Network congestion may occur naturally or intentionally caused by selfish nodes. Existing congestion control techniques designed by researchers for sensor-based networks have primarily focused on natural modes of congestion occurrence and ignored malevolent nodes' potential for purposeful congestion-like scenario creation. In light of this fact, a security attack-resistant congestion control method that takes into account both possible sources of congestion in sensor nodes has been developed. So firstly, a trust-based technique has been developed to get rid of selfish nodes' intentional attempts to cause congestion. After the elimination of malicious nodes, a congestion avoidance method has been applied which tries to prevent the natural way of congestion occurrence. For this purpose, we have applied a multi-criteria decision-making method as there are many factors responsible for congestion occurrence. The remaining energy, node potential value, node load factor, and traffic burst rate have been considered as decision factors. Simulation results show that our Security Aware Congestion Control technique using the AHP method (SACC-AHP) outperforms the existing relevant techniques LEACH, TCEER, TASRP, CARA and SACC in terms of energy efficiency, security, packet delivery ratio and network lifetime.
Citation: Divya Pandey, Vandana Kushwaha. The use of Analytical Hierarchy Process in sensor-based networks for security-aware congestion control[J]. Networks and Heterogeneous Media, 2023, 18(1): 244-274. doi: 10.3934/nhm.2023009
[1] | Dazhao Chen . Endpoint estimates for multilinear fractional singular integral operators on Herz and Herz type Hardy spaces. AIMS Mathematics, 2021, 6(5): 4989-4999. doi: 10.3934/math.2021293 |
[2] | Muhammad Asim, Ghada AlNemer . Boundedness on variable exponent Morrey-Herz space for fractional multilinear Hardy operators. AIMS Mathematics, 2025, 10(1): 117-136. doi: 10.3934/math.2025007 |
[3] | Shuhui Yang, Yan Lin . Multilinear strongly singular integral operators with generalized kernels and applications. AIMS Mathematics, 2021, 6(12): 13533-13551. doi: 10.3934/math.2021786 |
[4] | Pham Thi Kim Thuy, Kieu Huu Dung . Hardy–Littlewood maximal operators and Hausdorff operators on $ p $-adic block spaces with variable exponents. AIMS Mathematics, 2024, 9(8): 23060-23087. doi: 10.3934/math.20241121 |
[5] | Wanjing Zhang, Suixin He, Jing Zhang . Boundedness of sublinear operators on weighted grand Herz-Morrey spaces. AIMS Mathematics, 2023, 8(8): 17381-17401. doi: 10.3934/math.2023888 |
[6] | Jie Sun, Jiamei Chen . Weighted estimates for commutators associated to singular integral operator satisfying a variant of Hörmander's condition. AIMS Mathematics, 2023, 8(11): 25714-25728. doi: 10.3934/math.20231311 |
[7] | Kieu Huu Dung, Do Lu Cong Minh, Pham Thi Kim Thuy . Commutators of Hardy-Cesàro operators on Morrey-Herz spaces with variable exponents. AIMS Mathematics, 2022, 7(10): 19147-19166. doi: 10.3934/math.20221051 |
[8] | Yueping Zhu, Yan Tang, Lixin Jiang . Boundedness of multilinear Calderón-Zygmund singular operators on weighted Lebesgue spaces and Morrey-Herz spaces with variable exponents. AIMS Mathematics, 2021, 6(10): 11246-11262. doi: 10.3934/math.2021652 |
[9] | Babar Sultan, Mehvish Sultan, Qian-Qian Zhang, Nabil Mlaiki . Boundedness of Hardy operators on grand variable weighted Herz spaces. AIMS Mathematics, 2023, 8(10): 24515-24527. doi: 10.3934/math.20231250 |
[10] | Tianyang He, Zhiwen Liu, Ting Yu . The Weighted $ \boldsymbol{L}^{\boldsymbol{p}} $ estimates for the fractional Hardy operator and a class of integral operators on the Heisenberg group. AIMS Mathematics, 2025, 10(1): 858-883. doi: 10.3934/math.2025041 |
Network congestion may occur naturally or intentionally caused by selfish nodes. Existing congestion control techniques designed by researchers for sensor-based networks have primarily focused on natural modes of congestion occurrence and ignored malevolent nodes' potential for purposeful congestion-like scenario creation. In light of this fact, a security attack-resistant congestion control method that takes into account both possible sources of congestion in sensor nodes has been developed. So firstly, a trust-based technique has been developed to get rid of selfish nodes' intentional attempts to cause congestion. After the elimination of malicious nodes, a congestion avoidance method has been applied which tries to prevent the natural way of congestion occurrence. For this purpose, we have applied a multi-criteria decision-making method as there are many factors responsible for congestion occurrence. The remaining energy, node potential value, node load factor, and traffic burst rate have been considered as decision factors. Simulation results show that our Security Aware Congestion Control technique using the AHP method (SACC-AHP) outperforms the existing relevant techniques LEACH, TCEER, TASRP, CARA and SACC in terms of energy efficiency, security, packet delivery ratio and network lifetime.
Fractional calculus has emerged as a powerful tool to study complex phenomena in numerous scientific and engineering disciplines such as biology, physics, chemistry, economics, signal and image processing, control theory and so on. Fractional differential equations describe many real world process related to memory and hereditary properties of various materials more accurately as compared to classical order differential equations. For examples and applications see the monographs as [1,2,3,4,5,7,6,8].
In the literature, many authors focused on Riemann-Liouville and Caputo type derivatives in investigating fractional differential equations. A generalization of derivatives of both Riemann-Liouville and Caputo was given by R. Hilfer in [9], the known as the Hilfer fractional derivative of order α and a type β∈[0,1], which interpolates between the Riemann-Liouville and Caputo derivative, since it is reduced to the Riemann-Liouville and Caputo fractional derivatives when β=0 and β=1, respectively. Some properties and applications of the Hilfer fractional derivative are given in [10,11] and references cited therein.
Initial value problems involving Hilfer fractional derivatives were studied by several authors, see for example [12,13,14,15] and references therein. In [16] the authors initiated the study of nonlocal boundary value problems for Hilfer fractional derivative, by studying boundary value problem of Hilfer-type fractional differential equations with nonlocal integral boundary conditions
HDα,βx(t)=f(t,x(t)),t∈[a,b],1<α<2,0≤β≤1, | (1.1) |
x(a)=0,x(b)=m∑i=1δiIφix(ξi),φi>0,δi∈R,ξi∈[a,b], | (1.2) |
where HDα,β is the Hilfer fractional derivative of order α, 1<α<2 and parameter β, 0≤β≤1, Iφi is the Riemann-Liouville fractional integral of order φi>0, ξi∈[a,b], a≥0 and δi∈R. Several existence and uniqueness results were proved by using a variety of fixed point theorems.
In [17] the existence and uniqueness of solutions were studied, for a new class of system of Hilfer-Hadamard sequential fractional differential equations
{(HDα1,β11++k1HDα1−1,β11+)u(t)=f(t,u(t),v(t)), 1<α1≤2, t∈[1,e],(HDα2,β21++k2HDα2−1,β21+)v(t)=g(t,u(t),v(t)), 1<α2≤2, t∈[1,e], | (1.3) |
with two point boundary conditions
{u(1)=0, u(e)=A1,v(1)=0, v(e)=A2, | (1.4) |
where HDαi,βi is the Hilfer-Hadamard fractional derivative of order αi∈(1,2] and type βi∈[0,1] for i∈{1,2}, k1,k2,A1,A2∈R+ and f, g:[1,e]×R2→R are given continuous functions.
The fractional derivative with another function, in the Hilfer sense, called ψ-Hilfer fractional derivative, has been introduced in [18]. For some recent results on existence and uniqueness of initial value problems and results on Ulam-Hyers-Rassias stability see [19,20,21,22,23,24,25,26,27,28,29] and references therein. Recently, in [30] the authors extended the results in [16] to ψ-Hilfer nonlocal implicit fractional boundary value problems.
Recently in [31] the existence and uniqueness of solutions were studied, for a new class of boundary value problems of sequential ψ-Hilfer-type fractional differential equations with multi-point boundary conditions of the form
(HDα,β;ψ+kHDα−1,β;ψ)x(t)=f(t,x(t)),t∈[a,b], | (1.5) |
x(a)=0,x(b)=m∑i=1λix(θi), | (1.6) |
where HDα,β;ψ is the ψ-Hilfer fractional derivative of order α, 1<α<2 and parameter β, 0≤β≤1, f:[a,b]×R→R is a continuous function, a<b, k,λi∈R,i=1,2,…,m and a<θ1<θ2<…<θm<b.
In this paper, motivated by the research going on in this direction, we study a new class of boundary value problems of ψ-Hilfer fractional integro-differential equations with mixed nonlocal boundary conditions of the form
{HDα,ρ;ψ0+x(t)=f(t,x(t),Iϕ;ψ0+x(t)),t∈(0,T],x(0)=0,m∑i=1δix(ηi)+n∑j=1ωjIβj;ψ0+x(θj)+r∑k=1λkHDμk,ρ;ψ0+x(ξk)=κ, | (1.7) |
where HDu,ρ;ψ0+ is ψ-Hilfer fractional derivatives of order u={α,μk} with 1<μk<α≤2, 0≤ρ≤1, Iv;ψ0+ is ψ-Riemann-Liouville fractional integral of order v={ϕ,βj}, ϕ,βj>0 for j=1,2,…,n, κ,δi,ωj,λk∈R are given constants, the points ηi,θj,ξk∈J, i=1,2,…,m, j=1,2,…,n, k=1,2,…,r and f:J×R2→R is a given continuous function, and J:=[0,T], T>0. It is imperative to note that the problems addressed in this paper provide more insight into the study of ψ-Hilfer fractional differential equations involving mixed nonlocal boundary conditions. Our results are not only interesting from theoretical point of view, but also helpful in studying the applied problems containing the systems like the ones considered in this paper. Our nonlocal boundary conditions are also useful, since they are the most general mixed type. We emphasize that the mixed nonlocal boundary conditions include multi-point, fractional derivative multi-order and fractional integral multi-order boundary conditions.
This paper is organized as follows: In Section 2, we present some necessary definitions and preliminaries results that will be used to prove our main results. The existence and uniqueness of the solutions for the problem (1.7) are established in Section 3. Our methodology for obtaining the desired results is standard, but its application in the framework of the present problem is new. In Section 4, we discuss the Ulam's stability of the solutions of the problem (1.7) in the frame of Ulam-Hyers (UH) stability, generalized Ulam-Hyers (UH) stability, Ulam-Hyers-Rassias (UHR) stability and generalized Ulam-Hyers-Rassias (UHR) stability is investigated. Finally, examples are given in Section 5 to illustrate the theoretical results.
In this section, we introduce some notation, spaces, definitions and fundamental lemmas which are useful throughout this paper.
Let C=C(J,R) denote the Banach space of all continuous functions from J into R with the norm defined by
‖f‖=supt∈J{|f(t)|}. |
On the order hand, we have n-times absolutely continuous functions given by
ACn(J,R)={f:J→R;f(n−1)∈AC(J,R)}. |
Definition 2.1. [2] Let (a,b), (−∞≤a<b≤∞), be a finite or infinite interval of the half-axis R+ and α∈R+. Also let ψ(x) be an increasing and positive monotone function on (a,b], having a continuous derivative ψ′(x) on (a,b). The ψ-Riemann-Liouville fractional integral of a function f with respect to another function ψ on [a,b] is defined by
Iα;ψa+f(t)=1Γ(α)∫taψ′(s)(ψ(t)−ψ(s))α−1f(s)ds,t>a>0, | (2.1) |
where Γ(⋅) is represent the Gamma function.
Definition 2.2. [2] Let ψ′(t)≠0 and α>0, n∈N. The Riemann–Liouville derivatives of a function f with respect to another function ψ of order α correspondent to the Riemann–Liouville, is defined by
Dα;ψa+f(t)=(1ψ′(t)ddt)nIn−α;ψa+f(t)=1Γ(n−α)(1ψ′(t)ddt)n∫taψ′(s)(ψ(t)−ψ(s))n−α−1f(s)ds, | (2.2) |
where n=[α]+1, [α] is represent the integer part of the real number α.
Definition 2.3. [18] Let n−1<α<n with n∈N, [a,b] is the interval such that −∞≤a<b≤∞ and f,ψ∈Cn([a,b],R) two functions such that ψ is increasing and ψ′(t)≠0, for all t∈[a,b]. The ψ-Hilfer fractional derivative of a function f of order α and type 0≤ρ≤1, is defined by
HDα,ρ;ψa+f(t)=Iρ(n−α);ψa+(1ψ′(t)ddt)nI(1−ρ)(n−α);ψa+f(t)=Iγ−α;ψa+Dγ;ψa+f(t), | (2.3) |
where n=[α]+1, [α] represents the integer part of the real number α with γ=α+ρ(n−α).
Lemma 2.4. [2] Let α,β>0. Then we have the following semigroup property given by,
Iα;ψa+Iβ;ψa+f(t)=Iα+β;ψa+f(t),t>a. | (2.4) |
Next, we present the ψ-fractional integral and derivatives of a power function.
Proposition 2.5. [2,18] Let α≥0, υ>0 and t>a. Then, ψ-fractional integral and derivative of a power function are given by
(i) Iα;ψa+(ψ(s)−ψ(a))υ−1(t)=Γ(υ)Γ(υ+α)(ψ(t)−ψ(a))υ+α−1.
(ii) Dα,ρ;ψa+(ψ(s)−ψ(a))υ−1(t)=Γ(υ)Γ(υ−α)(ψ(t)−ψ(a))υ−α−1.
(iii) HDα,ρ;ψa+(ψ(s)−ψ(a))υ−1(t)=Γ(υ)Γ(υ−α)(ψ(t)−ψ(a))υ−α−1,υ>γ=α+ρ(2−α).
Lemma 2.6. Let m−1<α<m, n−1<β<n, n,m∈N, n≤m, 0≤ρ≤1 and α≥β+ρ(n−β). If h∈Cn(J,R), then
HDβ,ρ;ψa+Iα;ψa+h(t)=Iα−β;ψa+h(t). | (2.5) |
Proof. Let λ=β+ρ(n−β) with n−1<λ<n, we get
HDβ,ρ;ψa+(Iα;ψa+h(t))=Iλ−β;ψa+Dλ;ψa+(Iα;ψa+h(t))=Iλ−β;ψa+(1ψ′(t)ddt)nIn−λ;ψa+(Iα;ψa+h(t))=Iλ−β;ψa+(1ψ′(t)ddt)nIn−λ+α;ψa+h(t). |
By using Definition 2.1, we obtain
(1ψ′(t)ddt)In−λ+α;ψa+h(t)=1ψ′(t)ddt(1Γ(n−λ+α)∫taψ′(τ)(ψ(t)−ψ(τ))n+α−λ−1h(τ)dτ)=1Γ(n−λ+α)1ψ′(t)(∫ta(n+α−λ−1)ψ′(τ)ψ′(t)(ψ(t)−ψ(τ))n+α−λ−2h(τ)dτ)=1Γ(n−λ+α−1)∫taψ′(τ)(ψ(t)−ψ(τ))n+α−λ−2h(τ)dτ=In−λ+α−1;ψa+h(t), |
and
(1ψ′(t)ddt)2In−λ+α;ψa+h(t)=1ψ′(t)ddt(1Γ(n−λ+α−1)∫taψ′(τ)(ψ(t)−ψ(τ))n+α−λ−2h(τ)dτ)=1Γ(n−λ+α−1)1ψ′(t)(∫ta(n+α−λ−2)ψ′(τ)ψ′(t)(ψ(t)−ψ(τ))n+α−λ−3h(τ)dτ)=1Γ(n−λ+α−2)∫taψ′(τ)(ψ(t)−ψ(τ))n+α−λ−3h(τ)dτ=In−λ+α−2;ψa+h(t). |
Repeat the above process, we have
(1ψ′(t)ddt)nIn−λ+α;ψa+h(t)=1ψ′(t)ddt(1Γ(α−λ)∫taψ′(τ)(ψ(t)−ψ(τ))α−λ−1h(τ)dτ)=1Γ(α−λ+1)1ψ′(t)(∫ta(α−λ)ψ′(τ)ψ′(t)(ψ(t)−ψ(τ))α−λ−1h(τ)dτ)=1Γ(λ+α)∫taψ′(τ)(ψ(t)−ψ(τ))α−λ−1h(τ)dτ=Iα−λ;ψa+h(t), |
which implies that
HDβ,ρ;ψa+(Iα;ψa+h(t))=Iλ−β;ψa+Iα−λ;ψa+h(t)=Iα−β;ψa+h(t). |
This completes the proof.
Lemma 2.7. [18] If f∈Cn(J,R), n−1<α<n, 0≤ρ≤1 and γ=α+ρ(n−α) then
Iα;ψa+HDα,ρ;ψa+f(t)=f(t)−n∑k=1(ψ(t)−ψ(a))γ−kΓ(γ−k+1)f[n−k]ψI(1−ρ)(n−α);ψa+f(a), | (2.6) |
for all t∈J, where f[n]ψf(t):=(1ψ′(t)ddt)nf(t).
Fixed point theorems play a major role in establishing the existence theory for the problem (1.7). We collect here some well-known fixed point theorems used in this paper.
Lemma 2.8. (Banach contraction principle [32]). Let D be a non-empty closed subset of a Banach space E. Then any contraction mapping T from D into itself has a unique fixed point.
Lemma 2.9. (Krasnosel'ski⌣i's fixed point theorem [33]). Let M be a closed, bounded, convex, and nonempty subset of a Banach space. Let A,B be the operators such that (i) Ax+By∈M whenever x, y∈M; (ii) A is compact and continuous; (iii) B is contraction mapping. Then there exists z∈M such that z=Az+bz.
Lemma 2.10. (Leray-Schauder nonlinear alternative [32]). Let E be a Banach space, C a closed, convex subset of E,U an open subset of C and 0∈U. Suppose that D:¯U→C is a continuous, compact (that is, D(¯U) is a relatively compact subset of C) map. Then either
(i) D has a fixed point in ¯U, or
(ii) there is a x∈∂U (the boundary of U in C) and ν∈(0,1) with x=νD(x).
In order to transform the problem (1.7) into a fixed point problem, we must convert it into an equivalent Voltera integral equation. We provide the following auxiliary lemma, which is important in our main results and concern a linear variant of the boundary value problem (1.7).
Lemma 2.11. Let 1<μk<α≤2, 0≤ρ≤1, γ=α+ρ(2−α), k=1,2,…,r and Ω≠0. Suppose that h∈C. Then x∈C2 is a solution of the problem
{HDα,ρ;ψ0+x(t)=h(t),t∈(0,T],x(0)=0,m∑i=1δix(ηi)+n∑j=1ωjIβj;ψ0+x(θj)+r∑k=1λkHDμk,ρ;ψ0+x(ξk)=κ, | (2.7) |
if and only if x satisfies the integral equation
x(t)=Iα;ψ0+h(t)+(ψ(t)−ψ(0))γ−1ΩΓ(γ)[κ−(m∑i=1δiIα;ψ0+h(ηi)+n∑j=1ωjIα+βj;ψ0+h(s)(θj)+r∑k=1λkIα−μk;ψ0+h(s)(ξk))], | (2.8) |
where
Ω=m∑i=1δi(ψ(ηi)−ψ(0))γ−1Γ(γ)+n∑j=1ωj(ψ(θj)−ψ(0))γ+βj−1Γ(γ+βj)+r∑k=1λk(ψ(ξk)−ψ(0))γ−μk−1Γ(γ−μk). | (2.9) |
Proof. Let x∈C be a solution of the problem (1.7). By using Lemma 2.7, we have
x(t)=Iα;ψ0+h(t)+(ψ(t)−ψ(0))γ−1Γ(γ)c1+(ψ(t)−ψ(0))γ−2Γ(γ−1)c2, | (2.10) |
where c1,c2∈R are arbitrary constants.
For t=0, we get c2=0, and thus
x(t)=Iα;ψ0+h(t)+(ψ(t)−ψ(0))γ−1Γ(γ)c1. | (2.11) |
Taking the operators HDμk,ρ;ψ0+ and Iβj;ψ0+ into (2.10), we obtain
HDμk,ρ;ψ0+x(t)=Iα−μk;ψ0+h(t)+(ψ(t)−ψ(0))γ−μk−1Γ(γ−μk)c1,Iβj;ψ0+x(t)=Iα+βj;ψ0+h(t)+(ψ(t)−ψ(0))γ+βj−1Γ(γ+βj)c1. |
Applying the second boundary condition in (1.7), we have
c1[m∑i=1δi(ψ(ηi)−ψ(0))γ−1Γ(γ)+n∑j=1ωj(ψ(θj)−ψ(0))γ+βj−1Γ(γ+βj)+r∑k=1λk(ψ(ξk)−ψ(0))γ−μk−1Γ(γ−μk)]+m∑i=1δiIα;ψ0+h(ηi)+n∑j=1ωjIα+βj;ψ0+h(θj)+r∑k=1λkIα−μk;ψ0+h(ξk)=κ, |
from which we get
c1=1Ω[κ−(m∑i=1δiIα;ψ0+h(ηi)+n∑j=1ωjIα+βj;ψ0+h(θj)+r∑k=1λkIα−μk;ψ0+h(ξk))], |
where Ω is defined by (2.9). Substituting the value of c1 in (2.11), we obtain (2.8).
Conversely, it is easily to shown, by a direct calculation, that the solution x given by (2.8) satisfies the problem (2.7). The Lemma 2.11 is proved.
In this section, we present existence and uniqueness results to the considered problem (1.7).
For the sake of convenience, we use the following notations:
A(χ,ε)=(ψ(χ)−ψ(0))εΓ(ε+1), | (3.1) |
Λ0=1+A(T,ϕ), | (3.2) |
Λ1=A(T,α)+A(T,γ−1)|Ω|(m∑i=1|δi|A(ηi,α)+n∑j=1|ωj|A(θj,α+βj)+r∑k=1|λk|A(ξk,α−μk)). | (3.3) |
In view of Lemma 2.11, an operator Q:C→C is defined by
(Qx)(t)=Iα;ψ0+Fx(s)(t)+A(t,γ−1)Ω[κ−(m∑i=1δiIα;ψ0+Fx(s)(ηi)+n∑j=1ωjIα+βj;ψ0+Fx(s)(θj)+r∑k=1λkIα−μk;ψ0+Fx(s)(ξk))], | (3.4) |
where
Fx(t)=f(t,x(t),Iϕ;ψ0+x(t)),t∈J. |
Throughout this paper, the expression Iq,ρ0+Fx(s)(c) means that
Iu;ψ0+Fx(s)(c)=1Γ(u)∫c0ψ′(s)(ψ(c)−ψ(s))u−1Fx(s)ds, |
where u={ϕ,βj} and c={t,σ,θj}, j=1,2,…,n.
It should be noticed that the problem (1.7) has solutions if and only if the operator Q has fixed points.
In the first result, we establish the existence and uniqueness of solutions for the problem (1.7), by applying Banach's fixed point theorem.
Theorem 3.1. Assume that f:J×R2→R is a continuous function such that:
(H1) there exist a constant L1>0 such that
|f(t,u1,v1)−f(t,u2,v2)|≤L1(|u1−u2|+|v1−v2|) |
for any ui, vi∈R, i=1,2 and t∈J.
If
Λ0Λ1L1<1, | (3.5) |
where Λ0 and Λ1 are given by (3.2) and (3.3) respectively, then the problem (1.7) has a unique solution on J.
Proof. Firstly, we transform the problem (1.7) into a fixed point problem, x=Qx, where the operator Q is defined as in (3.4). Applying the Banach contraction mapping principle, we shall show that the operator Q has a unique fixed point, which is the unique solution of the problem (1.7)
Let supt∈J|f(t,0,0)|:=M1<∞. Next, we set Br1:={x∈C:‖x‖≤r1} with
r1≥Λ1M1+(|κ|A(T,γ−1))/|Ω|1−Λ0Λ1L1, | (3.6) |
where Ω, A(T,γ−1), Λ0, Λ1 are given by (2.9), (3.1)–(3.3), respectively. Observe that Br1 is a bounded, closed, and convex subset of C. The proof is divided into two steps:
Step I. We show that QBr1⊂Br1.
For any x∈Br1, we have
|(Qx)(t)|≤Iα;ψ0+|Fx(s)|(T)+A(T,γ−1)|Ω|(|κ|+m∑i=1|δi|Iα;ψ0+|Fx(s)|(ηi)+n∑j=1|ωj|Iα+βj;ψ0+|Fx(s)|(θj)+r∑k=1|λk|Iα−μk;ψ0+|Fx(s)|(ξk)). |
We note that
Iϕ;ψ0+|x(τ)|(s)=1Γ(ϕ)∫s0ψ′(τ)(ψ(s)−ψ(τ))ϕ−1|x(τ)|dτ≤A(s,ϕ)‖x‖. |
It follows from conditions (H1) that
|Fx(t)|≤|f(t,x(t),Iϕ;ψ0+x(s)(t))−f(t,0,0)|+|f(t,0,0)|≤L1(|x(t)|+Iϕ;ψ0+|x(s)|(t))+M1,≤L1(1+(ψ(T)−ψ(0))ϕΓ(ϕ+1))‖x‖+M1=L1[1+A(T,ϕ)]‖x‖+M1=L1Λ0‖x‖+M1. |
Then we have
|(Qx)(t)|≤(L1Λ0‖x‖+M1)(ψ(T)−ψ(0))αΓ(α+1)+A(T,γ−1)|Ω|[|κ|+(L1Λ0‖x‖+M1)(m∑i=1|δi|(ψ(ηi)−ψ(0))αΓ(α+1)+n∑j=1|ωj|(ψ(θj)−ψ(0))α+βjΓ(α+βj+1)+r∑k=1|λk|(ψ(ξk)−ψ(0))α−μkΓ(α−μk+1))]=L1Λ0[A(T,α)+A(T,γ−1)|Ω|(m∑i=1|δi|A(ηi,α)+n∑j=1|ωj|A(θj,α+βj)+r∑k=1|λk|A(ξk,α−μk))]‖x‖+[A(T,α)+A(T,γ−1)|Ω|(m∑i=1|δi|A(ηi,α)+n∑j=1|ωj|A(θj,α+βj)+r∑k=1|λk|A(ξk,α−μk))]M1+|κ|A(T,γ−1)|Ω|≤Λ0Λ1L1r1+Λ1M1+|κ|A(T,γ−1)|Ω|≤r1, |
which implies that QBr1⊂Br1.
Step II. We show that Q:C→C is a contraction.
For any x, y∈C and for each t∈J, we have
|(Qx)(t)−(Qy)(t)|≤Iα;ψ0+|Fx(s)−Fy(s)|(T)+A(T,γ−1)|Ω|(m∑i=1|δi|Iα;ψ0+|Fx(s)−Fy(s)|(ηi)+n∑j=1|ωj|Iα+βj;ψ0+|Fx(s)−Fy(s)|(θj)+r∑k=1|λk|Iα−μk;ψ0+|Fx(s)−Fy(s)|(ξk))≤{(ψ(T)−ψ(0))αΓ(α+1)+A(T,γ−1)|Ω|(m∑i=1|δi|(ψ(ηi)−ψ(0))αΓ(α+1)+n∑j=1|ωj|(ψ(θj)−ψ(0))α+βjΓ(α+βj+1)+r∑k=1|λk|(ψ(ξk)−ψ(0))α−μkΓ(α−μk+1))}L1Λ0‖x−y‖={A(T,α)+A(T,γ−1)|Ω|(m∑i=1|δi|A(ηi,α)+n∑j=1|ωj|A(θj,α+βj)+r∑k=1|λk|A(ξk,α−μk))}L1Λ0‖x−y‖=Λ0Λ1L1‖x−y‖, |
which implies that ‖Qx−Qy‖≤Λ0Λ1L1‖x−y‖. As Λ0Λ1L1<1, hence, the operator Q is a contraction. Therefore, by the Banach contraction mapping principle (Lemma 2.8) the operator Q has a fixed point, and hence the problem (1.7) has a unique solution on J. The proof is completed.
Next, we present an existence theorem by using Krasnosel'ski⌣i's fixed point theorem.
Theorem 3.2. Assume that f:J×R2→R is a continuous function satisfying (H1). In addition, we assume that:
(H2) |f(t,u,v)|≤σ(t), ∀(t,u,v)∈J×R2, and σ∈C(J,R+).
If
L1Λ0[Λ1−A(T,α)]<1, | (3.7) |
where Λ0, Λ1, A(T,α) are defined by (3.2), (3.3) and (3.1), respectively, then the problem (1.7) has at least one solution on J.
Proof. Let supt∈J|σ(t)|=‖σ‖ and Br2:={x∈C:‖x‖≤r2}, where
r2≥‖σ‖Λ1+|κ|A(T,γ−1)|Ω|. |
We define the operators Q1 and Q2 on Br2 by
(Q1x)(t)=Iα;ψ0+Fx(s)(t),t∈J,(Q2x)(t)=A(t,γ−1)Ω[κ−(m∑i=1δiIα;ψ0+Fx(s)(ηi)+n∑j=1ωjIα+βj;ψ0+Fx(s)(θj)+r∑k=1λkIα−μk;ψ0+Fx(s)(ξk))],t∈J. |
Note that Q=Q1+Q2. For any x,y∈Br2, we have
|(Q1x)(t)+(Q2y)(t)|≤supt∈J{Iα;ψ0+|Fx(s)|(t)+A(t,γ−1)|Ω|(|κ|+m∑i=1|δi|Iα;ψ0+|Fy(s)|(ηi)+n∑j=1|ωj|Iα+βj;ψ0+|Fy(s)|(θj)+r∑k=1|λk|Iα−μk;ψ0+|Fy(s)|(ξk))}≤‖σ‖{A(T,α)+A(T,γ−1)|Ω|(m∑i=1|δi|A(ηi,α)+n∑j=1|ωj|A(θj,α+βj)+r∑k=1|λk|A(ξk,α−μk))}+|κ|A(T,γ−1)|Ω|≤‖σ‖Λ1+|κ|A(T,γ−1)|Ω|≤r2. |
This implies that Q1x+Q2x∈Br2, which satisfies the assumption (i) of Lemma 2.9.
We show that the assumption (ii) of Lemma 2.9 is satisfied.
Let xn be a sequence such that xn→x in C. Then for each t∈J, we have
|(Q1xn)(t)−(Q1x)(t)|≤Iα;ψ0+|Fxn(s)−Fx(s)|(T)≤A(T,α)‖Fxn−Fx‖. |
Since f is continuous, this implies that the operator Fx is also continuous. Hence, we obtain
‖Fxn−Fx‖→0asn→∞. |
Thus, this shows that the operator Q1x is continuous. Also, the set Q1Br2 is uniformly bounded on Br2 as
‖Q1x‖≤A(T,α)‖σ‖. |
Next, we prove the compactness of the operator Q1. Let sup(t,u,v)∈J×B2r2|f(t,u,v)|=ˆf<∞, then for each t1,t2∈J with 0≤t1<t2≤T, we obtain
|(Q1x)(t2)−(Q1x)(t1)|=1Γ(α)|∫t10ψ′(s)[(ψ(t2)−ψ(s))α−1−(ψ(t1)−ψ(s))α−1]Fx(s)ds+∫t2t1ψ′(s)(ψ(t2)−ψ(s))α−1Fx(s)ds|≤ˆfΓ(α+1)[2(ψ(t2)−ψ(t1))α+|(ψ(t2)−ψ(0))α−(ψ(t1)−ψ(0))α|]. |
Obviously, the right hand side in the above inequality is independent of x and tends to zero as t2→t1. Therefore, the operator Q1 is equicontinuous. So Q1 is relatively compact on Br2. Then, by the Arzelá-Ascoli theorem, Q1 is compact on Br2.
Moreover, it is easy to prove, using condition (3.7), that the operator Q2 is a contraction and thus the assumption (iii) of Lemma 2.9 holds. Thus all the assumptions of Lemma 2.9 are satisfied. So the conclusion of Lemma 2.9 implies that the problem (1.7) has at least one solution on J. The proof is completed.
The Leray-Schauder's nonlinear alternative [32] is used to prove our last existence result.
Theorem 3.3. Assume that:
(H3) there exist a function q∈C(J,R+) and a continuous nondecreasing function Φ:[0,∞)→[0,∞) which is subhomogeneous (that is, Φ(μx)≤μΦ(x), for all μ≥1 and x∈C), such that
|f(t,u,v)|≤q(t)Φ(|u|+|v|)for each(t,u,v)∈J×R2; |
(H4) there exist a constant M2>0 such that
M2Λ0Λ1Φ(M2)‖q‖+(|κ|A(T,γ−1))/|Ω|>1, |
with Ω, A(T,α) Λ0 and Λ1 by (2.9), (3.1), (3.2) and (3.3).
Then, the problem (1.7) has at least one solution on J.
Proof. Let the operator Q be defined by (3.4). Firstly, we show that Q maps bounded sets (balls) into bounded set in C. For a constant r3>0, let Br3={x∈C:‖x‖≤r3} be a bounded ball in C. Then, for t∈J, we obtain
|(Qx)(t)|≤supt∈J{Iα;ψ0+|Fx(s)|(t)+A(t,γ−1)|Ω|(|κ|+m∑i=1|δi|Iα;ψ0+|Fx(s)|(ηi)+n∑j=1|ωj|Iα+βj;ψ0+|Fx(s)|(θj)+r∑k=1|λk|Iα−μk;ψ0+|Fx(s)|(ξk))}≤‖q‖Φ{(1+(ψ(T)−ψ(0))ϕΓ(ϕ+1))‖x‖}{Iα;ψ0+(1)(T)+A(T,γ−1)|Ω|×(m∑i=1|δi|Iα;ψ0+(1)(ηi)+n∑j=1|ωj|Iα+βj;ψ0+(1)(θj)+r∑k=1|λk|Iα−μk;ψ0+(1)(ξk))}+|κ|(ψ(T)−ψ(0))γ−1|Ω|Γ(γ)=‖q‖Φ(Λ0‖x‖){A(T,α)+A(T,γ−1)|Ω|(m∑i=1|δi|A(ηi,α)+n∑j=1|ωj|A(θj,α+βj)+r∑k=1|λk|A(ξk,α−μk))}+|κ|A(T,γ−1)|Ω|≤Λ0Λ1Φ(‖x‖)‖q‖+|κ|A(T,γ−1)|Ω|. |
Consequently
‖Qx||≤Λ0Λ1Φ(r3)‖q‖+|κ|A(T,γ−1)|Ω|. |
Next, we show that the operator Q maps bounded sets into equicontinuous sets of C. Let t1,t2∈J with t1<t2 and x∈Br3. Then we get
|(Qx)(t2)−(Qx)(t1)|≤1Γ(α)|∫t10ψ′(s)[(ψ(t2)−ψ(s))α−1−(ψ(t1)−ψ(s))α−1]Fx(s)ds+∫t2t1ψ′(s)(ψ(t2)−ψ(s))α−1Fx(s)ds|+(ψ(t2)−ψ(0))γ−1−(ψ(t1)−ψ(0))γ−1|Ω|Γ(γ)(|κ|+m∑i=1|δi|Iα;ψ0+|Fx(s)|(ηi)+n∑j=1|ωj|Iα+βj;ψ0+|Fx(s)|(θj)+r∑k=1|λk|Iα−μk;ψ0+|Fx(s)|(ξk))≤Λ0Φ(r3)‖q‖Γ(α+1)[2(ψ(t2)−ψ(t1))α+|(ψ(t2)−ψ(0))α−(ψ(t1)−ψ(0))α|]+|κ|+Λ0Λ1Φ(r3)‖q‖|Ω|Γ(γ)|(ψ(t2)−ψ(0))γ−1−(ψ(t1)−ψ(0))γ−1|. | (3.8) |
As t2−t1→0, the right hand side of (3.8) tends to zero independently of x∈Br3. Hence, by the Arzelá-Ascoli theorem, the operator Q is completely continuous.
The result will follow from the Leray-Schauder's nonlinear alternative once we have proved the boundedness of the set of all solutions to the equations x=ϱQx for ϱ∈(0,1).
Let x be a solution. Then, for t∈J, and following calculations similar to the first step, we obtain
|x(t)|=|ϱ(Qx)(t)|≤Λ0Λ1Φ(‖x‖)‖q‖+|κ|A(T,γ−1)|Ω|, |
which leads to
‖x‖Λ0Λ1Φ(‖x‖)‖q‖+(|κ|A(T,γ−1))/|Ω|≤1. |
In view of (H4), there exists a constant M2>0 such that ‖x‖≠M2. Let us set
K:={x∈C:‖x‖<M2}. |
We see that the operator Q:¯K→C is continuous and completely continuous. From the choice of ¯K, there is no x∈∂K such that x=ϱQx for some ϱ∈(0,1). Consequently, by the nonlinear alternative of Leray-Schauder type (Lemma 2.10), we deduce that the operator Q has a fixed point x∈¯K which is a solution of the problem (1.7). The proof is completed.
In this section, we are developing some results on the different types of Ulam's stability such as Ulam-Hyers (UH), generalized Ulam-Hyers (UH), Ulam-Hyers-Rassias (UHR) and generalized Ulam-Hyers-Rassias (UHR) stability for the proposed problem (1.7).
We start with needed definitions. Let ϵ>0 be a positive real number and Θ:J→R+ be a continuous function. We consider the following inequalities:
|HDα,ρ;ψ0+z(t)−f(t,z(t),Iϕ;ψ0+z(t))|≤ϵ, | (4.1) |
|HDα,ρ;ψ0+z(t)−f(t,z(t),Iϕ;ψ0+z(t))|≤ϵΘ(t), | (4.2) |
|HDα,ρ;ψ0+z(t)−f(t,z(t),Iϕ;ψ0+z(t))|≤Θ(t). | (4.3) |
Definition 4.1. [34] The problem (1.7) is said to be UH stable if there exists a real number Mf>0 such that for each ϵ>0 and for each solution z∈C of the inequality (4.1), there exists a solution x∈C of the problem (1.7) with
|z(t)−x(t)|≤Mfϵ,t∈J. | (4.4) |
Definition 4.2. [34] The problem (1.7) is said to be generalized UH stable if there exists a function Θ∈C(R+,R+) with Θ(0)=0 such that, for each solution z∈C of inequality (4.2), there exists a solution x∈C of the problem (1.7) with
|z(t)−x(t)|≤Θ(ϵ),t∈J. | (4.5) |
Definition 4.3. [34] The problem (1.7) is said to be UHR stable with respect to Θ∈C(J,R+) if there exists a real number Mf,Θ>0 such that for each ϵ>0 and for each solution z∈C of the inequality (4.2) there exists a solution x∈C of the problem (1.7) with
|z(t)−x(t)|≤Mf,ΘϵΘ(t),t∈J. | (4.6) |
Definition 4.4. [34] The problem (1.7) is said to be generalized UHR stable with respect to Θ∈C(J,R+) if there exists a real number Mf,Θ>0 such that for each solution z∈C of the inequality (4.3), there exists a solution x∈C of the problem (1.7) with
|z(t)−x(t)|≤Mf,ΘΘ(t),t∈J. | (4.7) |
Remark 4.5. It is clear that (i) Definition 4.1 ⇒ Definition 4.2; (ii) Definition 4.3 ⇒ Definition 4.4; (iii) Definition 4.3 for Θ(t)=1 ⇒ Definition 4.1.
Remark 4.6. A function z∈C(J,R) is a solution of the inequality (4.1) if and only if there exists a function w∈C(J,R) (which depends on z) such that:
(i) |w(t)|≤ϵ, ∀t∈J.
(ii) HDα,ρ;ψ0+z(t)=Fz(t)+w(t), t∈J.
Remark 4.7. A function z∈C is a solution of the inequality (4.2) if and only if there exists a function v∈C (which depends on z) such that:
(i) |v(t)|≤ϵΘ(t), ∀t∈J.
(ii) HDα,ρ;ψ0+z(t)=Fz(t)+v(t), t∈J.
Firstly, we present an important lemma that will be used in the proofs of UH stability and GUH stability.
Lemma 4.8. Let α∈(1,2], ρ∈[0,1). If z∈C is a solution of the inequality (4.1), then z is a solution of the following inequality
|z(t)−Rz−Iα;ψ0+Fz(s)(t)|≤Λ1ϵ, | (4.8) |
where
Rz=A(t,γ−1)Ω[κ−m∑i=1δiIα;ψ0+Fz(s)(ηi)−n∑j=1ωjIα+βj;ψ0+Fz(s)(θj)−r∑k=1λkIα−μk;ψ0+Fz(s)(ξk)], |
and Λ1 is given by (3.3).
Proof. Let z be a solution of the inequality (4.1). So, in view of Remark 4.6 (ii) and Lemma 2.11, we have
{HDα,ρ;ψ0+z(s)(t)=Fz(t)+w(t),t∈(0,T],z(0)=0,m∑i=1δiz(ηi)+n∑j=1ωjIβj;ψ0+z(s)(θj)+r∑k=1λkHDμk,ρ;ψ0+z(s)(ξk)=κ. | (4.9) |
Thus, the solution of (4.9) will be in the following term
z(t)=Iα;ψ0+Fz(s)(t)+A(t,γ−1)Ω(κ−m∑i=1δiIα;ψ0+Fz(ηi)−n∑j=1ωjIα+βj;ψ0+Fz(θj)−r∑k=1λkIα−μk;ψ0+Fz(ξk))+Iα;ψ0+w(s)(t)−A(t,γ−1)Ω(m∑i=1δiIα;ψ0+w(s)(ηi)+n∑j=1ωjIα+βj;ψ0+w(s)(θj)+r∑k=1λkIα−μk;ψ0+w(s)(ξk)). |
Then, by using Remark 4.6 (i), it follows that
|z(t)−Rz−Iα;ψ0+Fz(s)(t)|=|Iα;ψ0+w(s)(t)−A(t,γ−1)Ω(m∑i=1δiIα;ψ0+w(s)(ηi)+n∑j=1ωjIα+βj;ψ0+w(s)(θj)+r∑k=1λkIα−μk;ψ0+w(s)(ξk))|≤[A(T,α)+A(T,γ−1)|Ω|(m∑i=1|δi|A(ηi,α)+n∑j=1|ωj|A(θj,α+βj)+r∑k=1|λk|A(ξk,α−μk))]ϵ=Λ1ϵ, |
from which inequality (4.8) is obtained. The proof is completed.
Now, we prove UH stability and generalized UH stability results for the problem (1.7).
Theorem 4.9. Assume that the function f:J×R2→R is continuous and (H1) holds with Λ0A(T,α)L1<1. Then the problem (1.7) is UH stable on J and consequently generalized UH stable.
Proof. Let ϵ>0 and z∈C be any solution of the inequality (4.1). Let x∈C be the unique solution of the following problem (1.7)
{HDα,ρ;ψ0+x(s)(t)=Fx(t),t∈(0,T],x(0)=0,m∑i=1δix(ηi)+n∑j=1ωjIβj;ψ0+x(s)(θj)+r∑k=1λkHDμk,ρ;ψ0+x(s)(ξk)=κ. |
Using Lemma 2.11, we obtain
x(t)=Rx+Iα;ψ0+Fx(s)(t), |
where
Rx=A(t,γ−1)Ω(κ−m∑i=1δiIα;ψ0+Fx(s)(ηi)−n∑j=1ωjIα+βj;ψ0+Fx(s)(θj)−r∑k=1λkIα−μk;ψ0+Fx(ξk)). |
On the other hand, if x(0)=z(0), x(ηi)=z(ηi), Iβj;ψ0+x(s)(θj)=Iβj;ψ0+z(s)(θj) and HDμk,ρ;ψ0+x(s)(ξk)=HDμk,ρ;ψ0+z(s)(ξk), then Rx=Rz. Indeed, we have
|Rx−Rz|≤A(t,γ−1)|Ω|(m∑i=1|δi|Iα;ψ0+|Fx(s)−Fz(s)|(ηi)+n∑j=1|ωj|Iα+βj;ψ0+|Fx(s)−Fz(s)|(θj)+r∑k=1|λk|Iα−μk;ψ0+|Fx(s)−Fz(s)|(ξk))≤A(t,γ−1)|Ω|(m∑i=1|δi|Iα;ψ0+|x(s)−z(s)|(ηi)+n∑j=1|ωj|Iα+βj;ψ0+|x(s)−z(s)|(θj)+r∑k=1|λk|Iα−μk;ψ0+|x(s)−z(s)|(ξk))Λ0Λ1L1=0. |
Thus Rx=Rz. Now, by applying the triangle inequality, |u−v|≤|u|+|v|, and Lemma 4.8, for any t∈J, we have
|z(t)−x(t)|≤|z(t)−Rx−Iα;ψ0+Fx(s)(t)|≤|z(t)−Rz−Iα;ψ0+Fz(s)(t)|+Iα;ψ0+|Fz(s)−Fx(s)|(t)+|Rz−Rx|≤Λ1ϵ+Λ0A(T,α)L1|z(t)−x(t)|. |
This implies that
|z(t)−x(t)|≤Λ11−Λ0A(T,α)L1ϵ. |
By setting
Mf=Λ11−Λ0A(T,α)L1, |
we obtain
|z(t)−x(t)|≤Mfϵ. |
Hence, the problem (1.7) is UH stable. Further, if we set Θ(ϵ)=Mfϵ and Θ(0)=0 we have
|z(t)−x(t)|≤Θ(ϵ), |
which implies that the solution of the problem (1.7) is generalized UH stable. The proof is completed.
For the proof of our next lemma, we assume the following assumption:
(H3) There exists an increasing function Θ∈C(J,R+) and there exists nΘ>0, such that, for any t∈J, the following integral inequality
Iα;ψ0+Θ(t)≤nΘΘ(t). | (4.10) |
Next, we present an important lemma that will be used in the proofs of UHR and generalized UHR stability results.
Lemma 4.10. Let α∈(1,2], ρ∈[0,1]. If z∈C is a solution of the inequality (4.2), then z is a solution of the following inequality
|z(t)−Rz−Iα;ψ0+Fz(s)(t)|≤Λ2ϵnΘΘ(t), | (4.11) |
where
Λ2=1+A(T,γ−1)|Ω|(m∑i=1|δi|+n∑j=1|ωj|+r∑k=1|λk|). | (4.12) |
Proof. Let z be a solution of the inequality (4.2). So, in view of Remark 4.7 (ii) and Lemma 2.11, the solution of (4.9) can be written by
z(t)=Iα;ψ0+Fz(s)(t)+A(t,γ−1)Ω(κ−m∑i=1δiIα;ψ0+Fz(s)(ηi)−n∑j=1ωjIα+βj;ψ0+Fz(s)(θj)−r∑k=1λkIα−μk;ψ0+Fz(s)(ξk))+Iα;ψ0+v(s)(t)−A(t,γ−1)Ω(m∑i=1δiIα;ψ0+v(s)(ηi)+n∑j=1ωjIα+βj;ψ0+v(s)(θj)+r∑k=1λkIα−μk;ψ0+v(s)(ξk)). |
Then, by using Remark 4.7 (i) with (H3), we have the following estimation
|z(t)−Rz−Iα;ψ0+Fz(s)(t)|=|Iα;ψ0+v(s)(t)−A(t,γ−1)Ω(m∑i=1δiIα;ψ0+v(s)(ηi)+n∑j=1ωjIα+βj;ψ0+v(s)(θj)+r∑k=1λkIα−μk;ψ0+v(s)(ξk))|≤[1+A(T,γ−1)|Ω|(m∑i=1|δi|+n∑j=1|ωj|+r∑k=1|λk|)]ϵnΘΘ(t)=Λ2ϵnΘΘ(t), |
from which inequality (4.11) is obtained. The proof is completed.
Finally, we present UHR and generalized UHR stability results for the problem (1.7).
Theorem 4.11. Assume that the function f:J×R2→R is continuous and (H1) holds. Then the problem (1.7) is UHR stable on J and consequently generalized UHR stable.
Proof. Let ϵ>0 and z∈C be the solution of the inequality (4.3). Let x∈C be the unique solution of the problem (1.7). By using Lemma 2.11, we obtain
x(t)=Rx+Iα;ψ0+Fx(s)(t), |
where
Rx=A(t,γ−1)Ω(κ−m∑i=1δiIα;ψ0+Fx(s)(ηi)−n∑j=1ωjIα+βj;ψ0+Fx(s)(θj)−r∑k=1λkIα−μk;ψ0+Fx(s)(ξk)). |
On the other hand, if x(0)=z(0), x(ηi)=z(ηi), Iβj;ψ0+x(s)(θj)=Iβj;ψ0+z(s)(θj) and HDμk,ρ;ψ0+x(s)(ξk)=HDμk,ρ;ψ0+z(s)(ξk), then it is easy to see that Rx=Rz.
Now, by appying |u−v|≤|u|+|v| and Lemma 4.10, for any t∈J, we have
|z(t)−x(t)|≤|z(t)−Rx−Iα;ψ0+Fx(s)(t)|≤|z(t)−Rz−Iα;ψ0+Fz(s)(t)|+Iα;ψ0+|Fz(s)−Fx(s)|(t)+|Rz−Rx|≤Λ2ϵnΘΘ(t)+Λ0A(T,α)L1|z(t)−x(t)| |
This implies that
|z(t)−x(t)|≤Λ2nΘ1−Λ0A(T,α)L1ϵΘ(t). |
By setting
Mf,Θ=Λ2nΘ1−Λ0A(T,α)L1, |
we obtain
|z(t)−x(t)|≤Mf,ΘϵΘ(t). |
Therefore, the problem (1.7) is UHR stable. Further, in the same fashion, it is easy to check that the solution of the problem (1.7) is generalized UHR stable. This completes the proof.
This section presents some examples which illustrate the validity and applicability of our main results.
Example 5.1. Consider the following mixed nonlocal boundary problem of the form:
{HD85,14;et20+x(t)=f(t,x(t),I13;et20+x(t)),t∈(0,1],x(0)=0,3∑i=1(−ii+5)i+1x(i3)+2∑j=1(j+1j+2)Ij3;et20+x(j2)+4∑k=1(−kk+2)kHDk+88,14;et20+x(k4)=12. | (5.1) |
Here α=8/5, ρ=1/4, ϕ=1/3, T=1, κ=1/2, m=3, n=2, r=4, δi=((−i)/(i+5))(i+1), ωj=(j+1)/(j+2), λk=((−k)/(k+2))k, ηi=i/3, θj=j/2, ξk=k/4, βj=j/3, μk=(k+8)/8 for i=1,2,3, j=1,2 and k=1,2,3,4. From the given all data, we obtain that Ω≈0.5377547471≠0, Λ0≈1.96941831, Λ1≈2.131548185 and Λ2≈6.661728461.
(I) Consider the function
f(t,x(t),I13;et20+x(t)):=t2+1(3−sin2πt)2⋅|x(t)|2+|x(t)|+(2t−1)⋅|I13;et20+x(t)|9+|I13;et20+x(t)|. | (5.2) |
For x1, x2, y1, y2∈R and t∈[0,1], we have
|f(t,x1,y1)−f(t,x2,y2)|≤19(|x1−x2|+|y1−y2|). |
The assumptions (H1) is satisfied with L1=1/9. Hence
Λ0Λ1L1≈0.4664344471<1. |
Since, all the assumptions of Theorem 3.1 are satisfied, then the problem (5.1) has a unique solution on [0,1]. Further, we can also compute that
Mf=Λ11−Λ0A(T,α)L1≈2.30834181>1. |
Therefore, by Theorem 4.9, the problem (5.1) is both UH and generalized UH stable on [0.1]. In addition, by setting Θ(t)=ψ(t)−ψ(0) with Proposition 2.5 (i), it is easy to calculate that
Iα;ψ0+Θ(t)=1Γ(72)(ψ(t)−ψ(0))52Θ(t)≤4(e0.5−1)5215√πΘ(t). |
Thus, the inequality (4.10) is satisfied with nΘ=4(e0.5−1)5215√π>0. It follows that
Mf,Θ=Λ2nΘ1−Λ0A(T,α)L1≈0.3679010534>0. |
Hence, by Theorem 4.11, the problem (5.1), with f given by (5.2), is both UHR and also generalized UHR stable on [0.1].
(II) Consider the function
f(t,x(t),I13;et20+x(t)):=e−t+tan−1|x(t)|4+t+2sin|x(t)|4+t⋅|I13;et20+x(t)|2+|I13;et20+x(t)|. | (5.3) |
For x1, x2, y1, y2∈R and t∈[0,1], we have
|f(t,x1,y1)−f(t,x2,y2)|≤14+t(|x1−x2|+|y1−y2|)≤14(|x1−x2|+|y1−y2|). |
This means that the assumption (H1) is satisfied with L1=1/4. We obtain
L1Λ0(Λ1−A(T,α))≈0.8771522228<1, |
and
|f(t,x,y)|≤e−t+14+t(π2+1), |
which satisfy (3.7) and (H2), respectively. Using the Theorem 3.2, the problem (5.1), with f given by (5.3), has at least one solution on [0,1]
(III) Consider the function
f(t,x(t),I13;et20+x(t)):=e−t(4+t)2(|x5(t)|1+x4(t)+I13;et20+x6(t)1+|I13;et20+x5(t)|+1). | (5.4) |
Also, the nonlinear function can be expressed as
|f(t,x,y)|≤e−t(4+t)2(|x|+|y|+1). |
By (H3), we set q(t)=e−t/(4+t)2 and Φ(u)=u+1, then ‖q‖=1/16 and Φ(|x|+|y|)=|x|+|y|+1. Thus, we can compute that there exists a constant M2>1.527092217 satisfying inequality in (H4). Therefore, all conditions in Theorem 3.3 are fulfilled. Thus the problem (5.1) with f given by (5.4) has at least one solution on [0,1].
This paper discussed a new class of ψ-Hilfer fractional integro-differential equation supplemented with mixed nonlocal boundary condition which is a combination of multi-point, fractional derivative multi-order and fractional integral multi-order boundary conditions. Existence and uniqueness results are established. The uniqueness result is proved by applying the Banach's fixed point theorem, while the existence results are investigated via Krasnosel'ski⌣i's fixed point theorem and Larey-Schauder nonlinear alternative. Our results are not only new in the given setting but also provide some new special cases by fixing the parameters involved in the problem at hand. For instance, by fixing ωj=0,λk=0 for all j=1,2,…,n,k=1,2,…,r our results correspond to the ones for boundary value problems for ψ-Hilfer nonlinear fractional integro-differential equations supplemented with multi-point boundary conditions. In case we take δi=0,λk=0 for all i=1,2,…,m,k=1,2,…,r we obtain the results for boundary value problems for ψ-Hilfer nonlinear fractional integro-differential equations equipped with multi-term integral boundary conditions. Further, we studied different kinds of Ulam's stability such as UH, generalized UH, UHR and generalized UHR stability. In the end, we present examples to demonstrate the consistency to the theoretical findings.
The work accomplished in this paper is new and enrich the literature on boundary value problems for nonlinear ψ-Hilfer fractional differential equations.
The first author would like to thank King Mongkut's University of Technology North Bangkok and the Center of Excellence in Mathematics (CEM), CHE, Sri Ayutthaya Rd., Bangkok, 10400, Thailand for support this work. The second author would like to thank for funding this work through the Center of Excellence in Mathematics (CEM), CHE, Sri Ayutthaya Rd., Bangkok, 10400, Thailand and Barapha University.
On behalf of all authors, the corresponding author states that there is no conflict of interest.
[1] |
J. Yick, B. Mukherjee, D. Ghosal, Wireless sensor network survey, Comput. Netw., 52 (2008), 2292–2330. https://doi.org/10.1016/j.comnet.2008.04.002 doi: 10.1016/j.comnet.2008.04.002
![]() |
[2] |
C. F. Cheng, Y. C. Chen, J. C. W. Lin, A carrier-based sensor deployment algorithm for perception layer in the IoT architecture, IEEE. Sens. J., 20 (2020), 10295–10305. https://doi.org/10.1109/JSEN.2020.2989871 doi: 10.1109/JSEN.2020.2989871
![]() |
[3] |
M. Majid, S. Habib, A. R. Javed, M. Rizwan, G. Srivastava, T. R. Gadekallu, J. C. W. Lin, Applications of wireless sensor networks and internet of things frameworks in the industry revolution 4.0: a systematic literature review, Sensors-Basel., 22 (2022), 2087. https://doi.org/10.3390/s22062087 doi: 10.3390/s22062087
![]() |
[4] | N. Labraoui, M. Gueroui, L. Sekhri, On-off attacks mitigation against trust systems in wireless sensor networks, In: A. Amine, L. Bellatreche, Z. Elberrichi, E. J. Neuhold, and R. Wrembel, Computer Science and Its Applications, CIIA 2015, IFIP Advances in Information and Communication Technology, Springer, Cham, (2015), 406–415. |
[5] |
J. C. W. Lin, P. Fournier-Viger, L. Wu, W. Gan, Y. Djenouri, J. Zhang, PPSF: an open-source privacy-preserving and security mining framework, 2018 IEEE International Conference on Data Mining Workshops (ICDMW) (2018), 1459–1463. https://doi.org/10.1109/ICDMW.2018.00208 doi: 10.1109/ICDMW.2018.00208
![]() |
[6] | J. Zheng, A. Jamalipour, Wireless Sensor Networks: A Networking Perspective, John Wiley and Sons, Ltd, (2009). https://doi.org/10.1002/9780470443521 |
[7] |
D. Pandey, V. Kushwaha, An exploratory study of congestion control techniques in wireless sensor networks, Comput. Commun., 157 (2020), 257–283. https://doi.org/10.1016/j.comcom.2020.04.032 doi: 10.1016/j.comcom.2020.04.032
![]() |
[8] |
L. Q. Tao, F. Q. Yu, ECODA: enhanced congestion detection and avoidance for multiple class of traffic in sensor networks, IEEE. T. Consum. Electr., 56 (2010), 1387–1394. https://doi.org/10.1109/TCE.2010.5606274 doi: 10.1109/TCE.2010.5606274
![]() |
[9] |
F. Ren, T. He, S. K. Das, C. Lin, Traffic-aware dynamic routing to alleviate congestion in wireless sensor networks, IEEE. T. Parall. Distr., 22 (2011), 1585–1599. https://doi.org/10.1109/TPDS.2011.24 doi: 10.1109/TPDS.2011.24
![]() |
[10] |
C. Sergiou, V. Vassiliou, A. Paphitis, Hierarchical tree alternative path (HTAP) algorithm for congestion control in wireless sensor networks, Ad. Hoc. Netw., 11 (2013), 257–272. https://doi.org/10.1016/j.adhoc.2012.05.010 doi: 10.1016/j.adhoc.2012.05.010
![]() |
[11] |
A. A. Rezaee, M. H. Yaghmaee, A. M. Rahmani, A. H. Mohajerzadeh, HOCA: Healthcare aware optimized congestion avoidance and control protocol for wireless sensor networks, J. Netw. Comput. Appl., 37 (2014), 216–228. https://doi.org/10.1016/j.jnca.2013.02.014 doi: 10.1016/j.jnca.2013.02.014
![]() |
[12] | L. Tshiningayamwe, G. A. Lusilao-Zodi, M. E. Dlodlo, A priority rate-based routing protocol for wireless multimedia sensor networks, In: N. Pillay, A. P. Engelbrecht, A. Abraham, M. C. du Plessis, V. Snášel, and A. K. Muda, Advances in Nature and Biologically Inspired Computing, Advances in Intelligent Systems and Computing, Springer, Cham, (2016), 347–358. |
[13] |
P. K. Donta, T. Amgoth, C. S. R. Annavarapu, Congestion-aware data acquisition with q-learning for wireless sensor networks, 2020 IEEE International IOT, Electronics and Mechatronics Conference (IEMTRONICS) (2020), 1–6, https://doi.org/10.1109/IEMTRONICS51293.2020.9216379 doi: 10.1109/IEMTRONICS51293.2020.9216379
![]() |
[14] | K. Sohraby, D. Minoli, T. Znati, Wireless Sensor Networks: Technology, Protocols, and Applications, John wiley and sons, (2007). |
[15] |
A. Srivastava, P. K. Mishra, Multi-attributes based energy efficient clustering for enhancing network lifetime in WSN's, Peer. Peer. Netw. Appl., 15 (2022), 2670–2693. https://doi.org/10.1007/s12083-022-01357-w doi: 10.1007/s12083-022-01357-w
![]() |
[16] |
V. Kushwaha, Ratneshwer, A review of router based congestion control algorithms, Int. J. Comput. Netw. Inf. Secur., 1 (2014), 1–10. https://doi.org/10.5815/ijcnis.2014.01.01 doi: 10.5815/ijcnis.2014.01.01
![]() |
[17] | T. L. Saaty, Decision making with the analytic hierarchy process, Int. J. Serv. Sci., 1 (2008), 83–98. |
[18] | M. M. Momani, Bayesian Methods for Modelling and Management of Trust in Wireless Sensor Networks, University of Technology, Sydney, (2008). |
[19] |
P. Rodrigues, J. John, Joint trust: an approach for trust-aware routing in WSN, Wirel. Netw., 26 (2020), 3553–3568. https://doi.org/10.1007/s11276-020-02271-w doi: 10.1007/s11276-020-02271-w
![]() |
[20] |
Y. Tao, X. Xu, P. Li, T. Li, L. Pan, A secure routing of wireless sensor networks based on trust evaluation model, Procedia. Comput. Sci., 131 (2018), 1156–1163. https://doi.org/10.1016/j.procs.2018.04.289 doi: 10.1016/j.procs.2018.04.289
![]() |
[21] |
X. Yin, S. Li, Trust evaluation model with entropy-based weight assignment for malicious node's detection in wireless sensor networks, EURASIP. J. Wirel. Comm., 198 (2019). https://doi.org/10.1186/s13638-019-1524-z doi: 10.1186/s13638-019-1524-z
![]() |
[22] |
T. Kalidoss, L. Rajasekaran, K. Kanagasabai, G. Sannasi, A. Kannan, QoS aware trust based routing algorithm for wireless sensor networks, Wireless. Pers. Commun., 110 (2020), 1637–1658. https://doi.org/10.1007/s11277-019-06788-y doi: 10.1007/s11277-019-06788-y
![]() |
[23] |
A. Tajeddine, A. Kayssi, A. Chehab, I. Elhajj, W. Itani, CENTERA: a centralized trust-based efficient routing protocol with authentication for wireless sensor networks, Sensors-Basel., 15 (2015), 3299–3333. https://doi.org/10.3390/s150203299 doi: 10.3390/s150203299
![]() |
[24] |
T. Khan, K. Singh, M. H. Hasan, K. Ahmad, G. T. Reddy, S. Mohan, A. Ahmadian, ETERS: A comprehensive energy aware trust-based efficient routing scheme for adversarial WSNs, Future. Gener. Comp. Sy., 125 (2021), 921–943. https://doi.org/10.1016/j.future.2021.06.049 doi: 10.1016/j.future.2021.06.049
![]() |
[25] |
M. Selvi, K. Thangaramya, S. Ganapathy, K. Kulothungan, H. K. Nehemiah, A. Kannan, An energy aware trust based secure routing algorithm for effective communication in wireless sensor networks, Wireless. Pers. Commun., 105 (2019), 1475–1490. https://doi.org/10.1007/s11277-019-06155-x doi: 10.1007/s11277-019-06155-x
![]() |
[26] |
E. Thenmozhi, S. Audithan, Trust based cluster and secure routing scheme for wireless sensor network, Second International Conference on Current Trends In Engineering and Technology–ICCTET 2014, (2014), 489–494. https://10.1109/ICCTET.2014.6966345 doi: 10.1109/ICCTET.2014.6966345
![]() |
[27] |
N. A. Khalid, Q. Bai, A. Al-Anbuky, Adaptive trust-based routing protocol for large scale WSNs, IEEE. Access. 7 (2019), 143539–143549. https://10.1109/ACCESS.2019.2944648 doi: 10.1109/ACCESS.2019.2944648
![]() |
[28] |
D. C. Mehetre, S. E. Roslin, S. J. Wagh, Detection and prevention of Black Hole and selective forwarding attack in clustered WSN with active trust, Cluster. Comput., 22 (2019), 1313–1328. https://doi.org/10.1007/s10586-017-1622-9 doi: 10.1007/s10586-017-1622-9
![]() |
[29] |
W. Fang, W. Zhang, W. Yang, Z. Li, W. Cao, Y. Yang, Trust management-based and energy efficient hierarchical routing protocol in wireless sensor networks, Digit. Commun. Netw., 7 (2021), 470–478. https://doi.org/10.1016/j.dcan.2021.03.005 doi: 10.1016/j.dcan.2021.03.005
![]() |
[30] |
N. Dharini, N. Duraipandian, J. Katiravan, ELPC-trust framework for wireless sensor networks, Wireless. Pers. Commun., 113 (2020), 1709–1742. https://doi.org/10.1007/s11277-020-07288-0 doi: 10.1007/s11277-020-07288-0
![]() |
[31] |
M. Rathee, S. Kumar, A. H. Gandomi, K. Dilip, B. Balusamy, R. Patan, Ant colony optimization based quality of service aware energy balancing secure routing algorithm for wireless sensor networks, IEEE. T. Eng. Manage., 68 (2019), 170–182. https://doi.org/10.1109/TEM.2019.2953889 doi: 10.1109/TEM.2019.2953889
![]() |
[32] |
M. U. Ghazi, S. S. H. Naqvi, K. Yamin, O. Humayun, Congestion-aware routing algorithm based on traffic priority in wireless sensor networks, 2018 15th International Conference on Smart Cities: Improving Quality of Life Using ICT and IoT (HONET-ICT), (2018), 112–116. https://doi.org/10.1109/HONET.2018.8551337 doi: 10.1109/HONET.2018.8551337
![]() |
[33] |
J. Yan, B. Qi, CARA: a congestion-aware routing algorithm for wireless sensor networks, Algorithms., 14 (2021), 199. https://doi.org/10.3390/a14070199 doi: 10.3390/a14070199
![]() |
[34] |
H. S. Das, S. Bhattacharjee, A congestion aware routing for lifetime improving in grid-based sensor networks, J. High. Speed. Netw., 23 (2017), 1–14. https://doi.org/10.3233/JHS-170553 doi: 10.3233/JHS-170553
![]() |
[35] |
G. Sangeetha, M. Vijayalakshmi, S. Ganapathy, A. Kannan, An improved congestion-aware routing mechanism in sensor networks using fuzzy rule sets, Peer. Peer. Netw. Appl., 13 (2020), 890–904. https://doi.org/10.1007/s12083-019-00821-4 doi: 10.1007/s12083-019-00821-4
![]() |
[36] | S. S. Babu, A. Raha, M. K. Naskar, Geometric mean based trust management system for WSNs (GMTMS), in 2011 World Congress on Information and Communication Technologies, (2011), 444–449. https://doi.org/10.1109/WICT.2011.6141286 |
[37] | S. Ganguly, A. Chakraborty, M. K. Naskar, A trust-based framework for congestion-aware energy efficient routing in wireless multimedia sensor networks, (2013). https://doi.org/10.48550/arXiv.1312.4071 |
[38] |
J. Duan, D. Yang, H. Zhu, S. Zhang, J. Zhao, TSRF: a trust-aware secure routing framework in wireless sensor networks, Int. J. Distrib. Sens. N., 10 (2014), 209436. https://doi.org/10.1155/2014/209436 doi: 10.1155/2014/209436
![]() |
[39] |
X. Wu, J. Huang, J. Ling, L. Shu, BLTM: beta and LQI based trust model for wireless sensor networks, IEEE. Access., 7 (2019), 43679–43690. https://doi.org/10.1109/ACCESS.2019.2905550 doi: 10.1109/ACCESS.2019.2905550
![]() |
[40] |
M. Gholipour, A. T. Haghighat, M. R. Meybodi, Congestion avoidance in cognitive wireless sensor networks using TOPSIS and response surface methodology, Telecommun. Syst., 67 (2018), 519–537. https://doi.org/10.1007/s11235-017-0356-6 doi: 10.1007/s11235-017-0356-6
![]() |
[41] |
K. Sumathi, P. Pandiaraja, Dynamic alternate buffer switching and congestion control in wireless multimedia sensor networks, Peer. Peer. Netw. Appl., 13 (2020), 2001–2010. https://doi.org/10.1007/s12083-019-00797-1 doi: 10.1007/s12083-019-00797-1
![]() |
[42] |
S. Li, Q. Xu, J. Gaber, Z. Dou, J. Chen, Congestion control mechanism based on dual threshold DI-RED for WSNs, Wireless. Pers. Commun., 115 (2020), 2171–2195. https://doi.org/10.1007/s11277-020-07676-6 doi: 10.1007/s11277-020-07676-6
![]() |
[43] |
A. Beheshtiasl, A. Ghaffari, Secure and trust-aware routing scheme in wireless sensor networks, Wireless. Pers. Commun., 107 (2019), 1799–1814. https://doi.org/10.1007/s11277-019-06357-3 doi: 10.1007/s11277-019-06357-3
![]() |
[44] |
S. Qu, L. Zhao, Y. Chen, W. Mao, A discrete-time sliding mode congestion controller for wireless sensor networks, Optik., 225 (2021), 165727. https://doi.org/10.1016/j.ijleo.2020.165727 doi: 10.1016/j.ijleo.2020.165727
![]() |
[45] |
S. Sefati, M. Abdi, A. Ghaffari, Cluster‐based data transmission scheme in wireless sensor networks using black hole and ant colony algorithms, Int. J. Commun. Syst., 34 (2012), e4768. https://doi.org/10.1002/dac.4768 doi: 10.1002/dac.4768
![]() |
[46] |
W. R. Heinzelman, A. Chandrakasan, H. Balakrishnan, Energy-efficient communication protocol for wireless microsensor networks, Proceedings of The 33rd Annual Hawaii International Conference on System Sciences (2000). https://doi.org/10.1109/HICSS.2000.926982 doi: 10.1109/HICSS.2000.926982
![]() |
[47] |
G. Srivastava, J. C. W. Lin, M. Pirouz, Y. Li, U. Yun, A pre-large weighted-fusion system of sensed high-utility patterns, IEEE. Sens. J., 21 (2020), 15626–15634. https://doi.org/10.1109/JSEN.2020.2991045 doi: 10.1109/JSEN.2020.2991045
![]() |
[48] |
T. Khan, K. Singh, TASRP: a trust aware secure routing protocol for wireless sensor networks, Int. J. Innov. Comput. Appl., 12 (2021), 108–122. https://doi.org/10.1504/ijica.2021.113750 doi: 10.1504/ijica.2021.113750
![]() |
[49] |
P. Chanak, I. Banerjee, Congestion free routing mechanism for IoT-enabled wireless sensor networks for smart healthcare applications. IEEE. T. Consum. Electr., 66 (2020), 223–232. https://doi.org/10.1109/TCE.2020.2987433 doi: 10.1109/TCE.2020.2987433
![]() |
[50] |
T. Gao, R. C. Jin, J. Y. Song, T. B. Xu, L. D. Wang, Energy-efficient cluster head selection scheme based on multiple criteria decision making for wireless sensor networks, Wireless. Pers. Commun., 63 (2012), 871–894. https://doi.org/10.1007/s11277-010-0172-8 doi: 10.1007/s11277-010-0172-8
![]() |
[51] |
P. Mukherjee, P. K. Pattnaik, A. A. Al-Absi, D. K. Kang, Recommended system for cluster head selection in a remote sensor cloud environment using the fuzzy-based multi-criteria decision-making technique, Sustainability-Basel., 13 (2021), 10579. https://doi.org/10.3390/su131910579 doi: 10.3390/su131910579
![]() |
[52] |
R. K. Dewi, B. T. Hanggara, A. Pinandito, A comparison between AHP and hybrid AHP for mobile based culinary recommendation system, Int. J. Interact. Mob. Technol., 12 (2018), 133–140. https://doi.org/10.3991/ijim.v12i1.7561 doi: 10.3991/ijim.v12i1.7561
![]() |
1. | Surang Sitho, Sotiris K. Ntouyas, Ayub Samadi, Jessada Tariboon, Boundary Value Problems for ψ-Hilfer Type Sequential Fractional Differential Equations and Inclusions with Integral Multi-Point Boundary Conditions, 2021, 9, 2227-7390, 1001, 10.3390/math9091001 | |
2. | Djamal Foukrach, Soufyane Bouriah, Mouffak Benchohra, Erdal Karapinar, Some new results for $$\psi -$$Hilfer fractional pantograph-type differential equation depending on $$\psi -$$Riemann–Liouville integral, 2022, 30, 0971-3611, 195, 10.1007/s41478-021-00339-0 | |
3. | Chanakarn Kiataramkul, Sotiris K. Ntouyas, Jessada Tariboon, Maria L. Gandarias, Existence Results for ψ -Hilfer Fractional Integro-Differential Hybrid Boundary Value Problems for Differential Equations and Inclusions, 2021, 2021, 1687-9139, 1, 10.1155/2021/9044313 | |
4. | Sotiris K. Ntouyas, A Survey on Existence Results for Boundary Value Problems of Hilfer Fractional Differential Equations and Inclusions, 2021, 1, 2673-9321, 63, 10.3390/foundations1010007 | |
5. | Qing Yang, Chuanzhi Bai, Dandan Yang, Finite-time stability of nonlinear stochastic $ \psi $-Hilfer fractional systems with time delay, 2022, 7, 2473-6988, 18837, 10.3934/math.20221037 | |
6. | Amin Jajarmi, Dumitru Baleanu, Samaneh Sadat Sajjadi, Juan J. Nieto, Analysis and some applications of a regularized Ψ–Hilfer fractional derivative, 2022, 415, 03770427, 114476, 10.1016/j.cam.2022.114476 | |
7. | S. Naveen, R. Srilekha, S. Suganya, V. Parthiban, Controllability of damped dynamical systems modelled by Hilfer fractional derivatives, 2022, 16, 1658-3655, 1254, 10.1080/16583655.2022.2157188 | |
8. | Cholticha Nuchpong, Sotiris K. Ntouyas, Ayub Samadi, Jessada Tariboon, Boundary value problems for Hilfer type sequential fractional differential equations and inclusions involving Riemann–Stieltjes integral multi-strip boundary conditions, 2021, 2021, 1687-1847, 10.1186/s13662-021-03424-7 | |
9. | Thabet Abdeljawad, Pshtiwan Othman Mohammed, Hari Mohan Srivastava, Eman Al-Sarairah, Artion Kashuri, Kamsing Nonlaopon, Some novel existence and uniqueness results for the Hilfer fractional integro-differential equations with non-instantaneous impulsive multi-point boundary conditions and their application, 2023, 8, 2473-6988, 3469, 10.3934/math.2023177 | |
10. | Ramasamy Arul, Panjayan Karthikeyan, Kulandhaivel Karthikeyan, Palanisamy Geetha, Ymnah Alruwaily, Lamya Almaghamsi, El-sayed El-hady, On Ψ-Hilfer Fractional Integro-Differential Equations with Non-Instantaneous Impulsive Conditions, 2022, 6, 2504-3110, 732, 10.3390/fractalfract6120732 | |
11. | Abdulkafi M. Saeed, Mohammed S. Abdo, Mdi Begum Jeelani, Existence and Ulam–Hyers Stability of a Fractional-Order Coupled System in the Frame of Generalized Hilfer Derivatives, 2021, 9, 2227-7390, 2543, 10.3390/math9202543 | |
12. | Sotiris Ntouyas, Bashir Ahmad, Jessada Tariboon, Nonlocal ψ-Hilfer Generalized Proportional Boundary Value Problems for Fractional Differential Equations and Inclusions, 2022, 2, 2673-9321, 377, 10.3390/foundations2020026 | |
13. | Weerawat Sudsutad, Chatthai Thaiprayoon, Bounmy Khaminsou, Jehad Alzabut, Jutarat Kongson, A Gronwall inequality and its applications to the Cauchy-type problem under ψ-Hilfer proportional fractional operators, 2023, 2023, 1029-242X, 10.1186/s13660-023-02929-x | |
14. | Wasfi Shatanawi, Abdellatif Boutiara, Mohammed S. Abdo, Mdi B. Jeelani, Kamaleldin Abodayeh, Nonlocal and multiple-point fractional boundary value problem in the frame of a generalized Hilfer derivative, 2021, 2021, 1687-1847, 10.1186/s13662-021-03450-5 | |
15. | Chanakarn Kiataramkul, Sotiris K. Ntouyas, Jessada Tariboon, An Existence Result for ψ-Hilfer Fractional Integro-Differential Hybrid Three-Point Boundary Value Problems, 2021, 5, 2504-3110, 136, 10.3390/fractalfract5040136 | |
16. | Elkhateeb S. Aly, M. Latha Maheswari, K. S. Keerthana Shri, Waleed Hamali, A novel approach on the sequential type ψ-Hilfer pantograph fractional differential equation with boundary conditions, 2024, 2024, 1687-2770, 10.1186/s13661-024-01861-3 | |
17. | Weerawat Sudsutad, Wicharn Lewkeeratiyutkul, Chatthai Thaiprayoon, Jutarat Kongson, Existence and stability results for impulsive $ (k, \psi) $-Hilfer fractional double integro-differential equation with mixed nonlocal conditions, 2023, 8, 2473-6988, 20437, 10.3934/math.20231042 | |
18. | Naveen S., Parthiban V., Mohamed I. Abbas, Qualitative Analysis of RLC Circuit Described by Hilfer Derivative with Numerical Treatment Using the Lagrange Polynomial Method, 2023, 7, 2504-3110, 804, 10.3390/fractalfract7110804 | |
19. | Zainab Alsheekhhussain, Ahmad Gamal Ibrahim, Mohammed Mossa Al-Sawalha, Yousef Jawarneh, The Existence of Solutions for w-Weighted ψ-Hilfer Fractional Differential Inclusions of Order μ ∈ (1, 2) with Non-Instantaneous Impulses in Banach Spaces, 2024, 8, 2504-3110, 144, 10.3390/fractalfract8030144 | |
20. | Mohammed O. Mohammed, Ava Sh. Rafeeq, New Results for Existence, Uniqueness, and Ulam Stable Theorem to Caputo–Fabrizio Fractional Differential Equations with Periodic Boundary Conditions, 2024, 10, 2349-5103, 10.1007/s40819-024-01741-5 | |
21. | C. Kausika, P. Suresh Kumar, N. Annapoorani, Linearized asymptotic stability of implicit fractional integrodifferential system, 2023, 2195-268X, 10.1007/s40435-023-01334-y | |
22. | M. Lavanya, B. Sundara Vadivoo, Kottakkaran Sooppy Nisar, Controllability Analysis of Neutral Stochastic Differential Equation Using $$\psi $$-Hilfer Fractional Derivative with Rosenblatt Process, 2025, 24, 1575-5460, 10.1007/s12346-024-01178-7 | |
23. | Lamya Almaghamsi, Aeshah Alghamdi, Abdeljabbar Ghanmi, Existence of solution for a Langevin equation involving the $ \psi $-Hilfer fractional derivative: A variational approach, 2025, 10, 2473-6988, 534, 10.3934/math.2025024 | |
24. | M. Latha Maheswari, K. S. Keerthana Shri, Mohammad Sajid, 2025, Chapter 17, 978-3-031-58640-8, 253, 10.1007/978-3-031-58641-5_17 | |
25. | S. Naveen, K. Venkatachalam, V. Parthiban, Analysis of variable-order derivative with Mittag–Leffler kernel and integral boundary conditions for RLC circuit system, 2025, 0020-7160, 1, 10.1080/00207160.2025.2486409 |