Int. J. Anal. Appl. (2023), 21:82 On Existence and Attractivity of Ψ-Hilfer Hybrid Fractional-order Langevin Differential Equations Savita Rathee1, Yogeeta Narwal2,∗ 1Department of Mathematics, Maharshi Dayanand University, Rohtak, India 2Government College, Baund Kalan, Charkhi Dadri-127025, India ∗Corresponding author: yogeetawork@gmail.com Abstract. The work reported in this article studies the equivalence relationship between fractional integral equation and Ψ-Hilfer Hybrid Langevin Differential Equations of fractional order with nonlocal initial conditions, and then we use this relationship to establish the existence of the results by means of Banach algebra and Schauder’s fixed point theorem. We then demonstrate the uniform local attractiveness of all the solutions. 1. Introduction ODEs are extended to include fractional differential equations (FDEs), where the order of the derivative can be any positive number. For this reason, approaching the problem as an FDE typically allows us to model an experimental dynamic more effectively. Which fractional derivative (FD) is most appropriate at this point? The solution to this question typically depends on the problem and hence on the collected information. Consider using a definition of fractional operators that is more broad to get around the multitude of definitions for FDs. For better and more accurate simulations, we use Ψ-Hilfer fractional derivative(Ψ-HFD) and fixed point theory as an important tool to derive existence criterion of solutions. Kilbas et al. [1] introduced the notion of FD with respect to another function in the context of the RL FD. Similar to this, Almeida [15] proposed the Ψ- Caputo FD and looked at a variety of intriguing aspects of this operator. FD operator with two parameters was presented by Hilfer [16]. The Hilfer derivative unifies the RL FD and Caputo FD-based theories of FDEs. Sousa and Oliveira Received: Jun. 15, 2023. 2020 Mathematics Subject Classification. 26A33, 34A08, 34B15, 47H10. Key words and phrases. fractional Langevin differential equation; Ψ-Hilfer fractional derivative; Schauder fixed point theorem. https://doi.org/10.28924/2291-8639-21-2023-82 ISSN: 2291-8639 © 2023 the author(s). https://doi.org/10.28924/2291-8639-21-2023-82 2 Int. J. Anal. Appl. (2023), 21:82 presented the Hilfer FD with respect to another function in [9]; known as the Ψ-HFD. The Ψ-HFD’s significance stems from the fact that it uses a number of well-known FD operators as its specific cases, for example, RL [1], Caputo [1], Hadmard [1], Riesz [1], Erdeyl-Kober [1], Ψ-Caputo [15], Katugampola [21], Hilfer [16, 17] and so on. With this approach, it is possible to examine a wide range of properties of FDE solutions that employ several FD operators using a single FD operator. A number of researches have been conducted using Ψ-HFD [6,7,10,12–14,17–20]. “In 2021, Bachir et al. [8] proved the existence and attractivity of solutions for Ψ-Hilfer hybrid FDEs: Dλ,σ;Ψ u(t) v(t,u(t)) = w(t,u(t)); a.e. t ∈R+, (1.1) (Ψ(t) − Ψ(0))1−ζu(t)|t=0 = u0; u0 ∈R, (1.2) where R+ = [0,∞), 0 < λ < 1, 0 ≤ σ ≤ 1, ζ = λ + σ(1 −λ), Dλ,σ;Ψ is the Ψ-HFD of order λ and type σ, v : R+ ×R→R∗ and w : R+ ×R→R are given functions." “In 2022, Kucche et al. [11] established the existence of solutions in the weighted space for the following Ψ- Hilfer Hybrid FDE: Dµ,ν;Ψ y(t) f (t,y(t)) = g(t,y(t)); a.e. t ∈ (0,T ], (1.3) (Ψ(t) − Ψ(0))1−ξy(t)|t=0 = y0; y0 ∈R, (1.4) where 0 < µ < 1, 0 ≤ ν ≤ 1, ξ = µ + ν(1 − µ), Dµ,ν;Ψ is the Ψ-HFD of order µ and type ν, f ∈ C(I ×R,R−{0}) is bounded, I = [0,T ] and g ∈ C(I ×R,R)={h|the map w → h(t,w) is continuous for each t and the map t → h(t,w) is measurable for each w}." Motivated by [11], [8], the following IVP of the Ψ-Hilfer type fractional-order langevin equation with nonlocal initial conditions is explored, and the existence and attractivity results are obtained: Dν1,β1;Ψ 0+ ( Dν2,β2;Ψ 0+ κ(t) G(t,κ(t)) + pκ(t) ) = F(t,κ(t)) (1.5) κ(t)|t=0 = 0, (Ψ(t) − Ψ(0))1−γ1−ν2κ(t)|t=0 = κ0 (1.6) where Dνi,βi ;Ψ 0+ , i = 1, 2 is the Ψ-HFD of order νi, 0 < νi < 1 and type βi, 0 ≤ βi ≤ 1; 1 < ν1 + ν2 ≤ 2, F : I×R→R is a continuous function, p ∈R, t ∈ [0,ε] and γi = νi + βi (1 −νi ). Special Cases: (a) For β1,β2 = 0, Ψ(t) = t; we get nonlinear hybrid RL Langevin FDE for a.e. t ∈ (0,ε) of the form RLDν1 ( RLDν2 κ(t) G(t,κ(t)) + pκ(t) ) = F(t,κ(t)) κ(t)|t=0 = 0 Int. J. Anal. Appl. (2023), 21:82 3 For ν1 = 1,ν2 = 1, we generate hybrid differential equations of integer order d dt { d dt κ(t) G(t,κ(t)) + pκ(t) } = F(t,κ(t)) κ(t)|t=0 = 0 (b) For ν1 = 0, β2 = 0, p = 0, Ψ(t) = t, κ0 = 0 and for a.e. t ∈ (0,ε) we obtain the nonlocal hybrid FDEs of the form RLDν2 [ κ(t) G(t,κ(t)) ] = F(t,κ(t)) κ(t)|t=0 = 0 the existence results for which are obtained in [24]. For ν2 = 1 and t ∈ (0,ε) a.e., the investigations of [4] regarding the hybrid differential equation of integer order are incorporated into the results of the current work d dt [ κ(t) G(t,κ(t)) ] = F(t,κ(t)) κ(t)|t=0 = 0 (c) The outcomes are relevant to the below mentioned nonlinear Ψ-HFD for t ∈ (0,ε) and G = 1 Dν1,β1;Ψ 0+ ( Dν2,β2;Ψ 0+ + p ) κ(t) = F(t,κ(t)) κ(t)|t=0 = 0, (Ψ(t) − Ψ(0))1−γ1−ν2κ(t)|t=0 = κ0 When G is nothing but constant function with value equal to 1, β1,β2 = 0,ν1 = 0, Ψ(t) = t, i.e. γ1 = 0 the investigation of nonlinear FDEs involving RL FD is among the obtained outcomes of [22] for a.e. t ∈ (0,ε) RLDν2κ(t) = F(t,κ(t)) t1−ν2κ(t)|t0 = κ0 ∈R. With β1 = 0,β2 = 1,ν1 = 0, Ψ(t) = t, i.e.γ1 = 0, G = 1 the investigation of nonlinear FDEs utilising the Caputo FD is among the obtained outcomes [23] for a.e. t ∈ (0,ε) CDν2κ(t) = F(t,κ(t)) κ(t)|t=0 = κ0 The following is how the paper is organised: Section 2 goes through a few crucial foundational concepts from fractional calculus. Finding corresponding integral equations for the Hybrid-HFDE is the main topic of Section 3, along with the existence of solutions to the IVP (1.5)-(1.6) using FPT and uniform local attractiveness of the solution. Section 4 provides a summary of the results. 4 Int. J. Anal. Appl. (2023), 21:82 2. Auxiliary results Let Lp([0,ε],R) be the Banach space of all Lebesgue measurable functions from [0,ε] to R with ‖f‖Lp[0,ε] < ∞. Consider a function which is differentiable and increasing for all t ∈ [0,ε] = I, say Ψ ∈ C1(I,R), where (0 < ε < ∞) and Ψ ′ (t) 6= 0. We will be using, PσΨ(t,s) = (Ψ(t) − Ψ(s)) σ, WνΨ(t,s) = Ψ ′ (s)(Ψ(t) − Ψ(s))ν and γ1 + ν2 = σ throughout this paper to reduce the length of the equations. Here, we’ve listed some of the spaces used in this article: (a) C1−σ;Ψ([0,ε],R), the weighted Banach Space of all partially ordered functions with ||.||C1−σ;Ψ[0,ε] defined as: C1−σ;Ψ[0,ε] = {h : (0,ε] →R|P1−σΨ (t, 0)h(t) ∈C[0,ε]} where, ||h||C1−σ;Ψ[0,ε] = max t∈[0,ε] |P1−σΨ (t, 0)h(t)|. Let Ξ = ( C1−σ;Ψ(I,R),‖.‖C1−σ;Ψ(I,R) ) be a Banach algebra where (κy)(t) = κ(t)y(t), t ∈ I is how the product of vectors is defined. (b) Let BC = BC(R+) be the Banach space of all functions Φ : R+ to R which are bounded as well as continuous. (c) BC1−σ = BC1−σ(R+), denote the weighted space defined by BC1−σ = {φ : R+ →R : P1−σΨ (t, 0)Φ(t) ∈BC} of all bounded and continuous functions with the norm ‖Φ‖BC1−σ = sup t∈R+ |P1−σΨ (t, 0)Φ(t)|. Let’s revisit some fractional calculus definitions and characteristics. Definition 2.1. [1] “Let ν > 0, ν ∈ R, and g ∈ L1([0,ε],R).The Ψ-R-L fractional integral of a function g with respect to Ψ is defined by Iν;Ψ a+ = 1 Γ(ν) ∫ t a Ψ ′ (s)(Ψ(t) − Ψ(s))ν−1g(s)ds.” Definition 2.2. [9] “Let n − 1 < ν < n, n ∈ N and g ∈ Cn([0,ε],R). The Ψ-HFD HDν,β;Ψ a+ (.) of a function g of order ν and type 0 ≤ β ≤ 1 is defined by HDν,β;Ψ a+ g(t) = I β(n−ν);Ψ a+ ( 1 Ψ ′ (s) d dt )n I (1−β)(n−ν);Ψ a+ g(t).” Lemma 2.1. [9] “Let ν > 0 and δ > 0. Then (i) Iν;Ψ a+ Iν;Ψ a+ h(t) = Iν+ν;Ψ a+ h(t); (ii) Iν;Ψ a+ (Ψ(t) − Ψ(a))δ−1 = Γ(ν) Γ(ν+δ) (Ψ(t) − Ψ(a))ν+δ−1." And we observe that HDν,β;Ψ a+ (Ψ(t) − Ψ(a))(γ−1) = 0. Int. J. Anal. Appl. (2023), 21:82 5 Lemma 2.2. [9] “Let f ∈ L(0,ε),n−1 < ν ≤ n,n ∈N, 0 ≤ β ≤ 1, γ = ν + β(1−ν), I(1−β)(n−ν) a+ f ∈ ACk[0,ε]. Then( Iν;Ψ a+ HDν,β;Ψ a+ f ) (t) = f (t) − n∑ k=1 (Ψ(t) − Ψ(s))γ−k Γ(γ −k + 1) ( 1 Ψ ′ (s) d dt )n lim t→a+ (I (1−β)(n−ν) a+ f )(t).” Also, note that Dν1,β;Ψ a+ I ν2;Ψ a+ f (t) = I ν2−ν1 a+ f (t), if ν2 > ν1 and Dν1−ν2a+ f (t), if ν1 > ν2. For the readers’ convenience, we have included some of the Fixed Point Theorems (FPTs) that were utilised in this article. Lemma 2.3. [3]“Let S be a non-empty closed, convex and bounded subset of the Banach algebra Ξ and let A : Ξ → Ξ and B : S → Ξ be two operators such that (i) A is Lipschitzian with a Lipschitz constant α; (ii) B is completely continuous; (iii) y = AyBκ =⇒ y ∈ S for all κ ∈ S and (iv) αM < 1 where M = sup{‖Bκ‖ : κ ∈ S}. Then, the operator equation y = AyBy has a solution in S." Lemma 2.4. [8] “Solution of equation (K(κ))(t) = t are locally attractive if there exists a ball B(κ0,µ) in the space BC such that, for any solutions y = y(t) and σ = σ(t) of above equations that belong to B(κ0,µ) ∩ Λ, we can write lim t→∞ (y(t) −σ(t)) = 0. (2.1) If the limit (2.1) is uniform with respect to B(κ0,µ) ∩ Λ, where φ 6= Λ ⊂ BC, then the solutions are said to be uniformly locally attractive (or, equivalently, that the solutions are locally asymptotically stable)." Lemma 2.5. [5] “Let M ⊂ BC. Then M is relatively compact in BC if the following conditions are satisfied: (i) M is uniformly bounded in BC; (ii) the functions belonging to M are almost equicontinuous in R+, i.e., equicontinuous on every compact set in R+; (iii) the functions from M are equiconvergent, i.e. given � > 0, there exists L(�) > 0 such that |κ(t) − lim t→∞ κ(t)| < �, for any t ≥ L(�) and κ ∈ M." Theorem 2.1. [2] (Schauder Fixed-Point Theorem). “Let F be a Banach space, let U be a nonempty bounded convex and closed subset of F, and let K : U → U be a compact and continuous map. Then, K has at least one fixed point in U." 6 Int. J. Anal. Appl. (2023), 21:82 3. Main Results Here, we develop an auxiliary lemma showing the relationship between the fractional IVP (1.5)-(1.6) and a corresponding fractional IE. Lemma 3.1. The hybrid fractional IVP (1.5)-(1.6) for t ∈ [0,ε] is equivalent to the hybrid fractional IE κ(t) = G(t,κ(t)) { κ0 G(0,κ(0)) Pσ−1Ψ (t, 0) + I ν1+ν2;Ψ 0+ F(t,κ(t)) −pIν2;Ψ 0+ κ(t) } (3.1) and thus a function κ ∈C1−σ(I,R) is a solution of (1.5)-(1.6) iff it is a solution of (3.1). Proof. We shall establish that a solution of the IVP (1.5)-(1.6) is a solution of the fractional IE (3.1). Using Lemma (2.2) and the Ψ-R-L FI of order ν1 on equation (1.5), we obtain Dν2,β2;Ψ 0+ κ(t) G(t,κ(t)) + pκ(t) = Iν1;Ψ 0+ F(t,κ(t)) + c0 Γ(γ1) Pγ1−1 Ψ (t, 0). (3.2) Using Lemma (2.2) and the Ψ-R-L FI of order ν2 on equation (3.2), we get κ(t) G(t,κ(t)) = I ν1+ν2;Ψ 0+ F(t,κ(t)) −pIν2;Ψ 0+ κ(t) + c0 Γ(σ) Pσ−1Ψ (t, 0) + c1 Γ(γ2) Pγ2−1 Ψ (t, 0). (3.3) Using κ(t)|t=0 = 0, we get c1 = 0 for G(0,κ(0)) = 0 Thus, κ(t) = G(t,κ(t)) { c0 Γ(σ) Pσ−1Ψ (t, 0) + I ν1+ν2;Ψ 0+ F(t,κ(t)) −pIν2;Ψ 0+ κ(t) } . (3.4) Multiplying P1−σ Ψ (t, 0) on both sides of above equation, we get P1−σΨ (t, 0)κ(t) = c0 Γ(σ) G(t,κ(t)) + P1−σΨ (t, 0)G(t,κ(t))I ν1+ν2;Ψ 0+ F (t,κ(t)) −p G(t,κ(t))P1−σΨ (t, 0)I ν2;Ψ 0+ κ(t). Applying initial condition (1.6) and substituting t = 0, we obtain c0 = κ0Γ(σ) G(0,κ(0)) . Replacing c0 in eqn. (3.4), we get κ(t) = G(t,κ(t)) { κ0 G(0,κ(0)) Pσ−1Ψ (t, 0) + I ν1+ν2;Ψ 0+ F(t,κ(t)) −pIν2;Ψ 0+ κ(t) } . Conversely, a solution of the fractional IE (3.1) is also a solution of the IVP (1.5)-(1.6). Then, The aforementioned equation may be expressed as κ(t) G(t,κ(t)) = κ0 G(0,κ(0)) Pσ−1Ψ (t, 0) + I ν1+ν2;Ψ 0+ F(t,κ(t)) −pIν2;Ψ 0+ κ(t). (3.5) Operating the Ψ-HD, Dν2,β2;Ψ on both sides and using the Lemma (2.2) , we obtain Dν2,β2;Ψ κ(t) G(t,κ(t)) = κ0 G(0,κ(0)) Pγ1−1 Ψ (t, 0) + I ν1;Ψ 0+ F(t,κ(t)) −pκ(t). Int. J. Anal. Appl. (2023), 21:82 7 Again applying Dν1,β1;Ψ on above equation , we get Dν1,β1;Ψ ( Dν2,β2;Ψ κ(t) G(t,κ(t)) + pκ(t) ) = κ0 G(0,κ(0)) Dν1,β1;ΨPγ1−1 Ψ (t, 0) + F(t,κ(t)). Now, using the Lemma (2.1) Dν1,β1;ΨPγ1−1 Ψ (t, 0) = 0, we get Dν1,β1;Ψ ( Dν2,β2;Ψ κ(t) G(t,κ(t)) + pκ(t) ) = F(t,κ(t)). At t = 0 and F(0,κ(0)) = 0, the given equation simplifies to κ(t)|t=0 = 0 and from equation (3.5) and Lemma 2.1(ii), we get P1−σ Ψ (t, 0)κ(t)|t=0 = κ0. � In the next theorem, we utilise Banach algebra to demonstrate the existence of solution for (1.5)- (1.6). We require the following hypotheses on G and F in order to establish our conclusion: (A) G is a bounded function in C ( I×R,R−{0} ) such that: (i) κ → κG(t,κ) for t ∈ I a.e. is an increasing map in R ; (ii) For all κ,y ∈R, t ∈ I, such that G satisfies Lipchitz condition for second variable. (B) For all κ ∈R and t ∈ I a.e. ∃h1,h2 ∈C(I,R), such that |F(t,κ)| ≤ h1(t) and κ(t) ≤ h2(t). Theorem 3.1. If (A)-(B) holds. Then, ∃ a solution κ ∈C1−σ;Ψ(I,R) of the hybrid FDE (1.5)-(1.6) provided L {∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + ‖h1‖∞P1−γ1+ν1Ψ (ε, 0)Γ(ν1 + ν2 + 1) + p‖h2‖∞P 1−γ1 Ψ (ε, 0) Γ(ν2 + 1) } < 1. (3.6) Proof. Define, S = {κ ∈ Ξ : ‖κ‖C1−σ;Ψ(I,R) ≤ R} where R = K {∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + ‖h1‖∞P1−γ1+ν1Ψ (t, 0)Γ(ν1 + ν2 + 1) + p‖h2‖∞P 1−γ1 Ψ (t, 0) Γ(ν2 + 1) } and K is bound on G. It is evident that S is a bounded subset of Ξ which is closed and convex. Define A : Ξ → Ξ and B : S → Ξ as Aκ(t) =G(t,κ(t)), t ∈ I, Bκ(t) = κ0 G(0,κ(0)) Pσ−1Ψ (t, 0) + 1 Γ(ν1 + ν2) ∫ t 0 Wν1+ν2−1 Ψ (t,s)F(s,κ(s))ds −p 1 Γ(ν2) ∫ t 0 Wν2−1 Ψ (t,s)κ(s)ds Thus, equation (3.1) is nothing but κ = AκBκ, κ ∈ Ξ. We shall demonstrate that A and B meet all of the criteria of Lemma(2.3): Firstly, we shall prove that ‖Aκ −Ay‖C1−σ,Ψ(I,R) ≤L‖κ −y‖C1−σ,Ψ(I,R) (3.7) 8 Int. J. Anal. Appl. (2023), 21:82 i.e. A is an operator satisfying Lipschitz condition. From assumption (A)(ii), we observe that |P1−σΨ (t, 0)(Aκ(t) −Ay(t))| = ∣∣∣∣P1−σΨ (t, 0) ( G(t,κ(t)) −g(t,y(t)) )∣∣∣∣ ≤L|P1−σΨ (t, 0)(κ(t) −y(t))| ≤L‖κ −y‖C1−σ,Ψ(I,R) Next, we need to prove that B : S → Ξ is completely continuous. For this we shall prove that B is continuous, uniformly bounded and equicontinuous. For continuity of B, consider a sequence κn →κ in S. Then, ‖Bκn −Bκ‖C1−σ;Ψ(I,R) = max t∈I |P1−σΨ (t, 0) ( Bκn(t) −Bκ(t) ) | ≤ max t∈I { P1−σ Ψ (t, 0) Γ(ν1 + ν2) ∫ t 0 Wν1+ν2−1 Ψ (t,s)|F(s,κn(s)) −F(s,κ(s))|ds −p P1−σ Ψ (t, 0) Γ(ν2) ∫ t 0 Wν2−1 Ψ (t,s)|κn(s) −κ(s)|ds } . As n →∞, ‖Bκn −Bκ‖C1−σ;Ψ(I,R) → 0 by virtue of continuity of F and Lebesgue dominated conver- gence theorem. For any t ∈ I and κ ∈S, we shall exhibit that B(S) = {Bκ : κ ∈S} is uniformly bounded. |P1−σΨ (t, 0)Bκ(t)| ≤ ∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + P1−σΨ (t, 0)Γ(ν1 + ν2) ∫ t 0 Wν1+ν2−1 Ψ (t,s)|F(s,κ(s))|ds + p P1−σ Ψ (t, 0) Γ(ν2) ∫ t 0 Wν2−1 Ψ (t,s)|κ(s)|ds ≤ ∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + ‖h1‖∞P1−γ1+ν1Ψ (t, 0)Γ(ν1 + ν2 + 1) + p‖h2‖∞P 1−γ1 Ψ (t, 0) Γ(ν2 + 1) . Therefore, ‖Bκ‖C1−σ;Ψ(I,R) ≤ ∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + ‖h1‖∞P1−γ1+ν1Ψ (t, 0)Γ(ν1 + ν2 + 1) + p‖h2‖∞P 1−γ1 Ψ (t, 0) Γ(ν2 + 1) . (3.8) Now, for any κ ∈S and t1, t2 ∈ I with t1 < t2 we shall prove the equicontinuity of B(S). Making use of assumption (B), we have |P1−σΨ (t2, 0)Bκ(t2) −P 1−σ Ψ (t1, 0)Bκ(t1)| = ∣∣∣∣ { κ0 G(0,κ(0)) + P1−σ Ψ (t2, 0) Γ(ν1 + ν2) ∫ t2 0 Wν1+ν2−1 Ψ (t2, 0)F(s,κ(s))ds −p P1−σ Ψ (t2, 0) Γ(ν2) ∫ t2 0 Wν2−1 Ψ (t2,s)|κ(s)|ds } − { κ0 G(0,κ(0)) + P1−σ Ψ (t1, 0) Γ(ν1 + ν2) ∫ t1 0 Wν1+ν2−1 Ψ (t1,s)F(s,κ(s))ds Int. J. Anal. Appl. (2023), 21:82 9 −p P1−σ Ψ (t1, 0) Γ(ν2) ∫ t1 0 Wν2−1 Ψ (t1,s)|κ(s)|ds }∣∣∣∣ ≤ ∣∣∣∣P1−σΨ (t2, 0)Γ(ν1 + ν2) ‖h1‖∞ ∫ t2 0 Wν1+ν2−1 Ψ (t1, 0)ds− P1−σ Ψ (t1, 0) Γ(ν1 + ν2) ‖h1‖∞ ∫ t1 0 Wν1+ν2−1 Ψ (t1,s)ds −p P1−σ Ψ (t2, 0) Γ(ν2) ‖h2‖∞ ∫ t2 0 Wν2−1 Ψ (t2,s)ds + p P1−σ Ψ (t1, 0) Γ(ν2) ‖h2‖∞ ∫ t1 0 Wν2−1 Ψ (t2,s)ds ∣∣∣∣ ≤ ‖h1‖∞ Γ(ν1 + ν2) { P1−γ1+ν1 Ψ (t2, 0) −P 1−γ1+ν1 Ψ (t1, 0) } + p ‖h2‖∞ Γ(ν2) { P1−γ1 Ψ (t1, 0) −P 1−γ1 Ψ (t2, 0) } . Thus, the continuity of Ψ implies |P1−σ Ψ (t2, 0)Bκ(t2) −P1−σΨ (t1, 0)Bκ(t1)|→ 0 as |t1 − t2| → 0. Thus, Arzela-Ascoli theorem implies B(S) is relatively compact and hence a compact operator as a result. It is completely continuous from the continuity and compactness of B : S → Ξ. Now, we shall show that for any u ∈ Ξ, u = AuBκ =⇒ u ∈S for all κ ∈S. Let any u ∈ Ξ and κ ∈ S such that u = AuBκ. The function G being bounded and using the hypothesis (B), for any t ∈ I, we have |P1−σΨ (t, 0)u(t)| = |P 1−σ Ψ (t, 0)Au(t)Bκ(t)| ≤ ∣∣∣∣P1−σΨ (t, 0)G(t,u(t)) { κ0 G(0,κ(0)) Pσ−1Ψ (t, 0) + 1 Γ(ν1 + ν2) ∫ t 0 Wν1+ν2−1 Ψ (t,s)F(s,κ(s))ds−p 1 Γ(ν2) ∫ t 0 Wν2−1 Ψ (t,s)κ(s)ds }∣∣∣∣ ≤|G(t,u(t))| {∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + P1−σΨ (t, 0)Γ(ν1 + ν2) ∫ t 0 Wν1+ν2−1 Ψ (t,s)|F(s,κ(s))|ds + p P1−σ Ψ (t, 0) Γ(ν2) ∫ t 0 Wν2−1 Ψ (t,s)|κ(s)|ds } ≤K {∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + P1−σΨ (t, 0)Γ(ν1 + ν2 + 1)‖h1‖∞Pν1+ν2Ψ (t, 0) + pP 1−σ Ψ (t, 0) Γ(ν2 + 1) ‖h2‖∞Pν2Ψ (t, 0) } . i.e., ‖u‖C1−σ;Ψ(I,R) ≤ K {∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + ‖h1‖∞P1−γ1+ν1Ψ (t, 0)Γ(ν1 + ν2 + 1) + p‖h2‖∞P 1−γ1 Ψ (t, 0) Γ(ν2 + 1) } = R =⇒ u ∈S. In the end, we shall show that for M = sup{‖Bu‖C1−σ;Ψ(I,R) : u ∈ S}, we have αM < 1. Utilising inequality (3.8), we obtain M = sup { ‖Bκ‖C1−σ;Ψ(I,R) : κ ∈S } ≤ {∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + ‖h1‖∞P1−γ1+ν1Ψ (ε, 0)Γ(ν1 + ν2 + 1) + p‖h2‖∞P 1−γ1 Ψ (ε, 0) Γ(ν2 + 1) } 10 Int. J. Anal. Appl. (2023), 21:82 Making use of inequality (3.7), we get α = L. Therefore, as a consequence of the condition (3.6), we get the required αM ≤L {∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + ‖h1‖∞P1−γ1+ν1Ψ (ε, 0)Γ(ν1 + ν2 + 1) + p‖h2‖∞P 1−γ1 Ψ (ε, 0) Γ(ν2 + 1) } < 1. On applying Lemma (2.3), the solution for equation κ = AκBκ in S is obtained and thus for hybrid FDE (1.5)-(1.6). � Using Schauder’s FPT, we can now exhibit the existence and attractiveness of solutions. Assume the following: (C) For each κ ∈ BC1−σ, t → F(t,κ(t)) is measurable on R+ ; for a.e. t ∈ R+ the mapping κ →F(t,κ(t)) is continuous on BC1−σ and κ →G(t,κ(t)) is continuous and bounded. (D) For each κ ∈R and a.e. t ∈R+, ∃ T : R+ →R+ such that T is a continuous function and F(t,κ(t)) ≤ T (t) 1 + p|κ| , lim t→∞ P1−σΨ (t, 0)(I ν1+ν2;Ψ 0+ + pI ν2;Ψ 0+ )T (t) = 0. Set T∗ = sup t∈R+ P1−σΨ (t, 0)(I ν1+ν2;Ψ 0+ + pI ν2;Ψ 0+ )T (t) < ∞. Theorem 3.2. If (C)- (D) holds, then, ∃ at least one solution for problem (1.5) defined on R+ which is uniformly locally attractive. Proof. Define K for κ ∈BC1−σ (Kκ)(t) = G(t,κ(t)) { κ0Pσ−1Ψ (t, 0) G(0,κ(0)) + 1 Γ(ν1 + ν2) ∫ t 0 Wν1+ν2−1 Ψ (t,s)F(s,κ(s))ds −p 1 Γ(ν2) ∫ t 0 Wν2−1 Ψ (t,s)κ(s)ds } Let function G be bounded by N . Now, for κ ∈BC1−σ, t ∈R+ |P1−σΨ (t, 0)(Kκ)(t)| ≤|G(t,κ(t))| {∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + P1−σΨ (t, 0)Γ(ν1 + ν2) ∫ t 0 Wν1+ν2−1 Ψ (t,s)|F(s,κ(s))|ds + pP1−σ Ψ (t, 0) Γ(ν2) ∫ t 0 Wν2−1 Ψ (t,s)|κ(s)|ds } , ≤N {∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + T∗ } = R∗. ‖K(κ)‖BC ≤R∗, (3.9) implies, K(κ) ∈BC1−σ and K(BC1−σ) ⊂BC1−σ as a result of the continuity of K(κ) on R+; for any κ ∈BC1−σ. Consider BR∗ = B(0,R∗) = {G ∈BC1−σ : ‖G‖BC1−σ ≤ R∗}. Int. J. Anal. Appl. (2023), 21:82 11 Equation (3.9) implies K transforms the ball BR∗ into itself. From Lemma (3.1) the solutions of problem (1.5)-(1.6) are nothing but the fixed points of K(κ). We shall show that the operator K satisfies all the assumptions of Theorem (2.1). Step 1. Firstly, we shall prove the continuity of K. Consider a convergent sequence {κn}n∈N in BR∗ such that κn →κ . Then, for each t ∈R+, we have |Q1−σΨ (t, 0)(Kκn)(t) −Q 1−σ Ψ (t, 0)(Kκ)(t)| ≤ ∣∣∣∣G(t,κn(t)) { κ0 G(0,κ(0)) + Q1−σ Ψ (t, 0) Γ(ν1 + ν2) ∫ t 0 Wν1+ν2−1 Ψ (t,s)F(s,κn(s))ds−p Q1−σ Ψ (t, 0) Γ(ν2)∫ t 0 Wν2−1 Ψ (t,s)κn(s)ds } −G(t,κ(t)) { κ0 G(0,κ(0)) + Q1−σ Ψ (t, 0) Γ(ν1 + ν2) ∫ t 0 Wν1+ν2−1 Ψ (t,s)F(s,κ(s))ds −p Q1−σ Ψ (t, 0) Γ(ν2) ∫ t 0 Wν2−1 Ψ (t,s)κ(s)ds }∣∣∣∣ ≤ ∣∣∣∣G(t,κn(t)) { κ0 G(0,κ(0)) + Q1−σ Ψ (t, 0) Γ(ν1 + ν2) ∫ t 0 Wν1+ν2−1 Ψ (t,s)F(s,κn(s))ds−p Q1−σ Ψ (t, 0) Γ(ν2)∫ t 0 Wν2−1 Ψ (t,s)κn(s)ds } −G(t,κ(t)) { κ0 G(0,κ(0)) + Q1−σ Ψ (t, 0) Γ(ν1 + ν2) ∫ t 0 Wν1+ν2−1 Ψ (t,s)F(s,κn(s))ds −p Q1−σ Ψ (t, 0) Γ(ν2) ∫ t 0 Wν2−1 Ψ (t,s)κn(s)ds } + G(t,κ(t)) {∣∣∣∣ κ0G(0,κ(0)) + Q 1−σ Ψ (t, 0) Γ(ν1 + ν2)∫ t 0 Wν1+ν2−1 Ψ (t,s)F(s,κn(s))ds−p Q1−σ Ψ (t, 0) Γ(ν2) ∫ t 0 Wν2−1 Ψ (t,s)κn(s)ds } −G(t,κ(t)) { κ0 G(0,κ(0)) + Q1−σ Ψ (t, 0) Γ(ν1 + ν2) ∫ t 0 Wν1+ν2−1 Ψ (t,s)F(s,κ(s))ds−p Q1−σ Ψ (t, 0) Γ(ν2) ∫ t 0 Wν2−1 Ψ (t,s)κ(s)ds }∣∣∣∣ ≤ ∣∣∣∣G(t,κn(t)) −G(t,κ(t)) ∣∣∣∣ {∣∣∣∣ κ0G(0,κ(0)) + Q 1−σ Ψ (t, 0) Γ(ν1 + ν2) ∫ t 0 Wν1+ν2−1 Ψ (t,s)F(s,κn(s))ds −p Q1−σ Ψ (t, 0) Γ(ν2) ∫ t 0 Wν2−1 Ψ (t,s)κn(s)ds ∣∣∣∣ } + |G(t,κ(t))| {∣∣∣∣Q1−σΨ (t, 0)Γ(ν1 + ν2) ∫ t 0 Wν1+ν2−1 Ψ (t,s)( F(s,κn(s)) −F(s,κ(s)) ) ds−p Q1−σ Ψ (t, 0) Γ(ν2) ∫ t 0 Wν2−1 Ψ (t,s)(κn(s) −κ(s))ds ∣∣∣∣ } ≤ ∣∣∣∣G(t,κn(t)) −G(t,κ(t)) ∣∣∣∣ {∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + Q1−σΨ (t, 0)Γ(ν1 + ν2) ∫ t 0 Wν1+ν2−1 Ψ (t,s)|F(s,κn(s))|ds + p Q1−σ Ψ (t, 0) Γ(ν2) ∫ t 0 Wν2−1 Ψ (t,s)|κn(s)|ds } + N { Q1−σ Ψ (t, 0) Γ(ν1 + ν2) ∫ t 0 Wν1+ν2−1 Ψ (t,s) |F(s,κn(s)) −F(s,κ(s))|ds + p Q1−σ Ψ (t, 0) Γ(ν2) ∫ t 0 Wν2−1 Ψ (t,s)|κn(s) −κ(s)|ds } . Case I. For t ∈ [0,ε], by applying Lebesgue dominated convergence theorem and κn →κ as n →∞ on above equation along with the continuity of G and F, we get ‖K(κn)−K(κ)‖BC1−σ → 0 as n →∞. 12 Int. J. Anal. Appl. (2023), 21:82 Case II. For t ∈ (ε,∞), then, from the hypotheses and above equation, we have |P1−σΨ (t, 0)(Kκn)(t) −P 1−σ Ψ (t, 0)(Kκ)(t)| ≤ ∣∣∣∣G(t,κn(t)) −G(t,κ(t)) ∣∣∣∣ {∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + P1−σΨ (t, 0) ( I ν1+ν2;Ψ 0+ F(s,κn(s)) + pIν2;Ψ0+ κn(s) ) + N { P1−σΨ (t, 0) ( I ν1+ν2;Ψ 0+ |F(s,κn(s)) −F(s,κ(s))| + pIν2;Ψ 0+ |κn(s) −κ(s)| )} ≤ ∣∣∣∣G(t,κn(t)) −G(t,κ(t)) ∣∣∣∣ {∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + P1−σΨ (t, 0) ( I ν1+ν2;Ψ 0+ + pI ν2;Ψ 0+ ) T (t) } + 2NP1−σΨ (t, 0) ( I ν1+ν2;Ψ 0+ + pI ν2;Ψ 0+ ) T (t). Since, κn → κ as n → ∞, G is continuous and P1−σΨ (t, 0)(I ν1+ν2;Ψ 0+ + pI ν2;Ψ 0+ )T (t) → 0 as t → ∞, it follows from above equation that ‖K(κn) −K(κ)‖BC1−σ → 0 as n →∞. Step 2. On every compact subset [0,ε] of R+,ε > 0, we need to prove the uniform boundedness and equi-continuity of L(BR∗). Since BR∗ is bounded and L(BR∗) ⊂ BR∗, so L(BR∗) is uniformly bounded. For each κ ∈ BR∗ and t1, t2 ∈ [0,ε], t1 < t2, we have |P1−σΨ (t2, 0)(Kκ)(t2) −P 1−σ Ψ (t1, 0)(Kκ)(t1)| ≤ ∣∣∣∣G(t2,κ(t2)) { κ0 G(0,κ(0)) + P1−σΨ (t2, 0) Γ(ν1 + ν2) ∫ t2 0 Wν1 +ν2−1Ψ (t1,s)F(s,κ(s))ds−p P1−σΨ (t2, 0) Γ(ν2)∫ t2 0 Wν2−1Ψ (t2,s)κ(s)ds } −G(t1,κ(t1)) { κ0 G(0,κ(0)) + P1−σΨ (t1, 0) Γ(ν1 + ν2) ∫ t1 0 Wν1 +ν2−1Ψ (t1,s) F(s,κ(s))ds−p P1−σΨ (t1, 0) Γ(ν2) ∫ t1 0 Wν2−1Ψ (t2,s)κ(s)ds }∣∣∣∣ ≤ ∣∣∣∣G(t2,κ(t2)) { κ0 G(0,κ(0)) + P1−σΨ (t2, 0) Γ(ν1 + ν2) ∫ t2 0 Wν1 +ν2−1Ψ (t2,s)F(s,κ(s))ds−p P1−σΨ (t2, 0) Γ(ν2)∫ t2 0 Wν2−1Ψ (t2,s)κ(s)ds } −G(t1,κ(t1)) { κ0 G(0,κ(0)) + P1−σΨ (t2, 0) Γ(ν1 + ν2) ∫ t2 0 Wν1 +ν2−1Ψ (t2,s) F(s,κ(s))ds−p P1−σΨ (t2, 0) Γ(ν2) ∫ t2 0 Wν2−1Ψ (t2,s)κ(s)ds } + G(t1,κ(t1)) { κ0 G(0,κ(0)) + P1−σΨ (t2, 0) Γ(ν1 + ν2)∫ t2 0 Wν1 +ν2−1Ψ (t2,s)F(s,κ(s))ds−p P1−σΨ (t2, 0) Γ(ν2) ∫ t2 0 Wν2−1Ψ (t2,s)κ(s)ds } −G(t1,κ(t1)){ κ0 G(0,κ(0)) + P1−σΨ (t1, 0) Γ(ν1 + ν2) ∫ t1 0 Wν1 +ν2−1Ψ (t1,s)F(s,κ(s))ds−p P1−σΨ (t1, 0) Γ(ν2) ∫ t1 0 Wν2−1Ψ (t1,s)κ(s)ds }∣∣∣∣ ≤|G(t2,κ(t2)) −G(t1,κ(t1))| {∣∣∣∣ κ0G(0,κ(0)) + P 1−σ Ψ (t2, 0) Γ(ν1 + ν2) ∫ t2 0 Wν1 +ν2−1Ψ (t2,s)F(s,κ(s))ds + p P1−σΨ (t2, 0) Γ(ν2) ∫ t2 0 Wν2−1Ψ (t2,s)κ(s)ds ∣∣∣∣ } + |G(t1,κ(t1))| {∣∣∣∣P1−σΨ (t2, 0)Γ(ν1 + ν2) ∫ t1 0 Wν1 +ν2−1Ψ (t2,s)F(s,κ(s))ds + P1−σΨ (t2, 0) Γ(ν1 + ν2) ∫ t2 t1 Wν1 +ν2−1Ψ (t2,s)F(s,κ(s))ds− P1−σΨ (t1, 0) Γ(ν1 + ν2) ∫ t1 0 Wν1 +ν2−1Ψ (t1,s)F(s,κ(s))ds } Int. J. Anal. Appl. (2023), 21:82 13 + p { P1−σΨ (t2, 0) Γ(ν2) ∫ t1 0 Wν2−1Ψ (t2,s)κ(s)ds + P1−σΨ (t2, 0) Γ(ν2) ∫ t2 t1 Wν2−1Ψ (t2,s)κ(s)ds− P1−σΨ (t1, 0) Γ(ν2)∫ t1 0 Wν2−1Ψ (t1,s)κ(s)ds ∣∣∣∣ } ≤|G(t2,κ(t2)) −G(t1,κ(t1))| {∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + P1−σΨ (t2, 0)Γ(ν1 + ν2) ∫ t2 0 Wν1 +ν2−1Ψ (t1,s)|F(s,κ(s))|ds + p P1−σΨ (t2, 0) Γ(ν2) ∫ t2 0 Wν2−1Ψ (t2,s)|κ(s)|ds } + L {(∫ t1 0 ∣∣∣∣P1−σΨ (t2, 0)Γ(ν1 + ν2) Wν1 +ν2−1Ψ (t2,s) − P1−σΨ (t1, 0) Γ(ν1 + ν2) Wν1 +ν2−1Ψ (t1,s) ∣∣∣∣|F(s,κ(s))|ds + P1−σΨ (t2, 0)Γ(ν1 + ν2) ∫ t2 t1 Wν1 +ν2−1Ψ (t2,s)|F(s,κ(s))|ds ) + p (∫ t1 0 ∣∣∣∣P1−σΨ (t2, 0)Γ(ν2) Wν2−1Ψ (t2,s) − P 1−σ Ψ (t1, 0) Γ(ν2) Wν2−1Ψ (t1,s) ∣∣∣∣|κ(s)|ds + P1−σΨ (t2, 0) Γ(ν2) ∫ t2 t1 Wν2−1Ψ (t2,s)|κ(s)|ds )} ≤|G(t2,κ(t2)) −G(t1,κ(t1))| {∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + P1−σΨ (t2, 0)Γ(ν1 + ν2) ∫ t2 0 Wν1 +ν2−1Ψ (t1,s)T (s)ds } + L {(∫ t1 0 ∣∣∣∣P1−σΨ (t2, 0)Γ(ν1 + ν2) Wν1 +ν2−1Ψ (t2,s) − P 1−σ Ψ (t1, 0) Γ(ν1 + ν2) Wν1 +ν2−1Ψ (t1,s) ∣∣∣∣T (s)ds + P1−σΨ (t2, 0) Γ(ν1 + ν2) ∫ t2 t1 Wν1 +ν2−1Ψ (t1,s)T (s)ds )} . Given that T, G are continuous and setting T∗ = supt∈[0,ε] T (t), we obtain |P1−σΨ (t2, 0)(Kκ)(t2) −P 1−σ Ψ (t1, 0)(Kκ)(t1)| ≤|G(t2,κ(t2)) −G(t1,κ(t1))| {∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + T∗P1−σΨ (t2, 0)Γ(ν1 + ν2) ∫ t2 0 Wν1+ν2−1 Ψ (t1,s)ds } + LT∗(∫ t1 0 ∣∣∣∣P1−σΨ (t2, 0)Γ(ν1 + ν2) Wν1+ν2−1Ψ (t1,s) − P 1−σ Ψ (t1, 0) Γ(ν1 + ν2) Wν1+ν2−1 Ψ (t1,s) ∣∣∣∣ds + P1−σΨ (t2, 0)Γ(ν1 + ν2)∫ t2 t1 Wν1+ν2−1 Ψ (t1,s)ds ) . As t1 → t2, we have |P1−σΨ (t2, 0)(Kκ)(t2) −P 1−σ Ψ (t1, 0)(Kκ)(t1)|→ 0. Step 3. To prove the equiconvergence of L(BR). For any κ ∈ L(BR∗), |P1−σΨ (t, 0)(Kκ)(t)| ≤|G(t,κ(t))| {∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + ∣∣∣∣P1−σΨ (t, 0)Γ(ν1 + ν2) ∫ t 0 Wν1+ν2−1 Ψ (t,s)F(s,κ(s))ds −p P1−σ Ψ (t, 0) Γ(ν2) ∫ t 0 Wν2−1 Ψ (t,s)κ(s)ds ∣∣∣∣ } , t ∈R+ ≤L {∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + P1−σΨ (t, 0)(Iν1+ν2;Ψ0+ + pIν2;Ψ0+ )T (t) } . 14 Int. J. Anal. Appl. (2023), 21:82 Since P1−σ Ψ (t, 0)(I ν1+ν2;Ψ 0+ + I ν2;Ψ 0+ )T (t) → 0 as t →∞, we find |(Kκ(t)| ≤L {∣∣∣∣ κ0P1−σ Ψ (t, 0)G(0,κ(0)) ∣∣∣∣ + P1−σΨ (t, 0)(Iν1+ν2;Ψ0+ T )(t)P1−σ Ψ (t, 0) + p P1−σ Ψ (t, 0)(I ν2;Ψ 0+ T )(t) P1−σ Ψ (t, 0) } . Hence, |(Kκ)(t) − (Kκ)(∞)| → 0 as t → ∞. Thus, K : BR∗ → BR∗ is compact and continuous using Lemma (2.5). Applying Schauder FPT (2.1),the fixed point of K is a solution of (1.5) on R+. Step 4. Uniform local attractivity. Let κ∗ be a solution of hybrid FDE (1.5) and κ ∈ B ( κ∗, 2L {∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + 2T∗ }) , we have |P1−σΨ (t, 0)K(κ)(t) −P 1−σ Ψ (t, 0)(κ∗)(t)| ≤|P1−σΨ (t, 0)K(κ)(t) −P 1−σ Ψ (t, 0)K(κ∗)(t)| ≤ ∣∣∣∣G(t,κ(t)) −G(t,κ∗(t)) ∣∣∣∣ {∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + P1−σΨ (t, 0)Γ(ν1 + ν2) ∫ t 0 Wν1+ν2−1 Ψ (t,s)|F(s,κ(s))|ds + p P1−σ Ψ (t, 0) Γ(ν2) ∫ t 0 Wν2−1 Ψ (t,s)|κ(s)|ds } + L { P1−σ Ψ (t, 0) Γ(ν1 + ν2) ∫ t 0 Wν1+ν2−1 Ψ (t,s)|F(s,κ(s)) −F(s,κ∗(s))|ds + p P1−σ Ψ (t, 0) Γ(ν2) ∫ t 0 Wν2−1 Ψ (t,s)|κ(s) −κ∗(s)|ds } ≤2L {∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + P1−σΨ (t, 0)Γ(ν1 + ν2) ∫ t 0 Wν1+ν2−1 Ψ (t,s)|T (s)|ds } + 2L P1−σ Ψ (t, 0) Γ(ν1 + ν2) ∫ t 0 Wν1+ν2−1 Ψ (t,s)|T (s)|ds ≤2L {∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + 2T∗ } . Thus, we get ‖K(κ) −κ∗‖BC1−σ ≤ 2L {∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + 2T∗ } . This implies the continuity of K such that K ( B ( κ∗, 2L {∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + 2T∗ })) ⊂ ( B ( κ∗, 2L {∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + 2T∗ })) . Moreover, if κ is a solution of problem (1.5), then |κ(t) −κ∗(t)| = |Kκ(t) −Kκ∗(t)| ≤ |G(t,κ(t)) −G(t,κ∗(t))| { Pσ−1Ψ (t, 0) ∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + { 1 Γ(ν1 + ν2)∫ t 0 Wν1+ν2−1 Ψ (t,s)|F(s,κ(s)) −F(s,κ∗(s))|ds− p Γ(ν2) ∫ t 0 Wν2−1 Ψ (t,s)|κ(s) −κ∗(s)|ds } ≤ 2L { Pσ−1Ψ (t, 0) ∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + 2P1−σΨ (t, 0)(Iν1+ν2;Ψ0+ T )(t)P1−σ Ψ (t, 0) + 2p P1−σ Ψ (t, 0)(I ν2;Ψ 0+ T )(t) P1−σ Ψ (t, 0) } . Int. J. Anal. Appl. (2023), 21:82 15 Therefore, |κ(t) −κ∗(t)| =|(K(κ(t)) − (K(κ∗(t))| ≤ 2L { Pσ−1Ψ (t, 0) ∣∣∣∣ κ0G(0,κ(0)) ∣∣∣∣ + 2 P1−σ Ψ (t, 0)(I ν1+ν2;Ψ 0+ T )(t) P1−σ Ψ (t, 0) + 2p P1−σ Ψ (t, 0)(I ν2;Ψ 0+ T )(t) P1−σ Ψ (t, 0) } . (3.10) By using (3.10) and lim t→∞ P1−σΨ (t, 0)(I ν1+ν2;Ψ 0+ + I ν2;Ψ 0+ )T (t) = 0, we conclude lim t→∞ |κ(t) −κ∗(t)| = 0. The Lemma (2.4) indicates that solutions of IVP (1.5) has uniform local attractiveness. � 4. 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Appl. 62 (2011), 1312-1324. https://doi.org/10.1016/j.camwa.2011.03.041. https://doi.org/10.1007/s40314-022-01800-x https://doi.org/10.3934/math.2021477 https://doi.org/10.1016/j.camwa.2012.01.009 https://doi.org/10.1007/s40435-022-01005-4 https://doi.org/10.1016/j.cnsns.2016.09.006 https://doi.org/10.1016/s0301-0104(02)00670-5 https://doi.org/10.1007/s12346-022-00650-6 https://doi.org/10.4134/BKMS.B170887 https://doi.org/10.4134/BKMS.B170887 https://doi.org/10.1016/j.amc.2011.03.062 https://doi.org/10.1016/j.camwa.2011.03.041 1. Introduction 2. Auxiliary results 3. Main Results 4. Conclusion References