Abstract
In this paper, we investigate the existence and uniqueness results for sequential fractional integro-differential equations with impulsive conditions. The nonlinear term contains the integral terms which are used to represent in the thermal conductivity of the material problems. Our methods are based on the fixed point theorems such as Banach fixed point theorem and Krasnoselskii’s. We obtain the sufficient conditions of the existence of solutions. Examples are given to illustrate the results.
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1 Introduction
Fractional calculus is a mathematical branch investigating the properties of derivatives and integrals of non-integer orders. In the past two decades, fractional calculus has been a research focus and attracted the attention of many researchers all over the world. More than that, fractional calculus is more and more widely used in various disciplines, especially in fluid mechanics, physics, signal processing, materials science, electrochemistry, biology and so on. It is due to the further development of fractional calculus theory itself.
In recent years, fractional differential equations have been widely used in the mathematical modeling of real-world phenomena. These applications have motivated many researchers in the field of differential equations to investigate fractional differential equations with different fractional derivatives [1,2,3,4,5].
The dynamics of populations subject to abrupt changes, as well as other phenomena like harvesting, diseases, and other phenomena, have all been described using impulsive differential equations. In [6], Karthikeyan et al. discussed the existence results for impulsive fractional integro-differential equations involving integral boundary conditions. In [7], the authors studied almost sectorial operators on \(\Psi \)—Hilfer derivative fractional impulsive integro-differential equations. Renusumrit et al. [8] investigated the existence and stability results for impulsive fractional integro-differential equations involving the ABC derivative under integral boundary condition.
Sequential fractional differential equations have also received considerable attention. The concept of this kind of equations was assumed in the book [9], which includes an in-depth investigation of a particular class of sequential differential equations. Fascinated by this kind of problem, various authors have examined these equations covering numerous fractional derivative types [10,11,12]. In mathematical physics, there are many significant boundary value problems corresponding to the real-life problem. Specifically, in the studies of vibrations of a membrane, vibrations of a structure one have to solve a homogeneous boundary value problem. Recently, the existence of solutions to boundary value problems and boundary condition has received a great deal of interest; see reference [13,14,15,16].
In [17], the authors have analyzed the following nonlinear fractional integro-differential equations
where \(\alpha \in (1,2]\), \(\beta \in (0,1] \), \(f:[0,1] \times \mathbb {R}^{3} \rightarrow \mathbb {R}\) is continuous and
where \(\gamma , \ \lambda : [0,1] \times [0,1] \rightarrow [0,+\infty )\) are such that with \(\varphi ^*=\sup _{t\in [0,1]}(\int _{0}^{t}\lambda (t,s)\textrm{d}s){<}\infty \), and \(\psi ^*{=}\sup _{t\in [0,1]}(\int _{0}^{t}\gamma (t,s)\textrm{d}s)<\infty \).
In [18], the authors have examined the new existence results for nonlinear fractional integro-differential equations
where \(\alpha \in (1,2]\), \(\beta \in (0,2] \), \(f:[0,1] \times \mathbb {R}^{3} \rightarrow \mathbb {R}\) is continuous and
where \(\lambda , \ \delta : [0,1] \times [0,1] \rightarrow [0,+\infty )\) are such that with \(\phi ^{*} =\sup _{t\in [0,1]}(\int _{0}^{t}\lambda (t,s)\textrm{d}s)<\infty \), and \(\psi ^{*} = \sup _{t\in [0,1]}(\int _{0}^{t}\gamma (t,s)\textrm{d}s)<\infty \).
In [19], the authors have discussed the following sequential fractional differential equations with three-point boundary conditions,
By using the Banach’s contraction principle and Krasnoselskii’s fixed point theorem, they proved the existence and uniqueness Results.
In [20], the authors have studied the following nonlinear implicit neutral fractional differential equations with finite delay and impulses
where \(^{c} D_{t_{k}}^{\alpha }\) is the Caputo fractional derivative, \(f: [0,T] \times PC([0,T],\mathbb {R}) \times \mathbb {R} \rightarrow \mathbb {R} \), \(\phi :[0,T] \times PC([0,T],\mathbb {R}) \rightarrow \mathbb {R}\) are given functions with \(\phi (0,\varphi )=0\), \(I_{k}: PC([0,T],\mathbb {R}) \rightarrow \mathbb {R} \), \(\varphi \in PC([0,T],\mathbb {R})\), \(0=t_{0}<t_{1}<...<t_{m}<t_{m+1}=T\).
Inspired by the above works, we consider the sequential fractional integro-differential equations with impulsive conditions of the form:
\(\varpi \in (1,2]\), \(\vartheta \in (0,2] \), \(^{c} D^{\varpi },\ ^{c} D^{\vartheta }\) are the Caputo fractional derivatives, \(\check{f}: \Theta \times PC(\Theta ,\mathbb {R}) \times \mathbb {R} \rightarrow \mathbb {R} \), \(\check{I}_{\Bbbk }: PC(\Theta ,\mathbb {R}) \times \mathbb {R} \rightarrow \mathbb {R} \)
where \(\alpha , \beta : \Theta \times \Theta \rightarrow [0,+\infty )\) with \(\Psi ^* {=} \sup |\int _{0}^{\mathfrak {t}}\alpha (\mathfrak {t},\ell )\textrm{d}\ell |<\infty , \ \chi ^* = \sup |\int _{0}^{\mathfrak {t}}\beta (\mathfrak {t},\ell )\textrm{d}\ell |<\infty \). \(\Delta \upsilon (\mathfrak {t}_{\Bbbk })=\upsilon (\mathfrak {t}_{\Bbbk }^{+})-\upsilon (\mathfrak {t}_{\Bbbk }^{-})\) denotes the jump of \(\upsilon \) at \(\mathfrak {t}=\mathfrak {t}_{\Bbbk }\), \(\upsilon (\mathfrak {t}_{\Bbbk }^{+})\) and \(\upsilon (\mathfrak {t}_{\Bbbk }^{-})\) represent the right and left limits of \(\upsilon (\mathfrak {t})\) at \(\mathfrak {t}=\mathfrak {t}_{\Bbbk }\) respectively, \(\Bbbk =1,2,...,m\).
This paper is organized as follows: In Sect. 2, we introduce some preliminaries about fractional calculus, lemma and definitions. In Sect. 3, we present two main results: the first one is based on the Krasnoselskii’s the fixed point theorem, and the second one is based on Banach contraction principle. The results are illustrated by two examples in the last section.
2 Preliminaries
We introduce few definitions, notations, and lemmas of fractional calculus.
Definition 2.1
[21] The fractional integral of order \(\alpha > 0\) with the lower limit zero for a function f can be defined as
where \(\Gamma (.)\) is the Gamma function.
Definition 2.2
[21] The Caputo derivative of order \(\alpha > 0\) with the lower limit zero for a function f can be defined as
where \( \mathfrak {t} > 0, \ 0 \le n-1< \alpha < n, \ n \in \mathbb {N}\).
Lemma 2.3
[21] Let \(\alpha , \beta \le 0\); then, the following relation holds:
Lemma 2.4
[21] Let \(n \in \mathbb {N}\) and \(n-1< \alpha < n\). If f is a continuous function, then we have
Theorem 2.5
[8] Let M be a convex, closed, bounded and nonempty subset of a Banach space X. Let A and B be two operators such that
-
1.
\(Ax + By \in M\) whenever \(x, y \in M\)
-
2.
A is continuous and compact
-
3.
B is a contraction mapping.
Then, there exists \(z \in M \) such that \(z = Az + Bz\).
Lemma 2.6
Let \(\alpha \in (1,2]\), \(\beta \in (0,2] \) and \(h \in C(\Theta , \mathbb {R})\). A function x is a solution of the fractional integral equation
\(\Bbbk =1,..m\), if and only if x is a solution of the following fractional problem
Proof
By applying Lemma 2.4, we get
where \(a_{0},a_{1},a_{2},a_{3} \in \mathbb {R}\). So
And by using \(^{c} D^{\alpha } x(0)=x(0)=0\), we attain \(a_{0} = 0\) and \(a_{2 }= 0\). As a result of \(^{c} D^{\alpha } x(1)=0\), we have
Now, we use \(x(1) = 0\) to get
By substituting the value of \(a_0, a_1, a_2, a_3\), we obtain
If \(t \in (t_{1}, t_{2}]\),
If \(t \in (t_{2}, t_{3}]\),
Repeating this process, the solution x(t), for \(t \in (t_{\Bbbk }, t_{\Bbbk +1}]\), where \(\Bbbk =1,...,m\) could be expressed as
Conversely, by direct estimations, we get the desired result. \(\square \)
3 Main results
Consider the Banach space \(\check{X}\) of all continuous functions from \(\Theta \rightarrow \mathbb {R}\) endowed with the norms \(\Vert \mathfrak {y} \Vert = \sup \{|\mathfrak {y}(\mathfrak {t})|:\mathfrak {t} \in \Theta \}\). Also consider the space
\(\Vert \mathfrak {y} \Vert _v = \sup _{\mathfrak {t}\in \Theta } \left( \displaystyle {\frac{|\mathfrak {y}(\mathfrak {t})|}{e^{v\mathfrak {t}}}}\right) \) where \(v>\displaystyle {\frac{(1+\Phi ^{*}+\varphi ^{*})}{(\Gamma (\varpi +\vartheta ))}}\Vert \sigma \Vert \), and \(\sigma \in C(\Theta ; [0, \infty )) \cap L^1(\Theta ; [0, \infty ))\).
The following hypotheses are needed to prove the main results:
\((H_1)\) For all \(\mathfrak {t} \in \Theta \) and \(\upsilon _1, \upsilon _2, \upsilon _3, \mathfrak {y}_1, \mathfrak {y}_2, \mathfrak {y}_3 \in \mathbb {R}\), we have
with \( \sigma \in C(\Theta ; [0, \infty )) \)
\((H_2)\) \(|\check{f} (\mathfrak {t}, \upsilon , \mathfrak {y}, z)| \le \theta (\mathfrak {t}), \forall (\mathfrak {t}, \upsilon , \mathfrak {y}, z) \in \Theta \times \mathbb {R}^3\) with \(\theta \in C(\Theta ; \mathbb {R}^+)\).
\((H_3)\) There exists a constant \(l>0\) such that
and \(\vert \check{I}_{\Bbbk }(\upsilon ) \vert \le l \theta (\mathfrak {t}). \)
\((H_4)\) For all \(\mathfrak {t} \in \Theta \) and \(\upsilon _1, \upsilon _2, \upsilon _3, \mathfrak {y}_1, \mathfrak {y}_2, \mathfrak {y}_3 \in \mathbb {R}\), we have \(|\check{f}(\mathfrak {t}, \upsilon _1, \upsilon _2, \upsilon _3)-\check{f} (\mathfrak {t}, \mathfrak {y}_1, \mathfrak {y}_2, \mathfrak {y}_3| \le \sigma (\mathfrak {t}) |\upsilon _1 - \mathfrak {y}_1| + |\upsilon _2 - \mathfrak {y}_2| + |\upsilon _3 - \mathfrak {y}_3|\) with \( \sigma (\mathfrak {t}) \in L^1(\Theta ; [0, \infty )) \) and
\((H_5)\) There exists a constant \(l>0\) such that \(\vert \check{I}_{\Bbbk }(\upsilon )-\check{I}_{\Bbbk }(\mathfrak {y}) \vert \le l \vert \upsilon -\mathfrak {y} \vert \).
Theorem 3.1
Assume that hypotheses \((H_1)-(H_3)\) are satisfied. Then, problem (1.1) has at least one solution.
Proof
Let the ball \(B_\mathfrak {r} = \{\mathfrak {y} \in \check{X}: \Vert \mathfrak {y} \Vert _v \le \mathfrak {r}\}\) with
Define the operators \( \digamma _1, \digamma _2\) on \(B_\mathfrak {r}\), where
For \(\upsilon , \mathfrak {y} \in B_\mathfrak {r} \), we have
Therefore,
Then \(\digamma _1\upsilon +\digamma _2\mathfrak {y} \in B_\mathfrak {r}\).
Now, we prove that \(\digamma _1\) is a contraction. For \(\upsilon , \mathfrak {y} \in B_\mathfrak {r} \), we have
We arrive at the conclusion that \(\digamma _1\) is a contraction.
Next, we have to show that \(\digamma _2\) is continuous and compact. \(\check{f}\) is continuous that implies that the operator \(\digamma _2\) is continuous. Also, \(\digamma _2\) is bounded uniformly on \(B_\mathfrak {r}\) as
Assume \(0 \le \mathfrak {t}_1 < \mathfrak {t}_2 \le 1\). We get
Then, \(|\digamma _2\mathfrak {y}(\mathfrak {t}_2) - \digamma _2\mathfrak {y}(\mathfrak {t}_1)|\rightarrow 0\), as \(\mathfrak {t}_1\rightarrow \mathfrak {t}_2\) independently from \(\mathfrak {y} \in B_\mathfrak {r}\). This proves the operator \(\digamma _2\) is relatively compact on \(B_\mathfrak {r}\). Thus, by the Arzela Ascoli theorem, we obtain that \(\digamma _2\) is compact on \(B_\mathfrak {r}\). By the Krasnoselskii’s fixed point theorem, problem (1.1) has at least one solution on \(B_\mathfrak {r}\). \(\square \)
Theorem 3.2
Suppose that \(\check{f}:\Theta \times \mathbb {R}^3 \rightarrow \mathbb {R}\) is a continuous function satisfying hypotheses \((H_4)\) and \((H_5)\), then there exists a unique solution for problem (1.1) under the following condition: \(\mathfrak {r}_1<1\),
where
with \(\sigma ^{*} = \int _{0}^{1}\sigma (\mathfrak {t})\textrm{d}\mathfrak {t}.\)
Proof
Define the operator \( \digamma :\check{X} \rightarrow \check{X}\) by
Setting \(\sup |\check{f}(\mathfrak {t},0,0,0)|=P\).
Consider the set \(B_\mathfrak {r} = \{ \upsilon \in \check{X}: \Vert \upsilon \Vert \le \mathfrak {r}\}\), where \(\mathfrak {r}\ge \displaystyle {\frac{\mathfrak {r}_2}{1-\mathfrak {r}_1}}\), with
For each \(\mathfrak {t} \in \Theta \) and \(\upsilon \in B_\mathfrak {r}\), we have
Then,
Therefore,
Next, we have to prove that \(\digamma \) is a contraction mapping.
For \(\upsilon , \mathfrak {y} \in B_\mathfrak {r} \), we have
Since, \(\mathfrak {\mathfrak {r}}_{1} < 1\), then \(\digamma \) is a contraction. Therefore, system (1.1) has a unique solution. \(\square \)
4 Examples
We provide two examples in this section to demonstrate the applicability of our findings.
Example 4.1
Assume the following problem:
Here \(\vartheta =\frac{16}{11}\), \(\varpi =\frac{17}{11}, \ m=1\)
It follows that
Theorem 3.1 allows us to conclude that there is at least one solution to the given problem.
Example 4.2
Take the following problem:
here \(\vartheta =\frac{11}{7}\), \(\varpi =\frac{10}{7}, \ m=1\)
We conclude from Theorem 3.2 that the given problem has an unique solution.
5 Conclusion
In this paper, we have investigated sequential fractional integro-differential equations with impulsive conditions.. We proved the existence and uniqueness of solutions by using the Banach contraction mapping principle and Krasnoselskii’s fixed point theorem. Examples are given to highlight the main results. Furthermore, this can be extended with delay conditions over the infinite interval.
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References
Ahmad B, Nieto J (2012) Sequential fractional differential equations with three-point boundary conditions. Comput Math Appl 64(10):3046–3052
Benchohra M, Bouriah S, Henderson J (2015) Existence and stability results for nonlinear implicit neutral FDEs with finite delay and impulses. Comm Appl Nonlinear Anal 22(1):46–67
Bragdi A, Frioui A, Guezane Lakoud A (2020) Existence of solutions for nonlinear fractional integro-differential equations. Adv Differ Equ 2020(1). Article ID 418
Fuentes OM, Vázquez FM, Anaya GF, Aguilar JFG (2021) Analysis of fractional-order nonlinear dynamic systems with general analytic kernels: Lyapunov stability and inequalities. Mathematics 9(17):2084
Asma JFG, Aguilar G, Rahman and Javed M, (2021) Stability analysis for fractional order implicit \(\psi \)-Hilfer differential equations. Math Methods Appl Sci 45(5):2701–2712
Dhayal R, Aguilar JFG, Jimenez JT (2022) Stability analysis of Atangana-Baleanu fractional stochastic differential systems with impulses. Int J Syst Sci 53(16):3481–3495
Calderon AG, Cruz LXV, Hernández MAT, Aguilar JFG (2022) Assessment of the performance of the hyperbolic-NILT method to solve fractional differential equations. Math Comput Simul 206:375–390
Martinez HY, Aguilar JFG, Mustafa, (2023) New modified Atangana-Baleanu fractional derivative applied to solve nonlinear fractional differential equations. Physica Scripta 98(3):035202
Karthikeyan K, Karthikeyan P, Chalishajar DN, Senthil Raja D (2021) Analysis on \(\Psi \)-Hilfer fractional impulsive differential equations. Symmetry 13:1895
Krim S, Abbas S, Benchohra M (2021) Caputo-Hadamard implicit fractional differential equations with delay. Sao Paulo J Math Sci 15:463–484
Wattanakejorn V, Karthikeyan P, Poornima S, Karthikeyan K, Sitthiwirattham T (2022) Existence solutions for implicit fractional relaxation differential equations with impulsive delay boundary conditions. Axioms 611(11)
Karthikeyan K, Reunsumrit J, Karthikeyan P, Poornima S, Tamizharasan D, Sitthiwirattham T (2022) Existence results for impulsive fractional integrodifferential equations involving integral boundary conditions. Math Probl Eng 2022. Article ID 6599849
Karthikeyan K, Karthikeyan P, Baskonus HM, Ming-Chu Yu, Venkatachalam K (2021) Almost sectorial operators on \(\Psi \)- Hilfer derivative fractional impulsive integro-differential equations. Math Methods Appl Sci 45(3):8045–8059
Reunsumrit J, Karthikeyan P, Poornima S, Karthikeyan K, Sitthiwirattham T (2022) Analysis of existence and stability results for impulsive fractional Integro-Differential Equations Involving the Atangana-Baleanu-Caputo Derivative under Integral Boundary Conditions. Math Probl Eng 2022. Article ID 5449680
Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential equations. Wiley, New York
Ahmad B, Ntouyas SK, Agarwal RP, Alsaedi A (2016) Existence results for sequential fractional integro-differential equations with nonlocal multi-point and strip conditions. Boundary Value Probl 2016(1). Article ID 205
Bouaouid M, Hilal K, Melliani S (2019) Sequential evolution conformable differential equations of second order with nonlocal condition. Adv Differ Equ 21
Ahmad B, Nieto J (2013) Boundary value problems for a class of sequential integrodifferential equations of fractional order. J Function Spaces 2013. Article ID 149659
Ibnelazyz L, Guida K, Hilal K, Melliani S (2021) Existence results for nonlinear fractional integro-differential equations with integral and antiperiodic boundary conditions. Comput Appl Math 40(1), article 33
Ahmad B, Sivasundaram S (2010) On four-point nonlocal boundary value problems of nonlinear integro-differential equations of fractional order. Appl Math Comput 217(2):480–487
Karthikeyan P, Venkatachalam K (2021) Some results on multipoint integral boundary value problems for fractional integro-differential equations. Prog Fraction Differ Appl 7(2):127–136
Baleanu D, Etemad S, Rezapour S (2020) On a fractional hybrid integro-differential equation with mixed hybrid integral boundary value conditions by using three operators. Alex Eng J 59(5):3019–3027
Ibnelazyz L, Guida K, Hilal K, Melliani S (2021) New existence results for nonlinear fractional integrodifferential equations. Adv Math Phys 2021. Article ID 5525591
Podlubny L (1993) Fractional differential equations. Academic Press, New York
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Karthikeyan, P., Poornima, S. Existence results for sequential fractional integro-differential equations with impulsive conditions. Int. J. Dynam. Control 12, 227–236 (2024). https://doi.org/10.1007/s40435-023-01240-3
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DOI: https://doi.org/10.1007/s40435-023-01240-3