# Some inequality techniques in handling fixed point problems on unbounded sets via homotopy methods

• 249 Accesses

## Abstract

As is well known, fixed point theorems and problems play important roles in differential equations, mathematical programming, control, and so on. In this paper, by providing some unboundedness conditions and by using some inequality techniques, we can give constructive proofs of the existence of fixed points on unbounded non-convex sets via homotopy methods.

## Introduction

As is well known, fixed point theorems and problems play important roles in many areas and have attracted extensive attention of more and more researchers [1,2,3,4,5,6,7,8,9,10,11]. The homotopy method now has become an important tool in handling various fixed point theorems and problems, bilevel programming, and so on [12,13,14,15,16,17,18,19], because the general Brouwer fixed point theorem does not require the convexity of the subsets of $$R^{n}$$, which is a necessary condition for the constructive proofs of the Brouwer fixed point theorem given by the classical homotopy methods and is difficult to remove. Recently, to remove the convexity restriction, Yu et al. [20] introduced the normal cone condition and combined the interior point methods with the classical homotopy methods, therefore gave a constructive proof of the general Brouwer fixed point theorem on a class of non-convex sets.

In [21], we applied linear homotopy and Newton homotopy techniques and hence extended the results [20] from the cases only with inequality constraints to more general cases with equality constraints. In [22], we furthermore introduced $$C^{2}$$ mappings $$\xi _{i}(x,y_{i}) \in R^{n}$$, $$i=1,\ldots ,m$$, $$\eta _{j}(x,z_{j}) \in R^{n}$$, $$j=1,\ldots ,m$$. With these new mappings, we were able to handle the general Brouwer fixed point theorem in more general non-convex sets. Moreover, we applied the perturbation techniques to the equality and inequality constraints to expand the scope of the choice of initial points. This point improves the computational efficiency of the algorithm greatly. However, to our knowledge, the global convergence results in [21, 22] were obtained under the boundedness assumptions. In this paper, we give two sets of unbounded conditions, with which we extend the results in [21, 22] from the boundedness cases to the unboundedness ones, respectively. Our results may be helpful in dealing with important nonlinear problems because fixed point theorems are widely applied in differential equations, economics, and so on.

Throughout this paper, we need the following notations: $$\varOmega =\{x \in R^{n}: g_{i}(x)\leq 0, i=1,\ldots ,m,~h_{j}(x)=0, j=1,\ldots ,l\}$$, $$\varOmega ^{0}=\{x\in R^{n}: g_{i}(x)<0, i=1,\ldots ,m, h_{j}(x)=0, j=1, \ldots ,l\}$$ and $$\partial \varOmega = \varOmega \backslash \varOmega ^{0}$$. The index set is $$B(x)=\{i\in \{1,\ldots ,m\}: g_{i}(x)=0\}$$.

## Main results

In [21], we gave a constructive proof of the existence of fixed points on a class of non-convex set Ω. The assumptions are listed as follows:

$$(A_{1})$$ :

$$\varOmega ^{0}$$ is nonempty and Ω is bounded;

$$(A_{2})$$ :

For any $$x\in \varOmega$$, if

$$\sum_{i\in I(x, \mu )} y_{i} \nabla g_{i}(x)+\nabla h(x) z=0,\quad y _{i}\geq 0,$$

then $$y_{i}=0$$, $$\forall i\in I(x)$$, $$z=0$$;

$$(A_{3})$$ :

For any $$x\in \varOmega$$ and for all $$(x,y,z)\in \varOmega \times R_{+}^{m}\times R^{l}$$, we have

$$\biggl\{ x+\sum_{i\in B(x)} \nabla g_{i}(x)y_{i} + \nabla h(x)z\biggr\} \cap \varOmega = \{x\}.$$
Then the homotopy is constructed as follows:

\begin{aligned} H\bigl(w,w^{(0)},\mu \bigr)=\begin{pmatrix} (1-\mu )(x-F(x) + \nabla g(x)y) + \nabla h(x)z + \mu (x-x^{(0)}) \\ h(x) \\ Yg(x)-\mu Y^{(0)}g(x^{(0)}) \end{pmatrix}=0, \end{aligned}
(1)

where $$g(x)=(g_{1}(x),\ldots ,g_{m}(x))^{T}$$, $$h(x)=(h_{1}(x),\ldots ,h _{m}(x))^{T}$$, $$\nabla g(x)=(\nabla g_{1}(x), \ldots ,\nabla g_{m}(x)) \in R^{n\times m}$$, $$Y=\operatorname{diag}(y) \in R^{m\times m}$$, $$Y^{(0)}=\operatorname{diag}(y^{(0)})\in R^{m\times m}$$, and $$\mu \in (0, 1]$$.

In this paper, to extend the results in [21] to the unbounded non-convex sets, we use the idea of infinite solutions in [23, 24] and replace the boundedness condition $$(A_{1})$$ with the following unboundedness one:

$$(A_{1}^{\prime })$$ $$\varOmega ^{0}$$ is nonempty; for any given $$\eta \in \varOmega$$ and for any sequences $$\{{x}^{(k)}\} \subset \varOmega$$,

$$\bigl(x^{(k)}-\eta \bigr)^{T} \nabla h\bigl(x^{(k)} \bigr)\geq h\bigl(x^{(k)}\bigr)^{T} -h(\eta )^{T}.$$

Moreover, there are no sequences $$\{{x}^{(k)}\} \subset \varOmega$$, if $$\|x^{(k)}\| \rightarrow \infty$$ as $$k \rightarrow \infty$$, such that there exist $$y^{(k)} \in R_{+}^{m}$$ and $$z^{(k)} \in R^{l}$$ satisfying

$$\lim_{k \rightarrow \infty } \bigl(\eta -x^{(k)}\bigr)^{T} \bigl(x^{(k)}-F\bigl(x ^{(k)}\bigr) + \nabla g \bigl(x^{(k)}\bigr)y^{(k)} + \nabla h\bigl(x^{(k)} \bigr)z^{(k)}\bigr)\geq 0.$$

Using assumptions $$(A_{1}^{\prime })$$, $$(A_{2})$$, and $$(A_{3})$$, we show that a smooth curve $$\varGamma _{w^{0}}$$ starting from $$(w^{(0)}, 1)$$ exists by a similar analysis to that in [21].

### Lemma 2.1

Let assumptions $$(A_{1}^{\prime })$$, $$(A_{2})$$, and $$(A_{3})$$ hold. Then, for almost all $$w^{(0)}$$, the projection of the smooth curve $$\varGamma _{w ^{(0)}}$$ onto the x-plane is bounded.

### Proof

If the conclusion does not hold, there exists a sequence of points $$\{(x^{(k)}, y^{(k)}, z^{(k)}, \mu _{k})\}_{k=1}^{\infty }$$ such that $$\|x^{(k)}\|\rightarrow \infty$$ as $$k\rightarrow \infty$$.

The following conclusion is easily obtained by simple computation:

\begin{aligned} \bigl\Vert x^{(k)}-\eta \bigr\Vert ^{2}- \bigl\Vert x^{(0)}-\eta \bigr\Vert ^{2}\leq 2\bigl(x^{(k)}-\eta \bigr)^{T}\bigl(x ^{(k)}-x^{(0)}\bigr). \end{aligned}
(2)

Then we obtain

\begin{aligned} & (1-\mu _{k}) \bigl(x^{(k)}-F\bigl(x^{(k)}\bigr) + \nabla g\bigl(x^{(k)}\bigr)y^{(k)}\bigr) + \nabla h \bigl(x^{(k)}\bigr)z^{(k)} + \mu \bigl(x^{(k)}-x^{(0)} \bigr)=0, \end{aligned}
(3)
\begin{aligned} &h\bigl(x^{(k)}\bigr)=0, \end{aligned}
(4)
\begin{aligned} &Y^{(k)} g\bigl(x^{(k)}\bigr)-\mu _{k} Y^{(0)} g\bigl(x^{(0)}\bigr)=0. \end{aligned}
(5)

Multiplying (3) by $$(x^{(k)}-\eta )^{T}$$, we obtain

\begin{aligned} &(1-\mu _{k}) \bigl(x^{(k)}-\eta \bigr)^{T} \bigl(x^{(k)}-F\bigl(x^{(k)}\bigr) + \nabla g \bigl(x^{(k)}\bigr)y ^{(k)}\bigr) \\ &\quad {}+\bigl(x^{(k)}-\eta \bigr)^{T}\nabla h \bigl(x^{(k)}\bigr)z^{(k)}+\mu _{k} \bigl(x^{(k)}-\eta \bigr)^{T}\bigl(x ^{(k)}-x^{(0)} \bigr)=0, \end{aligned}
(6)

i.e.,

\begin{aligned} \mu _{k}\bigl(x^{(k)}-\eta \bigr)^{T} \bigl(x^{(k)}-x^{(0)}\bigr)= {}&{-}(1-\mu _{k}) \bigl(x^{(k)}- \eta \bigr)^{T}\bigl(x^{(k)}-F \bigl(x^{(k)}\bigr) + \nabla g\bigl(x^{(k)}\bigr)y^{(k)} \bigr) \\ &{}-\bigl(x^{(k)}-\eta \bigr)^{T}\nabla h \bigl(x^{(k)}\bigr)z^{(k)}. \end{aligned}
(7)

So

\begin{aligned} &\mu _{k}\bigl( \bigl\Vert x^{(k)}-\eta \bigr\Vert ^{2}- \bigl\Vert x^{(0)}-\eta \bigr\Vert ^{2} \bigr) \\ &\quad \leq 2\mu _{k}\bigl(x^{(k)}-\eta \bigr)^{T} \bigl(x^{(k)}-x^{(0)}\bigr) \\ &\quad =-2(1-\mu _{k}) \bigl(x^{(k)}-\eta \bigr)^{T}\bigl(x^{(k)}-F\bigl(x^{(k)}\bigr) + \nabla g\bigl(x ^{(k)}\bigr)y^{(k)}\bigr)-2\bigl(x^{(k)}-\eta \bigr)^{T}\nabla h\bigl(x^{(k)}\bigr)z^{(k)} \\ &\quad =-2(1-\mu _{k}) \bigl(x^{(k)}-\eta \bigr)^{T}\bigl(x^{(k)}-F\bigl(x^{(k)}\bigr) + \nabla g\bigl(x ^{(k)}\bigr)y^{(k)}\bigr) \\ & \qquad {}-2(1-\mu _{k}) \bigl(x^{(k)}-\eta \bigr)^{T}\nabla h\bigl(x^{(k)} \bigr)z^{(k)} +2\mu _{k}\bigl(\eta -x^{(k)} \bigr)^{T}\nabla h\bigl(x^{(k)}\bigr)z^{(k)} \\ &\quad \leq -2(1-\mu _{k}) \bigl(x^{(k)}-\eta \bigr)^{T}\bigl(x^{(k)}-F\bigl(x^{(k)}\bigr) + \nabla g\bigl(x ^{(k)}\bigr)y^{(k)}+\nabla h\bigl(x^{(k)} \bigr)z^{(k)}\bigr) \\ & \qquad {} +2\mu _{k}\bigl(h(\eta )^{T}-h \bigl(x^{(k)}\bigr)^{T}\bigr)z^{(k)} \\ &\quad =-2(1-\mu _{k}) \bigl(x^{(k)}-\eta \bigr)^{T}\bigl(x^{k}-F\bigl(x^{(k)}\bigr) + \nabla g\bigl(x^{(k)}\bigr)y ^{k}+\nabla h\bigl(x^{(k)} \bigr)z^{(k)}\bigr). \end{aligned}
(8)

From (8), we obtain

\begin{aligned} &\bigl(\eta -x^{(k)}\bigr)^{T}\bigl(x^{(k)}-F \bigl(x^{(k)}\bigr) + \nabla g\bigl(x^{(k)}\bigr)y^{(k)}+ \nabla h\bigl(x^{(k)}\bigr)z^{(k)}\bigr) \\ &\quad \geq \frac{\mu _{k}}{2(1-\mu _{k})}\bigl( \bigl\Vert x^{(k)}-\eta \bigr\Vert ^{2}- \bigl\Vert x^{(0)}- \eta \bigr\Vert ^{2}\bigr). \end{aligned}
(9)

When $$\|x^{(k)}\|\rightarrow \infty$$, from (9), we obtain

\begin{aligned} & \lim_{k \rightarrow \infty } \bigl(\eta -x^{(k)}\bigr)^{T} \bigl(x^{(k)}-F\bigl(x ^{(k)}\bigr) + \nabla g \bigl(x^{(k)}\bigr)y^{(k)}+\nabla h\bigl(x^{(k)} \bigr)z^{(k)}\bigr) \\ & \quad \geq \lim_{k \rightarrow \infty } \frac{\mu _{k}}{2(1-\mu _{k})}\bigl( \bigl\Vert x^{(k)}-\eta \bigr\Vert ^{2}- \bigl\Vert x^{(0)}- \eta \bigr\Vert ^{2}\bigr) \geq 0, \end{aligned}
(10)

which contradicts assumption $$(A_{1}^{\prime })$$. □

Then by a similar analysis to that in [21], we show that the projections of the smooth curve $$\varGamma _{w^{(0)}}$$ onto the y and z planes are also bounded, and we can also find a fixed point following the curve $$\varGamma _{w^{(0)}}$$. As for how to trace $$\varGamma _{w^{(0)}}$$, one can refer to [25, 26].

To give a constructive proof of the general Brouwer fixed point theorem in more general non-convex sets and to expand the scope of the choice of initial points of the homotopy method, in [22], we introduced continuous mappings $$\xi (x, u)=(\xi _{1}(x, u_{1}), \ldots , \xi _{m}(x, u_{m})) \in R^{n \times m}$$ and $$\eta (x, v)=(\eta _{1}(x, v_{1}), \ldots ,\eta _{l}(x, v_{l})) \in R^{n \times l}$$, then applied proper perturbations to the constrained functions $$g(x)$$ and $$h(x)$$ by using the following parameters:

\begin{aligned} &\gamma _{i}=\textstyle\begin{cases} 2 g_{i}(x^{(0)}), & g_{i}(x^{(0)})> 0, \\ 1, & g_{i}(x^{(0)})=0, \\ 0, & g_{i}(x^{(0)})< 0, \end{cases}\displaystyle \quad i=1,\ldots ,m, \\ & \theta _{j}=\textstyle\begin{cases} 1, & h_{j}(x^{(0)})\neq 0, \\ 0, & h_{j}(x^{(0)})= 0, \end{cases}\displaystyle \quad j=1,\ldots ,l. \end{aligned}

Then let

\begin{aligned} & e_{m}=(1,\ldots ,1)^{T} \in R^{m},\qquad \varUpsilon = \operatorname{diag} (\gamma _{1},\ldots ,\gamma _{m}) \in R^{m \times m},\\ & \varTheta =\operatorname{diag}( \theta _{1},\ldots , \theta _{l})^{T} \in R^{l \times l}, \\ & \varOmega (\mu )=\bigl\{ x\in R^{n}: g(x)-\mu \varUpsilon \bigl(g \bigl(x^{(0)}\bigr)+e_{m}\bigr) \leq 0, h(x)-\mu \varTheta h \bigl(x^{(0)}\bigr)=0\bigr\} , \\ & \varOmega ^{0}(\mu )=\bigl\{ x\in R^{n}: g(x)-\mu \varUpsilon \bigl(g\bigl(x^{(0)}\bigr)+e_{m}\bigr)< 0, h(x)-\mu \varTheta h \bigl(x^{(0)}\bigr)=0\bigr\} , \\ & I(x,\mu )=\bigl\{ i \in \{1,\ldots ,m\}:g_{i}(x)-\mu \gamma _{i} \bigl(g_{i}\bigl(x ^{(0)}\bigr)+1\bigr)=0\bigr\} . \end{aligned}

In [22], we made the following assumptions:

$$(C_{1})$$ :

$$\varOmega ^{0}(\mu )$$ is nonempty and $$\varOmega (\mu )$$ is bounded;

$$(C_{2})$$ :

$$\xi _{i}(x,0)=0$$, $$i=1,\ldots ,m$$, $$\eta _{j}(x,0)=0$$, $$j=1, \ldots ,l$$; besides, for any $$x\in \varOmega (\mu )$$, if $$\|(y, z, u, v) \|\rightarrow \infty$$, then

$$\biggl\Vert \sum_{i\in I(x,\mu )} \bigl(y_{i} \nabla g_{i}(x)+\xi _{i}(x,u _{i})\bigr) +\nabla h(x) z+\eta (x, v) \biggr\Vert \rightarrow \infty ;$$
$$(C_{3})$$ :

For any $$x\in \varOmega (\mu )$$, if

$$\sum_{i\in I(x,\mu )} \bigl(y_{i} \nabla g_{i}(x)+\xi _{i}(x, u_{i})\bigr) +\nabla h(x) z+\eta (x, v)=0,\quad y_{i}\geq 0, u_{i}\geq 0,$$

then $$y_{i}=0$$, $$u_{i}=0$$, $$\forall i\in I(x,\mu )$$, $$z=0$$, $$v=0$$;

$$(C_{4})$$ :

When $$\mu =0, 1$$, for any $$x\in \varOmega (\mu )$$, we have

$$\biggl\{ x+\sum_{i\in I(x, \mu )} \xi _{i}(x,u_{i})+ \eta (x, v):u_{i} \geq 0\mbox{ for }i\in I(x,\mu ), v\in R^{l}\biggr\} \cap \varOmega (\mu )=\{x\}.$$

Furthermore, we construct the following homotopy:

\begin{aligned} H\bigl(w,w^{(0)},\mu \bigr)&= \begin{pmatrix} (1-\mu )(x-F(x)+(1-\mu )\mu \nabla g(x)y)+\sum_{i=1}^{m}\xi _{i}(x, (1-\mu )y_{i}) \\ {}+(1-\mu ) \mu (\nabla h(x) z+\beta )+\eta (x, z) + \mu (x-x^{(0)}) \\ h(x)-\mu \varTheta h(x^{(0)}) \\ Y(g(x)-\mu \varUpsilon (g(x^{(0)})+e_{m}))-\mu Y^{(0)}(g(x^{(0)})-\varUpsilon (g(x^{(0)})+e_{m})) \end{pmatrix} \\ & =0, \end{aligned}
(11)

where $$\beta \in R^{n}$$ is a given constant vector.

In this paper, to extend the results in [22] to the unbounded sets, we replace the boundedness condition $$(C_{1})$$ with the following unboundedness one:

$$(C_{1}^{\prime })$$ :

$$\varOmega ^{0}(\mu )$$ is nonempty; for any given $$\alpha \in \varOmega (\mu )$$,

\begin{aligned} (\alpha -x)^{T} \xi _{i}(x, u_{i}) \leq& \bigl( \bigl(g_{i}(\alpha )-\mu \gamma _{i} \bigl(g_{i} \bigl(x^{(0)}\bigr)+1\bigr)\bigr)^{T}\\ &{}-\bigl(g_{i}(x)- \mu \gamma _{i} \bigl(g_{i}\bigl(x^{(0)}\bigr)+1 \bigr)\bigr)^{T}\bigr)u _{i},\quad i =1,\ldots ,m, \end{aligned}

and

$$(\alpha -x)^{T} \eta (x, v)\leq \bigl(h(x)-\mu \varTheta h \bigl(x^{(0)}\bigr)\bigr)^{T} v -\bigl(h _{j}(\alpha )-\mu \varTheta h\bigl(x^{(0)}\bigr)\bigr)^{T} v.$$

In addition, there are no sequences $$\{{x}^{(k)}\} \subset \varOmega$$, if $$\|x^{(k)}\| \rightarrow \infty$$ as $$k \rightarrow \infty$$, such that there exist $$u^{(k)} \in R_{+}^{m}$$ and $$v^{(k)} \in R^{l}$$ satisfying

$$\lim_{k \rightarrow \infty } \bigl(x^{(k)}-\alpha \bigr)^{T} \bigl(x^{(k)}-F\bigl(x ^{(k)}\bigr) + \nabla g \bigl(x^{(k)}\bigr)u^{(k)} + \nabla h\bigl(x^{(k)} \bigr)v^{(k)}+\mu \beta \bigr)\leq 0.$$

In the following, we are devoted to proving the boundedness of x-component of w.

### Lemma 2.2

Let H be defined as in (11), let $$g_{i}(x)$$, $$i=1, \ldots , m$$, and $$h_{j}(x)$$, $$j=1, \ldots , l$$ be $$C^{3}$$ functions, let assumptions $$(C_{1}^{\prime })$$, $$(C_{2})$$$$(C_{4})$$ hold, and let $$\xi _{i}(x, u_{i})$$, $$i=1,\ldots , m$$, and $$\eta _{j}(x, v_{j})$$, $$j=1, \ldots , l$$, be $$C^{2}$$ functions. Then, for almost all $$w^{(0)}$$, the x-component of w is bounded.

### Proof

If the conclusion does not hold, then there exists a sequence of points $$\{(x^{(k)}, y^{(k)}, z^{(k)}, \mu _{k})\}_{k=1} ^{\infty }$$ such that $$\|x^{(k)}\|\rightarrow \infty$$ as $$k\rightarrow \infty$$. From the first equation in (11), we obtain

\begin{aligned} &(1-\mu _{k}) \bigl(x^{(k)}-F\bigl(x^{(k)}\bigr)+ (1-\mu _{k})\mu _{k}\nabla g\bigl(x^{(k)}\bigr)y ^{(k)}\bigr)+\sum_{i=1}^{m}\xi _{i}\bigl(x^{(k)}, (1-\mu _{k})y_{i}^{(k)} \bigr) \\ &\quad {}+(1-\mu _{k}) \mu _{k} \bigl(\nabla h \bigl(x^{(k)}\bigr) z^{(k)}+\beta \bigr)+\eta \bigl(x, z ^{(k)}\bigr) + \mu _{k}\bigl(x^{(k)}-x^{(0)} \bigr) =0. \end{aligned}
(12)

Multiplying (12) by $$(x^{(k)}-\alpha )^{T}$$, we obtain

\begin{aligned} &(1-\mu _{k}) \bigl(x^{(k)}-\alpha \bigr)^{T} \bigl(x^{(k)}-F\bigl(x^{(k)}\bigr)+ (1-\mu _{k})\mu _{k}\nabla g\bigl(x^{(k)}\bigr)y^{(k)}\bigr) \\ &\quad {}+\sum_{i=1}^{m} \bigl(x^{(k)}-\alpha \bigr)^{T} \xi _{i} \bigl(x^{(k)}, (1-\mu _{k})y_{i}^{(k)} \bigr)+(1-\mu _{k}) \mu _{k}\bigl(x^{(k)}-\alpha \bigr)^{T} \bigl(\nabla h\bigl(x ^{(k)}\bigr) z^{(k)}+ \beta \bigr) \\ &\quad {}+\bigl(x^{(k)}-\alpha \bigr)^{T}\eta \bigl(x, z^{(k)}\bigr)+ \mu _{k}\bigl(x^{(k)}-\alpha \bigr)^{T}\bigl(x ^{(k)}-x^{(0)}\bigr)=0. \end{aligned}
(13)

Furthermore, rewrite (13) as

\begin{aligned} &\mu _{k}\bigl(x^{(k)}-\alpha \bigr)^{T} \bigl(x^{(k)}-x^{(0)}\bigr) \\ &\quad =-(1-\mu _{k}) \bigl(x^{(k)}-\alpha \bigr)^{T}\bigl(x^{(k)}-F\bigl(x^{(k)}\bigr)+ (1-\mu _{k}) \mu _{k}\nabla g\bigl(x^{(k)} \bigr)y^{(k)}\bigr) \\ &\qquad {} -\sum_{i=1}^{m} \bigl(x^{(k)}-\alpha \bigr)^{T} \xi _{i} \bigl(x^{(k)}, (1- \mu _{k})y_{i}^{(k)} \bigr)-(1-\mu _{k})\mu _{k}\bigl(x^{(k)}-\alpha \bigr)^{T} \bigl(\nabla h\bigl(x^{(k)}\bigr) z^{(k)}+ \beta \bigr) \\ &\qquad {} -\bigl(x^{(k)}-\alpha \bigr)^{T}\eta \bigl(x, z^{(k)}\bigr). \end{aligned}
(14)

By a simple computation, we derive

\begin{aligned} \bigl\Vert x^{(k)}-\alpha \bigr\Vert ^{2}- \bigl\Vert x^{(0)}-\alpha \bigr\Vert ^{2}\leq 2\bigl(x^{(k)}- \alpha \bigr)^{T}\bigl(x^{(k)}-x^{(0)}\bigr). \end{aligned}
(15)

Combining (14) and (15), we conclude

\begin{aligned} &\mu _{k}\bigl( \bigl\Vert x^{(k)}-\alpha \bigr\Vert ^{2}- \bigl\Vert x^{(0)}-\alpha \bigr\Vert ^{2} \bigr) \\ &\quad \leq 2\mu _{k}\bigl(x^{(k)}-\alpha \bigr)^{T}\bigl(x^{(k)}-x^{(0)}\bigr) \\ & \quad =-2(1-\mu _{k}) \bigl(x^{(k)}-\alpha \bigr)^{T}\bigl(x^{(k)}-F\bigl(x^{(k)}\bigr)+ (1-\mu _{k}) \mu _{k}\nabla g\bigl(x^{(k)} \bigr)y^{(k)}\bigr) \\ &\qquad {} -\sum_{i=1}^{m} 2 \bigl(x^{(k)}-\alpha \bigr)^{T} \xi _{i} \bigl(x^{(k)}, (1- \mu _{k})y_{i}^{(k)} \bigr) \\ &\qquad {}-2(1-\mu _{k}) \mu _{k}\bigl(x^{(k)}-\alpha \bigr)^{T} \bigl( \nabla h\bigl(x^{(k)}\bigr) z^{(k)}+ \beta \bigr) -2\bigl(x^{(k)}-\alpha \bigr)^{T}\eta \bigl(x, z^{(k)}\bigr) \\ &\quad =-2(1-\mu _{k}) \bigl(x^{(k)}-\alpha \bigr)^{T}\bigl(x^{(k)}-F\bigl(x^{(k)}\bigr) \\ &\qquad {}+(1-\mu _{k}) \mu _{k} \nabla g\bigl(x^{(k)} \bigr)y^{(k)}+\mu _{k}\bigl(\nabla h\bigl(x^{(k)} \bigr) z^{(k)}+ \beta \bigr)\bigr) \\ &\qquad {}-\sum_{i=1}^{m} 2 \bigl(x^{(k)}-\alpha \bigr)^{T} \xi _{i} \bigl(x^{(k)}, (1- \mu _{k})y_{i}^{(k)} \bigr)-2\bigl(x^{(k)}-\alpha \bigr)^{T}\eta \bigl(x, z^{(k)}\bigr). \end{aligned}
(16)

By assumption $$(C_{1}^{\prime })$$ and (16), we obtain

\begin{aligned} &\mu _{k}\bigl( \bigl\Vert x^{(k)}-\alpha \bigr\Vert ^{2}- \bigl\Vert x^{(0)}-\alpha \bigr\Vert ^{2} \bigr) \\ &\quad \leq -2(1-\mu _{k}) \bigl(x^{(k)}-\alpha \bigr)^{T}\bigl(x^{(k)}-F\bigl(x^{(k)}\bigr)+\nabla g \bigl(x ^{(k)}\bigr) \bigl((1-\mu _{k})\mu _{k}y^{(k)}\bigr) \\ &\qquad {}+\nabla h\bigl(x^{(k)}\bigr) \bigl( \mu _{k} z^{(k)}\bigr)+ \mu _{k}\beta \bigr) +\sum_{i=1}^{m}2\bigl( \bigl(g_{i}(\alpha )-\mu _{k}\gamma _{i} \bigl(g_{i}\bigl(x ^{(0)}\bigr)+1\bigr)\bigr)^{T} \\ &\qquad {}- \bigl(g_{i}\bigl(x^{(k)}\bigr)-\mu _{k}\gamma _{i} \bigl(g_{i}\bigl(x^{(0)}\bigr)+1\bigr) \bigr)^{T}\bigr) (1- \mu _{k})y_{i}^{(k)} \\ &\qquad {}+\bigl(h(\alpha )-\mu _{k}\varTheta h\bigl(x^{(0)} \bigr)\bigr)^{T} z^{(k)}-\bigl(h\bigl(x^{(k)}\bigr)- \mu _{k}\varTheta h\bigl(x^{(0)}\bigr)\bigr)^{T} z^{(k)}. \end{aligned}
(17)

Because $$g_{i}(\alpha )-\mu _{k}\gamma _{i} (g_{i}(x^{(0)})+1) \leq 0$$ and $$y_{i}^{(k)} \geq 0$$, $$i=1,\ldots ,m$$, then

\begin{aligned} &\mu _{k}\bigl( \bigl\Vert x^{(k)}-\alpha \bigr\Vert ^{2}- \bigl\Vert x^{(0)}-\alpha \bigr\Vert ^{2} \bigr) \\ &\quad \leq -2(1-\mu _{k}) \bigl(x^{(k)}-\alpha \bigr)^{T}\bigl(x^{(k)}-F\bigl(x^{(k)}\bigr)+\nabla g \bigl(x ^{(k)}\bigr) \bigl((1-\mu _{k})\mu _{k}y^{(k)}\bigr) \\ &\qquad {}+\nabla h\bigl(x^{(k)}\bigr) \bigl( \mu _{k} z^{(k)}\bigr)+ \mu _{k}\beta \bigr) \\ &\qquad {}-\sum_{i=1}^{m}2( \bigl(g_{i}\bigl(x^{(k)}\bigr)-\mu _{k}\gamma _{i} \bigl(g_{i}\bigl(x ^{(0)}\bigr)+1\bigr) \bigr)^{T}(1-\mu _{k})y_{i}^{(k)}. \end{aligned}
(18)

It follows from the third equation in (11) that

\begin{aligned} &\bigl(g_{i}\bigl(x^{(k)}\bigr)-\mu _{k}\gamma _{i} \bigl(g_{i}\bigl(x^{(0)}\bigr)+1\bigr) \bigr)y_{i}^{(k)}=-\mu _{k} (g_{i} \bigl(x^{(0)}\bigr)-\gamma _{i} \bigl(g_{i} \bigl(x^{(0)}\bigr)+1\bigr)y_{i}^{(0)}. \end{aligned}
(19)

Then (18) becomes

\begin{aligned} &\mu _{k}\bigl( \bigl\Vert x^{(k)}-\alpha \bigr\Vert ^{2}- \bigl\Vert x^{(0)}-\alpha \bigr\Vert ^{2} \bigr) \\ &\quad \leq -2(1-\mu _{k}) \bigl(x^{(k)}-\alpha \bigr)^{T}\bigl(x^{(k)}-F\bigl(x^{(k)}\bigr)+\nabla g \bigl(x ^{(k)}\bigr) \bigl((1-\mu _{k})\mu _{k}y^{(k)}\bigr) \\ &\qquad {}+\nabla h\bigl(x^{(k)}\bigr) \bigl( \mu _{k} z^{(k)}\bigr)+ \mu _{k}\beta \bigr) +\sum_{i=1}^{m}2(1-\mu _{k})\mu _{k}(\bigl(g_{i}\bigl(x^{(0)} \bigr)-\mu _{k} \gamma _{i} \bigl(g_{i} \bigl(x^{(0)}\bigr)+1\bigr)\bigr)^{T}y_{i}^{(0)} \\ &\quad \leq -2(1-\mu _{k}) \bigl(x^{(k)}-\alpha \bigr)^{T}\bigl(x^{(k)}-F\bigl(x^{(k)}\bigr)+\nabla g \bigl(x ^{(k)}\bigr) \bigl((1-\mu _{k})\mu _{k}y^{(k)}\bigr) \\ &\qquad {}+\nabla h\bigl(x^{(k)}\bigr) \bigl( \mu _{k} z^{(k)}\bigr)+ \mu _{k}\beta \bigr). \end{aligned}
(20)

Dividing two sides of (20) by the item $$2(1-\mu _{k})$$, one obtains

\begin{aligned} &\bigl(\alpha -x^{(k)}\bigr)^{T}\bigl(x^{(k)}-F \bigl(x^{(k)}\bigr)+\nabla g\bigl(x^{(k)}\bigr) \bigl((1-\mu _{k})\mu _{k}y^{(k)}\bigr)+\nabla h \bigl(x^{(k)}\bigr) \bigl(\mu _{k} z^{(k)}\bigr)+\mu _{k} \beta \bigr) \\ &\quad \geq \frac{\mu _{k}}{2(1-\mu _{k})}\bigl( \bigl\Vert x^{(k)}-\alpha \bigr\Vert ^{2}- \bigl\Vert x^{(0)}- \alpha \bigr\Vert ^{2}\bigr). \end{aligned}
(21)

From (21), as $$\|x^{(k)}\|\rightarrow \infty$$, one obtains

\begin{aligned} & \lim_{k \rightarrow \infty } \bigl(\alpha -x^{(k)} \bigr)^{T}\bigl(x^{(k)}-F\bigl(x ^{(k)}\bigr)+\nabla g \bigl(x^{(k)}\bigr) \bigl((1-\mu _{k})\mu _{k}y^{(k)} \bigr)+\nabla h\bigl(x^{(k)}\bigr) \bigl(\mu _{k} z^{(k)}\bigr)+\mu _{k}\beta \bigr) \\ &\quad \geq \lim_{k \rightarrow \infty } \frac{\mu _{k}}{2(1-\mu _{k})}\bigl( \bigl\Vert x^{(k)}-\alpha \bigr\Vert ^{2}- \bigl\Vert x^{(0)}-\alpha \bigr\Vert ^{2}\bigr)\geq 0. \end{aligned}

By assumption $$(C_{1}^{\prime })$$, this is not possible. □

The following analysis is similar to that in [22], so we omit it in this paper.

## References

1. 1.

Bollobas, B., Fulton, W., Katok, A., Kirwan, F., Sarnak, P.: Fixed Point Theory and Applications. Cambridge University Press, Cambridge (2004)

2. 2.

Eaves, B.C., Saigal, R.: Homotopies for computation of fixed points on unbounded regions. Math. Program. 3, 225–237 (1972)

3. 3.

Heikkila, S., Reffett, K.: Fixed point theorems and their applications to theory of Nash equilibria. Nonlinear Anal. TMA 64, 1415–1436 (2006)

4. 4.

Lin, L.J., Yu, Z.T.: Fixed point theorems and equilibrium problems. Nonlinear Anal. TMA 43, 987–999 (2001)

5. 5.

Park, P.: Fixed points and quasi-equilibrium problems. Math. Comput. Model. 32, 1297–1303 (2000)

6. 6.

Todd, M.J.: Improving the convergence of fixed point algorithms. Math. Program. 7, 151–179 (1978)

7. 7.

Yao, Y.H., Yao, J.-C., Liou, Y.-C., Postolache, M.H.: Iterative algorithms for split common fixed points of demicontractive operators without priori knowledge of operator norms. Carpath. J. Math. 34(3), 459–466 (2018)

8. 8.

Zegeye, H., Shahzad, N., Yao, Y.H.: Minimum-norm solution of variational inequality and fixed point problem in Banach spaces. Optimization 64(2), 453–471 (2015)

9. 9.

Yao, Y.H., Shahzad, N.: Strong convergence of a proximal point algorithm with general errors. Optim. Lett. 6(4), 621–628 (2012)

10. 10.

Yao, Y.H., Postolache, M.H., Zhu, Z.C.: Gradient methods with selection technique for the multiple-sets split feasibility problem. Optimization. In press

11. 11.

Zhu, Z.C., Yang, L.: A constraint shifting homotopy method for computing fixed points on nonconvex sets. J. Nonlinear Sci. Appl. 9(6), 3850–3857 (2016)

12. 12.

Kellogg, K.B., Li, T.Y., Yorke, J.A.: A constructive proof of the Brouwer fixed-point theorem and computational results. SIAM J. Numer. Anal. 13, 473–483 (1976)

13. 13.

Chow, S.N., Mallet-Paret, J., Yorke, J.A.: Finding zeros of maps: homotopy methods that are constructive with probability one. Math. Comput. 32, 887–899 (1978)

14. 14.

Garcia, C.B., Zangwill, W.I.: An approach to homotopy and degree theory. Math. Oper. Res. 4, 390–405 (1979)

15. 15.

Li, Y., Lin, Z.H.: A constructive proof of the Poincaré–Birkhoff theorem. Trans. Am. Math. Soc. 347, 2111–2126 (1995)

16. 16.

Xu, Q., Dang, C.Y., Zhu, D.L.: Generalizations of fixed point theorems and computation. J. Math. Anal. Appl. 354, 550–557 (2009)

17. 17.

Zhu, Z.C., Zhou, Z.Y., Liou, Y.-C., Yao, Y.H.: A globally convergent method for computing the fixed point of self-mapping on general nonconvex set. J. Nonlinear Convex Anal. 18, 1067–1078 (2017)

18. 18.

Zhu, Z.C., Yu, B.: Globally convergent homotopy algorithm for solving the KKT systems to the principal-agent bilevel programming. Optim. Methods Softw. 32(1), 69–85 (2017)

19. 19.

Zhu, Z.C., Yu, B.: A modified homotopy method for solving the principal-agent bilevel programming problem. Comput. Appl. Math. 37(1), 541–566 (2018)

20. 20.

Yu, B., Lin, Z.H.: Homotopy method for a class of nonconvex Brouwer fixed-point problems. Appl. Math. Comput. 74, 65–77 (1996)

21. 21.

Su, M.L., Liu, Z.X.: Modified homotopy methods to solve fixed points of self-mapping in a broader class of nonconvex sets. Appl. Numer. Math. 58, 236–248 (2008)

22. 22.

Su, M.L., Shang, Y.F.: Solving fixed-point problems with inequality and equality constraints via a non-interior point homotopy path-following method. Math. Probl. Eng. 2017, Article ID 3456834 (2017)

23. 23.

Xu, Q., Yu, B., Feng, G.C., Dang, C.Y.: Condition for global convergence of a homotopy method for variational inequality problems on unbounded sets. Optim. Methods Softw. 22, 587–599 (2007)

24. 24.

Xu, Q., Lin, Z.H.: The combined homotopy convergence in unbounded set. Acta Math. Appl. Sin. 27, 624–631 (2004)

25. 25.

Allgower, E.L., Georg, K.: Introduction to Numerical Continuation Methods. SIAM, Philadelphia (2003)

26. 26.

Lin, Z.H., Sheng, Z.P., Yang, L., Bai, Y.Z.: A predictor–corrector algorithm for tracing combined interior homotopy pathway. Math. Numer. Sin. 24, 405–416 (2002)

### Acknowledgements

The authors would like to thank the editor for kind help.

Not applicable.

### Funding

This work was supported by the Natural Science Foundation of China (No. 11671188).

## Author information

The main idea of this paper was proposed by MLS and MJL. MLS prepared the manuscript initially and performed all the steps of the proofs in this research. All authors read and approved the final manuscript.

Correspondence to Menglong Su.

## Ethics declarations

### Competing interests

The authors declare that they have no competing interests.

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Su, M., Liu, M. Some inequality techniques in handling fixed point problems on unbounded sets via homotopy methods. J Inequal Appl 2019, 170 (2019) doi:10.1186/s13660-019-2125-4

• 34B15
• 35A01
• 39B05
• 65H10
• 90C30

### Keywords

• Inequality techniques
• Constructive proofs
• Fixed point problems
• Homotopy methods