Abstract
By means of ς fractional sum operator, certain discrete fractional nonlinear inequalities are replicated in this text. Considering the methodology of discrete fractional calculus, we establish estimations of Gronwall type inequalities for unknown functions. These inequalities are of a new form comparative with the current writing discoveries up until this point and can be viewed as a supportive strategy to assess the solutions of discrete partial differential equations numerically. We show a couple of employments of the compensated inequalities to reflect the benefits of our work. The main outcomes might be demonstrated by the use of the examination procedure and the approach of the mean value hypothesis.
Similar content being viewed by others
1 Introduction
Fractional calculus consisting of a derivative and an integral component of noninteger order is a natural increase in the regular integer order calculus. With different analysts and experts devoting themselves to this area, fractional analytic is apparently widespread considering its intriguing applications concerning various fields of science, for instance, viscoelasticity, dispersion, nervous system science, control hypothesis, and statistics [1–9].
The justification for this paper is to implement discrete fractional sum equations in terms of creating a method for interpreting such equations and to derive the related Gronwall form of inequality. Particularly Gronwall’s inequality is pointed out as one of the central inequalities in the premise of differential form equations. Starting now and into the foreseeable future, various speculations and growth of these inequalities ended up being a bit of the composition. In 1969, Sugiyama [10] claimed to have turned up and created the discrete Gronwall inequality. In the associated structure he carried out the most precise and complete discrete module of Gronwall inequality as follows.
Theorem 1.1
Let \(h(t_{1})\) and \(r(t_{1})\) be real-valued functions defined for \(t_{1}\in \mathbb{N}_{0}\), and suppose that \(r(t_{1})>0\) for every \(t_{1}\in \mathbb{N}_{0}\). If
where \(\eta _{0}\) is a nonnegative constant, then
For difference and integral equations, Theorem 1.1 is as yet used to possess the integration of the discrete factor models.
Despite the existence of a systematic mathematical theory of continuous fractional calculus, the potential progress of the discrete fractional calculus (DFC) has been insufficient until very recently. The discrete counterpart of the hypothesis in the presence of a fractional sum of order \(\varsigma >0 \) was described by Miller and Ross [11] who discussed solutions to linear difference equation and checked some basic features of this operator. Additionally, Atici and Eloe [12] actualized a discrete Laplace transform strategy for a sequence of fractional difference equations. The triggers of the initial value in the discrete fractional calculus were established by Atici and Eloe [13]. Atici and Eloe [14] explored the layout of a discrete fractional calculus with the nabla operator. They generated exponential laws and the item rule for the forward fractional calculus. Atici and Sengul [15] set up the law of Leibniz and summation by parts equation in a discrete fractional principle. Bastos and Torres [16] created a more wide-ranging, discrete fractional operator, which has been calculated by delta and nabla fractional sums. Holm [17] introduced fractional sums and difference operators and extended this concept to resolve the issue of fractional initial value. Anastassiou [18] defined the privilege of the discrete nabla fractional Taylor equation. The science that came about because of this depiction was charming to a few perusers, and now it is a subject of extreme examination in various ways: existence and accuracy of discrete fractional equations, modeling of tumor growth [19], stability of tumor-based solutions to the order of Legendre’s derivative ς [20], Euler–Lagrange equation, and optimal status for calculus of variations problems [21]. The impression of discrete fractional calculus is being presented just more as of late, generally attributable to the blast of work in the analysis of fractional differential (see, for example, the books [22, 23]). Usually, within a particular fractional system, there are several derivative analogues, so experts choose those that are generally appropriate in a specific sense (see [24–28]).
Finite difference inequalities showing unique limits of unknown functions provide a thoroughly valuable and important method for improving the perception of finite differential equations. During the recent years, guided and motivated by their characterizations in different parts of difference equations, several of these inequalities have been linked up [29–34]. Hence, difference equations arise as logical constructs that describe such real-life situations, e.g., queueing problems, electrical networks, financial dimensions, etc., and this defense is sufficient to seek such a framework. There are many representations for these sorts of inequalities at the point where one wants to evaluate several properties of a differential equation. Basically reliant on the capacity of the above investigation, in this material, we can search for the discrete fractional nonlinear inequalities related to ς fractional sum operator that has been built up to explain fractional inequalities and integrate some proven literature trials.
To represent the theoretical aspects, it was seen that the inequalities delivered may be used to analyze various classes of discrete fractional differential equations. So as to investigate the uniqueness and boundedness of the usage of fractional sum difference equations, two theorems are secured throughout this manuscript.
Definitive parts of the record are situated as such. In Sect. 2, we portray significant real factors and fundamental hypotheses which can be key setups for our main impacts. Section 3 is devoted to abstract discussions of nonlinear discrete fractional inequalities with a few conducting remarks. The last bit is considered in fulfilling the theoretical examination necessities.
2 Material history
Throughout this endeavor, without lack of broad declaration, let \(C^{\alpha }(L,K)\) be the class of functions of continuously differentiable with all α times from a set L into a set K, P be a constant, \(\mathbb{N}_{t_{1}}= \{ t_{1},t_{1}+1, t_{1}+2,\ldots \} \), \(M_{t_{1}}=[t_{1}, P]\cap \mathbb{N}_{t_{1}}\), where \(P, t_{1}\in \mathbb{N}_{t_{1}}\), \(\sum_{t_{2}=c}^{\vartheta }r(t_{2})=0\), \(\mathbb{R}_{+}=[0,\infty )\) and difference operator of q be assigned as \(\Delta q(\vartheta )=q( \vartheta )-q( \vartheta -1), \vartheta \in \mathbb{N}_{t_{1}}\).
A portion of the basic necessities and theorems in the assessment of discrete fractional are accounted for as follows.
Definition 2.1
([15])
Let \(\varsigma >0\), l be any real number, and \(\sigma (t_{1})=t_{1}+1\), then ςth fractional sum of r is defined for \(t_{2}=l\) (mod 1) by
so that \(t_{1}^{\varsigma }= \frac{\Gamma (t_{1}+1)}{\Gamma (t_{1}+1-\varsigma )}\), \(\Delta _{l}^{-\varsigma } r\) is defined for \(t_{2}=l+\varsigma \) (mod1), and \(\Delta _{l}^{-\varsigma }:\mathbb{N}_{l}\rightarrow \mathbb{N}_{l+ \varsigma }\).
Definition 2.2
([15])
Let \(\delta >0\) and \(\gamma -1<\delta <\gamma \). Then the δth fractional difference of r is characterized as
where γ is a positive integer and \(-\varsigma =\delta -\gamma \).
Theorem 2.3
([13])
If a real-valued function r is prescribed on \(\mathbb{N}_{l}\), such that \(\delta, \varsigma >0\), then
Theorem 2.4
([13])
Let \(\varsigma >0\) and r be a function which is real valued on \(\mathbb{N}_{l}\), then
On the discrete fractional principle, the reader can turn their attention to further important properties to [13, 15].
In this article, based on the Riemann–Liouville definition of fractional difference pioneered by Miller and Ross [11] and generated by Atici and Eloe [13], we explore certain new nonlinear discrete fractional sum inequalities which lead to the generalizations of Gronwall–Bellman types.
3 Outcome assertion
Now we are going to round off the simple tests.
Theorem 3.1
Suppose that \(h\in \mathbb{N}_{\varsigma -1}\rightarrow \mathbb{R}_{+}, W: \mathbb{N}_{\varsigma }\rightarrow \mathbb{R}_{+}\) are functions, \(0< \varsigma \leq 1\), \(j> g> 0\) are constants, and η is a positive nondecreasing function defined on \(\mathbb{N}_{\varsigma -1}\). If
is satisfied, then
where
Proof
Since \(\eta (t_{1})\) is a positive nondecreasing function, from (1) we have
Defining
Inequality (5) with Definition 2.1 imply that
where \(W(t_{2},t_{1})\) is given as in (3). Now \(W(t_{2},t_{1})\), \(t_{1}^{\varsigma }\) by their definition and \(W(t_{2},t_{1}))\) is decreasing in \(t_{1}\) for each \(t_{2}\in \mathbb{N}_{0}\). Using straightforward computation, for \(t_{1}\in M_{\varsigma }\) and (6), we get
leads to
By the mean value theorem, it can be seen that
In view of (8) and (9), we conclude
Summing (10) from ς to \(t_{1}-1\) and \(y(\varsigma -1)=1\), we obtain
that is,
or
so that
the conclusion of (2) can be obtained from (6) and (11). □
Remark 3.2
By inserting \(\varsigma =1\), \(\Gamma (1)=1\) (the property of gamma function), \(\eta (t_{1})=\eta _{0}\), and \(j=g=1\) in (1), then Theorem 3.1 shifts to Theorem 1.1 [10].
Theorem 3.3
If \(T:\mathbb{N}_{\varsigma }\rightarrow \mathbb{R}_{+}\), \(j\neq 1\), \(j>1\) is a constant and the inequality
is satisfied under the same suppositions of h, W, ς, η, \(W(t_{2},t_{1})\) of Theorem 3.1, then
such that
Proof
Obviously, by the positive and nondecreasing nature of \(\eta (t_{1})\), inequality (12) takes the form
denoting
Utilizing Definition 2.1 to (16), we deduce
the last inequality with (17) turns out to be
where \(T(t_{2},t_{1})\) is given as in (14) and
As \(R(t_{1})\geq 0\) is nondecreasing and with the support of straightforward computation for \(t_{1}\in M_{\varsigma }\), the decreasing nature of \(W(t_{2},t_{1}), T(t_{2},t_{1})\) for \(t_{2}\in \mathbb{N}_{0}\), the definition of \(W(t_{2},t_{1})\), \(T(t_{2},t_{1})\), \(t_{1}^{\varsigma }\), and (18), we get
On the other hand, by the mean value theorem, we attain
for some \(\varrho (t_{1})\in [R(t_{1}-1),R(t_{1})]\). Furthermore
the above inequality, by summing from ς to \(t_{1}-1\) and with \(R(\varsigma -1)=1\), implies
Consider
We proceed from (19) and (20) to
where
and
Equation (23) with inequality (24) becomes
A similar analysis of the mean value theorem as before yields
gives
Substituting (26) in (25), we get
which with \(S(\varsigma -1)=1\) offers the estimation
Also,
From the last inequality and (22), we observe that
Summing (27) from ς to \(t_{1}-1\) and utilizing \(B(\varsigma -1)=1\), we acquire
or
The acquired bound in (13) can be carried out by substituting (28) in (21), (18), and (17) with \(t_{1}\in M_{\varsigma -1}\) simultaneously. □
Remark 3.4
If \(j=1\), \(\varsigma =1\), \(\Gamma (1)=1\), \(T(t_{2})=k(s,\sigma )\) and \(h^{2}(t_{1})=u(n)\), \(k(n, s)\), \(\Delta _{1}k(n, s)\), \(0 < s <n < 1\), \(n, s\in \mathbb{N}_{0}\) in (12), then Theorem 3.3 can be modified into [35] Theorem 2.3(c2).
Corollary 3.5
Suppose that h, ς, W, T, η, \(W(t_{2},t_{1})\), \(T(t_{2},t_{1})\) of Theorem 3.3and ĵ, g of Theorem 3.1with the inequality
are fulfilled. Then
Remark 3.6
Corollary 3.5 alters into [36], Lemma 2.5 \((\beta _{1})\) by letting \(j=1\), \(\varsigma =1\), \(\Gamma (1)=1\), \(T(t_{2})=0\), and \(w(t_{1})=b(s)\).
4 Boundedness and uniqueness
The boundedness and uniqueness of the discrete fractional inequalities can be evaluated by a relevant practice of Theorem 3.3 in this segment. Consider the IVP of fractional difference equation of the form
where \(K, V:\mathbb{N}_{0}\times \mathbb{R}\rightarrow \mathbb{R}\) are functions, \(h_{0}\) is a constant, and \(t_{1}\), ς, j, h are mentioned as in Theorem 3.3.
The ensuing theorem can illustrate the boundedness on the solutions of (29).
Theorem 4.1
Suppose that
If \(h(t_{1})\) is a solution of (29), then
Proof
Equation (29) is transformed into
Apparently, equation (29) with Definition 2.1 and the combination of (30) and (31) approaches to
where \(\frac{t_{1}^{\varsigma -1}}{\Gamma (\varsigma )}|h_{0}|\leq |\eta ^{j}(t_{1})|\). The remaining calculations can be done through the assumption of correct composition of Theorem 3.3 to get the required inequality (32). □
The uniqueness of solutions of (29) can be identified by the following theorem.
Theorem 4.2
Let
Then (29) has at most one solution.
Proof
IVP (29) with solutions \(h_{1}(t_{1})\) and \(h_{2}(t_{1})\) is restated as follows:
The last equation with speculations (33), (34) provides
The prior inequality by making a few modifications in the technique for Theorem 3.3 to \(|h_{1}^{ j}(t_{1})-h_{2}^{j}(t_{1})|\) induces
Subsequently \(h_{1}(t_{1})=h_{2}(t_{1})\), and one positive solution of fractional difference equation (29) exists. □
5 Concluding remarks
Fixated on the guideline of discrete fractional analytic and with the advantage of fractional sum inequalities, we suggested new varieties of discrete Gronwall fractional inequalities in this paper. Such inequalities can be seen not exclusively to remember explicit estimations for solutions of fractional difference equations in discrete type yet additionally in the investigation to the uniqueness and continuous dependency on the initial value for the solutions in the analysis.
Availability of data and materials
Not applicable.
References
Isaacs, G.: Exponential laws for fractional differences. Math. Comput. 35(151), 933–936 (1980)
Podlubny, I.: Matrix approach to discrete fractional calculus. Fract. Calc. Appl. Anal. 3(4), 359–386 (2000)
Khan, Z.: Hadamard type fractional differential equations for the system of integral inequalities on time scales. Integral Transforms Spec. Funct. 31(5), 412–423 (2020)
Heymans, N., Podlubny, I.: Physical interpretation of initial conditions for fractional differential equations with Riemann–Liouville fractional derivatives. Rheol. Acta 45, 765–771 (2006)
Khurshid, Y., Khan, M.A., Chu, Y.M., Khan, Z.: Hermite–Hadamard Fejér inequalities for conformal fractional integrals via preinvex functions. J. Funct. Spaces 2019, 1–10 (2019)
Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives—Theory and Applications. Gordon & Breach, Linghorne (1993)
Khan, Z.: Analysis on some powered integral inequalities with retarded argument and application. J. Taibah Univ. Sci. 14(1), 488–495 (2020)
Goodrich, C.: Solutions to a discrete right-focal fractional boundary value problem. Int. J. Differ. Equ. 5(2), 195–216 (2010)
Khan, M.A., Hanif, M., Khan, Z., Ahmad, K., Chu, Y.M.: Association of Jensen’s inequality for s-convex function with Csiszaár divergence. J. Inequal. Appl. 2019(162), 1 (2019)
Sugiyama, S.: On stability problem of difference equations. Bull. Sci. Engr. Res. Lab. Waseda Univ. 35, 140 (1969)
Miller, K.S., Ross, B.: Fractional difference calculus, in univalent functions, fractional calculus, and their applications. In: Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, pp. 139–152. Horwood, Chichester (1989)
Atici, F.M., Eloe, P.W.: A transform method in discrete fractional calculus. Int. J. Difference Equ. 2, 165–176 (2007)
Atici, F.M., Eloe, P.W.: Initial value problems in discrete fractional calculus. Proc. Am. Math. Soc. 137(3), 981–989 (2009)
Atici, F.M., Eloe, P.W.: Discrete fractional calculus with the nabla operator. Electron. J. Qual. Theory Differ. Equ. 2009, 3 (2009)
Atici, F.M., Sengul, S.: Modeling with fractional difference equations. J. Math. Anal. Appl. 369(1), 1–9 (2010)
Bastos, N.R.O., Torres, D.F.M.: Combined delta-nabla sum operator in discrete fractional calculus. Commun. Fract. Calc. 1, 41–47 (2010)
Holm, M.: Sum and difference compositions in discrete fractional calculus. CUBO 13(3), 153–184 (2011)
Anastassiou, G.A.: Right nabla discrete fractional calculus. Int. J. Difference Equ. 6(2), 91–104 (2011)
Atici, F.M., Eloe, P.W.: Two-point boundary value problems for finite fractional difference equations. J. Differ. Equ. Appl. 17, 445–456 (2011)
Goodrich, C.S.: Continuity of solutions to discrete fractional initial value problems. Comput. Math. Appl. 59(11), 3489–3499 (2010)
Bastos, N.R.O., Ferreira, R.A.C., Torres, D.F.M.: Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete Contin. Dyn. Syst. 29(2), 417–437 (2011)
Magin, R.L.: Fractional Calculus in Bioengineering. Begell House, Danbury (2006)
Milkelazde, S.E.: De la resolution numerique des equations integrales. Bull. Acad. Sci. URSS VII, 255–257 (1935) (in Russian)
Khan, Z.: Solvability for a class of integral inequalities with maxima on the theory of time scales and their applications. Bound. Value Probl. 2019, 146 (2019)
Abdeljawad, T.: On Riemann and Caputo fractional differences. Comput. Math. Appl. 62(3), 1602–1611 (2011)
Zheng, B.: Some new discrete fractional inequalities and their applications in fractional difference equations. J. Math. Inequal. 9(3), 823–839 (2015)
Khan, Z.: Study on the estimates of Gronwall Ou-Iang dynamic integral inequalities by means of diamond-α derivatives. Adv. Differ. Equ. 2020, 240 (2020)
Abdeljawad, T.: Fractional differences and integration by parts. J. Comput. Anal. Appl. 13(3), 574–582 (2011)
Ullah, S.Z., Khan, M.A., Khan, Z., Chu, Y.M.: Integral majorization type inequalities for the functions in the sense of strong convexity. J. Funct. Spaces 2019, 1–12 (2019)
Gray, H., Zhang, N.: On a new definition of the fractional difference. Math. Comput. 50(182), 513–529 (1988)
Khan, A., Syam, M.I., Zada, A., Khan, H.: Stability analysis of nonlinear fractional differential equations with Caputo and Riemann–Liouville derivatives. Eur. Phys. J. Plus 133, 1–16 (2018)
Khan, Z.: On some explicit bounds of integral inequalities related to time scales. Adv. Differ. Equ. 2019, 243 (2019)
Ullah, S.Z., Khan, M.A., Khan, Z., Chu, Y.M.: Coordinate strongly s-convex functions and related results. J. Math. Inequal. 14(3), 829–843 (2020)
Agarwal, R.P.: Difference Equations and Inequalities: Theory, Methods and Applications. CRC Press, Boca Raton (2000)
Pachpatte, B.G.: Bounds on certain integral inequalities. J. Inequal. Pure Appl. Math. 3(3), Article ID 47 (2002)
Pachpatte, B.G.: On some fundamental integral inequalities and their discrete analogues. J. Inequal. Pure Appl. Math. 2(2), Article ID 15 (2001)
Acknowledgements
The authors would like to express their sincere thanks to the editor and anonymous reviewers for their helpful comments and suggestions.
Funding
This research was funded by the Deanship of Scientific Research at Princess Nourah bint Abdulrahman University through the Fast-track Research Funding Program.
Author information
Authors and Affiliations
Contributions
All authors made equal contributions, read, and supported the last original copy.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Abbreviations
Not applicable.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Khan, Z.A., Jarad, F., Khan, A. et al. Nonlinear discrete fractional sum inequalities related to the theory of discrete fractional calculus with applications. Adv Differ Equ 2021, 100 (2021). https://doi.org/10.1186/s13662-021-03257-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-021-03257-4