## 1 Introduction

There are many works concerned with the existence of solutions for some fractional finite difference equations from different views by using the fixed point theory techniques (see for example, [17]). The readers can find more details as regards elementary notions and definitions of fractional finite difference equations in [815]. Also, much attention was devoted to the fractional differential inclusions (see for example, [9, 10, 1624]). To the best of our knowledge, there is no published research work about fractional finite difference inclusions.

In 2011, Goodrich [25] investigated the general discrete fractional boundary problem, namely

$$\left \{\textstyle\begin{array}{l} -\Delta^{\nu}y(t)=f(t+\nu-1,y(t+\nu-1)), \\ \alpha y(\nu-2)-\beta\Delta y(\nu-2)=0, \\ \gamma y(\nu+b)-\delta\Delta y(\nu+b)=0, \end{array}\displaystyle \right .$$

where $$t\in[0,b]_{\mathbb{N}_{0}}$$, $$\nu\in(1,2]$$, and $$\alpha\gamma+\alpha \delta+\beta\gamma\neq0$$ with $$\alpha,\beta,\gamma,\delta\geq0$$. In this paper, with this thought and motivation in our minds, we investigate the existence of solution for the fractional finite difference inclusion

$$\left \{\textstyle\begin{array}{l} \Delta^{\nu}x(t)\in F(t,x(t),\Delta x(t),\Delta^{2} x(t)), \\ \xi x(\nu-3)+\beta\Delta x(\nu-3)=0, \\ x(\eta)=0, \\ \gamma x(b+\nu)+\delta\Delta x(b+\nu)=0, \end{array}\displaystyle \right .$$

where $$\eta\in\mathbb{N}_{\nu-2}^{b+\nu-1}$$, $$2<\nu<3$$ and $$F:\mathbb {N}_{\nu-3}^{b+\nu+1}\times\mathbb{R}\times\mathbb{R}\times\mathbb {R}\to2^{\mathbb{R}}$$ is a compact valued multifunction.

## 2 Preliminaries

As is well known, the Gamma function has some properties as $$\Gamma (z+1)=z\Gamma(z)$$ and $$\Gamma(n)=(n-1)!$$ for all $$n\in\mathbb{N}$$. Define

$$t^{\underline{\nu}}=\frac{\Gamma(t+1)}{\Gamma(t+1-\nu)}$$

for all $$t,\nu\in\mathbb{R}$$ whenever the right-hand side is defined. If $$t+1-\nu$$ is a pole of the gamma function and $$t+1$$ is not a pole, then we define $$t^{\underline{\nu}}=0$$. One can verify that $$\nu ^{\underline{\nu}}=\nu^{\underline{\nu-1}}=\Gamma(\nu+1)$$ and $$t^{\underline{\nu+1}}=(t-\nu)t^{\underline{\nu}}$$. We use the notations $$\mathbb{N}_{a}=\{a, a+1, a+2, \ldots\}$$ for all $$a\in\mathbb {R}$$ and $$\mathbb{N}^{b}_{a}=\{a, a+1, a+2, \ldots, b\}$$ for all real numbers a and b whenever $$b-a$$ is a natural number.

Let $$\nu>0$$ be such that $$m-1<\nu\leq m$$ for some natural number m. Then the νth fractional sum of f based at a is defined by

$$\Delta^{-\nu}_{a}f(t)=\frac{1}{\Gamma(\nu)}\sum ^{t-\nu}_{k=a}\bigl(t-\sigma (k)\bigr)^{\underline{\nu-1}}f(k)$$

for all $$t\in\mathbb{N}_{a+\nu}$$. Similarly, we define

$$\Delta^{\nu}_{a}f(t)=\frac{1}{\Gamma(-\nu)}\sum ^{t+\nu}_{k=a}\bigl(t-\sigma (k)\bigr)^{\underline{-\nu-1}}f(k)$$

for all $$t\in\mathbb{N}_{a+m-\nu}$$.

### Lemma 2.1

[1]

Let $$h:\mathbb{N}_{\nu-3}^{b+\nu+1}\to\mathbb{R}$$ be a mapping and $$2<\nu\leq3$$. The general solution of the equation $$\Delta^{\nu}_{\nu -3}x(t)=h(t)$$ is given by

$$x(t)=\sum^{3}_{i=1}c_{i}t^{\underline{\nu-i}}+ \frac{1}{\Gamma(\nu)}\sum^{t-\nu}_{s=0}\bigl(t- \sigma(s)\bigr)^{\underline{\nu-1}}h(s),$$
(1)

where $$c_{1}$$, $$c_{2}$$, $$c_{3}$$ are arbitrary constants.

Since $$\Delta t^{\underline{\mu}}=\mu t^{\underline{\mu-1}}$$, we have

$$\Delta x(t)=\sum^{3}_{i=1}c_{i}( \nu-i)t^{\underline{\nu-i-1}}+\frac {1}{\Gamma(\nu-1)}\sum^{t-\nu+1}_{s=0} \bigl(t-\sigma(s)\bigr)^{\underline{\nu-2}}h(s)$$
(2)

Let $$(X,d)$$ be a metric space. Denote by $$2^{X}$$, $$\mathit{CB}(X)$$, and $$P_{\mathrm{cp}}(X)$$ the class of all nonempty subsets, the class of all closed and bounded subsets, and the class of all compact subsets of X, respectively. A mapping $$Q: X\to2^{X}$$ is called a multifunction on X and $$u\in X$$ is called a fixed point of Q whenever $$u\in Qu$$.

Consider the Hausdorff metric $$H_{d}: 2^{X}\times2^{X}\to[0,\infty)$$ by

$$H_{d}(A,B)=\max\Bigl\{ \sup_{a\in A}d(a,B), \sup _{b\in B}d(A,b)\Bigr\} ,$$

where $$d(A,b)=\inf_{a\in A}d(a, b)$$. Let $$(X,d)$$ be a metric space, $$\alpha: X\times X\to[0,\infty)$$ a map, and $$T:X\to2^{X}$$ a multifunction.

We say that X obeys the condition ($$C_{\alpha}$$) whenever for each sequence $$\{x_{n}\}$$ in X with $$\alpha(x_{n}, x_{n+1})\geq1$$ for all n and $$x_{n}\to x$$, there exists a subsequence $$\{x_{n_{k}}\}$$ of $$\{x_{n}\}$$ such that $$\alpha(x_{n_{k}},x)\geq1$$ for all k. The map T is said to be α-admissible whenever for each $$x\in X$$ and $$y\in Tx$$ with $$\alpha(x, y)\geq1$$, we have $$\alpha(y, z)\geq1$$ for all $$z\in Ty$$ [26]. Suppose that Ψ is the family of nondecreasing functions $$\psi:[0,\infty)\to[0,\infty)$$ such that $$\sum_{n=1}^{\infty }\psi^{n}(t)<\infty$$ for all $$t>0$$ (for more on this please see [26]).

By using the following fixed point result, we review the existence of solutions for the fractional finite difference inclusion

$$\Delta_{\nu-3}^{\nu}x(t)\in F\bigl(t,x(t),\Delta x(t), \Delta^{2} x(t)\bigr)$$

via the boundary conditions $$\xi x(\nu-3)+\beta\Delta x(\nu-3)=0$$, $$\gamma x(b+\nu)+\delta\Delta x(b+\nu)=0$$, and $$x(\eta)=0$$, where $$\eta \in\mathbb{N}_{\nu-2}^{b+\nu-1}$$, $$2<\nu<3$$, and $$F:\mathbb{N}_{\nu -3}^{b+\nu}\times\mathbb{R}\times\mathbb{R}\times\mathbb{R}\to 2^{\mathbb{R}}$$ is a compact valued multifunction.

### Lemma 2.2

[26]

Let $$(X,d)$$ be a complete metric space, $$\psi\in\Psi$$ a strictly increasing map, $$\alpha: X\times X\to[0,\infty)$$ a map and $$T:X\to \mathit{CB}(X)$$ an α-admissible multifunction such that $$\alpha (x,y)H(Tx,Ty)\leq\psi(d(x,y))$$ for all $$x,y \in X$$ and there exist $$x_{0}\in X$$ and $$x_{1}\in Tx_{0}$$ with $$\alpha(x_{0},x_{1})\geq1$$. If X obeys the condition ($$C_{\alpha}$$), then T has a fixed point.

## 3 Main result

In this section, we consider the fractional finite difference inclusion

$$\Delta_{\nu-3}^{\nu}x(t)\in F\bigl(t,x(t),\Delta x(t),\Delta^{2} x(t)\bigr)$$
(3)

via the boundary value conditions $$\xi x(\nu-3)+\beta\Delta x(\nu -3)=0$$, $$\gamma x(b+\nu)+\delta\Delta x(b+\nu)=0$$, and $$x(\eta)=0$$, where ξ, β, γ, δ are non-zero numbers, $$\eta\in \mathbb{N}_{\nu-2}^{b+\nu-1}$$, $$2<\nu<3$$, $$x:\mathbb{N}_{\nu-3}^{b+\nu +1}\to\mathbb{R}$$ and $$F:\mathbb{N}_{\nu-3}^{b+\nu+1}\times\mathbb {R}\times\mathbb{R}\times\mathbb{R}\to2^{\mathbb{R}}$$ is a compact valued multifunction.

### Lemma 3.1

Let $$y:\mathbb{N}_{0}^{b+1}\to\mathbb{R}$$ and $$2<\nu< 3$$. Then $$x_{0}$$ is a solution for the fractional finite difference equation $$\Delta_{\nu-3}^{\nu}x(t)=y(t)$$ via the boundary conditions $$\xi x(\nu -3)+\beta\Delta x(\nu-3)=0$$, $$x(\eta)=0$$, and $$\gamma x(b+\nu)+\delta \Delta x(b+\nu)=0$$ if and only if $$x_{0}$$ is a solution of the fractional sum equation $$x(t)=\sum_{s=0}^{b+1}G(t,s,\eta)y(s)$$, where

\begin{aligned} G(t,s,\eta) =& \biggl[\frac{[\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta+3-\nu )]t^{\underline{\nu-3}} -\theta[\gamma+\delta(\nu-1)]t^{\underline{\nu-1}}}{\theta\beta_{0}\mu \Gamma(\nu)(b+\nu)^{\underline{\nu-4}}} \\ &{}-\frac{[\xi-\beta(\nu-3)] [\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta+3-\nu)]t^{\underline{\nu -2}}}{\beta(\nu-2)\theta\beta_{0}\mu\Gamma(\nu)(b+\nu)^{\underline{\nu -4}}} \biggr] \\ &{}\times(b-s+2) \bigl(b+\nu-\sigma(s) \bigr)^{\underline{\nu-2}} + \biggl[\frac{[(\eta+2-\nu)(\eta+3-\nu)-\theta\beta_{0}]t^{\underline{\nu -3}}-\theta t^{\underline{\nu-1}}}{\beta_{0}\theta^{2}\eta^{\underline{\nu -3}}\Gamma(\nu)} \\ &{}+\frac{[-\xi+\beta(\nu-3)][(\eta+2-\nu)(\eta+3-\nu)-\theta\beta _{0}]t^{\underline{\nu-2}}}{\beta(\nu-2)\theta^{2}\beta_{0}\eta^{\underline {\nu-3}}\Gamma(\nu)} \biggr] \bigl(\eta-\sigma(s)\bigr)^{\underline{\nu-1}} \\ &{}+ \frac{(t-\sigma(s))^{\underline{\nu -1}}}{\Gamma(\nu)}, \end{aligned}

whenever $$0\leq s\leq t-\nu\leq b+1$$ and $$0\leq s\leq\eta-\nu\leq b+1$$,

\begin{aligned} G(t,s,\eta) =& \biggl[\frac{[\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta+3-\nu )]t^{\underline{\nu-3}}-\theta[\gamma+\delta(\nu-1)] t^{\underline{\nu-1}}}{\theta\beta_{0}\mu\Gamma(\nu)(b+\nu)^{\underline {\nu-4}}} \\ &{}-\frac{[\xi-\beta(\nu-3)] [\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta+3-\nu)]t^{\underline{\nu -2}}}{\beta(\nu-2)\theta\beta_{0}\mu\Gamma(\nu)(b+\nu)^{\underline{\nu -4}}} \biggr] \\ &{}\times(b-s+2) \bigl(b+\nu-\sigma(s) \bigr)^{\underline{\nu-2}} + \biggl[\frac{[(\eta+2-\nu)(\eta+3-\nu)-\theta\beta_{0}]t^{\underline{\nu -3}}-\theta t^{\underline{\nu-1}}}{\beta_{0}\theta^{2}\eta^{\underline{\nu -3}}\Gamma(\nu)} \\ &{}+\frac{[-\xi+\beta(\nu-3)][(\eta+2-\nu)(\eta+3-\nu)-\theta\beta _{0}]t^{\underline{\nu-2}}}{\beta(\nu-2)\theta^{2}\beta_{0}\eta^{\underline {\nu-3}}\Gamma(\nu)} \biggr] \bigl(\eta-\sigma(s)\bigr)^{\underline{\nu-1}}, \end{aligned}

whenever $$0\leq t-\nu< s\leq\eta-\nu\leq b+1$$,

\begin{aligned} G(t,s,\eta) =& \biggl[\frac{[\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta+3-\nu )]t^{\underline{\nu-3}}-\theta[\gamma+\delta(\nu-1)] t^{\underline{\nu-1}}}{\theta\beta_{0}\mu\Gamma(\nu)(b+\nu)^{\underline {\nu-4}}} \\ &{}-\frac{[\xi-\beta(\nu-3)][\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta +3-\nu)]t^{\underline{\nu-2}}}{\beta(\nu-2)\theta\beta_{0}\mu\Gamma(\nu )(b+\nu)^{\underline{\nu-4}}} \biggr] \\ &{}\times(b-s+2) \bigl(b+\nu-\sigma(s) \bigr)^{\underline{\nu-2}} +\frac{(t-\sigma(s))^{\underline{\nu-1}}}{\Gamma(\nu)}, \end{aligned}

whenever $$0\leq\eta-\nu< s\leq t-\nu\leq b+1$$ and

\begin{aligned} G(t,s,\eta) =& \biggl[\frac{[\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta+3-\nu )]t^{\underline{\nu-3}}-\theta[\gamma+\delta(\nu-1)]t^{\underline{\nu-1}}}{ \theta\beta_{0}\mu\Gamma(\nu)(b+\nu)^{\underline{\nu-4}}} \\ &{}-\frac{[\xi-\beta(\nu-3)][\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta +3-\nu)]t^{\underline{\nu-2}}}{\beta(\nu-2)\theta\beta_{0}\mu\Gamma(\nu )(b+\nu)^{\underline{\nu-4}}} \biggr] \\ &{}\times(b-s+2) \bigl(b+\nu-\sigma(s) \bigr)^{\underline{\nu-2}}, \end{aligned}

whenever $$0\leq t-\nu< s\leq b+1$$ and $$0\leq\eta-\nu< s\leq b+1$$. Here,

\begin{aligned}& \theta=\frac{\eta\beta\nu-\eta\xi-3\eta\beta-2\xi+\xi\nu-\beta\nu ^{2}+6\beta\nu-8\beta}{\beta(\nu-2)}, \\& \mu=\frac{b\xi\delta\nu-2b\delta\xi+\gamma\xi b^{2}+3b\gamma\xi+\beta b\nu ^{2}\delta+\delta b^{2}\beta\nu+\beta b\delta\nu-6\beta\delta b+3\beta \delta b^{2}+4\xi\delta\nu}{\beta(\nu-2)} \\& \hphantom{\mu=}{}+\frac{-8\delta\xi+4\gamma\xi b+12\gamma\xi+4\beta\nu^{2}\delta+7\gamma \beta\nu b+12\gamma\beta\nu+4\beta\delta\nu-24\beta\delta+21\beta\gamma b+36\beta\gamma}{\beta(\nu-2)} \end{aligned}

and

$$\beta_{0}=\frac{\theta[\delta(\nu-1)+\gamma(b+2)](b+3)(b+4)+\mu(\eta+2-\nu )(\eta+3-\nu)}{\theta\mu}.$$

### Proof

Let $$x_{0}$$ be a solution for the equation $$\Delta_{\nu-3}^{\nu}x(t)=y(t)$$ via the boundary conditions $$\xi x(\nu -3)+\beta\Delta x(\nu-3)=0$$, $$x(\eta)=0$$, and $$\gamma x(b+\nu)+\delta \Delta x(b+\nu)=0$$. Then by using (2) and Lemma 2.1, we get

$$x_{0}(t)=c_{1}t^{\underline{\nu-1}}+c_{2}t^{\underline{\nu -2}}+c_{3}t^{\underline{\nu-3}} +\frac{1}{\Gamma(\nu)}\sum_{s=0}^{t-\nu}\bigl(t- \sigma(s)\bigr)^{\underline{\nu-1}}y(s)$$

and

\begin{aligned} \Delta x_{0}(t) =&c_{1}(\nu-1)t^{\underline{\nu-2}}+c_{2}( \nu-2)t^{\underline {\nu-3}}+c_{3}(\nu-3)t^{\underline{\nu-4}} \\ &{}+\frac{1}{\Gamma(\nu-1)} \sum_{s=0}^{t-\nu+1}\bigl(t-\sigma(s) \bigr)^{\underline {\nu-2}}y(s), \end{aligned}

where $$c_{1},c_{2},c_{3}\in\mathbb{R}$$ are arbitrary constants. Now, by using the boundary condition

$$\xi x(\nu-3)+\beta\Delta x(\nu-3)=0,$$

we get $$\xi c_{3} +\beta[c_{2}(\nu-2)+c_{3}(\nu-3)]=0$$. Also, by using the condition $$x(\eta)=0$$ we obtain

\begin{aligned} c_{3} =&-(\eta+2-\nu) (\eta+3-\nu)c_{1}-(\eta+2- \nu)c_{2} \\ &{}-\frac{1}{\eta^{\underline{\nu-3}}\Gamma(\nu)}\sum_{s=0}^{\eta-\nu} \bigl(\eta -\sigma(s)\bigr)^{\underline{\nu-1}}y(s). \end{aligned}

Moreover, by using the boundary condition $$\gamma x(b+\nu)+\delta \Delta x(b+\nu)=0$$, we get

\begin{aligned}& c_{1}\bigl[\delta(\nu-1)+\gamma(b+2)\bigr](b+\nu)^{\underline{\nu-2}}+c_{2} \bigl[\delta(\nu -2)+\gamma(b+3)\bigr](b+\nu)^{\underline{\nu-3}} \\& \qquad {}+c_{3} \bigl[\delta(\nu-3)+\gamma(b+4)\bigr] (b+\nu)^{\underline{\nu-4}} \\& \quad =-\frac{\delta}{\Gamma(\nu-1)}\sum_{s=0}^{b+1} \bigl(b+\nu-\sigma (s)\bigr)^{\underline{\nu-2}}y(s)-\frac{\gamma}{\Gamma(\nu)}\sum _{s=0}^{b} \bigl(b+\nu-\sigma(s)\bigr)^{\underline{\nu-1}}y(s). \end{aligned}

Thus, by using a simple calculation, we get

\begin{aligned}& c_{1}=-\frac{1}{\beta_{0}\theta\eta^{\underline{\nu-3}}\Gamma(\nu)}\sum_{s=0}^{\eta-\nu} \bigl(\eta-\sigma(s)\bigr)^{\underline{\nu-1}}y(s) \\& \hphantom{c_{1}={}}{}-\frac{\gamma+\delta(\nu-1)}{\beta_{0}\mu\Gamma(\nu)(b+\nu)^{\underline {\nu-4}}}\sum _{s=0}^{b+1}(b-s+2) \bigl(b+\nu-\sigma(s) \bigr)^{\underline{\nu-2}}y(s), \\& c_{2}=\frac{[-\xi+\beta(\nu-3)][(\eta+2-\nu)(\eta+3-\nu)-\theta\beta _{0}]}{\beta(\nu-2)\theta^{2}\beta_{0}\eta^{\underline{\nu-3}}\Gamma(\nu)}\sum_{s=0}^{\eta-\nu} \bigl(\eta-\sigma(s)\bigr)^{\underline{\nu-1}}y(s) \\& \hphantom{c_{2}={}}{}-\frac{[\xi-\beta(\nu-3)][\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta +3-\nu)]}{\beta(\nu-2)\theta\beta_{0}\mu\Gamma(\nu)(b+\nu)^{\underline{\nu -4}}} \\& \hphantom{c_{2}={}}{}\times\sum_{s=0}^{b+1}(b-s+2) \bigl(b+\nu-\sigma(s)\bigr)^{\underline{\nu-2}}y(s) \end{aligned}

and

\begin{aligned} c_{3} =&\frac{(\eta+2-\nu)(\eta+3-\nu)-\theta\beta_{0}}{\theta^{2}\beta_{0}\eta ^{\underline{\nu-3}}\Gamma(\nu)}\sum_{s=0}^{\eta-\nu} \bigl(\eta-\sigma(s)\bigr)^{\underline{\nu-1}}y(s) \\ &{}+\frac{[\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta+3-\nu)]}{\theta\beta _{0}\mu\Gamma(\nu) (b+\nu)^{\underline{\nu-4}}}\sum_{s=0}^{b+1}(b-s+2) \bigl(b+\nu-\sigma (s)\bigr)^{\underline{\nu-2}}y(s). \end{aligned}

Hence,

\begin{aligned} x_{0}(t) =& \biggl[\frac{[\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta+3-\nu )]t^{\underline{\nu-3}}-\theta[\gamma+\delta(\nu-1)]t^{\underline{\nu-1}}}{ \theta\beta_{0}\mu\Gamma(\nu)(b+\nu)^{\underline{\nu-4}}} \\ &{}-\frac{[\xi-\beta(\nu-3)][\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta +3-\nu)]t^{\underline{\nu-2}}}{\beta(\nu-2)\theta\beta_{0}\mu\Gamma(\nu )(b+\nu)^{\underline{\nu-4}}} \biggr] \\ &{}\times \sum_{s=0}^{b+1}(b-s+2) \bigl(b+\nu-\sigma(s)\bigr)^{\underline{\nu-2}}y(s) \\ &{}+ \biggl[\frac{[(\eta+2-\nu)(\eta+3-\nu)-\theta\beta_{0}]t^{\underline{\nu -3}}-\theta t^{\underline{\nu-1}}}{\beta_{0}\theta^{2}\eta^{\underline{\nu -3}}\Gamma(\nu)} \\ &{}+\frac{[-\xi+\beta(\nu-3)][(\eta+2-\nu)(\eta+3-\nu)-\theta\beta _{0}]t^{\underline{\nu-2}}}{\beta(\nu-2)\theta^{2}\beta_{0}\eta^{\underline {\nu-3}}\Gamma(\nu)} \biggr] \sum_{s=0}^{\eta-\nu} \bigl(\eta-\sigma(s)\bigr)^{\underline{\nu-1}}y(s) \\ &{}+\sum_{s=0}^{t-\nu}\frac{(t-\sigma(s))^{\underline{\nu-1}}}{\Gamma(\nu)}y(s) =\sum_{s=0}^{b+1}G(s,t,\eta)y(s). \end{aligned}

Now, let $$x_{0}$$ be a solution for the equation $$x(t)=\sum_{s=0}^{b+1}G(s,t,\eta)y(s)$$. Then we have

\begin{aligned} x_{0}(t) =& \biggl[\frac{[\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta+3-\nu )]t^{\underline{\nu-3}}-\theta[\gamma+\delta(\nu-1)]t^{\underline{\nu-1}}}{ \theta\beta_{0}\mu\Gamma(\nu)(b+\nu)^{\underline{\nu-4}}} \\ &{}-\frac{[\xi-\beta(\nu-3)][\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta +3-\nu)] t^{\underline{\nu-2}}}{\beta(\nu-2)\theta\beta_{0}\mu\Gamma(\nu)(b+\nu )^{\underline{\nu-4}}} \biggr] \\ &{}\times\sum_{s=0}^{b+1}(b-s+2) \bigl(b+\nu-\sigma(s)\bigr)^{\underline{\nu-2}}y(s) \\ &{}+ \biggl[\frac{[(\eta+2-\nu)(\eta+3-\nu)-\theta\beta_{0}]t^{\underline{\nu -3}}-\theta t^{\underline{\nu-1}}}{\beta_{0}\theta^{2}\eta^{\underline{\nu -3}}\Gamma(\nu)} \\ &{}+\frac{[-\xi+\beta(\nu-3)][(\eta+2-\nu)(\eta+3-\nu)-\theta\beta _{0}]t^{\underline{\nu-2}}}{\beta(\nu-2)\theta^{2}\beta_{0}\eta^{\underline {\nu-3}}\Gamma(\nu)} \biggr] \\ &{}\times\sum_{s=0}^{\eta-\nu} \bigl(\eta-\sigma(s)\bigr)^{\underline{\nu-1}}y(s) +\sum_{s=0}^{t-\nu}\frac{(t-\sigma(s))^{\underline{\nu-1}}}{\Gamma(\nu)}y(s). \end{aligned}

Since $$(\nu-3)^{\underline{\nu-1}}=(\nu-3)^{\underline{\nu-2}}=0$$, $$(\nu -3)^{\underline{\nu-3}}=(\nu-3)^{\underline{\nu-4}}=\Gamma(\nu-2)$$, and

$$\sum_{s=0}^{-3}\bigl(\nu-3-\sigma(s) \bigr)^{\underline{\nu-1}}y(s)=\sum_{s=0}^{-2} \bigl(\nu-3-\sigma(s)\bigr)^{\underline{\nu-2}}y(s)=0,$$

we get $$\xi x_{0}(\nu-3)+\beta\Delta x_{0}(\nu-3)=0$$. A simple calculation shows us $$\gamma x_{0}(b+\nu)+\delta\Delta x_{0}(b+\nu)=0$$ and $$x_{0}(\eta)=0$$. On the other hand,

$$x_{0}(t)=c_{1}t^{\underline{\nu-1}}+c_{2}t^{\underline{\nu-2}} +c_{3}t^{\underline{\nu-3}}+\frac{1}{\Gamma(\nu)}\sum _{s=0}^{t-\nu }\bigl(t-\sigma(s)\bigr)^{\underline{\nu-1}}y(s)$$

is a solution for the equation $$\Delta^{\nu}_{\nu-3} x(t)=y(t)$$ and so $$\Delta^{\nu}_{\nu-3} x_{0}(t)=y(t)$$. □

A function $$x:\mathbb{N}_{\nu-3}^{b+\nu+1}\to\mathbb{R}$$ is a solution of the problem (3) whenever it satisfies the boundary conditions and there exists a function $$y:\mathbb{N}_{0}^{b+1}\to\mathbb {R}$$ such that

$$y(t)\in F\bigl(t, x(t),\Delta x(t),\Delta^{2} x(t)\bigr)$$

for all $$t\in\mathbb{N}_{0}^{b+1}$$ and

\begin{aligned} x(t) =& \biggl[\frac{[\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta+3-\nu )]t^{\underline{\nu-3}}-\theta[\gamma+\delta(\nu-1)]t^{\underline{\nu-1}}}{ \theta\beta_{0}\mu\Gamma(\nu)(b+\nu)^{\underline{\nu-4}}} \\ &{}-\frac{[\xi-\beta(\nu-3)][\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta +3-\nu)]t^{\underline{\nu-2}}}{\beta(\nu-2)\theta\beta_{0}\mu\Gamma(\nu )(b+\nu)^{\underline{\nu-4}}} \biggr] \\ &{}\times\sum_{s=0}^{b+1}(b-s+2) \bigl(b+\nu-\sigma(s)\bigr)^{\underline{\nu-2}}y(s) \\ &{}+ \biggl[\frac{[(\eta+2-\nu)(\eta+3-\nu)-\theta\beta_{0}]t^{\underline{\nu -3}}-\theta t^{\underline{\nu-1}}}{\beta_{0}\theta^{2}\eta^{\underline{\nu -3}}\Gamma(\nu)} \\ &{}+\frac{[-\xi+\beta(\nu-3)][(\eta+2-\nu)(\eta+3-\nu)-\theta\beta _{0}]t^{\underline{\nu-2}}}{\beta(\nu-2)\theta^{2}\beta_{0}\eta^{\underline {\nu-3}}\Gamma(\nu)} \biggr] \\ &{}\times\sum_{s=0}^{\eta-\nu} \bigl(\eta-\sigma(s)\bigr)^{\underline{\nu-1}}y(s) +\sum_{s=0}^{t-\nu}\frac{(t-\sigma(s))^{\underline{\nu-1}}}{\Gamma(\nu)}y(s). \end{aligned}

Let $$\mathcal{X}$$ be the set of all functions $$x:\mathbb{N}_{\nu -3}^{b+\nu+1}\to\mathbb{R}$$ endowed with the norm

$$\|x\|=\max_{t\in\mathbb{N}_{\nu-3}^{b+\nu+1}}\bigl\vert x(t)\bigr\vert +\max _{t\in\mathbb {N}_{\nu-3}^{b+\nu+1}}\bigl\vert \Delta x(t)\bigr\vert +\max _{t\in\mathbb{N}_{\nu-3}^{b+\nu +1}}\bigl\vert \Delta^{2} x(t)\bigr\vert .$$

We show that $$(\mathcal{X},\|\cdot\|)$$ is a Banach space. Let $$\{x_{n}\}$$ be a Cauchy sequence in $$\mathcal{X}$$ and $$\epsilon>0$$ be given. Choose a natural number N such that $$\|x_{n}-x_{m}\|<\epsilon$$ for all $$m,n>N$$. This implies that $$\max_{t\in\mathbb{N}_{\nu-3}^{b+\nu+1}}|x_{n}(t)-x_{m}(t)|<\epsilon$$, $$\max_{t\in\mathbb{N}_{\nu-3}^{b+\nu+1}}|\Delta x_{n}(t)-\Delta x_{m}(t)|<\epsilon$$ and

$$\max_{t\in\mathbb{N}_{\nu-3}^{b+\nu+1}}\bigl\vert \Delta^{2} x_{n}(t)-\Delta^{2} x_{m}(t)\bigr\vert < \epsilon.$$

Choose $$x(t), z(t), w(t)\in\mathbb{R}$$ such that $$x_{n}(t)\to x(t)$$, $$\Delta x_{n}(t)\to z(t)$$, and $$\Delta^{2} x_{n}(t)\to w(t)$$ for all $$t\in \mathbb{N}_{\nu-3}^{b+\nu+1}$$. Note that $$\Delta x_{n}(t)=x_{n}(t+1)-x_{n}(t)$$ and so $$\Delta x(t)=x(t+1)-x(t)=z(t)$$. Similarly, we get $$\Delta^{2} x(t)=w(t)$$. This implies that $$|x_{n}(t)-x(t)|<\frac{\epsilon}{3}$$, $$|\Delta x_{n}(t)-\Delta x(t)|<\frac {\epsilon}{3}$$, and $$|\Delta^{2} x_{n}(t)-\Delta^{2} x(t)|<\frac{\epsilon }{3}$$ for all $$t\in\mathbb{N}_{\nu-3}^{b+\nu+1}$$ and $$n>M$$ for some natural number M. Thus,

$$\|x_{n}-x\|=\max_{t\in\mathbb{N}_{\nu-3}^{b+\nu+1}}\bigl\vert x_{n}(t)-x(t)\bigr\vert +\max_{t\in\mathbb{N}_{\nu-3}^{b+\nu+1}}\bigl\vert \Delta x_{n}(t)-\Delta x(t)\bigr\vert +\max_{t\in\mathbb{N}_{\nu-3}^{b+\nu+1}} \bigl\vert \Delta^{2} x(t)-\Delta^{2} x(t)\bigr\vert < \epsilon.$$

Hence, $$(\mathcal{X},\|\cdot\|)$$ is a Banach space.

Let $$x\in\mathcal {X}$$. Define the set of selections of F by

$$S_{F,x}=\bigl\{ y:\mathbb{N}_{0}^{b+1}\to\mathbb{R} \mid y(t)\in F\bigl(t, x(t),\Delta x(t),\Delta^{2} x(t)\bigr) \mbox{ for all } t \in\mathbb {N}_{0}^{b+1}\bigr\} .$$

Since $$F(t, x(t),\Delta x(t),\Delta^{2} x(t))\neq\emptyset$$, the selection principle implies that $$S_{F,x}$$ is nonempty.

### Theorem 3.2

Suppose that $$\psi\in\Psi$$ and $$F: \mathbb{N}_{\nu-3}^{b+\nu+1}\times \mathbb{R} \times\mathbb{R}\times\mathbb{R}\to P_{\mathrm{cp}}(\mathbb{R})$$ is a multifunction such that

$$H_{d}\bigl(F(t,x_{1},x_{2},x_{3})-F(t,z_{1},z_{2},z_{3}) \bigr)\leq\psi\bigl(\vert x_{1}-z_{1}\vert +\vert x_{2}-z_{2}\vert +\vert x_{3}-z_{3} \vert \bigr)$$

for all $$t\in\mathbb{N}_{\nu-3}^{b+\nu+1}$$ and $$x_{1},x_{2},x_{3},z_{1},z_{2},z_{3}\in\mathbb{R}$$. Then the boundary value inclusion (3) has a solution.

### Proof

Choose $$y\in S_{F,x}$$ and put $$h(t)=\sum_{s=0}^{b+1}G(t,s,\eta)y(s)$$ for all $$t\in\mathbb{N}_{\nu-3}^{\nu+b+1}$$. Then $$h\in\mathcal{X}$$ and so the set

$$\Biggl\{ h\in\mathcal{X}: \mbox{there exists } y\in S_{F,x} \mbox{ such that } h(t)=\sum_{s=0}^{b+1}G(t,s,\eta)y(s) \mbox{ for all } t\in\mathbb {N}_{\nu-3}^{b+\nu+1} \Biggr\}$$

is nonempty. Now define $$\mathcal{F}: \mathcal{X}\to2^{\mathcal{X}}$$ by

\begin{aligned} \mathcal{F}(x) =& \Biggl\{ h\in\mathcal{X}: \mbox{there exists } y\in S_{F,x} \mbox{ such that } h(t)=\sum_{s=0}^{b+1}G(t,s, \eta)y(s) \\ &\mbox{for all } t\in\mathbb{N}_{\nu-3}^{b+\nu+1} \Biggr\} . \end{aligned}

We show that the multifunction $$\mathcal{F}$$ has a fixed point. First, we show that $$\mathcal{F}(x)$$ is closed subset of $$\mathcal{X}$$ for all $$x\in\mathcal{X}$$. Let $$x\in\mathcal{X}$$ and $$\{u_{n}\}_{n\geq1}$$ be a sequence in $$\mathcal{F}(x)$$ with $$u_{n}\to u$$. For each n, choose $$y_{n} \in S_{F,x}$$ such that

\begin{aligned} u_{n}(t) =& \biggl[\frac{[\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta+3-\nu )]t^{\underline{\nu-3}}-\theta[\gamma+\delta(\nu-1)] t^{\underline{\nu-1}}}{\theta\beta_{0}\mu\Gamma(\nu)(b+\nu)^{\underline {\nu-4}}} \\ &{}-\frac{[\xi-\beta(\nu-3)][\gamma+\delta(\nu-1)] [(\eta+2-\nu)(\eta+3-\nu)]t^{\underline{\nu-2}}}{\beta(\nu-2)\theta\beta _{0}\mu\Gamma(\nu)(b+\nu)^{\underline{\nu-4}}} \biggr] \\ &{}\times\sum_{s=0}^{b+1}(b-s+2) \bigl(b+\nu-\sigma(s)\bigr)^{\underline{\nu-2}}y_{n}(s) \\ &{}+ \biggl[\frac{[(\eta+2-\nu)(\eta+3-\nu)-\theta\beta_{0}]t^{\underline{\nu -3}}-\theta t^{\underline{\nu-1}}}{\beta_{0}\theta^{2}\eta^{\underline{\nu -3}}\Gamma(\nu)} \\ &{}+\frac{[-\xi+\beta(\nu-3)][(\eta+2-\nu)(\eta+3-\nu)-\theta\beta _{0}]t^{\underline{\nu-2}}}{\beta(\nu-2)\theta^{2}\beta_{0}\eta^{\underline {\nu-3}}\Gamma(\nu)} \biggr] \\ &{}\times\sum_{s=0}^{\eta-\nu} \bigl(\eta-\sigma(s)\bigr)^{\underline{\nu-1}}y_{n}(s) +\sum_{s=0}^{t-\nu}\frac{(t-\sigma(s))^{\underline{\nu-1}}}{\Gamma(\nu)}y_{n}(s) \end{aligned}

for all $$t\in\mathbb{N}_{\nu-3}^{b+\nu+1}$$. Since F has compact values, $$\{y_{n}\}_{n\geq1}$$ has a subsequence which converges to some $$y\in S_{F,x}$$. We denote this subsequence again by $$\{y_{n}\}_{n\geq 1}$$. So

\begin{aligned} u_{n}(t) \to& u(t) \\ =& \biggl[\frac{[\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta +3-\nu)]t^{\underline{\nu-3}}-\theta[\gamma+\delta(\nu-1)] t^{\underline{\nu-1}}}{\theta\beta_{0}\mu\Gamma(\nu)(b+\nu)^{\underline {\nu-4}}} \\ &{}-\frac{[\xi-\beta(\nu-3)][\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta +3-\nu)]t^{\underline{\nu-2}}}{\beta(\nu-2)\theta\beta_{0}\mu\Gamma(\nu )(b+\nu)^{\underline{\nu-4}}} \biggr] \\ &{}\times\sum_{s=0}^{b+1}(b-s+2) \bigl(b+\nu-\sigma(s)\bigr)^{\underline{\nu-2}}y(s) \\ &{}+ \biggl[\frac{[(\eta+2-\nu)(\eta+3-\nu)-\theta\beta_{0}]t^{\underline{\nu -3}}-\theta t^{\underline{\nu-1}}}{\beta_{0}\theta^{2}\eta^{\underline{\nu -3}}\Gamma(\nu)} \\ &{}+\frac{[-\xi+\beta(\nu-3)][(\eta+2-\nu)(\eta+3-\nu)-\theta\beta _{0}]t^{\underline{\nu-2}}}{\beta(\nu-2)\theta^{2}\beta_{0}\eta^{\underline {\nu-3}}\Gamma(\nu)} \biggr] \\ &{}\times\sum_{s=0}^{\eta-\nu} \bigl(\eta-\sigma(s)\bigr)^{\underline{\nu-1}}y(s) +\sum_{s=0}^{t-\nu}\frac{(t-\sigma(s))^{\underline{\nu-1}}}{\Gamma(\nu)}y(s) \end{aligned}

for all $$t\in\mathbb{N}_{\nu-3}^{b+\nu+1}$$. This implies that $$u\in \mathcal{F}(x)$$. Thus, the multifunction $$\mathcal{F}$$ has closed values. Since F is a compact multifunction, it is easy to check that $$\mathcal {F}(x)$$ is bounded set in $$\mathcal{X}$$ for all $$x\in\mathcal{X}$$. Let $$x,z\in\mathcal{X}$$, $$h_{1}\in\mathcal{F}(x)$$, and $$h_{2}\in\mathcal {F}(z)$$. Choose $$y_{1}\in S_{F,x}$$ and $$y_{2}\in S_{F,z}$$ such that

\begin{aligned} h_{1}(t) =& \biggl[\frac{[\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta+3-\nu )]t^{\underline{\nu-3}}-\theta[\gamma+\delta(\nu-1)] t^{\underline{\nu-1}}}{\theta\beta_{0}\mu\Gamma(\nu)(b+\nu)^{\underline {\nu-4}}} \\ &{}-\frac{[\xi-\beta(\nu-3)][\gamma+\delta(\nu-1)] [(\eta+2-\nu)(\eta+3-\nu)]t^{\underline{\nu-2}}}{\beta(\nu-2)\theta\beta _{0}\mu\Gamma(\nu)(b+\nu)^{\underline{\nu-4}}} \biggr] \\ &{}\times\sum_{s=0}^{b+1}(b-s+2) \bigl(b+\nu-\sigma(s)\bigr)^{\underline{\nu-2}}y_{1}(s) \\ &{}+ \biggl[\frac{[(\eta+2-\nu)(\eta+3-\nu)-\theta\beta_{0}]t^{\underline{\nu -3}}-\theta t^{\underline{\nu-1}}}{\beta_{0}\theta^{2}\eta^{\underline{\nu -3}}\Gamma(\nu)} \\ &{}+\frac{[-\xi+\beta(\nu-3)][(\eta+2-\nu)(\eta+3-\nu)-\theta\beta _{0}]t^{\underline{\nu-2}}}{\beta(\nu-2)\theta^{2}\beta_{0}\eta^{\underline {\nu-3}}\Gamma(\nu)} \biggr] \\ &{}\times\sum_{s=0}^{\eta-\nu} \bigl(\eta-\sigma(s)\bigr)^{\underline{\nu-1}}y_{1}(s) +\sum_{s=0}^{t-\nu}\frac{(t-\sigma(s))^{\underline{\nu-1}}}{\Gamma(\nu)}y_{1}(s) \end{aligned}

and

\begin{aligned} h_{2}(t) =& \biggl[\frac{[\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta+3-\nu )]t^{\underline{\nu-3}}-\theta[\gamma+\delta(\nu-1)] t^{\underline{\nu-1}}}{\theta\beta_{0}\mu\Gamma(\nu)(b+\nu)^{\underline {\nu-4}}} \\ &{}-\frac{[\xi-\beta(\nu-3)][\gamma+\delta(\nu-1)] [(\eta+2-\nu)(\eta+3-\nu)]t^{\underline{\nu-2}}}{\beta(\nu-2)\theta\beta _{0}\mu\Gamma(\nu)(b+\nu)^{\underline{\nu-4}}} \biggr] \\ &{}\times\sum_{s=0}^{b+1}(b-s+2) \bigl(b+\nu-\sigma(s)\bigr)^{\underline{\nu-2}}y_{2}(s) \\ &{}+ \biggl[\frac{[(\eta+2-\nu)(\eta+3-\nu)-\theta\beta_{0}]t^{\underline{\nu -3}}-\theta t^{\underline{\nu-1}}}{\beta_{0}\theta^{2}\eta^{\underline{\nu -3}}\Gamma(\nu)} \\ &{}+\frac{[-\xi+\beta(\nu-3)][(\eta+2-\nu)(\eta+3-\nu)-\theta\beta _{0}]t^{\underline{\nu-2}}}{\beta(\nu-2)\theta^{2}\beta_{0}\eta^{\underline {\nu-3}}\Gamma(\nu)} \biggr] \\ &{}\times\sum_{s=0}^{\eta-\nu} \bigl(\eta-\sigma(s)\bigr)^{\underline{\nu-1}}y_{2}(s) +\sum_{s=0}^{t-\nu}\frac{(t-\sigma(s))^{\underline{\nu-1}}}{\Gamma(\nu)}y_{2}(s) \end{aligned}

for all $$t\in\mathbb{N}_{\nu-3}^{b+\nu+1}$$. Since

\begin{aligned}& H_{d}\bigl(F\bigl(t,x(t),\Delta x(t),\Delta^{2} x(t) \bigr)-F\bigl(t,z(t),\Delta z(t),\Delta^{2} z(t)\bigr)\bigr) \\& \quad \leq \psi\bigl(\bigl\vert x(t)-z(t)\bigr\vert +\bigl\vert \Delta x(t)-\Delta z(t)\bigr\vert +\bigl\vert \Delta^{2} x(t)- \Delta^{2} z(t)\bigr\vert \bigr) \end{aligned}

for all $$x,z\in\mathcal{X}$$ and $$t\in\mathbb{N}_{\nu-3}^{b+\nu+1}$$, we get

$$\bigl\vert y_{1}(t)-y_{2}(t)\bigr\vert \leq\psi\bigl( \bigl\vert x(t)-z(t)\bigr\vert +\bigl\vert \Delta x(t)-\Delta z(t)\bigr\vert +\bigl\vert \Delta^{2} x(t)-\Delta^{2} z(t)\bigr\vert \bigr).$$

Now, put

\begin{aligned}& G_{1}=\max_{t\in\mathbb{N}_{\nu-3}^{b+1+\nu}} \Biggl\{ \biggl\vert \frac{[\gamma +\delta(\nu-1)][(\eta+2-\nu)(\eta+3-\nu)]t^{\underline{\nu-3}} -\theta[\gamma+\delta(\nu-1)]t^{\underline{\nu-1}}}{\theta\beta_{0}\mu \Gamma(\nu)(b+\nu)^{\underline{\nu-4}}} \\& \hphantom{G_{1}={}}{}-\frac{[\xi-\beta(\nu-3)] [\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta+3-\nu)]t^{\underline{\nu -2}}}{\beta(\nu-2)\theta\beta_{0}\mu\Gamma(\nu)(b+\nu)^{\underline{\nu -4}}}\biggr\vert \\& \hphantom{G_{1}={}}{}\times\sum _{s=0}^{b+1}(b-s+2) \bigl(b+\nu-\sigma(s) \bigr)^{\underline{\nu-2}} +\biggl\vert \frac{[(\eta+2-\nu)(\eta+3-\nu)-\theta\beta_{0}]t^{\underline{\nu -3}}-\theta t^{\underline{\nu-1}}}{\beta_{0}\theta^{2}\eta^{\underline{\nu -3}}\Gamma(\nu)} \\& \hphantom{G_{1}={}}{}+\frac{[-\xi+\beta(\nu-3)][(\eta+2-\nu)(\eta+3-\nu)-\theta\beta _{0}]t^{\underline{\nu-2}}}{\beta(\nu-2)\theta^{2}\beta_{0}\eta^{\underline {\nu-3}}\Gamma(\nu)}\biggr\vert \\& \hphantom{G_{1}={}}{}\times\sum _{s=0}^{\eta-\nu}\bigl(\eta-\sigma(s)\bigr)^{\underline{\nu-1}}+ \sum_{s=0}^{t-\nu}\frac{(t-\sigma(s))^{\underline{\nu-1}}}{\Gamma(\nu)} \Biggr\} , \\& G_{2}=\max_{t\in\mathbb{N}_{\nu-3}^{b+1+\nu}} \Biggl\{ \biggl\vert \frac{(\nu -3)[\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta+3-\nu)]t^{\underline{\nu-4}}}{\theta\beta _{0}\mu\Gamma(\nu)(b+\nu)^{\underline{\nu-4}}} \\& \hphantom{G_{2}={}}{}-\frac{(\nu-1)\theta[\gamma+\delta(\nu-1)]t^{\underline{\nu-2}}}{\theta\beta _{0}\mu\Gamma(\nu)(b+\nu)^{\underline{\nu-4}}} \\& \hphantom{G_{2}={}}{}-\frac{[\xi-\beta(\nu-3)] [\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta+3-\nu)]t^{\underline{\nu -3}}}{\beta\theta\beta_{0}\mu\Gamma(\nu)(b+\nu)^{\underline{\nu-4}}}\biggr\vert \\& \hphantom{G_{2}={}}{}\times\sum _{s=0}^{b+1}(b-s+2) \bigl(b+\nu-\sigma(s) \bigr)^{\underline{\nu-2}} \\& \hphantom{G_{2}={}}{}+\biggl\vert \frac{(\nu-3)[(\eta+2-\nu)(\eta+3-\nu)-\theta\beta _{0}]t^{\underline{\nu-4}}-\theta(\nu-1) t^{\underline{\nu-2}}}{\beta _{0}\theta^{2}\eta^{\underline{\nu-3}}\Gamma(\nu)} \\& \hphantom{G_{2}={}}{}+\frac{[-\xi+\beta(\nu-3)][(\eta+2-\nu)(\eta+3-\nu)-\theta\beta _{0}]t^{\underline{\nu-3}}}{\beta\theta^{2}\beta_{0}\eta^{\underline{\nu -3}}\Gamma(\nu)}\biggr\vert \\& \hphantom{G_{2}={}}{}\times\sum _{s=0}^{\eta-\nu}\bigl(\eta-\sigma(s)\bigr)^{\underline{\nu-1}}+ \sum_{s=0}^{t-\nu+1}\frac{(t-\sigma(s))^{\underline{\nu-2}}}{\Gamma(\nu -1)} \Biggr\} \end{aligned}

and

\begin{aligned} G_{3} =&\max_{t\in\mathbb{N}_{\nu-3}^{b+1+\nu}} \Biggl\{ \biggl\vert \frac{(\nu-3)(\nu-4)[\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta+3-\nu)] t^{\underline{\nu-5}}}{\theta\beta_{0}\mu\Gamma(\nu)(b+\nu )^{\underline{\nu-4}}} \\ &{}-\frac{(\nu-1)(\nu-2)\theta[\gamma+\delta(\nu -1)]t^{\underline{\nu-3}}}{\theta\beta_{0}\mu\Gamma(\nu)(b+\nu )^{\underline{\nu-4}}} \\ &{}-\frac{(\nu-3)[\xi-\beta(\nu-3)][\gamma+\delta(\nu-1)][(\eta+2-\nu )(\eta+3-\nu)]t^{\underline{\nu-4}}}{\beta\theta\beta_{0}\mu\Gamma(\nu) (b+\nu)^{\underline{\nu-4}}}\biggr\vert \\ &{}\times\sum_{s=0}^{b+1}(b-s+2) \bigl(b+\nu-\sigma (s)\bigr)^{\underline{\nu-2}} \\ &{}+\biggl\vert \frac{(\nu-3)(\nu-4)[(\eta+2-\nu)(\eta+3-\nu)-\theta\beta _{0}]t^{\underline{\nu-5}} -\theta(\nu-1)(\nu-2) t^{\underline{\nu-3}}}{\beta_{0}\theta^{2}\eta ^{\underline{\nu-3}}\Gamma(\nu)} \\ &{}+\frac{(\nu-3)[-\xi+\beta(\nu-3)][(\eta+2-\nu)(\eta+3-\nu)-\theta\beta _{0}]t^{\underline{\nu-4}}}{\beta\theta^{2}\beta_{0}\eta^{\underline{\nu -3}}\Gamma(\nu)}\biggr\vert \\ &{}\times\sum_{s=0}^{\eta-\nu} \bigl(\eta-\sigma(s)\bigr)^{\underline{\nu-1}} +\sum_{s=0}^{t-\nu+2}\frac{(t-\sigma(s))^{\underline{\nu-3}}}{\Gamma(\nu -2)} \Biggr\} . \end{aligned}

Then we have

\begin{aligned}& \bigl\vert h_{1}(t)-h_{2}(t)\bigr\vert \\& \quad = \Biggl\vert \biggl[\frac{[\gamma+\delta(\nu-1)][(\eta+2-\nu )(\eta+3-\nu)]t^{\underline{\nu-3}}-\theta[\gamma+\delta(\nu-1)] t^{\underline{\nu-1}}}{\theta\beta_{0}\mu\Gamma(\nu)(b+\nu)^{\underline {\nu-4}}} \\& \qquad {}-\frac{[\xi-\beta(\nu-3)][\gamma+\delta(\nu-1)] [(\eta+2-\nu)(\eta+3-\nu)]t^{\underline{\nu-2}}}{\beta(\nu-2)\theta\beta _{0}\mu\Gamma(\nu)(b+\nu)^{\underline{\nu-4}}} \biggr] \\& \qquad {}\times \sum_{s=0}^{b+1}(b-s+2) \bigl(b+\nu-\sigma(s)\bigr)^{\underline{\nu-2}}(y_{1}-y_{2}) (s) \\& \qquad {}+ \biggl[\frac{[(\eta+2-\nu)(\eta+3-\nu)-\theta\beta_{0}]t^{\underline{\nu -3}}-\theta t^{\underline{\nu-1}}}{\beta_{0}\theta^{2}\eta^{\underline{\nu -3}}\Gamma(\nu)} \\& \qquad {}+\frac{[-\xi+\beta(\nu-3)][(\eta+2-\nu)(\eta+3-\nu)-\theta\beta _{0}]t^{\underline{\nu-2}}}{\beta(\nu-2)\theta^{2}\beta_{0}\eta^{\underline {\nu-3}}\Gamma(\nu)} \biggr] \\& \qquad {}\times\sum_{s=0}^{\eta-\nu} \bigl(\eta-\sigma(s)\bigr)^{\underline{\nu-1}}(y_{1}-y_{2}) (s) +\sum_{s=0}^{t-\nu} \frac{(t-\sigma(s))^{\underline{\nu-1}}}{ \Gamma(\nu)}(y_{1}-y_{2}) (s)\Biggr\vert \\& \quad \leq \biggl\vert \frac{[\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta+3-\nu )]t^{\underline{\nu-3}}-\theta[\gamma+\delta(\nu-1)]t^{\underline{\nu-1}}}{ \theta\beta_{0}\mu\Gamma(\nu)(b+\nu)^{\underline{\nu-4}}} \\& \qquad {}-\frac{[\xi-\beta(\nu-3)][\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta +3-\nu)] t^{\underline{\nu-2}}}{\beta(\nu-2)\theta\beta_{0}\mu\Gamma(\nu)(b+\nu )^{\underline{\nu-4}}}\biggr\vert \\& \qquad {}\times\sum_{s=0}^{b+1}(b-s+2) \bigl(b+\nu-\sigma(s)\bigr)^{\underline{\nu-2}}\bigl\vert y_{1}(s)-y_{2}(s) \bigr\vert \\& \qquad {}+\biggl\vert \frac{[(\eta+2-\nu)(\eta+3-\nu)-\theta\beta_{0}]t^{\underline{\nu -3}}-\theta t^{\underline{\nu-1}}}{\beta_{0}\theta^{2}\eta^{\underline{\nu -3}}\Gamma(\nu)} \\& \qquad {}+\frac{[-\xi+\beta(\nu-3)][(\eta+2-\nu)(\eta+3-\nu)-\theta\beta _{0}]t^{\underline{\nu-2}}}{\beta(\nu-2)\theta^{2}\beta_{0}\eta^{\underline {\nu-3}}\Gamma(\nu)}\biggr\vert \\& \qquad {}\times\sum_{s=0}^{\eta-\nu} \bigl(\eta-\sigma(s)\bigr)^{\underline{\nu-1}} \bigl\vert y_{1}(s)-y_{2}(s) \bigr\vert +\sum_{s=0}^{t-\nu}\frac{(t-\sigma(s))^{\underline{\nu-1}}}{ \Gamma(\nu)} \bigl\vert y_{1}(s)-y_{2}(s)\bigr\vert \\& \quad \leq \max_{t\in\mathbb{N}_{0}^{b+1}}\bigl\vert y_{1}(t)-y_{2}(t) \bigr\vert \\& \qquad {}\times\max_{t\in\mathbb{N}_{\nu-3}^{b+1+\nu}} \Biggl\{ \biggl\vert \frac{[\gamma+\delta (\nu-1)][(\eta+2-\nu) (\eta+3-\nu)]t^{\underline{\nu-3}}-\theta[\gamma+\delta(\nu -1)]t^{\underline{\nu-1}}}{\theta\beta_{0}\mu\Gamma(\nu)(b+\nu )^{\underline{\nu-4}}} \\& \qquad {}-\frac{[\xi-\beta(\nu-3)][\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta +3-\nu)]t^{\underline{\nu-2}}}{\beta(\nu-2)\theta\beta_{0}\mu\Gamma(\nu) (b+\nu)^{\underline{\nu-4}}}\biggr\vert \\& \qquad {}\times\sum_{s=0}^{b+1}(b-s+2) \bigl(b+\nu-\sigma (s)\bigr)^{\underline{\nu-2}} +\biggl\vert \frac{[(\eta+2-\nu)(\eta+3-\nu)-\theta\beta_{0}]t^{\underline{\nu -3}}-\theta t^{\underline{\nu-1}}}{\beta_{0}\theta^{2}\eta^{\underline{\nu -3}}\Gamma(\nu)} \\& \qquad {}+\frac{[-\xi+\beta(\nu-3)][(\eta+2-\nu)(\eta+3-\nu)-\theta\beta _{0}]t^{\underline{\nu-2}}}{\beta(\nu-2)\theta^{2}\beta_{0}\eta^{\underline {\nu-3}}\Gamma(\nu)}\biggr\vert \\ & \qquad {}\times\sum_{s=0}^{\eta-\nu} \bigl(\eta-\sigma(s)\bigr)^{\underline{\nu-1}}+\sum_{s=0}^{t-\nu} \frac{(t-\sigma(s))^{\underline{\nu-1}}}{\Gamma(\nu)} \Biggr\} \\& \quad \leq \psi\bigl(\bigl\vert x(t)-z(t)\bigr\vert +\bigl\vert \Delta x(t)-\Delta z(t)\bigr\vert +\bigl\vert \Delta^{2} x(t)- \Delta^{2} z(t)\bigr\vert \bigr)\times G_{1}. \end{aligned}

Since

\begin{aligned} \Delta h_{1}(t) =& \biggl[\frac{(\nu-3)[\gamma+\delta(\nu-1)][(\eta+2-\nu )(\eta+3-\nu)]t^{\underline{\nu-4}}-(\nu-1)\theta[\gamma+\delta(\nu-1)] t^{\underline{\nu-2}}}{\theta\beta_{0}\mu\Gamma(\nu)(b+\nu)^{\underline {\nu-4}}} \\ &{}-\frac{[\xi-\beta(\nu-3)][\gamma+\delta(\nu-1)] [(\eta+2-\nu)(\eta+3-\nu)]t^{\underline{\nu-3}}}{\beta\theta\beta_{0}\mu \Gamma(\nu)(b+\nu)^{\underline{\nu-4}}} \biggr] \\ &{}\times\sum_{s=0}^{b+1}(b-s+2) \bigl(b+\nu-\sigma(s)\bigr)^{\underline{\nu-2}}y_{1}(s) \\ &{}+ \biggl[\frac{(\nu-3)[(\eta+2-\nu)(\eta+3-\nu)-\theta\beta _{0}]t^{\underline{\nu-4}}-\theta(\nu-1) t^{\underline{\nu-2}}}{\beta _{0}\theta^{2}\eta^{\underline{\nu-3}}\Gamma(\nu)} \\ &{}+\frac{[-\xi+\beta(\nu-3)][(\eta+2-\nu)(\eta+3-\nu)-\theta\beta _{0}]t^{\underline{\nu-3}}}{\beta\theta^{2}\beta_{0}\eta^{\underline{\nu -3}}\Gamma(\nu)} \biggr] \\ &{}\times\sum_{s=0}^{\eta-\nu} \bigl(\eta-\sigma(s)\bigr)^{\underline{\nu-1}}y_{1}(s) +\sum_{s=0}^{t-\nu+1}\frac{(t-\sigma(s))^{\underline{\nu-2}}}{\Gamma(\nu -1)}y_{1}(s), \end{aligned}

we get

\begin{aligned}& \bigl\vert \Delta h_{1}(t)-\Delta h_{2}(t)\bigr\vert \\& \quad \leq\biggl\vert \frac{(\nu-3)[\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta+3-\nu )]t^{\underline{\nu-4}}-(\nu-1)\theta[\gamma+\delta(\nu-1)] t^{\underline{\nu-2}}}{\theta\beta_{0}\mu\Gamma(\nu)(b+\nu)^{\underline {\nu-4}}} \\& \qquad {}-\frac{[\xi-\beta(\nu-3)][\gamma+\delta(\nu-1)][(\eta+2-\nu) (\eta+3-\nu)]t^{\underline{\nu-3}}}{\beta\theta\beta_{0}\mu\Gamma(\nu )(b+\nu)^{\underline{\nu-4}}}\biggr\vert \\& \qquad {}\times\sum_{s=0}^{b+1}(b-s+2) \bigl(b+\nu-\sigma(s)\bigr)^{\underline{\nu-2}}\bigl\vert y_{1}(s)-y_{2}(s) \bigr\vert \\& \qquad {}+\biggl\vert \frac{(\nu-3)[(\eta+2-\nu)(\eta+3-\nu)-\theta\beta _{0}]t^{\underline{\nu-4}}-\theta(\nu-1) t^{\underline{\nu-2}}}{\beta _{0}\theta^{2}\eta^{\underline{\nu-3}}\Gamma(\nu)} \\& \qquad {}+\frac{[-\xi+\beta(\nu-3)][(\eta+2-\nu)(\eta+3-\nu)-\theta\beta _{0}]t^{\underline{\nu-3}}}{\beta\theta^{2}\beta_{0}\eta^{\underline{\nu -3}}\Gamma(\nu)}\biggr\vert \\& \qquad {}\times\sum_{s=0}^{\eta-\nu} \bigl(\eta-\sigma(s)\bigr)^{\underline{\nu-1}} \bigl\vert y_{1}(s)-y_{2}(s) \bigr\vert +\sum_{s=0}^{t-\nu+1}\frac{(t-\sigma(s))^{\underline{\nu-2}}}{ \Gamma(\nu-1)} \bigl\vert y_{1}(s)-y_{2}(s)\bigr\vert \\& \quad \leq\max_{t\in\mathbb{N}_{0}^{b+1}}\bigl\vert y_{1}(t)-y_{2}(t) \bigr\vert \\ & \qquad {}\times \max_{t\in\mathbb{N}_{\nu-3}^{b+1+\nu}} \Biggl\{ \biggl\vert \frac{(\nu-3)[\gamma +\delta(\nu-1)] [(\eta+2-\nu)(\eta+3-\nu)]t^{\underline{\nu-4}}}{\theta\beta_{0}\mu\Gamma(\nu) (b+\nu)^{\underline{\nu-4}}} \\& \qquad {}-\frac{(\nu-1)\theta[\gamma +\delta(\nu-1)]t^{\underline{\nu-2}}}{\theta\beta_{0}\mu\Gamma(\nu) (b+\nu)^{\underline{\nu-4}}} \\& \qquad {}-\frac{[\xi-\beta(\nu-3)][\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta +3-\nu)]t^{\underline{\nu-3}}}{ \beta\theta\beta_{0}\mu\Gamma(\nu)(b+\nu)^{\underline{\nu-4}}}\biggr\vert \\& \qquad {}\times\sum_{s=0}^{b+1}(b-s+2) \bigl(b+\nu-\sigma(s)\bigr)^{\underline{\nu-2}} \\& \qquad {}+\biggl\vert \frac{(\nu-3)[(\eta+2-\nu)(\eta+3-\nu)-\theta\beta _{0}]t^{\underline{\nu-4}}-\theta(\nu-1) t^{\underline{\nu-2}}}{\beta _{0}\theta^{2}\eta^{\underline{\nu-3}}\Gamma(\nu)} \\& \qquad {}+\frac{[-\xi+\beta(\nu-3)][(\eta+2-\nu)(\eta+3-\nu)-\theta\beta _{0}]t^{\underline{\nu-3}}}{\beta\theta^{2}\beta_{0}\eta^{\underline{\nu -3}}\Gamma(\nu)}\biggr\vert \\& \qquad {}\times \sum_{s=0}^{\eta-\nu} \bigl(\eta-\sigma(s)\bigr)^{\underline{\nu-1}}+\sum_{s=0}^{t-\nu+1} \frac{(t-\sigma(s))^{\underline{\nu-2}}}{\Gamma(\nu -1)} \Biggr\} \\& \quad \leq\psi\bigl(\bigl\vert x(t)-z(t)\bigr\vert +\bigl\vert \Delta x(t)-\Delta z(t)\bigr\vert +\bigl\vert \Delta^{2} x(t)- \Delta^{2} z(t)\bigr\vert \bigr)\times G_{2}. \end{aligned}

Also, we have

\begin{aligned}& \bigl\vert \Delta^{2} h_{1}(t)-\Delta^{2} h_{2}(t)\bigr\vert \\& \quad \leq\biggl\vert \frac{(\nu-3)(\nu-4)[\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta+3-\nu )]t^{\underline{\nu-5}}}{\theta\beta _{0}\mu\Gamma(\nu)(b+\nu)^{\underline{\nu-4}}} \\& \qquad {}-\frac{(\nu-1) (\nu-2)\theta[\gamma+\delta(\nu-1)]t^{\underline{\nu-3}}}{\theta\beta _{0}\mu\Gamma(\nu)(b+\nu)^{\underline{\nu-4}}} \\& \qquad {}-\frac{(\nu-3)[\xi-\beta(\nu-3)] [\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta+3-\nu)]t^{\underline{\nu -4}}}{\beta\theta\beta_{0}\mu\Gamma(\nu)(b+\nu)^{\underline{\nu-4}}}\biggr\vert \\& \qquad {}\times\sum_{s=0}^{b+1}(b-s+2) \bigl(b+\nu-\sigma(s)\bigr)^{\underline{\nu-2}} \bigl\vert y_{1}(s)-y_{2}(s) \bigr\vert \\& \qquad {}+\biggl\vert \frac{(\nu-3)(\nu-4)[(\eta+2-\nu)(\eta+3-\nu)-\theta\beta _{0}]t^{\underline{\nu-5}} -\theta(\nu-1)(\nu-2) t^{\underline{\nu-3}}}{\beta_{0}\theta^{2}\eta ^{\underline{\nu-3}}\Gamma(\nu)} \\& \qquad {}+\frac{(\nu-3)[-\xi+\beta(\nu-3)][(\eta+2-\nu)(\eta+3-\nu)-\theta\beta _{0}]t^{\underline{\nu-4}}}{\beta\theta^{2}\beta_{0}\eta^{\underline{\nu -3}}\Gamma(\nu)}\biggr\vert \\& \qquad {}\times\sum_{s=0}^{\eta-\nu} \bigl(\eta-\sigma(s)\bigr)^{\underline{\nu-1}} \bigl\vert y_{1}(s)-y_{2}(s) \bigr\vert +\sum_{s=0}^{t-\nu+2}\frac{(t-\sigma(s))^{\underline{\nu-3}}}{ \Gamma(\nu-2)} \bigl\vert y_{1}(s)-y_{2}(s)\bigr\vert \\& \quad \leq\max_{t\in\mathbb{N}_{0}^{b+1}}\bigl\vert y_{1}(t)-y_{2}(t) \bigr\vert \\ & \qquad {}\times \max_{t\in \mathbb{N}_{\nu-3}^{b+1+\nu}} \Biggl\{ \biggl\vert \frac{(\nu-3) (\nu-4)[\gamma+\delta(\nu-1)][(\eta+2-\nu)(\eta+3-\nu)]t^{\underline{\nu -5}}}{ \theta\beta_{0}\mu\Gamma(\nu)(b+\nu)^{\underline{\nu-4}}} \\& \qquad {}-\frac{(\nu-1)(\nu-2)\theta[\gamma+\delta(\nu-1)]t^{\underline{\nu-3}}}{ \theta\beta_{0}\mu\Gamma(\nu)(b+\nu)^{\underline{\nu-4}}} \\& \qquad {}-\frac{(\nu-3)[\xi-\beta(\nu-3)][\gamma+\delta(\nu-1)][(\eta+2-\nu )(\eta+3-\nu)] t^{\underline{\nu-4}}}{\beta\theta\beta_{0}\mu\Gamma(\nu)(b+\nu )^{\underline{\nu-4}}}\biggr\vert \\& \qquad {}\times\sum_{s=0}^{b+1}(b-s+2) \bigl(b+\nu-\sigma (s)\bigr)^{\underline{\nu-2}} \\& \qquad {}+\biggl\vert \frac{(\nu-3)(\nu-4)[(\eta+2-\nu)(\eta+3-\nu)-\theta\beta _{0}]t^{\underline{\nu-5}} -\theta(\nu-1)(\nu-2) t^{\underline{\nu-3}}}{\beta_{0}\theta^{2}\eta ^{\underline{\nu-3}}\Gamma(\nu)} \\& \qquad {}+\frac{(\nu-3)[-\xi+\beta(\nu-3)][(\eta+2-\nu)(\eta+3-\nu)-\theta\beta _{0}]t^{\underline{\nu-4}}}{\beta\theta^{2}\beta_{0}\eta^{\underline{\nu -3}}\Gamma(\nu)}\biggr\vert \\& \qquad {}\times\sum_{s=0}^{\eta-\nu} \bigl(\eta-\sigma(s)\bigr)^{\underline{\nu-1}} +\sum_{s=0}^{t-\nu+2}\frac{(t-\sigma(s))^{\underline{\nu-3}}}{\Gamma(\nu -2)} \Biggr\} \\& \quad \leq\psi\bigl(\bigl\vert x(t)-z(t)\bigr\vert +\bigl\vert \Delta x(t)-\Delta z(t)\bigr\vert +\bigl\vert \Delta^{2} x(t)- \Delta^{2} z(t)\bigr\vert \bigr)\times G_{3}. \end{aligned}

Hence, we obtain

\begin{aligned} \|h_{1}-h_{2}\| =&\max_{t\in\mathbb{N}_{\nu-3}^{b+1+\nu}}\bigl\vert h_{1}(t)-h_{2}(t)\bigr\vert + \max_{t\in\mathbb{N}_{\nu-3}^{b+1+\nu}} \bigl\vert \Delta h_{1}(t)-\Delta h_{2}(t)\bigr\vert \\ &{}+ \max_{t\in\mathbb{N}_{\nu-3}^{b+1+\nu}}\bigl\vert \Delta^{2} h_{1}(t)- \Delta^{2} h_{2}(t)\bigr\vert \\ \leq&\psi\bigl(\bigl\vert x(t)-z(t)\bigr\vert +\bigl\vert \Delta x(t)- \Delta z(t)\bigr\vert +\bigl\vert \Delta^{2} x(t)- \Delta^{2} z(t)\bigr\vert \bigr) (G_{1}+G_{2}+G_{3}) \\ \leq&(G_{1}+G_{2}+G_{3})\psi\bigl(\Vert x-z \Vert \bigr) \end{aligned}

for all $$x,z\in\mathcal{X}$$, $$h_{1}\in\mathcal{F}(x)$$, and $$h_{2}\in \mathcal{F}(z)$$. So $$H_{d}(\mathcal{F}(x),\mathcal{F}(z))\leq(G_{1}+G_{2}+G_{3})\psi(\|x-z\|)$$ for all $$x,z\in\mathcal{X}$$.

Define the function α on $$\mathcal{X}\times\mathcal{X}$$ by $$\alpha(x,z)=1$$ whenever $$G_{1}+G_{2}+G_{3}< 1$$ and $$\alpha(x,z)=\frac{1}{G_{1}+G_{2}+G_{3}}$$ otherwise. Thus,

$$\alpha(x,z) H_{d}\bigl(\mathcal{F}(x),\mathcal{F}(z)\bigr)\leq\psi \bigl(\Vert x-z\Vert \bigr)$$

for all $$x,z\in\mathcal{X}$$. Let $$\{x_{n}\}$$ be a sequence in $$\mathcal {X}$$ with $$\alpha(x_{n}, x_{n+1})\geq1$$ for all n and $$x_{n}\to x$$. Then it is easy to check that there exists a subsequence $$\{x_{n_{k}}\}$$ of $$\{ x_{n}\}$$ such that $$\alpha(x_{n_{k}},x)\geq1$$ for all k. This implies that $$\mathcal{X}$$ obeys the condition ($$C_{\alpha}$$). If $$x\in\mathcal {X}$$ and $$y\in\mathcal{F}(x)$$ with $$\alpha(x, y)\geq1$$, then it is easy to see that $$\alpha(y, z)\geq1$$ for all $$z\in\mathcal{F}(y)$$. Thus, $$\mathcal{F}$$ is an α-admissible α-ψ-contractive multifunction. Hence by using Theorem 2.2, there exists $$x^{*}\in\mathcal{X}$$ such that $$x^{*}\in\mathcal{F}(x^{*})$$. One can check that $$x^{*}$$ is a solution for the problem (3). □

### Example 3.1

Consider the fractional finite difference inclusion

$$\Delta^{2.5}_{-0.5}x(t)\in \biggl[1 , e^{t^{2}}+2+\frac{\sin x(t)}{e^{2|t|}}+\sinh^{2} t+\frac{|\Delta x(t)|}{4|t|}+ \frac {3}{6t^{2}-1}+\frac{|\Delta^{2}x(t)|}{\cosh|3t|} \biggr]$$
(4)

via the boundary value conditions $$\xi x(-0.5)+\beta\Delta x(-0.5)=0$$, $$\gamma x(6.5)+\delta\Delta x(6.5)=0$$, and $$x(3.5)=0$$, where ξ, β, γ, δ are non-zero numbers. In fact, this problem is a special case of the problem (3), where $$\nu=2.5$$, $$\eta =3.5$$, $$b=4$$, and

$$F(t,x_{1},x_{2},x_{3})= \biggl[1 , e^{t^{2}}+2+\frac{\sin x_{1}}{e^{2|t|}}+\sinh^{2} t+\frac{|x_{2}|}{4|t|}+ \frac{3}{6t^{2}-1}+\frac {|x_{3}|}{\cosh|3t|} \biggr].$$

Note that $$e^{t^{2}}+2+\frac{\sin x_{1}}{e^{2|t|}}+\sinh^{2} t+\frac {|x_{2}|}{4|t|}+\frac{3}{6t^{2}-1}+\frac{|x_{3}|}{\cosh|3t|}>1$$ for all $$t\in \mathbb{N}_{-0.5}^{7.5}$$ and $$x_{1},x_{2},x_{3}\in\mathbb{R}$$. Also, $$e^{2|t|}\geq2$$, $$4|t|\geq2$$, and $$\cosh|3t|\geq2$$ for all $$t\in \mathbb{N}_{-0.5}^{7.5}$$ and F is a compact valued multifunction on $$\mathbb{N}_{-0.5}^{7.5}\times\mathbb{R}\times\mathbb{R}\times\mathbb {R}$$. Now, define $$\psi\in\Psi$$ by $$\psi(z)=\frac{z}{2}$$ for all $$z\geq0$$. Since

\begin{aligned}& H_{d}\bigl(F(t,x_{1},x_{2},x_{3}),F(t,z_{1},z_{2},z_{3}) \bigr) \\& \quad \leq\biggl\vert \frac{\sin x_{1}}{e^{2|t|}}-\frac{x_{2}}{4|t|}+ \frac{x_{3}}{\cosh|3t|}-\frac{\sin z_{1}}{e^{2|t|}}+\frac{z_{2}}{4|t|}-\frac{z_{3}}{\cosh|3t|}\biggr\vert \\& \quad \leq\frac{|x_{1}-z_{1}|+|x_{2}-z_{2}|+|x_{3}-z_{3}|}{2} \\& \quad =\psi\bigl(\vert x_{1}-z_{1} \vert +\vert x_{2}-z_{2}\vert +\vert x_{3}-z_{3}\vert \bigr) \end{aligned}

for all $$t\in\mathbb{N}_{-0.5}^{7.5}$$ and $$x_{1},x_{2},x_{3},z_{1},z_{2},z_{3}\in \mathbb{R}$$, by using Theorem 3.2 the problem (4) has at least one solution.

## 4 Conclusions

In this manuscript, based on a fixed point theorem, we provided the existence result for a fractional finite difference inclusion in the presence of the general boundary conditions. An example illustrates our result.