammad Shoaib Saleem, Waqas Nazeer, Ahsan Fareed Shah . New Hermite-Hadamard inequalities in fuzzy-interval fractional calculus via exponentially convex fuzzy interval-valued function. AIMS Mathematics, 2021, 6(11): 12260-12278. doi: 10.3934/math.2021710 |
[10] | Jamshed Nasir, Saber Mansour, Shahid Qaisar, Hassen Aydi . Some variants on Mercer's Hermite-Hadamard like inclusions of interval-valued functions for strong Kernel. AIMS Mathematics, 2023, 8(5): 10001-10020. doi: 10.3934/math.2023506 |
Consider the second order nonlinear partial differential equation:
∂2tu=F(u,∂xu),(t,x)∈C×R.
Given small analytic data, Yamane was able to obtain the order of the lifespan of the solution with respect to the smallness parameter ε. On the other hand, Gourdin and Mechab studied the lifespan of the solution given small Gevrey data, but under the assumption that F is independent of u. In this paper, we considered non-vanishing Gevrey data and used the method of successive approximations to obtain a solution and constructively estimate its lifespan.
We initiate our study by the conception of concavity of functions which is stated as follows: Any function E:[σ1,δ1]→R is referred to as a convex function if,
E((1−κ)ϱ+κω1)≤(1−κ)E(ϱ)+κE(ω1),∀ϱ,ω1∈[σ1,δ1], | (1.1) |
where κ∈[0,1]. Due to the several important geometrical and analytical aspects, convex functions are discussed from multiple approaches and directions in the literature. They have an immense amount of applications and generalizations to investigate non-convex problems. Their role in the emergence of inequalities is unprecedented because this concept itself is closely linked with inequalities. One can easily relate the that almost fundamental results in the theory of inequalities having a governing role can be achieved directly or indirectly by considering the convex functions. Now we provide a significant consequence of convex functions:
Let E:[σ1,δ1]→R be a convex function. Then,
E(σ1+δ12)≤1δ1−σ1∫δ1σ1E(ϱ)dϱ≤E(σ1)+E(δ1)2. |
From this inequality, one can conclude that this result computes the bounds for the average mean value integral of convex functions. Moreover, this result serves as a criteria to discuss the concavity of functions. Also, this result has several applications in the theory of means, special functions, error analysis probability theory, information theory, etc. Due to the effective range of implications, several variants of trapezium-type inequalities have been developed and assessed through various classes of convex functions along with applications. For recent developments, consult [1,2,3,4,5,6].
Furthermore, we recover the essential prelude relative to interval analysis to attain the required result. Suppose that the space of all non-empty compact intervals and positive non-degenerate compact intervals of the real line R are specified by RI and R+I. For any A,A2∈RI such that A1=[σ1⋆,σ1⋆] and A2=[δ1⋆,δ1⋆] and λ∈R, then
A1+A2=[σ1⋆,σ1⋆]+[δ1⋆,δ1⋆]=[σ1⋆+δ1⋆,σ1⋆+δ1⋆], |
and
λ[σ1⋆,σ1⋆]={[λσ1⋆,λσ1⋆],λ>0,[λσ1⋆,λσ1⋆],λ<0,0,λ=0. |
The concept of generalized Hukuhara(gh) difference was developed in [7].
Let A1,A2∈RI. The gh difference is illustrated as:
A1⊖gA2=[min{σ1⋆−δ1⋆,σ1⋆−δ1⋆},max{σ1⋆−δ1⋆,σ1⋆−δ1⋆}]. |
Also, for A1∈RI, the length of interval is computed by w(A1)=σ1⋆−σ1⋆. Then, for all A1,A2∈RI, we have
A1⊖gA2={[σ1∗−δ1∗,σ1⋆−δ1⋆],w(A1)≥w(A2),[σ1⋆−δ1⋆,σ1⋆−δ1⋆],w(A1)≤w(A2). |
Some properties linked with the gH-difference are described as follows:
(1) A1⊖gA1={0},A1⊖g{0}=A1,{0}⊖gA1=−A1,
(2) A1⊖gA2=(−A2)⊖g(−A1)=−(A2⊖gA1),
(3) A1⊖g(−A2)=A2⊖g(−A1)=−(A2⊖gA1),
(4) (A1+A2)⊖gA2=A1,
(5) λA1⊖gλA2=λ(A1⊖gA2).
For any A1,A2,C,D∈RI, consider κ1=w(A1)−w(C),κ2=w(A2)−w(D),κ3=w(A1)−w(A2), and κ4=w(C)−w(D). Then:
(1) (A1+A2)⊖g(C+D)={(A1⊖gC)+(A2⊖gD),κ1κ2≥0,(A1⊖gC)⊖g(−A2⊖gD),κ1κ2<0.
(2) (A1⊖gA2)+(C⊖gD)={(A1⊖g(−C))⊖g(A2⊖g(−D)),κ1κ2≥0,κ3κ4<0,(A1⊖g(−C))+(−A2⊖gD),κ1κ2<0,κ3κ4<0,(A1+C)⊖g(A2+D),κ3κ4≥0.
(3) (A1⊖gA2)⊖g(C⊖gD)={(A1⊖gC)⊖g(A2⊖gD),κ1κ2≥0,κ3κ4≥0,(A1⊖gC)+(−(A2⊖gD)),κ1κ2<0,κ3κ4≥0,(A1+(−C))⊖g(A2+(−D)),κ3κ4<0.
Any function E:[σ1,δ1]→RI is regarded as an I.V function if E(ϱ)=[E⋆(ϱ),E⋆(ϱ)] such that E⋆(ϱ)≤E⋆(ϱ),∀ϱ∈[σ1,δ1]. It is crucial to understand that limϱ→ϱ0E(ϱ) exist ⇔ limϱ→ϱ0E⋆(ϱ) and limϱ→ϱ0E⋆(ϱ).
Additionally, an I.V function E is assumed to be continuous ⇔ both E⋆ and E⋆ are continuous. Presume that E,g:[σ1,δ1]→RI. Then, (E⊖gg)(ϱ)=E(ϱ)⊖gg(ϱ):[σ1,δ1]→RI, and
limϱ→ϱ0(E⊖gg)(ϱ)=A⊖gδ1 |
exists, if limϱ→ϱ0E(ϱ)=A and limϱ→ϱ0g(ϱ)=δ1.
One can notice that if E,g:[σ1,δ1]→RI obey the continuous property of functions, then E⊖gg is referred to as a continuous function.
Consequently, we come up with differentiability concepts based on the ⊖g difference.
Definition 1.1. Presume that E:[σ1,δ1]→R is an I.V function. Then,
E′(ϱ)=limh→0E(ϱ+h)⊖gE(ϱ)h |
is termed as ⊖g derivative at ϱ∈[σ1,δ1]. E is differentiable on (σ1,δ1) if it is differentiable almost everywhere in the domain.
Now, we discuss the concept of I.V convex functions.
Definition 1.2. Let E:[σ1,δ1]→R be an I.V function satisfying E(ϱ)=[E⋆(ϱ),E⋆(ϱ)]. Then, it is considered as an I.V convex, if
E((1−κ)σ1+κδ1)⊇(1−κ)E(σ1)+κE(δ1),κ∈[0,1]. |
The next result ensures the convexity of functions in the interval domain.
Theorem 1.1. Let E:[σ1,δ1]→R be an I.V function such that E(ϱ)=[E⋆(ϱ),E⋆(ϱ)]. Then, E is an I.V convex function ⇔ E⋆ is a convex function and E⋆ is a concave function.
The I.V analogues of the trapezium inequality through containments ordering relation is given as follows:
Let E:[σ1,δ1]→R+I be an I.V convex function. Then,
E(σ1+δ12)⊇1δ1−σ1∫δ1σ1E(ϱ)dϱ⊇E(σ1)+E(δ1)2. |
For, comprehensive review, see [8].
q-Symmetric Calculus. Throughout the investigation, let I=[σ1,δ1] be any arbitrary subset of R such that 0∈I and q∈(0,1). Then, the q-geometric set is expressed as Iq={qϱ|ϱ∈I}. Taking the benefit of this set, the classical symmetric quantum difference operator and left and right quantum symmetric derivatives are defined as follows:
Definition 1.3 ([9]). Let E:I→R. Then the symmetric quantum variant derivative operator is
DsqE(κ)=E(q−1κ)−E(qκ)(q−1−q)κ,κ≠0. |
And, DsqE(0)=E′(0), where κ=0 provided that E is differentiable at κ=0. If E is a differentiable function at κ∈Iq, then limq→1DsqE(κ)=E′(κ).
The quantum symmetric number is described as
nq,s=q−n−qnq−1−q. |
Recently, Bilal et al. [10] proposed the conception of quantum symmetric operators over finite intervals. Presume that I=[σ1,δ1]⊂R, 0∈I and 0<q<1. Then the left sided quantum symmetric derivative is given as
Definition 1.4 ([10]). Presume that E:J→R is a continuous function. Then
σ1DsqE(κ)=E(q−1κ+(1−q−1)σ1)−E(qκ+(1−q)σ1)(q−1−q)(κ−σ1),κ≠σ1. |
And, σ1DsqE(σ1)=limq→1σ1DsqE(κ), if the limit exists. If σ1=0, then σ1DsqE=DsqE.
Consequently, the corresponding integral operator is stated as
Definition 1.5. Presume that E:J→R is a continuous function. Then
∫δ1σ1E(κ)σ1dsqκ=(δ1−σ1)(q−1−q)∞∑n=0q2n+1E(q2n+1δ1+(1−q2n+1)σ1)=(δ1−σ1)(1−q2)∞∑n=0q2nE(q2n+1δ1+(1−q2n+1)σ1). |
If σ1=0, then it coincides with the symmetric quantum integrals operator in [9]. For more detail concerning quantum symmetric differences, see [11].
Definition 1.6 ([12]). Let E:I→R be a continuous function. Then,
δ1DsqE(κ)=E(qκ+(1−q)δ1)−E(q−1κ+(1−q−1)δ1)(q−1−q)(δ1−κ),κ≠δ1. |
And, δ1DsqE(δ1)=limq→1δ1DsqE(κ), if the limit exists. If δ1=0, then δ1DsqE=DsqE.
Definition 1.7 ([12]). Presume that E:I→R is a continuous function. Then,
∫δ1σ1E(κ)δ1dsqκ=(δ1−σ1)(q−1−q)∞∑n=0q2n+1E(q2n+1σ1+(1−q2n+1)δ1)=(δ1−σ1)(1−q2)∞∑n=0q2nE(q2n+1σ1+(1−q2n+1)δ1). |
Clearly, a function is said to be right quantum symmetric integrable if ∑∞n=0q2n+1E(q2n+1κ+(1−q2n+1)δ1) converges. Remember that the above-discussed operators do not coincide with classical Jackson q operators.
Recently, Cortez et al. [13] investigated the interval-valued quantum symmetric operators in interval space based on Hukuhara differences, which are described as
Definition 1.8. Suppose E:I→RI is continuous I.V function. Then, the left interval-valued quantum symmetric difference operator is defined as
σ1Diq,s={E(q−1ϱ+(1−q−1)σ1)⊖gE(qϱ+(1−q)σ1)(q−1−q)(ϱ−σ1),ϱ≠σ1,limϱ→σ1σ1Diq,s=E(q−1ϱ)⊖gE(qϱ)ϱ(q−1−q),ϱ=σ1. |
And the corresponding integral operator is expressed as
Definition 1.9. Let E∈C([σ1,δ1],RI). Then the σ1Iiq,s-integral operator is described as
∫δ1σ1E(ϱ)σ1dsqϱ=(q−1−q)(δ1−σ1)∞∑n=0q2n+1E(q2n+1δ1+(1−q2n+1)σ1)=(1−q2)(δ1−σ1)∞∑n=0q2nE(q2n+1δ1+(1−q2n+1)σ1). |
One can easily observe that a function is considered to be I.V left q-symmetric integrable if ∑∞n=0q2nE(q2n+1δ1+(1−q2n+1)σ1) converges.
In 2013, Tariboon and Ntouyas [14] noticed some limitations of q-Jackson operators over finite intervals in impulsive difference equations and developed the q operators over finite interval connected with left point and explored its several implications in both impulsive difference equations and inequalities as well. This development paved another way to investigate integral inequalities. Alp and his coauthors [15] noticed that the trapezoidal inequality established in [16] is not correct and provided the new proof by using the concept of the support line of differentiable convex functions, and also established several midpoint quantum estimates. The authors of [17] analyzed the error estimates of the Milne rule by utilizing the Jensen-Mercer inequality and quantum calculus along with applications. Nosheen et al. [18] concluded some new quantum symmetric counterparts of basic inequality results, such as Hölder's inequality and error inequalities of one point rule associated with s-convex functions. To prove the Hermite-Hadamard inequality analytically, it was necessary to introduce the right quantum operators. In 2022, Kunt et al. [19] introduced the right sided quantum operators and provided their applications in integral inequalities. For a complete investigation, see [20,21,22,23,24,25,26,27].
In [28,29] the authors computed the estimates of one one-point integration rule incorporated with I.V functions and delivered applications as well. Following the idea of the previously discussed paper, Costa and Roman Flores [30] computed several fuzzy integral inequalities. In 2018, Zhao et al. [31,32] delivered the Jensen's and trapezium type inequalities associated with I.V-h-convex functions and I.V Chebyshev kinds of inequalities, respectively. Budak et al. [33] initiated the development of fractional versions of the trapezium inequalities for I.V convex functions defined by the means of containment ordering. In [34] the authors presented the class of I.V two-dimensional harmonic convex functions and derived several fractional integral inequalities by taking into account Raina's integral operators. Lou et al. [35] devoted their efforts to develop the notion of I.V quantum calculus based on generalized Hukuhara differences and provided the applications to inequalities. Motivated by the technique of [35], Kalsoom et al.[36] worked on I.V general quantum calculus by the means of Hukuhara differences and explored some applications to the Hermite-Hadamard inequality. Ali et al. [37] extended the idea presented in [36] for the right I.V-(p,q) operators and presented their key properties. Bin-Mohsin et al. [38] stated another class of generalized convexity based on a left-right ordering relation named as LR-fuzzy bi-convex function, and delivered Hermite-Hadamard and its weighted forms type results. In 2022, Duo and Zhou [39] examined the coordinated integral inequalities by considering the two-dimensional convex functions generalized integral operator having a non-singular kernel. In [40] the authors reported the parametric fractional versions of inequalities through convex functions and presented some visuals to support their outcomes.
The principal intent of this article is to examine the right sided quantum symmetric operators in the frame work of I.V functions along with their implications. We structured our study into two portions: In the initial portion of the study, we recover some vital details, background, and inspiration of the research. In the next part, we introduce the I.V right symmetric quantum derivative operators based on generalized Hukuhara difference and discuss their key properties. Based on the newly developed quantum operators, a new antiderivative operator is developed. Also, some crucial results, including the fundamental theorem of calculus and several other properties will be provided. Later on, several Hermite-Hadamard-type inequalities will be proved by both graphical and analytical methods essentially utilizing the convexity of the function. We will also present a visual breakdown in the support of principal findings.
In the current part of the study, interval-valued quantum symmetric operators are based on ⊖g differences. First, we introduce the notion of interval-valued right symmetric quantum difference and integral operators and their properties.
Now, we investigate the right I.V q-symmetric difference and integral operators and their properties. The space of left q-symmetric differentiable operators is denoted by δ1diq,s.
Definition 2.1. Suppose E:I→RI is a continuous I.V function. Then, the right I.V q-symmetric difference operator is defined as
δ1Diq,s={E(q−1ϱ+(1−q−1)δ1)⊖gE(qϱ+(1−q)δ1)(q−1−q)(ϱ−δ1),ϱ≠δ1,limϱ→σ1δ1Diq,s=E(q−1ϱ)⊖gE(qϱ)ϱ(q−1−q),ϱ=δ1. |
Example 2.1. Assume that E:[0,1]→RI is an I.V function such that E(ϱ)=[−4ϱ,5ϱ]. Then, we compute δ1Diq,sE(ϱ) by implementing Definition 2.1, and we have
δ1Diq,sE(ϱ)=[−4q−1ϱ−4(1−q−1)δ1,5q−1ϱ+5(1−q−1)δ1]⊖g[−4qϱ−4(1−q)δ1,5qϱ+5(1−q)δ1](q−1−q)(ϱ−δ1)=[−4q−1ϱ−4(1−q−1)δ1+4qϱ+4(1−q)δ1,5q−1ϱ+5(1−q−1)δ1−5qϱ−5(1−q)δ1](q−1−q)(ϱ−δ1)=[−4(q−1−q)(ϱ−δ1),5(q−1−q)(ϱ−δ1)](q−1−q)(ϱ−δ1)=[−4,5]. |
Theorem 2.1. A function E:[σ1,δ1]→RI is I.V right q-symmetric differentiable at ϱ∈[σ1,δ1] ⇔ E⋆ and E⋆ are right quantum symmetric differentiable at ϱ∈[σ1,δ1], and
δ1Diq,sE(ϱ)=[min{δ1Dq,sE⋆(ϱ),δ1Dq,sE⋆(ϱ)},max{δ1Dq,sE⋆(ϱ),δ1Dq,sE⋆(ϱ)}]. | (2.1) |
Proof. Assume that E is an I.V right q-symmetric differentiable function. Then, there exists g⋆ and g⋆ such that δ1Diq,sE(ϱ)=[g⋆,g⋆], and by considering Definition 1.8, we that
g⋆(ϱ)=min{E⋆(q−1ϱ+(1−q−1)δ1)−E⋆(qϱ+(1−q)δ1)(q−1−q)(ϱ−δ1),E⋆(q−1ϱ+(1−q−1)δ1)−E⋆(qϱ+(1−q)δ1)(q−1−q)(ϱ−δ1)}, |
and
g⋆(ϱ)=max{E⋆(q−1ϱ+(1−q−1)δ1)−E⋆(qϱ+(1−q)δ1)(q−1−q)(ϱ−δ1),E⋆(q−1ϱ+(1−q−1)δ1)−E⋆(qϱ+(1−q)δ1)(q−1−q)(ϱ−δ1)} |
exist. Then, δ1Dq,sE⋆(ϱ) and δ1Dq,sE⋆(ϱ) exist, and (2.1) is straight forward.
Conversely, suppose E⋆ and E⋆ are right q-symmetric differentiable at ϱ. If δ1Dq,sE⋆(ϱ)≤δ1Dq,sE⋆(ϱ), then
[δ1Dq,sE⋆(ϱ),δ1Dq,sE⋆(ϱ)]=[E⋆(q−1ϱ+(1−q−1)δ1)−E⋆(qϱ+(1−q)δ1)(q−1−q)(ϱ−δ1),E⋆(q−1ϱ+(1−q−1)δ1)−E⋆(qϱ+(1−q)δ1)(q−1−q)(ϱ−δ1)]=E(q−1ϱ+(1−q−1)δ1)⊖gE(qϱ+(1−q)δ1)(q−1−q)(ϱ−δ1). |
Similarly, if δ1Dq,sE⋆(ϱ)≤δ1Dq,sE⋆(ϱ), then δ1Diq,sE(ϱ)=[δ1Dq,sE⋆(ϱ),δ1Dq,sE⋆(ϱ)].
Now we provide another characterization of the right q-symmetric derivative based on the monotonic property of functions.
Theorem 2.2. Let E:[σ1,δ1]→RI and if it is I.V right q-symmetric differentiable on [σ1,δ1], then
(1) δ1Diq,sE(ϱ)=[δ1Dq,sE⋆(ϱ),δ1Dq,sE⋆(ϱ)], if E is l-increasing.
(2) δ1Diq,sE(ϱ)=[δ1Dq,sE⋆(ϱ),δ1Dq,sE⋆(ϱ)], if E is l-decreasing.
Proof. Suppose that E is an l-increasing and right q-symmetric differentiable function on [σ1,δ1], and we observe that q−1ϱ+(1−q−1)δ1>qϱ+(1−q)δ1 for q∈(0,1). Since l(E)=E⋆−E⋆ is increasing,
[E⋆(q−1ϱ+(1−q−1)δ1)−E⋆(q−1ϱ+(1−q−1)δ1)]−[E⋆(qϱ+(1−q)δ1)−E⋆(qϱ+(1−q)δ1)]>0,E⋆(q−1ϱ+(1−q−1)δ1)−E⋆(qϱ+(1−q)δ1)>E⋆(q−1ϱ+(1−q−1)δ1)−E⋆(qϱ+(1−q)δ1). |
Thus,
δ1Diq,sE(ϱ)=[[E⋆(q−1ϱ+(1−q−1)δ1),E⋆(q−1ϱ+(1−q−1)δ1)]⊖g[E⋆(qϱ+(1−q)δ1)−E⋆(qϱ+(1−q)δ1)](q−1−q)(ϱ−δ1)]=[E⋆(q−1ϱ+(1−q−1)δ1)−E⋆(qϱ+(1−q)δ1)(q−1−q)(ϱ−δ1),E⋆(q−1ϱ+(1−q−1)δ1)−E⋆(qϱ+(1−q)δ1)(q−1−q)(ϱ−δ1)]=[δ1Dq,sE⋆(ϱ),δ1Dq,sE⋆(ϱ)]. |
Hence, the required result is achieved. By a similar process, we can prove the 2nd result.
Remark 2.1. If υ∈(σ1,δ1) and E is l-increasing on [σ1,υ) and l-decreasing on (υ,δ1], then δ1Diq,sE(ϱ)=[δ1Dq,sE⋆(ϱ),δ1Dq,sE⋆(ϱ)] on [σ1,υ) and δ1Diq,sE(ϱ)=[δ1Dq,sE⋆(ϱ),δ1Dq,sE⋆(ϱ)] on (υ,δ1].
Theorem 2.3. Let E:[σ1,δ1]→RI be a left symmetric q-differentiable function. Then, for any υ=[υ⋆,υ⋆]∈RI and α∈R, the mappings E+υ and αE are also right q-symmetric differentiable on [σ1,δ1]. Then:
(1) δ1Diq,s(E(ϱ)+υ)=δ1Diq,sE(ϱ).
(2) δ1Diq,s(αE)(ϱ)=αδ1Diq,sE(ϱ).
Proof. For ϱ∈[σ1,δ1] and from Definition 1.8, we have
δ1Diq,s(E(ϱ)+υ)=[E(q−1ϱ+(1−q−1)δ1)+υ]⊖g[E(qϱ+(1−q)δ1)+υ](q−1−q)(ϱ−δ1)=E(q−1ϱ+(1−q−1)δ1)⊖gE(qϱ+(1−q)δ1)(q−1−q)(ϱ−δ1)=δ1Diq,sE(ϱ). |
We leave the second proof for interested readers.
Theorem 2.4. Let E:[σ1,δ1]→RI be a right q-symmetric differentiable function. For any υ=[υ⋆,υ⋆]∈RI, if l(E)−l(υ) has a constant sign over [σ1,δ1], then E⊖gυ is right q-symmetric differentiable.
Proof. Take ϱ∈[σ1,δ1]. Then,
δ1Diq,s(E⊖gυ)(ϱ)=(E(q−1ϱ+(1−q−1)δ1)⊖gυ)⊖g(E(qϱ+(1−q)δ1)⊖gυ)(q−1−q)(ϱ−δ1)=E(q−1ϱ+(1−q−1)δ1)⊖gE(qϱ+(1−q)δ1)(q−1−q)(ϱ−δ1). |
Theorem 2.5. Let E,g:[σ1,δ1]→R be right q-symmetric differentiable mappings. Then, the sum E+g is a right q-symmetric differentiable if one of the following cases hold:
(1) If E,g are equally l-monotonic on [σ1,δ1], then
δ1Diq,s(E(ϱ)+g(ϱ))=δ1Diq,sE(ϱ)+δ1Diq,sg(ϱ). |
(2) If E,g are differently l-monotonic on [σ1,δ1], then
δ1Diq,s(E(ϱ)+g(ϱ))=δ1Diq,sE(ϱ)⊖g(−1)δ1Diq,sg(ϱ). |
Moreover, in both cases, δ1Diq,s(E(ϱ)+g(ϱ))⊆δ1Diq,sE(ϱ)+δ1Diq,sg(ϱ).
Proof. Suppose E,g are I.V right q-symmetric differentiable and l-increasing on [σ1,δ1]. Then, E⋆,E⋆,g⋆ and g⋆ are left q-symmetric differentiable, and δ1Dq,sE⋆(ϱ)≤δ1Dq,sE⋆(ϱ), δ1Dq,sg⋆(ϱ)≤δ1Dq,sg⋆(ϱ). Then, E⋆+g⋆ and E⋆+g⋆ are q-symmetric and
δ1Diq,s(E(ϱ)+g(ϱ))=[min{δ1Dq,sE⋆(ϱ)+δ1Dq,sg⋆(ϱ),δ1Dq,sE⋆(ϱ)+δ1Dq,sg⋆(ϱ)},max{δ1Dq,sE⋆(ϱ)+δ1Dq,sg⋆(ϱ),δ1Dq,sE⋆(ϱ)+δ1Dq,sg⋆(ϱ)}]=[δ1Dq,sE⋆(ϱ)+δ1Dq,sg⋆(ϱ),δ1Dq,sE⋆(ϱ)+δ1Dq,sg⋆(ϱ)]=δ1Diq,sE(ϱ)+δ1Diq,sg(ϱ). |
By similar proceedings, we can obtain that the result for both E and g are l-decreasing.
Now, suppose that E is l increasing and g is l-decreasing. Then δ1Dq,sE⋆(ϱ)≤δ1Dq,sE⋆(ϱ), δ1Dq,sg⋆(ϱ)≤δ1Dq,sg⋆(ϱ). Also,
δ1Diq,s(E(ϱ)+g(ϱ))=[min{δ1Dq,sE⋆(ϱ)+δ1Dq,sg⋆(ϱ),δ1Dq,sE⋆(ϱ)+δ1Dq,sg⋆(ϱ)},max{δ1Dq,sE⋆(ϱ)+δ1Dq,sg⋆(ϱ),δ1Dq,sE⋆(ϱ)+δ1Dq,sg⋆(ϱ)}]. | (2.2) |
And,
δ1Diq,sE(ϱ)⊖g(−1)δ1Diq,sg(ϱ)=[δ1Dq,sE⋆(ϱ),δ1Dq,sE⋆(ϱ)]⊖g(−1)[δ1Dq,sg⋆(ϱ),δ1Dq,sg⋆(ϱ)]=[δ1Dq,sE⋆(ϱ),δ1Dq,sE⋆(ϱ)]⊖g[−δ1Dq,sg⋆(ϱ),−δ1Dq,sg⋆(ϱ)]=[min{δ1Dq,sE⋆(ϱ)+δ1Dq,sg⋆(ϱ)},max{δ1Dq,sE⋆(ϱ)+δ1Dq,sg⋆(ϱ)}]. | (2.3) |
By comparing (2.2) and (2.3), we conclude that δ1Diq,s(E(ϱ)+g(ϱ))=δ1Diq,sE(ϱ)⊖g(−1)δ1Diq,sg(ϱ).
Since E+g is l-increasing and decreasing, we get δ1Diq,s(E(ϱ)+g(ϱ))⊆δ1Diq,sE(ϱ)+δ1Diq,sg(ϱ), and the opposite case can be obtained by a similar procedure.
Hence, the required results are achieved.
Theorem 2.6. Let E,g:[σ1,δ1]→R be right q-symmetric differentiable mappings and l(E)−l(g) have constant sign over domain. Then, E⊖gg is a right q-symmetric differentiable if one of the following cases hold
(1) If E,g are equally l-monotonic on [σ1,δ1], then
δ1Diq,s(E(ϱ)⊖gg(ϱ))=δ1Diq,sE(ϱ)⊖gδ1Diq,sg(ϱ). |
(2) If E,g are differently l-monotonic on [σ1,δ1], then
δ1Diq,s(E(ϱ)⊖gg(ϱ))=δ1Diq,sE(ϱ)+(−1)δ1Diq,sg(ϱ). |
Proof. Assume that l(E)≥l(g) and E⊖gg=[E⋆−g⋆,E⋆−g⋆].
Suppose E,g are I.V right q-symmetric differentiable and l-increasing on [σ1,δ1]. Then, E⋆,E⋆,g⋆ and g⋆ are left q-symmetric differentiable, and δ1Dq,sE⋆(ϱ)≤δ1Dq,sE⋆(ϱ), δ1Dq,sg⋆(ϱ)≤δ1Dq,sg⋆(ϱ). Then, E⋆−g⋆ and E⋆−g⋆ are q-symmetric differentiable, and thus E⊖gg is a right q-symmetric differentiable function such that
δ1Diq,s(E(ϱ)⊖gg(ϱ))=[min{δ1Dq,sE⋆(ϱ)−δ1Dq,sg⋆(ϱ),δ1Dq,sE⋆(ϱ)−δ1Dq,sg⋆(ϱ)},max{δ1Dq,sE⋆(ϱ)−δ1Dq,sg⋆(ϱ),δ1Dq,sE⋆(ϱ)−δ1Dq,sg⋆(ϱ)}]=[δ1Dq,sE⋆(ϱ)−δ1Dq,sg⋆(ϱ),δ1Dq,sE⋆(ϱ)−δ1Dq,sg⋆(ϱ)]=[δ1Dq,sE⋆(ϱ),δ1Dq,sE⋆(ϱ)]⊖g[δ1Dq,sg⋆(ϱ),δ1Dq,sg⋆(ϱ)]=δ1Diq,sE(ϱ)⊖gδ1Diq,sg(ϱ). |
By similar proceedings, we can obtain the result that both E and g are l-decreasing.
Now, suppose that E is l increasing and g is l-decreasing. Then, δ1Dq,sE⋆(ϱ)≤δ1Dq,sE⋆(ϱ), δ1Dq,sg⋆(ϱ)≤δ1Dq,sg⋆(ϱ). Also,
δ1Diq,s(E(ϱ)⊖gg(ϱ))=[min{δ1Dq,sE⋆(ϱ)−δ1Dq,sg⋆(ϱ),δ1Dq,sE⋆(ϱ)−δ1Dq,sg⋆(ϱ)},max{δ1Dq,sE⋆(ϱ)−δ1Dq,sg⋆(ϱ),δ1Dq,sE⋆(ϱ)−δ1Dq,sg⋆(ϱ)}]=[δ1Dq,sE⋆(ϱ)−δ1Dq,sg⋆(ϱ),δ1Dq,sE⋆(ϱ)−δ1Dq,sg⋆(ϱ)]. | (2.4) |
And,
δ1Diq,sE(ϱ)+(−1)δ1Diq,sg(ϱ)=[δ1Dq,sE⋆(ϱ),δ1Dq,sE⋆(ϱ)]+(−1)[δ1Dq,sg⋆(ϱ),δ1Dq,sg⋆(ϱ)]=[δ1Dq,sE⋆(ϱ),δ1Dq,sE⋆(ϱ)]+[−δ1Dq,sg⋆(ϱ),−δ1Dq,sg⋆(ϱ)]=[min{δ1Dq,sE⋆(ϱ)−δ1Dq,sg⋆(ϱ)},max{δ1Dq,sE⋆(ϱ)−δ1Dq,sg⋆(ϱ)}]=[δ1Dq,sE⋆(ϱ)−δ1Dq,sg⋆(ϱ),δ1Dq,sE⋆(ϱ)−δ1Dq,sg⋆(ϱ)]. | (2.5) |
Comparing (2.4) and (2.5) yields the required result.
In the current part of the study, we introduce the concept of I.V right q-symmetric integral operator and its essential characterization. For our convenience, we specify the space of I.V right q-symmetric integrable mappings and the space of all continuous I.V mappings by δ1Iiq,s and υ([σ1,δ1],RI) respectively.
Definition 2.2. Let E∈υ([σ1,δ1],RI). Then, the δ1Iiq,s-integral operator is described as:
∫δ1σ1E(ϱ)δ1dsqϱ=(q−1−q)(δ1−σ1)∞∑n=0q2n+1E(q2n+1σ1+(1−q2n+1)δ1)=(1−q2)(δ1−σ1)∞∑n=0q2nE(q2n+1σ1+(1−q2n+1)δ1). |
One can easily observe that a function is considered to be I.V right q-symmetric integrable if ∑∞n=0q2nE(q2n+1σ1+(1−q2n+1)δ1) converges.
Theorem 2.7. Let E∈υ([σ1,δ1],RI). Then, E∈δ1Iiq,s ⇔ both E⋆ and E⋆ are right q-symmetric integrable mappings. Also,
∫δ1σ1E(κ)δ1dsqκ=[∫δ1σ1E⋆(κ)δ1dsqκ,∫δ1σ1E⋆(κ)δ1dsqκ]. |
Proof. Consider E∈δ1Iiq,s. Then,
∫δ1σ1E(κ)δ1dsqκ=(1−q2)(δ1−σ1)∞∑n=0q2nE(q2n+1σ1+(1−q2n+1)δ1)=[(1−q2)(δ1−σ1)∞∑n=0q2nE⋆(q2n+1σ1+(1−q2n+1)δ1),(1−q2)(δ1−σ1)∞∑n=0q2nE⋆(q2n+1σ1+(1−q2n+1)δ1)]. |
This implies that both E⋆ and E⋆ are right q-symmetric integrable mappings.
Conversely, suppose that E⋆ and E⋆ are q-symmetric integrable mappings and E⋆≤E⋆. Then, the result is obvious.
So, the result is proven.
Example 2.2. Let E:[0,2]→RI such that E(κ)=[2κ,3κ2]. Then,
∫10E(κ)1δ1dsqκ=[∫102κδ1dsqκ,∫103κ2δ1dsqκ]=[2(1−q2)∞∑n=0q2n(1−q2n+1),3(1−q2)∞∑n=0q2n(1−q2n+1)2]=[2(1−q2)∞∑n=0(q2n−q4n+1),3(1−q2)∞∑n=0(q2n+q6n+2−2q4n+1)]=[2(1+q2−q)1+q2,3(1+q2−2q)1+q2+31+q2+q4]. |
Theorem 2.8. Let E,g∈υ([σ1,δ1],RI) and (σ1,ϱ)⊆[σ1,δ1]. Then, for α∈RI :
(1) ∫δ1σ1[E(κ)+g(κ)]δ1dsqκ=∫δ1σ1E(κ)δ1dsqκ+∫δ1σ1g(κ)δ1dsqκ.
(2) ∫δ1σ1(αE)(κ)δ1dsqκ=α∫δ1σ1E(κ)δ1dsqκ.
Proof. From Definition 2.2,
∫δ1σ1[E(κ)+g(κ)]δ1dsqκ=(1−q2)(δ1−σ1)∞∑n=0q2n[E⋆(q2n+1σ1+(1−q2n+1)δ1)+g⋆(q2n+1σ1+(1−q2n+1)δ1),E⋆(q2n+1σ1+(1−q2n+1)δ1)+g⋆(q2n+1σ1+(1−q2n+1)δ1)]=(1−q2)(δ1−σ1)∞∑n=0q2n[E⋆(q2n+1σ1+(1−q2n+1)δ1),E⋆(q2n+1σ1+(1−q2n+1)δ1)]+(1−q2)(δ1−σ1)∞∑n=0q2n[g⋆(q2n+1σ1+(1−q2n+1)δ1),g⋆(q2n+1σ1+(1−q2n+1)δ1)]=∫δ1σ1E(κ)δ1dsqκ+∫δ1σ1g(κ)δ1dsqκ. |
Hence, the first proof is obtained. The proof of the second result is obvious.
Theorem 2.9. Let E,g∈υ([σ1,δ1],RI) and (σ1,ϱ)⊆[σ1,δ1]. Then,
∫δ1σ1E(κ)δ1dsqκ⊖g∫δ1σ1g(κ)δ1dsqκ⊆∫δ1σ1E(κ)⊖gg(κ)δ1dsqκ. |
Furthermore, l(E)−l(g) has constant sign on [σ1,δ1], thus
∫δ1σ1E(κ)δ1dsqκ⊖g∫δ1σ1g(κ)δ1dsqκ=∫δ1σ1E(κ)⊖gg(κ)δ1dsqκ. |
Proof. We observe that
∫δ1σ1min{E⋆−g⋆,E⋆−g⋆}δ1dsqκ≤min∫δ1σ1{E⋆−g⋆,E⋆−g⋆}δ1dsqκ≤max∫δ1σ1{E⋆−g⋆,E⋆−g⋆}δ1dsqκ≤∫δ1σ1max{E⋆−g⋆,E⋆−g⋆}δ1dsqκ. | (2.6) |
Also, we have
∫δ1σ1E(κ)δ1dsqκ⊖g∫δ1σ1g(κ)δ1dsqκ=[min{∫δ1σ1(E⋆−g⋆)δ1dsqκ,∫δ1σ1(E⋆−g⋆)δ1dsqκ},max{∫δ1σ1(E⋆−g⋆)δ1dsqκ,∫δ1σ1(E⋆−g⋆)δ1dsqκ}]⊆[∫δ1σ1min{(E⋆−g⋆),E⋆−g⋆}δ1dsqκ,∫δ1σ1max{(E⋆−g⋆),E⋆−g⋆}δ1dsqκ]=∫δ1σ1E(κ)⊖g(κ)δ1dsqκ. | (2.7) |
Comparison of (2.6) and (2.7) results in the desired inequality.
By the notion of generalized Hukuhara difference, if l(E)≥l(g), then E⊖gg=[E⋆−g⋆,E⋆−g⋆], and if l(E)≤l(g), then [E⋆−g⋆,E⋆−g⋆]. We suppose that l(E)≥l(g) on [σ1,δ1] such that E⊖gg=[E⋆−g⋆,E⋆−g⋆]. This implies that ∫δ1σ1(E⋆−g⋆)δ1dsqκ≤∫δ1σ1(E⋆−g⋆)δ1dsqκ. Now,
∫δ1σ1E⊖ggδ1dsqκ=[∫δ1σ1min{E⋆−g⋆,E⋆−g⋆}δ1dsqκ,∫δ1σ1max{E⋆−g⋆,E⋆−g⋆}δ1dsqκ]=[∫δ1σ1E⋆δ1dsqκ,∫δ1σ1E⋆δ1dsqκ]⊖g[∫δ1σ1g⋆δ1dsqκ,∫δ1σ1g⋆δ1dsqκ]=∫δ1σ1E(κ)δ1dsqκ⊖g∫δ1σ1g(κ)δ1dsqκ. |
Hence, the desired result is obtained.
Theorem 2.10. Let E:[σ1,δ1]→RI be right a q-symmetric differentiable function on (σ1,δ1) and δ1Diq,s(E(ϱ)∈υ([σ1,δ1],RI). If E is l-monotone on [s,κ], then
E(ϱ)⊖gE(υ)=∫ϱυδ1Diq,sE(κ)δ1dsqκ,υ,ϱ∈(σ1,δ1). | (2.8) |
Proof. If E is a I.V right q-symmetric differentiable function, then E⋆ and E⋆ are right q-symmetric differentiable mappings, so σ1Dq,sE⋆ and σ1Dq,sE⋆ are right q-symmetric integrable mappings. Therefore, E is also a I.V right q-symmetric integrable function. If E is l-increasing on [σ1,δ1], then δ1Diq,s(E(κ)=[δ1Diq,sE⋆(κ),δ1Diq,sE⋆(κ)], and then
E⋆(ϱ)−E⋆(υ)=∫ϱυδ1Dq,sE⋆(κ)δ1dsqκ. | (2.9) |
E⋆(ϱ)−E⋆(υ)=∫ϱυδ1Dq,sE⋆(κ)δ1dsqκ. | (2.10) |
Since E⋆≤E⋆ and from (2.9) and (2.10), we acquire
E(δ1)=E(υ)+∫δ1υδ1Diq,sE(κ)δ1dsqκ. |
Now, from the notion of gh-difference, then
E(ϱ)⊖gE(υ)=∫δ1υδ1Dq,sE(κ)δ1dsqκ. |
If E is l-decreasing on [σ1,δ1], then δ1Diq,s(E(κ))=[δ1Diq,sE⋆(κ),δ1Diq,sE⋆(κ)], and
∫δ1υδ1Diq,sE(κ)δ1dsqκ=[∫δ1υδ1Dq,sE⋆(κ)δ1dsqκ,∫δ1υδ1Dq,sE⋆(κ)δ1dsqκ]=[E⋆(δ1)−E⋆(υ),E⋆(δ1)−E⋆(υ)]=[E⋆(ϱ),E⋆(ϱ)]⊖g[E⋆(υ),E⋆(υ)]=E(ϱ)⊖gE(υ). |
Hence, the required result is acquired.
Remark 2.2. If E is l-increasing, then (2.8) can be interpreted as
E(ϱ)=E(υ)+∫δ1υδ1Dq,sE(κ)δ1dsqκ. |
If E is l-decreasing, then (2.8) can be interpreted as
E(ϱ)=E(υ)⊖g(−1)∫δ1υδ1Dq,sE(κ)δ1dsqκ. |
It is interesting to observe that the above result does not hold, if E is not l-monotone.
Now we develop the trapezium type inequalities by utilizing the I.V right q-symmetric integral operator.
Theorem 2.11. Suppose E:[σ1,δ1]→RI is an I.V right q-symmetric differentiable function. Then
E(σ1q2+δ11+q2)−q1+q2δ1Dq,sE(σ1q2+δ11+q2)⊇1δ1−σ1∫δ1σ1E(ϱ)δ1dq(ϱ)⊇qE(σ1)+(1+q2−q)E(δ1)1+q2. |
Proof. Since E is an I.V right q-symmetric differentiable function on (σ1,δ1), there exist two tangents at σ1q2+δ11+q2 given as
h⋆(ϱ)=E⋆(σ1q2+δ11+q2)+δ1Dq,sE⋆(σ1q2+δ11+q2)(ϱ−σ1q2+δ11+q2), | (2.11) |
and
h⋆(ϱ)=E⋆(σ1q2+δ11+q2)+δ1Dq,sE⋆(σ1q2+δ11+q2)(ϱ−σ1q2+δ11+q2). | (2.12) |
Since E is an I.V convex function, H1(ϱ)⊆E(ϱ), and then applying the I.V right q-symmetric integration, we have
∫δ1σ1H1(σ1,δ1)δ1dq(ϱ)=∫δ1σ1[E(σ1q2+δ11+q2)+δ1Dq,sE(σ1q2+δ11+q2)(ϱ−σ1q2+δ11+q2)]δ1dq(ϱ)=(δ1−σ1)E(σ1q2+δ11+q2)+δ1Dq,sE(σ1q2+δ11+q2)(∫δ1σ1ϱδ1dq(ϱ)−(δ1−σ1)σ1q2+δ11+q2)=(δ1−σ1)E(σ1q2+δ11+q2)+(δ1−σ1)δ1Dq,sE(σ1q2+δ11+q2)((1−q2)∞∑n=0q2n(q2n+1σ1+(1−q2n+1)δ1)−σ1q2+δ11+q2)=(δ1−σ1)[E(σ1q2+δ11+q2)+δ1Dq,sE(σ1q2+δ11+q2)(q(δ1−σ1)+σ1(1+q2)1+q2−σ1q2+δ11+q2)]=(δ1−σ1)[E(σ1q2+δ11+q2)−q1+q2δ1Dq,sE(σ1q2+δ11+q2)]⊇∫δ1σ1E(ϱ)δ1dq(ϱ). |
Now we establish the proof of second containments.
Moreover, the secant line through (σ1,E(σ1)) and (δ1,E(δ1)) can be expressed as:
ω1⋆(ϱ)=E⋆(σ1)+E⋆(δ1)−E⋆(σ1)δ1−σ1(κ−σ1), |
and
ω1⋆(ϱ)=E⋆(σ1)+E⋆(δ1)−E⋆(σ1)δ1−σ1(κ−σ1). |
Since E(ϱ)=[E⋆(ϱ),E⋆(ϱ)] is an I.V convex mapping and ω1(ϱ)⊆E(ϱ), then
∫δ1σ1ω1(ϱ)δ1dq(ϱ)=∫δ1σ1[E(σ1)+E(δ1)−E(σ1)δ1−σ1(κ−σ1)]δ1dq(ϱ)=(δ1−σ1)E(σ1)+E(δ1)−E(σ1)δ1−σ1(∫δ1σ1κδ1dq(ϱ)−(δ1−σ1)σ1)=(δ1−σ1)E(σ1)+(E(δ1)−E(σ1))((1−q2)∞∑n=0q2n(q2n+1σ1+(1−q2n+1)δ1)−σ1)=(δ1−σ1)E(σ1)+(E(δ1)−E(σ1))(q(δ1−σ1)+(1+q2)σ11+q2−σ1)=(δ1−σ1)[qE(σ1)+(1+q2−q)E(δ1)1+q2]⊆∫δ1σ1E(ϱ)δ1dq(ϱ). |
Example 2.3. Let E:[0,2]→RI be an I.V right q-symmetric differentiable function such that E(ϱ)=[2ϱ2,−2ϱ2+20]. Then, from Theorem 2.11, we obtain
E(σ1q2+δ11+q2)=E(21+q2)=[8(1+q2)2,−8(1+q2)2+20]. |
Next, the I.V right q-symmetric derivative of E is given as δ1Dq,sE(ϱ)=[−4(2−q),4(2−q)].
Furthermore, the I.V right q-symmetric integral of E is given as
12∫20E(ϱ)2dsqx=∫20[2ϱ2,−2ϱ2+20]2dsqx=[8+8q21+q2+q4−16q1+q2,12−8q21+q2+q4+16q1+q2]. | (2.13) |
From the above computations, we have the following containment:
[8(1+q2)2−8q(2−q)1+q2,20−8(1+q2)2+8q(2−q)1+q2]⊇[8+8q21+q2+q4−16q1+q2,12−8q21+q2+q4+16q1+q2]⊇[8(1+q2−q)1+q2,12q2+12+8q1+q2]. | (2.14) |
For q=13 in (2.14), we have
[2.48,17.52]⊇[3.99121,16.0088]⊇[6.91358,14.4]. |
For graphical validation (Figure 1), we take q∈(0,1) in (2.14).
Theorem 2.12. Suppose E:[σ1,δ1]→RI is an I.V right q-symmetric differentiable function. Then,
E(σ1+δ1q21+q2)+q(1−q)(δ1−σ1)1+q2δ1Dq,sE(σ1+δ1q21+q2)⊇1δ1−σ1∫δ1σ1E(ϱ)δ1dq(ϱ)⊇qE(σ1)+(1+q2−q)E(δ1)1+q2. |
Proof. Since E is an I.V right q-symmetric differentiable function on (σ1,δ1), there exist two tangents at σ1+δ1q21+q2 given as
2h⋆(ϱ)=E⋆(σ1+δ1q21+q2)+δ1Dq,sE⋆(σ1+δ1q21+q2)(ϱ−σ1+δ1q21+q2), |
and
2h⋆(ϱ)=E⋆(σ1+δ1q21+q2)+δ1Dq,sE⋆(σ1+δ1q21+q2)(ϱ−σ1+δ1q21+q2). |
Since E is an I.V convex function, H2(ϱ)⊆E(ϱ), and then applying the I.V right q-symmetric integration, we have
∫δ1σ1H2(σ1,δ1)δ1dq(ϱ)=∫δ1σ1[E(σ1+δ1q21+q2)+δ1Dq,sE(σ1+δ1q21+q2)(ϱ−σ1+δ1q21+q2)]δ1dq(ϱ)=(δ1−σ1)E(σ1+δ1q21+q2)+δ1Dq,sE(σ1+δ1q21+q2)(∫δ1σ1ϱδ1dq(ϱ)−(δ1−σ1)σ1+δ1q21+q2)=(δ1−σ1)E(σ1+δ1q21+q2)+(δ1−σ1)δ1Dq,sE(σ1+δ1q21+q2) | (2.15) |
((1−q2)∞∑n=0q2n(q2n+1σ1+(1−q2n+1)δ1)−σ1+δ1q21+q2)=(δ1−σ1)[E(σ1+δ1q21+q2)+δ1Dq,sE(σ1+δ1q21+q2)(q(δ1−σ1)+σ1(1+q2)1+q2−σ1+δ1q21+q2)]=(δ1−σ1)[E(σ1+δ1q21+q2)+(1−q)(δ1−σ1)1+q2δ1Dq,sE(σ1+δ1q21+q2)]⊇∫δ1σ1E(ϱ)δ1dq(ϱ). | (2.16) |
Example 2.4. Let E:[0,2]→RI be an I.V right q-symmetric differentiable function such that E(ϱ)=[2ϱ2,−2ϱ2+20]. Then, from Theorem 2.12, we obtain:
E(σ1+δ1q21+q2)=[8q4(1+q2)2,−8q4(1+q2)2+20]. | (2.17) |
From (2.17) and (2.13), we obtain the following expression for Theorem 2.12:
[8q4(1+q2)2−8(1−q)(2q−1)1+q2,20−8q4(1+q2)2+8(1−q)(2q−1)1+q2]⊇[8+8q21+q2+q4−16q1+q2,12−8q21+q2+q4+16q1+q2]⊇[8(1+q2−q)1+q2,12q2+12+8q1+q2]. | (2.18) |
For q=13 in (2.18), we have
[1.68,18.32]⊇[3.99121,16.0088]⊇[6.91358,14.4] |
For graphical validation (Figure 2), we take q∈(0,1) in (2.18).
Theorem 2.13. Let E:[σ1,δ1]→R+I be an I.V convex function. Then,
E(σ1+δ12)⊇1δ1−σ1[∫δ1σ1E(ϱ)σ1δ1dsqϱ+∫δ1σ1E(ϱ)δ1δ1dsqϱ]⊇E(σ1)+E(δ1)2. |
Proof. Since E is an I.V convex function, then
E(σ1+δ12)⊇12[E(ϱσ1+(1−ϱ)δ1)+E((1−ϱ)σ1+ϱδ1)]. | (2.19) |
Applying quantum symmetric integration on (2.19) with respect to ′ϱ′ over [0,1], we have
E(σ1+δ12)⊇12[∫10E(ϱσ1+(1−ϱ)δ1)δ1dsqϱ+∫10E((1−ϱ)σ1+ϱδ1)δ1dsqϱ], | (2.20) |
and
∫10E(ϱσ1+(1−ϱ)δ1)δ1dsq=∞∑n=0q2nE(q2n+1σ1+(1−q2n+1)δ1)=1δ1−σ1∫δ1σ1E(ϱ)δ1δ1dsqϱ. | (2.21) |
Similarly,
∫10E((1−ϱ)σ1+ϱδ1)δ1dsqϱ=1δ1−σ1∫δ1σ1E(ϱ)σ1δ1dsqϱ. | (2.22) |
The combination of (2.20)–(2.22), results in the first containment. To prove our second containment, we employ the convexity of E. Hence, the result is completed.
Example 2.5. Let E:[0,2]→RI be an I.V right q-symmetric differentiable function such that E(ϱ)=[2ϱ2,−2ϱ2+20]. Then, from Theorem 2.13, we obtain:
[8q4(1+q2)2−8(1−q)(2q−1)1+q2,20−8q4(1+q2)2+8(1−q)(2q−1)1+q2]⊇[8+8q21+q2+q4−16q1+q2,12−8q21+q2+q4+16q1+q2]⊇[8(1+q2−q)1+q2,12q2+12+8q1+q2]. | (2.23) |
For q=13 in (2.23), we have
[0.666667,6]⊇[0.79707,5.8696]⊇[1.33333,5.33333]. |
For graphical validation (Figure 3), we take q∈(0,1) in (2.23).
In the realm of mathematical analysis, one of the key research aims is how to find the derivative of absolute functions, which are not differentiable at certain points. To handle such kinds of problems, symmetric calculus plays a vital role. In this paper, we have developed the concept of right interval-valued quantum symmetric operators and examined numerous properties of operators in the setting of interval analysis. We have discussed the utility of these operators in inequalities. In favor of our findings, some visuals have been provided. It is important to observe that the derivative of I.V functions involving absolute functions can be computed from our proposed operators. We hope these operators and techniques will create new avenues of research. Based on these operators, several kinds of inequalities can be obtained. Also, by utilizing these operators, several results of optimization theory can be updated. In the future, we will try to establish some error bounds of numerical formulas in the interval-domain associated with the developed theory.
Yuanheng Wang: conceptualization, investigation, writing-review and editing, visualization, validation; Muhammad Zakria Javed: conceptualization, software, formal analysis, investigation, writing-original draft preparation, writing-review and editing, visualization; Muhammad Uzair Awan: conceptualization, software, validation, formal analysis, investigation, writing-review and editing, visualization, supervision; Bandar Bin-Mohsin: conceptualization, validation, investigation, writing-review and editing, visualization; Badreddine Meftah: conceptualization, software, validation, formal analysis, investigation, writing-review and editing, visualization; Savin Treanta: conceptualization, software, investigation, writing-review and editing. All authors have read and agreed to the published version of the manuscript.
This research is supported by "The National Natural Science Funds of China (No. 12171435)". Prof Bandar is thankful to King Saud University, Riyadh, Saudi Arabia for the project "Researchers Supporting Project number (RSP2024R158), King Saud University, Riyadh, Saudi Arabia". The authors are thankful to the editor and the anonymous reviewers for their valuable comments and suggestions.
The authors declare no conflict of interest.
[1] | P. D'Ancona, S. Spagnolo, On the life span of the analytic solutions to quasilinear weakly hyperbolic equations, Indiana Univ. Math. J., 40 (1991), 71–99. |
[2] |
D. Gourdin, M. Mechab, Problème de Cauchy pour des équations de Kirchhoff généralisées, Commun. Partial Differ. Equ., 23 (1998), 761–776. https://doi.org/10.1080/03605309808821364 doi: 10.1080/03605309808821364
![]() |
[3] |
D. Gourdin, M. Mechab, Problème de Cauchy non linéaire à données initiales lentement oscillantes, Bull. Sci. Math., 125 (2001), 371–394. https://doi.org/10.1016/S0007-4497(01)01082-X doi: 10.1016/S0007-4497(01)01082-X
![]() |
[4] |
H. Kubo, M. Ohta, On systems of semilinear wave equations with unequal propagation speeds in three space dimensions, Funkcial. Ekvac., 48 (2005), 65–98. https://doi.org/10.1619/fesi.48.65 doi: 10.1619/fesi.48.65
![]() |
[5] | H. Kubo, M. Ohta, On the global behavior of classical solutions to coupled systems of semilinear wave equations, In: New trends in the theory of hyperbolic equations, Basel: Birkhäuser, 2005,113–211. https://doi.org/10.1007/3-7643-7386-5_2 |
[6] |
H. Kubo, M. Ohta, Blowup for systems of semilinear wave equations in two space dimensions, Hokkaido Math. J., 35 (2006), 697–717. https://doi.org/10.14492/hokmj/1285766425 doi: 10.14492/hokmj/1285766425
![]() |
[7] |
J. E. C. Lope, M. P. F. Ona, Local solvability of a system of equations related to Ricci-flat Kähler metrics, Funkcial. Ekvac., 59 (2016), 141–155. https://doi.org/10.1619/fesi.59.141 doi: 10.1619/fesi.59.141
![]() |
[8] |
J. E. C. Lope, H. Tahara, On the analytic continuation of solutions to nonlinear partial differential equations, J. Math. Pures Appl., 81 (2002), 811–826. https://doi.org/10.1016/S0021-7824(02)01257-6 doi: 10.1016/S0021-7824(02)01257-6
![]() |
[9] | J. P. O. Soto, J. E. C. Lope, M. P. F. Ona, A constructive approach to obtaining the lifespan of solutions to Cauchy problems with small data, Matimyás Mat., 46 (2023), 22–32. |
[10] |
M. A. C. Tolentino, D. B. Bacani, H. Tahara, On the lifespan of solutions to nonlinear Cauchy problems with small analytic data, J. Differ. Equ., 260 (2016), 897–922. https://doi.org/10.1016/j.jde.2015.09.013 doi: 10.1016/j.jde.2015.09.013
![]() |
[11] | C. Wagschal, Le problème de Goursat non-linéaire, J. Math. Pures Appl., 58 (1979), 309–337. |
[12] | H. Yamane, Nonlinear Cauchy problems with small analytic data, Proc. Amer. Math. Soc., 134 (2006), 3353–3361. |
[13] | H. Yamane, Local existence for nonlinear Cauchy problems with small analytic data, J. Math. Sci. Univ. Tokyo, 18 (2011), 51–65. |
1. | Munaza Chaudhry, Muhammad Abdul Basit, Muhammad Imran, Madeeha Tahir, Aiedh Mrisi Alharthi, Jihad Younis, Numerical analysis of mathematical model of nanofluid flow through stagnation point involving thermal radiation, activation energy, and living organisms, 2025, 15, 2158-3226, 10.1063/5.0249122 | |
2. | Muhammad Zakria Javed, Muhammad Uzair Awan, Loredana Ciurdariu, Omar Mutab Alsalami, Pseudo-ordering and δ1-level mappings: A study in fuzzy interval convex analysis, 2025, 10, 2473-6988, 7154, 10.3934/math.2025327 |