Abstract
The existence of a solution for a system of two nonlinear high-order fractional differential equations including the Atangana-Baleanu-Caputo derivative with integral boundary conditions, is proved. Simultaneously, we discuss the existence of a solution by applying the Schauder fixed point theorem and a generalized Darbo fixed point theorem, which involves the concept of measure of noncompactness. The paper also contains some examples that illustrate the application of the main result.
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1 Introduction
The partial differential equations of fractional order, which are considered as an extension of the partial differential equations of integer order, play a crucial role in the mathematical modeling of real objects see [1, 2, 5, 15, 23, 31, 33, 37,38,39, 43,44,45, 47, 48].
Since the derivative of fractional order, is a generalization of the derivative with integer order, it makes mathematical modeling more accurate. For example in [29] the authors show that the fractional derivative plays a crucial role in a numerical method used to simulate an earthquake response in buildings and in [42] the authors study why fractional dynamics is needed in machine learning and optimal randomness.
So far, different types of fractional derivatives have been defined, such as the Riemann-Liouville [39], Hadamard [34], Grunwald-Letnikov [46], Weyl [53], Caputo [28] and Caputo-Fabrizio [27] derivative. Although the Caputo-Fabrizio derivative, with its exponential Kernel, aimed to give a better description of the dynamics of systems with memory effect than other classic fractional derivatives, its associated integral was not fractional. To fix this defect, Atangana and Baleanu launched a new fractional operator in the Caputo and Riemann-Liouville sense [18]. Atangana-Baleanu derivative by the nonlocal and nonsingular kernel, Mittag-Leffler function, attracts more interest in applying in different fields. In the field of optimal control, the fractional derivative, in particular the Atangana-Baleanu yields improved results compared to the classical derivative (see [13, 19, 20, 32, 36]). In [26], the authors applied successfully the Caputo and Atangana-Baleanu fractional derivatives in data fitting.
Several studies have been devoted to the existence of the solutions of the fractional differential equations via various approaches, see [6, 7, 12, 17, 24, 40, 41, 51, 52]. Many latest studies in the existence theory focus on the fractional equations with the integral boundary conditions, which improve the classical conditions in the development of mathematical modeling, [11, 17, 35].
This paper which draws inspiration from the aforementioned works, investigates the existence of a solution to the following high-order fractional boundary value problem
in which \(t\in \left[ a,b\right]\) and completed by two integral boundary conditions of the form
where \(\zeta ,\gamma \in (n-1,n]\) for some \(n\in {\mathbb {N}}\), \(\lambda ,\mu \in {\mathbb {R}}\), \(f,g:[a,b]\times {\mathbb {R}}^{2n}\times {\mathbb {R}}^{2n}\rightarrow {\mathbb {R}}\) are continuous functions, and \({}^{ABC}_{a}{D}^{\zeta }\) denotes the \(\zeta\) order of Atangana-Baleanu fractional derivative in the (left) Caputo sense.
To investigate the existence of the solutions in the space \(C^{n-1}([a,b])\), which is equipped with the measure of noncompactness introduced in [16], we use the Schauder fixed point theorem [50] and the generalized Darbo fixed point theorem [30].
This paper is organized as follows. In section two, the needed preliminaries from the Atangana-Baleanu fractional calculus are stated. In the third section which is the main section, firstly by setting the appropriate solution space, we provide a modified version of the measure of noncompactness on the solution space. So by introducing the main conditions of the problem (1, 2), through some preliminary lemmas and propositions we prove the main result. Also two examples have been studied in order to show the application of the main theorem.
2 Preliminaries of Fractional Calculus
Firstly, we recall the Sobolev space \(H^1(a,b)\) where \(a<b\) and
Here \(u'\) is taken in the weak (distributional) sense; see [25] for more details.
Let us refer to [3, 18], from which most of the contents in this section has been adapted.
Definition 2.1
Let \(f \in H^{1} (a,b)\), \(\alpha \in [0,1]\), \(E_\alpha (t)=\sum _{k=0}^\infty \frac{t^k}{\Gamma (\alpha k+1)}\) is Mittag-Leffler function and \(B(\alpha )>0\) is a normalization function satisfying \(B(0)=B(1)=1\).
The Atangana-Baleanu fractional derivative with Mittag-Leffler nonlocal and nonsingular kernel in the (left) Caputo sense is defined by
and \(({}^{ABC}_{a}{D}^{1} f)(t)=f'(t)\). The associated fractional integral is
where \(({}_{a}I^{\alpha }f)(t)=\frac{1}{\Gamma (\alpha )}\int _{a}^{t}(t-s)^{\alpha -1}f(s)ds\) for \(\alpha >0\) and \(({}_{a}I^{0}f)(t)=f\).
Lemma 2.2
[3, 4, 18] If \(\alpha \in (0,1)\) and \(f \in H^{1}(a,b)\), then
The high-order ABC-derivative and integral are defined in [4] as follows.
Definition 2.3
Let \(\alpha \in (n-1,n]\) for some \(n\in {\mathbb {N}}\) and f be such that \(f^{(n-1)}\in H^{1}(a,b)\). Then the AB-derivative of \(\alpha\)-order in the (left) Caputo sense is defined by
Moreover, the associated fractional integral of \(\alpha\)-order is defined by
Remark 2.4
Obviously, for \(\alpha \in (0,1]\), since \(n=1\), (8) becomes (4). Also, since \(f^{0}(t)=f(t)\), thus the formula (7) generalizes the formula (3).
The following proposition explains the action of the AB- fractional Integral on the ABC-derivative and vice versa.
Proposition 2.5
Let \(\alpha \in (n-1,n]\) for some \(n\in {\mathbb {N}}\) and f be such that \(f ^{n-1}\in H^{1}(a,b)\). Then we have
Proof
The proof of (9) can be found in [4]. The proof of (10) is based on the definition, and one can check that easily. So we omit it for the sake of brevity. \(\square\)
3 Main Results
In this paper we discuss the existence of solution for system (1)-(2). Due to its boundary conditions we set \(C^{n-1}([a,b]) \times C^{n-1}([a,b])\) as the Banach framework for our investigation; where \(C^{n-1}([a,b])\) denotes the space of \(n-1\) times continuously differentiable functions on [a, b] with the standard norm, i.e.;
where \(\Vert g\Vert _\infty =\max \{\vert g(t)\vert , t\in [a,b]\}\)
Let \(\zeta ,\gamma \in (n-1,n]\) for some \(n\in {\mathbb {N}}\), \(\lambda ,\mu \in {\mathbb {R}}\) and \(\phi , \psi\) as two real functions on [a, b]. Suppose the system
with the following boundary conditions:
and
Lemma 3.1
Let \(\phi , \psi \in C([a,b])\) and \(\Delta := \lambda \mu -(\frac{(b-a)^{n}}{n!})^{2}\ne 0\). The solution \((x,y)\in C^{n-1}[a,b]\times C^{n-1}[a,b]\) of the boundary value system (11)-(16) is given by
where
and
in which \(c_{1}=\dfrac{n-\zeta }{B(\zeta -n+1)}\), \(c_{2}=\dfrac{\zeta -n+1}{B(\zeta -n+1)}\), \(d_{1}=\dfrac{n-\gamma }{B(\gamma -n+1)}\) and \(d_{2}=\dfrac{\gamma -n+1}{B(\gamma -n+1)}\).
Proof
In view of Proposition 2.5, by exerting the fractional integral \({}_{a}^{AB}I^{\zeta }\) on (11) under the condition (13), we obtain
Similarly, by exerting the fractional integral \({}_{a}^{AB}I^{\zeta }\) on equation (12) with (14), we obtain
By differentiating \(n-1\) times of (21) at \(t=b\) and regarding the condition (15), we derive
Similarly, from (22), regarding the condition (16), we derive
Now, by inserting the formulation of y(s) from (22) into (23), we deduce
Similarly, by inserting the formulation of x(s) from (21) into (24), we deduce
Obviously, from the system (25) and (26), one can obtain the quantities \(x^{(n-1)}(a)\) and \(y^{(n-1)}(a)\). Since \(x^{(n-1)}(a), y^{(n-1)}(a)\) are dependent to \(\phi ,\psi\), we denote \(x^{(n-1)}(a)\) and \(y^{(n-1)}(a)\) by \(s_{1}(\phi ,\psi )\) and \(s_{2}(\phi ,\psi )\), respectively. Thus, x(t) and y(t) have a form according to (17) and (18), respectively. \(\square\)
We consider two operators \(\Theta _{1}:C^{n-1}([a,b])\times C^{n-1}([a,b]) \rightarrow C^{n-1}([a,b])\) and \(\Theta _{2}:C^{n-1}([a,b])\times C^{n-1}([a,b]) \rightarrow C^{n-1}([a,b])\), where for each \((x,y) \in C^{n-1}([a,b])\times C^{n-1}([a,b])\), \(\Theta _{1}(x,y)\) and \(\Theta _{2}(x,y)\) are given by
where
and \(s_{1},s_{2},c_{1},c_{2},d_{1},d_{2}\) are introduced in Lemma 3.1. Hence regarding to the Lemma 3.1 evidently we deduce the following Corollary.
Corollary 3.2
Suppose that \(\Delta := \lambda \mu -(\frac{(b-a)^{n}}{n!})^{2}\ne 0\). Then \((x,y)\in C^{n-1}([a,b])\times C^{n-1}([a,b])\) is a solution of system (1)-(2) if and only if \(\Theta _1(x,y)=x\) and \(\Theta _2(x,y)=y\).
Let us define the operator \(\Theta : C^{n-1}([a,b]) \times C^{n-1}([a,b]) \rightarrow C^{n-1}([a,b]) \times C^{n-1}([a,b])\) by \(\Theta (x,y) = (\Theta _{1}(x,y), \Theta _{2}(x,y))\). Hence, due to Corollary 3.2, we derive the following Corollary.
Corollary 3.3
\((x,y)\in C^{n-1}([a,b])\times C^{n-1}([a,b])\) is a solution of system (1)-(2) if and only if (x, y) be a fixed point of \(\Theta\).
The various measures of noncompactness are defined on the known spaces such as bounded continuous functions \(BC(R^{+})\), [22]; K-times continuously differentiable functions \(C^{k}(\Omega )\), [16]; Lebesgue spaces \(L^{p}(R^{n})\), [10]; Holder spaces \(C^{k,\gamma }(\Omega )\), [49]; Sobolev spaces \(W^{n,p}([0,T])\), [14].
Let us recall, in details a measure of noncompactness on k-times continuously differentiable functions \(C^{k}([a,b])\), according to [16].
Theorem 3.4
[16] Suppose Y is a bounded subset of \(C^{k}([a,b])\). For \(g\in Y, \varepsilon > 0, i\in \lbrace 0,\ldots k \rbrace\), let \(\omega (g,\varepsilon , i)= max \lbrace \vert g^{(i)}(t_{2}) - g^{(i)}(t_{1})\vert ; \ t_{1},t_{2}\in ([a,b]), \ \vert t_{1}-t_{2}\vert < \varepsilon \rbrace\), \(\omega _{k}(g,\varepsilon )= max \lbrace \omega (g,\varepsilon , i); \ 0\le i \le k\rbrace\), \(\omega _{k}(Y,\varepsilon )= sup \lbrace \omega _{k}(g,\varepsilon ); \ g\in Y\rbrace\) Then, \({\overline{\omega }}_{k}: {\mathfrak {M}}_{C^{k}([a,b])}\rightarrow [0,\infty )\) which is given by \({\overline{\omega }}_{k}(Y)= \lim \limits _{\varepsilon \rightarrow 0} \omega _{k} (Y,\varepsilon )\) is a measure of noncompactness on \(C^{k}([a,b])\).
By the following Lemma, we may compute the measure of noncompactness \({\overline{\omega }}_{k}\) on \(C^{k}([a,b])\) with a more simple formulation.
Lemma 3.5
Every bounded subset of \((C^{k}([a,b]),\Vert . \Vert _{k})\) is relatively compact in \((C^{k-1}([a,b]),\Vert . \Vert _{k-1})\).
Proof
Firstly, we prove that every bounded subset of \((C^{k}([a,b]),\Vert . \Vert _{k})\) is uniformly equicontinuous in \((C^{k-1}([a,b]),\Vert . \Vert _{k-1})\). Let Y be as a bounded subset of \(C^{k}([a,b])\) and \(r > 0\) such that for each \(g\in Y, \ \Vert g \Vert _{k}\le r\). For each \(\varepsilon > 0\), if \(\vert t_{2}-t_{1}\vert < \varepsilon\), then by mean value theorem, we have
Hence, for each \(\varepsilon > 0\), by letting \(\delta =\dfrac{\varepsilon }{r}\), if \(\vert t_{2}-t_{1}\vert < \delta\) then \(\Vert g(t_{2})- g(t_{1})\Vert _{k-1}< \varepsilon\), for every \(g\in Y\). This means that Y is equicontinuous on \(C^{k-1}([a,b])\). Thus, if \(\lbrace g_{n}\rbrace \subset Y\) be an arbitrary sequence, and \(\varepsilon > 0\), by the uniform equicontinuity of \(\lbrace g_{n}\rbrace\) on \(C^{k-1}([a,b])\), there is a \(\delta > 0\) such that for all n, \(\Vert g_{n}(u) - g_{n}(v)\Vert _{k-1} < \dfrac{\varepsilon }{3}\), for all \(u,v\in [a,b]\) with \(\vert u-v\vert < \delta\).
Since [a, b] is compact, there is a finite number of points \(t_{1},\ldots ,t_{m}\) in [a, b] for which [a, b] is converted by the family of balls, centered in \(\lbrace t_{i}\rbrace _{i=1}^{m}\) with radius \(\delta\).
For each \(i\in \lbrace 1,\ldots ,m\rbrace\), \(\lbrace g_{n}(t_{i})\rbrace _{n=1}^{\infty }\) which is a bounded sequence in \({\mathbb {R}}\) has a convergent subsequence \(\lbrace g_{n_{j}}(t_i)\rbrace _{j=1}^{\infty }\) of \(\lbrace g_{n}(t_i)\rbrace _{n=1}^{\infty }\), that is convergent and as a result, Cauchy, which means \(\vert g_{n_{s}}(t_{i}) - g_{n_{l}}(t_{i})\vert < \dfrac{\varepsilon }{3}\) for \(i\in \lbrace 1,\ldots ,m\rbrace\), and \(n_{s},n_{l}\ge N\). Therefore, for any \(t\in [a,b]\), there is \(i\in \lbrace 1,\ldots ,m\rbrace\), such that \(\vert t-t_{i}\vert < \delta\) and so for \(n_{s},n_{l}\ge N\),
Thus \(\lbrace g_{n_{j}}\rbrace _{j=1}^{\infty }\) is Cauchy in \(C^{k-1}[a,b]\) and since \(C^{k-1}[a,b]\) is a complete normed space, \(\lbrace g_{n_{j}}\rbrace _{j=1}^{\infty }\) converges to a function in \(C^{k-1}([a,b])\).
Consequently, it has been proved that any bounded subset of \(C^{k}([a,b])\) is sequentially relatively compact and, thus, relatively compact in \(C^{k-1}([a,b])\). \(\square\)
Corollary 3.6
For every bounded subset Y of \(C^{k}([a,b])\), we have \({\overline{\omega }}_{k-1}(Y)=0\).
Corollary 3.7
For every bounded subset Y of \(C^{k}([a,b])\), we have \({\overline{\omega }}_{k}(Y)= \lim \limits _{\varepsilon \rightarrow 0} \sup \limits _{g\in Y} \ \omega (g,\varepsilon ,k)\).
Proof
For any \(g\in Y\), we have \(\omega _{k}(g,\varepsilon )= max\lbrace \omega _{k-1}(g,\varepsilon ), \omega (g,\varepsilon ,k)\rbrace\). Thus, \(\omega _{k}(Y,\varepsilon )= max\lbrace \omega _{k-1}(Y,\varepsilon ), \sup \limits _{g\in Y} \omega (g,\varepsilon ,k)\rbrace\).
Then,\({\overline{\omega }}(Y)= max \lbrace {\overline{\omega }}_{k-1}(Y), \lim \limits _{\varepsilon \rightarrow 0} \sup \limits _{g\in Y} \ \omega (g,\varepsilon ,k)\rbrace =\lim \limits _{\varepsilon \rightarrow 0} \sup \limits _{g\in Y} \ \omega (g,\varepsilon ,k).\) \(\square\)
In the sequel, we set the following conditions on the system (1)–(2).
(C0) \(\Delta := \lambda \mu -(\frac{(b-a)^{n}}{n!})^{2}\ne 0\).
(C1) \(f,g: [a,b] \times {\mathbb {R}}^{2n}\rightarrow {\mathbb {R}}\) are continuous functions satisfying:
and
where \(\varphi : [0,\infty )^{2}\rightarrow [0,\infty )\) is a non-decreasing and upper semicontinuous function with respect to it’s components and for every \(t > 0, \ \varphi (t,t) < t\).
(C2) \({\mathcal {K}}_{0}:= \max \lbrace \vert f(t,0,\ldots ,0)\vert ,\vert g(t,0,\ldots ,0)\vert ; \ t\in [a,b] \rbrace < \infty\).
For \(r>0\), denote \(B_r:=\{x\in C^{n-1}([a,b]); \Vert x\Vert _{n-1}\le r\}\).
Lemma 3.8
For every \((x,y)\in B_{r}\times B_{r}\), we have
Proof
Similarly, the estimate for \(\psi _{x,y}\) will derived. \(\square\)
Lemma 3.9
For any \(m \ge 0\) and \((x,y)\in B_{r} \times B_{r}\), we have
and
Proof
Similarly, the estimate for \({}_{a}I^{m}\psi _{x,y}\) will derived. \(\square\)
Proposition 3.10
For each \(m > 0\), \(\lbrace {}_{a}I^{m}\phi _{x,y}\rbrace _{(x,y)\in ( B_{r} \times B_{r})}\) and \(\lbrace {}_{a}I^{m}\psi _{x,y}\rbrace _{(x,y)\in ( B_{r} \times B_{r})}\) are equicontinuous subsets of \(C([a,b],\Vert . \Vert _{\infty })\).
Proof
For each \(\varepsilon > 0\) and \(t_{1},t_{2}\in [a,b]\), with \(\vert t_{1}-t_{2}\vert < \varepsilon\), we have
Since the right hand side of the last inequality tends to zero, uniformly, with respect to \((x,y)\in B_{r} \times B_{r}\) when \(\varepsilon \rightarrow 0\), we can find that \(\delta = \delta (\varepsilon ) > 0\), where if \(\vert t_{1}-t_{2} \vert < \delta\), then \(\Vert ({}_{a}I^{m}\phi _{x,y})(t_{2}) - {}_{a}I^{m}\phi _{x,y})(t_{1})\Vert < \varepsilon\), for all \((x,y)\in B_{r} \times B_{r}\). \(\square\)
Lemma 3.11
For every \((x,y) \in B_{r}\times B_{r}\), we have \(\vert s_{1} (\phi _{x,y},\psi _{x,y})\vert \le \varsigma _{1}(\varphi (r,r)+{\mathcal {K}}_{0})\) and \(\vert s_{2} (\phi _{x,y},\psi _{x,y})\vert \le \varsigma _{2}(\varphi (r,r)+{\mathcal {K}}_{0});\) where
and
in which \(c_1, c_2, d_1, d_2\) are introduced in the Lemma 3.1.
Proof
By the definition of the Riemann-Liouville fractional integral of order \(\alpha\), which is
and since \(\Vert \phi _{x,y}\Vert _\infty , \Vert \psi _{x,y}\Vert _\infty\) are
and by using Lemma 3.8, we have
Similarly, it can be shown that \(\vert s_{2} (\phi _{x,y},\psi _{x,y})\vert \le \varsigma _{2}(\varphi (r,r)+{\mathcal {K}}_{0})\). \(\square\)
Proposition 3.12
There exists \(r > 0\) such that \(T_{1},T_{2}\) maps \(B_{r}\times B_{r}\) onto \(B_{r}\), provided \(\max \{\eta _{1},\eta _{2}\} < 1\), where
and
in which \(\varsigma _{1},\varsigma _{2}\) are introduced in the Lemma 3.11.
Proof
For every \(m\in \lbrace 0,\ldots , n-2\rbrace\), we have
and
Thus, by Lemma 3.9
Hence, by Lemma 3.11
Thus, \(\Theta _{1}\) is a self map on \(B_{r}\times B_{r}\) provided \(\eta _{1}\varphi (r,r)+ \xi _{1} \le r\). Since \(\varphi (r,r) < r\) and due to the assumption, \(\eta _{1} < 1\), the desired result will be obtained for \(r > \dfrac{\xi _{1}}{1- \eta _{1}}\). Similarly, we obtain
Thus, for \(r >\dfrac{\xi _{2}}{1- \eta _{2}}\), \(\Theta _{2}\) maps \(B_{r}\times B_{r}\) onto \(B_{r}\). Hence, by choosing \(r > \max \lbrace \dfrac{\xi _{1}}{1-\eta _{1}}, \dfrac{\xi _{2}}{1-\eta _{2}} \rbrace\), both \(\Theta _{1}\) and \(\Theta _{2}\) are self maps on \(B_{r}\times B_{r}\). \(\square\)
Proposition 3.13
\(\lbrace \Theta _{1}(x,y)\rbrace _{(x,y)\in B_{r} \times B_{r}}\) and \(\lbrace \Theta _{2}(x,y)\rbrace _{(x,y)\in B_{r} \times B_{r}}\) are equicontinuous subsets of \(C^{n-1}([a,b])\), provided \(\varphi (t,s)\le {\overline{\varphi }}(t)\) for some functions \({\overline{\varphi }}: [0,\infty )\rightarrow [0,\infty )\) which is non-decreasing and upper semicontinuous and for \(t > 0\), \({\overline{\varphi }}(t) < t\).
Proof
Regarding Lemma 3.5, we shall only check that \(\lbrace \dfrac{d^{n-1}\Theta _{1}(x,y)}{dt^{n-1}}\rbrace _{(x,y) \in B_{r}\times B_{r}}\) be equicontinuous on [a, b]. Since
and according to Proposition 3.10, \(\lbrace ({}_{a}I^{\zeta -n+1})\rbrace \phi _{(x,y)\in B_{r}\times B_{r} }\) is equicontinuous. Thus it remains that we show \(\lbrace \phi _{x,y} \rbrace\) is equicontinuous on \(B_{r}\times B_{r}\). Indeed,
in which
Since \(\lim \limits _{\delta \rightarrow 0}\Lambda (\delta ) = 0\), thus for an arbitrary \(\varepsilon > 0\) we can choose \(\delta _1 > 0\) where for \(\delta <\delta _{1}\), we have \(\Lambda (\delta ) < \dfrac{\varepsilon }{2}\) and by taking \(\delta \le min\lbrace \delta _{1},\dfrac{\varepsilon }{2r} \rbrace\), we have
\(\square\)
Theorem 3.14
Assume the conditions (C0)–(C2). Then the boundary value system (1)–(2) has at least one solution in \(C^{n-1}([a,b])\), provided \(\max \{\eta _{1},\eta _{2} \}< 1\), where \(\eta _{1},\eta _{2}\) are introduced in the proposition 3.12.
Proof
Let the well defined operator \(\Theta : C^{n-1}([a,b]) \times C^{n-1}([a,b]) \rightarrow C^{n-1}([a,b]) \times C^{n-1}([a,b])\) by \(\Theta (x,y) = (\Delta _{1}(x,y), \Theta _{2}(x,y))\), where \(\Theta _1, \Theta _2\) are introduced in (27) and (28); respectively. For \(r > 0\) introduced in the Proposition 3.12, we have \(\Theta (B_{r} \times B_{r})\subset B_{r} \times B_{r}\). Let us prove the assertion in two cases. If \(\varphi\) satisfies the condition of Proposition 3.13, \(\Theta _{1}\) and \(\Theta _{2}\) are equicontinuous on \(B_{r} \times B_{r}\), where regarding the Arzela-Ascoli Theorem, \(\Theta _1\), \(\Theta _2\) are compact operators on \(B_r\times B_r\). Therefore, by Schauder fixed point theorem [8], \(\Theta\) has at least one fixed point in \(B_{r} \times B_{r}\), and due to the Corollary 3.3, the boundary value system (1)–(2) has at least one solution in \(B_{r} \times B_{r}\).
In the case that \(\varphi\) does not satisfy the condition of Proposition 3.13, since \(\varphi\) can not be replaced by \({\overline{\varphi }}\), which is independent of s we can not necessarily deduce that \(\Theta _{1},\Theta _{2}\) are compact operators, whereas by using the concept of measure of noncompactness on \(C^{n-1}([a,b])\) and applying a generalization of Schauder fixed point theorem, named generalized Darbo theorem (see [9], Corollary 2.2) we deduce the existence of a fixed point for the operator T. Firstly, we claim that
To this end, let \(\varepsilon > 0, t_{1},t_{2}\in [a,b]\) and \(\vert t_{1}-t_{2}\vert \le \varepsilon\). Then we have
Moreover,
where
Hence
Thus,
By uniform continuity of f on \([a,b] \times [-r,r]^{2n-2}\), \(\Lambda (\varepsilon ) \rightarrow 0\), as \(\varepsilon \rightarrow 0\). Further, from Proposition 3.10 we deduce that \(\omega ( {}_{a}I^{\zeta -n+1} \phi _{x,y},\varepsilon ,0)\rightarrow 0\) as \(\varepsilon \rightarrow 0\) uniformly with respect to \((x,y)\in B_{r}\times B_{r}\). Therefore,
Similarly, we have
If \(\rho : {\mathfrak {M}}_{C^{n-1}([a,b]) \times C^{n-1}([a,b])}\rightarrow [0,\infty )\) is defined by \(\rho (Y_{1} \times Y_{2})= max\lbrace {\overline{\omega }}_{n-1}(Y_{1}),{\overline{\omega }}_{n-1}(Y_{2}) \rbrace\), it is known that \(\rho\) is a measure of noncompactness on \(C^{n-1}([a,b]) \times C^{n-1}([a,b])\), based on [21]. Further, since, \(\Theta (B_{r}\times B_{r})\subseteq \Theta _{1}(B_{r}\times B_{r})\times \Theta _{2}(B_{r}\times B_{r})\), we have
Indeed \(\max \lbrace c_{1},d_{1}\rbrace<\max \lbrace \eta _{1},\eta _{2}\rbrace < 1\) Thus, by applying Corollary 2.2 of [9], for \(\Upsilon (t)= \max \lbrace c_1,d_1\rbrace \varphi (t)\) which is admissible, \(\Theta\) has at least one fixed point in \(C^{n-1}([a,b]) \times C^{n-1}([a,b])\); which, due to the Corollary 3.3, means that the boundary value system (1),(2) has at least one solution in \(B_{r} \times B_{r}\). \(\square\)
Example 3.15
Consider
for \(t\in [0,1]\), completed by two integral boundary conditions of the form
This system is an example of the system (1)-(2) with \([a,b]=[0,1]\), \(\zeta =\frac{10}{6},\gamma =\frac{7}{4},\) and thus \(n=2\). Moreover, \(\lambda =1, \mu =2\),
and
evidently, according to the condition (C0) we have \(\Delta = \dfrac{7}{4} \ne 0\). Further, we have
Moreover,
Therefor, with \(\varphi (t,s):=\frac{1}{2}t+\frac{9}{20}s\) condition (C1) is fulfilled. Also according to condition (C2), \(\kappa _0=\frac{1}{10}\) and using maple, we obtain \(\varsigma _1=0.5749179761\), \(\varsigma _2=0.5881253745\) and so \(\eta _1=0.7892822650\) and \(\eta _2=0.8013351624\). Hence due to the Theorem 3.14, the system (31)-(32) has a solution in \(C^1([0,1])\).
Here, we applied the following normalization function, that is involved in the Definition of Atangana-Baleanu fractional derivative,
Example 3.16
Consider
for \(t\in [1,2]\), completed by two integral boundary conditions of the form
This system is an example of the system (1)-(2) with \([a,b]=[1,2]\), \(\zeta =\frac{19}{7},\gamma =\frac{7}{3},\) and thus \(n=3\). Moreover, \(\lambda =\frac{1}{4}, \mu =3\),
and
evidently, according to the condition (C0) we have \(\Delta = \dfrac{13}{18} \ne 0\). Further, we have
The last inequality is due to the the monotonicity of \(\ln (1+t)\) and the boundedness of the derivative of \(e^{\cos (t)}\). Moreover, we have
where the last inequality comes from the boundedness of the derivative of the functions \(\frac{t^2}{1+t^2}\), \(\sqrt{1+t}\), \(\sin t\) and \(e^{-t}\).
Since
Therefor, by considering
condition (C1) is fulfilled. Also according to condition (C2), \(\kappa _0=\frac{1}{10}\) and using maple, we obtain \(\varsigma _1=0.7570714840\), \(\varsigma _2=0.2708016058\) and so \(\eta _1=0.9709550528\) and \(\eta _2=0.4787913740\). Hence due to the Theorem 3.14, the system (33, 34) has a solution in \(C^1([0,1])\).
Data Availibility Statement
Not applicable.
References
AlAhmad, R., AlAhmad, Q., Abdelhadi, A.: Solution of fractional autonomous ordinary differential equations. J. Math. Comput. Sci. 27(1), 59–64 (2022)
Abdeljawad, T., Agarwal, R.P., Karapınar, E., Kumari, P.S.: Solutions of the nonlinear integral equation and fractional differential equation using the technique of a fixed point with a numerical experiment in extended b-metric space. Symmetry 11(5), 686 (2019)
Abdeljawad, T., Baleanu, D.: Discrete fractional differences with nonsingular discrete Mittag-Leffler kernels. Adv. Differ Equ. 232, 1–18 (2016)
Abdeljawad, T.: Lyapunov type inequality for fractional operators with nonsingular Mittag-Leffler kernel. J. Inequal. Appl. 1, 1–11 (2017)
Abdolrazaghi, F., Razani, A.: A unique weak solution for a kind of coupled system of fractional Schrodinger equations. Opusc. Math. 40(3), 313–322 (2020)
Abdo, M.S., Panchal, S.K.: Fractional integro-differential equations involving \(\Psi\) -Hilfer fractional derivative. Adv. Appl. Math. Mech. 11(2), 338–359 (2019)
Abdo, M.S., et al.: On fractional boundary value problems involving fractional derivatives with Mittag-Leffler kernel and nonlinear integral conditions. Adv Differ Equ. 2021(1), 1–21 (2021)
Agarwal, R., Meehan, M., O’regan, D.: Fixed point theory and applications. Cambridge University Press, Cambridge (2004). p. 141
Aghajani, A., Allahyari, R., Mursaleen, M.: A generalization of Darbos theorem with application to the solvability of systems of integral equations. J. Comput. Appl. Math. 260, 68–77 (2014)
Aghajani, A., O’Regan, D., Shole Haghighi, A.: Measures of noncompactness on \(L^{p}({\mathbb{R} }^{n})\) and applications. CUBO A Math J 17(1), 85–97 (2015)
Ahmad, B., Ntouyas, S.K.: A fully Hadamard type integral boundary value problem of a coupled system of fractional differential equations. Fract. Calc. Appl. Anal. 17(2), 348–360 (2014)
Ahmad, B., Ntouyas, S.K., Alsaedi, A.: Sequential fractional differential equations and inclusions with semi-periodic and nonlocal integro-multipoint boundary conditions. J. King Saud Univ-Sci 31(2), 184–193 (2019)
Alkahtani, B.S.T., Koca, I., Atangana, A.: Analysis of a new model of HINI spread: model obtained via Mittag-Leffler function. Adv Mech Eng. 9(8), 1–8 (2017)
Allahyari, R., Arab, R., Shole Haghighi, A.: Measure of noncompactness in a Sobolev space and integro-differential equations. Bull. Aust. Math. Soc 3, 497–506 (2016)
Alqahtani, B., Aydi, H., Karapınar, E., Rakocevi, V.: A solution for Volterra fractional integral equations by hybrid contractions. Mathematics 7(8), 694 (2019)
Arab, R., Allahyari, R., Shole Haghighi, A.: Construction of measures of noncompactness of \(C^ k (Omega)\) and \(C^ k_0\) and their application to functional integral-differential equations.". Bull Iran Math Soc 43(1), 53–67 (2017)
Atangana, A., Baleanu, D.: Caputo-Fabrizio derivative applied to groundwater flow within confined aquifer. J Eng Mech. 143(5), 1–5 (2017)
Atangana, A., Baleanu, D.: New fractional derivative with non-local and non-singular kernel. Theory and application to heat transfer model. Therm. Sci. 20(2), 1–7 (2016)
Abdullah, T.Q.S., Xiao, H., Huang, G., Al-Sadi, W.: Stability and existence results for a system of fractional differential equations via Atangana-Baleanu derivative with\(\phi _{p}\)-Laplacian operator. J. Math. Comput. SC. 27(2), 184–195 (2022)
Bokhari, A., Baleanu, D., Belgacem, R.: Application of Shehu transform to Atangana-Baleanu derivatives. J. Math. Comput. SC. 20(2), 101–107 (2020)
Banaś, J., Goebel, K.: Measures of noncompactness in Banach spaces. Lecture notes in pure and applied mathematics, p. 60. Dekker, New York (1980)
Banaś, J.: On measures of noncompactness in Banach spaces. Commentationes Mathematicae Universitatis Carolinae. 21(1), 131–143 (1980)
Behboudi, F., Razani, A., Oveisiha, M.: Existence of a mountain pass solution for a nonlocal fractional (p, q)-Laplacian problem. Bound. Value Probl. 2020, 149 (2020)
Benchohra, M., Bouriah, S., Nieto, J.J.: Terminal value problem for differential equations with Hilfer-Katugampola fractional derivative. Symmetry 11(5), 672 (2019)
Functional analysis, Sobolev spaces and partial differentialequations. Springer, New York. (2011)
Can, N.H., Jafari, H., Ncube, M.N.: Fractional calculus in data fitting. Alexandria Eng J. 59(5), 3269–3274 (2020)
Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Progr. Fract. Differ. Appl. 1(2), 73–85 (2015)
Caputo, M.: Linear models of dissipation whose Q is almost frequency independent. Ann Geophys. 19(4), 383–393 (1966)
Dadkhah, E., Shiri, B., Ghaffarzadeh, H., Baleanu, D.: Visco-elastic dampers in structural buildings and numerical solution with spline collocation methods. J Appl Math Comput. 63(1), 29–57 (2020)
Darbo, G.: Punti uniti in trasformazioni a codominio non compatto. Rendiconti del Seminario matematico della Universitá di Padova. 24, 84–92 (1955)
Fallahgoul, H.A., Focardi, S.M., Fabozzi, F.J.: Fractional calculus and fractional processes with applications to financial economics. Theory and application. Elsevier Academic Press, London (2017)
Gómez-Aguilar, J.F., Atangana, A., Morales-Delgado, N., Fabian, V.: Electrical circuits RC, LC and RL described by Atangana-Baleanu fractional derivatives. Int J Circt Theory Appl. 45(11), 1514–33 (2017)
Golbabai, A., Nikan, O., Molavai Arabshai, M.: Numerical approximation of time fractional advection-dispersion model arising from solute transport in rivers. TWMS J. Pure Appl. Math. 10(1), 1–17 (2019)
Hadamard, J.: Essai sur l’étude des fonctions, données par leur développement de Taylor. J. Mat. Pure Appl. 8, 101–186 (1892)
Henderson, J., Luca, R., Tudorache, A.: On a system of fractional differential equations with coupled integral boundary conditions. Fract. Calc. Appl. Anal. 18, 361–386 (2015)
Jajarmi, A., Baleanu, D.: On the fractional optimal control problems with a general derivative operator. Asian J Control. 23(2), 1062–1071 (2021)
Karapınar, E., Abdeljawad, T., Jarad, F.: Applying new fixed point theorems on fractional and ordinary differential equations. Adv. Differ. Equ. 421, 1–25 (2019)
Keten, A., Yavuz, M., Baleanu, D.: Nonlocal Cauchy problem via a fractional operator involving power Kernel in Banach Spaces. Fractal Fract. 3(2), 27 (2019)
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and applications of fractional differential equations. North-Holland mathematics studies, p. 204. Elsevier, Amsterdam (2006)
Mehmood, N., et al.: Existence results for ABC-fractional differential equations with non-separated and integral type of boundary conditions. Fractals. 29(5), 2140016 (2021)
Mei, Z.D., Peng, J.G., Gao, J.H.: Existence and uniqueness of solutions for nonlinear general fractional differential equations in Banach spaces. Indag. Math. 26(4), 669–678 (2015)
Niu, H., Chen, Y.: Why do big data and machine learning entail the fractional dynamics. Entropy 23(3), 297 (2021)
Nikan, O., Molavi-Arabshai, S.M., Jafari, H.: Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete Contin. Dyn. Syst. S 14(10), 3685–3701 (2021)
Nyamoradi, N., Razani, A.: Existence to fractional critical equation with Hardy-Littlewood Sobolev nonlinearities. Acta Math Sci. 41(4), 1321–1332 (2021)
Oderinu, R.A., Owolabi, J., Taiwo, M.: Approximate solutions of linear time-fractional differential equations. J. Math. Comput. Sci. 29(1), 60–72 (2023)
Podlubn, I.: Fractional Differential Equations. Academic Press, San Diego-London (1999)
Prasad, R., Kumar, K., Dohare, R.: Caputo fractional order derivative model of Zika virus transmission dynamics. J. Math. Comput. Sci. 28(2), 145–157 (2023)
Sabatier, J., Agrawal, R.P., Machado, J.T.: Advances in fractional calculus. Theoretical developments and applications in physics and engineering. Springer, Dordrecht. 4(9 )(2007)
Saiedinezhad, S.: On a measure of noncompactness in the Holder space \(C^{k,\gamma }(\Omega )\) and its application. J Comput Appl Math 346, 566–571 (2019)
Schauder, J.: Der Fixpunktsatz in funktionalraümen. Studia math. 2(1), 171–180 (1930)
Srivastava, H.M.: Remarks on some families of fractional-order differential equations. Integral Transforms Spec. Funct. 28(7), 560–564 (2017)
Sutar, S.T., Kucche, K.D.: Existence and data dependence results for fractional differential equations involving atangana-baleanu derivative. Rendiconti del Circolo Matematico di Palermo Series. 71(2), 647–663 (2022)
Weyl, H.: Bemerkungen zum Begriff des Differentialquotienten gebrochener Ordnung. Zurich. Naturf. Ges. 62(1–2), 296–302 (1917)
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The authors profoundly thank the anonymous referees for their careful reading of the manuscript and giving valuable suggestions that significantly improved the presentation of the paper.
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MS: participated in the conceptualization, data curation, investigation, methodology, implementation, validation, visualization and writing the original draft. SS: participated in the conceptualization, administration, validation, visualization and writing of the manuscript. MBG: participated in the formal analysis, investigation, supervision, review and editing of the manuscript. All authors read and approved the final manuscript.
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Sangi, M., Saiedinezhad, S. & Ghaemi, M.B. A System of High-Order Fractional Differential Equations with Integral Boundary Conditions. J Nonlinear Math Phys 30, 699–718 (2023). https://doi.org/10.1007/s44198-022-00103-3
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DOI: https://doi.org/10.1007/s44198-022-00103-3
Keywords
- Atangana-Baleanu-Caputo fractional derivative
- Integral boundary condition
- Measure of noncompactness
- Fixed point theory