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Existence of solutions for equations and inclusions of multiterm fractional q-integro-differential with nonseparated and initial boundary conditions
Journal of Inequalities and Applications volume 2019, Article number: 273 (2019)
Abstract
The goal of this paper is to investigate existence of solutions for the multiterm nonlinear fractional q-integro-differential \({}^{c}D_{q}^{\alpha } u(t)\) in two modes equations and inclusions of order \(\alpha\in(n -1, n]\), with non-separated boundary and initial boundary conditions where the natural number n is more than or equal to five. We consider a Carathéodory multivalued map and use Leray–Schauder and Covitz–Nadler famous fixed point theorems for finding solutions of the inclusion problems. Besides, we present results whenever the multifunctions are convex and nonconvex. Lastly, we give some examples illustrating the primary effects.
1 Introduction
Fractional calculus and q-calculus are the significant branches in mathematical analysis. The field of fractional calculus has countless applications, and the subject of fractional differential equations ranges from the theoretical views of existence and uniqueness of solutions to the analytical and mathematical methods for finding solutions (for instance, see [1,2,3,4]). There has been an intensive development in fractional differential equations and inclusion (for example, see [5,6,7,8,9,10,11]). During the last two decades, the fractional differential equations and inclusions, both differential and q-differential, were developed intensively by many authors for a variety of subjects (for instance, consider [12,13,14,15,16,17,18,19,20]). In recent years, there are many published papers about differential and integro-differential equations and inclusions which are valuable tools in the modeling of many phenomena in various fields of science (for more details, see [21,22,23,24,25,26,27,28,29,30,31,32] and references therein).
In this article, motivated by [7, 21, 27], among these achievements and following results, we are working to stretch out the analytical and computational methods of checking of positive solutions for fractional q-integro-differential equation
for almost all \(t \in\overline{J} = [0,\delta]\), with \(\delta> 0\), and the inclusion case
for each \(t \in I =[0,1]\), with the initial and antiperiodic boundary conditions as follows:
where \({}^{c}D_{q}^{\alpha}\) denotes the Caputo fractional q-derivative, \(\alpha\in(n-1,n]\), with the natural number n more than or equal to five, \(\beta_{1j_{1}}\), \(\beta_{2j_{2}}\), \(\beta_{3j_{3}}\), in problems (1) and (2), belonging to \(J_{0}=(0,1)\), \(J_{1}=(1,2)\), \(J_{2}=(2,3)\), for \(j_{1}\in N_{k_{1}}\), \(j_{2} \in N_{k_{2}}\), \(j_{3} \in N_{k_{3}}\), respectively, with \(N_{k}=\{1, 2, \dots, k\}\), while the map \(\varphi_{i}\), in problem (1) and (2), is defined by
where the real-valued functions \(\mu_{i}\), \(\theta_{i}\) defined on \(\overline{J}^{2} \), \(\overline{J}^{2}\times\mathbb{R}^{7}\), respectively, are continuous, for \(i=1,2\), \(\gamma_{i1}\), \(\gamma_{i2}\), \(\gamma _{i3}\) belong to \(J_{0}\), \(J_{1}\), \(J_{2}\), respectively, a continuous function f maps \(\overline{J} \times\mathcal{R}^{m}\) to \(\mathbb{R}\), a map \(T : I \times\mathcal{R}^{m} \to P(\mathbb{R})\) is a multifunction, here \(\mathcal{R}^{m}=\mathbb{R}^{6+k_{1}+k_{2}+k_{3}}\) and \(P(\mathbb{R})= \{A \subseteq\mathbb{R} \mid A \neq\emptyset \}\), in (4), \(a_{1} + a_{2} \neq0\), and the constants \(p_{1}\), \(p_{2}\), \(p_{3} \) in (5) belong to \(J_{0}\), \(J_{1}\), \(J_{2}\), respectively.
As before, we remind some of the previous works briefly. In 1910, the subject of q-difference equations was introduced by Jackson [33,34,35]. After that, at the beginning of the last century, studies on q-difference equations appeared in many works, especially in Carmichael [36], Mason [37], Adams [38], Trjitzinsky [39], Agarwal [40]. An excellent account in the study of fractional differential and q-differential equations can be found in [1, 3, 41,42,43]. In 2009, Su and Zhang investigated the problem
for all \(t \in(0,1)\), where \(a_{1} b_{1} + a_{1} b_{2} + a_{2}b_{1} >0\), \(\eta \in(1, 2]\), \(\nu\in(0, 1]\), for \(i=1, 2\), \({a_{i}, b_{i} \geq0}\), a continuous function f maps \(I=[0,1] \times \mathbb{R}^{2} \) to \(\mathbb{R,}\) and \({}^{c}D_{0^{+}}^{\eta}\) is the Caputo’s fractional derivative [19]. In 2011, Agarwal, O’regan and Staněk investigated the problem \({}^{c}D^{\alpha}f (x) + T (x, f(x), f'(x), {}^{c}D^{\mu}f(x) )=0\), \(f(1)= f'(0) =0\), for all \(t \in[0,1]\), where \(\mu\in(0,1)\), and as always \({}^{c}D^{\alpha}\) is the Caputo fractional derivative of order α with \(\alpha\in(1,2)\), positive function T is a scalar \(L^{\kappa}\)-Carathéodory on \(I \times E\) with \(E= (0,\infty)^{3}\) and \(\kappa(\alpha-1)> 1\), such that \(T(t, x_{1}, x_{2}, x_{3})\) may be singular at 0 in one dimension of its space variables \(x_{1}\), \(x_{2}\), and \(x_{3}\) [44]. In 2012, Ahmad et al. discussed the existence and uniqueness of solutions for the fractional q-difference equations \({}^{c}D_{q}^{\alpha}u(t)= T ( t, u(t) ) \), \(\alpha_{1} u(0) - \beta_{1} D_{q} u(0) = \gamma_{1} u(\eta_{1})\) and \(\alpha_{2} u(1) - \beta_{2} D_{q} u(1) = \gamma_{2} u(\eta_{2})\), for \(t \in I\), where \(\alpha\in(1, 2]\), \(\alpha_{i}, \beta _{i}, \gamma_{i}, \eta_{i} \in\mathbb{R}\), for \(i=1,2\) and \(T \in C([0,1] \times\mathbb{R}, \mathbb{R})\) [13].
In 2013, Baleanu, Rezapour and Mohammadi et al., by using fixed-point methods, studied the existence and uniqueness of a solution for the nonlinear fractional differential equation boundary-value problem \(D^{\alpha}u(t) = f(t,u(t))\) with a Riemann–Liouville fractional derivative via the different boundary-value conditions: \(u(0)=u(\delta )\), as well as the three-point boundary condition \(u(0) =\beta_{1} u(\eta )\) and \(u(\delta) = \beta_{2} u(\eta)\), where \(\delta>0\), \(t \in I=[0,\delta]\), \(\alpha\in(0,1)\) \(\eta\in(0, \delta)\) and \(0<\beta _{1} < \beta_{2} < 1\) [12]. In 2016, Ahmad et al. investigated solutions of the problem
for each \(x \in I\), where \(\eta\in(2, 3]\), \(\nu\in[0, 3]\), \({}^{c}D_{q}^{\eta}\) denotes Caputo fractional q-derivative, \(q \in(0,1)\), and F mapping \(I \times A\) to \(\mathcal{P} (\mathbb{R}) \) is a multivalued map, here \(\mathcal{P} (\mathbb{R})\) is a power set of \(\mathbb{R}\) and \(A=\mathbb{R}^{3}\) [15]. In 2017, Baleanu, Mousalou and Rezapour presented a new method to investigate some fractional integro-differential equations involving the Caputo–Fabrizio derivative
where \(t\in[0,1]\), \(M(\alpha)\) is a normalization constant depending on α such that \(M(0) =M(1) = 1\), and proved the existence of approximate solutions for these problems [10]. Also in the same year, they introduced a new operator called the infinite coefficient-symmetric Caputo–Fabrizio fractional derivative and applied it to investigate the approximate solutions for two infinite coefficient-symmetric Caputo–Fabrizio fractional integro-differential problems [11]. In addition to this, Akbari and Rezapour, by using the shifted Legendre and Chebyshev polynomials, discussed the existence of solutions for a sum-type fractional integro-differential problem under the Caputo differentiation [6]. Over the past three years, Baleanu, Rezapour and many others, by using the Caputo–Fabrizio derivative, achieved innovative and remarkable results for solutions of fractional differential equations [22, 23, 25, 28, 30, 32]. In the following year, Rezapour and Hedayati investigated the existence of solutions for the inclusion \({}^{c}D^{\alpha}x(t) \in F (x, f(x), {}^{c}D^{\beta}f(x), f' (x) )\) for each \(x\in I\) with the conditions \({}^{c}D^{\beta} f(0) -\int_{0}^{\eta_{1}} f(r) \,dr= f(0) + f' (0)\) and \({}^{c}D^{\beta} f(1) - \int_{0}^{\eta_{2}} f(r) \,dr = f(1) + f' (1)\), where multifunction F maps \([0,1] \times\mathbb{R}^{3} \) to \(2^{ \mathbb {R}}\) and is compact-valued, while \({}^{c}D^{\alpha}\) is the Caputo differential operator [16]. In 2019, Samei et al. discussed the fractional hybrid q-differential inclusions \({}^{c}D_{q}^{\alpha}( x / F ( t, x, I_{q}^{\alpha_{1}} x, \dots, I_{q}^{\alpha_{n}} x ) ) \in T ( t, x, I_{q}^{\beta_{1}} x, \dots, I_{q}^{\beta_{k}} x )\), with the boundary conditions \(x(0) =x_{0}\) and \(x(1)=x_{1}\), where \(1 < \alpha\leq2\), \(q \in(0,1)\), \(x_{0}, x_{1} \in\mathbb{R}\), \(\alpha_{i} >0\), for \(i=1, 2, \ldots, n\), \(\beta_{j} > 0\), for \(j=1, 2, \ldots, k\), \(n, k\in\mathbb{N}\), \({}^{c}D_{q}^{\alpha}\) denotes Caputo type q-derivative of order α, \(I_{q}^{\beta}\) denotes Riemann–Liouville type q-integral of order β, \(F: J \times\mathbb{R}^{n} \to(0,\infty)\) is continuous, and T mapping \(J\times\mathbb{R}^{k}\) to \(P (\mathbb{R})\) is a multifunction [17].
2 Preliminaries
As before, we point out some of the fundamental facts on the fractional q-calculus which are needed in the next sections (for more information, consider [1,2,3, 33]). Then, some well-known fixed point theorems and definitions are presented.
The proposed method for calculating \((a-b)_{q}^{(\alpha)}\)
The proposed method for calculating \(\varGamma_{q}(x)\)
The proposed method for calculating \((D_{q} f)(x)\)
The proposed method for calculating \((I_{q}^{\alpha}f)(x)\)
The proposed method for calculating \(\varLambda_{2}\)
Assume that \(q \in(0,1)\) and \(a \in\mathbb{R}\). Define \([a]_{q}=\frac {1-q^{a}}{1-q}\) and consider the power function \((a-b)_{q}^{(n)}= \prod_{k=0}^{n-1} (a - bq^{k})\) whenever \(n \in\mathbb{N}\) and \((a-b)_{q}^{(n)}=1\) where \(n=0\), and \(a, b \in\mathbb{R}\) [1, 3, 33]. Also, for \(\alpha\in \mathbb{R}\) and \(a \neq0\), we define [33]
If \(b=0\), then it is clear that \(a^{(\alpha)}= a^{\alpha}\) (Algorithm 1). The q-Gamma function is defined by \(\varGamma_{q}(x) = ((1-q)^{(x-1)})/((1-q)^{x -1}) \), where \(x \in\mathbb{R} \setminus\{ 0, -1, -2, \dots\}\) and satisfies \(\varGamma_{q} (x+1) = [x]_{q} \varGamma_{q} (x)\) [3, 33, 45, 46]. The value of q-Gamma function, \(\varGamma_{q}(x)\), for input values q and x will have a counting number of sentences n in summation by simplifying analysis. For this design, we prepare a pseudo-code description of the technique for estimating q-Gamma function of order n which is shown in Algorithm 2. For any positive numbers α and β, the q-Beta function is defined by [41]
The q-derivative of function f is defined by \((D_{q} f)(x) = \frac {f(x) - f(qx)}{(1- q)x}\) and \((D_{q} f)(0) = \lim_{x \to0} (D_{q} f)(x)\), which is shown in Algorithm 3 [3, 38, 41]. Also, the higher order q-derivative of a function f is defined by \((D_{q}^{n} f)(x) = D_{q}(D_{q}^{n-1} f)(x)\) for all \(n \geq1\), where \((D_{q}^{0} f)(x) = f(x)\) [38, 41]. The q-integral of a function f defined in the interval \([0,b]\) is defined by \(I_{q} f(x) = \int_{0}^{x} f(s) \,d_{q} s = x(1- q) \sum_{k=0}^{\infty} q^{k} f(x q^{k})\), for \(0 \leq x \leq b\), provided that the sum converges absolutely [38, 47]. If \(a \in[0, b]\), then
whenever the series exists. The operator \(I_{q}^{n}\) is given by \((I_{q}^{0} f)(x) = f(x) \) and \((I_{q}^{n} f)(x) = (I_{q} (I_{q}^{n-1} f)) (x) \) for all \(n \geq1\) [3, 38, 47]. It has been proved that \((D_{q} (I_{q} f))(x) = f(x) \) and \((I_{q} (D_{q} f))(x) = f(x) - f(0)\) whenever f is continuous at \(x =0\) [38, 48]. The fractional Riemann–Liouville type q-integral of the function f on \([0,1]\), of \(\alpha\geq0\) is given by \((I_{q}^{0} f)(t) = f(t) \) and
for \(t \in[0,1]\) and \(\alpha>0\) [15, 18, 45]. Also, the fractional Caputo type q-derivative of the function f is given by
for \(t \in[0,1]\), \(\alpha>0\), and \([\alpha]\) denotes the smallest integer greater or equal to α [3, 15, 18, 45]. It has been proved that \(( I_{q}^{\beta} (I_{q}^{\alpha} f) ) (x) = ( I_{q}^{\alpha+ \beta} f ) (x)\) and \((D_{q}^{\alpha} (I_{q}^{\alpha} f) ) (x) = f(x)\), where \(\alpha, \beta\geq0\) [18]. By employing Algorithm 2, we can calculate \((I_{q}^{\alpha}f)(x)\); this is shown in Algorithm 4.
Let us consider a normed space \((\mathcal{X},\|\cdot\|) \). We denote the set of all nonempty subsets, all nonempty closed subsets, all nonempty bounded subsets, all nonempty compact subsets, and all nonempty compact and convex subsets of \(\mathcal{X}\), by \(P(\mathcal {X})\), \(P_{cl}( \mathcal{X})\), \(P_{b}( \mathcal{X})\), \(P_{cp}( \mathcal {X})\), and \(P_{cp,c}(\mathcal{X})\), respectively. We say that a multivalued map \(\varTheta: \mathcal{X} \to P(\mathcal{X})\) is convex(closed)-valued whenever for any \(x\in\mathcal{X}\), \(\varTheta (\mathcal{X})\) is convex (closed) [29]. If for all \(\mathcal{A} \in P_{b}(\mathcal{X})\), we have \(\varTheta(\mathcal{A}) = \bigcup_{a\in\mathcal{A}} \varTheta(a)\) is a bounded subset of \(\mathcal {X}\), then multifunction Θ is called bounded on bounded sets, where \(\sup_{a\in\mathcal{A}} \{ \sup\{|b|: b\in\varTheta(a)\} \} \) is finite [29]. We use the concepts of upper semicontinuous, compact, completely continuous for the multifunction \(\varTheta: \mathcal{X}\to P(\mathcal{X})\) as in [49, 50]. For investigating the nonlinear problem (1) and (2) under conditions (3), (4), and (5), we need the following lemma, which can be found in [51] and [52].
Lemma 1
The general solution of the fractional q-differential equation \({}^{c}D_{q}^{\alpha} u(t) =0\) is given by \(u(t) = b_{0} + b_{1} t + b_{2} t^{2} + \cdots+ b_{n-1} t^{n-1}\), for \(\alpha>0\), where \(b_{i} \in\mathbb{R}\) for \(i=N_{n-1}\) and \(n = [\alpha] + 1\).
In fact, by using Lemma 1, for the solution of the fractional q-differential equation \({}^{c}D_{q}^{\alpha} u(t) = 0\) we have \({I_{q}^{ \alpha}}{}^{c}D_{q}^{\alpha} u(t) = u(t) + b_{0} + b_{1} t + b_{2} t^{2} + \cdots+ b_{n-1}t^{n-1}\). Now, we prove the next key result.
Lemma 2
Consider the boundary value problem with the antiperiodic conditions
for \(n-1<\alpha\leq n\), with the natural number n being more than or equal to five, \(q \in(0,1)\), \(v \in L^{1} (\overline{J} , \mathbb{R} )\) and each \(t\in[0, \delta]\) with \(\delta> 0\), where \(p_{1}\), \(p_{2}\), \(p_{3}\) belong to \((0,1)\), \((1,2)\), and \((2,3)\), respectively. Then, the problem has at least one solution, namely, \(u(t) = \int_{0}^{\delta}G(t,qs) v(s) \, d_{q}s\), where
whenever \(s\leq t\) or \(t\leq s\), respectively, here
Proof
We assume that u is one of the solutions of (9). By applying Lemma 1, there exist \(c_{i} \in\mathbb{R}\) for \(i \in N_{n-1}\) such that
By using conditions (3) for problem (1), we obtain \(b_{4} = \cdots= b_{n-1} =0\). Since \({}^{c}D_{q}^{p_{1}} k = 0\) for all constant k, \({}^{c}D_{q}^{p_{1}} t\), \({}^{c}D_{q}^{p_{1}} t^{2}\), \({}^{c}D_{q}^{p_{1}} t^{3}\) are equal to \(\frac{ t^{ 1 - p_{1}}}{\varGamma_{q} (2 - p_{1})}\), \(\frac{2 t^{ 2 - p_{1}}}{ \varGamma_{q}( 3 - p_{1})}\), \(\frac{ 6t^{ 3 - p_{1}}}{ \varGamma_{q}( 4 - p_{1})}\), respectively, \({}^{c}D_{q}^{p_{2}} t\), \({}^{c}D_{q}^{p_{2}} t^{2}\), \({}^{c}D_{q}^{p_{2}} t^{3}\) are equal to 0, \(\frac{2 t^{ 2 - p_{1}}}{ \varGamma_{q}( 3 - p_{2})}\), \(\frac{ 6 t^{ 3 - p_{2}}}{ \varGamma_{q} ( 4 - p_{2})}\), respectively, \({}^{c}D_{q}^{p_{3}} t= {}^{c}D_{q}^{p_{3}} t^{2} =0\), \({}^{c}D_{q}^{p_{3}} t^{3} = \frac{ 6 t^{ 3 - p_{3}}}{ \varGamma_{q}( 4 - p_{3})}\) and \({}^{c}D_{q}^{p_{i}} I_{q}^{\alpha} v(t) = I_{q}^{\alpha- p_{i}} v(t)\), for \(i=1,2,3\), we get
By applying conditions (4) and (5), we obtain
Thus, substituting the values of \(c_{i}\), for \(i \in N_{n-1}\) in condition (9), we get the unique solution of the problem. □
3 Main results
At present, we are ready, by using the above results and basic definitions, to investigate positive solutions of problems (1) and (2) with conditions (3), (4), and (5) in the subsequent two subsections. For brevity, we denote the space of all \(x \in C^{3}(\overline{J})\) by \(\mathcal{X}\). We consider the norm
on \(\mathcal{X}\). As we know, \((\mathcal{X}, \|\cdot\|)\) is a Banach space.
3.1 Positive solutions for problem (1)
We first give the following theorem which can be found in [26].
Theorem 3
The completely continuous operator Θ defined on a Banach space A has a fixed point in A whenever the set of all \(a \in A\) such that \(a = \lambda\varTheta(a)\) is bounded, for \(0 < \lambda<1\).
Theorem 4
The operator \(\varTheta: \mathcal{X}\to \mathcal{X}\) defined by
is completely continuous, where
and
Proof
To begin, consider a sequence \(\{ u_{n} \}\) in \(\mathcal{X}\) such that \(u_{n} \) tends to \(u_{0}\) and \(\beta_{1j} \in(0,1)\) for \(j \in N_{k_{1}}\). By using assumptions, we get
Since \(\| u_{n} - u\| \to0\), \(\lim_{n\to\infty} {}^{c}D_{q}^{\beta_{1j}} u_{n} (t) = {}^{c}D_{q}^{\beta_{1j} } u_{0} (t)\) uniformly on J̅. Again with the same method, we have \(\lim_{n \to\infty} {}^{c}D_{q}^{\beta_{2j} } u_{n} (t) = {}^{c}D_{q}^{\beta_{2j}} u_{0}(t)\) and
uniformly on J̅ for \(j \in N_{k_{2}}\) and \(j \in N_{k_{3}}\), respectively. Also, we obtain \(\lim_{n\to\infty} {}^{c}D_{q}^{ \gamma _{i1}} u_{n} (t) = {}^{c}D_{q}^{\gamma_{i1}} u_{0} (t)\), \(\lim_{n \to \infty} {}^{c}D_{q}^{\gamma_{2i} } u_{n} (t) = {}^{c}D_{q}^{\gamma_{i2}} u_{0} (t)\), and
uniformly on J̅ for \(i \in N_{2}\). On the other hand,
Thus, by employing the continuity of f, \(\theta_{1}\), \(\theta_{2}\), we conclude that \(\| \varTheta u_{n} -\varTheta u \| \to0\). Therefore, Θ is continuous on \(\mathcal{X}\). At present, suppose that \(\mathcal{B} \subseteq \mathcal{X}\) is bounded. So there exists \(L\in(0, \infty)\) such that \(|\widetilde{f} (t, u(t) ) | \leq L\) for each t and u belonging to J̅ and \(\mathcal{B}\), respectively. Due to the assumptions, we get
for almost all \(u\in\mathcal{B}\). Hence, we have
where
Equation (10) implies that \(\varTheta(\mathcal{B} )\) is a bounded set. Now, we demonstration that the sets of all Θu, \((\varTheta u)'\), \((\varTheta u)''\), and \((\varTheta u)'''\) are equicontinuous on J̅ for all \(u \in\mathcal{B}\). Let \(t_{1}\) and \(t_{2}\) in J̅. If \(t_{1} \leq t_{2}\), then we get
Again, by using a similar technique, we have
If \(t_{2} \to t_{1}\), then right-hand sides of all inequalities (12)–(15) tend to zero, and so Θ is completely continuous. This completes the proof. □
Theorem 5
Problem (1) under conditions (3), (4), and (5), has at least one solution whenever function f mapping \(\overline{J} \times\mathcal{R}^{m} \) into \(\mathbb{R}\) is continuous and the following assumptions hold for each \(t, s \in\overline{J}\), \({}_{i}x_{j} \in\mathbb{R}\):
-
(1)
There exists positive constants \(d_{0}>0\) and \({}_{0}d_{j_{0}}\), \({}_{1}d_{j_{1}}\), \({}_{2}d_{j_{2}}\), \({}_{3}d_{j_{3}} \in[0, \infty)\) such that
$$\begin{aligned}& \bigl\vert f (t, {}_{0}x_{1}, {}_{0}x_{2}, {}_{0}x_{3}, {}_{0}x_{4}, {}_{0}x_{5}, {}_{0}x_{6}, {}_{1}x_{1}, {}_{1}x_{2}, \dots, \\& \qquad {}_{1}x_{k_{1}}, {}_{2}x_{1}, {}_{2}x_{2}, \dots, {}_{2}x_{k_{2}}, {}_{3}x_{1}, {}_{3}x_{2}, \dots, {}_{3}x_{k_{3}}) \bigr\vert \\& \quad \leq d_{0} + \sum_{j=1}^{6} {}_{0}d_{j} \vert {}_{0}x_{j} \vert + \sum_{j=1}^{k_{1}} {}_{1}d_{j} \vert {}_{1}x_{j} \vert + \sum_{j=1}^{k_{2}} {}_{2}d_{j} \vert {}_{2}x_{j} \vert + \sum_{j=1}^{k_{3}} {}_{3}d_{j} \vert {}_{3}x_{j} \vert , \end{aligned}$$for \(j_{0}\), \(j_{1}\), \(j_{2}\), \(j_{3}\) belonging to \(N_{6}\), \(N_{k_{1}}\), \(N_{k_{2}}\), and \(N_{k_{3}}\), respectively.
-
(2)
There exist constants \({}_{0}c_{1}\), \({}_{0}c_{2}\) in \((0, \infty)\) and \({}_{i}\eta_{j} \in[0, \infty)\) such that
$$\bigl\vert \theta_{i} (t, s, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}) \bigr\vert \leq{}_{0}c_{i} + \sum _{j=1}^{7} {}_{i} \eta_{j} \vert x_{j} \vert , $$for each \(x_{i}\in\mathbb{R}\), where \(i=1,2\) and \(j \in N_{7}\). In addition,
$$\begin{aligned} \varLambda'_{1} ={}& \varLambda_{1} \Biggl[ \Biggl( \sum_{j=1}^{4} {}_{0}d_{j} \Biggr) \\ &+ {}_{0}d_{5} \gamma_{1}^{0} \Biggl( \Biggl( \sum_{j=1}^{4} {}_{1}c_{j} \Biggr) + {}_{1}c_{5} \frac{\delta^{1 - \gamma_{11}}}{\varGamma_{q}( 2 -\gamma_{11})} + {}_{1}c_{6} \frac{\delta^{2 -\gamma_{12}}}{\varGamma_{q}( 3 -\gamma_{12})} + {}_{1}c_{7} \frac{\delta^{ 3 - \gamma_{13}}}{\varGamma_{q}(4 - \gamma_{13})} \Biggr) \\ & + {}_{0}d_{6}\gamma_{2}^{0} \Biggl( \Biggl( \sum_{j=1}^{4} {}_{2}c_{j} \Biggr) + {}_{2}c_{5} \frac{\delta^{1 - \gamma_{21}}}{\varGamma_{q}( 2 -\gamma_{21})} + {}_{2}c_{6} \frac{\delta^{ 2 -\gamma_{22}}}{\varGamma_{q}( 3 -\gamma _{22})} + {}_{2}c_{7} \frac{\delta^{ 3 -\gamma_{23}}}{\varGamma_{q}( 4 -\gamma _{23})} \Biggr) \\ & + \sum_{j=1}^{k_{1}} {}_{1}d_{j} \frac{\delta^{ 1 -\beta_{1j}}}{ \varGamma_{q}( 2 -\beta_{1j})} + \sum_{j=1}^{k_{2}} {}_{2}d_{j} \frac{\delta^{ 2 - \beta_{2j}}}{ \varGamma_{q}( 3 -\beta_{2j})} + \sum _{j=1}^{k_{3}} {}_{3}d_{j} \frac{\delta^{ 3 -\beta _{3j}}}{\varGamma_{q}( 4 -\beta_{3j})} \Biggr] \\ < {}&1, \end{aligned}$$where \(\gamma_{i}^{0}= \sup_{t\in\overline{J}} \int_{0}^{t} |\mu_{i} (t,s)|\, ds\) and \(\varLambda_{1}\) is defined in Eq. (11) for \(i=1,2\).
Proof
Thus, Theorem 4 implies that operator Θ of \(\mathcal {X} \) to itself is completely continuous. Now, we prove that \(\mathcal {B} \subset \mathcal{X}\) which contains all \(u \in\mathcal{X}\) such that \(u= \lambda\varTheta(u)\) where \(\lambda\in(0,1)\) is bounded. Let \(u \in\mathcal{B}\) and \(t\in \overline{J}\). Then, we obtain
Therefore, we have
Thus, we conclude that
Hence, \((1 -\varLambda'_{1}) \|u\| \leq\varLambda_{1}( d_{0} + {}_{0}d_{5} \gamma_{1}^{0} {}_{0}c_{1} +{}_{0}d_{6} \gamma_{2}^{0} {}_{0}c_{2})\). Therefore, the set \(\mathcal{B}\) is bounded. At present, by employing Theorem 3, the operator Θ has at least one fixed point. By a simple review, we conclude that each fixed point of the operator Θ is a solution for problem (1). □
Theorem 6
Assume that the real-valued functions f and \(\theta_{i}\), defined on \(\overline{J} \times\mathcal{R}^{m}\) and \({\overline{J}^{2}\times\mathbb {R}^{7}}\), respectively, are continuous. Then problem (1) under conditions (3), (4), and (5) has a unique solution whenever the following assumptions hold for each \(t, s \in \overline{J}\), \({}_{i}x_{j} \in\mathbb{R}\):
-
(1)
There exists constants \({}_{0}\eta_{j} > 0\) and \({}_{1}\eta_{j_{0}}, {}_{2}\eta_{j_{0}} , {}_{3}\eta_{j_{0}} \geq0\) such that
$$\begin{aligned}& \bigl\vert f (t, {}_{0}x_{1}, {}_{0}x_{2}, {}_{0}x_{3}, {}_{0}x_{4}, {}_{0}x_{5}, {}_{0}x_{6}, \\& \qquad{}_{1}x_{1}, {}_{1}x_{2}, \dots, {}_{1}x_{k_{1}}, {}_{2}x_{1}, {}_{2}x_{2}, \dots, {}_{2}x_{k_{2}}, {}_{3}x_{1}, {}_{3}x_{2}, \dots, {}_{3}x_{k_{3}}) \\& \qquad{} - f \bigl(t, {}_{0}x'_{1}, {}_{0}x'_{2}, {}_{0}x'_{3}, {}_{0}x'_{4}, {}_{0}x'_{5}, {}_{0}x'_{6}, \\& \qquad {}_{1}x'_{1}, {}_{1}x'_{2}, \dots, {}_{1}x'_{k_{1}}, {}_{2}x'_{1}, {}_{2}x'_{2}, \dots, {}_{2}x'_{k_{2}}, {}_{3}x'_{1}, {}_{3}x'_{2}, \dots, {}_{3}x'_{k_{3}} \bigr) \bigr\vert \\& \quad \leq\sum_{j=1}^{6} {}_{0} \eta_{j} \bigl\vert {}_{0}x_{j} - {}_{0}x'_{j} \bigr\vert + \sum _{j=1}^{k_{1}} {}_{1}\eta_{j} \bigl\vert {}_{1}x_{j} - {}_{1}x'_{j} \bigr\vert \\& \quad\quad{}+ \sum_{j=1}^{k_{2}} {}_{2} \eta_{j} \bigl\vert {}_{2}x_{j} - {}_{2}x'_{j} \bigr\vert + \sum _{j=1}^{k_{3}} {}_{3}\eta_{j} \bigl\vert {}_{3}x_{j} - {}_{3}x'_{j} \bigr\vert , \end{aligned}$$for \(j_{0}\), \(j_{1}\), \(j_{2}\), \(j_{3}\) belonging to \(N_{6}\), \(N_{k_{1}}\), \(N_{k_{2}}\), and \(N_{k_{3}}\), respectively.
-
(2)
There exist constants \({}_{0}c_{i}\) in \((0, \infty)\) and \({}_{i}\eta _{j} \in[0, \infty)\) such that
$$\begin{aligned}& \bigl\vert \theta_{i} (t, s, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}) - \theta_{i} \bigl(t, s, x'_{1}, x'_{2}, x'_{3}, x'_{4}, x'_{5}, x'_{6}, x'_{7} \bigr) \bigr\vert \\& \quad \leq \sum_{j=1}^{7} {}_{i}c_{j} \bigl\vert x_{j} - x'_{j} \bigr\vert , \end{aligned}$$for each \(i=1,2\), \(x_{j}, x'j\in\mathbb{R}\), where \(j \in N_{7}\) and
$$\begin{aligned} \varLambda'_{1} ={}& \varLambda_{1} \Biggl[ \Biggl( \sum_{j=1}^{4} {}_{0} \eta_{j} \Biggr) \\ & + {}_{0}\eta_{5} \gamma_{1}^{0} \Biggl( \Biggl( \sum_{j=1}^{4} {}_{1}c_{j} \Biggr)+ {}_{1}c_{5} \frac{\delta^{1 - \gamma_{11}}}{\varGamma_{q}( 2 -\gamma_{11})} + {}_{1}c_{6} \frac{\delta^{2 -\gamma_{12}}}{\varGamma_{q}( 3 -\gamma_{12})} + {}_{1}c_{7} \frac{\delta^{ 3 -\gamma_{13}}}{\varGamma_{q}(4 - \gamma_{13})} \Biggr) \\ & + {}_{0}\eta_{6}\gamma_{2}^{0} \Biggl( \Biggl( \sum_{j=1}^{4} {}_{2}c_{j} \Biggr) + {}_{2}c_{5} \frac{\delta^{1 - \gamma_{21}}}{\varGamma_{q}( 2 -\gamma _{21})} + {}_{2}c_{6} \frac{\delta^{ 2 -\gamma_{22}}}{\varGamma_{q}( 3 -\gamma _{22})}+ {}_{2}c_{7} \frac{\delta^{ 3 -\gamma_{23}}}{\varGamma_{q}( 4 -\gamma _{23})} \Biggr) \\ &+ \sum_{j=1}^{k_{1}} {}_{1} \eta_{j} \frac{\delta^{ 1 -\beta _{1j}}}{ \varGamma_{q}( 2 -\beta_{1j})} + \sum_{j=1}^{k_{2}} {}_{2} \eta_{j} \frac{\delta^{ 2 - \beta_{2j}}}{ \varGamma_{q}( 3 -\beta_{2j})} + \sum _{j=1}^{k_{3}} {}_{3}\eta_{j} \frac{\delta^{ 3 -\beta_{3j}}}{ \varGamma_{q}( 4 -\beta_{3j})} \Biggr] \\ < {}&1. \end{aligned}$$
Proof
We choose a positive constant r such that
where \(\eta_{0} = \sup_{ t \in\overline{J}} |f(t, 0, 0, \dots, 0)|\), \(\vartheta_{j} = \sup_{ t, s \in\overline{J}} |\theta_{j} (t, s, 0,0, \dots , 0)| \) are finite for \(j=1,2\). We claim that \(\varTheta(\mathcal{B}_{r}) \subseteq\mathcal{B}_{r}\), where \(\mathcal{B}_{r}\) is the set of all \(u \in X\) such that \(\|u\|\leq r\). In this case, considering \(u \in\mathcal{B}_{r}\), we get
In a similar manner, we conclude that
Therefore,
Indeed, \(\| \varTheta u\| \leq r\). On the other hand, we obtain
for each \(t \in\overline{J}\) and each \(u, v \in\mathcal{X}\). By considering similar arguments, we obtain
for each \(t \in\overline{J}\) and each \(u, v \in \mathcal{X}\). Hence, we conclude that
Therefore, Θ is a contraction, because \(\varLambda'_{1} < 1\), and so, by employing the Banach contraction principle, Θ has a unique fixed point, which is a solution of problem (1). □
3.2 Positive solutions for inclusion problem (2)
In the second section of main results, we look into the positive solutions for the inclusion problem (2) with the antiperiodic boundary conditions (3), (4), and (5). Now, we recall some definitions and concepts which are needed in the sequel, and also we use the same definitions of the previous section. As one knows, a multivalued map \(T: \overline{J} \times\mathcal{R}^{m} \to P(\mathbb{R})\) is said to be Carathéodory whenever the map \(t \mapsto T(t, r_{1}, r_{2}, \dots, r_{m})\) is measurable and the map \((r_{1}, r_{2}, \dots, r_{m}) \mapsto T(t, r_{1},r_{2}, \dots, r_{m})\) is upper semicontinuous, and we say that a Carathéodory function T is \(L^{1}\)-Carathéodory whenever for each \(l>0\) there exists \(\varphi_{l} \in L^{1}(\overline{J}, \mathbb{R}^{+} )\) such that
where \(|r_{i}| \leq l\), for each \(r_{i}\in\mathbb{R}\) with \(i \in N_{m}\), each \(t\in\overline{J}\), respectively [29, 50]. One can find the following lemma in [31].
Lemma 7
The composite operator \(N \circ S_{G} : C (\overline{J}, \mathcal{A})\to P_{cp,c}( C (\overline{J}, \mathcal{A}))\) defined by \(N\circ S_{G}(r) = N(S_{G,r})\) is a closed-graph operator, whenever \(G: \overline{J} \times\mathcal{A} \to P_{cp,c}(\mathcal{A})\) is an \(L^{1}\)-Carathéodory multifunction and N is a linear continuous mapping from \(L^{1}(\overline{J},\mathcal{A})\) to \(C(\overline{J},\mathcal{A})\), where \(\mathcal{A}\) is a Banach space and \(S_{G,r} \) is the set of all \(w \in L^{1}(\overline{J},\mathcal{A})\) such that \(w(t)\in G( t, x(t))\) for each \(t\in\overline{J}\).
The multivalued map \(G: \overline{J}\times\mathcal{A}\to P_{cp}(\mathcal{A})\) is said to be of lower semicontinuous type whenever \(S_{G}: C(\overline{J}, \mathcal{A})\to P(L^{1}(\overline{J},\mathcal{A}))\) is lower semicontinuous and has nonempty closed and decomposable values [51]. Also, one can see the following lemma in [51].
Lemma 8
The lower semicontinuous multivalued map \(N: \mathcal{A}\to P(L^{1}(\overline{J}, \mathbb{R}))\) has a continuous selection, i.e., there exists a continuous mapping \(H: \mathcal{A} \to L^{1}(\overline{J}, \mathbb{R})\) such that \(H(a)\in N(a)\) for each \(a\in\mathcal{A}\), whenever N has closed decomposable values, where \(\mathcal{A}\) be a separable metric space.
Theorem 9
([53])
Suppose that \((\mathcal{A}, \rho)\) be a complete metric space. Then each contraction multivalued map \(T: \mathcal{A}\to P_{cl}(\mathcal{A})\) has a fixed point.
Theorem 10
([54])
Let \(\mathcal {C}\) be a closed and convex subset of a Banach space \(\mathcal{A}\) and \(\mathcal{O}\) be an open subset of \(\mathcal{C}\) such that \(0\in \mathcal{O}\). Then either T has a fixed point in \(\overline{\mathcal {O}}\) or there are \(a\in\partial\mathcal{O}\) and \(\kappa\in(0,1)\) such that \(a\in\kappa T(a)\), whenever \(T :\overline{\mathcal{O} } \to P_{cp,c}( \mathcal{C})\) is an upper semicontinuous compact map.
Theorem 11
If a multivalued map T mapping \(\overline{J} \times\mathcal{R}^{m} \) into \(P_{cp,c}(\mathbb{R})\) is Carathéodory, then problem (2) has at least one positive solution whenever the following assumptions are hold for each \(t, s \in\overline{J}\), \({}_{i}x_{j} \in \mathbb{R}\):
-
(1)
There exist positive real-valued and continuous nondecreasing functions \(\phi_{j_{0}}\) and \(\psi_{1j_{1}}\), \(\psi_{2j_{2}}\), \(\psi_{3j_{3}}\) defined on \([0, \infty)\) and nonnegative functions \(g_{0j_{0}}\), \(g_{1j_{1}}\), \(g_{2j_{2}}\), \(g_{3j_{3}}\) in \(L^{1}(\overline{J})\) such that
$$\begin{aligned}& \bigl\Vert T (t, {}_{0}x_{1}, {}_{0}x_{2}, {}_{0}x_{3}, {}_{0}x_{4}, {}_{0}x_{5}, {}_{0}x_{6}, \\& \quad\quad{}_{1}x_{1}, {}_{1}x_{2}, \dots, {}_{1}x_{k_{1}}, {}_{2}x_{1}, {}_{2}x_{2}, \dots, {}_{2}x_{k_{2}}, {}_{3}x_{1}, {}_{3}x_{2}, \dots, {}_{3}x_{k_{3}}) \bigr\Vert _{p} \\& \quad = \sup \bigl\{ \vert x \vert : x \in T ( t, {}_{0}w_{1}, {}_{0}w_{2}, {}_{0}w_{3}, {}_{0}w_{4}, {}_{0}w_{5}, {}_{0}w_{6}, {}_{1}w_{1}, {}_{1}w_{2}, \dots, \\& \qquad {}_{1}w_{k_{1}}, {}_{2}w_{1}, {}_{2}w_{2}, \dots, {}_{2}w_{k_{2}}, {}_{3}w_{1}, {}_{3}w_{2}, \dots, {}_{3}w_{k_{3}}) \bigr\} \\& \quad \leq \sum_{j=1}^{6} g_{0j} (t) \phi_{j} \bigl( \vert {}_{0}x_{j} \vert \bigr) + \sum_{j=1}^{k_{1}} g_{1j}(t) \psi_{1j} \bigl( \vert {}_{1}x_{j} \vert \bigr) \\& \qquad{}+ \sum_{j=1}^{k_{2}} g_{2j}(t) \psi_{2j} \bigl( \vert {}_{2}x_{j} \vert \bigr) + \sum_{j=1}^{k_{3}} g_{3j}(t) \psi_{3j} \bigl( \vert {}_{3}x_{j} \vert \bigr), \end{aligned}$$for \(j_{0}\), \(j_{1}\), \(j_{2}\) and \(j_{3}\) in \(N_{6}\), \(N_{k_{1}}\), \(N_{k_{2}}\), \(N_{k_{3}}\), respectively.
-
(2)
There exist constants \({}_{0}c_{1}\), \({}_{0}c_{2}\) in \((0, \infty)\) and \({}_{i}\eta_{j} \in[0, \infty)\) such that
$$\bigl\vert \theta_{i} (t, s, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}) \bigr\vert \leq{}_{0}c_{i} + \sum _{j=1}^{7} {}_{i} \eta_{j} \vert x_{j} \vert , $$for each \(x_{i}\in\mathbb{R}\), where \(i=1,2\) and \(j \in N_{7}\).
-
(3)
There exists a constant \(\Delta>0\) such that \(\varLambda_{2} A(\Delta) < \Delta\), where
$$\begin{aligned} A (\Delta) = {}& \Biggl( \sum_{j=1}^{4} \Vert g_{0j} \Vert _{1} \phi_{j}(\Delta) \Biggr) \\ & + \Vert g_{05} \Vert _{1} \phi_{5} \Biggl( {}_{0}c_{1} \gamma_{1}^{0} + \Delta \gamma _{1}^{0} \Biggl[ \Biggl( \sum _{j=1}^{4} {}_{1}\eta_{j} \Biggr) \\ & + {}_{1}\eta_{5} \frac{\delta^{ 1 - \gamma_{11}}}{ \varGamma_{q}( 2 -\gamma_{11})} + {}_{1} \eta_{6} \frac{\delta^{ 2 - \gamma_{12}}}{ \varGamma_{q}( 3 -\gamma_{12})} + {}_{1}\eta_{7} \frac{\delta^{ 3 -\gamma_{13}}}{ \varGamma _{q}( 4 -\gamma_{13})} \Biggr] \Biggr) \\ & + \Vert g_{06} \Vert _{1} \phi_{6} \Biggl( {}_{0}c_{2} \gamma_{2}^{0} + \Delta \gamma _{2}^{0} \Biggl[ \Biggl(\sum _{j=1}^{4} {}_{2}\eta_{j} \Biggr) \\ & + {}_{2}\eta_{5} \frac{\delta^{ 1 -\gamma_{21}}}{ \varGamma_{q}( 2 -\gamma_{21})} + {}_{2} \eta_{6} \frac{\delta^{ 2 -\gamma_{22}}}{ \varGamma_{q}( 3 -\gamma_{22})} + {}_{2}\eta_{7} \frac{\delta^{ 3 - \gamma_{23}}}{ \varGamma _{q}( 4 - \gamma_{23})} \Biggr] \Biggr) \\ & + \sum_{j=1}^{k_{1}} \Vert g_{1j} \Vert _{1} \psi_{1j} \biggl( \frac{ \delta ^{ 1 - \beta_{1j}}}{ \varGamma_{q}( 2 - \beta_{1j})} \Delta \biggr) \\ & + \sum_{j=1}^{k_{2}} \Vert g_{2j} \Vert _{1} \psi_{2j} \biggl( \frac{\delta^{ 2 -\beta_{2j}}}{ \varGamma_{q}( 3 -\beta_{2j})} \Delta \biggr) \\ & + \sum_{j=1}^{k_{3}} \Vert g_{3j} \Vert _{1} \psi_{3j} \biggl( \frac{ \delta ^{ 3 -\beta_{3j}}}{ \varGamma_{q}( 4 -\beta_{3j})} \Delta \biggr) , \end{aligned}$$(16)and
$$\begin{aligned} \varLambda_{2} ={}& \biggl[ \frac{ \vert a_{1} \vert + 2 \vert a_{2} \vert }{ \vert a_{1} + a_{2} \vert \varGamma_{q}( \alpha) } + \frac{( \vert a_{1} \vert + 2 \vert a_{2} \vert ) \varGamma_{q}( 2 - p_{1})}{ \vert a_{1} + a_{2} \vert \varGamma_{q}(\alpha- p_{1})} \\ & + \frac{( \vert a_{2} \vert p_{1} + \vert a_{1} + a_{2} \vert ( 4 - p_{1})) \varGamma_{q} (3-p_{2})}{ 2 \vert a_{1} + a_{2} \vert ( 2 -p_{1}) \varGamma_{q}( \alpha- p_{2})} \\ & + \frac{ \vert a_{2} \vert [ 6 (p_{2} -p_{1}) + ( 2 -p_{1})( 3 -p_{1}) p_{2}]\varGamma_{q}( 4 -p_{3}) }{6 \vert a_{1} + a_{2} \vert ( 2 -p_{1})( 3 -p_{1}) ( 3 -p_{2}) \varGamma_{q}( \alpha- p_{3})} \\ & + \frac{ [6 (p_{2} - p_{1}) + ( 2 - p_{1})( 3 -p_{1})( 6 -p_{2})] \varGamma _{q}( 4 -p_{3})}{ 6 ( 2 -p_{1})( 3- p_{1})( 3- p_{2}) \varGamma_{q}( \alpha- p_{3})} \biggr] \delta^{\alpha- 1} \\ & + \biggl[ \frac{1}{ \varGamma_{q}( \alpha- 1)} + \frac{ \varGamma_{q}( 2 -p_{1})}{ \varGamma_{q}( \alpha- p_{1}) } + \frac{ (3 - p_{1}) \varGamma_{q}( 3 -p_{2})}{( 2 -p_{1}) \varGamma_{q}( \alpha- p_{2})} \\ & + \frac{ [2 ( p_{2} -p_{1}) + (2 - p_{1})( 3 -p_{1})( 5- p_{2}) ] \varGamma _{q}( 4 - p_{3})}{ 2 ( 2 - p_{1})( 3 - p_{1})( 3-p_{2} ) \varGamma_{q}( \alpha-p_{3})} \biggr] \delta^{\alpha- 2} \\ & + \biggl[ \frac{1}{ \varGamma_{q}( \alpha- 2)} + \frac{ \varGamma_{q}( 3 -p_{2})}{ \varGamma_{q}( \alpha-p_{2})} + \frac{( 4 -p_{2}) \varGamma_{q}( 4 -p_{3})}{( 3 - p_{2}) \varGamma_{q}( \alpha- p_{3})} \biggr] \delta^{ \alpha- 3} \\ & + \biggl[ \frac{1}{ \varGamma_{q}( \alpha- 3)} + \frac{ \varGamma_{q}( 4 - p_{3})}{ \varGamma_{q}( \alpha- p_{3})} \biggr] \delta^{ \alpha- 4} . \end{aligned}$$(17)
Proof
To begin, we define the set of selections of T for an arbitrary element \(u\in\mathcal{X}\) which contains all \(v \in L^{1}(\overline {J}, \mathbb{R})\) such that \(v(t)\) belongs to the multifunction \(\widetilde{T}(t, u(t))\) for each \(t\in\overline{J}\) and is denoted by \(S_{T, u}\), where
By considering the first property of the multifunction T and using Theorem 1.3.5 in [8], we know that \(S_{T,u}\) is nonempty. Defining an operator \(H: X \to P(X)\) on the set of all \(h \in X\) for which there exists \(v \in S_{T,u}\) such that \(h(t) = T_{v} (t)\) for \(t\in\overline{J}\) and denoting by \(H(x)\) where
we claim that \(H(x)\) is convex for all \(u\in\mathcal{X}\). Assume that \(h_{1}, h_{2} \in H(x)\) and \(\tau\in[0,1]\). Choose \(v_{1}, v_{2}\in S_{T,u}\) such that
for each t in J̅. Then, we obtain
Since T has convex values, by simple calculation, we can see that \(S_{T,u}\) is convex and so \(\tau h_{1} + ( 1 - \tau) h_{2} \in H(x)\). At present, we prove that H maps bounded sets into bounded sets in \(\mathcal{X}\). Suppose that \(B_{r}\) is the set of all \(u \in\mathcal{X}\) such that \(\|u\|\) is less than or equal to r, \(u \in B_{r}\) and \(h\in H(x)\). We select \(v \in S_{T,u}\) such that
for any \(t\in\overline{J}\). Thus, similarly as for inequality (18), we get
Thus, from inequalities (18), (19), (20), and (22), we obtain
Thus, we conclude that H maps bounded sets into bounded sets in \(\mathcal{X}\). Let \(\tau_{1}, \tau_{2} \in\overline{J}\) with \(\tau_{1} < \tau_{2}\), \(u\in B_{r}\) and \(h\in H(x)\). Then, we have
where
Similarly, from inequality (23), we have
Therefore, since \(u\in B_{r}\), when \(t_{2} - t_{1} \to0\), the above inequalities (23)–(26) tend to zero. Therefore, by employing Arzelà–Ascoli theorem, we get that \(H: \mathcal{X} \to P(\mathcal{X})\) is a compact multivalued map. Let \(u_{n} \to u^{*}\), \(h_{n} \in H(u_{n})\) for all n and \(h_{n}\to h^{*}\). We show that \(h^{*} \in H(u^{*})\). Since \(h_{n}\in H(u_{n})\) for all n, there exists \(v_{n} \in S_{T,u_{n}}\) such that
for all \(t\in\overline{J}\). We claim that there exists \(v^{*} \) belonging to \(S_{T,u^{*}}\) such that
for each t belonging to J̅. In this case, we consider the linear operator \(\varOmega: L^{1}(\overline{J}, \mathbb{R})\to\mathcal{X}\) defined by \(v \mapsto\varOmega(v)(t)\), where Ω is continuous and
for all t in J̅. On the other hand, Ω is a linear continuous map and by applying Lemma 7, we obtain \(\varOmega \circ S_{T, u}\) is a closed-graph operator. Note that \(h_{n}\in\varOmega \circ S_{T, u_{n}}\) for all n. Since \(u_{n}\to u^{*}\) and \(h_{n}\to h^{*}\), there exists \(v^{*}\in S_{T, u^{*}}\) such that
If \(0<\kappa<1\) and \(u\in\kappa H(x)\), then there exists \(v \in S_{T,u}\) such that
for any \(t\in\overline{J}\). Hence,
Indeed, \(\|u\| \leq\varLambda_{2} A(\|u\|)\). On the other hand, the operator \(\varPhi: \overline{D} \to P_{cp,c}(\mathcal{X})\) is upper semicontinuous and compact, where \(D = \{u\in\mathcal{X}: \| u\| < \Delta\}\). By considering the choice of D, there is no \(u\in \partial D\) such that \(u\in\kappa H(u)\) for some \(\kappa\in(0,1)\) and so H has a fixed point \(u\in\overline{D}\) due to Theorem 10. Therefore H satisfies the assumptions of the nonlinear alternative of the Leray–Schauder-type result. It is easy to check that each fixed point of H is a solution of problem (2). This completes the proof. □
In the next case we will show that convex-valued condition of T is not necessary.
Theorem 12
If T defined on \(\overline{J} \times\mathcal{R}^{m} \) to \(P_{cp,c}(\mathbb{R})\) is a multifunction such that the map \((t, x_{1}, x_{2}, \dots, x_{m})\mapsto T( t, x_{1}, x_{2}, \dots, x_{m})\) is both \(L(\overline{J} ) \otimes\mathcal{B}(R)\) measurable and lower semicontinuous for each \(t\in\overline{J}\) where \(m=6+k_{1}+k_{2}+k_{3}\) and \(\mathcal{B}(R) =\bigotimes_{j=1}^{m} B(\mathbb{R})\), then problem (2) has at least one positive solution whenever the assumptions (1), (2), and (3) in Theorem 11 hold.
Proof
By using the assumptions and Lemma 4.1 in [55], we conclude that T is lower semicontinuous. Also, Lemma 8 implies that there exists a continuous function \(N: \mathcal{X} \to L^{1}(\overline{J}, \mathbb{R})\) such that \(N(u) \in S_{T, u}\) for all \(u\in\mathcal{X}\). Now consider the problem
with the boundary conditions (23)–(26). Obviously, each solution of problem (27) is a solution of problem (2). Define the operator \(\overline{H}: \mathcal{X}\to\mathcal{X}\) by
for each \(t\in\overline{J}\). Similar to proof of the last result, it can be shown that H̅ is continuous, completely continuous, and satisfies all conditions of the nonlinear alternative of Leray–Schauder type for single-valued maps. Again by using a similar argument as for the last result, one can find a solution for problem (27). This completes the proof. □
Theorem 13
If a multivalued T mapping \(\overline{J} \times\mathcal{R}^{m} \) into \(P_{cp}(\mathbb{R})\) is measurable and bounded for each \(t\in\overline {J}\), then problem (2) has at least one solution whenever the following assumptions hold for each \(t, s \in\overline{J}\), \({}_{i}x_{j} , {}_{i}x'_{j} \in\mathbb{R}\):
-
(1)
There exist nonnegative functions \(m_{ij} \in L^{1}(\overline {J})\) such that
$$\begin{aligned}& d_{H} \bigl( T (t, {}_{0}x_{1}, {}_{0}x_{2}, {}_{0}x_{3}, {}_{0}x_{4}, {}_{0}x_{5}, {}_{0}x_{6}, \\& \qquad {}_{1}x_{1}, {}_{1}x_{2}, \dots, {}_{1}x_{k_{1}}, {}_{2}x_{1}, {}_{2}x_{2}, \dots, {}_{2}x_{k_{2}}, {}_{3}x_{1}, {}_{3}x_{2}, \dots, {}_{3}x_{k_{3}}), \\& \qquad T \bigl(t, {}_{0}x'_{1}, {}_{0}x'_{2}, {}_{0}x'_{3}, {}_{0}x'_{4}, {}_{0}x'_{5}, {}_{0}x'_{6}, \\& \qquad {}_{1}x'_{1}, {}_{1}x'_{2}, \dots, {}_{1}x'_{k_{1}}, {}_{2}x'_{1}, {}_{2}x'_{2}, \dots, {}_{2}x'_{k_{2}}, {}_{3}x'_{1}, {}_{3}x'_{2}, \dots, {}_{3}x'_{k_{3}} \bigr) \bigr) \\& \quad \leq\sum_{j=1}^{6} m_{0j} (t) \bigl\vert {}_{0}x_{j} - {}_{0}x'_{j} \bigr\vert + \sum_{j=1}^{k_{1}} m_{1j}(t) \bigl\vert {}_{1}x_{j} - {}_{1}x'_{j} \bigr\vert \\& \qquad{} + \sum_{j=1}^{k_{2}} m_{2j}(t) \bigl\vert {}_{2}x_{j} - {}_{2}x'_{j} \bigr\vert + \sum_{j=1}^{k_{3}} m_{3j}(t) \bigl\vert {}_{3}x_{j} - {}_{3}x'_{j} \bigr\vert . \end{aligned}$$ -
(2)
There exist \({}_{i}c_{j}\geq0\) such that
$$\begin{aligned}& \bigl\vert \theta_{i} (t, s, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7})- \theta_{i} \bigl(t, s, x'_{1}, x'_{2}, x'_{3}, x'_{4}, x'_{5}, x'_{6}, x'_{7} \bigr) \bigr\vert \\& \quad\leq \sum_{j=1}^{7} {}_{i}c_{j} \bigl\vert x_{j} - x'_{j} \bigr\vert , \end{aligned}$$for \(i=1,2\), \(x_{j}, x'j \in\mathbb{R}\) where \(j \in N_{7}\) and
$$\begin{aligned} \varLambda'_{2} ={}& \varLambda_{2} \Biggl[ \Biggl( \sum_{j=1}^{4} \bigl\Vert m_{0j} (t) \bigr\Vert _{1} \Biggr) + \bigl\Vert m_{05} (t) \bigr\Vert _{1} \gamma_{1}^{0} \Biggl( \Biggl( \sum_{j=1}^{4} {}_{1}c_{j} \Biggr) \\ & + {}_{1}c_{5} \frac{\delta^{1 - \gamma_{11}}}{\varGamma_{q}( 2 - \gamma _{11})} + {}_{1}c_{6} \frac{\delta^{2 -\gamma_{12}}}{\varGamma_{q}( 3 - \gamma _{12})} + {}_{1}c_{7} \frac{\delta^{ 3 -\gamma_{13}}}{\varGamma_{q}(4 - \gamma _{13})} \Biggr) \\ & + \bigl\Vert m_{06} (t) \bigr\Vert _{1} \gamma_{2}^{0} \Biggl( \Biggl( \sum _{j=1}^{4} {}_{2}c_{j} \Biggr) \\ & + {}_{2}c_{5} \frac{\delta^{1 - \gamma_{21}}}{\varGamma_{q}( 2 -\gamma _{21})} + {}_{2}c_{6} \frac{\delta^{ 2 -\gamma_{22}}}{\varGamma_{q}( 3 -\gamma _{22})} + {}_{2}c_{7} \frac{\delta^{ 3 -\gamma_{23}}}{\varGamma_{q}( 4 -\gamma _{23})} \Biggr) \\ & + \sum_{j=1}^{k_{1}} \bigl\Vert m_{1j} (t) \bigr\Vert _{1} \frac{\delta^{ 1 -\beta _{1j}}}{ \varGamma_{q}( 2 -\beta_{1j})} + \sum _{j=1}^{k_{2}} \bigl\Vert m_{2j} (t) \bigr\Vert _{1} \frac{\delta^{ 2 - \beta_{2j}}}{ \varGamma_{q}( 3 -\beta_{2j})} \\ & + \sum_{j=1}^{k_{3}} \bigl\Vert m_{3j} (t) \bigr\Vert _{1} \frac{\delta^{ 3 -\beta _{3j}}}{ \varGamma_{q}( 4 -\beta_{3j})} \Biggr] \\ < {}&1. \end{aligned}$$
Proof
By applying the hypothesis and Theorem III-6 (the measurable selection theorem in [52]), T admits a measurable selection \(v: \overline{J}\to\mathbb{R}\). Since T is integrable and bounded, \(v\in L^{1}(\overline{J}, \mathbb{R})\) and so \(S_{T,u} \neq \emptyset\) for each \(u\in\mathcal{X}\). We claim that the operator H satisfies the assumptions of Theorem 9. In this case, we prove that \(H(u)\in P_{cl}(\mathcal{X})\) for any \(u\in\mathcal{X}\). In this case, consider the sequence \(\{u_{n}\} \subset H(u)\) such that \(u_{n} \to u^{*}\) for some \(u^{*} \in\mathcal{X}\). For each n, choose \(w_{n} \in S_{T,u}\) such that
for all \(t\in\overline{I}\). Hence, there exists a subsequence of \(\{ w_{n}\}\) that converges to w in \(L^{1}(\overline{I}, \mathbb{R})\), because T has compact values. We denote this subsequence again by \(\{ w_{n}\}\). Thus, \(w\in S_{T,u}\) and \(u_{n}(t)\) tends to \(u^{*}(t)\), where
for any \(t\in\overline{I}\). Indeed, \(u^{*}\in H(u)\). Now, we show that there exists \(\varLambda'_{2} < 1\) such that \(d_{H} (H(v), H(\tilde{v})) \leq \varLambda'_{2} \|v - \tilde{v}\|\), for all \(v, \tilde{v}\in\mathcal{X}\). Let \(v, \tilde{v}\in\mathcal{X}\) and \(h_{1}\in H(v)\). Choose \(v_{1} \) belonging to \(S_{T,v}\) such that
for almost all \(t\in\overline{I}\). On the other hand, we get
for each \(t\in\overline{J}\). Hence, there exists \(f_{t}\in\widetilde {T}( t, \tilde{v}(t))\) such that \(|v_{1}(t) - f_{t}| < \varLambda'_{t}\), where
for almost all \(t\in\overline{J}\). Define \(N: \overline{J}\to P(\mathbb {R})\) by \(N(t)=\{ x \in\mathbb{R}: |v_{1}(t) - x| \leq\varLambda'_{t} \}\) for all \(t\in\overline{J}\). By employing Theorem III-41 in [52], we get that N is measurable. Since the multivalued operator \(t \mapsto N(t) \cap\widetilde{T} ( t, \tilde {v}(t) )\) is measurable (Proposition III-4 in [52]), there exists a function \(v_{2}\in S_{F,\tilde {z}}\) such that
for almost all \(t\in\overline{J}\). Define
for all \(t\in\overline{J}\). Then, we have
By interchanging the roles of v and ṽ, we get \(d_{H} (H(v), H(\tilde{v}) ) \leq \varLambda'_{2} \|v - \tilde{v}\|\). Since \(\varLambda'_{2} <1\), H is a contraction and so by using Theorem 9, H has a fixed point. It is easy to check that each fixed point of H is a solution of problem (2). □
4 Examples and numerical check technique for the problems
In this part, we give complete computational techniques for checking of the existence of solutions for the inclusion problem (1) in Theorems 11, 13, which cover all similar problems and present numerical examples for solving perfectly. Foremost, we show that a simplified analysis can be executed to calculate the value of q-Gamma function, \(\varGamma_{q} (x) \), for input values q and x by counting the number of sentences n in the summation. To this aim, we consider a pseudo-code description of the method for calculating q-Gamma function of order n in Algorithm 2 (for more details, see https://en.wikipedia.org/wiki/Q-gamma_function).
Table 1 shows that when q is constant, the q-Gamma function is an increasing function. Also, for smaller values of x, an approximate result is obtained with smaller values of n. It has been shown by underlined rows. Table 2 shows that the q-Gamma function for values of q near 1 is obtained with more values of n in comparison with other columns. They have been underlined in line 8 of the first column, line 17 of the second column, and line 29 of the third column of Table 2. Also, Table 3 is the same as Table 2, but x values increase in Table 3.
Note that all routines are written in MATLAB software with the variable Digits set to 16 (This environment variable controls the number of digits in MATLAB) and work on a PC with 2.90 GHz of Core 2 CPU and 4 GB of RAM. Furthermore, we provided Algorithm 3 which calculates \((D_{q}^{\alpha}f) (x)\).
Here, we give two examples to illustrate the inclusion problems (2) in Theorems 11 and 13.
Example 1
Consider the fractional q-differential inclusion
for \(t\in\overline{J}= [0,1]\) (\(\delta=1\)), with the conditions \(u^{(4)} (0) =u^{(5)} (0)=0\), \(\frac{1}{4} u(0) + \frac{2}{3} u(1)=0\) and
where
Put \(\alpha=\frac{16}{3} \in(5, 6]\), when \(n=6\),
\({}_{0}c_{1} =\frac{10}{17} \in(0, \infty)\),
where each \({}_{1}\eta_{i} \) in \([0, \infty)\) and \(\gamma_{1}^{0} = \frac{\sqrt {e}-1}{25}\). Define the multifunction \(T: \overline{J} \times{\mathbb {R}}^{9}\to P(\mathbb{R})\) by
where
On the other hand, we get
for all \(t\in\overline{J}\) and \({}_{i}x_{j} \in\mathbb{R}\). It is obvious that T has convex and compact values and is of Carathéodory type. Put \(g_{0j}(t)=1\) here \(j\in N_{5}\), \(\phi_{1}({}_{0}x_{1})= \frac{4}{5}\), \(\phi_{2}({}_{0}x_{2})= \frac{1}{2}\), \(\phi_{3}({}_{0}x_{3})= \frac{1}{5}\), \(\phi_{4}({}_{0}x_{4})= \frac{1}{2}\), \(\phi_{5}({}_{0}x_{5})= \frac{1}{119} ( |{}_{0}x_{5}|+1 )\), and
for each \(t\in\overline{J}\) and \({}_{i}x_{j} \in\mathbb{R}\). Hence,
for all \(t\in\overline{J}\) and \({}_{i}x_{j}\in\mathbb{R}\). According to data values of problem (28), we have
Table 4 shows the some numerical values of \(\varLambda_{2}\) from Eq. (17), for five examples of \(q \in \frac{1}{8}, \frac{1}{5}, \frac{1}{2}, \frac{3}{4}, \frac{8}{9} \) which yield 12.891036, 11.28628, 7.188979, 5.412507, 4.758008, respectively, which that have been shown by underlined rows. On the other hand,
With the right choice for Δ from Eq. (16), the conditions of Theorem 11 hold and so problem (28) has at least one solution.
Example 2
Consider the fractional differential inclusion
for \(t\in[0,1]\), with boundary value conditions \(u^{(4)}(0) = u^{(5)}(0) =0\), \(u(0) - 3 u(1) =0\) and
where
Put \(\alpha= \frac{16}{3}\),
\(a_{1}=1\), \(a_{2}=-3\) (\(a_{1} + a_{2} \neq0\)), \(\gamma_{1}^{0}=\frac{e-1}{4050}\),
Define the multifunction \(T: [0,1] \times{\mathbb{R}}^{9}\to P(\mathbb {R})\) by
for all \(t\in[0,1]\) and \({}_{i}x_{j} \in\mathbb{R}\), where
It is clear that T has compact values. From the above assumptions, we have
for all \(t\in[0,1]\) and \({}_{i}x_{j} \in\mathbb{R}\). According to data values of problem (29), we have
Table 5 shows the some numerical values of \(\varLambda_{2}\) from Eq. (17), for five examples of \(q =\frac{1}{8}, \frac {1}{5}, \frac{1}{2}, \frac{3}{4}, \frac{8}{9}\) that yield 15.400781, 11.914180, 4.521162, 2.316636, 1.704692, respectively, which have been shown by the underlined rows. Also
Put
With the right choice Δ, we get
Now, by using Theorem 13, the inclusion problem (29) has at least one solution.
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Samei, M.E., Khalilzadeh Ranjbar, G. & Hedayati, V. Existence of solutions for equations and inclusions of multiterm fractional q-integro-differential with nonseparated and initial boundary conditions. J Inequal Appl 2019, 273 (2019). https://doi.org/10.1186/s13660-019-2224-2
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DOI: https://doi.org/10.1186/s13660-019-2224-2