The paper investigated the exact controllability of delayed fractional evolution systems of order α∈(1,2) in abstract spaces. At first, the exact controllability result is obtained when the nonlinear term f is locally Lipschitz continuous. Then, the certain compactness conditions and the measure of noncompactness conditions were applied to demonstrate the exact controllability of the concerned problem. The discussion was based on the fixed point theorems and the cosine family theory.
Citation: Lijuan Qin. On the controllability results of semilinear delayed evolution systems involving fractional derivatives in Banach spaces[J]. AIMS Mathematics, 2024, 9(7): 17971-17983. doi: 10.3934/math.2024875
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The paper investigated the exact controllability of delayed fractional evolution systems of order α∈(1,2) in abstract spaces. At first, the exact controllability result is obtained when the nonlinear term f is locally Lipschitz continuous. Then, the certain compactness conditions and the measure of noncompactness conditions were applied to demonstrate the exact controllability of the concerned problem. The discussion was based on the fixed point theorems and the cosine family theory.
Fractional calculus generalized the classical calculus to an arbitrary (non-integer) order. The history of this theory goes back to mathematicians like Leibniz (1646–1716), Liouville (1809–1882), Riemann (1826–1866), Letnikov (1837–1888), Grünwald (1838–1920) and others [16,19]. For the last three centuries, fractional calculus is getting famous, like all fields of science. It is one of the most intensively developed fields of mathematical assessment. Because of its various applications in designing, financial aspects, account, geography, likelihood and measurements, compound designing, physical science, splines, thermodynamics and neural organizations [5,9,22].
There are some recent developments in fractional calculus and its applications. In [8], the authors studied the fractional second and third-order nonlinear Schrödinger equations. They studied symmetric and antisymmetric solutions and analyzed the influence of the Lêvy index on different solutions. Lu et al. [14] solved the fractional discrete coupled nonlinear Schrödinger equations on account of the modified Riemann-Liouville fractional derivative and Mittag Leffler function. In [13], Li et al. studied the existence, bifurcation and stability of two-dimensional optical solutions in the framework of fractional nonlinear Schrödinger equation.
From the literature review reader can see several definitions of fractional operators like Riemann-Liouville, Caputo, Grünwald-Letnikov, Weyl, Hadamard, Marchaud and Riesz [12,15]. These types of derivatives do not satisfy the fundamental formulas of differentiation like the product rule, the quotient rule, the chain rule, etc. Khalil [12] introduced a new well-behaved simple fractional derivative known as the conformable fractional derivative (CFD) based on the derivative's basic limit concept: Suppose Φ:[0,∞)⟶R be a function, then for all υ>0 and α∈(0,1],
Φ(α)(υ)=lim∈→0Φ(υ+ϵυ1−α)−Φ(υ)ϵ, |
where Φ(α)(υ) is known as the CFD of Φ of order α. The definition of CFD introduced by Khalil retaines all the classical characteristics of the derivative, and satisfy the chain rule. This new definition attracted many researchers, some results have been obtained for the fundamental properties of the CFD in [1].
In 1988, Hilger in his Ph. D. thesis introduced the time scale theory that has recently received a great deal of attraction to integrate and extend the discrete and continuous analysis [10]. The investigation of dynamic equations on arbitrary time scales show such distinction and helps avoid proving outcomes twice: Once for the differential equation and once for the difference equation. The basic idea is to prove a result for a dynamic equation where the domain of the function is a so-called time scale T, which is an arbitrary nonempty closed subset of reals [7].
Bastos in his Ph. D. thesis developed fractional calculus on time [18]. The theory of fractional (non-integer order) calculus on time scales is a topic of great interest for researchers nowadays. In [6], Benkhettou et al. developed a conformable fractional calculus theory on an arbitrary time scale, which extends the conformable fractional calculus and the fundamental techniques for fractional differentiation and integration on time scales.
Ahmed [2] discussed the class of non-local stochastic differential equations involving conformable fractional time derivative operator and the existence of mild solution for the non-local conformable stochastic differential equation. In [3], Ahmed discussed the class of conformable fractional stochastic differential equations driven by the Rosenblatt process. In [4], Ahmed studied a non-instantaneous impulsive conformable fractional stochastic delay integro-differential system driven by the Rosenblatt process.
The exponential function on the time scale introduced by Hilger in [11] and has been devoted to solving the first-order linear dynamic equations, and second-order linear dynamic equations with constant and variable coefficients [11]. Euler-Cauchy dynamic equations on time scales have been solved using an exponential function. Exponential function for conformable fractional calculus has been defined in [17]. The conformable exponential function with respect to operator Δα has been defined [20]. This definition is implicit. In this paper, we define the generalized conformable fractional exponential function by following the conformable calculus outline and also developing its fundamental characteristics. However, our definition is explicit. The generalized exponential function is consistent with the exponential function on time scales for α=1 and consistent with CF exponential function if T=R. The generalized exponential function and these theorems are the generalizations of the exponential function on time scales discussed by Bohner et al. [7] and the conformable fractional exponential function in [17].
In [24], the global existence, extension, boundedness and stabilities of solutions have been discussed for the following conformable fractional differential equation:
x(α)(t)=f(t,x(t)), t∈[a,∞), 0<α<1, |
corresponding to the local and non-local initial condition x(a)=xa and x(a)+g(x)=xa respectively. We generalized the theory established in [24] to the conformable fractional dynamic equation:
ψ(α)(s)=k(s,ψ(s)), s∈[0,∞)T, 0<α<1. |
We consider the global existence, extension, boundedness and stability of the solutions corresponding to the conformable fractional dynamic equation:
ψ(α)(s)=k(s,ψ(s)), s∈[0,∞)T, 0<α<1, | (1.1) |
according to the local initial condition
ψ(0)=ψ0, | (1.2) |
and non-local initial condition
ψ(0)+g(ψ)=ψ0, | (1.3) |
where ψ(α)(s) refers to the CFD of order α∈(0,1] for functions defined on arbitrary time scales T, k:[0,∞)T×R→R. Also, k is right dense continuous and on a suitable function space g is functional.
We structure the remaining of the paper: Section 2 recalls some essential definitions and results. Section 3 aims to define the generalized exponential function and develop its fundamental characteristics. In Section 4, we generalize the Grönwall type inequalities in the conformable setting. In Section 5, we set some rules for the global existence, extension and boundedness of solutions to the local initial value problem and then discuss the stability of solutions. Section 6 examines the existence of solutions to the nonlocal initial value problem. Finally, we conclude our findings in the last section.
Here, we recall some basic definitions and results which are essential to the sequel [21]. Throughout this manuscript, let us denote the time scale by T and the set of all rd-continuous functions by Crd.
Lemma 2.1. If h∈Crd and u∈Tk, then
∫σ(u)uh(τ)Δτ=μ(u)h(u). |
Definition 2.1. Assume that k>0, the Hilger complex number is defined by
Ck={z∈C:z≠−1k}. |
Definition 2.2. For k>0, the strip is defined by
Zk={z∈C:−πk<Im(z)<πk}. |
Definition 2.3. For k>0, the cylindrical transformation ξk:Ck→Zk is defined by
ξk(z)=1klog(1+zk), |
where log is the principal logarithm function. Notice that
ξ−1k(z)=exp(zk)−1k. |
Let us denote by Cα(T,R) the set of all functions whose conformable fractional differentiable of order α is continuous, where α∈(0,1]. The following lemmas give some meaningful connections concerning the conformable fractional derivative on time scales [6].
Lemma 2.2. Suppose l∈C(T,R) and assume v∈Tk. If l is continuous at v where v isright-scattered, implies that l∈Cα(T,R) at v such that
(l)(α)(v)=l(σ(v))−l(v)μ(v)v1−α=lΔ(v)v1−α, |
where (l)(α)(⋅)represents the conformable fractional derivative of order α∈(0,1].
Lemma 2.3. Suppose h,l∈Cα(T,R). Then
(1) If h, l∈Cα(T,R), then the product hl∈Cα(T,R) with
(hl)(α)=(h)(α)l+(h∘σ)(l)(α)=(h)(α)(l∘σ)+h(l)(α)=(hl)Δ(s)s1−α. |
(2) If h∈Cα(T,R), then 1h∈Cα(T,R) such that
(1h)(α)=−(h)(α)h(h∘σ)=(1h)Δ(s)s1−α, |
applicable at each points s∈Tk for which h(s)h(σ(s))≠0.
(3) If h, l∈Cα(T,R), then hl∈Cα(T,R) such that
(hl)(α)=(h)(α)l−h(l)(α)l(l∘σ)=(hl)Δ(s)s1−α, |
applicable at each points s∈Tk for which l(s)l(σ(s))≠0.
Lemma 2.4. If (g(s))(α) is continuous on [c,d]T, then
Iα(g(s))(α)=g(s)−g(0), |
where Iα represents the conformable fractional integral of order α∈(0,1].
Lemma 2.5. Let α∈(0,1]. Then for allright-dense continuous function g:T⟶R, a functionGα:T→R exists in such a way that
[(Gα)(s)](α)=(Iαg(s))(α)=g(s), |
for each s∈Tk. Function Gα is called an α-antiderivative of g.
In this section, we use Definition 2.3 to define a generalized exponential function. Let us generalize the regressive concept in a conformable setting.
Definition 3.1. A function l:T→R is said to be "α-regressive" provided
1+μ(u)l(u)uα−1≠0, ∀ u∈Tk, |
holds. The set of all α-regressive and right-dense continuous functions l:T→R [7,Definition 1.58] is referred to as Rα.
Definition 3.2. In Rα "α-circle plus" addition ⊕α is defined as below:
(k⊕αϝ)(ω)=k(ω)+ϝ(ω)+μ(ω)k(ω)ϝ(ω)ωα−1, ∀ ω∈Tk. |
Definition 3.3. For h∈Rα, define
(⊖αh)(v)=−h(v)1+μ(v)h(v)vα−1, ∀ v∈Tk. |
Definition 3.4. Define "α-circle minus" subtraction ⊖α on Rα as below:
(l⊖αk)(u)=(l⊕α(⊖αk))(u), ∀ v∈Tk. |
For k,l∈Rα, we have
(l⊖αk)(u)=l(u)−k(u)1+μ(u)k(u)uα−1. |
Lemma 3.1. Show that (Rα,⊕α) is an Abelian group.
Proof. The proof is straight-forward, so it is left as an exercise.
The group (Rα,⊕α) is also called the α-regressive group.
Remark 3.1. For α=1, it becomes regressive group [7].
Definition 3.5. The set Rα+ of all positively α-regressive elements of Rα is defined by
Rα+=Rα+(T,R)={f:f∈Rα, 1+μ(t)f(t)tα−1>0, ∀ t∈Tk}. |
Theorem 3.1. Suppose h,k∈Rα. Then
(1) (h⊖αh)(τ)=0,
(2) ⊖α(⊖αh)(τ)=φ(τ),
(3) (k⊖αh)(τ)∈Rα,
(4) ⊖α(h⊖αk)(τ)=(k⊖αh)(τ),
(5) ⊖α(k⊕αh)(τ)=[(⊖αk)⊕α(⊖αh)](τ),
(6) [h⊕αk1+μhτα−1](τ)=h(τ)+k(τ).
Proof. (1) By using Definitions 3.2–3.4 respectively, it follows that
(h⊖αh)(τ)=(h⊕α(⊖αh))(τ)=h(τ)⊕α(−h(τ)1+μ(τ)h(τ)τα−1)=h(τ)−h(τ)1+μ(τ)h(τ)τα−1−h2(τ)μ(τ)τα−11+μ(τ)h(τ)τα−1=h(τ)+h2(τ)μ(τ)τα−1−h(τ)−h2(τ)μ(τ)τα−11+μ(τ)h(τ)τα−1=0. |
(2) Definition 3.3 yields that
⊖α(⊖αh)(τ)=⊖α(−h(τ)1+μ(τ)h(τ)τα−1)=h(τ)1+μ(τ)h(τ)τα−11−μ(τ)h(τ)τα−11+μ(τ)h(τ)τα−1=h(τ)1+μ(τ)h(τ)τα−1−μ(τ)h(τ)τα−1=h(τ). |
(3) By using Definitions 3.1 and 3.4 respectively, we have
1+μ(τ)(k⊖αh)(τ)τα−1=1+μ(τ)(k(τ)−h(τ)1+μ(τ)h(τ)τα−1)τα−1=1+μ(τ)k(τ)τα−1−μ(τ)h(τ)τα−11+μ(τ)h(τ)τα−1=1+μ(τ)h(τ)τα−1+μ(τ)k(τ)τα−1−μ(τ)h(τ)τα−11+μ(τ)h(τ)τα−1=1+μ(τ)k(τ)τα−11+μ(τ)h(τ)τα−1≠0. |
We note that k(τ)−h(τ)1+μ(τ)h(τ)τα−1 is right-dense continuous. Therefore, (k⊖αh)(τ)∈Rα.
(4) Using Definitions 3.3 and 3.4, it implies that
⊖α[h⊖αk](τ)=−[h(τ)−k(τ)1+μ(τ)k(τ)τα−1]1+μ(τ)(h(τ)−k(τ)1+μ(τ)k(τ)τα−1)τα−1=k(τ)−h(τ)1+μ(τ)h(τ)τα−1=[k⊝αh](τ). |
(5) By using Definitions 3.2 and 3.3 respectively, we get
⊖α[k⊕αh](τ)=⊖α(k(τ)+h(τ)+μ(τ)k(τ)h(τ)τα−1)=−[k(τ)+h(τ)+μ(τ)k(τ)h(τ)τα−1]1+μ(τ)[k(τ)+h(τ)+μ(τ)k(τ)h(τ)τα−1]τα−1=−[k(τ)+h(τ)+μ(τ)k(τ)h(τ)τα−1]1+μ(τ)k(τ)τα−1+μ(τ)h(τ)τα−1+μ2(τ)k(τ)h(τ)τ2α−2=−[k(τ)+h(τ)+μ(τ)k(τ)h(τ)τα−1][1+μ(τ)k(τ)τα−1][1+μ(τ)h(τ)τα−1]. |
On the other side, using Definitions 3.2 and 3.3 respectively, we obtain
[(⊖αk)⊕α(⊖αh)](τ)=(−k(τ)1+μ(τ)k(τ)τα−1)⊕α(−h(τ)1+μ(τ)h(τ)τα−1)=−k(τ)1+μ(τ)k(τ)τα−1−h(τ)1+μ(τ)h(τ)τα−1+μ(τ)k(τ)h(τ)τα−1[1+μ(τ)k(τ)τα−1][1+μ(τ)h(τ)τα−1]=−(k(τ)+h(τ)+μ(τ)k(τ)h(τ)τα−1)[1+μ(τ)k(τ)τα−1][1+μ(τ)h(τ)τα−1]. |
Hence,
⊖α(k⊕αh)(τ)=[(⊖αk)⊕α(⊖αh)](τ). |
(6) From Definition 3.2, we have
[h⊕αk1+μhτα−1](τ)=h(τ)+k(τ)1+μ(τ)h(τ)τα−1+μ(τ)h(τ)k(τ)τα−11+μ(τ)h(τ)τα−1=h(τ)+μ(τ)h2(τ)τα−1+k(τ)+μ(τ)h(τ)k(τ)τα−11+μ(τ)h(τ)τα−1=h(τ)[1+μ(τ)h(τ)τα−1]+k(τ)[1+μ(τ)h(τ)τα−1]1+μ(τ)h(τ)τα−1=h(τ)+k(τ). |
Next, we define the generalized exponential function.
Definition 3.6. Let h∈Rα, then the generalized exponential function is defined by
Eh(r,0)=exp(∫r0ξμ(τ)(h(τ)τα−1)Δτ), ∀ 0,r∈T. | (3.1) |
To be more precise, using the definition for the cylindrical transformation Definition 2.3 we obtain
Eh(r,0)=exp(∫r01μ(τ)log(1+μ(τ)h(τ)τα−1)Δτ), ∀ 0, r∈T. |
Example 3.1. When T=R, then
Eh(r,0)=exp(∫r0ξ0(h(τ)τα−1)dτ)=exp(∫r0h(τ)dατ). |
Example 3.2. When T=Z, then
Eh(r,0)=exp(∫r0ξ1(h(τ)τα−1)Δτ)=exp(r−1∑τ=0ξ(h(τ)τα−1))=r−1∏τ=0[1+h(τ)τα−1]. |
Example 3.3. When T=qN, q>1, then
Eϝ(r,0)=exp(∫r0ξ(q−1)ω(ϝ(ω)ωα−1)Δω)=exp(r−1∑ω=0[log(1+(q−1)ϝ(ω)ωα)])=r−1∏ω=0[1+(q−1)ϝ(ω)ωα]. |
Remark 3.2. The exponential function on arbitrary time scale [7] is obtained by choosing α=1.
Remark 3.3. Definition 2.3 in [17] is obtained by choosing T=R.
Theorem 3.2. (Semigroup property) If f∈Rα, then
Ef(t,0)Ef(0,r)=Ef(t,r), ∀t,0,r∈T. |
Proof. By using Definition 3.6, we have
Ef(t,0)Ef(0,r)=exp(∫t0ξμ(τ)(f(τ)τα−1)Δτ)exp(∫0rξμ(τ)(f(τ)τα−1)Δτ)=exp(∫trξμ(τ)(f(τ)τα−1)Δτ)=Ef(t,r). |
Theorem 3.3. Suppose k∈Rα. Then
EΔk(v,0)=k(v)Ek(v,0)vα−1 |
and
E(α)k(v,0)=k(v)Ek(v,0). |
Proof. Let σ(v)>v. Then
EΔk(v,0)=Ek(σ(v),0)−Ek(v,0)μ(v)=exp(∫v0ξμ(τ)(k(τ)τα−1)Δτ+∫σ(v)vξμ(τ)(k(τ)τα−1)Δτ)−exp(∫v0ξμ(τ)(k(τ)τα−1)Δτ)μ(v)=[exp(∫σ(v)vξμ(τ)(k(τ)τα−1)Δτ)−1]Ek(v,0)μ(v). |
From Lemma 2.1, it implies that
EΔk(v,0)=[exp(μ(v)ξμ(v)(k(v)vα−1))−1]Ek(v,0)μ(v). |
From Definition 2.3, it follows that
EΔk(v,0)=k(v)Ek(v,0)vα−1. |
Hence we have
E(α)k(v,0)=k(v)Ek(v,0). |
Corollary 3.1. Suppose h∈Rα. Then Eh(u,0) is a solution of the following Cauchy problem:
x(α)(u)=h(u)x(u), x(0)=1. | (3.2) |
Proof. Let x(⋅)=Eh(⋅,0) be a solution of Eq (3.2). First note that
x(0)=Eh(0,0)=1. |
It remains to show that Eh(u,0) satisfies Eq (3.2). By Theorem 3.3, it follows that
(Eh(u,0))(α)=h(u)Eh(u,0). |
Therefore,
(x(u))(α)=h(u)x(u). |
Hence Eh(u,0) is a solution to the Cauchy problem (3.2).
Corollary 3.2. Suppose h∈Rα. Then Eh(v,0) is the unique solution of the IVP (3.2).
Proof. Suppose x(⋅) be any solution of the IVP (3.2). Then by using part (3) of Lemma 2.3, it follows that
(x(v)Eh(v,0))(α)=x(α)(v)Eh(v,0)−x(v)E(α)h(v,0)Eh(σ(v),0)Eh(v,0). |
According to our assumption and Theorem 3.3, it implies that
(x(v)Eh(v,0))(α)=h(v)x(v)Eh(v,0)−x(v)h(v)Eh(v,0)Eh(σ(v),0)Eh(v,0)=0. |
Consequently, x(v)=bEh(v,0), where b is a constant. Thus,
1=x(0)=bEh(0,0)=b. |
Hence, x(v)=Eh(v,0) is the unique solution.
Theorem 3.4. Suppose h∈Rα. Then
Eh(σ(u),0)=(1+μ(u)h(u)uα−1)Eh(u,0). |
Proof. Since for right-dense points (σ(u)=u), the case is trivial. And for right-scattered points (σ(u)>u), by Lemma 2.2, we have
Eh(σ(u),0)−Eh(u,0)=E(α)h(u,0)μ(u)uα−1. |
Theorem 3.3 implies that
Eh(σ(u),0)=(1+μ(u)h(u)uα−1)Eh(u,0), |
which proves the desired result.
Theorem 3.5. If l∈Rα, then
El(0,u)=1El(u,0)=E⊖αl(u,0). |
Proof. Let us consider the following IVP:
x(α)(u)=⊖αl(u)x(u), x(0)=1. |
First note that
x(0)=E⊖αl(0,0)=1. |
Differentiating
(x)(α)(u)=(1El(u,0))(α)=(1El(u,0))Δu1−α. |
By part (2) of Lemma 2.3, Theorems 3.3, 3.4 and Definition 3.3 we obtain
(E⊖αl(u,0))(α)=(−EΔl(u,0)El(σ(u),0)El(u,0))u1−α=(⊖αl)(u)E⊖αl(u,0). |
Hence,
(x)(α)(u)=(⊖αl)(u)x(u), |
which proves the required result
1El(u,0)=E⊖αl(u,0). |
Theorem 3.6. If h,p∈Rα, then
Eh(ω,0)Ep(ω,0)=Eh⊕αp(ω,0). |
Proof. Let us consider the IVP:
(x)(α)(ω)=(h⊕αp)(ω)x(ω), x(0)=1. |
We show that its solution is Eh(ω,0)Ep(ω,0). We have
x(0)=Eh(0,0)Ep(0,0)=1. |
Now by using part (1) of Lemma 2.3 it implies that
(x)(α)(ω)=(Eh(ω,0)Ep(ω,0))(α)=(Eh(ω,0))(α)Ep(σ(ω),0)+Eh(ω,0)(Ep(ω,0))(α). |
By Theorems 3.3, 3.4 and Definition 3.2, we have
(x)(α)(ω)={h(ω)[1+μ(ω)p(ω)ωα−1]+p(ω)}Eh(ω,0)Ep(ω,0)=(h⊕αp)(ω)Eh(ω,0)Ep(ω,0). |
Therefore,
(x)(α)(ω)=(h⊕αp)(ω)x(ω), |
which proves the desired result.
Theorem 3.7. Assume h,p∈Rα. Then
Eh(ω,0)Ep(ω,0)=Eh⊖αp(ω,0). |
Proof. By using Theorems 3.5 and 3.6, we have
Eh(ω,0)Ep(ω,0)=Eh⊕α(⊖αp)(ω,0)=Eh⊖αp(ω,0). |
By using Definitions 3.2–3.4, we have
(h⊕α(⊖αp))(ω)=h(ω)+(⊖αp)(ω)+μ(ω)h(ω)(⊖αp)(ω)=(h⊖αp)(ω), |
which proves the required result.
Theorem 3.8. Suppose h∈Rα. Then
(1Eh(v,0))(α)=−h(v)Eh(σ(v),0). |
Proof. By taking α-conformable fractional derivative on time scales, we have
(1Eh(v,0))(α)=(1Eh(v,0))Δv1−α. |
According to Theorems 3.3 and 3.5, it follows that
(1Eh(v,0))(α)=(⊖αh)E⊖αh(v,0). |
By using Definition 3.3 and Theorems 3.4, 3.5, we can obtain
(1Eh(v,0))(α)=−h(v)Eh(σ(v),0), |
which proves the required result.
Theorem 3.9. Let h,l∈Rα. Then
E(α)h⊖αl(v,0)=[h(v)−l(v)]Eh(v,0)El(σ(v),0). |
Proof. By taking α-conformable fractional derivative on time scales, we have
E(α)h⊖αl(v,0)=[EΔh⊖αl(v,0)]v1−α. |
Theorem 3.7 and using part (3) of Lemma 2.3 implies that
E(α)h⊖αl(v,0)=[EΔh(v,0)El(v,0)−Eh(v,0)EΔl(v,0)El(v,0)El(σ(v),0)]v1−α. |
By applying Theorem 3.3, we obtain
E(α)h⊖αl(v,0)=(h(v)−l(v))Eh(v,0)El(σ(v),0), |
which proves the required result.
Theorem 3.10. Suppose h∈Rα and c,d,0∈T.Then
(Eh(0,v))(α)=−h(v)Eh(0,σ(v)) |
and
∫dch(v)Eh(0,σ(v))Δαv=Eh(0,c)−Eh(0,d). |
Proof. Firstly, according to Theorems 3.4 and 3.5, we obtain
h(v)Eh(0,σ(v))=h(v)[1+μ(v)(⊖αh)(v)vα−1]E⊖αh(v,0). |
According to Definition 3.3, Theorems 3.3 and 3.5, it follows that
E(α)h(0,v)=−h(v)Eh(0,σ(v)), |
which proves our first part.
Now, taking both sides the α-conformable fractional integral, we have
∫dch(v)Eh(0,σ(v))Δαv=Eh(0,c)−Eh(0,d), |
which gives the desired result.
Theorem 3.11. Let k∈Rα. Then
E⊖αk(σ(v),u)=(1+(⊖αk)(v)μ(v)vα−1)E⊖αk(v,u). |
Proof. By Lemma 2.2, it implies that
E⊖αk(σ(v),u)−E⊖αk(v,u)=E(α)⊖αk(v,u)μ(v)vα−1. |
From Theorem 3.3, we obtain
E⊖αk(σ(v),u)=[1+(⊖αk)(v)μ(v)vα−1]E⊖αk(v,u). |
Similarly, we find
Ek(u,σ(v))=[1+k(v)μ(v)vα−1]Ek(u,v). |
Theorem 3.12. Let k∈Rα. Then
(Ek(η,ω))(α)=−k(ω)Ek(η,σ(ω)) |
and
∫η0k(η)Ek(η,σ(ω))Δαω=Ek(η,0)−1. |
Proof. From Theorems 3.5 and 3.11, we obtain
k(ω)Ek(η,σ(ω))=k(ω)(1+⊖αk(ω)μ(ω)ωα−1)E⊖αk(ω,η). |
Definition 3.3, Theorems 3.3 and 3.5 implies that
E(α)k(η,ω)=−k(ω)Ek(η,σ(ω)), |
which proves our first part.
Therefore, conformable α-fractional integration on time scales implies
∫η0k(η)Ek(η,σ(ω))Δαω=Ek(η,0)−1, |
which proves the required result.
Theorem 4.1. Let y,f∈Crd and p∈Rα+. Then
(y)(α)(t)≤p(t)y(t)+f(t),for all t∈Tk, |
implies
y(t)≤y(0)Ep(t,0)+∫t0Ep(t,σ(τ))f(τ)Δατ, ∀t∈T. |
Proof. By using part (1) of Lemma 2.3, Theorems 3.3 and 3.4, we have
[yE⊖αp(t,0)](α)(t)=y(α)(t)E⊖αp(σ(t),0)+((⊖αp)(t)1+μ(t)(⊖αp)(t)tα−1)y(t)E⊖αp(σ(t),0). |
Definition 3.3 implies that
[yE⊖αp(t,0)](α)(t)=[y(α)(t)−p(t)y(t)]E⊖αp(σ(t),0). |
Now taking the α-conformable fractional integral on time scales, and by using Lemma 2.4, it implies that
y(t)E⊖αp(t,0)−y(0)E⊖αp(0,0)=∫t0[y(α)(τ)−p(τ)y(τ)]E⊖αp(σ(τ),0)Δατ. |
By given assumption and using Theorem 3.5, we obtain
y(t)≤y(0)Ep(t,0)+∫t0f(τ)E⊖αp(σ(τ),0)E⊖αp(t,0)Δατ. |
And hence the assertion follows by applying Theorem 3.2:
y(t)≤y(0)Ep(t,0)+∫t0f(τ)Ep(t,σ(τ))Δατ, |
which proves the required result.
Theorem 4.2. Let y,f∈Crd and p∈Rα+, p≥0. Then
y(t)≤f(t)+∫t0y(τ)p(τ)Δατ, ∀t∈T, |
implies that
y(t)≤f(t)+∫t0Ep(t,σ(τ))f(τ)p(τ)Δατ, ∀t∈T. |
Proof. Define
z(t):=∫t0y(τ)p(τ)Δατ, ∀ t∈T. |
Then z(0)=0 and
y(t)≤f(t)+z(t). | (4.1) |
By using Lemma 2.5 and Eq (4.1), we obtain
z(α)(t)=y(t)p(t)≤f(t)p(t)+p(t)z(t). |
Theorem 4.1 yields
z(t)≤∫t0Ep(t,σ(τ))p(τ)f(τ)Δατ. |
And hence the claim follows because of Eq (4.1). Therefore,
y(t)≤f(t)+∫t0Ep(t,σ(τ))f(τ)p(τ)Δατ, |
which completes the proof.
Theorem 4.3. (Grönwall's inequality) Let y,p∈Crd, p∈Rα+and λ≥0 such that
y(t)≤λ+∫t0y(τ)p(τ)Δατ, ∀t∈[0,b]T, |
then
y(t)≤λEp(t,0), ∀t∈[0,b]T. | (4.2) |
Proof. Let f(t)=λ. Then by Theorem 4.2, it follows that
y(t)≤λ[1+∫t0Ep(t,σ(τ))p(τ)Δατ]. |
From Theorem 3.12, we obtain
y(t)=λ[1+Ep(t,0)−Ep(t,t)]=λEp(t,0). |
Therefore,
y(t)≤λEp(t,0), |
which completes the proof.
Remark 4.1. For α=1 and T=R, we obtained the Grönwall's inequalities in classical calculus.
We will develop some conditions for the global existence, extension and boundedness of solutions related to the local initial value problem (LIVP) in the following section.
The following assumptions will be needed throughout the following section.
Suppose Ω=[0,∞)T×R.
(H1) The mapping k:Ω→R is right-dense continuous.
(H2) A positive constant K>0 exists in such manner that, for all (τ,x), (τ,x) in Ω,
|k(τ,x)−k(τ,y)|≤K|x−y|. |
(H3) A nonnegative mapping l≥0 exists in such manner that, for all (τ,x) in Ω,
|k(τ,x)|≤l(τ)|x|, |
for which
∫τ0ξμ(r)(l(r))Δαr |
is bounded on [0,∞)T.
(H4) A nonnegative mapping h≥0 and a positive constant K>0 exists in such manner that, for all (τ,x), (τ,y) in Ω,
|k(τ,x)−k(τ,y)|≤h(τ)|x−y|≤K|x−y|, |
for which
∫τ0ξμ(r)(h(r))Δαr |
is bounded on [0,∞)T.
Through Lemmas 2.4 and 2.5, the LIVP (1.1) and (1.2) is simply transformed into an Integral Equation (IE).
Lemma 5.1. If (H1) holds, then a function ψ in C([0,b]T) is a solution of local initial value problem (1.1) and (1.2) if and only if ψ is acontinuous solution of the following integral equation:
ψ(τ)=ψ0+∫τ0k(r,ψ(r))Δαr, τ∈[0,a]T. | (5.1) |
We can now demonstrate the existence and uniqueness of the solution to the LIVP (1.1) and (1.2) as a consequence of Definition 3.6.
Theorem 5.1. The local initial value problem (1.1) and (1.2)has unique solution defined on [0,a]T wheneverthe assumptions (H1) and (H2) hold.
Proof. The claim will be verified via Banach's contraction principle on C([0,a]T). Let k>0 be a constant and k∈Rα+ and let ‖⋅‖ denote the Euclidean norm on Rn. Define the interval [0,a]T. Let us denote by C([0,a]T) the space of continuous functions along with a suitable norm. Define the term "TZ-norm"
‖ψ‖k=supτ∈[0,σ(a)]T‖ψ(τ)‖Ek(τ,0), |
where Ek(τ,0) in Defintion 3.6. The well-known sup-norm
‖ψ‖0=supτ∈[0,σ(a)]T‖ψ(τ)‖. |
It is simple to prove that ‖⋅‖k is equivalent to ‖⋅‖0. Hence (C([0,a]T),‖⋅‖k) is Banach space.
Define an operator
T:(C([0,a]T),‖⋅‖k)⟶(C([0,a]T),‖⋅‖k) |
as
T(ψ(τ))=ψ(0)+∫τ0k(r,ψ(r))Δαr. |
Lemma 5.1 assures that the fixed points of the operator T are the solutions of local IVP (1.1) and (1.2).
For any ψ,φ∈(C([0,a]T),‖⋅‖k), then
‖T(ψ)−T(φ)‖k=supτ∈[0,σ(a)]T‖T(ψ(τ))−T(φ(τ))‖Ek(τ,0)≤supτ∈[0,σ(a)]T[1Ek(τ,0)∫τ0‖k(r,ψ(r))−k(r,φ(r))‖Δαr]. |
By (H2), it follows that
‖T(ψ)−T(φ)‖k≤supτ∈[0,σ(a)]T[1Ek(τ,0)∫τ0K‖ψ(r)−φ(r)‖Δαr]=Ksupτ∈[0,σ(a)]T[1Ek(τ,0)∫τ0Ek(r,0)‖ψ(r)−φ(r)‖Ek(r,0)Δαr]≤Ksupr∈[0,σ(a)]T‖ψ(r)−φ(r)‖Ek(r,0)×supτ∈[0,σ(a)]T[1Ek(τ,0)∫τ0Ek(r,0)Δαr]=K‖ψ−φ‖ksupτ∈[0,σ(a)]T[1Ek(τ,0)∫τ0Ek(r,0)Δαr]. |
Thus, it implies that
‖T(ψ)−T(φ)‖k<K‖ψ−φ‖ksupτ∈[0,σ(a)]T[1Ek(τ,0)∫τ0Ek(r,0)Δαr]. | (5.2) |
Now we have to find
∫τ0Ek(r,0)Δαr. |
By Theorem 3.3, it implies that
Ek(r,0)=1kE(α)k(r,0), |
where k>0 be a positive constant. Then by taking α-conformable fractional integral, it follows that
∫τ0Ek(r,0)Δαr=∫τ01kE(α)k(r,0)Δαr. |
By using Lemma 2.4, we obtain
∫τ0Ek(r,0)Δαr=1k[Ek(τ,0)−1]. | (5.3) |
Now, using Eq (5.3) in Eq (5.2), it follows that
‖T(ψ)−T(φ)‖k<K‖ψ−φ‖ksupτ∈[0,σ(a)]T[1kEk(τ,0)[Ek(τ,0)−1]]<Kk‖ψ−φ‖ksupτ∈[0,σ(a)]T[1−1Ek(τ,0)]<Kk‖ψ−φ‖ksupτ∈[0,σ(a)]T[1−1Ek(σ(a),0)]. |
Therefore,
‖T(ψ)−T(φ)‖k<Kk‖ψ−φ‖k. |
As 0<Kk<1, we observe that T is a contractive map and the Banach contraction principle assures that there exists only one solution ψ in C([0,a]T) such that T(ψ)=ψ, and therefore the LIVP (1.1) and (1.2) has unique ψ in C([0,a]T). This completes the proof.
Next, we investigate the expansion to the right of the solutions of LIVP (1.1) and (1.2).
Lemma 5.2. Suppose ψ(τ) is a solution to thelocal IVP (1.1) and (1.2) defined on [0,τ+)T such that τ+≠∞. If limτ→τ+ψ(τ) exists, then ψ(τ) can be expanded to [0,τ+]T provided the hypothesis (H1) holds.
Proof. Here, τ+ is a right-dense point. Let limτ⟶τ+ψ(τ)=ψ+. Now suppose J=[0,τ+)T and define a function ˜ψ(τ) by
˜ψ(τ)={ψ(τ),τ∈[0,τ+)T,ψ+,τ=τ+. |
By [6,part (ⅰ) of Theorem 4] and since limτ⟶τ+ψ(τ)=ψ+, therefore the function ˜ψ(τ) is surely continuous on [0,τ+]T. We next demonstrate that the function ˜ψ(τ) is also a solution of the LIVP (1.1) and (1.2) defined on [0,τ+]T, and obviously, it is sufficient to prove
˜ψ(α)(τ+)=k(τ+,˜ψ(τ+)). |
Note that the equation
˜ψ(α)(τ)=k(τ,˜ψ(τ)), τ∈[0,τ+)T. |
And the continuities of ˜ψ and k gives that
limτ→τ+˜ψ(α)(τ)=k(τ+,˜ψ(τ+)). | (5.4) |
Moreover, using mean value theorem [23,Theorem 15], we see that for all τ in [0,τ+)T, ∃ a point ζ in [τ,τ+]kT such that
˜ψ(α)(ζ)=˜ψ(τ+)−˜ψ(τ)τ+−τζ1−α, ζ∈[τ,τ+]kT. |
Now taking the limτ→τ+ on both sides and using Eq (5.4), we obtain
k(τ+,˜ψ(τ+))=(limτ→τ+˜ψ(τ+)−˜ψ(τ)τ+−τ)(limτ→τ+τ1−α)=˜ψΔ(τ+)(τ+)1−α. |
By Lemma 2.2, we conclude that
˜ψ(α)(τ+)=k(τ+,˜ψ(τ+)). |
Hence, we have shown that the function ˜ψ(τ) is also a solution of the LIVP (1.1) and (1.2) defined on [0,τ+]T, and it is an extension of the solution ψ(τ) to [0,τ+]T. Therefore, the required result follows.
Definition 5.1. Suppose I is the maximal existence interval of the solution ψ(τ) of the LIVP (1.1) and (1.2), then ψ(τ) is said to be come arbitrarily close to the boundary of Ω=[0,∞)T×R to the right if it is not possible for every closed and bounded domain Ω0 in Ω, the point (τ,ψ(τ)) always remains in Ω0 for all τ in I.
Theorem 5.2. If (H1) and (H2) hold, then the solution of the local initial value problem (1.1) and (1.2) comes arbitrarily close to the boundary of Ω=[0,∞)T×R to the right.
Proof. The local IVP (1.1) and (1.2) has a unique solution by Theorem 5.1, and refers to the solution by ψ(τ). Suppose \boldsymbol{I} refers to the maximal existence interval of \psi\left(\tau\right) . Again, we conclude that using Theorem 5.1, {\bf{I}} = \left[0, \infty\right) _{\mathbb{T}} or \left[0, \tau^{+}\right) _{\mathbb{T}} with \tau^{+}\neq\infty . The required result is clear when {\bf{I}} = \left[0, \infty\right) _{\mathbb{T}} . Now, assume the case {\bf{I}} = \left[0{\bf{, }}\tau ^{+}\right) _{\mathbb{T}} with \tau^{+}\neq\infty . Conversely, suppose that the desired result is not true. That is, the solution \psi\left(\tau\right) of the LIVP (1.1) and (1.2) does not go arbitrarily near to the boundary of \Omega = \left[0, \infty\right) _{\mathbb{T}}\times\mathbb{R} to the right implies that \Omega_{0} \subset\Omega exists such that \Omega_{0} is closed and bounded with \left(\tau, \psi\left(\tau\right) \right) \in\Omega_{0} , \forall \tau\in{\bf{I}} . Because k on \Omega_{0}\subset\Omega is continuous, a positive number C must exist such that
\begin{equation} \begin{array} [c]{c} \left\vert k\left( \tau, \psi\left( \tau\right) \right) \right\vert \leq C\ {\rm{ for\ all }}\ \tau\in{\bf{I}}{\rm{.}} \end{array} \end{equation} | (5.5) |
Furthermore, mean value theorem [23] ensures that, for any \tau_{1}, \tau_{2} in {\bf{I}} with \tau_{1} < \tau_{2} , a point \xi exists in \left[\tau_{1}, \tau_{2}\right] _{\mathbb{T}}^{k} such that
\begin{array} [c]{c} \psi\left( \tau_{2}\right) -\psi\left( \tau_{1}\right) = \left[ \xi^{\alpha-1}\left( k\right) ^{\left( \alpha\right) }\left( \xi\right) \right] \left( \tau_{2}-\tau_{1}\right) . \end{array} |
From Eqs (1.1) and (5.5), it follows that
\begin{array} [c]{ll} \left\vert \psi\left( \tau_{2}\right) -\psi\left( \tau_{1}\right) \right\vert & = \dfrac{\left\vert k\left( \xi, \psi\left( \xi\right) \right) \right\vert }{\xi^{1-\alpha}}\left[ \left\vert \tau_{2}-\tau _{1}\right\vert \right] \\ & \leq\dfrac{C}{\xi^{1-\alpha}}\left[ \left\vert \tau_{2}-\tau_{1}\right\vert \right] . \end{array} |
Therefore,
\begin{array} [c]{l} \left\vert \psi\left( \tau_{2}\right) -\psi\left( \tau_{1}\right) \right\vert < \epsilon\ {\rm{ whenever }}\ \left\vert \tau_{2}-\tau_{1}\right\vert < \dfrac{\epsilon\xi^{1-\alpha}}{C} = \delta, \\ \left\vert \psi\left( \tau_{2}\right) -\psi\left( \tau_{1}\right) \right\vert < \epsilon\ {\rm{ whenever }}\ \left\vert \tau_{2}-\tau_{1}\right\vert < \delta, \end{array} |
which shows that \psi\left(\tau\right) is uniformly continuous on {\bf{I}} , and hence \lim\limits_{\tau\longrightarrow\tau^{+}}\psi\left(\tau\right) exists. And thus by using Lemma 5.2, the solution \psi\left(\tau\right) can be extended to the closed interval \left[0, \tau^{+}\right] _{\mathbb{T}} , it violates the statement that \left[0, \tau^{+}\right) _{\mathbb{T}} is the maximal existence interval of \psi\left(\tau\right) . Therefore, it follows the required result.
By the use of Theorems 5.1, 5.2 and Grönwall's inequality (4.3), we now present a result to ensure that the solution of Eq (1.1) with the local initial condition (1.2) is defined and bounded on \left[0, \infty\right) _{\mathbb{T}} .
Theorem 5.3. The solution of the local IVP (1.1) and (1.2) is defined and bounded on \left[0, \infty\right)_{\mathbb{T}} whenever (H1)–(H3) hold.
Proof. The LIVP has only one solution, according to Theorem 5.1. Define the solution as \psi\left(\tau\right) , with \left[0, \tau^{+}\right) _{\mathbb{T}} as its maximal existence interval. Now we have to show that \tau^{+} = \infty and \psi\left(\tau\right) is bounded on \left[0, \tau^{+}\right) _{\mathbb{T}} . Based on assumption (H3), Eq (5.1) follows that
\begin{equation} \begin{array} [c]{c} \left\vert \psi\left( \tau\right) \right\vert \leq\left\vert \psi\left( 0\right) \right\vert +I_{\alpha}\left[ k\left( \tau\right) \left\vert \psi\left( \tau\right) \right\vert \right] . \end{array} \end{equation} | (5.6) |
And so, Eq (5.6) follows using Grönwall's inequality (4.3):
\begin{array} [c]{ll} \left\vert \psi\left( \tau\right) \right\vert & \leq\left\vert \psi _{0}\right\vert E_{k}\left( \tau, 0\right) , {\rm{ }}\ \forall{\rm{ }}\ \tau \in\left[ 0, \tau^{+}\right) _{\mathbb{T}}, \\ & \leq\left\vert \psi_{0}\right\vert \left[ \exp\left( { \int_{0}^{\tau}} \xi_{\mu\left( \nu\right) }\left( k\left( \nu\right) \right) v^{\alpha-1}\Delta\nu\right) \right] . \end{array} |
It should be noted that the hypothesis of boundedness of { \int_{0}^{\tau}} \xi_{\mu\left(\nu\right) }\left(k\left(\nu\right) \right) \nu ^{\alpha-1}\Delta\nu would imply that a positive number C exists so that { \int_{0}^{\tau}} \xi_{\mu\left(\nu\right) }\left(k\left(\nu\right) \right) \nu ^{\alpha-1}\Delta\nu\leq C . Therefore,
\begin{array} [c]{c} \left\vert \psi\left( \tau\right) \right\vert \leq\left\vert \psi _{0}\right\vert \left[ e^{C}\right] {\rm{, }}\ \forall{\rm{ }}\ \tau\in\left[ 0, \tau^{+}\right) _{\mathbb{T}}. \end{array} |
Thus \psi\left(\tau\right) is bounded on \left[0, \tau^{+}\right) _{\mathbb{T}} . When \tau^{+}\neq\infty , then by Theorem 5.2 clearly follows that
\begin{array} [c]{c} \lim\limits_{\tau\rightarrow \tau^{+}}\psi\left( \tau\right) = \infty. \end{array} |
This is in contradiction with the boundedness of \psi\left(\tau\right) on \left[0, \tau^{+}\right) _{\mathbb{T}} . Therefore, \tau^{+} = \infty , and consequently follows the required result.
Through the Grönwall's inequality (4.3), we are further investigating the stability of the solutions to the LIVP (1.1) and (1.2).
Definition 5.2. Assume \psi\left(\tau\right) be a solution to Eq (1.1) defined on \left[0, \infty\right) _{\mathbb{T}} with \psi\left(0\right) = \psi_{0} . The solution \psi\left(\tau\right) is called stable if for all positive number \epsilon , \exists a positive number \delta such that each solution \varphi\left(\tau\right) with \left\vert \varphi\left(0\right) -\psi\left(0\right) \right\vert < \delta holds for any \tau\geq0 and satisfies the inequality
\begin{array} [c]{c} \left\vert \varphi\left( \tau\right) -\psi\left( \tau\right) \right\vert < \epsilon{, \ for \ }\tau\geq0. \end{array} |
Theorem 5.4. Every solution of the LIVP (1.1) and (1.2) is always stable whenever the hypothesis (H1), (H3) and (H4) holds.
Proof. By Theorem 5.3, the solution of LIVP (1.1) and (1.2) always exists and is defined on \left[0, \infty\right)_{\mathbb{T}} . Let \psi\left(\tau\right) be a solution with \psi\left(0\right) = \psi_{0} and \varphi\left(\tau\right) asolution with \varphi\left(0\right) = \varphi_{0} . Then
\begin{array} [c]{c} \psi\left( \tau\right) = \psi_{0}+I_{\alpha}k\left( \tau, \psi\left( \tau\right) \right) \end{array} |
and
\begin{array} [c]{c} \varphi\left( \tau\right) = \varphi\left( 0\right) +I_{\alpha}k\left( \tau, \varphi\left( \tau\right) \right) . \end{array} |
By (H4), it follows that
\begin{equation} \begin{array} [c]{c} \left\vert \varphi\left( \tau\right) -\psi\left( \tau\right) \right\vert \leq\left\vert \varphi_{0}-\psi_{0}\right\vert +I_{\alpha}\left[ h\left( \tau\right) \left\vert \varphi\left( \tau\right) -\psi\left( \tau\right) \right\vert \right] . \end{array} \end{equation} | (5.7) |
And Eq (5.7) follows, using Grönwall's inequality(4.3) and the hypothesis of boundedness of { \int_{0}^{\tau}}\xi_{\mu\left(\nu\right) }\left(h\left(\nu\right) \right)\Delta^{\alpha}\nu would imply that a positive number C exists so that { \int_{0}^{\tau}}\xi_{\mu\left(\upsilon\right) }\left(h\left(\upsilon\right) \right)\Delta^{\alpha}\upsilon\leq C . Therefore,
\begin{array} [c]{c} \left\vert \varphi\left( \tau\right) -\psi\left( \tau\right) \right\vert \leq\left\vert \varphi_{0}-\psi_{0}\right\vert e^{C}\mathit{\rm{, }}\ \forall\mathit{\rm{}}\tau\in\left[ 0, \infty\right) _{\mathbb{T}}. \end{array} |
By Definition 5.2, it follows that
\begin{array} [c]{c} \left\vert \varphi\left( \tau\right) -\psi\left( \tau\right) \right\vert < \epsilon. \end{array} |
Hence \psi\left(\tau\right) is stable on \left[0, \infty\right)_{\mathbb{T}} .
The existence of solutions to the non-local initial value problem (NLIVP) is addressed in this section. Then, in order to prove the main result, we recall a fixed point theorem that will be used in this section.
Lemma 6.1. (Fixed point theorem) U is an open set in a Banach space B 's closed, convex set C . Suppose 0\in U . It is also assumed that A\left(\bar{U}\right) is bounded and that A:\bar{U}\rightarrow C isgiven by
\begin{array} [c]{c} A = A_{1}+A_{2}, \end{array} |
in which
\begin{array} [c]{c} A_{1}:\bar{U}\mathbb{\rightarrow }B\ {{is\ completely\ continuous}}, \end{array} |
and
\begin{array} [c]{c} A_{2}:\bar{U}\mathbb{\rightarrow }B\ {{is\ a\ nonlinear\ contraction.}} \end{array} |
That is, a nonnegative nondecreasing function \phi:\left[0, \infty\right)_{\mathbb{T}}\mathbb{\rightarrow }\left[0, \infty\right) _{\mathbb{T}} exists satisfying \phi\left(z\right) < z for z > 0 , such that
\begin{array} [c]{c} \left\Vert A_{2}\left( x\right) -A_{2}\left( y\right) \right\Vert _{k} \leq\phi\left( \left\Vert x-y\right\Vert _{k}\right) , \end{array} |
for any x, y\in\bar{U} . Then either (C1) A has a fixed point u\in\bar{U} ; or (C2) there exist a point u\in\partial U and \lambda\in\left(0, 1\right) such that u = \lambda A\left(u\right) , where \bar{U} and \partial U refers to the closure and boundary of U , respectively.
Further we need the following hypothesis.
(H5) k is a right-dense continuous function defined on \left[0, a\right] _{\mathbb{T}}\times\mathbb{R}.
(H6) There exists a positive constant \gamma in \left(0, 1\right) and a nonnegative and nondecreasing function \phi in C\left(\left[0, \infty\right) _{\mathbb{T}}\right) with \phi\left(\varsigma\right) < \gamma\varsigma , \varsigma > 0 and \left\vert g\left(\tau\right) -g\left(\nu\right) \right\vert _{k}\leq\phi\left(\left\Vert \tau -\nu\right\Vert _{k}\right) for all \tau, \nu in C\left(\left[0, a\right] _{\mathbb{T}}\right) .
(H7) There is a nonnegative function \varphi\in C\left(\left[0, a\right] _{\mathbb{T}}\right) such that \varphi > 0 on a subinterval of \left[0, a\right] _{\mathbb{T}} , as well as a nonnegative and nondecreasing function \Psi\in C\left(\left[0, \infty\right) _{\mathbb{T}}\right) ,
\begin{array} [c]{c} \left\vert k\left( t, u\right) \right\vert \leq\varphi\left( t\right) \Psi\left( \left\vert u\right\vert \right) {, } \end{array} |
for each \left(t, u\right) in \left[0, a\right] _{\mathbb{T}} \times\mathbb{R} and
\begin{array} [c]{c} \sup\limits_{r\in\left( 0, \infty\right) }\dfrac{r}{\left\vert \psi _{0}\right\vert +\Psi\left( r\right) I_{\alpha}\varphi\left( a\right) } > \dfrac{1}{1-\gamma}. \end{array} |
The following lemma is easy to verify by Lemmas 2.4 and 2.5.
Lemma 6.2. A function \psi in C\left(\left[0, a\right]_{\mathbb{T}}\right) satisfies the NLIVP (1.1) and (1.3)provided that assumption (H5) holds iff \psi is a continuous as well assolution of IE:
\begin{equation} \begin{array} [c]{c} \psi\left( \tau\right) = \psi_{0}-g\left( \psi\right) +I_{\alpha}k\left( \tau, \psi\left( \tau\right) \right) , \mathit{\rm{}}\tau\in\left[ 0, a\right] _{\mathbb{T}}. \end{array} \end{equation} | (6.1) |
We first define some sets of functions in C\left(\left[0, a\right] _{\mathbb{T}}\right) and operators in order to use FPT to address the existence of solutions to the NLIVP.
Given a positive number r and k > 0 be a positive constant, define the subset {\large u}_{r} of C\left(\left[0, a\right] _{\mathbb{T}}\right) by
\begin{equation} \begin{array} [c]{c} {\Large u}_{r} = \left\{ u\in C\left( \left[ 0, a\right] _{\mathbb{T} }\right) :\left\Vert u\right\Vert _{k} < r\right\} . \end{array} \end{equation} | (6.2) |
Similarly, we define
\begin{equation} \begin{array} [c]{c} {\Large \bar{u}}_{r} = \left\{ u\in C\left( \left[ 0, a\right] _{\mathbb{T} }\right) :\left\Vert u\right\Vert _{k}\leq r\right\} . \end{array} \end{equation} | (6.3) |
Also, define three operators from the space C\left(\left[0, a\right] _{\mathbb{T}}\right) to itself, respectively, by
\begin{equation} \begin{array} [c]{c} A_{1}\psi\left( t\right) = I_{\alpha}k\left( t, \psi\left( t\right) \right) , \end{array} \end{equation} | (6.4) |
\begin{equation} \begin{array} [c]{c} A_{2}\psi\left( t\right) = \psi_{0}-g\left( \psi\right) , \end{array} \end{equation} | (6.5) |
\begin{equation} \begin{array} [c]{c} A\psi\left( t\right) = A_{1}\psi\left( t\right) +A_{2}\psi\left( t\right) . \end{array} \end{equation} | (6.6) |
Using the standard arguments, the complete continuity of the operator A_{1}:{\large \bar{u}}_{r}\mapsto C\left(\left[0, a\right] _{\mathbb{T} }\right) can be verified, and it is also easy to check that the operator A_{2}:{\large \bar{u}}_{r}\mapsto C\left(\left[0, a\right] _{\mathbb{T} }\right) is a nonlinear contraction under the condition (H6). Here we omit their proofs.
Lemma 6.3. The operator A_{1}:{\large \bar{u}}_{r}\mapsto C\left(\left[0, a\right] _{\mathbb{T}}\right) is completely continuous provided(H5) holds.
Lemma 6.4. The operator A_{2}:{\large \bar{u}}_{r}\mapsto C\left(\left[0, a\right] _{\mathbb{T}}\right) is a nonlinear contractionprovided (H6) holds.
In this section, we are now presenting the main result.
Theorem 6.1. There exists at least one solution defined on \left[0, a\right] _{\mathbb{T}} of the NLIVP (1.1) and (1.3) whenever (H5)–(H7) holds.
Proof. A positive number r exists in view of the hypothesis of supremum in (H7), such that
\begin{equation} \begin{array} [c]{c} \dfrac{r}{\left\vert \psi_{0}\right\vert +\Psi\left( r\right) I_{\alpha }\varphi\left( a\right) } > \dfrac{1}{1-\gamma}. \end{array} \end{equation} | (6.7) |
And then we define the set {\Large u}_{r} by
\begin{array} [c]{c} {\Large u}_{r} = \left\{ u\in C\left( \left[ 0, a\right] _{\mathbb{T} }\right) :\left\Vert u\right\Vert _{k} < r\right\} . \end{array} |
We first show that the operators A , A_{1} , A_{2} satisfy the corresponding conditions of FPT and owing to Lemmas 6.3 and 6.4, we just need to demonstrate the ßoundedness of A\left({\Large \bar{u}}_{r}\right) . Infact, for each \psi in {\Large \bar{u} }_{r} it implies under the assumptions (H5) and (H6) that
\begin{equation} \begin{array} [c]{c} \left\vert A_{1}\psi\left( t\right) \right\vert \leq I_{\alpha}\left\vert k\left( t, \psi\left( t\right) \right) \right\vert \leq\dfrac{a^{\alpha} }{\alpha}\sup\left\{ \left\vert k\left( t, u\right) \right\vert :t\in\left[ 0, a\right] _{\mathbb{T}}{\rm{, }}\ \left\vert u\right\vert \leq r\right\} \end{array} \end{equation} | (6.8) |
and
\begin{equation} \begin{array} [c]{c} \left\vert A_{2}\psi\left( t\right) \right\vert = \left\vert \psi _{0}-g\left( \psi\right) \right\vert \leq\left\vert \psi_{0}\right\vert +\left\vert g\left( \psi\right) \right\vert . \end{array} \end{equation} | (6.9) |
According to Eq (6.3), it implies that
\begin{equation} \begin{array} [c]{c} \left\vert g\left( \psi\right) \right\vert \leq\phi\left( \left\Vert \psi\right\Vert _{k}\right) \leq\gamma\left\Vert \psi\right\Vert _{k} \leq\gamma r. \end{array} \end{equation} | (6.10) |
By using Eq (6.10) in Eq (6.9), it follows that
\begin{equation} \begin{array} [c]{c} \left\vert A_{2}\psi\left( t\right) \right\vert \leq\left\vert \psi _{0}\right\vert +\gamma r. \end{array} \end{equation} | (6.11) |
Hence, according the definition of the operator A and using Eqs (6.8) and (6.11), we have
\begin{array} [c]{c} \left\Vert A\psi\right\Vert _{k}\leq\left\vert \psi_{0}\right\vert +\gamma r+\dfrac{a^{\alpha}}{\alpha}\sup\left\{ \left\vert k\left( t, u\right) \right\vert :t\in\left[ 0, a\right] _{\mathbb{T}}{\rm{, }}\ \left\vert u\right\vert \leq r\right\}, \end{array} |
which justifies the uniform boundedness of the set A\left({\Large \bar{u} }_{r}\right) .
Lastly, it remains to be shown that the case (C2) does not occur in the FPT (6.1). We claim, by contradiction. Suppose (C2) holds implies that \lambda\in\left(0, 1\right) and \psi\in\partial{\Large u}_{r} exists with \psi = \lambda A\left(\psi\right) , that is,
\begin{array} [c]{c} \psi\left( t\right) = \lambda\left[ \psi_{0}-g\left( \psi\right) +I_{\alpha}k\left( t, \psi\left( t\right) \right) \right] . \end{array} |
Under the hypothesis (H6) and (H7), it further follows that
\begin{equation} \begin{array} [c]{c} \left\vert \psi\left( t\right) \right\vert \leq\lambda\left[ \left\vert \psi_{0}\right\vert +\left\vert g\left( \psi\right) \right\vert +I_{\alpha }\left\vert k\left( t, \psi\left( t\right) \right) \right\vert \right] . \end{array} \end{equation} | (6.12) |
By using Eq (6.3), it implies that
\begin{equation} \begin{array} [c]{c} \left\vert k\left( t, \psi\left( t\right) \right) \right\vert \leq \varphi\left( t\right) \Psi\left( \left\vert \psi\left( t\right) \right\vert \right) \leq\varphi\left( t\right) \Psi\left( r\right) . \end{array} \end{equation} | (6.13) |
Using Eqs (6.10) and (6.13) in Eq (6.12), it becomes
\begin{array} [c]{c} \left\vert \psi\left( t\right) \right\vert \leq\lambda\left[ \left\vert \psi_{0}\right\vert +\gamma r+I_{\alpha}\varphi\left( t\right) \Psi\left( r\right) \right] . \end{array} |
Hence,
\begin{array} [c]{c} \left\vert \psi\left( t\right) \right\vert \leq\left[ \left\vert \psi _{0}\right\vert +\gamma r+\Psi\left( r\right) I_{\alpha}\varphi\left( t\right) \right] . \end{array} |
But
\begin{array} [c]{c} r\leq\sup\limits_{t\in\left[ 0, a\right] _{\mathbb{T}}}\left[ \left\vert \psi_{0}\right\vert +\gamma r+\Psi\left( r\right) I_{\alpha}\varphi\left( t\right) \right] \leq\left\vert \psi_{0}\right\vert +\gamma r+\Psi\left( r\right) I_{\alpha}\varphi\left( a\right) . \end{array} |
This implies that
\begin{array} [c]{c} \dfrac{r}{\left\vert \psi_{0}\right\vert +\Psi\left( r\right) I_{\alpha }\varphi\left( a\right) }\leq\dfrac{1}{1-\gamma}, \end{array} |
which is in contradiction with inequality (6.7). We have therefore shown that the operators A, \; A_{1} and A_{2} meet all the conditions in FPT (6.1), and hence we deduce that the operator A has at least one fixed point \psi in {\Large \bar{u}}_{r} , which satisfies the NLIVP.
Remark 6.1 For \alpha = 1 and \mathbb{T = R} , the classical results corresponding to ordinary differential equations will be yielded.
Example 6.2 Assume \mathbb{T = R} , r_{0} = 0 and p\left(s\right) = -1 , then E_{p}\left(s, s_{0}\right) = e^{-\frac{s^{\alpha}}{\alpha}} . Also let \Omega{ = }\left[0, \infty\right) \times\mathbb{R} , k\left(s, x\right) = e^{-\frac{s^{\alpha}}{\alpha}}\left(x+\sin x\right) , l\left(s\right) = h\left(s\right) = 2e^{-\frac{s^{\alpha}}{\alpha}} and K = 2.
(I) Local initial value problems.
For all \left(s, x\right) , \left(s, y\right) \in\Omega ,
\begin{array} [c]{c} \left\vert k\left( s, x\right) -k\left( s, y\right) \right\vert = \left\vert e^{-\frac{s^{\alpha}}{\alpha}}\left( x+\sin x\right) -e^{-\frac{s^{\alpha} }{\alpha}}\left( y+\sin y\right) \right\vert \leq\left\vert e^{-\frac {s^{\alpha}}{\alpha}}\right\vert \left[ \left\vert x-y\right\vert +\left\vert \sin x-\sin y\right\vert \right] . \end{array} |
It is easy to see that \sin x is Lipschitz: \left\vert \sin x-\sin y\right\vert \leq\left\vert x-y\right\vert with Lipschitz constant L = 1 . It implies that
\begin{array} [c]{c} \left\vert k\left( s, x\right) -k\left( s, y\right) \right\vert \leq\left\vert e^{-\frac{s^{\alpha}}{\alpha}}\right\vert \left[ \left\vert x-y\right\vert +\left\vert x-y\right\vert \right] = 2\left\vert e^{-\frac {s^{\alpha}}{\alpha}}\right\vert \left\vert x-y\right\vert \leq\left( 1\right) \left[ 2\left\vert x-y\right\vert \right] . \end{array} |
So
\begin{array} [c]{c} \left\vert k\left( s, x\right) -k\left( s, y\right) \right\vert \leq K\left\vert x-y\right\vert, \end{array} |
and
\begin{array} [c]{c} \left\vert k\left( s, x\right) \right\vert = \left\vert e^{-\frac{s^{\alpha} }{\alpha}}\left( x+\sin x\right) \right\vert \leq\left\vert e^{-\frac {s^{\alpha}}{\alpha}}\right\vert \left( \left\vert x\right\vert +\left\vert \sin x\right\vert \right) . \end{array} |
Since \left\vert \sin x\right\vert \leq\left\vert x\right\vert , therefore,
\begin{array} [c]{c} \left\vert k\left( s, x\right) \right\vert \leq\left\vert e^{-\frac {s^{\alpha}}{\alpha}}\right\vert \left( \left\vert x\right\vert +\left\vert x\right\vert \right) = 2\left\vert e^{-\frac{s^{\alpha}}{\alpha}}\right\vert \left\vert x\right\vert = l\left( s\right) \left\vert x\right\vert . \end{array} |
Thus,
\begin{array} [c]{c} \left\vert k\left( s, x\right) \right\vert \leq l\left( s\right) \left\vert x\right\vert {\rm{.}} \end{array} |
Since \mathbb{T = R} , therefore \mu\left(s\right) = 0 and \xi_{\mu\left(s\right) }\left[l\left(s\right) \right] = l\left(s\right) , for which
\begin{array} [c]{c} { \int_{0}^{t}} \xi_{\mu\left( s\right) }\left[ l\left( s\right) \right] d^{\alpha}s = 2 { \int_{0}^{t}} e^{-\frac{s^{\alpha}}{\alpha}}d^{\alpha}s. \end{array} |
Since we know that
\begin{array} [c]{c} \left( e^{-\frac{s^{\alpha}}{\alpha}}\right) ^{\left( \alpha\right) } = \left( -1\right) e^{-\frac{s^{\alpha}}{\alpha}}. \end{array} |
Hence, we can write
\begin{array} [c]{c} { \int_{0}^{t}} \xi_{\mu\left( s\right) }\left[ l\left( s\right) \right] d^{\alpha}s = -2 { \int_{0}^{t}} \left( e^{-\frac{s^{\alpha}}{\alpha}}\right) ^{\left( \alpha\right) }d^{\alpha}s = 2-2e^{-\frac{t^{\alpha}}{\alpha}} = 2-l\left( t\right), \end{array} |
which implies that
\begin{array} [c]{c} { \int_{0}^{t}} \xi_{\mu\left( s\right) }\left[ l\left( s\right) \right] d^{\alpha }s = 2-l\left( t\right) \leq2. \end{array} |
Thus, for the above-mentioned functions and variables, hypotheses (H1)–(H3) in Theorem 5.3 are met, indicating that the solution to the LIVP (1.1) and (1.2) is defined and bounded on \left[0, \infty\right) _{\mathbb{T}} .
(II) Stabilities.
We observe that, analogous to the case (I), for each \left(s, x\right) , \left(s, y\right) \in\Omega ,
\begin{array} [c]{c} \left\vert k\left( s, x\right) -k\left( s, y\right) \right\vert \leq h\left( s\right) \left\vert x-y\right\vert \leq K\left\vert x-y\right\vert, \end{array} |
for which I_{\alpha}h\left(s\right) \leq2 . Hence, all the requirements are satisfied for Theorem 5.4. We can derive that each solution to the LIVP (1.1) and (1.2) is always stable using Theorem 5.4.
(III) Non-local initial value problems.
Select \left[0, a\right] such that 2 > e^{\frac{a^{\alpha}}{\alpha}} . Let us define the functions
\begin{array} [c]{c} \varphi\left( s\right) = 2e^{-\tfrac{s^{\alpha}}{\alpha}}, \; \Psi\left( \tau\right) = \tau\ { and }\ \phi\left( \tau\right) = \dfrac{\gamma}{2}\tau, \end{array} |
such that 0 < \gamma < \varphi\left(a\right) -1 . For u\in C\left(\left[0, a\right] _{\mathbb{T}}\right) , define the functional
\begin{array} [c]{c} g\left( u\right) = \dfrac{\gamma}{2a} { \int\limits_{0}^{a}} u\left( s\right) ds, \end{array} |
and then it's simple to determine whether g is a contraction:
\begin{array} [c]{c} \left\vert g\left( u\right) -g\left( v\right) \right\vert _{k}\leq \dfrac{\gamma}{2a} { \int\limits_{0}^{a}} \left\vert u\left( t\right) -v\left( t\right) \right\vert _{k}dt\leq \dfrac{\gamma}{2a} { \int\limits_{0}^{a}} \left\Vert u-v\right\Vert _{k}dt\leq\phi\left( \left\Vert u-v\right\Vert _{k}\right) . \end{array} |
Moreover, observe that
\begin{array} [c]{c} \left\vert k\left( s, x\right) \right\vert \leq\varphi\left( s\right) \Psi\left( \left\vert x\right\vert \right) , \end{array} |
for any \left(s, x\right) \in\left[0, a\right] \times\mathbb{R} as well as the fact that a direct calculation yields
\begin{array} [c]{c} \sup\limits_{t\in\left( 0, \infty\right) }\dfrac{t}{\left\vert u_{0} \right\vert +\Psi\left( t\right) I_{\alpha}\varphi\left( a\right) } = \sup\limits_{t\in\left( 0, \infty\right) }\dfrac{t}{tI_{\alpha} \varphi\left( a\right) } = \dfrac{1}{I_{\alpha}\varphi\left( a\right) } = \dfrac{1}{2-\varphi\left( a\right) }. \end{array} |
As
\begin{array} [c]{rr} \gamma < \varphi\left( a\right) -1, \\ \dfrac{1}{2-\varphi\left( a\right) } > \dfrac{1}{1-\gamma}. \end{array} |
Thus,
\begin{array} [c]{c} \sup\limits_{t\in\left( 0, \infty\right) }\dfrac{t}{\left\vert u_{0} \right\vert +\Psi\left( t\right) I_{\alpha}\varphi\left( a\right) } = \dfrac{1}{2-\varphi\left( a\right) } > \dfrac{1}{1-\gamma}. \end{array} |
We can deduce that the corresponding non-local IVP (1.1) and(1.3) has at least one solution defined on \left[0, a\right] because assumptions (H5)–(H7) in Theorem 6.1 are fulfilled for the aforementioned functions, functionals and parameters.
Remark 6.2. For \alpha = 1 and \mathbb{T = R} , the classical results corresponding to ordinary differential equations will be yielded.
Based on the theory of conformable fractional calculus on time scales, we defined generalized exponential function. Also, we proved some of its fundamental properties. Furthermore, we introduced Grönwall's type inequalities in the considered frame. Through Grönwall's inequality, we investigate the stability of the solution to the LIVP. In addition, some conditions for the global existence, extension and boundedness of LIVP's solutions as well as their stabilities are established by using conformable fractional calculus on time scales and FPT. Moreover, we obtained the existence result of the nonlocal initial value problem.
The author M. A. Alqudah was supported by Princess Nourah bint Abdulrahman University Researchers Supporting Project (No. PNURSP2022R14). The author T. Abdeljawad would like to thank Prince Sultan University for the support through the TAS research lab.
The authors declare no conflicts of interest regarding this article.
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