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Nonlocal boundary value problems for Hilfer-type pantograph fractional differential equations and inclusions
Advances in Difference Equations volume 2020, Article number: 279 (2020)
Abstract
In this paper, we study boundary value problems, involving the Hilfer fractional derivative, for pantograph fractional differential equations and inclusions supplemented by nonlocal integral boundary conditions. Existence and uniqueness results are obtained by using well-known fixed point theorems for single and multi-valued functions. Examples illustrating our results are also presented.
1 Introduction
In recent years, the theory of fractional differential equations has played a very important role in a new branch of applied mathematics, which has been utilized for mathematical models in engineering, physics, chemistry, signal analysis, etc. For details and applications we refer the reader to the classical reference texts such as [1–7]. Fractional differential equations are considered valuable tools to model many real world problems. Boundary value problems of differential equations represent an important class of applied analysis. Most of the researchers have given attention to study fractional differential equations by taking Caputo or Riemann–Liouville derivatives. Engineers and scientists have developed some new models that involve fractional differential equations for which the Riemann–Liouville derivative is not considered appropriate. Therefore certain modifications were introduced to avoid the difficulties and some new types of fractional order derivative operators were introduced in the literature by authors like Caputo, Hadamard, and Erdely-Kober. A generalization of derivatives of both Riemann–Liouville and Caputo was given by Hilfer in [8], known as the Hilfer fractional derivative of order α and a type \(\beta\in [0, 1]\), which can be reduced to the Riemann–Liouville and Caputo fractional derivatives when \(\beta=0\) and \(\beta=1\), respectively. Such derivative interpolates between the Riemann–Liouville and Caputo derivative. Fractional differential equations involving the Hilfer derivative have many applications; see [9–12] and the references cited therein.
An important class of differential equations containing proportional delays are called pantograph equations. This important class was named after Ockendon and Tayler [13]. Numbers of applications have been studied by many researchers of these equations in applied sciences including biology, physics, economics, and electrodynamics. For more details as regards the aforesaid equations, we refer to [14, 15].
Initial value problems involving Hilfer fractional derivatives were studied by several authors; see for example [16–18] and the references therein. Nonlocal boundary value problems for the Hilfer fractional derivative were studied in [19]. Initial value problems for pantograph equations with the Hilfer fractional derivative were studied in [15, 20].
To the best of our knowledge, there is no work on boundary value problems for pantograph equations with the Hilfer fractional derivative in the literature. This paper comes to fill this gap, by introducing a new class of boundary value problems of pantograph equations with Hilfer-type fractional differential equations and nonlocal integral boundary conditions, of the form
where \({}^{H}D^{\alpha,\beta}\) is the Hilfer fractional derivative of order α, \(1<\alpha<2\) and parameter β, \(0\leq\beta\leq1\), \(f: [a,b]\times{\mathbb {R}}\times{\mathbb {R}} \to{\mathbb {R}}\) is a continuous function, \(I^{\delta}\) is the Riemann–Liouville fractional integral of order \(\delta>0\), \(a\geq0\), \(A, B, c\in{\mathbb {R}}\) and \(0<\lambda<1\).
Existence and uniqueness results are proved by using well-known fixed point theorems. We make use of Banach’s fixed point theorem to obtain the uniqueness result, while the nonlinear alternative of Leray–Schauder type [21] and Krasnoselskii’s fixed point theorem [22] are applied to obtain the existence results for the problem (1.1)–(1.2).
After that we study the multi-valued version of the problem (1.1)–(1.2) by considering the inclusion problem
where \(F: [a,b]\times{\mathbb {R}}\times{\mathbb {R}} \to{\mathcal {P}}({\mathbb {R}})\) is a multi-valued function (\({\mathcal {P}}({\mathbb {R}})\) is the family of all nonempty subsets of \({\mathbb {R}}\)).
For the problem (1.3)–(1.4) we prove existence results for both cases, convex valued (upper semicontinuous case) and non-convex valued (Lipschitz case) multifunctions. For the case when the multi-valued F has convex values we use the Bohnenblust–Karlin fixed point theorem, Martelli’s fixed point theorem and the nonlinear alternative for Kakutani maps. For the lower semicontinuous case the existence result is based in nonlinear alternative of Leray–Schauder type together with a selection theorem for lower semicontinuous maps with decomposable values. Finally in the case of possible non-convex valued multi-valued map we apply a fixed point theorem for contractive multi-valued maps due to Covitz and Nadler.
The outline of the paper is as follows: We present our main work for single-valued case in Sect. 3, and for multi-valued case in Sect. 4, while Sect. 2 contains some preliminary concepts related to our problem. Examples are constructed to illustrate the main results.
2 Preliminaries
In this section, we introduce some notations and definitions of fractional calculus and multi-valued analysis and present preliminary results needed in our proofs later [2, 5].
Definition 2.1
The Riemann–Liouville fractional integral of order \(\alpha>0\) of a continuous function \(u:[a,\infty)\to{\mathbb{R}}\), is defined by
provided the right-hand side exists on \((a,\infty)\).
Definition 2.2
The Riemann–Liouville fractional derivative of order \(\alpha>0\) of a continuous function u, is defined by
where \(n=[\alpha]+1\), \([\alpha]\) denotes the integer part of real number α, provided the right-hand side is point-wise defined on \((a,\infty)\).
Definition 2.3
The Caputo fractional derivative of order \(\alpha>0\) of a continuous function u, is defined by
provided the right-hand side is point-wise defined on \((a,\infty)\).
In [8] (see also [9]) another new definition of the fractional derivative was suggested. The generalized Riemann–Liouville fractional derivative is defined as follows.
Definition 2.4
The generalized Riemann–Liouville fractional derivative or the Hilfer fractional derivative of order α and parameter β of a function u is defined by
where \(n-1<\alpha< n \), \(0\leq\beta\leq1\), \(t>a \), \(D= \frac{d}{dt}\).
Remark 2.5
When \(\beta=0\) the Hilfer fractional derivative corresponds to the Riemann–Liouville fractional derivative
while when \(\beta=1\) the Hilfer fractional derivative corresponds to the Caputo fractional derivative
In the following lemma we present the compositional property of Riemann–Liouville fractional integral operator with the Hilfer fractional derivative operator.
Lemma 2.6
([9])
Let\(f\in L(a,b)\), \(n-1<\alpha\le n\), \(n\in{\mathbb {N}}\), \(0\le\beta\le 1\), \(I^{(n-\alpha)(1-\beta)}f\in AC^{k}[a,b]\). Then
The following lemma deals with a linear variant of the boundary value problem (1.1)–(1.2).
Lemma 2.7
Let\(a\geq0\), \(1< \alpha<2\), \(\gamma=\alpha+2\beta -\alpha\beta\), \(h\in C([a,b], {\mathbb {R}})\)and
Then the functionxis a solution of the boundary value problem
if and only if
Proof
Assume that x is a solution of the nonlocal (2.2)–(2.3). Operating the fractional integral \(I^{\alpha}\) on both sides of equation (2.2) and using Lemma 2.6, we obtain
since \((1-\beta)(2-\alpha)=2-\gamma\), where \(c_{0}\) and \(c_{1}\) are some real constants.
From the first boundary condition \(x(a)=0\) we can obtain \(c_{0}=0\), since \(\lim_{t\to a}(t-a)^{\gamma-2}=\infty\). Then we get
From \(Ax(b)+B I^{\delta}x(\eta)=c\) we found
Substituting the values of \(c_{1}\) in (2.5), we obtain the solution (2.4). The converse follows by direct computation. This completes the proof. □
3 Main results for the single-valued problem (1.1)–(1.2)
In view of Lemma 2.7, we define an operator \(\mathcal{A}:\mathcal {C}\rightarrow\mathcal{C}\) by
It should be noticed that problem (1.1)–(1.2) has solutions if and only if the operator \(\mathcal{A}\) has fixed points.
In the following, for the sake of convenience, we set
We prove existence as well as existence and uniqueness results, for the boundary value problem (1.1)–(1.2) by using well-known fixed point theorems.
Our existence and uniqueness result is based on Banach’s fixed point theorem.
Theorem 3.1
Assume that:
- \((H_{1})\):
there exists a constant\(L>0\)such that
$$\bigl\vert f(t,x_{1}, x_{2})-f(t,y_{1}, y_{2}) \bigr\vert \leq L\bigl( \vert x_{1}-y_{1} \vert + \vert x_{2}-y_{2} \vert \bigr) $$for each\(t\in[a,b]\)and\(x_{i},y_{i}\in\mathbb{R}\), \(i=1,2\).
If
whereΩis defined by (3.2), then the boundary value problem (1.1)–(1.2) has a unique solution on\([a,b]\).
Proof
We transform the boundary value problem (1.1)–(1.2) into a fixed point problem, \(x=\mathcal{A}x\), where the operator \(\mathcal{A}\) is defined as in (3.1). Observe that the fixed points of the operator \(\mathcal{A}\) are solutions of problem (1.1)–(1.2). Applying the Banach contraction mapping principle, we shall show that \(\mathcal{A}\) has a unique fixed point.
We let \(\sup_{t\in[a,b]}|f(t,0,0)|=M<\infty\), and choose
Now, we show that \(\mathcal{A}B_{r} \subset B_{r}\), where \(B_{r}=\{x\in \mathcal{C}:\|x\|\leq r\}\). For any \(x\in B_{r}\), we have
which implies that \(\mathcal{A}B_{r}\subset B_{r}\).
Next, we let \(x,y\in\mathcal{C}\). Then, for \(t\in[a,b]\), we have
which implies that \(\|\mathcal{A}x-\mathcal{A}y\| \leq2L\varOmega\| x-y\|\). As \(2L\varOmega<1\), \(\mathcal{A}\) is a contraction. Therefore, we deduce by the Banach contraction mapping principle that \(\mathcal {A}\) has a fixed point which is the unique solution of the boundary value problem (1.1)–(1.2). The proof is completed. □
Example 3.2
Consider the nonlocal boundary value problem for the Hilfer-type pantograph fractional differential equation of the form
Here \(\alpha=3/2\), \(\beta=2/5\), \(\lambda=1/2\), \(a=1/2\), \(b=5/2\), \(A=2/3\), \(B=3/4\), \(\delta=1/2\), \(\eta=3/2\), \(c=4/5\). The setting yields \(\gamma=17/10\), \(\varLambda=1.872599119\) and \(\varOmega=4.129461300\). Now, we put
which satisfies \((H_{1})\) as
Setting \(L=1/9\), we obtain \(2L\varOmega\approx0.9176580667<1\) which shows that inequality (3.3) is true. Then, by the conclusion of Theorem 3.1, we deduce that the boundary value problem (3.5) has a unique solution on \([1/2, 5/2]\).
Next we present two existence results. The first is based on the well-known Krasnoselskii fixed point theorem ([22]).
Theorem 3.3
Let\(f : [a,b]\times\mathbb{R}\times\mathbb{R} \rightarrow\mathbb {R}\)be a continuous function satisfying\((H_{1})\). In addition we assume that:
- \((H_{2})\):
\(|f(t,x,y)| \leq\varphi(t)\), \(\forall(t,x,y) \in [a,b] \times\mathbb{R}\times\mathbb{R}\), and\(\varphi\in C([a,b],\mathbb{R}^{+})\).
Then the boundary value problem (1.1)–(1.2) has at least one solution on\([a,b]\), provided
Proof
Setting \(\sup_{t \in[a,b]} \varphi(t)= \|\varphi\| \) and choosing
(where Ω is defined by (3.2)), we consider \(B_{\rho}= \{ x\in\mathcal{C} : \|x\| \leq\rho\}\). We define the operators \(\mathcal {A}_{1}\), \(\mathcal{A}_{2}\) on \(B_{\rho}\) by
and
For any \(x,y \in B_{\rho}\), we have
This shows that \(\mathcal{A}_{1}x+\mathcal{A}_{2}y \in B_{\rho}\). It is easy to see, using (3.6), that \(\mathcal{A}_{2}\) is a contraction mapping.
Continuity of f implies that the operator \(\mathcal{A}_{1}\) is continuous. Also, \(\mathcal{A}_{1}\) is uniformly bounded on \(B_{\rho}\) as
Now we prove the compactness of the operator \(\mathcal{A}_{1}\).
We define \(\sup_{(t,x)\in[a,b]\times B_{\rho}\times B_{\rho}} |f(t,x,y)| = \bar{f} <\infty\), and consequently, for any \(t_{1}, t_{2}\in[a,b]\) with \(t_{1}< t_{2}\), we have
which is independent of x and tends to zero as \(t_{2} - t_{1} \rightarrow 0\). Thus, \(\mathcal{A}_{1}\) is equicontinuous. So \(\mathcal{A}_{1}\) is relatively compact on \(B_{\rho}\). Hence, by the Arzelá–Ascoli theorem, \(\mathcal{A}_{1}\) is compact on \(B_{\rho}\). Thus all the assumptions of Krasnoselskii’s fixed point theorem ([22]) are satisfied. So the conclusion of Krasnoselskii’s fixed point theorem implies that the boundary value problem (1.1)–(1.2) has at least one solution on \([a,b]\). □
Example 3.4
Consider the nonlocal boundary value problem for the Hilfer-type pantograph fractional differential equation of the form
Here \(\alpha=5/3\), \(\beta=1/2\), \(\lambda=1/4\), \(a=1/3\), \(b=5/3\), \(A=3/5\), \(B=1/4\), \(\delta=3/2\), \(\eta=2/3\) and \(c=3/4\). Then we compute that \(\gamma=11/6\), \(\varLambda\approx0.8175877260\), \(\varOmega\approx 2.139687890\) and
From (3.8), we can find that
This means that the condition \((H_{1})\) is satisfied with \(L=9/10\). We get \(L\varOmega_{1}\approx 0.9595352934<1\) and
which satisfy conditions (3.6) and \((H_{2})\), respectively. Applying Theorem 3.3, the boundary value problem (3.8) has at least one solution on \([1/3,5/3]\).
Remark 3.5
In Theorem 3.1, the existence and uniqueness of solutions for the given problem is established by means of the Banach contraction mapping principle. In Theorem 3.3 the existence of solutions is established via the Krasnoselskii fixed point theorem. The proof of this result is based on the idea of splitting the operator \(\mathcal{A}\) into the sum of two operators \(\mathcal{A}_{1}\) and \(\mathcal{A}_{2}\) such that \(\mathcal{A}_{1}\) is contractive and \(\mathcal{A}_{2}\) is compact. One can notice that the entire operator \(\mathcal{A}\) is not required to be contractive. On the other hand, Theorem 3.1 deals with the existence of a unique solution of the given problem via the Banach contraction mapping principle, in which the entire operator \(\mathcal{A}\) is shown to be contractive. Thus, the linkage between contractive conditions imposed in Theorems 3.3 and 3.1 provides a precise estimate to pass onto a unique solution from the existence of a solution for the problem at hand. In Example 3.8 above we note that Theorem 3.1 is not applicable because \(2L\varOmega\approx3.851438202>1\).
The Leray–Schauder’s nonlinear alternative ([21]) is used for our next existence result.
Theorem 3.6
Assume that:
- (\(H_{3}\)):
there exist a continuous nondecreasing function\(\psi: [0,\infty) \rightarrow(0,\infty)\)and a function\(p \in C([a,b],\mathbb {R}^{+})\)such that
$$\bigl\vert f(t,u,v) \bigr\vert \leq p(t)\psi\bigl( \vert u \vert + \vert v \vert \bigr) \quad\textit{for each }(t,u,v) \in [a,b]\times \mathbb{R}\times\mathbb{R}; $$- (\(H_{4}\)):
there exists a constant\(M>0\)such that
$$\frac{M}{\psi(2M) \Vert p \Vert \varOmega+ \vert c \vert (b-a)^{\gamma-1}/ \vert \varLambda \vert \varGamma(\gamma)}>1, $$whereΩis defined by (3.2).
Then the boundary value problem (1.1)–(1.2) has at least one solution on\([a,b]\).
Proof
Let the operator \(\mathcal{A}\) be defined by (3.1). Firstly, we shall show that \(\mathcal{A}\) maps bounded sets (balls) into bounded set in \({\mathcal {C}}\). For a number \(r>0\), let \(B_{r} = \{ x\in{\mathcal {C}} : \|x\| \leq r\}\) be a bounded ball in \({\mathcal {C}}\). Then for \(t \in[a,b]\) we have
and, consequently,
Next we will show that \(\mathcal{A}\) maps bounded sets into equicontinuous sets of \({\mathcal {C}}\). Let \(\tau_{1},\tau_{2} \in[a,b]\) with \(\tau_{1} < \tau_{2}\) and \(x \in B_{r}\). Then we have
As \(\tau_{2}-\tau_{1} \rightarrow0\), the right-hand side of the above inequality tends to zero independently of \(x \in B_{r}\). Therefore, by the Arzelá–Ascoli theorem, the operator \(\mathcal{A} : {\mathcal {C}} \rightarrow{\mathcal {C}}\) is completely continuous.
The result will follow from the Leray–Schauder nonlinear alternative ([21]) once we have proved the boundedness of the set of all solutions to equations \(x=\nu{\mathcal {A}} x\) for \(\nu\in(0,1)\).
Let x be a solution. Then, for \(t \in[a,b]\), and following computations similar to the first step, we have
which leads to
In view of \((H_{4})\), there exists M such that \(\|x\| \neq M\). Let us set
We see that the operator \(\mathcal{A} : \bar{U} \rightarrow{\mathcal {C}}\) is continuous and completely continuous. From the choice of U, there is no \(x \in\partial U\) such that \(x = \nu\mathcal{A}x\) for some \(\nu\in(0,1)\). Consequently, by the nonlinear alternative of Leray–Schauder type ([21]), we deduce that \(\mathcal{A}\) has a fixed point \(x\in\bar{U}\) which is a solution of the boundary value problem (1.1)–(1.2). This completes the proof. □
Example 3.7
Consider the nonlocal boundary value problem for the Hilfer-type pantograph fractional differential equation of the form
Here \(\alpha=7/4\), \(\beta=4/5\), \(\lambda=1/5\), \(a=1/4\), \(b=3/4\), \(A=1/3\), \(B=3/7\), \(\delta=5/2\), \(\eta=1/2\), \(c=2/5\). Now, we find constants \(\gamma=39/20\), \(\varLambda\approx0.1764175022\) and \(\varOmega\approx0.3694499725\). Also, the nonlinear function can be expressed as
Setting \(p(t)=(e^{(1/4)-t})/(7+4t)\) and \(\psi(u)=u^{2}+1\), we have \(\|p\| =1/8\) and \(\psi(|x|+|y|)=(|x|+|y|)^{2}+1\). Thus we can compute that there exists a constant \(M\in(1.937052574, 3.476400259)\) satisfying inequality in \((H_{4})\). Therefore, all conditions in Theorem 3.6 are fulfilled. Thus the boundary value problem (3.9) has at least one solution on \([1/4,3/4]\).
4 Existence results for the multi-valued problem (1.3)–(1.4)
By \(\mathcal{C}=C([a,b],{\mathbb {R}})\) we denote the Banach space of all continuous functions from \([a,b]\) into \({\mathbb {R}}\) with the norm
Also by \(L^{1}([a,b],\mathbb{R})\) we denote the space of functions \(x : [a,b] \to\mathbb{R}\) such that \(\|x\|_{L^{1}} = \int_{a}^{b} |x(t)|\,dt\).
For a normed space \((X, \|\cdot\|)\), we define \({\mathcal {P}}_{q}(X)=\{Y \in{\mathcal{P}}(X) : Y \mbox{ has the property }q\}\). Thus, for example, \({\mathcal {P}}_{\mathrm{cl}, b}(X)=\{Y \in{\mathcal{P}}(X) : Y \mbox{ is closed and bounded}\}\), \({\mathcal {P}}_{\mathrm{cp}, c}(X)=\{Y \in{\mathcal{P}}(X) : Y \mbox{ is compact and convex}\}\).
For each \(y \in C([a,b], \mathbb{R})\), define the set of selections of F by
The following lemma will be used in the sequel.
Lemma 4.1
([23])
LetXbe a separable Banach space. Let\(F : [a,b] \times\mathbb{R}\times\mathbb{R} \to{\mathcal {P}}_{\mathrm{cp},c}(X)\)be an\(L^{1}\)-Carathéodory multi-valued map and letΘbe a linear continuous mapping from\(L^{1}([a,b],X)\)to\(C([a,b], X)\). Then the operator
is a closed graph operator in\(C([a,b], X)\times C([a,b], X)\).
Before stating and proving our main existence results for problem (1.3)–(1.4), we will give the definition of its solution.
Definition 4.2
A function \(x\in AC^{2}([a,b], {\mathbb {R}})\) is said to be a solution of the problem (1.3)–(1.4) if \(x(a)=0\), \(Ax(b)+B I^{\delta}x(\eta)=c\), and there exists a function \(v\in L^{1}([a,b],\mathbb{R})\) with \(v\in F(t,x,y)\) a.e. on \([a,b]\) such that
4.1 The upper semicontinuous case
Consider first the case when F has convex values. Our first result is based on Bohnenblust–Karlin fixed point theorem.
Lemma 4.3
((Bohnenblust–Karlin) [24])
LetXbe a Banach space, Da nonempty subset ofX, witch is bounded, closed and convex. Suppose\(G:[a,b] \times {\mathbb{R}}\rightarrow{\mathcal {P}}({\mathbb{R}})\)is upper semicontinuous with closed, convex values, and\(G(D)\subset D\)and\(\overline{G(D)}\)is compact. ThenGhas a fixed point.
Theorem 4.4
Assume that:
- \((A_{1})\):
\(F : [a,b]\times\mathbb{R} \times\mathbb{R} \to{\mathcal{P}_{\mathrm{cp},c}}(\mathbb{R})\)is\(L^{1}\)-Carathéodory; i.e.
- (i)
\(t \longmapsto F(t,x,y)\)is measurable for each\(x,y \in\mathbb{R}\);
- (ii)
\((x,y) \longmapsto F(t,x,y)\)is upper semicontinuous for almost all\(t\in [a,b]\);
- (iii)
for each\(\rho> 0\), there exists\(\varphi_{\rho} \in L^{1}( [a,b],\mathbb{R}^{+})\)such that
$$\bigl\Vert F (t, x,y) \bigr\Vert = \sup \bigl\{ \vert v \vert : v \in F (t, x,y)\bigr\} \le\varphi_{\rho} (t) $$for all\(x,y\in{\mathbb {R}}\)with\(\|x\|, \|y\| \le\rho\)and for a.e. \(t \in [a,b]\);
- (i)
- \((A_{2})\):
- $$ \liminf_{{\rho}\to\infty}\frac{1}{{\rho}} \int_{a}^{b} \phi_{{\rho }}(t)\,dt=\mu. $$(4.1)
Then the boundary problem (1.3)–(1.4) has at least one solution on\([a,b]\)provided that
Proof
In order to transform the problem (1.3)–(1.4) into a fixed point problem, we consider the multi-valued map: \(N:C([a,b], {\mathbb {R}})\rightarrow{\mathcal {P}}(C([a,b], {\mathbb {R}}))\) defined by
It is clear that fixed points of N are solutions of problem (1.3)–(1.4). In turn, we need to show that the operator N satisfies all condition of Lemma 4.3. The proof is constructed in several steps.
Step 1. \(N(x)\) is convex for each \(x \in C([a,b], \mathbb{R})\) .
Indeed, if \(h_{1}\), \(h_{2}\) belongs to \(N(x)\), then there exist \(v_{1}, v_{2}\in S_{F,x}\) such that, for each \(t\in[a,b]\), we have
Let \(0\leq\theta\leq1\). Then, for each \(t\in[a,b]\), we have
Since F has convex values, that is, \(S_{F,x}\) is convex, we have
Step 2. \(N(x)\) maps bounded sets (balls) into bounded sets in \(C([a,b], \mathbb{R})\) .
For a positive number ρ, let \(B_{\rho} = \{x \in C([a,b], \mathbb{R}): \|x\| \le{\rho} \}\) be a bounded ball in \(C([a,b], \mathbb{R})\). We shall prove that there exists a positive number \({\rho }'\) such that \(N(B_{{\rho}'})\subseteq B_{{\rho}'}\). If not, for each positive number ρ, there exists a function \(x_{{\rho}}(\cdot)\in B_{{\rho}}\), \(\|N(x_{{\rho}})\|>\rho\) for some \(t\in[a,b]\) and
for some \(v_{\rho}\in S_{F,x_{\rho}}\). However, on the other hand, we have
Dividing both sides by ρ and taking the lower limit as \({\rho}\rightarrow\infty\), we get
which contradicts (4.2). Hence there exists a positive number ρ such that \(N(B_{\rho})\subseteq B_{\rho}\).
Step 3. \(N(x)\) maps bounded sets into equicontinuous sets of \(C([a,b], \mathbb{R})\) .
Let x be any element in \(B_{\rho}\) and \(h\in N(x)\), then there exists a function \(v\in S_{F,x}\) such that, for each \(t\in[a,b]\), we have
Let \(\tau_{1}, \tau_{2}\in[a,b]\), \(\tau_{1}<\tau_{2}\). Thus
The right-hand side of the above inequality clearly tends to zero independently of \(x\in B_{\rho}\) as \(\tau_{1}\rightarrow\tau_{2}\). As a consequence of Steps 1–3 together with the Arzelá–Ascoli theorem, we conclude that \(N: C([a,b], \mathbb{R}) \to{\mathcal{P}}(C([a,b], \mathbb{R}))\) is completely continuous.
Step 4. \(N(x)\) is closed for each \(x \in C([a,b], \mathbb{R})\) .
Let \(\{u_{n}\}_{n \ge0} \in N(x)\) be such that \(u_{n} \to u\) (\(n \to\infty\)) in \(C([a,b], \mathbb{R})\). Then \(u \in C([a,b], \mathbb{R})\) and there exists \(v_{n} \in S_{F,x_{n}}\) such that, for each \(t \in[a,b]\),
As F has compact values, we pass onto a subsequence (if necessary) to find that \(v_{n}\) converges to v in \(L^{1} ([a,b],\mathbb{R})\). Thus \(v \in S_{F,x}\) and for each \(t \in [a,b]\), we have
Hence, \(u \in N(x)\).
Next we show that the operator N is upper semicontinuous. In order to do so, it is enough to establish that N has a closed graph ([25, Proposition 1.2]).
Step 5. N has a closed graph.
Let \(x_{n}\rightarrow x_{*}\), \(h_{n}\in N(x_{n})\) and \(h_{n}\rightarrow h_{*}\). We need to show that \(h_{*}\in N(x_{*})\). Now \(h_{n}\in N(x_{n})\) implies that there exists \(v_{n}\in S_{F,x_{n}}\) such that, for each \(t\in[a,b]\),
We must show that there exists \(v_{*}\in S_{F,x_{*}}\) such that, for each \(t\in[a,b]\),
Consider the continuous linear operator \(\varTheta:L^{1}([a,b],{\mathbb{R}})\rightarrow C([a,b], {\mathbb {R}})\) by
Observe that \(\|h_{n}(t)-h_{*}(t)\|\to0\) as \(n\to\infty\) and, thus, it follows from Lemma 4.1 that \(\varTheta\circ S_{F,x}\) is a closed graph operator. Moreover, we have
Since \(x_{n}\rightarrow x_{*}\), Lemma 4.1 implies that
for some \(v_{*}\in S_{F,x_{*}}\)
Hence, we conclude that N is a compact multi-valued map, u.s.c. with convex closed values. As a consequence of Lemma 4.3, we deduce that N has a fixed point which is a solution of the boundary problem (1.3)–(1.4). This completes the proof. □
Next, we give an existence result based upon the following form of fixed point theorem which is applicable to completely continuous operators [26].
Lemma 4.5
LetXa Banach space, and\(T: X \rightarrow{\mathcal {P}}_{b,\mathrm{cl},c}(X)\)be a completely continuous multi-valued map. If the set\({\mathcal{E}}=\{x\in X : \nu x\in T(x), \nu>1\} \)is bounded, thenThas a fixed point.
Theorem 4.6
Assume that the following hypotheses hold:
- \((A_{3})\):
\(F:[a,b]\times\mathbb{R}\times{\mathbb{R}}\rightarrow {\mathcal {P}}_{b,\mathrm{cl},c}(\mathbb{R})\)is a\(L^{1}\)-Carathéodory multi-valued map;
- \((A_{4})\):
there exists a function\(h\in C([a,b], \mathbb{R})\)such that
$$\bigl\Vert F(t,x,y) \bigr\Vert \leq h(t) \quad\textit{for a.e. } t\in[a,b] \textit{ and each } x,y\in\mathbb{R}. $$
Then the problem (1.3)–(1.4) has at least one solution on\([a,b]\).
Proof
Consider N defined in the proof of Theorem 4.4. As in Theorem 4.4, we can show that N is convex and completely continuous. It remains to show that the set
is bounded. Let \(x\in{\mathcal{E}}\), then \(\nu x\in N(x)\) for some \(\nu>1\) and there exists a function \(v\in S_{F,x}\) such that
For each \(t\in[a,b]\), we have
Taking the supremum over \(t\in[a,b]\), we get
Hence the set \(\mathcal{E}\) is bounded. As a consequence of Lemma 4.5 we deduce that N has at least one fixed point which implies that the problem (1.3)–(1.4) has a solution on \([a,b]\). □
Our final existence result in this subsection is based on the Leray–Schauder nonlinear alternative for Kakutani maps ([21]).
Theorem 4.7
Assume that\((H_{4})\)and\((A_{1})\)hold. In addition we assume that:
- \((A_{5})\):
there exists a continuous nondecreasing function\(\psi: [0,\infty) \to (0,\infty)\)and a function\(p \in C([a,b], \mathbb{R}^{+})\)such that\(\|F(t,x,y)\|_{\mathcal{P}}:=\sup\{|v|: v \in F(t,x,y)\}\le p(t)\psi(|x|)\)for each\((t,x,y) \in[a,b]\times\mathbb{R} \times\mathbb{R}\).
Then the boundary value problem (1.3)–(1.4) has at least one solution on\([a,b]\).
Proof
Consider the operator N defined in the proof of Theorem 4.4. Let \(x\in\nu N(x)\) for some \(\nu\in(0,1)\). We show there exists an open set \(U\subseteq C([a,b], {\mathbb {R}})\) with \(x\notin N (x)\) for any \(\nu\in(0,1)\) and all \(x\in\partial U\). Let \(\nu\in (0,1)\) and \(x\in\nu N (x)\). Then there exists \(v \in L^{1}([a,b], \mathbb{R})\) with \(v \in S_{F,x}\) such that, for \(t \in[a,b]\), we have
In view of \((A_{5})\), we have, for each \(t\in[a,b]\),
Consequently, we have
In view of \((H_{4})\), there exists M such that \(\|x\| \ne M\). Let us set
Proceeding as in the proof of Theorem 4.4, we claim that the operator \(N :\overline{U} \to\mathcal{P}(C([a,b], \mathbb{R}))\) is a compact, upper semicontinuous multi-valued map with convex closed values. From the choice of U, there is no \(x \in\partial U\) such that \(x \in\nu N (x)\) for some \(\nu\in(0,1)\). Consequently, by the nonlinear alternative of Leray–Schauder type ([21]), we deduce that N has a fixed point \(x \in\overline{U}\) which is a solution of the boundary value problem (1.3)–(1.4). This completes the proof. □
4.2 The lower semicontinuous case
Here we study the case when F is not necessarily convex valued, by applying the nonlinear alternative of Leray–Schauder type together with the selection theorem of Bressan and Colombo [27] for lower semicontinuous maps with decomposable values.
Theorem 4.8
Assume that\((H_{4})\), \((A_{5})\)and the following condition hold:
- \((A_{6})\):
\(F : [a,b] \times\mathbb{R}\times\mathbb{R} \to{\mathcal{P}}(\mathbb{R})\)is a nonempty compact-valued multi-valued map such that\((t,x,y) \longmapsto F(t,x,y)\)is\(\mathcal{L}\otimes \mathcal{B}\otimes \mathcal{B}\)measurable and\(x \longmapsto F(t,x,y)\)is lower semicontinuous for each\(t \in[a,b]\).
Then the boundary value problem (1.3)–(1.4) has at least one solution on\([a,b]\).
Proof
It follows from \((A_{5})\) and \((A_{6})\) that F is of l.s.c. type [28]. Then, by the selection theorem of Bressan and Colombo [27], there exists a continuous function \(v : AC^{1}([a,b],\mathbb{R}) \to L^{1}([a,b],\mathbb{R})\) such that \(v (x) \in \mathcal{F}(x)\) for all \(v \in C([a,b], \mathbb{R})\), where \(\mathcal{F} : C([a,b] \times \mathbb{R}) \to {\mathcal{P}}(L^{1}([a,b],\mathbb{R}))\) is the Nemytskii operator associated with F, defined as
Consider the problem
Observe that x is a solution to the boundary value problem (1.3)–(1.4) if \(x \in AC^{2}([a,b],\mathbb {R})\) is a solution of the problem (4.3)–(4.4). In order to transform the problem (4.3)–(4.4) into a fixed point problem, we define an operator N̅ as
It can easily be shown that N̅ is continuous and completely continuous. The remaining part of the proof is similar to that of Theorem 4.7. So we omit it. This completes the proof. □
4.3 The Lipschitz case
In this subsection, we prove the existence of solutions for the boundary value problem (1.3)–(1.4) with a non-convex valued right-hand side by applying a fixed point theorem for multi-valued maps due to Covitz and Nadler [29].
Theorem 4.9
Assume that the following conditions hold:
- \((B_{1})\):
\(F : [a,b] \times\mathbb{R}\times\mathbb{R} \to{\mathcal {P}}_{\mathrm{cp}}(\mathbb{R})\)is such that\(F(\cdot,x,y) : [a,b] \to{\mathcal {P}}_{\mathrm{cp}}(\mathbb{R})\)is measurable for each\(x,y \in\mathbb{R}\).
- \((B_{2})\):
\(H_{d}(F(t,x,y), F(t,\bar{x}),\bar{y})\le m(t)(|x-\bar{x}|+|y-\bar{y}|)\)for almost all\(t \in[a,b]\)and\(x, \bar{x}, y, \bar{y} \in\mathbb {R}\)with\(m \in C([a,b], \mathbb{R}^{+})\)and\(d(0,F(t,0,0))\le m(t)\)for almost all\(t \in[a,b]\).
Then the boundary value problem (1.3)–(1.4) has at least one solution on\([a,b]\)if
Proof
We transform the boundary value problem (1.3)–(1.4) into a fixed point problem by considering the operator \(N: C([a,b], \mathbb{R})\to{\mathcal {P}}(C([a,b], \mathbb{R}))\) defined at the beginning of the proof of Theorem 4.4. We show that the operator N satisfies the assumptions of Covitz and Nadler Theorem [29] in two steps.
Step I.N is nonempty and closed for every \(v\in S_{F,x}\).
Note that the set-valued map \(F(\cdot, x(\cdot))\) is measurable by the measurable selection theorem (e.g., [30, Theorem III.6]) and it admits a measurable selection \(v: [a,b] \to\mathbb{R}\). Moreover, by the assumption \((B_{1})\), we have
i.e. \(v \in L^{1}([a,b], \mathbb{R})\) and hence F is integrably bounded. Therefore, \(S_{F,x} \ne\emptyset\). Moreover \(N(x) \in{\mathcal {P}}_{\mathrm{cl}}(C([a,b], \mathbb{R}))\) for each \(x \in C([a,b], \mathbb{R})\), as proved in Step 4 of Theorem 4.4.
Step II. Next we show that there exists \(\theta<1\) such that
Let \(x, \bar{x}, y, \bar{y} \in AC^{2}([a,b], \mathbb{R})\) and \(h_{1} \in N(x)\). Then there exists \(v_{1}(t) \in F(t,x(t), y(t))\) such that, for each \(t \in[a,b]\),
By \((B_{2})\), we have
So, there exists \(w(t) \in F(t,\bar{x}(t), \bar{y}(t))\) such that
Define \(U : [a,b] \to\mathcal{P}(\mathbb{R})\) by
Since the multi-valued operator \(U(t)\cap F(t,\bar{x}(t),\bar{y}(t))\) is measurable (Proposition III.4 [30]), there exists a function \(v_{2}(t)\) which is a measurable selection for U. So \(v_{2}(t) \in F(t,\bar{x}(t),\bar{y}(t))\) and for each \(t \in[a,b]\), we have \(|v_{1}(t)-v_{2}(t)|\le m(t)(|x(t)-\bar{x}(t)|+|y(t)-\bar{y}(t)|)\).
For each \(t \in[a,b]\), let us define
Thus,
Hence
Analogously, interchanging the roles of x and x̅, we obtain
Since N is a contraction, it follows by Covitz and Nadler’s lemma ([29]) that N has a fixed point x which is a solution of (1.3)–(1.4). This completes the proof. □
Example 4.10
Consider the nonlocal boundary value problem for the Hilfer-type pantograph fractional differential inclusion of the form
Here \(\alpha=8/5\), \(\beta=2/3\), \(\lambda=1/2\), \(a=3/5\), \(b=9/5\), \(A=1/2\), \(B=3/5\), \(\delta=7/2\), \(\eta=7/5\), \(c=2/3\). Next, by direct computations, we have \(\gamma=28/15\), \(\varLambda\approx0.6212581095\), \(\varOmega\approx1.867435349\) and \(\varOmega_{1}\approx0.9310211429\).
(i) Let \(F(t, x(t), x(t/2))\) be defined by
It is easy to see that the condition \((A_{1})\) in Theorem 4.4 is satisfied. Indeed, we obtain from (4.6) that \(\|F(t,x,y)\|\leq (2\rho+1)e^{-t}\) for all \(x,y\in{\mathbb{R}}\) with \(\|x\|\), \(\|y\|\leq \rho\) and for a.e.\(t\in[3/5,9/5]\). Next, we can find that
which leads to
Therefore, by Theorem 4.4, the boundary value problem (4.5) with (4.6) has at least one solution on \([3/5, 9/5]\).
(ii) Let \(F(t, x(t), x(t/2))\) be defined by
It is clear that F, defined in (4.7), is measurable for all \(x,y\in{\mathbb{R}}\). Next, we can compute that
By setting \(m(t)=(t+1)/41\), we get \(\|m\|=14/205\). Also, we have \(d(0,F(t,0,0))\leq m(t)\), \(t\in[3/5,9/5]\). Hence we get \(2\varOmega\| m\|\approx0.2550643403<1\). Thus, applying the conclusion of Theorem 4.9, we deduce that the boundary value problem (4.5) with (4.7) has at least one solution on \([3/5, 9/5]\).
5 Conclusion
In this paper we initiated the study of a new class of boundary value problems, involving the Hilfer fractional derivative, for pantograph fractional differential equations and inclusions supplemented by nonlocal integral boundary conditions. Existence and uniqueness results are proved in the single-valued case. Banach’s fixed point theorem is used to obtain the uniqueness result, while the nonlinear alternative of Leray–Schauder type and Krasnoselskii’s fixed point theorem are applied to obtain the existence results. For the multi-valued problem we prove existence results for both convex valued and non-convex valued multifunctions. For the case when the multi-valued F has convex values we use the Bohnenblust–Karlin fixed point theorem, Martelli’s fixed point theorem and the nonlinear alternative for Kakutani maps. For the lower semicontinuous case the existence result is based on the nonlinear alternative of Leray–Schauder type together with a selection theorem for lower semicontinuous maps with decomposable values. Finally in the case of a possible non-convex valued multi-valued map we use a fixed point theorem for contractive multi-valued maps due to Covitz and Nadler. Examples illustrating the obtained results are also presented.
The results presented in this paper are more general and correspond to several new results corresponding to specific values of the parameters involved in the problem (1.1)–(1.2. For instance, the nonlocal boundary condition given by (1.2) with \(A=0\), \(c=0\), \(B\ne0\) can be conceived of as a conserved boundary condition as the sum of the values of the continuous unknown function over the given interval of arbitrary length is zero. In other words, our problem becomes an “average type” boundary value problem for Hilfer-type fractional differential equations or inclusions. With \(B=1\), \(A=-1\), \(c=0\) our boundary condition (1.2) becomes \(x(b)=BI^{\delta}x(\eta)\), etc. In our future work we plan to investigate the existence of solutions for boundary value problems for other kinds of fractional differential equations and boundary conditions.
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Wongcharoen, A., Ntouyas, S.K. & Tariboon, J. Nonlocal boundary value problems for Hilfer-type pantograph fractional differential equations and inclusions. Adv Differ Equ 2020, 279 (2020). https://doi.org/10.1186/s13662-020-02747-1
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DOI: https://doi.org/10.1186/s13662-020-02747-1
MSC
- 26A33
- 34A08
- 34A60
- 34B15
Keywords
- Hilfer fractional derivative
- Pantograph fractional differential equations
- Riemann–Liouville fractional derivative
- Caputo fractional derivative
- Boundary value problems
- Existence and uniqueness
- Fixed point theory