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A modified Halpern-proximal point method for approximating solutions of mixed equilibrium and fixed point problems in Hadamard spaces

Abstract

In this paper, we introduce and study a modified Halpern-type proximal point algorithm which comprises a finite family of resolvents of mixed equilibrium problems and a finite family of k-demimetric mappings. We prove that the algorithm converges strongly to a common solution of a finite family of mixed equilibrium problems, which is also a common fixed point of a finite family of k-demimetric mappings in a Hadamard space. Furthermore, we give a numerical example of our algorithm to show the applicability of our algorithm.

Introduction

The Equilibrium Problem (EP) and its several generalizations are very important optimization problems that have greatly influenced many mathematical problems in pure and applied sciences as well as social sciences. It is well known that EP provides a novel and unified framework for a wide class of problems that arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity, and optimization. Also, the EP covers other optimization problems such as minimization problem (MP), variational inequalities problems (VIP), monotone inclusion problems (MIP), fixed point problems (FPP), Nash equilibrium problems, complementarity problems, and game theory as special cases; see [9, 10, 38, 42] and other references therein. The classical EP was first introduced by Blum and Oettli [7]. Given a nonempty set C and a bifunction \(f:C\times C\rightarrow {\mathbb {R}},\) the EP is formulated as follows:

$$\begin{aligned} \text{ Find }~x^*\in C~\text{ such } \text{ that }~f(x^*,y)\ge 0,~\forall ~y\in C. \end{aligned}$$
(1.1)

The point \(x^*\) that satisfies (1.1) is called the equilibrium point (or stable point) of f. The set of solutions to the problem (1.1) is denoted as EP(fC). The EP has been extensively studied by numerous researchers in Hilbert spaces, Banach spaces,and complete Riemannian spaces [5, 11, 14, 15, 20, 34]. Recently, the EP (1.1) has been extended and studied independently by Khatibzadeh and Mohebbi [28], Kimura and Kishi [29] in Hadamard spaces. Kimura and Kishi [29] studied (1.1) under the assumption that the Hadamard space is equipped with the Convex Hull Finite Property (CHFP). They showed that the resolvent associated with the bifunction of EP is well defined. Based on the idea of [29], Kumam and Chaipunya [30] studied EP (1.1) in Hadamard spaces without the CHFP. They established the existence of an equilibrium point of a bifunction satisfying some convexity, continuity, and coercivity assumptions, and they also established some fundamental properties of the resolvent of the bifunction. Since then, researchers begin to develop the study of EP (1.1) in Hadamard spaces (see for example [2, 21, 22] and other references therein).

An important generalization of (1.1) is the Mixed Equilibrium Problem (MEP). The MEP was introduced in the framework of Hilbert spaces by Ceng and Yao [8]. They formulated the problem as follows: let H be a real Hilbert space and C be a nonempty closed convex subset of H. Let \(\Psi : C \rightarrow {\mathbb {R}}\) be a real-valued function and \(f: C \times C \rightarrow {\mathbb {R}}\) be an equilibrium bifunction.The MEP is to find \(x^*\in C\),such that

$$\begin{aligned} f(x^*,y) + \Psi (y) - \Psi (x^*)\ge 0,~\forall ~y\in C. \end{aligned}$$
(1.2)

In particular, if \(\Psi \equiv 0\) in (1.2), the MEP reduces to the EP (1.1). Also, if \(f\equiv 0\) in (1.2), the MEP reduces to the classical MP. The MEP (1.2) has extensively been studied both in Hilbert and Banach spaces; see for example [1, 33, 41]. Very recently, Izuchukwu et al. [23] extended the MEP (1.2) to the framework of Hadamard spaces. They established the existence of a solution of the MEP (1.2) under some basic assumptions in Hadamard spaces. Furthermore, following the idea of Kumam and Chaipunya [30], they introduced a perturbed MEP in Hadamard spaces as follows: let X be an Hadamard space and C be a nonempty subset of X. Let \(f : C \times C\rightarrow {\mathbb {R}}\) be a bifunction, \(\Psi : C \rightarrow {\mathbb {R}}\) be a real-valued function, \(\bar{x}\in X,\) and \(\lambda > 0;\) then, the perturbation \(\bar{f}_{\bar{x}} : C \times C \rightarrow {\mathbb {R}}\) of \(\bar{f}_{\bar{x}}\) and \(\Psi \) is given by

$$\begin{aligned} \bar{f}_{\bar{x}}:= f(x^*,y) + \Psi (y) - \Psi (x^*)+\frac{1}{\lambda }\langle \overrightarrow{xy},\overrightarrow{\bar{x}x}\rangle ,~\forall ~x,y\in C. \end{aligned}$$
(1.3)

They established the following existence result of the perturbed MEP (1.3).

Theorem 1.1

[23] Let C be a nonempty closed and convex subset of a Hadamard space X. Let \(\Psi : C \rightarrow {\mathbb {R}}\) be a convex function and \(f : C \times C \rightarrow {\mathbb {R}}\) be a bifunction, such that the following assumptions hold:

  1. (A1)

    \(f(x, x)\ge 0,~\forall ~x\in C,\)

  2. (A2)

    f is monotone, i.e., \(f(x, y) + f(y, x) \le 0,~\forall ~x,y\in C,\)

  3. (A3)

    \(f(x, .) : C \rightarrow {\mathbb {R}}\) is convex \(\forall ~x\in C,\)

  4. (A4)

    for each \(x\in X\) and \(\lambda >0,\) there exists a compact subset \(D_{\bar{x}}\subset C\) containing a point \(y_{\bar{x}}\in D_{\bar{x}}\), such that

    $$\begin{aligned} \bar{f}_{\bar{x}}:= f(x^*,y) + \Psi (y) - \Psi (x^*)+\frac{1}{\lambda }\langle \overrightarrow{xy_{\bar{x}}},\overrightarrow{\bar{x}x}\rangle <0, \end{aligned}$$

    whenever \(x\in C/D_{\bar{x}}.\)

Then, (1.3) has a unique solution

Furthermore, they introduced the resolvent operator for the MEP (1.3) associated with the bifunction f and the convex function \(\Psi \) of order \(\lambda .\) Let C be a nonempty subset of a Hadamard space X,  and the resolvent operator associated with f and \(\Psi \) is a set-valued mapping \(J^\Psi _{\lambda f}:X\rightarrow 2^C\) defined by

$$\begin{aligned} J^\Psi _{\lambda f}(x):=MEP(C,\bar{f}_{\bar{x}}) =\big \{z\in C: f(z,y) + \Psi (y) - \Psi (z)+\frac{1}{\lambda }\langle \overrightarrow{zy},\overrightarrow{xz}\rangle \ge 0\big \},~\forall ~x\in X. \end{aligned}$$
(1.4)

The authors [23] showed that the resolvent operator \(J^\Psi _{\lambda f}\) (1.4) is well defined. Its uniqueness and other fundamental properties such as firmly nonexpansivity and monotonicity of the resolvent operator were also established in Hadamard spaces.

On the other hand, the Proximal Point Algorithm (PPA) is an effective method for solving optimization problems such as EP, VIP, MIP, MIP, and their generalizations. The PPA was first introduced by Martinet [32] and was further developed by Rockafellar [37] in real Hilbert spaces. In 2013, Bacak [4] introduced and studied the PPA in the framework of Hadamard space. Since then, researchers have adopted the PPA to approximate several optimization problems in Hadamard spaces. Izuchukwu et al. [23] also employed PPA to approximate solutions of MEP (1.2) in Hadamard spaces. Specifically, they proposed the following Picard and Halpern types PPAs in Hadamard spaces:

$$\begin{aligned} {\left\{ \begin{array}{ll} x_1\in X\\ x_{n+1} = J^{\Psi }_{\lambda _nf}x_n,~\forall ~n\ge 1, \end{array}\right. } \end{aligned}$$
(1.5)

and

$$\begin{aligned} {\left\{ \begin{array}{ll} u,x_1\in X\\ x_{n+1} = \alpha _nu\oplus (1-\alpha _n)J^{\Psi }_{\lambda _nf}x_n,~\forall ~n\ge 1, \end{array}\right. } \end{aligned}$$
(1.6)

where \(\{\lambda _n\}\) is a sequence in \((0,\infty ),\) \(f : C \times C \rightarrow {\mathbb {R}}\) is a bifunction, and \(\Psi : C \rightarrow {\mathbb {R}}\) is a convex function satisfying Theorem 1.1. They established the \(\Delta \)-convergence result of Algorithm (1.5) and the strong convergence result of the Algorithm (1.6).

Let C be a nonempty subset of a Hadamard space X. A point \(x\in X\) is called a fixed point of a nonlinear mapping \(T : C \rightarrow X,\) if \(x = Tx.\) We denote by F(T) the set of fixed points of T. The mapping T is said to be:

  1. (i)

    L-Lipschitz, if there exists \(L > 0\), such that

    $$\begin{aligned} \mathrm{{d}}(Tx,Ty)\le L\mathrm{{d}}(x,y),~\forall ~x,y\in X, \end{aligned}$$
  2. (ii)

    firmly nonexpansive, if for all \(x, y \in C\)

    $$\begin{aligned} \mathrm{{d}}^2(Tx, Ty) \le \langle \overrightarrow{TxTy},\overrightarrow{xy}\rangle , \end{aligned}$$
  3. (iii)

    nonexpansive, if for all \(x, y \in C\)

    $$\begin{aligned} \mathrm{{d}}(Tx, Ty) \le \mathrm{{d}}(x, y), \end{aligned}$$
  4. (iv)

    quasi-nonexpansive, if \(F(T)\ne \emptyset \) and for \(y\in F(T), x\in C,\) we have

    $$\begin{aligned} \mathrm{{d}}(Tx, y) \le \mathrm{{d}}(x, y), \end{aligned}$$
  5. (v)

    k-demicontractive, if \(F(T)\ne \emptyset \) and there exists \(k\in [0, 1),\) such that

    $$\begin{aligned} \mathrm{{d}}^2(Tx,y) \le \mathrm{{d}}^2(x,y) + k\mathrm{{d}}^2(Tx, x)~\forall ~x\in C, y\in F(T). \end{aligned}$$

Approximation of fixed points of different types of nonlinear mappings has recently gained attention in Hadamard spaces. The work of Kirk [26] initiated the research in this direction. After that, other researchers have continued to obtain interesting results on fixed point theory in Hadamard spaces; see [12, 17, 25, 35, 39] and other references therein. In 2018, Aremu et al. [3] introduced the notion of k-demimetric mappings in Hadamard spaces, which is defined as follows: let X be a Hadamard space and C be a nonempty closed and convex subset of X. A mapping \(T : C \rightarrow X\) is said to be k-demimetric if F\((T)\ne \emptyset \) and there exists \(k\in (-\infty , 1),\) such that

$$\begin{aligned} \langle \overrightarrow{xy},\overrightarrow{xTx}\rangle \ge \frac{1-k}{2} \mathrm{{d}}^2(x, Tx),~\forall ~x\in X, y\in F(T). \end{aligned}$$
(1.7)

Clearly, the class of k-demimetric mappings with \(k\in (-\infty , 1)\) contains the class of k-demicontractive mappings with \(k\in [0, 1).\) It was established in [3] that F(T) of k-demimetric mappings is closed in convex. Furthermore, they considered the following iterative algorithm for finding a common minimizer of a finite family of proper convex and lower semicontinuous functions and common fixed points of a finite family of k demimetric mappings in Hadamard spaces:

$$\begin{aligned} {\left\{ \begin{array}{ll} v_n = (1-t_n)x_n\oplus t_nu\\ y_n = J_{r_nh_1}\circ J_{r_nh_2}\circ \cdots \circ J_{r_nh_N}v_n,\\ z_n = P_C(\beta _n^{(0)}v_n\oplus \beta _n^{(1)}y_n\oplus \cdots \oplus \beta _n^{(N)}y_n)\\ w_n = \gamma _n^{(0)}z_n\oplus \gamma _n^{(1)}T_{1\xi }z_n\oplus \gamma _n^{(2)}T_{2\xi }z_n\oplus \cdots \oplus \gamma _n^{(N)}T_{N\xi }z_n\\ x_{n+1} = \alpha _n v_n \oplus (1 - \alpha _n)w_n,~\forall ~n\ge 1, \end{array}\right. } \end{aligned}$$
(1.8)

where \(T_{i\xi }x = \xi x \oplus (1 -\xi )T_ix,\) such that \(T_{i\xi }\) are \(\Delta \)-demiclosed for each \(i = 1,2,\ldots ,N,\) \(\{r_n\},~\{t_n\},~\{\alpha _n\},~\{\beta _n^{(i)}\},\) \(\{\gamma _n^{(i)}\}\) are sequences in [0, 1], such that \(0< a \le \alpha _n,~\beta _n^{(i)},~\gamma _n^{(i)} \le b < 1,\) \(\sum \nolimits _{i=0}^N\beta _n^{(i)} = 1\) and \(\sum \nolimits _{i=0}^N\gamma _n^{(i)} = 1\) for all \(n\ge 1,\) \(h_i,~i=1,2,\ldots ,N\) is a finite family of lower semicontinuous functions and \(\{r_n^{(i)}\}\) is a sequence, such that \(r_n^{(i)}\le r^{(i)} >0\) for all \(n \ge 1, i = 1, 2,\ldots ,N.\) They showed that the sequence \(\{x_n\}\) generated by Algorithm (1.8) converges strongly to a common solution of a finite family of minimization problems and fixed point problems of a finite family of k-demimetric mappings in Hadamard spaces.

Motivated by the works of Izuchukwu et al. [23], Aremu et al. [3], and current research interest in this direction, we introduce a modified Halpern-proximal point method which comprises of a finite family of MEPs and a finite family of k-demimetric mappings. We prove that the algorithm converges strongly to a common zero of a finite family of MEPs, which is also a common fixed point of a finite family of k-demimetric mappings in a Hadamard space. Furthermore, we give a numerical example to show the applicability of our algorithm.

Preliminaries

In this section, we recall some results and definitions that will be needed in the proof of our main results.

Let X be a metric space and \(x, y \in X\). A geodesic from x to y is a map (or a curve) c from the closed interval \([0, \mathrm{{d}}(x, y)] \subset {\mathbb {R}}\) to X, such that \(c(0) = x,~ c(\mathrm{{d}}(x, y)) = y\) and \(\mathrm{{d}}(c(t), c(t')) = |t - t'|\) for all \(t, t' \in [0, \mathrm{{d}}(x, y)].\) The image of c is called a geodesic segment joining from x to y. When it is unique, this geodesic segment is denoted by [xy]. The space (Xd) is said to be a geodesic space if every two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each \(x, y \in X.\) A subset D of a geodesic space X is said to be convex if for any two points \(x, y \in D,\) the geodesic joining x and y is contained in D,  that is, if \(c : [0, \mathrm{{d}}(x, y)] \rightarrow X\) is a geodesic, such that \(x = c(0)\) and \(y = c(\mathrm{{d}}(x, y)),\) then \(c(t) \in D~~\forall ~ t \in [0, \mathrm{{d}}(x, y)].\) A geodesic triangle \(\Delta (x_1,x_2,x_3)\) in a geodesic metric space (Xd) consists of three vertices (points in X) with unparameterized geodesic segments between each pair of vertices. For any geodesic triangle, there is comparison (Alexandrov) triangle \(\bar{\Delta }\subset {\mathbb {R}}^2\), such that \(\mathrm{{d}}(x_i,x_j)=\mathrm{{d}}_{{\mathbb {R}}^2}(\bar{x}_i,\bar{x}_j)\), for \(i,j\in \{1,2,3\}\). A geodesic space X is a CAT(0) space if the distance between an arbitrary pair of points on a geodesic triangle \(\Delta \) does not exceed the distance between its corresponding pair of points on its comparison triangle \(\bar{\Delta }\). If \(\Delta \) and \(\bar{\Delta }\) are geodesic and comparison triangles in X, respectively, then \(\Delta \) is said to satisfy the CAT(0) inequality for all points xy of \(\Delta \) and \(\bar{x},\bar{y}\) of \(\bar{\Delta }\) if

$$\begin{aligned} \mathrm{{d}}(x,y)=d_{{\mathbb {R}}^2}(\bar{x},\bar{y}). \end{aligned}$$
(2.1)

Let xyz be points in X and \(y_0\) be the midpoint of the segment [yz], and then the CAT(0) inequality implies

$$\begin{aligned} \mathrm{{d}}^2(x,y_0)\le \frac{1}{2}\mathrm{{d}}^2(x,y)+\frac{1}{2}\mathrm{{d}}^2(x,z)-\frac{1}{4}\mathrm{{d}}(y,z). \end{aligned}$$
(2.2)

Berg and Nikolaev [6] introduced the notion of quasi-linearization in a CAT(0) space as follows: let a pair \((a, b) \in X \times X\) denoted by \(\overrightarrow{ab}\), be called a vector. Then, the quasi-linearization map \(\langle \cdot ,\cdot \rangle :(X\times X)\times (X\times X)\rightarrow {\mathbb {R}}\) is defined by

$$\begin{aligned} \langle \overrightarrow{ab},\overrightarrow{cd}\rangle =\frac{1}{2}(\mathrm{{d}}^2(a,d)+\mathrm{{d}}^2(b,c)-\mathrm{{d}}^2(a,c)-\mathrm{{d}}^2(b,d)),~~\text{ for } \text{ all }~a,b,c,d \in X. \end{aligned}$$
(2.3)

It is easy to see that \(\langle \overrightarrow{ab}, \overrightarrow{ab}\rangle =\mathrm{{d}}^2(a, b),~\langle \overrightarrow{ba}, \overrightarrow{cd}\rangle =-\langle \overrightarrow{ab}, \overrightarrow{cd}\rangle ,~\langle \overrightarrow{ab}, \overrightarrow{cd}\rangle =\langle \overrightarrow{ae}, \overrightarrow{cd}\rangle +\langle \overrightarrow{eb}, \overrightarrow{cd}\rangle \) and \(\langle \overrightarrow{ab}, \overrightarrow{cd}\rangle =\langle \overrightarrow{cd}, \overrightarrow{ab}\rangle \), for all \(a,b,c,d,e\in X\). Furthermore, a geodesic space X is said to satisfy the Cauchy–Schwartz inequality, if

$$\begin{aligned} \langle \overrightarrow{ab},\overrightarrow{cd}\rangle \le \mathrm{{d}}(a,b)\mathrm{{d}}(c,d), \end{aligned}$$

for all \(a,b,c,d \in X\). It is well known that a geodesically connected space is a CAT(0) space if and only if it satisfies the Cauchy-Schwartz inequality [17]. Also, it is known that complete CAT(0) spaces are called Hadamard spaces.

Let \(\{x_n\}\) be a bounded sequence in a metric space X and \(r(\cdot ,\{x_n\}) : X \rightarrow [0, \infty )\) be a continuous functional defined by \(r(x,\{x_n\}) = \limsup \nolimits _{n \rightarrow \infty } \mathrm{{d}}(x, x_n).\) The asymptotic radius of \(\{x_n\}\) is given by \(r(\{x_n\}) := \inf \{r(x, \{x_n\}) : x \in X\},\) while the asymptotic center of \(\{x_n\}\) is the set \(A(\{x_n\}) = \{x \in X : r(x, \{x_n\}) = r(\{x_n\})\}.\) A sequence \(\{x_n\}\) in X is said to be \(\Delta \)-convergent to a point \(x \in X\) if \(A(\{x_{n_k}\}) = \{x\}\) for every subsequence \(\{x_{n_k}\}\) of \(\{x_{n}\}\). In this case, we say that x is the \(\Delta \)-limit of \(\{x_{n}\}\) (see [16, 27]). The notion of \(\Delta \)-convergence in metric spaces was introduced and studied by Lim [31], and it is known as analogue of the notion of weak convergence in Banach spaces.

Definition 2.1

Let X be a Hadamard space. A function \(\Psi :X\rightarrow (-\infty , \infty ]\) is said to be

  1. (i)

    convex, if

    $$\begin{aligned} \Psi (\lambda x\oplus (1-\lambda ) y)\le \lambda \Psi (x)+(1-\lambda )\Psi (y)~\forall x, y\in X,~\lambda \in (0, 1), \end{aligned}$$
  2. (ii)

    proper, if \(D:=\{x\in X:\Psi (x)<+\infty \}\ne \emptyset \), where D denotes the domain of \(\Psi ,\)

  3. (iii)

    lower semicontinuous (or upper semicontinuous) at a point \(x\in D,\) if

    $$\begin{aligned} \Psi (x)\le \liminf _{n\rightarrow \infty } \Psi (x_n)~\big (\text{ or }~ \Psi (x)\ge \lim \nolimits _{n\rightarrow \infty } \Psi (x_n)\big ), \end{aligned}$$
    (2.4)

    for each sequence \(\{x_n\}\) in D, such that \(\lim \nolimits _{n\rightarrow \infty }N\beta _n^{(i)} = 1\)

  4. (iv)

    lower semicontinuous (or upper semicontinuous) on D,  if it is lower semicontinuous (or upper semicontinuous) at every point in D.

Proposition 2.2

[23] Let C be a nonempty closed and convex subset of a Hadamard space X. Let \(\Psi :C\rightarrow {\mathbb {R}}\) be a convex function and \(f:C\rightarrow C\rightarrow {\mathbb {R}}\) be a bifunction satisfying assumptions (A1)-(A4) of Theorem 1.1. For \(\lambda >0,\) we have that \(J^{\Psi }_{\lambda f}\) is single valued. Moreover, if \(C\subset D(J^{\Psi }_{\lambda f}),\) then

  1. (i)

    \(J^{\Psi }_{\lambda f}\) is firmly nonexpansive restricted to C

  2. (ii)

    for \(F(J^{\Psi }_{\lambda f})\ne \emptyset ,\) we have

    $$\begin{aligned} \mathrm{{d}}^2(J^{\Psi }_{\lambda f}x, x) \le \mathrm{{d}}^2(x,v) - \mathrm{{d}}^2(J^{\Psi }_{\lambda f}x, v),~\forall ~x\in C, \forall ~v\in F(J^{\Psi }_{\lambda f}), \end{aligned}$$
  3. (iii)

    for \(0<\lambda \le \mu ,\) we have \(\mathrm{{d}}(J^{\Psi }_{\mu f}x, J^{\Psi }_{\lambda f}x)\le \sqrt{1-(\frac{\lambda }{\mu })}\mathrm{{d}}(x,J^{\Psi }_{\lambda f}),\) which implies that

    $$\begin{aligned} \mathrm{{d}}(x, J^{\Psi }_{\lambda f}x)\le 2\mathrm{{d}}(x, J^{\Psi }_{\mu f}x),~\forall ~\in C, \end{aligned}$$
  4. (iv)

    \(F(J^{\Psi }_{\lambda f})=MEP(C, f,\Psi ).\)

Remark 2.3

[23] It follows from the Cauchy–Schwartz inequality that firmly nonexpansive mappings are nonexpansive, and it is well known that the set of fixed points of nonexpansive mappings is closed and convex. Thus, by (i) and (iv) of Theorem 2.2, we have that \(MEP(C,f,\Psi )\) is closed and convex.

Lemma 2.4

[19] Let X be a Hadamard space and \(T:X\rightarrow X\) be a nonexpansive mapping. Then, T is \(\Delta \)-demiclosed.

Lemma 2.5

[17] Every bounded sequence in a Hadamard space has a \(\triangle -\)convergent subsequence.

The following two lemmas are some fixed point properties of k-demimetric mappings in Hadamard spaces.

Lemma 2.6

[3] Let X be a Hadamard space and \(T:X\rightarrow X\) be a k-demimetric mapping with \(k\in (-\infty , 1),\) such that F(T) is nonempty. Then, F(T) is closed and convex.

Lemma 2.7

[3] Let X be a Hadamard space and \(T:X\rightarrow X\) be a k-demimetric mapping with \(k\in (-\infty , \xi ]\) and \(\xi \in (0, 1),\) such that F(T) is nonempty. Suppose that \(T_{\xi }x = \xi x\oplus (1-\xi )Tx.\) Then, \(T_\xi \) is quasi-nonexpansive and \(F(T_\xi ) = F(T).\)

Lemma 2.8

Let X be a Hadamard space, \(x,y,z\in X\) and \(t, s\in [0,1]\). Then

  1. (i)

    \(\mathrm{{d}}(t x\oplus (1-t)y,z)\le t \mathrm{{d}}(x,z)+(1-t)\mathrm{{d}}(y,z)\) (see [17]).

  2. (ii)

    \(\mathrm{{d}}^2(t x\oplus (1-t)y,z)\le t \mathrm{{d}}^2(x,z)+(1-t)\mathrm{{d}}^2(y,z)-t(1-t)\mathrm{{d}}^2(x,y)\) (see [17]).

  3. (iii)

    \(\mathrm{{d}}^2(t x\oplus (1-t)y,z)\le t^{2} \mathrm{{d}}^2(x,z)+(1-t)^{2}\mathrm{{d}}^2(y,z) +2t(1-t)\langle \overrightarrow{xz},\overrightarrow{yz}\rangle \) (see [18]).

Lemma 2.9

[13] Let X be a CAT(0) space and \(z \in X\). Let \(x_1,\ldots ,x_N \in X\) and \(\gamma _1,\ldots ,\gamma _N\) be real numbers in [0, 1], such that \(\sum \nolimits _{i=1}^{N}\gamma _i=1\). Then, the following inequality holds:

$$\begin{aligned} \mathrm{{d}}^2\big (\sum \limits _{i=1}^{N}\oplus \gamma _{i}x_i, z\big )\le \sum _{i=1}^{N}\gamma _{i}\mathrm{{d}}^2(x_i,z)-\sum _{i,j=1, i\ne j}^{N}\gamma _i\gamma _j\mathrm{{d}}^2(x_i,x_j). \end{aligned}$$

Lemma 2.10

[24] Let X be a Hadamard space, \(\{x_n\}\) be a sequence in X and \(x \in X.\) Then, \(\{x_n\}\) \(\Delta \)-converges to x if and only if

$$\begin{aligned} \limsup \limits _{n\rightarrow \infty }\langle \overrightarrow{x_nx},\overrightarrow{yx}\rangle \le 0,~~\forall ~y\in X. \end{aligned}$$

Definition 2.11

Let C be a nonempty, closed, and convex subset of an Hadamard space X. A mapping \(T : C \rightarrow C\) is said to be \(\Delta \)-demiclosed, if for any bounded sequence \(\{x_n\}\) in X,  such that \(\Delta -\lim \nolimits _{n\rightarrow \infty }x_n = x\) and \(\lim \nolimits _{n\rightarrow \infty }\mathrm{{d}}(x_n, Tx_n) = 0,\) then \(x = Tx\).

Lemma 2.12

[36] Let X be a Hadamard space and \(\{x_n\}\) be a sequence in X. If there exists a nonempty subset D in which

  1. (i)

    \(\lim \nolimits _{n\rightarrow \infty }\mathrm{{d}}(x_n,z)\) exists for every \(z \in D,\) and

  2. (ii)

    if \(\{x_{n_k}\}\) is a subsequence of \(\{x_n\}\) which is \(\Delta \)-convergent to x,  then \(x \in D.\)

Then, there is a \(p \in D\), such that \(\{x_n\}\) is \(\Delta \)-convergent to p in X.

Lemma 2.13

[40] Let \(\{s_n\}\) be a sequence of nonnegative real numbers, \(\{\alpha _n\}\) be a sequence of real numbers in (0, 1) with \(\sum \nolimits _{n\rightarrow 0}^{\infty }\alpha _n = \infty \) and \(\{t_n\}\) be a sequence of real numbers. Suppose that

$$\begin{aligned} s_{n+1} \le (1-\alpha _n)s_n + \alpha _nt_n,~\forall ~n\ge 0. \end{aligned}$$

If \(\limsup \nolimits _{k\rightarrow \infty } t_{n_k} \le 0,\) then, for every subsequence \(\{s_{n_k}\}\) of \(\{s_n\}\) satisfying \(\liminf \nolimits _{k\rightarrow \infty }(s_{n_{k}+1}- s_{n_k})\ge 0,\) it holds \(\lim \nolimits _{n\rightarrow \infty }s_n = 0.\)

Main results

In this section, we present the main result of this paper. We begin with the following lemma which is very handy in establishing our main result.

Lemma 3.1

Let X be a Hadamard space. Then, for all \(v, w, x, y, z \in X\) and \(\alpha , \beta , \gamma \in (0, 1),\) with \(\alpha + \beta + \gamma = 1,\) we have

$$\begin{aligned} \mathrm{{d}}^2(\alpha x \oplus \beta y \oplus \gamma z, v) \le \alpha \mathrm{{d}}^2(x, v) + \beta \mathrm{{d}}^2(y, v) + \gamma \mathrm{{d}}^2(z, v). \end{aligned}$$

Proof

Let \(w = \alpha x \oplus \beta y \oplus \gamma z.\) We can rewrite w as \(w = \alpha x \oplus (1 - \alpha )\big (\frac{\beta }{1 - \alpha }y\oplus \frac{\gamma }{1 - \alpha }z\big ).\) Then, from Lemma 2.8(ii), we have

$$\begin{aligned} \mathrm{{d}}^2(w, v)&= \mathrm{{d}}^2\Big ( \alpha x \oplus (1 - \alpha )d\big (\frac{\beta }{1 - \alpha }y\oplus \frac{\gamma }{1 - \alpha }z\big ), v\Big )\\&\le \alpha \mathrm{{d}}^2(x, v) + (1 - \alpha ) \mathrm{{d}}^2\Big ( \big (\frac{\beta }{1 - \alpha }y\oplus \frac{\gamma }{1 - \alpha }z\big ), v \Big ) - \alpha (1 - \alpha )\mathrm{{d}}^2\Big (x, \big (\frac{\beta }{1 - \alpha }y\oplus \frac{\gamma }{1 - \alpha }z\big ) \Big )\\&\le \alpha \mathrm{{d}}^2(x, v) + (1 - \alpha ) \Big [\frac{\beta }{1 - \alpha } \mathrm{{d}}^2(y, v) + \frac{\gamma }{1 - \alpha }\mathrm{{d}}^2(z, v) - \frac{\beta \gamma }{(1 - \alpha )^2}\mathrm{{d}}^2(y,z) \Big ]\\&\le \alpha \mathrm{{d}}^2(x, v) + \beta \mathrm{{d}}^2(y, v) + \gamma \mathrm{{d}}^2(z, v). \end{aligned}$$

\(\square \)

Theorem 3.2

Let C be a nonempty closed and convex subset of a Hadamard space X\(\Psi _i : C \rightarrow {\mathbb {R}}\) be a finite family of convex and lower semicontinuous functions and \(f_i : C \times C \rightarrow {\mathbb {R}},~i=1,2,\ldots ,N\) be a finite family of bifunctions satisfying (A1–A4) of Theorem 1.1. Let \(T_i : C \rightarrow C,~ i = 1,2,\ldots ,N\) be a finite family of \(k_i\)-demimetric mappings with \(k_i\in (-\infty , \xi ]\) and \(\xi \in (0,1).\) Suppose that \(\Gamma := \bigcap \nolimits _{i=1}^{N}MEP(C,f_i,\Psi _i)\cap \bigcap \nolimits _{i=1}^{N}F(T_i)\) is nonempty and \(\{x_n\}\) is a sequence generated for arbitrary \(u, x_1 \in X\) by

$$\begin{aligned} {\left\{ \begin{array}{ll} y_{n} = J_{\lambda _n{f_{N}}}^{\Psi _N}\circ J_{\lambda _n{f_{N-1}}}^{\Psi _{N-1}}\circ \cdots \circ J_{\lambda _n{f_{2}}}^{\Psi _2}\circ J_{\lambda _n{f_{1}}}^{\Psi _1}x_n,\\ w_n = \alpha _n u \oplus \beta _n x_{n} \oplus \gamma _n y_{n},\\ x_{n+1} = \delta _{n,0} w_n \oplus \sum \limits _{i=1}^N\oplus \delta _{n,i}T_{i\xi }w_n,~\forall ~n\ge 1, \end{array}\right. } \end{aligned}$$
(3.1)

where \(T_{i\xi }=\xi x \oplus (1-\xi )T_ix,\) such that \(T_{i\xi }\) are \(\Delta \)-demiclosed and Lipschitz for each \(i=1,2,\ldots ,N.\) Suppose that \(\{\alpha _n\}, \{\beta _n\}, \{\gamma _n\}, \{\lambda _n\}\) and \(\{\delta _{n,i}\}\) are sequences in [0, 1], such that the following conditions are satisfied:

  1. (C1)

    \(\lim \nolimits _{n\rightarrow \infty }\alpha _n=0\) and \(\sum \nolimits _{i=1}^\infty \alpha _{n}=\infty ,\)

  2. (C2)

    \(\alpha _n + \beta _n + \gamma _n = 1\) for \(n\ge 1,\)

  3. (C3)

    \(\delta _{n,i}\in (0,1)\) and \(\sum \nolimits _{i=0}^N\delta _{n,i}=1,\)

  4. (C4)

    \(\lambda _{n}> \lambda > 0,\) for all \(n\ge 1.\)

Then, \(\{x_n\}\) converges strongly to a point \(x^*\in \Gamma .\)

Proof

Let \(p\in \Gamma .\) We have from Proposition 2.2(i) that \(J_{\lambda _n{f_{N}}}^{\Psi _N}\) is nonexpansive and \(p=J_{\lambda _n{f_{N}}}^{\Psi _N}p.\) Then, from (3.1), we obtain that

$$\begin{aligned} \mathrm{{d}}(y_{n}, p)^2&= \mathrm{{d}}(J_{\lambda _n{f_{N}}}^{\Psi _N}\circ J_{\lambda _n{f_{N-1}}}^{\Psi _{N-1}}\circ \cdots \circ J_{\lambda _n{f_{2}}}^{\Psi _2}\circ J_{\lambda _n{f_{1}}}^{\Psi _1}x_n, p)^2\nonumber \\&\le \mathrm{{d}}(J_{\lambda _n{f_{N-1}}}^{\Psi _{N-1}}\circ \cdots \circ J_{\lambda _n{f_{2}}}^{\Psi _2}\circ J_{\lambda _n{f_{1}}}^{\Psi _1}x_n, p)^2\nonumber \\&\vdots&\nonumber \\&\le \mathrm{{d}}(J_{\lambda _n{f_{1}}}^{\Psi _1}x_n, p)^2\nonumber \\&\le \mathrm{{d}}(x_n, p)^2. \end{aligned}$$
(3.2)

From Lemma 2.9, Lemma 2.7, Lemma 3.1, (3.1), (3.2) and (C3), we have

$$\begin{aligned} \mathrm{{d}}(x_{n+1}, p)^2&= \mathrm{{d}}^2(\delta _{n,0} w_n \oplus \sum \limits _{i=1}^N\oplus \delta _{n,i}T_{i\xi }w_n, p)\nonumber \\&\le \delta _{n,0}\mathrm{{d}}(w_n, p)^2 + \sum \limits _{i=1}^N\delta _{n,i}\mathrm{{d}}^2(T_{i\xi }w_n,p) - \sum \limits _{i=1}^N\delta _{n,0}\delta _{n,i}\mathrm{{d}}^2(T_{i\xi }w_n,w_n)\nonumber \\&\le \mathrm{{d}}^2(w_n, p) - \sum \limits _{i=1}^N\delta _{n,0}\delta _{n,i}\mathrm{{d}}^2(T_{i\xi }w_n,w_n)\nonumber \\&\le \alpha _n\mathrm{{d}}^2(u, p) + \beta _n\mathrm{{d}}^2(x_n, p) + \gamma _n\mathrm{{d}}^2(y_n,p)\nonumber \\&\le \alpha _n\mathrm{{d}}(u, v)^p + (\beta _n + \gamma _n)\mathrm{{d}}^2(x_n, p)\nonumber \\&= \alpha _n\mathrm{{d}}^2(u, v) + (1 - \alpha _n)\mathrm{{d}}^2(x_n,p)\nonumber \\&\le \max \{\mathrm{{d}}^2(u,p), \mathrm{{d}}^2(x_n, p)\}, \end{aligned}$$
(3.3)

which implies from induction that

$$\begin{aligned} \mathrm{{d}}^2(x_{n+1},p)\le \max \{\mathrm{{d}}^2(u,p), \mathrm{{d}}^2(x_1, p)\},~\forall ~n\ge 1. \end{aligned}$$

Hence, \(\{x_n\}\) is bounded. Consequently, \(\{y_n\}\) and \(\{w_n\}\) are bounded.

Let \(c_n = \frac{\beta _n}{1-\alpha _n}x_n\oplus \frac{\gamma _n}{1-\alpha _n}y_n.\) Then, by Lemma 2.8(ii) and (3.2), we have that

$$\begin{aligned} \mathrm{{d}}^2(c_n, p)&\le \frac{\beta _n}{1-\alpha _n} \mathrm{{d}}^2(x_n, p) + \frac{\gamma _n}{1-\alpha _n} \mathrm{{d}}^2(y_{n}, p)\nonumber \\&\le \mathrm{{d}}^2(x_n,p). \end{aligned}$$
(3.4)

By rewriting \(w_n\) in (3.1) as \(w_n = \alpha _n u \oplus (1-\alpha _n)c_n,\) then (3.3) becomes

$$\begin{aligned} \mathrm{{d}}^2(x_{n+1},p)&\le \mathrm{{d}}^2(\alpha _n u \oplus (1-\alpha _n)c_n, p) - \sum \limits _{i=1}^N\delta _{n,0}\delta _{n,i}\mathrm{{d}}^2(T_{i\xi }w_n,w_n)\end{aligned}$$
(3.5)
$$\begin{aligned}&\le \alpha _n \mathrm{{d}}^2(u,p) + (1 - \alpha _n)\mathrm{{d}}^2(c_n, p) - \alpha _n(1 - \alpha _n)\mathrm{{d}}^2(u,c_n) - \sum \limits _{i=1}^N\delta _{n,0}\delta _{n,i}\mathrm{{d}}^2(T_{i\xi }w_n,w_n)\nonumber \\&\le (1 - \alpha _n)\mathrm{{d}}^2(c_n, p) + \alpha _n\big [\mathrm{{d}}^2(u,p) - (1 - \alpha _n)\mathrm{{d}}^2(u,c_n)\big ]. \end{aligned}$$
(3.6)

By Lemma 2.13, it is sufficient to show that

$$\begin{aligned} \limsup \limits _{k\rightarrow \infty }\big [\mathrm{{d}}^2(u,p) - (1 - \alpha _{n_k})\mathrm{{d}}^2(u,c_{n_k})\big ]\le 0, \end{aligned}$$

for every subsequence \(\{\mathrm{{d}}(x_{n_k},p)\}\) of \(\{\mathrm{{d}}(x_{n},p)\}\) satisfying

$$\begin{aligned} \liminf \limits _{k\rightarrow \infty }\big [\mathrm{{d}}^2(x_{n_{k}+1},p) - \mathrm{{d}}^2(x_{n_{k}},p)\big ]\ge 0, \end{aligned}$$
(3.7)

From (3.4), (3.6), (3.7) and (C1), we have that

$$\begin{aligned} 0&\le \liminf \limits _{k\rightarrow \infty }\big [\mathrm{{d}}^2(x_{n_{k}+1},p) - \mathrm{{d}}^2(x_{n_{k}},p)\big ]\\&\le \liminf \limits _{k\rightarrow \infty }\big [\alpha _{n_k} \mathrm{{d}}^2(u,p) + (1 - \alpha _{n_k})\mathrm{{d}}^2(c_{n_k}, p) - \alpha _{n_k}(1 - \alpha _{n_k})\mathrm{{d}}^2(u,c_{n_k}) - \mathrm{{d}}^2(x_{n_{k}},p)\big ]\\&\le \liminf \limits _{k\rightarrow \infty }\big [\alpha _{n_k} \mathrm{{d}}^2(u,p) + (1 - \alpha _{n_k})\mathrm{{d}}^2(c_{n_k}, p) - \mathrm{{d}}^2(x_{n_{k}},p)\big ]\\&\le \limsup \limits _{k\rightarrow \infty }\big [\alpha _{n_k}\big (\mathrm{{d}}^2(u,p) - \mathrm{{d}}^2(c_{n_k}, p)\big ) - \mathrm{{d}}^2(x_{n_{k}},p)\big ] + \liminf \limits _{k\rightarrow \infty }\big [ \mathrm{{d}}^2(c_{n_k}, p) - \mathrm{{d}}^2(x_{n_{k}},p)\big ]\\&=\liminf \limits _{k\rightarrow \infty }\big [ \mathrm{{d}}^2(c_{n_k}, p) - \mathrm{{d}}^2(x_{n_{k}},p)\big ]\\&\le \limsup \limits _{k\rightarrow \infty }\big [ \mathrm{{d}}^2(c_{n_k}, p) - \mathrm{{d}}^2(x_{n_{k}},p)\big ]\le 0. \end{aligned}$$

Thus, we get

$$\begin{aligned} \lim \limits _{k\rightarrow \infty }\big [ \mathrm{{d}}^2(c_{n_k}, p) - \mathrm{{d}}^2(x_{n_{k}},p)\big ]= 0. \end{aligned}$$
(3.8)

It implies from (3.4) and (3.6) that

$$\begin{aligned} \sum \limits _{i=1}^N\delta _{n,0}\delta _{n,i}\liminf \limits _{k\rightarrow \infty }\mathrm{{d}}^2(T_{i\xi }w_{n_k},w_{n_k})&\le \liminf \limits _{k\rightarrow \infty }\alpha _{n_k}\big [ \mathrm{{d}}^2(u,p) - \mathrm{{d}}^2(c_{n_k}, p) - (1 - \alpha _{n_k})\mathrm{{d}}^2(u,c_{n_k})\big ]\\&\quad +\liminf \limits _{k\rightarrow \infty }\big [\mathrm{{d}}^2(c_{n_k}, p) - \mathrm{{d}}^2(x_{n_{k}+1},p)\big ]\\&\le \liminf \limits _{k\rightarrow \infty }\alpha _{n_k}\big [ \mathrm{{d}}^2(u,p) - \mathrm{{d}}^2(c_{n_k}, p) - (1 - \alpha _{n_k})\mathrm{{d}}^2(u,c_{n_k})\big ]\\&+\liminf \limits _{k\rightarrow \infty }\big [\mathrm{{d}}^2(x_{{n_k}+1}, p) - \mathrm{{d}}^2(x_{n_{k}},p)\big ]. \end{aligned}$$

Hence, by (C1) and (3.7), we obtain that

$$\begin{aligned} \lim \limits _{k\rightarrow \infty }\mathrm{{d}}(T_{i\xi }w_{n_k},w_{n_k})=0. \end{aligned}$$
(3.9)

From (3.1) and (3.9), we have

$$\begin{aligned} \lim \limits _{k\rightarrow \infty }\mathrm{{d}}(x_{{n_k}+1},w_{n_k})=0. \end{aligned}$$
(3.10)

Now, let \(\Phi ^i_n = J_{\lambda _n{f_{N}}}^{\Psi _N}\circ J_{\lambda _n{f_{N-1}}}^{\Psi _{N-1}}\circ \cdots \circ J_{\lambda _n{f_{2}}}^{\Psi _2}\circ J_{\lambda _n{f_{1}}}^{\Psi _1},\) for \(1\le i \le N\) and \(n\in \mathbb {N}.\) Then, \(y_n = \Phi ^i_nx_n\) with the assumption that \(\Phi ^0_n = I\) is an identity mapping. From (3.1) and (3.2), we obtain that

$$\begin{aligned} \limsup \limits _{n\rightarrow \infty }\big [ \mathrm{{d}}^2(\Phi ^i_Nx_n, p) - \mathrm{{d}}^2(x_n,p)\big ]\le 0. \end{aligned}$$
(3.11)

From Lemma 2.8(ii), (3.6) and (C2), we have

$$\begin{aligned} \mathrm{{d}}^2(x_{n+1},p)&\le \alpha _n\big (\mathrm{{d}}^2(u,p) - \mathrm{{d}}^2(c_n,p)\big ) + \mathrm{{d}}^2(c_n, p)\\&\le \alpha _n\big (\mathrm{{d}}^2(u,p) - \mathrm{{d}}^2(c_n,p)\big ) +\frac{\beta _n}{1-\alpha _n} \mathrm{{d}}^2(x_n, p) + \frac{\gamma _n}{1-\alpha _n} \mathrm{{d}}^2(y_n, p)\\&= \alpha _n\big (\mathrm{{d}}^2(u,p) - \mathrm{{d}}^2(c_n,p)\big ) +\frac{\beta _n+\gamma _n}{1-\alpha _n} \mathrm{{d}}^2(x_n, p) + \frac{\gamma _n}{1-\alpha _n} \big (\mathrm{{d}}^2(y_n, p)-\mathrm{{d}}^2(x_n, p)\big ). \end{aligned}$$

This implies that

$$\begin{aligned} \mathrm{{d}}^2(x_{n+1},p)-\mathrm{{d}}^2(x_n, p)&\le \alpha _n\big (\mathrm{{d}}^2(u,p) - \mathrm{{d}}^2(c_n,p)\big ) + \big (\mathrm{{d}}^2(y_n, p)-\mathrm{{d}}^2(x_n, p)\big ). \end{aligned}$$
(3.12)

By (3.7), (3.12) and (C1), for \(1\le i \le N,\) we have

$$\begin{aligned} 0\le \liminf \limits _{k\rightarrow \infty }\big [ \mathrm{{d}}^2(\Phi ^i_{n_k}x_{n_k}, p) - \mathrm{{d}}^2(x_{n_k},p)\big ]. \end{aligned}$$

Thus, by (3.11) and (3.12), we obtain that

$$\begin{aligned} \lim \limits _{k\rightarrow \infty }\big [ \mathrm{{d}}^2(\Phi ^i_{n_k}x_{n_k}, p) - \mathrm{{d}}^2(x_{n_k},p)\big ]=0. \end{aligned}$$
(3.13)

For \(1\le i \le N,\) we have from Proposition 2.2(ii) that

$$\begin{aligned} \mathrm{{d}}^2(J_{\lambda _{n_k}{f_{N}}}^{\Psi _N}(\Phi ^{i-1}_{n_k}x_{n_k}), \Phi ^{i-1}_{n_k}x_{n_k})&\le \mathrm{{d}}^2(\Phi ^{i-1}_{n_k}x_{n_k}, p) - \mathrm{{d}}^2(\Phi ^{i}_{n_k}x_{n_k}, p)\\&\le \mathrm{{d}}^2(x_{n_k}, p) - \mathrm{{d}}^2(\Phi ^{i}_{n_k}x_{n_k}, p). \end{aligned}$$

We obtain from (3.13) that

$$\begin{aligned} \lim \limits _{k\rightarrow \infty }\mathrm{{d}}(\Phi ^{i}_{n_k}x_{n_k}, \Phi ^{i-1}_{n_k}x_{n_k})=0. \end{aligned}$$
(3.14)

By triangular inequality, we have

$$\begin{aligned} \mathrm{{d}}(x_{n_k}, \Phi ^{i}_{n_k}x_{n_k})&\le \mathrm{{d}}(\Phi ^{0}_{n_k}x_{n_k}, \Phi ^{1}_{n_k}x_{n_k}) + \mathrm{{d}}(\Phi ^{1}_{n_k}x_{n_k}, \Phi ^{2}_{n_k}x_{n_k})+\cdots \\&\quad + \mathrm{{d}}(\Phi ^{i-2}_{n_k}x_{n_k}, \Phi ^{i-1}_{n_k}x_{n_k})+\mathrm{{d}}(\Phi ^{i-1}_{n_k}x_{n_k}, \Phi ^{i}_{n_k}x_{n_k}), \end{aligned}$$

which implies from (3.14) that

$$\begin{aligned} \lim \limits _{k\rightarrow \infty }\mathrm{{d}}(x_{n_k}, \Phi ^{i}_{n_k}x_{n_k})=0. \end{aligned}$$
(3.15)

From Proposition 2.2(iii) and (C4), we have

$$\begin{aligned} \mathrm{{d}}(J_{\lambda {f_{N}}}^{\Psi _N}(\Phi ^{i-1}_{n_k}x_{n_k}), \Phi ^{i}_{n_k}x_{n_k})&\le \mathrm{{d}}(J_{\lambda {f_{N}}}^{\Psi _N}(\Phi ^{i-1}_{n_k}x_{n_k}), \Phi ^{i-1}_{n_k}x_{n_k}) + \mathrm{{d}}(\Phi ^{i-1}_{n_k}x_{n_k}, \Phi ^{i}_{n_k}x_{n_k})\\&\le 2\mathrm{{d}}(J_{\lambda _{n_k}{f_{N}}}^{\Psi _N}(\Phi ^{i-1}_{n_k}x_{n_k}), \Phi ^{i-1}_{n_k}x_{n_k}) + \mathrm{{d}}(\Phi ^{i-1}_{n_k}x_{n_k}, \Phi ^{i}_{n_k}x_{n_k})\\&\le 3\mathrm{{d}}(\Phi ^{i-1}_{n_k}x_{n_k}, \Phi ^{i}_{n_k}x_{n_k}). \end{aligned}$$

Therefore, from (3.14), we obtain that

$$\begin{aligned} \lim \limits _{k\rightarrow \infty }\mathrm{{d}}(J_{\lambda {f_{N}}}^{\Psi _N}(\Phi ^{i-1}_{n_k}x_{n_k}), \Phi ^{i}_{n_k}x_{n_k})=0. \end{aligned}$$
(3.16)

Now, for \(1\le i\le N,\) we have

$$\begin{aligned} \mathrm{{d}}(J_{\lambda {f_{N}}}^{\Psi _N}x_{n_k}, x_{n_k})&\le \mathrm{{d}}(J_{\lambda {f_{N}}}^{\Psi _N}x_{n_k}, J_{\lambda {f_{N}}}^{\Psi _N}(\Phi ^{i-1}_{n_k}x_{n_k})) + \mathrm{{d}}(J_{\lambda {f_{N}}}^{\Psi _N}(\Phi ^{i-1}_{n_k}x_{n_k}),\Phi ^{i}_{n_k}x_{n_k}) +\mathrm{{d}}(\Phi ^{i}_{n_k}x_{n_k}, x_{n_k})\\&\le \mathrm{{d}}(x_{n_k}, \Phi ^{i-1}_{n_k}x_{n_k}) + \mathrm{{d}}(J_{\lambda {f_{N}}}^{\Psi _N}(\Phi ^{i-1}_{n_k}x_{n_k}),\Phi ^{i}_{n_k}x_{n_k}) +\mathrm{{d}}(\Phi ^{i}_{n_k}x_{n_k}, x_{n_k}). \end{aligned}$$

Then, from (3.14), (3.15) and (3.16), we obtain that

$$\begin{aligned} \lim \limits _{k\rightarrow \infty }\mathrm{{d}}(J_{\lambda {f_{N}}}^{\Psi _N}x_{n_k}, x_{n_k})=0. \end{aligned}$$
(3.17)

From Lemma 2.8(i) and (3.1), we have that

$$\begin{aligned} \mathrm{{d}}(w_n, x_n) \le \alpha _n \mathrm{{d}}(u,x_n) + \gamma _n \mathrm{{d}}(x_n, y_n). \end{aligned}$$

Then, it implies from (3.15) and (C1) that

$$\begin{aligned} \lim \limits _{k\rightarrow \infty }\mathrm{{d}}(w_{n_k}, x_{n_k})=0. \end{aligned}$$
(3.18)

Also, from (3.10) and (3.18), we get

$$\begin{aligned} \lim \limits _{k\rightarrow \infty }\mathrm{{d}}(x_{{n_k}+1}, x_{n_k})=0. \end{aligned}$$
(3.19)

Finally, by triangular inequality and the fact that \(T_{i\xi }\) are Lipschtiz, for each \(1\le i \le N,\) we get

$$\begin{aligned} \mathrm{{d}}(T_{i\xi }x_{n_k},x_{n_k})&\le \mathrm{{d}}(T_{i\xi }x_{n_k}, T_{i\xi }w_{n_k}) + \mathrm{{d}}(T_{i\xi }w_{n_k},x_{{n_k}+1}) + \mathrm{{d}}(x_{{n_k}+1}, x_{n_k})\\&\le L \mathrm{{d}}(x_{n_k}, w_{n_k}) + \mathrm{{d}}(T_{i\xi }w_{n_k},x_{{n_k}+1}) + \mathrm{{d}}(x_{{n_k}+1}, x_{n_k}), \end{aligned}$$

which implies from (3.9), (3.10), (3.18) and (3.19) that

$$\begin{aligned} \lim \limits _{k\rightarrow \infty }\mathrm{{d}}(T_{i\xi }x_{n_k},x_{n_k})=0. \end{aligned}$$
(3.20)

Now, since \(\{x_{n_k}\}\) is bounded, it follows from Lemma 2.5 that there exists a subsequence \(\{x_{n_{k_j}}\}\) of \(\{x_{n_k}\}\), such that \(\{x_{n_{k_j}}\}\) \(\Delta \)-converges to \(x^*.\) It follows from (3.15) that there exists a subsequence \(\{y_{n_{k_j}}\}\) of \(\{y_{n_{k}}\}\), such that \(\{y_{n_{k_j}}\}\) \(\Delta \)-converges to \(x^*\) Similarly, from (3.20), we have that \(\{T_{i\xi }x_{n_{k_j}}\}\) \(\Delta \)-converges to \(x^*.\) Since \(T_{i\xi }\) is \(\Delta \)-demiclosed for each \(i=1,2,\ldots ,N\) it also follows from (3.20) and Lemma 2.7 that \(x^*\in \bigcap \nolimits _{i=1}^{N}F(T_{i\xi })=\bigcap \nolimits _{i=1}^{N}F(T_i).\) Since \(J_{\lambda {f_{i}}}^{\Psi _i}\) is nonexpansive, for each \(i=1,2,\ldots ,N,\) we obtain from (3.17) and Lemma 2.12 that \(x^*\in \bigcap \nolimits _{i=1}^{N}F(J_{\lambda {f_{i}}}^{\Psi _i}) = \bigcap \nolimits _{i=1}^{N}MEP(C,f_i,\Psi _i).\) Hence, \(x^*\in \Gamma = \bigcap \nolimits _{i=1}^{N}MEP(C,f_i,\Psi _i)\cap \bigcap \nolimits _{i=1}^{N}F(T_i).\)

Next, we prove that \(\{x_n\}\) converges strongly to \(x^*.\) For an arbitrary \(u\in X,\) we have by Lemma 2.10 that

$$\begin{aligned} \limsup \limits _{n\rightarrow \infty }\langle \overrightarrow{ux^*},\overrightarrow{x^*x_{n}}\rangle \le 0. \end{aligned}$$
(3.21)

Furthermore, by quasi-linearization properties and Cauchy–Schwartz inequality, we obtain

$$\begin{aligned} \langle \overrightarrow{ux^*},\overrightarrow{x^*c_{n}}\rangle&=\langle \overrightarrow{ux^*},\overrightarrow{x_{n}c_n}\rangle + \langle \overrightarrow{ux^*},\overrightarrow{x^*x_{n}}\rangle \\&\le \mathrm{{d}}(u,x^*)\mathrm{{d}}(x_n,c_n) + \langle \overrightarrow{ux^*},\overrightarrow{x^*x_{n}}\rangle . \end{aligned}$$

From (3.1) and (3.15), we have

$$\begin{aligned} \lim \limits _{k\rightarrow \infty }\mathrm{{d}}(c_{n_k},x_{n_k}) \le \frac{\gamma _{n_k}}{1-\alpha _{n_k}}\lim \limits _{k\rightarrow \infty }\mathrm{{d}}(x_{n_k}, y_{n_k})=0. \end{aligned}$$
(3.22)

Thus, by (3.21) and (3.22), we obtain that

$$\begin{aligned} \limsup \limits _{k\rightarrow \infty }\langle \overrightarrow{ux^*},\overrightarrow{x^*c_{n_k}}\rangle \le 0. \end{aligned}$$
(3.23)

From (3.4), (3.5) and Lemma 2.8(iii), we have

$$\begin{aligned} \mathrm{{d}}^2(x_{{n_k}+1},x^*)&\le \mathrm{{d}}^2(\alpha _{n_k} u \oplus (1-\alpha _{n_k})c_{n_k}, x^*)\nonumber \\&\le \alpha _{n_k}^2\mathrm{{d}}^2(u,x^*) + (1-\alpha _{n_k})\mathrm{{d}}^2(c_{n_k},x^*)+2\alpha _{n_k}(1-\alpha _{n_k})\langle \overrightarrow{ux^*},\overrightarrow{x^*c_{n_k}}\rangle \nonumber \\&\le (1-\alpha _{n_k})\mathrm{{d}}^2(c_{n_k},x^*) + \alpha _{n_k}\big (\alpha _{n_k}\mathrm{{d}}^2(u,x^*) + 2(1-\alpha _{n_k})\langle \overrightarrow{ux^*},\overrightarrow{x^*c_{n_k}}\rangle \big ) \nonumber \\&\le (1-\alpha _{n_k})\mathrm{{d}}^2(x_{n_k},x^*) + \alpha _{n_k}b_{n_k}, \end{aligned}$$
(3.24)

where

$$b_{n_k} = \big (\alpha _{n_k}\mathrm{{d}}^2(u,x^*) + 2(1-\alpha _{n_k})\langle \overrightarrow{ux^*},\overrightarrow{x^*c_{n_k}}\rangle \big ).$$

It is sufficient from (C1) and (3.23) that

$$\begin{aligned} \limsup \limits _{k\rightarrow \infty }b_{n_k}\le 0. \end{aligned}$$
(3.25)

Hence, applying Lemma 2.13 to (3.24), we have that \(\{\mathrm{{d}}(x_{n},x^*)\}\rightarrow 0,\) as \(n\rightarrow \infty .\) Therefore, \(x^*\in \Gamma .\) \(\square \)

In what follows, we give some consequences of our main theorem. If \(N=1\) in Theorem 3.2, we obtain the following result:

Corollary 3.3

Let C be a nonempty closed and convex subset of a Hadamard space X\(\Psi : C \rightarrow {\mathbb {R}}\) be a convex and lower semicontinuous function and \(f : C \times C \rightarrow {\mathbb {R}}\) be a bifunction satisfying (A1-A4) of Theorem 1.1. Let \(T : C \rightarrow C,\) be a k-demimetric mapping with \(k\in (-\infty , \xi ]\) and \(\xi \in (0,1).\) Suppose that \(\Gamma := MEP(C,f,\Psi )\cap F(T)\) is nonempty and \(\{x_n\}\) is a sequence generated for arbitrary \(u, x_1 \in X\) by

$$\begin{aligned} {\left\{ \begin{array}{ll} y_{n} = J_{\lambda _n f}^{\Psi }x_n,\\ w_n = \alpha _n u \oplus \beta _n x_{n} \oplus \gamma _n y_{n},\\ x_{n+1} = \delta _{n,0} w_n \oplus (1-\delta _{n,1})T_{\xi }w_n,~\forall ~n\ge 1, \end{array}\right. } \end{aligned}$$
(3.26)

where \(T_{\xi }=\xi x \oplus (1-\xi )Tx,\) such that \(T_{\xi }\) is \(\Delta \)-demiclosed and Lipschitz. Suppose that \(\{\alpha _n\}, \{\beta _n\}, \{\gamma _n\}, \{\lambda _n\}\) and \(\{\delta _{n,i}\}\) are sequences in [0, 1], such that the following conditions are satisfied:

  1. (C1)

    \(\lim \nolimits _{n\rightarrow \infty }\alpha _n=0\) and \(\sum \nolimits _{i=1}^\infty \alpha _{n}=\infty ,\)

  2. (C2)

    \(\alpha _n + \beta _n + \gamma _n = 1\) for \(n\ge 1,\)

  3. (C3)

    \(\delta _{n,i}\in (0,1)\) and \(\sum \nolimits _{i=0}^1\delta _{n,i}=1,\)

  4. (C4)

    \(\lambda _{n}> \lambda > 0,\) for all \(n\ge 1.\)

Then, \(\{x_n\}\) converges strongly to a point \(x^*\in \Gamma .\)

If \(\Psi _i\equiv 0,\) for all \(i=1,2,\ldots ,N\) in Theorem 3.2, we obtain the following result for a finite family of equilibrium problems and fixed point problems of a finite family of \(k_i\) demimetric mappings:

Corollary 3.4

Let C be a nonempty closed and convex subset of a Hadamard space X and \(f_i : C \times C \rightarrow {\mathbb {R}},~i=1,2,\ldots ,N\) be a finite family of bifunctions satisfying (A1-A4) of Theorem 1.1. Let \(T_i : C \rightarrow C,~ i = 1,2,\ldots ,N\) be a finite family of \(k_i\)-demimetric mappings with \(k_i\in (-\infty , \xi ]\) and \(\xi \in (0,1).\) Suppose that \(\Gamma := \bigcap \nolimits _{i=1}^{N}EP(C,f_i)\cap \bigcap \nolimits _{i=1}^{N}F(T_i)\) is nonempty and \(\{x_n\}\) is a sequence generated for arbitrary \(u, x_1 \in X\) by

$$\begin{aligned} {\left\{ \begin{array}{ll} y_{n} = J_{\lambda _n}^{f_N}\circ J_{\lambda _n}^{f_{N-1}}\circ \cdots \circ J_{\lambda _n}^{f_2}\circ J_{\lambda _n}^{f_1}x_n,\\ w_n = \alpha _n u \oplus \beta _n x_{n} \oplus \gamma _n y_{n},\\ x_{n+1} = \delta _{n,0} w_n \oplus \sum \limits _{i=1}^N\oplus \delta _{n,i}T_{i\xi }w_n,~\forall ~n\ge 1, \end{array}\right. } \end{aligned}$$
(3.27)

where \(T_{i\xi }=\xi x \oplus (1-\xi )T_ix,\) such that \(T_{i\xi }\) are \(\Delta \)-demiclosed and Lipschitz for each \(i=1,2,\ldots ,N.\) Suppose that \(\{\alpha _n\}, \{\beta _n\}, \{\gamma _n\}, \{\lambda _n\}\) and \(\{\delta _{n,i}\}\) are sequences in [0, 1], such that the following conditions are satisfied:

  1. (C1)

    \(\lim \nolimits _{n\rightarrow \infty }\alpha _n=0\) and \(\sum \nolimits _{i=1}^\infty \alpha _{n}=\infty ,\)

  2. (C2)

    \(\alpha _n + \beta _n + \gamma _n = 1\) for \(n\ge 1,\)

  3. (C3)

    \(\delta _{n,i}\in (0,1)\) and \(\sum \nolimits _{i=0}^N\delta _{n,i}=1,\)

  4. (C4)

    \(\lambda _{n}> \lambda > 0,\) for all \(n\ge 1.\)

Then, \(\{x_n\}\) converges strongly to a point \(x^*\in \Gamma .\)

If \(f_i\equiv 0,\) for all \(i=1,2,\ldots ,N\) in Theorem 3.2, we obtain the following result for a finite family of minimization problems and fixed point problems of a finite family of \(k_i\) demimetric mappings:

Corollary 3.5

Let C be a nonempty closed and convex subset of a Hadamard space X and \(\Psi _i : C \rightarrow {\mathbb {R}},~i=1,2,\ldots ,N\) be a finite family of convex and lower semicontinuous functions. Let \(T_i : C \rightarrow C,~ i = 1,2,\ldots ,N\) be a finite family of \(k_i\)-demimetric mappings with \(k_i\in (-\infty , \xi ]\) and \(\xi \in (0,1).\) Suppose that \(\Gamma := \bigcap \nolimits _{i=1}^{N}\arg \min \nolimits _{y\in X}\Psi _i(y)\cap \bigcap \nolimits _{i=1}^{N}F(T_i)\) is nonempty and \(\{x_n\}\) is a sequence generated for arbitrary \(u, x_1 \in X\) by

$$\begin{aligned} {\left\{ \begin{array}{ll} y_{n} = J_{\lambda _n}^{\Psi _N}\circ J_{\lambda _n}^{\Psi _{N-1}}\circ \cdots \circ J_{\lambda _n}^{\Psi _2}\circ J_{\lambda _n}^{\Psi _1}x_n,\\ w_n = \alpha _n u \oplus \beta _n x_{n} \oplus \gamma _n y_{n},\\ x_{n+1} = \delta _{n,0} w_n \oplus \sum \nolimits _{i=1}^N\oplus \delta _{n,i}T_{i\xi }w_n,~\forall ~n\ge 1, \end{array}\right. } \end{aligned}$$
(3.28)

where \(T_{i\xi }=\xi x \oplus (1-\xi )T_ix,\) such that \(T_{i\xi }\) are \(\Delta \)-demiclosed and Lipschitz for each \(i=1,2,\ldots ,N.\) Suppose that \(\{\alpha _n\}, \{\beta _n\}, \{\gamma _n\}, \{\lambda _n\}\) and \(\{\delta _{n,i}\}\) are sequences in [0, 1], such that the following conditions are satisfied:

  1. (C1)

    \(\lim \nolimits _{n\rightarrow \infty }\alpha _n=0\) and \(\sum \nolimits _{i=1}^\infty \alpha _{n}=\infty ,\)

  2. (C2)

    \(\alpha _n + \beta _n + \gamma _n = 1\) for \(n\ge 1,\)

  3. (C3)

    \(\delta _{n,i}\in (0,1)\) and \(\sum \nolimits _{i=0}^N\delta _{n,i}=1,\)

  4. (C4)

    \(\lambda _{n}> \lambda > 0,\) for all \(n\ge 1.\)

Then, \(\{x_n\}\) converges strongly to a point \(x^*\in \Gamma .\)

Numerical example

In this section, we give a numerical example to demonstrate the applicability of Algorithm (3.1).

Example 4.1

Let \(X = {\mathbb {R}},\) endowed with the usual metric and \(C = [0, 1].\) For \(N = 2,\) define \(T_i : C \rightarrow C,\) by \(T_ix = x-x^i,~i = 1, 2.\) Then, T is (-1)-demimetric, (see [3, Example 3.3]). Let \(f_1,f_2:C\times C\rightarrow {\mathbb {R}}\) be defined by

$$\begin{aligned} f_1(x,y) = -x^2-x + xy +2y,~\forall ~x,~y\in {\mathbb {R}}, \end{aligned}$$

and

$$\begin{aligned} f_2(x,y) = -7x^2 + xy + 6y^2,~\forall ~x,~y\in {\mathbb {R}}. \end{aligned}$$

Let the mapping \(\Psi _1,\Psi _2:{\mathbb {R}}\rightarrow {\mathbb {R}}\) be defined by \(\Psi _1x = \frac{x}{5}~\forall ~x\in {\mathbb {R}}\) and \(\Psi _2x = x^2~\forall ~x\in {\mathbb {R}}\), respectively. Then, \(f_1, f_2, \Psi ~\text{ and }~\Psi _2\) satisfy assumptions (i)-(iv) in Theorem 1.1. Let \(\lambda _n = 1~\forall ~n\ge 1.\) Then, by (1.4) and Theorem 2.2, we compute for \(v_n = J^{\Psi _1}_{\lambda _n f_1}x_n\)

$$\begin{aligned} 0&\le f_1(z_1, y) + \Psi _1(y) - \Psi _1(z_1) + \frac{1}{\lambda _n}\langle \overrightarrow{z_1y},\overrightarrow{x^*z_1}\rangle \\&= -z_1^2-z_1+z_1y+y^2 + \frac{1}{5}(y-z_1)+\mathrm{{d}}(z_1,y)\mathrm{{d}}(x^*,z_1)\\&=(z_1+2)(y-z_1)+\frac{1}{5}(y-z_1)+\mathrm{{d}}(z_1,y)\mathrm{{d}}(x^*,z_1)\\&=(z_1+2)+(z_1-x^*)+\frac{1}{5}, \end{aligned}$$
Fig. 1
figure 1

Errors vs iteration numbers (n): Case 1 (top left); Case 2 (top right); Case 3 (bottom left); Case 4 (bottom right)

which implies \((z_1+2)+(z_1-x^*)+\frac{1}{5}=0\) and \(z_1=\frac{1}{2}(x^*-\frac{11}{5}).\) Now, we proceed to compute \(x^* = J^{\Psi _2}_{\lambda _n f_2}v_n.\) Again, by (1.4) and Theorem 2.2, we have

$$\begin{aligned} 0&\le f_2(z_2, y) + \Psi _2(y) - \Psi _2(z_2) + \frac{1}{\lambda _n}\langle \overrightarrow{z_2y},\overrightarrow{x^*z_2}\rangle \nonumber \\&= -7z_2^2 + z_2y +6y^2+(y^2-z^2_2)+\mathrm{{d}}(y,z_2)\mathrm{{d}}(z_2, x^*)\nonumber \\&= -8z_2^2 + z_2y +7y^2+(y-z_2)(z_2-x^*)\nonumber \\&= -9z_2^2 + 2z_2y +7y^2-x^*y+x^*z_2\nonumber \\&=7y^2 + (2_2-x^*)y+(x^*z_2-9z^2_2). \end{aligned}$$
(4.1)

Suppose (4.1) is a quadratic inequality, such that \(a = 7,~b = (2z_2-x^*)~\text{ and }~c = (x^*z-9z^2_2).\) Then, the discriminant \(\Delta = b^2-4ac\) is

$$\begin{aligned} \Delta&=(2z_2-x^*)^2 - 28(x^*z-9z^2_2)\\&= 256z^2_2-32z_2x^*+x^*\\&=\big (16z_2-x^*)^2. \end{aligned}$$

This implies that \(z_2 = \frac{x^*}{16}.\) Take \(\alpha _n=\frac{1}{11n^2+5},~\beta _n=\frac{5n^2+4}{11n^2+5},~\gamma _n=\frac{6n^2}{11n^2+5}~\text{ and }~\delta _{n,0}=\frac{7n^2}{7n^2+9n+1},~ \delta _{n,1}=\frac{9n}{7n^2+9n+1},~\delta _{n,2}=\frac{1}{7n^2+9n+1}.\) Let \(\{v_n\},~\{y_n\},~\{w_n\}\) and \(\{x_n\}\) be generated by Algorithm 3.1 as follows:

$$\begin{aligned} {\left\{ \begin{array}{ll} v_{n} = J_{\lambda _n{f_{1}}}^{\Psi _1}x_n,\\ y_{n} = J_{\lambda _n{f_{2}}}^{\Psi _2}v_n,\\ w_n = \alpha _n u \oplus \beta _n x_{n} \oplus \gamma _n y_{n},\\ x_{n+1} = \delta _{n,0} w_n \oplus \sum \limits _{i=1}^2\oplus \delta _{n,i}T_{i\xi }w_n,~\forall ~n\ge 1. \end{array}\right. } \end{aligned}$$
(4.2)

Case 1: \(x_1=10,~u=0.5\) and Case 2: \(x_1=10,~ -0.5,\)

Case 3: \(x_1=0.5,~u=1\) and Case 4: \(x_1=0.5,~u=-0.5.\)

Conclusion

In this article, we presented a modified Halpern-proximal point method for solving families of mixed equilibrium problems and fixed point problems of k-demimetric mappings in Hadamard spaces. We propose an algorithm and establish its strong convergence for approximating the common solution of the aforementioned problems in Hadamard spaces. We demonstrated with a numerical example the efficiency of our iterative method in a Hadamard space to support the convergence result stated in this article.

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Aremu, K.O., Aphane, M., Ibrahim, A.H. et al. A modified Halpern-proximal point method for approximating solutions of mixed equilibrium and fixed point problems in Hadamard spaces. Arab. J. Math. (2022). https://doi.org/10.1007/s40065-022-00378-w

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Mathematics Subject Classification

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  • 47H10
  • 49J20
  • 49J40