Abstract

It is well known that the gradient-projection algorithm (GPA) is very useful in solving constrained convex minimization problems. In this paper, we combine a general iterative method with the gradient-projection algorithm to propose a hybrid gradient-projection algorithm and prove that the sequence generated by the hybrid gradient-projection algorithm converges in norm to a minimizer of constrained convex minimization problems which solves a variational inequality.

1. Introduction

Let š» be a real Hilbert space and š¶ a nonempty closed and convex subset of š». Consider the following constrained convex minimization problem: minimizeš‘„āˆˆš¶š‘“(š‘„),(1.1) where š‘“āˆ¶š¶ā†’ā„ is a real-valued convex and continuously FrĆ©chet differentiable function. The gradient āˆ‡š‘“ satisfies the following Lipschitz condition: ā€–āˆ‡š‘“(š‘„)āˆ’āˆ‡š‘“(š‘¦)ā€–ā‰¤šæā€–š‘„āˆ’š‘¦ā€–,āˆ€š‘„,š‘¦āˆˆš¶,(1.2) where šæ>0. Assume that the minimization problem (1.1) is consistent, and let š‘† denote its solution set.

It is well known that the gradient-projection algorithm is very useful in dealing with constrained convex minimization problems and has extensively been studied ([1ā€“5] and the references therein). It has recently been applied to solve split feasibility problems [6ā€“10]. Levitin and Polyak [1] consider the following gradient-projection algorithm: š‘„š‘›+1āˆ¶=Projš¶ī€·š‘„š‘›āˆ’šœ†š‘›ī€·š‘„āˆ‡š‘“š‘›ī€øī€ø,š‘›ā‰„0.(1.3) Let {šœ†š‘›}āˆžš‘›=0 satisfy 0<liminfš‘›ā†’āˆžšœ†š‘›ā‰¤limsupš‘›ā†’āˆžšœ†š‘›<2šæ.(1.4) It is proved that the sequence {š‘„š‘›} generated by (1.3) converges weakly to a minimizer of (1.1).

Xu proved that under certain appropriate conditions on {š›¼š‘›} and {šœ†š‘›} the sequence {š‘„š‘›} defined by the following relaxed gradient-projection algorithm: š‘„š‘›+1=ī€·1āˆ’š›¼š‘›ī€øš‘„š‘›+š›¼š‘›Projš¶ī€·š‘„š‘›āˆ’šœ†š‘›ī€·š‘„āˆ‡š‘“š‘›ī€øī€ø,š‘›ā‰„0,(1.5) converges weakly to a minimizer of (1.1) [11].

Since the Lipschitz continuity of the gradient of š‘“ implies that it is indeed inverse strongly monotone (ism) [12, 13], its complement can be an averaged mapping. Recall that a mapping š‘‡ is nonexpansive if and only if it is Lipschitz with Lipschitz constant not more than one, that a mapping is an averaged mapping if and only if it can be expressed as a proper convex combination of the identity mapping and a nonexpansive mapping, and that a mapping š‘‡ is said to be šœˆ-inverse strongly monotone if and only if āŸØš‘„āˆ’š‘¦,š‘‡š‘„āˆ’š‘‡š‘¦āŸ©ā‰„šœˆā€–š‘‡š‘„āˆ’š‘‡š‘¦ā€–2forallš‘„,š‘¦āˆˆš», where the number šœˆ>0. Recall also that the composite of finitely many averaged mappings is averaged. That is, if each of the mappings {š‘‡š‘–}š‘š‘–=1 is averaged, then so is the composite š‘‡1ā‹Æš‘‡š‘ [14]. In particular, an averaged mapping is a nonexpansive mapping [15]. As a result, the GPA can be rewritten as the composite of a projection and an averaged mapping which is again an averaged mapping.

Generally speaking, in infinite-dimensional Hilbert spaces, GPA has only weak convergence. Xu [11] provided a modification of GPA so that strong convergence is guaranteed. He considered the following hybrid gradient-projection algorithm: š‘„š‘›+1=šœƒš‘›ā„Žī€·š‘„š‘›ī€ø+ī€·1āˆ’šœƒš‘›ī€øProjš¶ī€·š‘„š‘›āˆ’šœ†š‘›ī€·š‘„āˆ‡š‘“š‘›ī€øī€ø.(1.6)

It is proved that if the sequences {šœƒš‘›} and {šœ†š‘›} satisfy appropriate conditions, the sequence {š‘„š‘›} generated by (1.6) converges in norm to a minimizer of (1.1) which solves the variational inequality š‘„āˆ—āˆˆš‘†,āŸØ(š¼āˆ’ā„Ž)š‘„āˆ—,š‘„āˆ’š‘„āˆ—āŸ©ā‰„0,š‘„āˆˆš‘†.(1.7)

On the other hand, Ming Tian [16] introduced the following general iterative algorithm for solving the variational inequality š‘„š‘›+1=š›¼š‘›ī€·š‘„š›¾š‘“š‘›ī€ø+ī€·š¼āˆ’šœ‡š›¼š‘›š¹ī€øš‘‡š‘„š‘›,š‘›ā‰„0,(1.8) where š¹ is a šœ…-Lipschitzian and šœ‚-strongly monotone operator with šœ…>0, šœ‚>0 and š‘“ is a contraction with coefficient 0<š›¼<1. Then, he proved that if {š›¼š‘›} satisfying appropriate conditions, the {š‘„š‘›} generated by (1.8) converges strongly to the unique solution of variational inequality āŸØ(šœ‡š¹āˆ’š›¾š‘“)Ģƒš‘„,Ģƒš‘„āˆ’š‘§āŸ©ā‰¤0,š‘§āˆˆFix(š‘‡).(1.9)

In this paper, motivated and inspired by the research work in this direction, we will combine the iterative method (1.8) with the gradient-projection algorithm (1.3) and consider the following hybrid gradient-projection algorithm: š‘„š‘›+1=šœƒš‘›ī€·š‘„š›¾ā„Žš‘›ī€ø+ī€·š¼āˆ’šœ‡šœƒš‘›š¹ī€øProjš¶ī€·š‘„š‘›āˆ’šœ†š‘›ī€·š‘„āˆ‡š‘“š‘›ī€øī€ø,š‘›ā‰„0.(1.10)

We will prove that if the sequence {šœƒš‘›} of parameters and the sequence {šœ†š‘›} of parameters satisfy appropriate conditions, then the sequence {š‘„š‘›} generated by (1.10) converges in norm to a minimizer of (1.1) which solves the variational inequality (š‘‰š¼)š‘„āˆ—āˆˆš‘†,āŸØ(šœ‡š¹āˆ’š›¾ā„Ž)š‘„āˆ—,š‘„āˆ’š‘„āˆ—āŸ©ā‰„0,āˆ€š‘„āˆˆš‘†,(1.11) where š‘† is the solution set of the minimization problem (1.1).

2. Preliminaries

This section collects some lemmas which will be used in the proofs for the main results in the next section. Some of them are known; others are not hard to derive.

Throughout this paper, we write š‘„š‘›ā‡€š‘„ to indicate that the sequence {š‘„š‘›} converges weakly to š‘„, š‘„š‘›ā†’š‘„ implies that {š‘„š‘›} converges strongly to š‘„. šœ”š‘¤(š‘„š‘›)āˆ¶={š‘„āˆ¶āˆƒš‘„š‘›š‘—ā‡€š‘„} is the weak šœ”-limit set of the sequence {š‘„š‘›}āˆžš‘›=1.

Lemma 2.1 (see [17]). Assume that {š‘Žš‘›}āˆžš‘›=0 is a sequence of nonnegative real numbers such that š‘Žš‘›+1ā‰¤ī€·1āˆ’š›¾š‘›ī€øš‘Žš‘›+š›¾š‘›š›æš‘›+š›½š‘›,š‘›ā‰„0,(2.1) where {š›¾š‘›}āˆžš‘›=0 and {š›½š‘›}āˆžš‘›=0 are sequences in [0,1] and {š›æš‘›}āˆžš‘›=0 is a sequence in ā„ such that(i)āˆ‘āˆžš‘›=0š›¾š‘›=āˆž; (ii)either limsupš‘›ā†’āˆžš›æš‘›ā‰¤0 or āˆ‘āˆžš‘›=0š›¾š‘›|š›æš‘›|<āˆž;(iii)āˆ‘āˆžš‘›=0š›½š‘›<āˆž. Then limš‘›ā†’āˆžš‘Žš‘›=0.

Lemma 2.2 (see [18]). Let š¶ be a closed and convex subset of a Hilbert space š», and let š‘‡āˆ¶š¶ā†’š¶ be a nonexpansive mapping with Fixš‘‡ā‰ āˆ…. If {š‘„š‘›}āˆžš‘›=1 is a sequence in š¶ weakly converging to š‘„ and if {(š¼āˆ’š‘‡)š‘„š‘›}āˆžš‘›=1 converges strongly to y, then (š¼āˆ’T)š‘„=š‘¦.

Lemma 2.3. Let š» be a Hilbert space, and let š¶ be a nonempty closed and convex subset of š». ā„Žāˆ¶š¶ā†’š¶ a contraction with coefficient 0<šœŒ<1, and š¹āˆ¶š¶ā†’š¶ a šœ…-Lipschitzian continuous operator and šœ‚-strongly monotone operator with šœ…,šœ‚>0. Then, for 0<š›¾<šœ‡šœ‚/šœŒ, āŸØš‘„āˆ’š‘¦,(šœ‡š¹āˆ’š›¾ā„Ž)š‘„āˆ’(šœ‡š¹āˆ’š›¾ā„Ž)š‘¦āŸ©ā‰„(šœ‡šœ‚āˆ’š›¾šœŒ)ā€–š‘„āˆ’š‘¦ā€–2,āˆ€š‘„,š‘¦āˆˆš¶.(2.2) That is, šœ‡š¹āˆ’š›¾ā„Ž is strongly monotone with coefficient šœ‡šœ‚āˆ’š›¾šœŒ.

Lemma 2.4. Let š¶ be a closed subset of a real Hilbert space š», given š‘„āˆˆš» and š‘¦āˆˆš¶. Then, š‘¦=š‘ƒš¶š‘„ if and only if there holds the inequality āŸØš‘„āˆ’š‘¦,š‘¦āˆ’š‘§āŸ©ā‰„0,āˆ€š‘§āˆˆš¶.(2.3)

3. Main Results

Let š» be a real Hilbert space, and let š¶ be a nonempty closed and convex subset of š» such that š¶Ā±š¶āŠ‚š¶. Assume that the minimization problem (1.1) is consistent, and let š‘† denote its solution set. Assume that the gradient āˆ‡š‘“ satisfies the Lipschitz condition (1.2). Since š‘† is a closed convex subset, the nearest point projection from š» onto š‘† is well defined. Recall also that a contraction on š¶ is a self-mapping of š¶ such that ā€–ā„Ž(š‘„)āˆ’ā„Ž(š‘¦)ā€–ā‰¤šœŒā€–š‘„āˆ’š‘¦ā€–,forallš‘„,š‘¦āˆˆš¶, where šœŒāˆˆ[0,1) is a constant. Let š¹ be a šœ…-Lipschitzian and šœ‚-strongly monotone operator on š¶ with šœ…,šœ‚>0. Denote by Ī  the collection of all contractions on š¶, namely, Ī ={ā„Žāˆ¶ā„Žisacontractiononš¶}.(3.1) Now given ā„ŽāˆˆĪ  with 0<šœŒ<1, š‘ āˆˆ(0,1). Let 0<šœ‡<2šœ‚/šœ…2,ā€‰ā€‰ 0<š›¾<šœ‡(šœ‚āˆ’(šœ‡šœ…2)/2)/šœŒ=šœ/šœŒ. Assume that šœ†š‘  with respect to š‘  is continuous and, in addition, šœ†š‘ āˆˆ[š‘Ž,š‘]āŠ‚(0,2/šæ). Consider a mapping š‘‹š‘  on š¶ defined by š‘‹š‘ (š‘„)=š‘ š›¾ā„Ž(š‘„)+(š¼āˆ’š‘ šœ‡š¹)Projš¶ī€·š¼āˆ’šœ†š‘ ī€øāˆ‡š‘“(š‘„),š‘„āˆˆš¶.(3.2) It is easy to see that š‘‹š‘  is a contraction. Setting š‘‰š‘ āˆ¶=Projš¶(š¼āˆ’šœ†š‘ āˆ‡š‘“). It is obvious that š‘‰š‘  is a nonexpansive mapping. We can rewrite š‘‹š‘ (š‘„) as š‘‹š‘ (š‘„)=š‘ š›¾ā„Ž(š‘„)+(š¼āˆ’š‘ šœ‡š¹)š‘‰š‘ (š‘„).(3.3) First observe that for š‘ āˆˆ(0,1), we can get ā€–ā€–(š¼āˆ’š‘ šœ‡š¹)š‘‰š‘ (š‘„)āˆ’(š¼āˆ’sšœ‡š¹)š‘‰š‘ ā€–ā€–(š‘¦)2=ā€–ā€–š‘‰š‘ (š‘„)āˆ’š‘‰š‘ ī€·(š‘¦)āˆ’š‘ šœ‡š¹š‘‰š‘ (š‘„)āˆ’š¹š‘‰š‘ ī€øā€–ā€–(š‘¦)2=ā€–ā€–š‘‰š‘ (š‘„)āˆ’š‘‰š‘ ā€–ā€–(š‘¦)2āˆ’2š‘ šœ‡āŸØš‘‰š‘ (š‘„)āˆ’š‘‰š‘ (š‘¦),š¹š‘‰š‘ (š‘„)āˆ’š¹š‘‰š‘ (š‘¦)āŸ©+š‘ 2šœ‡2ā€–ā€–š¹š‘‰š‘ (š‘„)āˆ’š¹š‘‰š‘ ā€–ā€–(š‘¦)2ā‰¤ā€–š‘„āˆ’š‘¦ā€–2ā€–ā€–š‘‰āˆ’2š‘ šœ‡šœ‚š‘ (š‘„)āˆ’š‘‰š‘ ā€–ā€–(š‘¦)2+š‘ 2šœ‡2šœ…2ā€–ā€–š‘‰š‘ (š‘„)āˆ’š‘‰š‘ ā€–ā€–(š‘¦)2ā‰¤ī€·ī€·1āˆ’š‘ šœ‡2šœ‚āˆ’š‘ šœ‡šœ…2ī€øī€øā€–š‘„āˆ’š‘¦ā€–2ā‰¤īƒ©ī€·1āˆ’š‘ šœ‡2šœ‚āˆ’š‘ šœ‡šœ…2ī€ø2īƒŖ2ā€–š‘„āˆ’š‘¦ā€–2ā‰¤ī‚µī‚µ1āˆ’š‘ šœ‡šœ‚āˆ’šœ‡šœ…22ī‚¶ī‚¶2ā€–š‘„āˆ’š‘¦ā€–2=(1āˆ’š‘ šœ)2ā€–š‘„āˆ’š‘¦ā€–2.(3.4) Indeed, we have ā€–ā€–š‘‹š‘ (š‘„)āˆ’š‘‹š‘ ā€–ā€–=ā€–ā€–(š‘¦)š‘ š›¾ā„Ž(š‘„)+(š¼āˆ’š‘ šœ‡š¹)š‘‰š‘ (š‘„)āˆ’š‘ š›¾ā„Ž(š‘¦)āˆ’(š¼āˆ’š‘ šœ‡š¹)š‘‰š‘ ā€–ā€–ā€–ā€–(š‘¦)ā‰¤š‘ š›¾ā€–ā„Ž(š‘„)āˆ’ā„Ž(š‘¦)ā€–+(š¼āˆ’š‘ šœ‡š¹)š‘‰š‘ (š‘„)āˆ’(š¼āˆ’š‘ šœ‡š¹)š‘‰š‘ ā€–ā€–(š‘¦)ā‰¤š‘ š›¾šœŒā€–š‘„āˆ’š‘¦ā€–+(1āˆ’š‘ šœ)ā€–š‘„āˆ’š‘¦ā€–=(1āˆ’š‘ (šœāˆ’š›¾šœŒ))ā€–š‘„āˆ’š‘¦ā€–.(3.5) Hence, š‘‹š‘  has a unique fixed point, denoted š‘„š‘ , which uniquely solves the fixed-point equation š‘„š‘ ī€·š‘„=š‘ š›¾ā„Žš‘ ī€ø+(š¼āˆ’š‘ šœ‡š¹)š‘‰š‘ ī€·š‘„š‘ ī€ø.(3.6) The next proposition summarizes the properties of {š‘„š‘ }.

Proposition 3.1. Let š‘„š‘  be defined by (3.6).(i){š‘„š‘ }isboundedforš‘ āˆˆ(0,(1/šœ)). (ii)limš‘ ā†’0ā€–š‘„š‘ āˆ’Projš¶(š¼āˆ’šœ†š‘ āˆ‡š‘“)(š‘„š‘ )ā€–=0. (iii)š‘„š‘ deļ¬nesacontinuouscurvefrom(0,1/šœ)intoš».

Proof. (i) Take a š‘„āˆˆš‘†, then we have ā€–ā€–š‘„š‘ āˆ’š‘„ā€–ā€–=ā€–ā€–ī€·š‘„š‘ š›¾ā„Žš‘ ī€ø+(š¼āˆ’š‘ šœ‡š¹)Projš¶ī€·š¼āˆ’šœ†š‘ š‘„āˆ‡š‘“ī€øī€·š‘ ī€øāˆ’š‘„ā€–ā€–=ā€–ā€–(š¼āˆ’š‘ šœ‡š¹)Projš¶ī€·š¼āˆ’šœ†š‘ š‘„āˆ‡š‘“ī€øī€·š‘ ī€øāˆ’(š¼āˆ’š‘ šœ‡š¹)Projš¶ī€·š¼āˆ’šœ†š‘ āˆ‡š‘“ī€øī€·š‘„ī€øī€·ī€·š‘„+š‘ š›¾ā„Žš‘ ī€øāˆ’šœ‡š¹Projš¶ī€·š¼āˆ’šœ†š‘ āˆ‡š‘“ī€øī€·š‘„ā€–ā€–ā€–ā€–š‘„ī€øī€øā‰¤(1āˆ’š‘ šœ)š‘ āˆ’š‘„ā€–ā€–ā€–ā€–ī€·š‘„+š‘ š›¾ā„Žš‘ ī€øī€·āˆ’šœ‡š¹š‘„ī€øā€–ā€–ā‰¤ā€–ā€–š‘„(1āˆ’š‘ šœ)š‘ āˆ’š‘„ā€–ā€–ā€–ā€–š‘„+š‘ š›¾šœŒš‘ āˆ’š‘„ā€–ā€–ā€–ā€–ī€·+š‘ š›¾ā„Žš‘„ī€øī€·āˆ’šœ‡š¹š‘„ī€øā€–ā€–.(3.7) It follows that ā€–ā€–š‘„š‘ āˆ’š‘„ā€–ā€–ā‰¤ā€–ā€–ī€·š›¾ā„Žš‘„ī€øī€·āˆ’šœ‡š¹š‘„ī€øā€–ā€–.šœāˆ’š›¾šœŒ(3.8) Hence, {š‘„š‘ } is bounded.
(ii) By the definition of {š‘„š‘ }, we have ā€–ā€–š‘„š‘ āˆ’Projš¶ī€·š¼āˆ’šœ†š‘ š‘„āˆ‡š‘“ī€øī€·š‘ ī€øā€–ā€–=ā€–ā€–ī€·š‘„š‘ š›¾ā„Žš‘ ī€ø+(š¼āˆ’š‘ šœ‡š¹)Projš¶ī€·š¼āˆ’šœ†š‘ š‘„āˆ‡š‘“ī€øī€·š‘ ī€øāˆ’Projš¶ī€·š¼āˆ’šœ†š‘ š‘„āˆ‡š‘“ī€øī€·š‘ ī€øā€–ā€–ā€–ā€–ī€·š‘„=š‘ š›¾ā„Žš‘ ī€øāˆ’šœ‡š¹Projš¶ī€·š¼āˆ’šœ†š‘ š‘„āˆ‡š‘“ī€øī€·sī€øā€–ā€–āŸ¶0,(3.9){š‘„š‘ } is bounded, so are {ā„Ž(š‘„š‘ )} and {š¹Projš¶(š¼āˆ’šœ†š‘ āˆ‡š‘“)(š‘„š‘ )}.
(iii) Take š‘ , š‘ 0āˆˆ(0,1/šœ), and we have ā€–ā€–š‘„š‘ āˆ’š‘„š‘ 0ā€–ā€–=ā€–ā€–ī€·š‘„š‘ š›¾ā„Žš‘ ī€ø+(š¼āˆ’š‘ šœ‡š¹)Projš¶ī€·š¼āˆ’šœ†š‘ š‘„āˆ‡š‘“ī€øī€·š‘ ī€øāˆ’š‘ 0ī€·š‘„š›¾ā„Žš‘ 0ī€øāˆ’ī€·š¼āˆ’š‘ 0ī€øšœ‡š¹Projš¶ī€·š¼āˆ’šœ†š‘ 0š‘„āˆ‡š‘“ī€øī€·š‘ 0ī€øā€–ā€–ā‰¤ā€–ā€–ī€·š‘ āˆ’š‘ 0ī€øī€·š‘„š›¾ā„Žš‘ ī€ø+š‘ 0š›¾ī€·ā„Žī€·š‘„š‘ ī€øī€·š‘„āˆ’ā„Žš‘ 0ā€–ā€–+ā€–ā€–ī€·ī€øī€øš¼āˆ’š‘ 0ī€øšœ‡š¹Projš¶ī€·š¼āˆ’šœ†š‘ 0š‘„āˆ‡š‘“ī€øī€·š‘ ī€øāˆ’ī€·š¼āˆ’š‘ 0ī€øšœ‡š¹Projš¶ī€·š¼āˆ’šœ†š‘ 0š‘„āˆ‡š‘“ī€øī€·š‘ 0ī€øā€–ā€–+ā€–ā€–(š¼āˆ’š‘ šœ‡š¹)Projš¶ī€·š¼āˆ’šœ†š‘ š‘„āˆ‡š‘“ī€øī€·š‘ ī€øāˆ’ī€·š¼āˆ’š‘ 0ī€øšœ‡š¹Projš¶ī€·š¼āˆ’šœ†š‘ 0š‘„āˆ‡š‘“ī€øī€·š‘ ī€øā€–ā€–ā‰¤ā€–ā€–ī€·š‘ āˆ’š‘ 0ī€øī€·š‘„š›¾ā„Žš‘ ī€ø+š‘ 0š›¾ī€·ā„Žī€·š‘„š‘ ī€øī€·š‘„āˆ’ā„Žš‘ 0ā€–ā€–+ā€–ā€–ī€·ī€øī€øš¼āˆ’š‘ 0ī€øšœ‡š¹Projš¶ī€·š¼āˆ’šœ†š‘ 0š‘„āˆ‡š‘“ī€øī€·š‘ ī€øāˆ’ī€·š¼āˆ’š‘ 0ī€øšœ‡š¹Projš¶ī€·š¼āˆ’šœ†š‘ 0š‘„āˆ‡š‘“ī€øī€·š‘ 0ī€øā€–ā€–+ā€–ā€–(š¼āˆ’š‘ šœ‡š¹)Projš¶ī€·š¼āˆ’šœ†š‘ š‘„āˆ‡š‘“ī€øī€·š‘ ī€øāˆ’(š¼āˆ’š‘ šœ‡š¹)Projš¶ī€·š¼āˆ’šœ†š‘ 0š‘„āˆ‡š‘“ī€øī€·š‘ ī€øā€–ā€–+ā€–ā€–(š¼āˆ’š‘ šœ‡š¹)Projš¶ī€·š¼āˆ’šœ†š‘ 0š‘„āˆ‡š‘“ī€øī€·š‘ ī€øāˆ’ī€·š¼āˆ’š‘ 0ī€øšœ‡š¹Projš¶ī€·š¼āˆ’šœ†š‘ 0š‘„āˆ‡š‘“ī€øī€·š‘ ī€øā€–ā€–ā‰¤||š‘ āˆ’š‘ 0||š›¾ā€–ā€–ā„Žī€·š‘„š‘ ī€øā€–ā€–+š‘ 0ā€–ā€–š‘„š›¾šœŒš‘ āˆ’š‘„š‘ 0ā€–ā€–+ī€·1āˆ’š‘ 0šœī€øā€–ā€–š‘„š‘ āˆ’š‘„š‘ 0ā€–ā€–+||šœ†š‘ āˆ’šœ†š‘ 0||ā€–ā€–ī€·š‘„āˆ‡š‘“š‘ ī€øā€–ā€–+ā€–ā€–š‘ šœ‡š¹Projš¶ī€·š¼āˆ’šœ†š‘ 0š‘„āˆ‡š‘“ī€øī€·š‘ ī€øāˆ’š‘ 0šœ‡š¹Projš¶ī€·š¼āˆ’šœ†š‘ 0š‘„āˆ‡š‘“ī€øī€·š‘ ī€øā€–ā€–=||š‘ āˆ’š‘ 0||š›¾ā€–ā€–ā„Žī€·š‘„š‘ ī€øā€–ā€–+š‘ 0ā€–ā€–š‘„š›¾šœŒš‘ āˆ’š‘„š‘ 0ā€–ā€–+ī€·1āˆ’š‘ 0šœī€øā€–ā€–š‘„š‘ āˆ’š‘„š‘ 0ā€–ā€–+||šœ†š‘ āˆ’šœ†š‘ 0||ā€–ā€–ī€·š‘„āˆ‡š‘“š‘ ī€øā€–ā€–+||š‘ āˆ’š‘ 0||ā€–ā€–šœ‡š¹Projš¶ī€·š¼āˆ’šœ†š‘ 0š‘„āˆ‡š‘“ī€øī€·š‘ ī€øā€–ā€–=ī€·š›¾ā€–ā€–ā„Žī€·š‘„š‘ ī€øā€–ā€–ā€–ā€–+šœ‡š¹Projš¶ī€·š¼āˆ’šœ†š‘ 0š‘„āˆ‡š‘“ī€øī€·š‘ ī€øā€–ā€–ī€ø||š‘ āˆ’š‘ 0||+š‘ 0ā€–ā€–š‘„š›¾šœŒš‘ āˆ’š‘„š‘ 0ā€–ā€–+ī€·1āˆ’š‘ 0šœī€øā€–ā€–š‘„š‘ āˆ’š‘„š‘ 0ā€–ā€–+||šœ†š‘ āˆ’šœ†š‘ 0||ā€–ā€–ī€·š‘„āˆ‡š‘“š‘ ī€øā€–ā€–.(3.10) Therefore, ā€–ā€–š‘„š‘ āˆ’š‘„š‘ 0ā€–ā€–ā‰¤š›¾ā€–ā€–ā„Žī€·š‘„š‘ ī€øā€–ā€–ā€–ā€–+šœ‡š¹Projš¶ī€·š¼āˆ’šœ†š‘ 0š‘„āˆ‡š‘“ī€øī€·š‘ ī€øā€–ā€–š‘ 0||(šœāˆ’š›¾šœŒ)š‘ āˆ’š‘ 0||+ā€–ā€–ī€·š‘„āˆ‡š‘“š‘ ī€øā€–ā€–š‘ 0||šœ†(šœāˆ’š›¾šœŒ)š‘ āˆ’šœ†š‘ 0||.(3.11) Therefore, š‘„š‘ ā†’š‘„š‘ 0 as š‘ ā†’š‘ 0. This means š‘„š‘  is continuous.

Our main result in the following shows that {š‘„š‘ } converges in norm to a minimizer of (1.1) which solves some variational inequality.

Theorem 3.2. Assume that {š‘„š‘ } is defined by (3.6), then š‘„š‘  converges in norm as š‘ ā†’0 to a minimizer of (1.1) which solves the variational inequality āŸØ(šœ‡š¹āˆ’š›¾ā„Ž)š‘„āˆ—,Ģƒš‘„āˆ’š‘„āˆ—āŸ©ā‰„0,āˆ€Ģƒš‘„āˆˆš‘†.(3.12) Equivalently, we have Projš‘ (š¼āˆ’(šœ‡š¹āˆ’š›¾ā„Ž))š‘„āˆ—=š‘„āˆ—.

Proof. It is easy to see that the uniqueness of a solution of the variational inequality (3.12). By Lemma 2.3, šœ‡š¹āˆ’š›¾ā„Ž is strongly monotone, so the variational inequality (3.12) has only one solution. Let š‘„āˆ—āˆˆš‘† denote the unique solution of (3.12).
To prove that š‘„š‘ ā†’š‘„āˆ—(š‘ ā†’0), we write, for a given Ģƒš‘„āˆˆš‘†, š‘„š‘ ī€·š‘„āˆ’Ģƒš‘„=š‘ š›¾ā„Žš‘ ī€ø+(š¼āˆ’š‘ šœ‡š¹)Projš¶ī€·š¼āˆ’šœ†š‘ š‘„āˆ‡š‘“ī€øī€·š‘ ī€øī€·ī€·š‘„āˆ’Ģƒš‘„=š‘ š›¾ā„Žš‘ ī€øī€øāˆ’šœ‡š¹Ģƒš‘„+(š¼āˆ’š‘ šœ‡š¹)Projš¶ī€·š¼āˆ’šœ†š‘ š‘„āˆ‡š‘“ī€øī€·š‘ ī€øāˆ’(š¼āˆ’š‘ šœ‡š¹)Projš¶ī€·š¼āˆ’šœ†š‘ ī€øāˆ‡š‘“(Ģƒš‘„).(3.13) It follows that ā€–ā€–š‘„š‘ ā€–ā€–āˆ’Ģƒš‘„2ī«ī€·š‘„=š‘ š›¾ā„Žš‘ ī€øāˆ’šœ‡š¹Ģƒš‘„,š‘„š‘ ī¬+ī«āˆ’Ģƒš‘„(š¼āˆ’š‘ šœ‡š¹)Projš¶ī€·š¼āˆ’šœ†š‘ š‘„āˆ‡š‘“ī€øī€·š‘ ī€øāˆ’(š¼āˆ’š‘ šœ‡š¹)Projš¶ī€·š¼āˆ’šœ†š‘ ī€øāˆ‡š‘“(Ģƒš‘„),š‘„š‘ ī¬ā€–ā€–š‘„āˆ’Ģƒš‘„ā‰¤(1āˆ’š‘ šœ)š‘ ā€–ā€–āˆ’Ģƒš‘„2ī«ī€·š‘„+š‘ š›¾ā„Žš‘ ī€øāˆ’šœ‡š¹Ģƒš‘„,š‘„š‘ ī¬.āˆ’Ģƒš‘„(3.14) Hence, ā€–ā€–š‘„š‘ ā€–ā€–āˆ’Ģƒš‘„2ā‰¤1šœī«ī€·š‘„š›¾ā„Žš‘ ī€øāˆ’šœ‡š¹Ģƒš‘„,š‘„š‘ ī¬ā‰¤1āˆ’Ģƒš‘„šœī‚†ā€–ā€–š‘„š›¾šœŒš‘ ā€–ā€–āˆ’Ģƒš‘„2+āŸØš›¾ā„Ž(Ģƒš‘„)āˆ’šœ‡š¹Ģƒš‘„,š‘„š‘ ī‚‡.āˆ’Ģƒš‘„āŸ©(3.15) To derive that ā€–ā€–š‘„š‘ ā€–ā€–āˆ’Ģƒš‘„2ā‰¤1šœāˆ’š›¾šœŒāŸØš›¾ā„Ž(Ģƒš‘„)āˆ’šœ‡š¹Ģƒš‘„,š‘„š‘ āˆ’Ģƒš‘„āŸ©.(3.16) Since {š‘„š‘ } is bounded as š‘ ā†’0, we see that if {š‘ š‘›} is a sequence in (0,1) such that š‘ š‘›ā†’0 and š‘„š‘ š‘›ā‡€š‘„, then by (3.16), š‘„š‘ š‘›ā†’š‘„. We may further assume that šœ†š‘ š‘›ā†’šœ†āˆˆ[0,2/šæ] due to condition (1.4). Notice that Projš¶(š¼āˆ’šœ†āˆ‡š‘“) is nonexpansive. It turns out that ā€–ā€–š‘„š‘ š‘›āˆ’Projš¶(š¼āˆ’šœ†āˆ‡š‘“)š‘„š‘ š‘›ā€–ā€–ā‰¤ā€–ā€–š‘„š‘ š‘›āˆ’Projš¶ī€·š¼āˆ’šœ†š‘ š‘›ī€øš‘„āˆ‡š‘“š‘ š‘›ā€–ā€–+ā€–ā€–Projš¶ī€·š¼āˆ’šœ†š‘ š‘›ī€øš‘„āˆ‡š‘“š‘ š‘›āˆ’Projš¶(š¼āˆ’šœ†āˆ‡š‘“)š‘„š‘ š‘›ā€–ā€–ā‰¤ā€–ā€–š‘„š‘ š‘›āˆ’Projš¶ī€·š¼āˆ’šœ†š‘ š‘›ī€øš‘„āˆ‡š‘“š‘ š‘›ā€–ā€–+ā€–ā€–ī€·šœ†āˆ’šœ†š‘ š‘›ī€øī€·š‘„āˆ‡š‘“š‘ š‘›ī€øā€–ā€–=ā€–ā€–š‘„š‘ š‘›āˆ’Projš¶ī€·š¼āˆ’šœ†š‘ š‘›ī€øš‘„āˆ‡š‘“š‘ š‘›ā€–ā€–+||šœ†āˆ’šœ†š‘ š‘›||ā€–ā€–ī€·š‘„āˆ‡š‘“š‘ š‘›ī€øā€–ā€–.(3.17) From the boundedness of {š‘„š‘ } and limš‘ ā†’0ā€–Projš¶(š¼āˆ’šœ†š‘ āˆ‡š‘“)š‘„š‘ āˆ’š‘„š‘ ā€–=0, we conclude that limš‘›ā†’āˆžā€–ā€–š‘„š‘ š‘›āˆ’Projš¶(š¼āˆ’šœ†āˆ‡š‘“)š‘„š‘ š‘›ā€–ā€–=0.(3.18) Since š‘„š‘ š‘›ā‡€š‘„, by Lemma 2.2, we obtain š‘„=Projš¶(š¼āˆ’šœ†āˆ‡š‘“)š‘„.(3.19) This shows that š‘„āˆˆš‘†.
We next prove that š‘„ is a solution of the variational inequality (3.12). Since š‘„š‘ ī€·š‘„=š‘ š›¾ā„Žš‘ ī€ø+(š¼āˆ’š‘ šœ‡š¹)Projš¶ī€·š¼āˆ’šœ†š‘ š‘„āˆ‡š‘“ī€øī€·š‘ ī€ø,(3.20) we can derive that ī€·š‘„(šœ‡š¹āˆ’š›¾ā„Ž)š‘ ī€ø1=āˆ’š‘ ī€·š¼āˆ’Projš¶ī€·š¼āˆ’šœ†š‘ š‘„āˆ‡š‘“ī€øī€·š‘ ī€øī€·š¹ī€·š‘„+šœ‡š‘ ī€øāˆ’š¹Projš¶ī€·š¼āˆ’šœ†š‘ š‘„āˆ‡š‘“ī€øī€·š‘ .ī€øī€ø(3.21) Therefore, for Ģƒš‘„āˆˆš‘†, ī«ī€·š‘„(šœ‡š¹āˆ’š›¾ā„Ž)š‘ ī€ø,š‘„š‘ ī¬1āˆ’Ģƒš‘„=āˆ’š‘ ī«ī€·š¼āˆ’Projš¶ī€·š¼āˆ’šœ†š‘ š‘„āˆ‡š‘“ī€øī€øī€·š‘ ī€øāˆ’ī€·š¼āˆ’Projš¶ī€·š¼āˆ’šœ†š‘ āˆ‡š‘“ī€øī€ø(Ģƒš‘„),š‘„š‘ ī¬ī«š¹ī€·š‘„āˆ’Ģƒš‘„+šœ‡š‘ ī€øāˆ’š¹Projš¶ī€·š¼āˆ’šœ†š‘ š‘„āˆ‡š‘“ī€øī€·š‘ ī€ø,š‘„š‘ ī¬ī«š¹ī€·š‘„āˆ’Ģƒš‘„ā‰¤šœ‡š‘ ī€øāˆ’š¹Projš¶ī€·š¼āˆ’šœ†š‘ š‘„āˆ‡š‘“ī€øī€·š‘ ī€ø,š‘„š‘ ī¬.āˆ’Ģƒš‘„(3.22) Since Projš¶(š¼āˆ’šœ†š‘ āˆ‡š‘“) is nonexpansive, we obtain that š¼āˆ’Projš¶(š¼āˆ’šœ†š‘ āˆ‡š‘“) is monotone, that is, ī«ī€·š¼āˆ’Projš¶ī€·š¼āˆ’šœ†š‘ š‘„āˆ‡š‘“ī€øī€øī€·š‘ ī€øāˆ’ī€·š¼āˆ’Projš¶ī€·š¼āˆ’šœ†š‘ āˆ‡š‘“ī€øī€ø(Ģƒš‘„),š‘„š‘ ī¬āˆ’Ģƒš‘„ā‰„0.(3.23) Taking the limit through š‘ =š‘ š‘›ā†’0 ensures that š‘„ is a solution to (3.12). That is to say ī«ī€·(šœ‡š¹āˆ’š›¾ā„Ž)š‘„ī€ø,ī¬š‘„āˆ’Ģƒš‘„ā‰¤0.(3.24) Hence š‘„=š‘„āˆ— by uniqueness. Therefore, š‘„š‘ ā†’š‘„āˆ— as š‘ ā†’0. The variational inequality (3.12) can be written as āŸØ(š¼āˆ’šœ‡š¹+š›¾ā„Ž)š‘„āˆ—āˆ’š‘„āˆ—,Ģƒš‘„āˆ’š‘„āˆ—āŸ©ā‰¤0,āˆ€Ģƒš‘„āˆˆš‘†.(3.25) So, by Lemma 2.4, it is equivalent to the fixed-point equation š‘ƒš‘†(š¼āˆ’šœ‡š¹+š›¾ā„Ž)š‘„āˆ—=š‘„āˆ—.(3.26)

Taking š¹=š“, šœ‡=1 in Theorem 3.2, we get the following

Corollary 3.3. We have that {š‘„š‘ } converges in norm as š‘ ā†’0 to a minimizer of (1.1) which solves the variational inequality āŸØ(š“āˆ’š›¾ā„Ž)š‘„āˆ—,Ģƒš‘„āˆ’š‘„āˆ—āŸ©ā‰„0,āˆ€Ģƒš‘„āˆˆš‘†.(3.27) Equivalently, we have Projš‘ (š¼āˆ’(š“āˆ’š›¾ā„Ž))š‘„āˆ—=š‘„āˆ—.

Taking š¹=š¼, šœ‡=1, š›¾=1 in Theorem 3.2, we get the following.

Corollary 3.4. Let š‘§š‘ āˆˆš» be the unique fixed point of the contraction š‘§ā†¦š‘ ā„Ž(š‘§)+(1āˆ’š‘ )Projš¶(š¼āˆ’šœ†š‘ āˆ‡š‘“)(š‘§). Then, {š‘§š‘ } converges in norm as š‘ ā†’0 to the unique solution of the variational inequality āŸØ(š¼āˆ’ā„Ž)š‘„āˆ—,Ģƒš‘„āˆ’š‘„āˆ—āŸ©ā‰„0,āˆ€Ģƒš‘„āˆˆš‘†.(3.28)

Finally, we consider the following hybrid gradient-projection algorithm, ī‚»š‘„0š‘„āˆˆš¶arbitrarily,š‘›+1=šœƒš‘›ī€·š‘„š›¾ā„Žš‘›ī€ø+ī€·š¼āˆ’šœ‡šœƒš‘›š¹ī€øProjš¶ī€·š‘„š‘›āˆ’šœ†š‘›ī€·š‘„āˆ‡š‘“š‘›ī€øī€ø,āˆ€š‘›ā‰„0.(3.29) Assume that the sequence {šœ†š‘›}āˆžš‘›=0 satisfies the condition (1.4) and, in addition, that the following conditions are satisfied for {šœ†š‘›}āˆžš‘›=0 and {šœƒš‘›}āˆžš‘›=0āŠ‚[0,1]:(i)šœƒš‘›ā†’0; (ii)āˆ‘āˆžš‘›=0šœƒš‘›=āˆž; (iii)āˆ‘āˆžš‘›=0|šœƒš‘›+1āˆ’šœƒš‘›|<āˆž; (iv)āˆ‘āˆžš‘›=0|šœ†š‘›+1āˆ’šœ†š‘›|<āˆž.

Theorem 3.5. Assume that the minimization problem (1.1) is consistent and the gradient āˆ‡š‘“ satisfies the Lipschitz condition (1.2). Let {š‘„š‘›} be generated by algorithm (3.29) with the sequences {šœƒš‘›} and {šœ†š‘›} satisfying the above conditions. Then, the sequence {š‘„š‘›} converges in norm to š‘„āˆ— that is obtained in Theorem 3.2.

Proof. (1) The sequence {š‘„š‘›}āˆžš‘›=0 is bounded. Setting š‘‰š‘›āˆ¶=Projš¶ī€·š¼āˆ’šœ†š‘›ī€øāˆ‡š‘“.(3.30) Indeed, we have, for š‘„āˆˆš‘†, ā€–ā€–š‘„š‘›+1āˆ’š‘„ā€–ā€–=ā€–ā€–šœƒš‘›ī€·š‘„š›¾ā„Žš‘›ī€ø+ī€·š¼āˆ’šœ‡šœƒš‘›š¹ī€øš‘‰š‘›š‘„š‘›āˆ’š‘„ā€–ā€–=ā€–ā€–šœƒš‘›ī€·ī€·š‘„š›¾ā„Žš‘›ī€øī€·āˆ’šœ‡š¹š‘„+ī€·ī€øī€øš¼āˆ’šœ‡šœƒš‘›š¹ī€øš‘‰š‘›š‘„š‘›āˆ’ī€·š¼āˆ’šœ‡šœƒš‘›š¹ī€øš‘‰š‘›š‘„ā€–ā€–ā‰¤ī€·1āˆ’šœƒš‘›šœī€øā€–ā€–š‘„š‘›āˆ’š‘„ā€–ā€–+šœƒš‘›ā€–ā€–š‘„šœŒš›¾š‘›āˆ’š‘„ā€–ā€–+šœƒš‘›ā€–ā€–ī€·š›¾ā„Žš‘„ī€øī€·āˆ’šœ‡š¹š‘„ī€øā€–ā€–=ī€·1āˆ’šœƒš‘›(ī€øā€–ā€–š‘„šœāˆ’š›¾šœŒ)š‘›āˆ’š‘„ā€–ā€–+šœƒš‘›ā€–ā€–ī€·š›¾ā„Žš‘„ī€øī€·āˆ’šœ‡š¹š‘„ī€øā€–ā€–ī‚»ā€–ā€–š‘„ā‰¤maxš‘›āˆ’š‘„ā€–ā€–,1ā€–ā€–ī€·šœāˆ’š›¾šœŒš›¾ā„Žš‘„ī€øī€·āˆ’šœ‡š¹š‘„ī€øā€–ā€–ī‚¼,āˆ€š‘›ā‰„0.(3.31) By induction, ā€–ā€–š‘„š‘›āˆ’š‘„ā€–ā€–īƒÆā€–ā€–š‘„ā‰¤max0āˆ’š‘„ā€–ā€–,ā€–ā€–ī€·š›¾ā„Žš‘„ī€øī€·āˆ’šœ‡š¹š‘„ī€øā€–ā€–īƒ°šœāˆ’š›¾šœŒ.(3.32) In particular, {š‘„š‘›}āˆžš‘›=0 is bounded.
(2) We prove that ā€–š‘„š‘›+1āˆ’š‘„š‘›ā€–ā†’0 as š‘›ā†’āˆž. Let š‘€ be a constant such that īƒÆš‘€>maxsupš‘›ā‰„0š›¾ā€–ā€–ā„Žī€·š‘„š‘›ī€øā€–ā€–,supšœ…,š‘›ā‰„0šœ‡ā€–ā€–š¹š‘‰šœ…š‘„š‘›ā€–ā€–,supš‘›ā‰„0ā€–ā€–ī€·š‘„āˆ‡š‘“š‘›ī€øā€–ā€–īƒ°.(3.33) We compute ā€–ā€–š‘„š‘›+1āˆ’š‘„š‘›ā€–ā€–=ā€–ā€–šœƒš‘›ī€·š‘„š›¾ā„Žš‘›ī€ø+ī€·š¼āˆ’šœ‡šœƒš‘›š¹ī€øš‘‰š‘›š‘„š‘›āˆ’šœƒš‘›āˆ’1ī€·š‘„š›¾ā„Žš‘›āˆ’1ī€øāˆ’ī€·š¼āˆ’šœ‡šœƒš‘›āˆ’1š¹ī€øš‘‰š‘›āˆ’1š‘„š‘›āˆ’1ā€–ā€–=ā€–ā€–šœƒš‘›š›¾ī€·ā„Žī€·š‘„š‘›ī€øī€·š‘„āˆ’ā„Žš‘›āˆ’1ī€·šœƒī€øī€ø+š›¾š‘›āˆ’šœƒš‘›āˆ’1ī€øā„Žī€·š‘„š‘›āˆ’1ī€ø+ī€·š¼āˆ’šœ‡šœƒš‘›š¹ī€øš‘‰š‘›š‘„š‘›āˆ’ī€·š¼āˆ’šœ‡šœƒš‘›š¹ī€øš‘‰š‘›š‘„š‘›āˆ’1+ī€·š¼āˆ’šœ‡šœƒš‘›š¹ī€øš‘‰š‘›š‘„š‘›āˆ’1āˆ’ī€·š¼āˆ’šœ‡šœƒš‘›āˆ’1š¹ī€øš‘‰š‘›āˆ’1š‘„š‘›āˆ’1ā€–ā€–=ā€–ā€–šœƒš‘›š›¾ī€·ā„Žī€·š‘„š‘›ī€øī€·š‘„āˆ’ā„Žš‘›āˆ’1ī€·šœƒī€øī€ø+š›¾š‘›āˆ’šœƒš‘›āˆ’1ī€øā„Žī€·š‘„š‘›āˆ’1ī€ø+ī€·š¼āˆ’šœ‡šœƒš‘›š¹ī€øš‘‰š‘›š‘„š‘›āˆ’ī€·š¼āˆ’šœ‡šœƒš‘›š¹ī€øš‘‰š‘›š‘„š‘›āˆ’1+ī€·š¼āˆ’šœ‡šœƒš‘›š¹ī€øš‘‰š‘›š‘„š‘›āˆ’1āˆ’ī€·š¼āˆ’šœ‡šœƒš‘›š¹ī€øš‘‰š‘›āˆ’1š‘„š‘›āˆ’1+ī€·š¼āˆ’šœ‡šœƒš‘›š¹ī€øš‘‰š‘›āˆ’1š‘„š‘›āˆ’1āˆ’ī€·š¼āˆ’šœ‡šœƒš‘›āˆ’1š¹ī€øš‘‰š‘›āˆ’1š‘„š‘›āˆ’1ā€–ā€–ā‰¤šœƒš‘›ā€–ā€–š‘„š›¾šœŒš‘›āˆ’š‘„š‘›āˆ’1ā€–ā€–||šœƒ+š›¾š‘›āˆ’šœƒš‘›āˆ’1||ā€–ā€–ā„Žī€·š‘„š‘›āˆ’1ī€øā€–ā€–+ī€·1āˆ’šœƒš‘›šœī€øā€–ā€–š‘„š‘›āˆ’š‘„š‘›āˆ’1ā€–ā€–+ā€–ā€–š‘‰š‘›š‘„š‘›āˆ’1āˆ’š‘‰š‘›āˆ’1š‘„š‘›āˆ’1ā€–ā€–||šœƒ+šœ‡š‘›āˆ’šœƒš‘›āˆ’1||ā€–ā€–š¹š‘‰š‘›āˆ’1š‘„š‘›āˆ’1ā€–ā€–ā‰¤šœƒš‘›ā€–ā€–š‘„š›¾šœŒš‘›āˆ’š‘„š‘›āˆ’1ā€–ā€–||šœƒ+š‘€š‘›āˆ’šœƒš‘›āˆ’1||+ī€·1āˆ’šœƒš‘›šœī€øā€–ā€–š‘„š‘›āˆ’š‘„š‘›āˆ’1ā€–ā€–+ā€–ā€–š‘‰š‘›š‘„š‘›āˆ’1āˆ’Vš‘›āˆ’1š‘„š‘›āˆ’1ā€–ā€–||šœƒ+š‘€š‘›āˆ’šœƒš‘›āˆ’1||=ī€·1āˆ’šœƒš‘›ī€øā€–ā€–š‘„(šœāˆ’š›¾šœŒ)š‘›āˆ’š‘„š‘›āˆ’1ā€–ā€–||šœƒ+2š‘€š‘›āˆ’šœƒš‘›āˆ’1||+ā€–ā€–š‘‰š‘›š‘„š‘›āˆ’1āˆ’š‘‰š‘›āˆ’1š‘„š‘›āˆ’1ā€–ā€–,ā€–ā€–š‘‰(3.34)š‘›š‘„š‘›āˆ’1āˆ’š‘‰š‘›āˆ’1š‘„š‘›āˆ’1ā€–ā€–=ā€–ā€–Projš¶ī€·š¼āˆ’šœ†š‘›ī€øš‘„āˆ‡š‘“š‘›āˆ’1āˆ’Projš¶ī€·š¼āˆ’šœ†š‘›āˆ’1ī€øš‘„āˆ‡š‘“š‘›āˆ’1ā€–ā€–ā‰¤ā€–ā€–ī€·š¼āˆ’šœ†š‘›ī€øš‘„āˆ‡š‘“š‘›āˆ’1āˆ’ī€·š¼āˆ’šœ†š‘›āˆ’1ī€øš‘„āˆ‡š‘“š‘›āˆ’1ā€–ā€–=||šœ†š‘›āˆ’šœ†š‘›āˆ’1||ā€–ā€–ī€·š‘„āˆ‡š‘“š‘›āˆ’1ī€øā€–ā€–||šœ†ā‰¤š‘€š‘›āˆ’šœ†š‘›āˆ’1||.(3.35) Combining (3.34) and (3.35), we can obtain ā€–ā€–š‘„š‘›+1āˆ’š‘„š‘›ā€–ā€–ā‰¤ī€·1āˆ’(šœāˆ’š›¾šœŒ)šœƒš‘›ī€øā€–ā€–š‘„š‘›āˆ’š‘„š‘›āˆ’1ā€–ā€–ī€·||šœƒ+2š‘€š‘›āˆ’šœƒš‘›āˆ’1||+||šœ†š‘›āˆ’šœ†š‘›āˆ’1||ī€ø.(3.36) Apply Lemma 2.1 to (3.36) to conclude that ā€–š‘„š‘›+1āˆ’š‘„š‘›ā€–ā†’0 as š‘›ā†’āˆž.
(3) We prove that šœ”š‘¤(š‘„š‘›)āŠ‚š‘†. Let Ģ‚š‘„āˆˆšœ”š‘¤(š‘„š‘›), and assume that š‘„š‘›š‘—ā‡€Ģ‚š‘„ for some subsequence {š‘„š‘›š‘—}āˆžš‘—=1 of {š‘„š‘›}āˆžš‘›=0. We may further assume that šœ†š‘›š‘—ā†’šœ†āˆˆ[0,2/šæ] due to condition (1.4). Set š‘‰āˆ¶=Projš¶(š¼āˆ’šœ†āˆ‡š‘“). Notice that š‘‰ is nonexpansive and Fixš‘‰=š‘†. It turns out that ā€–ā€–š‘„š‘›š‘—āˆ’š‘‰š‘„š‘›š‘—ā€–ā€–ā‰¤ā€–ā€–š‘„š‘›š‘—āˆ’š‘‰š‘›š‘—š‘„š‘›š‘—ā€–ā€–+ā€–ā€–š‘‰š‘›š‘—š‘„š‘›š‘—āˆ’š‘‰š‘„š‘›š‘—ā€–ā€–ā‰¤ā€–ā€–š‘„š‘›š‘—āˆ’š‘„š‘›š‘—+1ā€–ā€–+ā€–ā€–š‘„š‘›š‘—+1āˆ’š‘‰š‘›š‘—š‘„š‘›š‘—ā€–ā€–+ā€–ā€–š‘‰š‘›š‘—š‘„š‘›š‘—āˆ’š‘‰š‘„š‘›š‘—ā€–ā€–ā‰¤ā€–ā€–š‘„š‘›š‘—āˆ’š‘„š‘›š‘—+1ā€–ā€–+šœƒš‘›š‘—ā€–ā€–ī‚€š‘„š›¾ā„Žš‘›š‘—ī‚āˆ’šœ‡š¹š‘‰š‘›š‘—š‘„š‘›š‘—ā€–ā€–+ā€–ā€–Projš¶ī‚€š¼āˆ’šœ†š‘›š‘—ī‚š‘„āˆ‡š‘“š‘›š‘—āˆ’Projš¶(š¼āˆ’šœ†āˆ‡š‘“)š‘„š‘›š‘—ā€–ā€–ā‰¤ā€–ā€–š‘„š‘›š‘—āˆ’š‘„š‘›š‘—+1ā€–ā€–+šœƒš‘›š‘—ā€–ā€–ī‚€š‘„š›¾ā„Žš‘›š‘—ī‚āˆ’šœ‡š¹š‘‰š‘›š‘—š‘„š‘›š‘—ā€–ā€–+|||šœ†āˆ’šœ†š‘›š‘—|||ā€–ā€–ī‚€š‘„āˆ‡š‘“š‘›š‘—ī‚ā€–ā€–ā‰¤ā€–ā€–š‘„š‘›š‘—āˆ’š‘„š‘›š‘—+1ā€–ā€–ī‚€šœƒ+2š‘€š‘›š‘—+|||šœ†āˆ’šœ†š‘›š‘—|||ī‚āŸ¶0asš‘—āŸ¶āˆž.(3.37) So Lemma 2.2 guarantees that šœ”š‘¤(š‘„š‘›)āŠ‚Fixš‘‰=š‘†.
(4) We prove that š‘„š‘›ā†’š‘„āˆ— as š‘›ā†’āˆž, where š‘„āˆ— is the unique solution of the š‘‰š¼ (3.12). First observe that there is some Ģ‚š‘„āˆˆšœ”š‘¤(š‘„š‘›)āŠ‚š‘† Such that limsupš‘›ā†’āˆžāŸØ(šœ‡š¹āˆ’š›¾ā„Ž)š‘„āˆ—,š‘„š‘›āˆ’š‘„āˆ—āŸ©=āŸØ(šœ‡š¹āˆ’š›¾ā„Ž)š‘„āˆ—,Ģ‚š‘„āˆ’š‘„āˆ—āŸ©ā‰„0.(3.38)
We now compute ā€–ā€–š‘„š‘›+1āˆ’š‘„āˆ—ā€–ā€–2=ā€–ā€–šœƒš‘›ī€·š‘„š›¾ā„Žš‘›ī€ø+ī€·š¼āˆ’šœ‡šœƒš‘›š¹ī€øProjš¶ī€·š¼āˆ’šœ†š‘›š‘„āˆ‡š‘“ī€øī€·š‘›ī€øāˆ’š‘„āˆ—ā€–ā€–2=ā€–ā€–šœƒš‘›ī€·ī€·š‘„š›¾ā„Žš‘›ī€øāˆ’šœ‡š¹š‘„āˆ—ī€ø+ī€·š¼āˆ’šœ‡šœƒš‘›š¹ī€øš‘‰š‘›ī€·š‘„š‘›ī€øāˆ’ī€·š¼āˆ’šœ‡šœƒš‘›š¹ī€øš‘‰š‘›š‘„āˆ—ā€–ā€–2=ā€–ā€–šœƒš‘›š›¾ī€·ā„Žī€·š‘„š‘›ī€øī€·š‘„āˆ’ā„Žāˆ—+ī€·ī€øī€øš¼āˆ’šœ‡šœƒš‘›š¹ī€øš‘‰š‘›ī€·š‘„š‘›ī€øāˆ’(š¼āˆ’šœ‡šœƒš‘›š¹)š‘‰š‘›š‘„āˆ—+šœƒš‘›ī€·ī€·š‘„š›¾ā„Žāˆ—ī€øāˆ’šœ‡š¹š‘„āˆ—ī€øā€–ā€–2ā‰¤ā€–ā€–šœƒš‘›š›¾ī€·ā„Žī€·š‘„š‘›ī€øī€·š‘„āˆ’ā„Žāˆ—+ī€·ī€øī€øš¼āˆ’šœ‡šœƒš‘›š¹ī€øš‘‰š‘›ī€·š‘„š‘›ī€øāˆ’ī€·š¼āˆ’šœ‡šœƒš‘›š¹ī€øš‘‰š‘›š‘„āˆ—ā€–ā€–2+2šœƒš‘›ī«(š›¾ā„Žāˆ’šœ‡š¹)š‘„āˆ—,š‘„š‘›+1āˆ’š‘„āˆ—ī¬=ā€–ā€–šœƒš‘›š›¾ī€·ā„Žī€·š‘„š‘›ī€øī€·š‘„āˆ’ā„Žāˆ—ā€–ā€–ī€øī€ø2+ā€–ā€–ī€·š¼āˆ’šœ‡šœƒš‘›š¹ī€øš‘‰š‘›ī€·š‘„š‘›ī€øāˆ’ī€·š¼āˆ’šœ‡šœƒš‘›š¹ī€øš‘‰š‘›š‘„āˆ—ā€–ā€–2+2šœƒš‘›ī€·š‘„š›¾āŸØā„Žš‘›ī€øī€·š‘„āˆ’ā„Žāˆ—ī€ø,ī€·š¼āˆ’šœ‡šœƒš‘›š¹ī€øš‘‰š‘›ī€·š‘„š‘›ī€øāˆ’ī€·š¼āˆ’šœ‡šœƒš‘›š¹ī€øš‘‰š‘›š‘„āˆ—āŸ©+2šœƒš‘›āŸØ(š›¾ā„Žāˆ’šœ‡š¹)š‘„āˆ—,š‘„š‘›+1āˆ’š‘„āˆ—āŸ©ā‰¤šœƒ2š‘›š›¾2šœŒ2ā€–ā€–š‘„š‘›āˆ’š‘„āˆ—ā€–ā€–2+ī€·1āˆ’šœƒš‘›šœī€ø2ā€–ā€–š‘„š‘›āˆ’š‘„āˆ—ā€–ā€–2+2šœƒš‘›ī€·š›¾šœŒ1āˆ’šœƒš‘›šœī€øā€–ā€–š‘„š‘›āˆ’š‘„āˆ—ā€–ā€–2+2šœƒš‘›ī«(š›¾ā„Žāˆ’šœ‡š¹)š‘„āˆ—,š‘„š‘›+1āˆ’š‘„āˆ—ī¬=ī‚€šœƒ2š‘›š›¾2šœŒ2+ī€·1āˆ’šœƒš‘›šœī€ø2+2šœƒš‘›ī€·š›¾šœŒ1āˆ’šœƒš‘›šœī€øī‚ā€–ā€–š‘„š‘›āˆ’š‘„āˆ—ā€–ā€–2+2šœƒš‘›ī«(š›¾ā„Žāˆ’šœ‡š¹)š‘„āˆ—,š‘„š‘›+1āˆ’š‘„āˆ—ī¬ā‰¤ī€·šœƒš‘›š›¾2šœŒ2+1āˆ’2šœƒš‘›šœ+šœƒš‘›šœ2+2šœƒš‘›ī€øā€–ā€–š‘„š›¾šœŒš‘›āˆ’š‘„āˆ—ā€–ā€–2+2šœƒš‘›ī«(š›¾ā„Žāˆ’šœ‡š¹)š‘„āˆ—,š‘„š‘›+1āˆ’š‘„āˆ—ī¬=ī€·1āˆ’šœƒš‘›ī€·2šœāˆ’š›¾2šœŒ2āˆ’šœ2ā€–ā€–š‘„āˆ’2š›¾šœŒī€øī€øš‘›āˆ’š‘„āˆ—ā€–ā€–2+2šœƒš‘›ī«(š›¾ā„Žāˆ’šœ‡š¹)š‘„āˆ—,š‘„š‘›+1āˆ’š‘„āˆ—ī¬.(3.39) Applying Lemma 2.1 to the inequality (3.39), together with (3.38), we get ā€–š‘„š‘›āˆ’š‘„āˆ—ā€–ā†’0 as š‘›ā†’āˆž.

Corollary 3.6 (see [11]). Let {š‘„š‘›} be generated by the following algorithm: š‘„š‘›+1=šœƒš‘›ā„Žī€·š‘„š‘›ī€ø+ī€·1āˆ’šœƒš‘›ī€øProjš¶ī€·š‘„š‘›āˆ’šœ†š‘›ī€·š‘„āˆ‡š‘“š‘›ī€øī€ø,āˆ€š‘›ā‰„0.(3.40) Assume that the sequence {šœ†š‘›}āˆžš‘›=0 satisfies the conditions (1.4) and (iv) and that {šœƒš‘›}āŠ‚[0,1] satisfies the conditions (i)ā€“(iii). Then {š‘„š‘›} converges in norm to š‘„āˆ— obtained in Corollary 3.4.

Corollary 3.7. Let {š‘„š‘›} be generated by the following algorithm: š‘„š‘›+1=šœƒš‘›ī€·š‘„š›¾ā„Žš‘›ī€ø+ī€·š¼āˆ’šœƒš‘›š“ī€øProjš¶ī€·š‘„š‘›āˆ’šœ†š‘›ī€·š‘„āˆ‡š‘“š‘›ī€øī€ø,āˆ€š‘›ā‰„0.(3.41) Assume that the sequences {šœƒš‘›} and {šœ†š‘›} satisfy the conditions contained in Theorem 3.5, then {š‘„š‘›} converges in norm to š‘„āˆ— obtained in Corollary 3.3.

Acknowledgments

Ming Tian is Supported in part by The Fundamental Research Funds for the Central Universities (the Special Fund of Science in Civil Aviation University of China: No. ZXH2012ā€‰K001) and by the Science Research Foundation of Civil Aviation University of China (No. 2012KYM03).