1 Introduction

Let H be a real Hilbert space with inner product \(\langle\cdot,\cdot \rangle\) and norm \(\Vert \cdot \Vert \). Let C be a nonempty closed convex subset of H. Let \(\mathbb{N}\) and \(\mathbb{R}\) denote the sets of positive integers and real numbers. Suppose that f is a contraction on H with coefficient \(0<\alpha<1\). A nonlinear operator \(T:H\rightarrow H\) is nonexpansive if \(\Vert Tx-Ty\Vert \leq \Vert x-y\Vert \) for all \(x,y\in H\). We use \(\operatorname {Fix}(T)\) to denote the fixed point of T.

Firstly, consider the constrained convex minimization problem:

$$ \min_{x\in C}g(x), $$
(1.1)

where \(g:C\rightarrow\mathbb{R}\) is a real-valued convex function. Assume that the constrained convex minimization problem (1.1) is solvable, let U denote its solution set. The gradient-projection algorithm (GPA) is an effective method for solving the constrained convex minimization problem (1.1). A sequence \(\{x_{n}\}\) generated by the following recursive formula:

$$ x_{n+1}=P_{C}(I-\lambda\nabla g)x_{n}, \quad \forall n \geq0, $$
(1.2)

where the parameter λ is real positive number. In general, if the gradient ∇g is L-Lipschitz continuous and η-strongly monotone, \(0<\lambda<\frac{2\eta}{L^{2}}\), the sequence \(\{x_{n}\}\) generated by (1.2) converges strongly to a minimizer of (1.1). However, if the gradient ∇g is only to be \(\frac{1}{L}\)-ism with \(L>0\), \(0<\lambda<\frac{2}{L}\), the sequence \(\{x_{n}\}\) generated by (1.2) converges weakly to a minimizer of (1.1).

Recently, many authors combined the constrained convex minimization problem with a fixed point problem [1–3] and proposed composited iterative algorithms to find a solution of the constrained convex minimization problem [4–7].

In 2000, Moudafi [8] introduced the viscosity approximation method for nonexpansive mappings.

$$ x_{n+1}=\alpha_{n}f(x_{n})+(1- \alpha_{n})Tx_{n}, \quad \forall n\geq0. $$
(1.3)

In 2001, Yamada [9] introduced the so-called hybrid steepest-descent algorithm:

$$ x_{n+1}=Tx_{n}-\mu\lambda_{n}FTx_{n},\quad \forall n\geq0, $$
(1.4)

where F is Lipschitzian and strongly monotone operator. In 2006, Marino and Xu [10] considered a generative algorithm:

$$ x_{n+1}=\alpha_{n}\gamma f(x_{n})+(I- \alpha_{n}A)Tx_{n}, \quad \forall n\geq0, $$
(1.5)

where A is a strongly positive operator. In 2010, Tian [11] combined the iterative algorithm of (1.4), (1.5), and proposed a new iterative algorithm:

$$ x_{n+1}=\alpha_{n}\gamma f(x_{n})+(I-\mu \alpha_{n}F)Tx_{n}, \quad \forall n\geq0. $$
(1.6)

In 2010, Tian [12] generalized (1.6), obtained the following iterative algorithm:

$$ x_{n+1}=\alpha_{n}\gamma Vx_{n}+(I-\mu \alpha_{n}F)Tx_{n}, \quad \forall n\geq0, $$
(1.7)

where V is Lipschitzian operator. Based on these iterative algorithms, some authors combined GPA with averaged operator to solve the constrained convex minimization problem [13, 14].

In 2011, Ceng et al. [1] proposed a sequence \(\{x_{n}\}\) generated by the following iterative algorithm:

$$ x_{n+1}=P_{C} \bigl[\theta_{n}rh(x_{n})+(I- \theta_{n}\mu F)T_{n}(x_{n}) \bigr],\quad \forall n \geq0, $$
(1.8)

where \(h:C\rightarrow H\) is an l-Lipschitzian mapping with a constant \(l>0\), and \(F:C\rightarrow H\) is a k-Lipschitzian and η-strongly monotone operator with constants \(k, \eta>0\). \(\theta_{n}=\frac{2-\lambda_{n}L}{4}\), \(P_{C}(I-\lambda_{n}\nabla g)=\theta_{n}I+(1-\theta_{n})T_{n}\), \(\forall n\geq0\). Then a sequence \(\{ x_{n}\}\) generated by (1.8) converges strongly to a minimizer of (1.1).

On the other hand, Xu [15] proposed that regularization can be used to find the minimum-norm solution of the minimization problem.

Consider the following regularized minimization problem:

$$ \min_{x\in C}g_{\beta}(x):=g(x)+\frac{\beta}{2}\Vert x \Vert ^{2}, $$

where the regularization parameter \(\beta>0\). g is a convex function and the gradient ∇g is \(\frac{1}{L}\)-ism with \(L>0\). Then the sequence \(\{x_{n}\}\) generated by the following formula:

$$ x_{n+1}=P_{C}(I-\lambda\nabla g_{\beta_{n}})x_{n}=P_{C} \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}, \quad \forall n \geq0, $$
(1.9)

where the regularization parameters \(0<\beta_{n}<1\), \(0<\lambda<\frac {2}{L}\) converges weakly. But, if a sequence \(\{x_{n}\}\) defined by

$$ x_{n+1}=P_{C}(I-\lambda_{n} \nabla g_{\beta_{n}})x_{n}=P_{C} \bigl(I-\lambda_{n}( \nabla g+\beta_{n}I) \bigr)x_{n},\quad \forall n\geq0, $$
(1.10)

where the initial guess \(x_{0}\in C\), \(\{\lambda_{n}\}\), \(\{\beta_{n}\} \) satisfy the following conditions:

  1. (i)

    \(0<\lambda_{n}\leq\frac{\beta_{n}}{(L+\beta_{n})^{2}}\), \(\forall n\geq 0\),

  2. (ii)

    \(\beta_{n}\rightarrow0\) (and \(\lambda_{n}\rightarrow 0\)) as \(n\rightarrow\infty\),

  3. (iii)

    \(\sum_{n=1}^{\infty}\lambda_{n}\beta_{n} = \infty\),

  4. (iv)

    \(\frac{(\vert \lambda_{n}-\lambda_{n-1}\vert +\vert \lambda_{n}\beta_{n}-\lambda_{n-1}\beta_{n-1}\vert )}{(\lambda _{n}\beta_{n})^{2}}\rightarrow0\) as \(n\rightarrow\infty\).

Then the sequence \(\{x_{n}\}\) generated by (1.10) converges strongly to \(x^{*}\), which is the minimum-norm solution of (1.1) [15].

Secondly, Yu et al. [16] proposed a strong convergence theorem with a regularized-like method to find an element of the set of solutions for a monotone inclusion problem in a Hilbert space.

Theorem 1.1

[16]

Let H be a real Hilbert space and C be a nonempty closed and convex subset of H. Let \(L>0\), F is a \(\frac{1}{L}\)-ism mapping of C into H. Let B be a maximal monotone mapping on H and let G be a maximal monotone mapping on H such that the domains of B and G are included in C. Let \(J_{\rho}=(I+\rho B)^{-1}\) and \(T_{r}=(I+rG)^{-1}\) for each \(\rho>0\) and \(r>0\). Suppose that \((F+B)^{-1}(0)\cap G^{-1}(0)\neq\emptyset\). Let \(\{x_{n}\}\subset H\) defined by

$$ x_{n+1}=J_{\rho} \bigl(I-\rho(F+\beta_{n}I) \bigr)T_{r}x_{n}, \quad\forall n>0, $$
(1.11)

where \(\rho\in(0,\infty)\), \(\beta_{n}\in(0,1)\), \(r\in(0,\infty)\). Assume that

  1. (i)

    \(0< a\leq\rho<\frac{2}{2+L}\),

  2. (ii)

    \(\lim_{n\rightarrow\infty}\beta_{n}=0\), \(\sum_{n=1}^{\infty}\beta _{n}=\infty\).

Then the sequence \(\{x_{n}\}\) generated by (1.11) converges strongly to x̅, where \(\overline{x}= P_{(F+B)^{-1}(0)\cap G^{-1}(0)}(0)\).

From the article of Yu et al. [16], we obtain a new condition of parameter ρ, \(0<\rho<\frac{2}{L+2}\), which is used widely in our article. Motivated and inspired by Lin, when \(0<\lambda<\frac{2}{L+2}\), \(\{\beta_{n}\}\) satisfy certain conditions, a sequence \(\{x_{n}\}\) generated by the iterative algorithm (1.9):

$$ x_{n+1}=P_{C} \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}, \quad\forall n\geq0, $$

converges strongly to a point \(q\in U\), where \(q=P_{U}(0)\) is the minimum-norm solution of the constrained convex minimization problem.

Finally, we give concrete example and the numerical results to illustrate our algorithm is with fast convergence.

2 Preliminaries

In this part, we introduce some lemmas that will be used in the rest part. Let H be a real Hilbert space and C be a nonempty closed convex subset of H. We use ‘→’ to denote strong convergence of the sequence \(\{x_{n}\}\) and use ‘⇀’ to denote weak convergence.

Recall \(P_{C}\) is the metric projection from H into C, then to each point \(x\in H\), the unique point \(P_{C}\in C\) satisfy the property:

$$ \Vert x-P_{C}x\Vert =\inf_{y\in C}\Vert x-y \Vert =: d(x,C). $$

\(P_{C}\) has the following characteristics.

Lemma 2.1

[17]

For a given \(x\in H\):

  1. (1)

    \(z=P_{C}x \Longleftrightarrow\langle x-z,z-y\rangle\geq0\), \(\forall y\in C\);

  2. (2)

    \(z=P_{C}x \Longleftrightarrow \Vert x-z\Vert ^{2}\leq \Vert x-y\Vert ^{2}-\Vert y-z\Vert ^{2}\), \(\forall y\in C\);

  3. (3)

    \(\langle P_{C}x-P_{C}y, x-y\rangle\geq \Vert P_{C}x-P_{C}y\Vert ^{2}\), \(\forall x,y\in H\).

From (3), we can derive that \(P_{C}\) is nonexpansive and monotone.

Lemma 2.2

Demiclosed principle [18]

Let \(T : C\rightarrow C\) be a nonexpansive mapping with \(F(T)\neq\emptyset\). If \(\{x_{n}\}\) is a sequence in C weakly converging to x and if \(\{ (I-T)x_{n}\}\) converges strongly to y, then \((I-T)x = y\). In particular, if \(y = 0\), then \(x\in F(T)\).

Lemma 2.3

[19]

Let \(\{a_{n}\}\) is a sequence of nonnegative real numbers such that

$$a_{n+1} \leq(1-\alpha_{n})a_{n} + \alpha_{n}\delta_{n}, \quad n \geq0, $$

where \(\{\alpha_{n}\}_{n=0}^{\infty}\) and \(\{\delta_{n}\}_{n=0}^{\infty }\) are sequences of real numbers in \((0,1)\) and such that

  1. (i)

    \(\sum_{n=0}^{\infty}\alpha_{n} = \infty\);

  2. (ii)

    \(\limsup_{n\rightarrow\infty}\delta_{n} \leq0\) or \(\sum_{n=0}^{\infty}\alpha_{n}\vert \delta_{n}\vert < \infty\).

Then \(\lim_{n\rightarrow\infty}a_{n} = 0\).

3 Main results

Let H be a real Hilbert space and C be a nonempty closed convex subset of H. Assume that \(g:C\rightarrow\mathbb{R}\) is real-valued convex function and the gradient ∇g is \(\frac{1}{L}\)-ism with \(L>0\). Suppose that the minimization problem (1.1) is consistent and let U denote its solution set. Let \(0<\lambda<\frac{2}{L+2}\), \(0<\beta _{n}<1\). Consider the following mapping \(G_{n}\) on C defined by

$$ G_{n}x=P_{C} \bigl(I-\lambda(\nabla g+ \beta_{n}I) \bigr)x, \quad \forall x\in C, n\in\mathbb{N}. $$

We have

$$\begin{aligned} \Vert G_{n}x-G_{n}y\Vert ^{2} =& \bigl\Vert P_{C} \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x-P_{C} \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)y \bigr\Vert ^{2} \\ \leq& \bigl\Vert \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x- \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)y \bigr\Vert ^{2} \\ =&(1-\lambda\beta_{n})^{2}\Vert x-y\Vert ^{2}+\lambda^{2} \bigl\Vert \nabla g(x)-\nabla g(y) \bigr\Vert ^{2} \\ &{} -2\lambda(1-\lambda\beta_{n}) \bigl\langle x-y,\nabla g(x)- \nabla g(y) \bigr\rangle \\ \leq&(1-\lambda\beta_{n})^{2}\Vert x-y\Vert ^{2}+\lambda^{2} \bigl\Vert \nabla g(x)-\nabla g(y) \bigr\Vert ^{2} \\ &{} -\frac{2}{L}\lambda(1-\lambda\beta_{n}) \bigl\Vert \nabla g(x)-\nabla g(y) \bigr\Vert ^{2} \\ \leq&(1-\lambda\beta_{n})^{2}\Vert x-y\Vert ^{2}-\lambda \biggl(\frac{2}{L}(1-\lambda)-\lambda \biggr) \bigl\Vert \nabla g(x)-\nabla g(y) \bigr\Vert ^{2} \\ \leq&(1-\lambda\beta_{n})^{2}\Vert x-y\Vert ^{2}. \end{aligned}$$

That is,

$$ \Vert G_{n}x-G_{n}y\Vert \leq(1-\lambda \beta_{n})\Vert x-y\Vert . $$

Since \(0<1-\lambda\beta_{n}<1\), it follows that \(G_{n}\) is a contraction. Therefore, by the Banach contraction principle, \(G_{n}\) has a unique fixed point \(x_{n}\), such that

$$ x_{n}=P_{C} \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}. $$

Next, we prove that the sequence \(\{x_{n}\}\) converges strongly to \(q\in U\), which also solves the variational inequality

$$ \langle-q, p-q\rangle\leq0, \quad \forall p\in U. $$
(3.1)

Equivalently, \(q=P_{U}(0)\), that is, q is the minimum-norm solution of the constrained convex minimization problem.

Theorem 3.1

Let C be a nonempty closed convex subset of a real Hilbert space H. Let \(g:C\rightarrow\mathbb{R}\) is real-valued convex function and assume that the gradient ∇g is \(\frac{1}{L}\)-ism with \(L>0\). Assume that \(U \neq\emptyset\). Let \(\{x_{n}\}\) be a sequence generated by

$$ x_{n}=P_{C} \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}, \quad\forall n\in\mathbb{N}. $$
(3.2)

Let λ, \(\{\beta_{n}\}\) satisfy the following conditions:

  1. (i)

    \(0<\lambda<\frac{2}{2+L}\),

  2. (ii)

    \(\{\beta_{n}\}\subset(0,1)\), \(\lim_{n\rightarrow\infty}\beta _{n}=0\), \(\sum_{n=1}^{\infty}\beta_{n} = \infty\).

Then \(\{x_{n}\}\) converges strongly to a point \(q\in U\), where \(q=P_{U}(0)\), which is the minimum-norm solution of the minimization problem (1.1) and also solves the variational inequality (3.1).

Proof

First, we claim that \(\{x_{n}\}\) is bounded. Indeed, pick any \(p\in U\), then we have

$$\begin{aligned} \Vert x_{n}-p\Vert =& \bigl\Vert P_{C} \bigl(I- \lambda(\nabla g+\beta_{n}I) \bigr)x_{n}-P_{C}(I- \lambda\nabla g)p \bigr\Vert \\ \leq& \bigl\Vert \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}- \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)p \bigr\Vert \\ &{}+ \bigl\Vert \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)p-(I- \lambda\nabla g)p \bigr\Vert \\ \leq&(1-\lambda\beta_{n})\Vert x_{n}-p\Vert +\lambda \beta_{n}\Vert p\Vert . \end{aligned}$$

Then we derive that

$$ \Vert x_{n}-p\Vert \leq \Vert p\Vert , $$

and hence \(\{x_{n}\}\) is bounded.

Next, we claim that \(\Vert x_{n}-P_{C}(I-\lambda\nabla g )x_{n}\Vert \rightarrow0\). Indeed

$$\begin{aligned} \bigl\Vert x_{n}-P_{C}(I-\lambda\nabla g)x_{n} \bigr\Vert =& \bigl\Vert P_{C} \bigl(I-\lambda( \nabla g+\beta_{n}I) \bigr)x_{n}-P_{C}(I-\lambda \nabla g)x_{n} \bigr\Vert \\ \leq& \bigl\Vert \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}-(I-\lambda\nabla g)x_{n} \bigr\Vert \\ \leq&\lambda\beta_{n}\Vert x_{n}\Vert . \end{aligned}$$

Since \(\{x_{n}\}\) is bounded, \(\beta_{n}\rightarrow0\) (\(n\rightarrow \infty\)), we obtain

$$ \bigl\Vert x_{n}-P_{C}(I-\lambda\nabla g)x_{n} \bigr\Vert \rightarrow0. $$

∇g is \(\frac{1}{L}\)-ism. Consequently, \(P_{C}(I-\lambda\nabla g)\) is a nonexpansive self-mapping on C. As a matter of fact, we have for each \(x,y\in C\)

$$\begin{aligned}& \bigl\Vert P_{C}(I-\lambda\nabla g)x-P_{C}(I-\lambda \nabla g)y \bigr\Vert ^{2} \\& \quad \leq \bigl\Vert (I-\lambda\nabla g)x-(I- \lambda\nabla g)y \bigr\Vert ^{2} \\& \quad = \bigl\Vert x-y-\lambda \bigl(\nabla g(x)-\nabla g(y) \bigr) \bigr\Vert ^{2} \\& \quad = \Vert x-y\Vert ^{2}-2\lambda \bigl\langle x-y, \nabla g(x)- \nabla g(y) \bigr\rangle +\lambda^{2} \bigl\Vert \nabla g(x)-\nabla g(y) \bigr\Vert ^{2} \\& \quad \leq \Vert x-y\Vert ^{2}-\lambda \biggl(\frac{2}{L}-\lambda \biggr) \bigl\Vert \nabla g(x)-\nabla g(y) \bigr\Vert ^{2} \\& \quad \leq \Vert x-y\Vert ^{2}. \end{aligned}$$

\(\{x_{n}\}\) is bounded, consider a subsequence \(\{x_{n_{i}}\}\) of \(\{ x_{n}\}\). Since \(\{x_{n_{i}}\}\) is bounded, there exists a subsequence \(\{x_{n_{i_{j}}}\}\) of \(\{x_{n_{i}}\}\) which converges weakly to z. Without loss of generality, we can assume that \(x_{n_{i}}\rightharpoonup z\). Then by Lemma 2.2, we obtain \(z\in U\).

On the other hand

$$\begin{aligned} \Vert x_{n}-z\Vert ^{2} =& \bigl\Vert P_{C} \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}-P_{C}(I-\lambda\nabla g)z \bigr\Vert ^{2} \\ \leq& \bigl\langle \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}-(I-\lambda\nabla g)z, x_{n}-z \bigr\rangle \\ =& \bigl\langle \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}- \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)z, x_{n}-z \bigr\rangle \\ &{} +\langle-\lambda\beta_{n}z, x_{n}-z\rangle \\ \leq&(1-\lambda\beta_{n})\Vert x_{n}-z\Vert ^{2}+\lambda\beta_{n}\langle-z, x_{n}-z\rangle. \end{aligned}$$

Thus

$$\Vert x_{n}-z\Vert ^{2}\leq\langle-z, x_{n}-z\rangle. $$

In particular

$$\Vert x_{n_{i}}-z\Vert ^{2}\leq\langle-z, x_{n_{i}}-z\rangle. $$

Since \(x_{n_{i}}\rightharpoonup z\). Then we derive that \(x_{n_{i}}\rightarrow z\) as \(i\rightarrow\infty\).

Let q be the minimum-norm solution of U, that is, \(q=P_{U}(0)\). Since \(\{x_{n}\}\) is bounded, there exists a subsequence \(\{x_{n_{i}}\} \) of \(\{x_{n}\}\) such that \(x_{n_{i}}\rightharpoonup z\). As the above proof, we know that \(x_{n_{i}}\rightarrow z\), \(z\in U\).

Then we derive that

$$\begin{aligned} \Vert x_{n}-q\Vert ^{2} =& \bigl\Vert P_{C} \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}-q \bigr\Vert ^{2} \\ \leq& \bigl\langle \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}-(I-\lambda\nabla g)q, x_{n}-q \bigr\rangle \\ =& \bigl\langle \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}- \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)q, x_{n}-q \bigr\rangle \\ &{}+\langle-\lambda\beta_{n}q, x_{n}-q\rangle \\ \leq&(1-\lambda\beta_{n})\Vert x_{n}-q\Vert ^{2}+\lambda\beta_{n}\langle-q, x_{n}-q\rangle. \end{aligned}$$

Thus

$$\Vert x_{n}-q\Vert ^{2}\leq\langle-q, x_{n}-q\rangle. $$

In particular

$$\Vert x_{n_{i}}-q\Vert ^{2}\leq\langle-q, x_{n_{i}}-q\rangle. $$

Since \(x_{n_{i}}\rightarrow z\), \(z\in U\),

$$\Vert z-q\Vert ^{2}\leq\langle-q, z-q\rangle\leq0. $$

So, we have \(z=q\). From the arbitrariness of \(z\in U\), it follows that \(q\in U\) is a solution of the variational inequality (3.1). By the uniqueness of solution of the variational inequality (3.1), we conclude that \(x_{n}\rightarrow q\) as \(n\rightarrow\infty\), where \(q=P_{U}(0)\). □

Theorem 3.2

Let C be a nonempty closed convex subset of a real Hilbert space H and \(g:C\rightarrow\mathbb{R}\) is real-valued convex function and assume that the gradient ∇g is \(\frac{1}{L}\)-ism with \(L>0\). Assume that \(U\neq\emptyset\). Let \(\{x_{n}\}\) be a sequence generated by \(x_{1}\in C\) and

$$ x_{n+1}=P_{C} \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}, \quad\forall n\in\mathbb{N}, $$
(3.3)

where λ and \(\{\beta_{n}\}\) satisfy the following conditions:

  1. (i)

    \(0<\lambda<\frac{2}{L+2}\);

  2. (ii)

    \(\{\beta_{n}\}\subset(0,1)\), \(\lim_{n\rightarrow\infty}\beta _{n}=0\), \(\sum_{n=1}^{\infty}\beta_{n} = \infty\), \(\sum_{n=1}^{\infty }\vert \beta_{n+1}-\beta_{n}\vert <\infty\).

Then \(\{x_{n}\}\) converges strongly to a point \(q\in U\), where \(q=P_{U}(0)\), which is the minimum-norm solution of the minimization problem (1.1) and also solves the variational inequality (3.1).

Proof

First, we claim that \(\{x_{n}\}\) is bounded. Indeed, pick any \(p\in U\), then we know that, for any \(n\in\mathbb{N}\),

$$\begin{aligned} \Vert x_{n+1}-p\Vert \leq& \bigl\Vert P_{C} \bigl(I- \lambda(\nabla g+\beta_{n}I) \bigr)x_{n}-P_{C} \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)p \bigr\Vert \\ &{}+ \bigl\Vert P_{C} \bigl(I-\lambda(\nabla g+ \beta_{n}I) \bigr)p-P_{C}(I-\lambda\nabla g)p \bigr\Vert \\ \leq&(1-\lambda\beta_{n})\Vert x_{n}-p\Vert +\lambda \beta_{n}\Vert p\Vert \\ \leq&\max \bigl\{ \Vert x_{n}-p\Vert ,\Vert p\Vert \bigr\} . \end{aligned}$$

By the introduction

$$\begin{aligned} \Vert x_{n}-p\Vert \leq \max \bigl\{ \Vert x_{1}-p \Vert ,\Vert p\Vert \bigr\} , \end{aligned}$$

and hence \(\{x_{n}\}\) is bounded.

Next, we show that \(\Vert x_{n+1}-x_{n}\Vert \rightarrow0\).

$$\begin{aligned} \Vert x_{n+1}-x_{n}\Vert =& \bigl\Vert P_{C} \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}-P_{C} \bigl(I-\lambda(\nabla g+ \beta_{n-1}I) \bigr)x_{n-1} \bigr\Vert \\ \leq& \bigl\Vert \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}- \bigl(I-\lambda(\nabla g+\beta_{n-1}I) \bigr)x_{n-1} \bigr\Vert \\ =& \bigl\Vert \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}- \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n-1} \\ &{} -\lambda\beta_{n}x_{n-1}+ \lambda\beta_{n-1}x_{n-1} \bigr\Vert \\ \leq&(1-\lambda\beta_{n})\Vert x_{n}-x_{n-1} \Vert +\lambda \vert \beta_{n}-\beta_{n-1}\vert \cdot \Vert x_{n-1}\Vert \\ \leq&(1-\lambda\beta_{n})\Vert x_{n}-x_{n-1} \Vert +\lambda \vert \beta_{n}-\beta_{n-1}\vert \cdot M, \end{aligned}$$

where \(M=\sup\{\Vert x_{n}\Vert :n\in\mathbb{N}\}\). Hence, by Lemma 2.3, we have

$$\Vert x_{n+1}-x_{n}\Vert \rightarrow0. $$

Then we claim that \(\Vert x_{n}-P_{C}(I-\lambda\nabla g)x_{n}\Vert \rightarrow0\).

$$\begin{aligned} \bigl\Vert x_{n}-P_{C}(I-\lambda\nabla g)x_{n} \bigr\Vert =& \bigl\Vert x_{n}-x_{n+1}+x_{n+1}-P_{C}(I- \lambda\nabla g)x_{n} \bigr\Vert \\ \leq&\Vert x_{n}-x_{n+1}\Vert + \bigl\Vert P_{C} \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}-P_{C}(I-\lambda\nabla g)x_{n} \bigr\Vert \\ \leq&\Vert x_{n}-x_{n+1}\Vert +\lambda \beta_{n}\cdot \Vert x_{n}\Vert \\ \leq&\Vert x_{n}-x_{n+1}\Vert +\lambda \beta_{n}\cdot M, \end{aligned}$$

since \(\beta_{n}\rightarrow0\) and \(\Vert x_{n+1}-x_{n}\Vert \rightarrow0\), we have

$$\begin{aligned} \bigl\Vert x_{n}-P_{C}(I-\lambda\nabla g)x_{n} \bigr\Vert \rightarrow0. \end{aligned}$$

Next, we show that

$$\begin{aligned} \limsup_{n\rightarrow\infty}\langle-q, x_{n}-q\rangle\leq0. \end{aligned}$$
(3.4)

Let q be the minimum-norm solution of U, that is, \(q=P_{U}(0)\). Since \(\{x_{n}\}\) is bounded, without loss of generality, we assume that \(x_{n_{j}}\rightharpoonup z\). By the same argument as in the proof of Theorem 3.1, we have \(z\in U\).

$$\begin{aligned} \limsup_{n\rightarrow\infty}\langle-q, x_{n}-q\rangle=\lim _{j\rightarrow\infty}\langle-q, x_{n_{j}}-q\rangle=\langle-q, z-q \rangle\leq0. \end{aligned}$$

Then

$$\begin{aligned} \Vert x_{n+1}-q\Vert ^{2} =& \bigl\Vert P_{C} \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}-P_{C}(I-\lambda\nabla g)q \bigr\Vert ^{2} \\ =& \bigl\langle P_{C} \bigl(I-\lambda(\nabla g+\beta_{n}I) \bigr)x_{n}-P_{C} \bigl(I-\lambda(\nabla g+ \beta_{n}I) \bigr)q, x_{n+1}-q \bigr\rangle \\ &{}+ \bigl\langle P_{C} \bigl(I-\lambda(\nabla g+ \beta_{n}I) \bigr)q-P_{C}(I-\lambda\nabla g)q, x_{n+1}-q \bigr\rangle \\ \leq&(1-\lambda\beta_{n})\Vert x_{n}-q\Vert \cdot \Vert x_{n+1}-q\Vert +\lambda\beta_{n}\langle-q, x_{n+1}-q\rangle \\ \leq&\frac{1-\lambda\beta_{n}}{2}\Vert x_{n}-q\Vert ^{2}+ \frac{1}{2}\Vert x_{n+1}-q\Vert ^{2}+\lambda \beta_{n}\langle-q, x_{n+1}-q\rangle. \end{aligned}$$

It follows that

$$\begin{aligned} \Vert x_{n+1}-q\Vert ^{2} \leq&(1-\lambda \beta_{n})\Vert x_{n}-q\Vert ^{2}+2\lambda \beta_{n}\langle-q, x_{n+1}-q\rangle \\ =&(1-\lambda\beta_{n})\Vert x_{n}-q\Vert ^{2}+2\lambda\beta_{n}\delta_{n}, \end{aligned}$$

where \(\delta_{n}=\langle-q, x_{n+1}-q\rangle\).

It is easy to see that \(\lim_{n\rightarrow\infty}\lambda\beta_{n}=0\), \(\sum_{n=1}^{\infty}\lambda\beta_{n} = \infty\) and \(\limsup_{n\rightarrow \infty}\delta_{n}\leq0\). Hence, by Lemma 2.3, the sequence \(\{x_{n}\}\) converges strongly to q, where \(q=P_{U}(0)\). This completes the proof. □

4 Application

In this part, we will illustrate the practical value of our algorithm in the split feasibility problem. In 1994, Censor and Elfving [20] came up with the split feasibility problem. The SFP is formulated as finding a point x with the property:

$$ x\in C \quad \mbox{and} \quad Ax\in Q, $$
(4.1)

where C and Q are nonempty closed and convex subset of real Hilbert spaces \(H_{1}\) and \(H_{2}\), \(A:H_{1}\rightarrow H_{2}\) is bounded linear operator.

Next, we consider the constrained convex minimization problem:

$$ \min_{x\in C}g(x)=\min_{x\in C}\frac{1}{2} \Vert Ax-P_{Q}Ax\Vert ^{2}. $$
(4.2)

If \(x^{*}\) is a solution of SFP, then \(Ax^{*}\in Q\) and \(Ax^{*}-P_{Q}Ax^{*}=0\), \(x^{*}\) is the solution of the minimization problem (4.2). The gradient of g is ∇g, where \(\nabla g=A^{*}(I-P_{Q})A\). Applying Theorem 3.2, we obtain the following theorem.

Theorem 4.1

Assume that the SFP (4.1) is consistent. Let C be a nonempty closed convex subset of a real Hilbert space H. Assume that \(A:H_{1}\rightarrow H_{2}\) is bounded linear operator, \(W\neq\emptyset \), where W denotes the solution set of SFP (4.1). Let \(\{x_{n}\}\) be a sequence generated by \(x_{1}\in C\) and

$$ x_{n+1}=P_{C} \bigl(I-\lambda \bigl(A^{*}(I-P_{Q})A+ \beta_{n}I \bigr) \bigr)x_{n}, \quad\forall n\in\mathbb{N}. $$
(4.3)

Let λ and \(\{\beta_{n}\}\) satisfy the following conditions:

  1. (i)

    \(0<\lambda<\frac{2}{2+\Vert A\Vert ^{2}}\);

  2. (ii)

    \(\{\beta_{n}\}\subset(0,1)\), \(\lim_{n\rightarrow\infty}\beta _{n}=0\), \(\sum_{n=1}^{\infty}\beta_{n} = \infty\), \(\sum_{n=1}^{\infty }\vert \beta_{n+1}-\beta_{n}\vert <\infty\).

Then \(\{x_{n}\}\) converges strongly to a point \(q\in W\), where \(q=P_{W}(0)\).

Proof

We only need to show that ∇g is \(\frac{1}{\Vert A\Vert ^{2}}\)-ism, then Theorem 4.1 can be obtained by Theorem 3.2.

$$\begin{aligned} \nabla g=A^{*}(I-P_{Q})A. \end{aligned}$$

Since \(P_{Q}\) is firmly nonexpansive, so \(P_{Q}\) is \(\frac {1}{2}\)-averaged mapping, then \(I-P_{Q}\) is 1-ism, for any \(x,y\in C\), we derive that

$$\begin{aligned} \bigl\langle \nabla g(x)-\nabla g(y), x-y \bigr\rangle =& \bigl\langle A^{*}(I-P_{Q})Ax-A^{*}(I-P_{Q})Ay, x-y \bigr\rangle \\ =& \bigl\langle (I-P_{Q})Ax-(I-P_{Q})Ay, Ax-Ay \bigr\rangle \\ \geq& \bigl\Vert (I-P_{Q})Ax-(I-P_{Q})Ay \bigr\Vert ^{2} \\ =&\frac{1}{\Vert A\Vert ^{2}}\cdot \bigl\Vert A^{*} \bigl((I-P_{Q})Ax-(I-P_{Q})Ay \bigr) \bigr\Vert ^{2} \\ =&\frac{1}{\Vert A\Vert ^{2}}\cdot \bigl\Vert \nabla g(x)-\nabla g(y) \bigr\Vert ^{2}. \end{aligned}$$

So, ∇g is \(\frac{1}{\Vert A\Vert ^{2}}\)-ism. □

5 Numerical result

In this part, we use the algorithm in Theorem 4.1 to solve a system of linear equations. Then we calculate the \(4\times4\) system of linear equations.

Example 1

Let \(H_{1}=H_{2}=\mathbb{R}^{4}\). Take

$$\begin{aligned}& A=\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} 1&-1&2&-1\\ 2&-2&3&-3\\ 1&1&1&0\\ 1&-1&4&3 \end{array}\displaystyle \right ), \end{aligned}$$
(5.1)
$$\begin{aligned}& b=\left ( \textstyle\begin{array}{c} -2\\ -10\\ 6\\ 18 \end{array}\displaystyle \right ). \end{aligned}$$
(5.2)

Then the SFP can be formulated as the problem of finding a point \(x^{*}\) with the property

$$x^{*}\in C \quad \mbox{and} \quad Ax^{*}\in Q, $$

where \(C=\mathbb{R}^{4}\), \(Q=\{b\}\). That is, \(x^{*}\) is the solution of the system of linear equations \(Ax=b\), and

$$ x^{*}=\left ( \textstyle\begin{array}{c} 1\\ 3\\ 2\\ 4 \end{array}\displaystyle \right ).$$
(5.3)

Take \(P_{C}=I\), where I denotes the \(4\times4\) identity matrix. Given the parameters \(\beta_{n}=\frac{1}{(n+2)^{2}}\) for \(n\geq0\), \(\lambda =\frac{3}{200}\). Then by Theorem 4.1, the sequence \(\{x_{n}\}\) is generated by

$$x_{n+1} =x_{n}-\frac{3}{200}A^{*}Ax_{n}+ \frac{3}{200}A^{*}b-\frac{3}{200(n+2)^{2}}x_{n}. $$

As \(n\rightarrow\infty\), we have \(\{x_{n}\}\rightarrow x^{*}=(1,3,2,4)^{T}\).

From Table 1, we can easily see that with iterative number increasing \(x_{n}\) approaches to the exact solution \(x^{*}\) and the errors gradually approach zero.

Table 1 Numerical results as regards Example  1
Full size table

In Tian and Jiao [21], they use another iterative algorithm to calculate the same example.

Compare Table 1 with Table 2, we find that if the parameters \(\beta _{n}\) are the same, when \(\lambda\rightarrow\frac{2}{L+2}\), our algorithm is with fast convergence.

Table 2 Numerical results as regards Example  1
Full size table

6 Conclusion

In a real Hilbert space, there are many methods to solve the constrained convex minimization problem. However, most of them cannot find the minimum-norm solution. In this article, we use the regularized gradient-projection algorithm to find the minimum-norm solution of the constrained convex minimization problem, where \(0<\lambda<\frac {2}{L+2}\). Then under some suitable conditions, new strong convergence theorems are obtained. Finally, we apply this algorithm to the split feasibility problem and use a concrete example and numerical results to illustrate that our algorithm has fast convergence.