A regularization projection algorithm for various problems with nonlinear mappings in Hilbert spaces

Let C be a closed convex subset of a real Hilbert space H. Let A be an α-inverse-strongly monotone mapping of C into H, and let S be a nonexpansive mapping of C into itself such that \(F(S)\cap \operatorname{VI}(C,A)\neq\emptyset\). Suppose \(x_{1}=x\in C\) and \(\{x_{n}\}\) is given by for every \(n=1,2,\ldots\) , where \(\{\alpha_{n}\}\) is a sequence in \([0,1)\) and \(\{\lambda_{n}\}\) is a sequence in \([0,2\alpha]\). If \(\{\alpha_{n}\}\) and \(\{\lambda_{n}\}\) are chosen so that \(\lambda_{n}\in [a,b]\) for some a, b with \(0< a <b<2\alpha\), \(\lim_{n\rightarrow\infty}\alpha_{n}=0\), \(\sum_{n=1}^{\infty}\alpha_{n}=\infty \), \(\sum_{n=1}^{\infty}|\alpha_{n+1}-\alpha_{n}|<\infty\) and \(\sum_{n=1}^{\infty}|\lambda_{n+1}-\lambda_{n}|<\infty\), then \(\{x_{n}\}\) converges strongly to \(P_{F(S)\cap \operatorname{VI}(C,A)}x\).

Let C be a nonempty closed convex subset of a real Hilbert space H. Let F be a bifunction from \(C\times C\) to R satisfying (A1)-(A4), and let S be a nonexpansive mapping of C into H such that \(F(S)\cap \operatorname{EP}(F)\neq\emptyset\). Let f be a contraction of H into itself, and let \(\{x_{n}\}\) and \(\{u_{n}\}\) be sequences generated by \(x_{1}\in H\) and for every \(n=1,2,\ldots\) , where \(\{\alpha_{n}\}\subset[0,1]\) and \(\{r_{n}\} \subset(0,\infty)\) satisfy \(\lim_{n\rightarrow\infty}\alpha_{n}=0\), \(\sum_{n=1}^{\infty}\alpha_{n}=\infty \), \(\sum_{n=1}^{\infty}|\alpha_{n+1}-\alpha_{n}|<\infty\), \(\liminf_{n\rightarrow\infty}r_{n}>0\) and \(\sum_{n=1}^{\infty}|r_{n+1}-r_{n}|<\infty \). Then \(\{x_{n}\}\) and \(\{u_{n}\}\) converge strongly to \(z\in F(S)\cap \operatorname{EP}(F)\), where \(z=P_{F(S)\cap \operatorname{EP}(F)}f(z)\).

[21]

Let C be a nonempty closed convex subset of H. Let A be a monotone mapping of C into H, and let \(N_{C}v\) be a normal cone to C at \(v\in C\), i.e., and define a mapping T on C by Then T is maximal monotone and \(0\in Tv\) if and only if \(\langle Av,u-v\rangle\geq0\) for all \(u\in C\).

[2]

Let C be a nonempty closed convex subset of H, and let \(S:C\rightarrow H\) be a κ-strictly pseudocontractive mapping. Then \(I-S\) is demi-closed at zero.

[1]

Let C be a nonempty closed convex subset of H, and let \(F:C\times C\rightarrow\mathbb{R}\) be a bifunction satisfying (A1)-(A4). Then, for any \(r>0\) and \(x\in H\), there exists \(z\in C\) such that Further, define for all \(r>0\) and \(x\in H\). Then the following hold:

[2]

Let C be a nonempty closed convex subset of H, and let \(S:C\rightarrow H\) be a κ-strictly pseudocontractive mapping. Define a mapping T by \(T=\delta I+(1-\delta )S\), where δ is a constant in \((0,1)\). If \(\delta\in[\kappa ,1)\) then T is nonexpansive and \(F(T)=F(S)\).

[22]

Let \(\{x_{n}\}\) and \(\{y_{n}\}\) be bounded sequences in H, and let \(\{\beta_{n}\}\) be a sequence in \([0,1]\) with \(0<\liminf_{n\rightarrow\infty}\beta_{n}\leq\limsup_{n\rightarrow\infty }\beta_{n}<1\). Suppose \(x_{n+1}=(1-\beta_{n})y_{n}+\beta_{n}x_{n}\) for all integers \(n\geq 0\) and Then \(\lim_{n\rightarrow\infty}\|y_{n}-x_{n}\|=0\).

[23]

Assume that \(\{\alpha_{n}\}\) is a sequence of nonnegative real numbers such that where \(\{\gamma_{n}\}\) is a sequence in \((0,1)\) and \(\{\mu_{n}\}\), \(\{\delta _{n}\}\) are real sequences such that Then \(\lim_{n\rightarrow\infty}\alpha_{n}=0\).

Let C be a closed convex subset of a real Hilbert space H. Let \(A:C\rightarrow H\) be an α-inverse-strongly monotone mapping, and let F be a bifunction from \(C\times C\) to ℝ which satisfies (A1)-(A4). Let \(S:C\rightarrow H\) be a κ-strictly pseudocontractive mapping, and let f be a β-contraction on H. Assume that \(\Omega=F(S)\cap \operatorname{VI}(C,A)\cap \operatorname{EP} (F)\neq\emptyset\). Let \(\{ r_{n}\}\) and \(\{s_{n}\}\) be positive real number sequences. Let \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), \(\{\gamma_{n}\}\) and \(\{\delta_{n}\}\) be real number sequences in \((0,1)\) such that \(\alpha_{n}+\beta_{n}+\gamma_{n}=1\). Let \(\{x_{n}\}\) be a sequence generated in the following process: where \(\{e_{n}\}\) is a sequence in H. Assume that the control sequences satisfy the following conditions: where δ, s, \(s'\) are real constants. Then \(\{x_{n}\}\) converges strongly to q, which is also a unique solution to the variational inequality

For any \(x,y\in C\), we see that By using condition (e), we see that \(\|(I-s_{n}A)x-(I-s_{n}A)y\|\leq\|x-y\|\). This proves that \(I-s_{n}A\) is nonexpansive. Put \(S_{n}=\delta_{n}I+(1-\delta_{n})S\). It follows from Lemma 1.4 that \(S_{n}\) is nonexpansive and \(F(S_{n})=F(S)\). Let \(p\in\Omega\) be fixed arbitrarily. Hence, we have It follows that \(\|x_{n}-p\|\leq\max\{\|x_{1}-p\|,\frac{\|f(p)-p\|}{1-\beta}\}+\sum_{n=1}^{\infty}\|e_{n}\|\). This shows that \(\{x_{n}\}\) is bounded, so are \(\{y_{n}\}\) and \(\{z_{n}\}\). Let \(\lambda_{n}=\frac{x_{n+1}-\beta_{n}x_{n}}{1-\beta_{n}}\). It follows that Hence, we have Since \(z_{n}=T_{r_{n}}x_{n}\), we find that \(F(z_{n},z)+\frac{1}{r_{n}}\langle z-z_{n},z_{n}-x_{n}\rangle\geq0\), \(\forall z\in C \) and \(F(z_{n+1},z)+\frac{1}{r_{n+1}}\langle z-z_{n+1},z_{n+1}-x_{n+1}\rangle\geq0\), \(\forall z\in C\). It follows that \(F(z_{n},z_{n+1})+\frac{1}{r_{n}}\langle z_{n+1}-z_{n},z_{n}-x_{n}\rangle\geq0 \) and \(F(z_{n+1},z_{n})+\frac{1}{r_{n+1}}\langle z_{n}-z_{n+1},z_{n+1}-x_{n+1}\rangle\geq0\), \(\forall z\in C\). By using condition (A2), we find that \(\langle z_{n+1}-z_{n},\frac{z_{n}-x_{n}}{r_{n}}-\frac {z_{n+1}-x_{n+1}}{r_{n+1}}\rangle\geq0\). Hence, we have This implies that \(\langle z_{n+1}-z_{n}, x_{n+1}-x_{n} +(1-\frac {r_{n}}{r_{n+1}})(z_{n+1}-x_{n+1})\rangle\geq\|z_{n+1}-z_{n}\|^{2}\). Hence, we have It follows that This implies that Substituting (2.2) into (2.1), we find that It follows from conditions (a)-(e) that By using Lemma 1.5, we see that \(\lim_{n\rightarrow\infty}\|\lambda_{n}-x_{n}\|=0\), which in turn implies that Since \(T_{r_{n}}\) is firmly nonexpansive, we find that That is, It follows that where \(f_{n}=\|e_{n}\|(\|e_{n}\|+2\|z_{n}-p\|)\). This further implies that By using conditions (a) and (b), we find from (2.3) that Since A is α-inverse-strongly monotone, we find that It follows that This yields that By using (2.3), we find from conditions (a), (c) and (d) that Since \(P_{C}\) is firmly nonexpansive, we find that which yields that It follows that This in turn leads to By use of (2.3) and (2.5), we find from restrictions (a), (b), (c) and (e) that On the other hand, we have By using conditions (a) and (b), we find from (2.3) that Since \(S_{n}\) is nonexpansive, we find that It follows from (2.4), (2.6) and (2.7) that Notice that By using condition (c), we find that Now, we are in a position to show \(\limsup_{n\rightarrow\infty}\langle f(q)-q,x_{n}-q\rangle\leq0\), where \(q=P_{\Omega}f(q)\). To show it, we can choose a subsequence \(\{x_{n_{i}}\}\) of \(\{x_{n}\}\) such that Since \(\{x_{n_{i}}\}\) is bounded, we can choose a subsequence \(\{x_{n_{i_{j}}}\}\) of \(\{x_{n_{i}}\}\) which converges weakly to some point \(\bar{x}\). We may assume, without loss of generality, that \(\{x_{n_{i}}\}\) converges weakly to \(\bar{x}\).

Next, we prove \(\bar{x}\in\Omega\). First, we show \(x\in \operatorname{EP}(F)\). Notice that By using condition (A2), we see that \(\frac{1}{r_{n}}\langle y-y_{n},y_{n}-x_{n}\rangle\geq F(y,y_{n})\), \(\forall y\in C\). Replacing n by \(n_{i}\), we arrive at By using (2.4) and (2.6), we find that \(\{y_{n_{i}}\}\) converges weakly to \(\bar{x}\). It follows that \(0\geq F(y,\bar{x})\). For each t with \(0< t\leq1\), let \(z_{t}=tz+(1-t)\bar{x}\), where \(z\in C\). It follows that \(z_{t}\in C\). Hence, we have \(F(z_{t}, \bar{x})\leq0\). It follows that which yields that \(F(z_{t},z)\geq0\), \(\forall z\in C\). Letting \(t\downarrow0\), we obtain from condition (A3) that \(F(\bar{x},z)\geq0\), \(\forall z\in C\). This implies that \(\bar{x}\in \operatorname{EP}(F)\).

Next, we show that \(\bar{x}\in \operatorname{VI}(C,A)\). Let T be a maximal monotone mapping defined by For any given \((x,y)\in \operatorname{Graph}(T)\), we have \(y-Ax\in N_{C}x\). Since \(y_{n}\in C\), we have \(\langle x-y_{n}, y-Ax\rangle\geq0\). Since \(y_{n}=P_{C}(z_{n}-s_{n}Az_{n}+e_{n})\), we see that \(\langle x-y_{n}, y_{n}-(I-s_{n}A)z_{n}-e_{n}\rangle\geq0 \) and hence It follows that Since A is Lipschitz continuous, we see that \(\langle x-\bar{x},y\rangle\geq0\). Notice that T is maximal monotone and hence \(0\in T\bar{x}\). This shows that \(\bar{x}\in \operatorname{VI}(C,A)\). By using Lemma 1.2, we find that \(\bar {x}\in F(S)\). It follows that \(\limsup_{n\rightarrow\infty}\langle f(q)-q,x_{n}-q\rangle\leq0\), It follows that where \(K=\sup_{n\geq1}\{\|x_{n}-q\|\}\). By using Lemma 1.6, we find that \(\lim_{n\rightarrow\infty}\|x_{n}-q\| =0\). This completes the proof. □

Let C be a closed convex subset of a real Hilbert space H. Let \(A:C\rightarrow H\) be an α-inverse-strongly monotone mapping, and let F be a bifunction from \(C\times C\) to ℝ which satisfies (A1)-(A4). Let \(S:C\rightarrow H\) be nonexpansive, and let f be a β-contraction on H. Assume that \(\Omega=F(S)\cap \operatorname{VI}(C,A)\cap \operatorname{EP} (F)\neq\emptyset\). Let \(\{ r_{n}\}\) and \(\{s_{n}\}\) be positive real number sequences. Let \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\) and \(\{\gamma_{n}\}\) be real number sequences in \((0,1)\) such that \(\alpha_{n}+\beta_{n}+\gamma_{n}=1\). Let \(\{x_{n}\}\) be a sequence generated in the following process: where \(\{e_{n}\}\) is a sequence in H. Assume that the control sequences satisfy the following conditions: where s, \(s'\) are real constants. Then \(\{x_{n}\}\) converges strongly to q, which is also a unique solution to the variational inequality

Let C be a closed convex subset of a real Hilbert space H. Let \(A:C\rightarrow H\) be an α-inverse-strongly monotone mapping, and let F be a bifunction from \(C\times C\) to ℝ which satisfies (A1)-(A4). Let f be a β-contraction on H. Assume that \(\Omega= \operatorname{VI}(C,A)\cap \operatorname{EP} (F)\neq\emptyset\). Let \(\{r_{n}\}\) and \(\{s_{n}\}\) be positive real number sequences. Let \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\) and \(\{\gamma_{n}\}\) be real number sequences in \((0,1)\) such that \(\alpha_{n}+\beta_{n}+\gamma_{n}=1\). Let \(\{x_{n}\}\) be a sequence generated in the following process: where \(\{e_{n}\}\) is a sequence in H. Assume that the control sequences satisfy the following conditions: where s, \(s'\) are real constants. Then \(\{x_{n}\}\) converges strongly to q, which is also a unique solution to the variational inequality

Let C be a closed convex subset of a real Hilbert space H. Let \(A:C\rightarrow H\) be an α-inverse-strongly monotone mapping. Let \(S:C\rightarrow H\) be a κ-strictly pseudocontractive mapping, and let f be a β-contraction on H. Assume that \(\Omega=F(S)\cap \operatorname{VI}(C,A)\neq\emptyset\). Let \(\{s_{n}\}\) be a positive real number sequence. Let \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), \(\{\gamma_{n}\}\) and \(\{\delta_{n}\}\) be real number sequences in \((0,1)\) such that \(\alpha_{n}+\beta_{n}+\gamma_{n}=1\). Let \(\{x_{n}\}\) be a sequence generated in the following process: where \(\{e_{n}\}\) is a sequence in H. Assume that the control sequences satisfy the following conditions: where δ, s, \(s'\) are real constants. Then \(\{x_{n}\}\) converges strongly to q, which is also a unique solution to the variational inequality

Let C be a closed convex subset of a real Hilbert space H. Let F be a bifunction from \(C\times C\) to ℝ which satisfies (A1)-(A4). Let \(S:C\rightarrow H\) be a κ-strict pseudocontraction, and let f be a β-contraction on H. Assume that \(\Omega=F(S)\cap \operatorname{EP} (F)\neq\emptyset\). Let \(\{r_{n}\}\) be a positive real number sequence. Let \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\), \(\{\gamma_{n}\}\) and \(\{\delta_{n}\}\) be real number sequences in \((0,1)\) such that \(\alpha_{n}+\beta_{n}+\gamma_{n}=1\). Let \(\{x_{n}\}\) be a sequence generated in the following process: where \(\{e_{n}\}\) is a sequence in H. Assume that the control sequences satisfy the following conditions: where δ is a real constant. Then \(\{x_{n}\}\) converges strongly to q, which is also a unique solution to the variational inequality

Comparing Theorem 2.1, Corollaries 2.2-2.5 with Theorems IT, TT and the corresponding results in [6‐20], we have the following.

Let C be a closed convex subset of a real Hilbert space H. Let \(A:C\rightarrow H\) be an α-inverse-strongly monotone mapping, and let f be a β-contraction on H. Assume that \(\operatorname{VI}(C,A)\neq\emptyset\). Let \(\{s_{n}\}\) be a positive real number sequence. Let \(\{\alpha_{n}\}\), \(\{\beta_{n}\}\) and \(\{\gamma_{n}\}\) be real number sequences in \((0,1)\) such that \(\alpha_{n}+\beta_{n}+\gamma_{n}=1\). Let \(\{x_{n}\}\) be a sequence generated in \(x_{1} \in C\), \(x_{n+1} = \alpha_{n} f(x_{n}) + \beta_{n} x_{n} + \gamma_{n} y_{n}\), where \(y_{n} = P_{C}(x_{n} -s_{n} Ax_{n} + e_{n})\), where \(\{e_{n}\}\) is a sequence in H. Assume that the control sequences satisfy the following conditions: where s and \(s'\) are real constants. Then \(\{x_{n}\}\) converges strongly to \(q=P_{\operatorname{VI}(C,A)}f(q)\).

To construct a mathematical model which is as close as possible to a real complex problem, we often have to use more than one constraint. Solving such problems, we have to obtain some solution which is simultaneously the solution of two or more subproblems. This is a common problem in diverse areas of mathematics and physical sciences. It consists of trying to find a solution satisfying certain constraints. In this paper, we investigate the problem of solving common solutions of an equilibrium problem, a variational inequality problem and a fixed point problem of a strictly pseudocontractive mapping based on a regularization projection algorithm. Possible computation errors are also taken into account. Strong convergence theorems are established without compact assumptions and additional metric projections in the framework of real Hilbert spaces.