A new relaxed proximal gradient algorithm is introduced for approximating a common solution of a minimization problem and a fixed point problem of \(\delta\)-demimetric mappings in a real Hilbert space.
Let \(H\) be a real Hilbert space, and \(C\) be a nonempty closed convex subset of \(H\). The following is the main result of this paper.
Theorem. Let \(g,h:H\to R\cup\{\infty\}\) be two proper, convex, lower semicontinuous functions such that \(h\) is nonsmooth and \(\nabla g\) is \((1/L)\)-ism with \(L>0\). Let \(f:C\to C\) be a Meir-Keeler contraction mapping, \(B:C\to H\) be a strongly positive bounded linear operator with coefficient \(\tau>0\) such that \(0<\xi<\tau/2\) and \(T:C\to C\) be a \(\delta\)-demimetric mapping for \(\delta\in(-\infty,1)\) and \(\widehat{F}(T)=F(T)\). Suppose that \(\Gamma:=\Omega\cap F(T)\ne \emptyset\), and let \(\alpha_n\in[0,1]\), \(\beta_n\in[0,1)\), \(w_n,\theta_n\in(0,1)\), \(\gamma_n>0\). Choose initial points \(x_0,x_1\in H\) arbitrarily and let \((x_n)\), \((y_n)\), \((u_n)\) be introduced by the algorithm

(A1)

\(y_n=x_n+\beta_n(x_n-x_{n-1})\),

(A2)

\(u_n=(1-w_n)y_n+w_n\mathrm{prox}_{\gamma_nh}(y_n-\gamma_n\nabla g(y_n))\),

(A3)

\(x_{n+1}=P_C(\alpha_n\xi f(x_n)+\theta_nx_n+((1-\theta_n)I-\alpha_n B)T_{\lambda_n}u_n)\), \(n\ge 1\), where \(T_{\lambda_n}=(1-\lambda_n)I+\lambda_nT\), for \(\lambda_n\in(0,1)\).

Assume that the following conditions are satisfied:

(C1)

\(\lim_n\alpha_n=0\) and \(\sum_{n\ge 1}\alpha_n=\infty\),

(C2)

\(\lim_n(\beta_n/\alpha_n)||x_n-x_{n-1}||=0\),

(C3)

\(0<\liminf_nw_n\le\limsup_nw_n<1\),

(C4)

\(0<\liminf_n\gamma_n\le\limsup_n\gamma_n<2/L\),

(C5)

\(0<\liminf_n\lambda_n\le\limsup_n\lambda_n<1-\delta\).

Then \((x_n)\) converges strongly to a point \(x^*\), where

(i)

\(x^*\) is a fixed point of \(P_\Gamma(I-B+\xi f)(\cdot)\),

(ii)

\(x^*\) is the unique solution of the variational inequality \(\langle(B-\xi f)x^*,x^*-y\rangle\le 0\), \(y\in\Gamma\).

Further aspects occasioned by these developments are also discussed.