On generalized Φ-strongly monotone mappings and algorithms for the solution of equations of Hammerstein type
aa r X i v : . [ m a t h . F A ] A ug ON GENERALIZED Φ -STRONGLY MONOTONE MAPPINGS ANDALGORITHMS FOR THE SOLUTION OF EQUATIONS OF HAMMERSTEINTYPE M.O. AIBINU AND O.T. MEWOMO Abstract.
In this paper, we consider the class of generalized Φ-strongly monotone mappings and themethods of approximating a solution of equations of Hammerstein type. Auxiliary mapping is defined fornonlinear integral equations of Hammerstein type. The auxiliary mapping is the composition of boundedgeneralized Φ-strongly monotone mappings which satisfy the range condition. Suitable conditions areimposed to obtain the boundedness and to show that the auxiliary mapping is a generalized Φ-stronglywhich satisfies the range condition. A sequence is constructed and it is shown that it converges stronglyto a solution of equations of Hammerstein type. The results in this paper improve and extend somerecent corresponding results on the approximation of a solution of equations of Hammerstein type. Introduction
Let E be a real normed linear space and E ∗ denotes its corresponding dual space. We denote thevalue of the functional x ∗ ∈ E ∗ at x ∈ E by h x ∗ , x i , domain of A by D ( A ) , range of A by R ( A )and N ( A ) denotes the set of zeros of A (cid:0) i.e., N ( A ) = { x ∈ D ( A ) : 0 ∈ Ax } = A − (cid:1) . A multivaluedmapping A : E → E ∗ from E into 2 E ∗ is said to be monotone if for each x, y ∈ E , the followinginequality holds: h µ − ν, x − y i ≥ ∀ µ ∈ Ax, ν ∈ Ay.
A single-valued mapping A : D ( A ) ⊂ E → E ∗ is monotone if h Ax − Ay, x − y i ≥ , ∀ x, y ∈ D ( A ) . Fora linear mapping A , the above definition reduces to h Au, u i ≥ ∀ u ∈ D ( A ) . Multivalued mapping A is said to be generalized Φ-strongly monotone if there exists a strictly increasing function Φ : [0 , ∞ ) → [0 , ∞ ) with Φ(0) = 0 such that for each x, y ∈ D ( A ) , h µ − ν, x − y i ≥ Φ( k x − y k ) ∀ µ ∈ Ax, ν ∈ Ay.
Given that H is a real Hilbert space, a mapping A : H → H is said to be monotone if for each x, y ∈ H, h µ − ν, x − y i ≥ ∀ µ ∈ Ax, ν ∈ Ay.
Let A be a monotone mapping defined on H. It is well known (see e.g., Zeidler [28]) that many physicallysignificant problems can be modelled by initial-value problems of the form(1.1) u ′ ( t ) + Au ( t ) = 0 , u (0) = u . Heat, wave and Schr¨ o dinger equations are typical examples where such evolution equations occur. Atan equilibrium state (that is, if u ( t ) is independent of t ), then (1.1) reduces to(1.2) Au = 0 . Therefore, considerable research efforts have been devoted, especially within the past 40 years or so,to methods of finding approximate solutions (when they exist) of (1.2). One important generalization
Key words and phrases.
Generalized Φ-strongly monotone, Hammerstein equation, Strong convergence.
Mathematics Subject Classification : 47H06, 47J05, 47J25, 47H09.
This article should be cited as : M.O. Aibinu, O.T. Mewomo, On generalized Φ-strongly monotone mappings and al-gorithms for the solution of equations of Hammerstein type, International Journal of Nonlinear Analysis and Applications,(2019), DOI:10.22075/ijnaa.2019.16797.1894. AND O.T. MEWOMO of (1.2) is the so-called equation of Hammerstein type (see, e.g., Hammerstein [15]), where a nonlinearintegral equation of Hammerstein type is one of the form(1.3) u ( x ) + Z Ω k ( x, y ) f ( y, u ( y )) dy = h ( x ) , where dy stands for a σ -finite measure on the measure space Ω, the kernel k is defined on Ω × Ω, f is a real-valued function defined on Ω × R and is in general nonlinear, h is a given function on Ω and u is the unknown function defined on Ω. Let g be a function from Ω × R n into R . We denote by F ( X, Y ), the set of all maps from X to Y . The Nemystkii operator associated to g is the operator N g : F (Ω , R n ) → F (Ω , R ) defined by u N g ( u )where ( N g u )( x ) = g ( x, u ( x )) ∀ u ∈ F (Ω , R n ) , ∀ x ∈ Ω . For simplicity, we shall write N g u ( x ) insteadof ( N g u )( x ). Example 1.1.
Given a map g : R × R → R defined by g ( x, s ) = | s | ∀ ( x, s ) ∈ R × R , the Nemystkii operator associated to g is the expression N g u ( x ) = | u ( x ) | for any map u : R → R andfor any x ∈ R . Example 1.2.
Given a map g : R × R → R defined by g ( x, s ) = xe s ∀ ( x, s ) ∈ R × R , the Nemystkii operator associated to g is the expression N g u ( x ) = xe u ( x ) for any map u : R → R andfor any x ∈ R .Observe that by the continuity of g , N g maps the set of real-valued continuous function on Ω; C (Ω)into itself. Moreover, it maps the set of real-valued measurable function into itself. Define the operator K : F (Ω , R ) → F (Ω , R ) by Kv ( x ) = Z Ω k ( x, y ) v ( y ) dy for almost all x ∈ Ω , and the Nemystkii operator F : F (Ω , R ) → F (Ω , R ) associated with f by F u ( x ) = f ( x, u ( x )) for almost all x ∈ Ω , then the integral (1.3) can be put in functional equation form as follows:(1.4) u + KF u = 0 , where without loss of generality, we have taken h ≡
0. Also, Hammerstein equations play crucialroles in solving several problems that arise in differential equations (see, e.g., Pascali and Sburlan [21],Chapter IV, p. 164) and applicable in theory of optimal control systems and in automation and networktheory (see, e.g., Dolezale [14]). Several authors have proved existence and uniqueness theorems forequations of the Hammerstein type (see, e.g., Br´ e zis and F. E. Browder ([5, 6, 7]); Browder and Gupta[8]; Chepanovich [9]; De Figueiredo and Gupta [13]).Let C be a nonempty closed convex subset of a real Banach space E. A self-mapping T : C → C issaid to be nonexpansive if k T x − T y k ≤ k x − y k ∀ x, y ∈ C. If E is smooth, T : C → E is said to befirmly nonexpansive type (see e.g., [19]), if h T x − T y, J T x − J T y i ≤ h
T x − T y, J x − J y i for all x, y ∈ C, where J : E → E ∗ is the normalized duality mapping defined in Section 2.For the iterative approximation of solutions of (1.2), the monotonicity of A is crucial. A mapping A : E → E ∗ is said to be maximal monotone if it is monotone and R ( J + tA ) is all of E ∗ for some t > . Given that A is monotone and R ( J + tA ) = E ∗ for all t > , then A is said to satisfy therange condition. Let E be a uniformly smooth and uniformly convex Banach space and A, a maximal N GENERALIZED Φ-STRONGLY MONOTONE MAPPINGS AND ALGORITHMS 3 monotone or (a monotone mapping which satisfies the range condition). Then, one can define for all t > , the resolvent J t : C → D ( A ) by J t x = { z ∈ E : J x ∈ J z + tAz } for all x ∈ C, where C is a closed convex subset of E. The fact that F ( J t ) = A − F ( J t ) is the set of fixed points of J t (see e.g., [20, 22, 23]).In this present work, it is shown that if A is a multivalued generalized Φ-strongly monotone mappingand such that R ( J p + t A ) = E ∗ for some t > , then R ( J p + tA ) = E ∗ for all t > , where J p , p > ∈ Ax exists, it is necessarily unique. Our results generalize and improvesome important and recent results of Chidume and Idu [10].2. Preliminaries
Let S := { x ∈ E : k x k = 1 } denotes a unit sphere of a Banach space E with dimension greater than orequal to two. The space E is said to be Gˆ a teaux differentiable (or is smooth) if the limitlim t → k x + ty k − k x k t exists for each x, y ∈ S. If E is smooth and the limit is attained uniformly for each x, y ∈ S, then it issaid to be uniformly smooth. A Banach space E is said to be strictly convex if k x k = k y k = 1 , x = y ⇒ k x + y k < . The space E is said to be uniformly convex if, for each ǫ ∈ (0 , , there exists a δ := δ ( ǫ ) > x, y ∈ S, k x − y k ≥ δ implies that k x − y k ≤ − δ. E is reflexive if and only if the naturalembedding of E into E ∗∗ is onto. It is known that a uniformly convex Banach space is reflexive andstrictly convex. Also, if E is a reflexive Banach space, then, it is strictly convex (respectively smooth)if and only if E ∗ is smooth (respectively strictly convex).Let ϕ : [0 , ∞ ) → [0 , ∞ ) be a strictly increasing continuous function such that ϕ (0) = 0 and ϕ ( t ) → ∞ as t → ∞ , ϕ is called a gauge function. We associate to ϕ, the duality mapping J ϕ : E → E ∗ whichis defined as J ϕ ( x ) = { f ∈ E ∗ : h x, f i = k x kk f k , k f k = ϕ ( k x k ) } , where E ∗ denotes the dual space of E and h ., . i denotes the generalized duality pairing. If ϕ ( t ) = t p − , p > , the duality mapping J ϕ = J p is called generalized duality mapping. The duality mappingwith guage ϕ ( t ) = t (i.e. p = 2) is denoted by J and is referred to as the normalized duality mapping.It follows from the definition that J ϕ ( x ) = ϕ ( k x k ) k x k J ( x ) for each x = 0 and J p ( x ) = k x k p − J ( x ) , p > .J ϕ is single-valued if E is smooth and if E is a reflexive strictly convex Banach space with strictlyconvex dual space E ∗ , J p : E → E ∗ and J q : E ∗ → E being the duality mappings with gauge functions ϕ ( t ) = t p − and ϕ ( s ) = s q − , p + q = 1 , respectively, then J − p = J q . For a Banach space E and E ∗ asits dual space, the following properties of the generalized duality mapping have also been established(see e.g., Alber and Ryazantseva [4], Cioranescu [12], p. 25-77, Xu and Roach [26], Zˇ a linescu [27]):(i) If E is smooth, then J p is single-valued and norm-to-weak ∗ continuous;(ii) If E is strictly convex, then J p is strictly monotone (injective, in particular, i.e, if x = y, then J p x ∩ J p y = ∅ );(iii) If E is reflexive, then J p is onto;(iv) The expression h J p x, x i is naturally regarded as having power p as h J p x, x i = k x k p ; M.O. AIBINU AND O.T. MEWOMO (v) If E is uniformly smooth, then J q : E ∗ → E is a generalized duality mapping on E ∗ , J − p = J q , J p J q = I E ∗ and J q J p = I E , where I E and I E ∗ are the identity mappings on E and E ∗ respectively. Definition 2.1.
Let E be a smooth real Banach space with dual space E ∗ , the followings were intro-duced by Aibinu and Mewomo [2].(i) The function φ p : E × E → R is defined by φ p ( x, y ) = pq k x k q − p h x, J p y i + k y k p , for all x, y ∈ E, where J p is the generalized duality map from E to E ∗ , p and q are real numbers such that q ≥ p > p + q = 1. Notice that taking p = 2 in (2.1), it reduces to φ ( x, y ) = k x k − h x, J y i + k y k , for all x, y ∈ E, which was introduced by Alber [3].(ii) The mapping V p : E × E ∗ → R is defined by V p ( x, x ∗ ) = pq k x k q − p h x, x ∗ i + k x ∗ k p ∀ x ∈ E, x ∗ ∈ E ∗ such that q ≥ p > , p + 1 q = 1 . Remark 2.2.
These remarks follow from Definition 2.1:(i) It is obvious from the definition of the function φ p that(2.1) ( k x k − k y k ) p ≤ φ p ( x, y ) ≤ ( k x k + k y k ) p for all x, y ∈ E. (ii) Clearly, we also have that(2.2) V p ( x, x ∗ ) = φ p ( x, J − x ∗ ) ∀ x ∈ E, x ∗ ∈ E ∗ . In the sequel, we shall need the following lemmas.
Lemma 2.3.
Aibinu and Mewomo [2] . Let E be a smooth uniformly convex real Banach space with E ∗ as its dual. Then (2.3) V p ( x, x ∗ ) + p (cid:10) J − x ∗ − x, y ∗ (cid:11) ≤ V p ( x, x ∗ + y ∗ ) for all x ∈ E and x ∗ , y ∗ ∈ E ∗ . Lemma 2.4.
Aibinu and Mewomo [2] . Let E be a smooth uniformly convex real Banach space. For d > , let B d (0) := { x ∈ E : k x k ≤ d } . Then for arbitrary x, y ∈ B d (0) , k x − y k p ≥ φ p ( x, y ) − pq k x k q , q ≥ p > , p + 1 q = 1 . Lemma 2.5.
Aibinu and Mewomo [2] . Let E be a reflexive strictly convex and smooth real Banachspace with the dual E ∗ . Then (2.4) φ p ( y, x ) − φ p ( y, z ) ≥ p h z − y, J x − J z i for all x, y, z ∈ E. Lemma 2.6. Xu [25] . Let { a n } be a sequence of nonnegative real numbers satisfying the followingrelations: a n +1 ≤ (1 − α n ) a n + α n σ n + γ n , n ∈ N , where (i) { α } n ⊂ (0 , , ∞ X n =1 α n = ∞ ; (ii) lim sup { σ } n ≤ ; (iii) γ n ≥ , ∞ X n =1 γ n < ∞ . N GENERALIZED Φ-STRONGLY MONOTONE MAPPINGS AND ALGORITHMS 5
Then, a n → as n → ∞ . Lemma 2.7.
Chidume and Idu [10] . For a real number p > , let X, Y be real uniformly convexand uniformly smooth Banach spaces. Let W := X × Y with the norm k w k W = (cid:0) k u k pX + k v k pY (cid:1) p for arbitrary w := ( u, v ) ∈ W . Let W ∗ := X ∗ × Y ∗ denotes the dual space of Z . For arbitrary z = ( u, v ) ∈ Z , define the map j Zp : Z → Z ∗ by j Wp ( z ) = j Wp ( u, v ) = (cid:0) j Xp ( u ) , j Yp ( v ) (cid:1) , such that for arbitrary w = ( u , v ) , w = ( u , v ) in Z , the duality pairing h ., . i is given by (cid:10) w , j Wp ( w ) (cid:11) = (cid:10) u , j Xp ( u ) (cid:11) + (cid:10) v , j Yp ( v ) (cid:11) . Then, (i) W is uniformly smooth and uniformly convex, (ii) j Wp is single-valued duality mapping on W . Lemma 2.8.
Chidume and Idu [10] . Let E be a uniformly convex and uniformly smooth real Banachspace. Let F : E → E ∗ and K : E ∗ → E be monotone mappings with D ( F ) = R ( K ) = E . Let T : E × E ∗ → E ∗ × E be defined by T ( u, v ) = ( J u − F u + v, J − v − Kv − u ) for all ( u, v ) ∈ E × E ∗ ,then T is J -pseudocontractive. Moreover, if the Hammerstein equation u + KF u = 0 has a solution in E , then u ∗ is a solution of u + KF u = 0 if and only if ( u ∗ , v ∗ ) ∈ F JE ( T ) , where v ∗ = F u ∗ . Lemma 2.9. Z ˇ a linescu [27] . Let ψ : R + → R + be increasing with lim t →∞ ψ ( t ) = ∞ . Then J − ψ is single-valued and uniformly continuous on bounded sets of E ∗ if and only if E is a uniformly convex Banachspace. Theorem 2.10. Xu [24] . Let E be a real uniformly convex Banach space. For arbitrary r > , let B r (0) := { x ∈ E : k x k ≤ r } . Then, there exists a continuous strictly increasing convex function g : [0 , ∞ ) → [0 , ∞ ) , g (0) = 0 , such that for every x, y ∈ B r (0) , j p ( x ) ∈ J p ( x ) , j p ( y ) ∈ J p ( y ) , the following inequalities hold: (i) k x + y k p ≥ k x k p + p h y, j p ( x ) i + g ( k y k ) ; (ii) h x − y, j p ( x ) − j p ( y ) i ≥ g ( k x − y k ) . Lemma 2.11.
B. T. Kien [18] . The dual space E ∗ of a Banach space E is uniformly convex if andonly if the duality mapping J p is a single-valued map which is uniformly continuous on each boundedsubset of E . Lemma 2.12.
Kamimura and Takahashi [16] . Let E be a smooth uniformly convex real Banach spaceand let { x n } and { y n } be two sequences from E. If either { x n } or { y n } is bounded and φ ( x n , y n ) → as n → ∞ , then k x n − y n k → as n → ∞ . Theorem 2.13.
Kido [17] . Let E ∗ be a real strictly convex dual Banach space with a Fr ´ e chet dif-ferentiable norm and A a maximal monotone operator from E into E ∗ such that A − = ∅ . Let J t x := ( J + tA ) − x be the resolvent of A and P be the nearest point retraction of E onto A − . Then,for every x ∈ E , J t x converges strongly to P x as t → ∞ . Main Results
We give and prove the following lemmas which are useful in establishing our main result.
Lemma 3.1.
Suppose E is a Banach space with the dual E ∗ . Let F : E → E ∗ and K : E ∗ → E bemappings such that D ( K ) = R ( F ) and the following conditions hold: (i) For each u , u ∈ E, there exists a strictly increasing function Φ : [0 , ∞ ) → [0 , ∞ ) with Φ (0) = 0 such that h F u − F u , u − u i ≥ Φ ( k u − u k ); M.O. AIBINU AND O.T. MEWOMO (ii) For each v , v ∈ E ∗ , there exists a strictly increasing function Φ : [0 , ∞ ) → [0 , ∞ ) with Φ (0) = 0 such that h Ku − Ku , v − v i ≥ Φ ( k v − v k );(iii) Φ i ( t ) ≥ r i t for t ∈ [0 , ∞ ) and r i > , i = 1 , . Let W := E × E ∗ with norm k w k W := k u k E + k v k E ∗ for w = ( u, v ) ∈ W. Define a mapping A : W → W ∗ by Aw := ( F u − v, u + Kv ) . (i) Then for each w , w ∈ W, there exists a strictly increasing function Φ : [0 , ∞ ) → [0 , ∞ ) with Φ(0) = 0 such that h Aw − Aw , w − w i ≥ Φ( k w − w k );(ii) Suppose that F and K are bounded mappings, then A is a bounded map.Proof. (i) Define Φ : [0 , ∞ ) → [0 , ∞ ) by Φ( t ) := min { r , r } t for each t ∈ [0 , ∞ ) . Clearly, Φ isa strictly increasing function with Φ(0) = 0 . For w = ( u , v ) , w = ( u , v ) ∈ W , we have Aw = ( F u − v , Kv + u ) and Aw = ( F u − v , Kv + u ) such that Aw − Aw = ( F u − F u − ( v − v ) , Kv − Kv + ( u − u )) . Therefore, the following estimate follows from the properties of F and K. h Aw − Aw , w − w i = h F u − F u − ( v − v ) , u − u i + h Kv − Kv + ( u − u ) , v − v i = h F u − F u , u − u i − h v − v , u − u i + h Kv − Kv , v − v i + h u − u , v − v i≥ Φ ( k u − u k ) + Φ ( k v − v k ) ≥ r k u − u k + r k v − v k≥ min { r , r } ( k u − u k + k v − v k )= Φ( k w − w k ) . (ii) By the definition of A, it is a bounded map since F and K are bounded mappings. (cid:3) Remark 3.2.
Recall that a mapping A : E → E ∗ is said to be strongly monotone if there exists aconstant k ∈ (0 ,
1) such that h Ax − Ay, x − y i ≥ k k x − y k ∀ x, y ∈ D ( A ) . Therefore for a strongly monotone mapping, it is required that the norm on W be defined as k w k W := k u k E + k v k E ∗ . An analogue of Lemma , Chidume and Djitte, [11], which was proved in a Hilbert space is givenbelow in a uniformly smooth and uniformly convex Banach space.
Lemma 3.3.
Let E be a uniformly smooth and uniformly convex Banach space with dual E ∗ . Suppose D ( A ) = E and A : E → E ∗ is a multivalued generalized Φ -strongly monotone mapping such that R ( J p + t A ) = E ∗ for some t > . Then A satisfies the range condition, that is, R ( J p + tA ) = E ∗ forall t > . Proof.
By the strict convexity of E, we obtain for every x ∈ E, there exist unique x t ∈ E and suchthat J p x ∈ J p x t + t Ax t . N GENERALIZED Φ-STRONGLY MONOTONE MAPPINGS AND ALGORITHMS 7
Taking J p t ( x ) = x t , one can define a single-valued mapping J p t : E → D ( F ) by J p t := ( J p + t A ) − J p .J p t is called the resolvent of A . It is known that ( J p + t A ) is a bijection since it is monotone and R ( J p + t A ) = E ∗ . Since E is a smooth and strictly convex Banach space and A : E → E ∗ is suchthat R ( J p + t A ) = E ∗ , for each t >
0, one can verify that the resolvent J p t of A, defined by J p t ( x ) = { z ∈ E : J p x ∈ J p z + t Az } = n ( J p + t A ) − J p x o for all x ∈ E is a firmly nonexpansive type map. Infact, for x , x ∈ E and t > , and for every J p t ( x ) , J p t ( x ) ∈ D ( F ) , we have that J p x − J p ( J pt ( x )) t , J p x − J p ( J pt ( x )) t ∈ A, and generalized Φ-strongly monotonicity property of A gives, * J p x − J p ( J p t ( x )) t − J p x − J p ( J p t ( x )) t , J p t ( x ) − J p t ( x ) + ≥ Φ( k J p t ( x ) − J p t ( x ) k ) ≥ . consequently, D J p ( J p t ( x )) − J p ( J p t ( x )) , J p t ( x ) − J p t ( x ) E ≤ D J p x − J p x , J p t ( x ) − J p t ( x ) E . (3.1)Thus, the resolvent J p t is a firmly nonexpansive type map. A simple computation from (3.1) showsthat for x, y ∈ E, k J p t ( x ) − J p t ( y ) | ≤ k x − y k . (3.2)We claim that R ( J p + tA ) = E ∗ for any t > t . Indeed, let t > t , for every x ∈ E, we solve the equation(3.3) J p x + tAx = w ∗ , x ∗ ∈ E ∗ . Notice that x ∈ E is a solution of (3.3) provided that J p x + t Ax = t t w ∗ + (1 − t t ) J p x, which is equivalent to x = J p t (cid:18) t t w ∗ + (1 − t t ) J p x (cid:19) . By the contraction mapping principle, Eq.(3.3) has a unique solution since | − t t | < A is a monotone mapping and R ( J p + t A ) = E ∗ for some t > . By the claim, it follows that R ( J p + tA ) = E ∗ for any t > t . By induction, we therefore have that R ( J p + tA ) = E ∗ for any t > t n and any n ∈ N . Thus, R ( J p + tA ) = E ∗ for any t > . (cid:3) Lemma 3.4.
Let E be a uniformly smooth and uniformly convex real Banach space and denote thedual space by E ∗ . Suppose F : E → E ∗ is a generalized Φ -strongly monotone mapping such that R ( J p + t F ) = E ∗ for all t > and K : E ∗ → E is a generalized Φ -strongly monotone mapping suchthat R ( J q + t K ) = E for all t > . Let W := E × E ∗ with norm k w k W := k u k E + k v k E ∗ ∀ w =( u, v ) ∈ W and define a map A : W → W ∗ by (3.4) Aw = ( F u − v, Kv + u ) , ∀ w = ( u, v ) ∈ W, then R ( J p + tA ) = W ∗ for all t > . Proof.
We show that R ( J p + tA ) = W ∗ for all t >
0. Indeed, let t be such that 0 < t < J p t : E → D ( F ) of F by J p t := ( J p + t F ) − J p and J q t : E ∗ → D ( K ) of M.O. AIBINU AND O.T. MEWOMO K by J q t = ( J q + t K ) − J q . J p t and J q t are firmly nonexpansive type maps and hence (3.2) holds.Therefore, for h := ( h , h ) ∈ X ∗ , define G : W → W by Gw = (cid:16) J p t ( h − t u ) , J q t ( h + t v ) (cid:17) , ∀ w = ( u, v ) ∈ W. From the fact that (3.2) holds for J p t and J q t , we have k Gw − Gw k ≤ t k w − w k ∀ w , w ∈ W. Therefore G is a contraction and by Banach contraction mapping principle, G has a unique fixed point w ∗ := ( u ∗ , v ∗ ) ∈ W , that is Gw ∗ = w ∗ or equivalently u ∗ = J p t ( h − t u ∗ ) , v ∗ = J q t ( h + t v ∗ ). Theseimply that ( J p + t A ) w = h . Lemma 3.1 gives that A is a generalized Φ-strongly monotone mappingand by Lemma 3.3, R ( J p + tA ) = W ∗ for all t > (cid:3) Theorem 3.5.
Let E be a uniformly smooth and uniformly convex real Banach space and denotethe dual space by E ∗ . Let F : E → E ∗ be a generalized Φ -strongly monotone mapping such that R ( J p + t F ) = E ∗ for all t > and K : E ∗ → E be a generalized Φ -strongly monotone mappingsuch that R ( J q + t K ) = E for all t > . Suppose F and K are bounded mappings such that D ( K ) = R ( F ) = E ∗ . Define { u n } and { v n } iteratively for arbitrary u ∈ E and v ∈ E ∗ by (3.5) u n +1 = J q ( J p u n − λ n ( F u n − v n + θ n ( J p u n − J p u ))) , n ∈ N , (3.6) v n +1 = J p ( J q v n − λ n ( Kv n + u n + θ n ( J q v n − J q v ))) , n ∈ N , where J p is the generalized duality mapping from E to E ∗ and J q is the generalized duality mappingfrom E ∗ to E. Let the real sequences { λ n } and { θ n } in (0 , be such that, (i) lim θ n = 0 and { θ n } is decreasing; (ii) ∞ X n =1 λ n θ n = ∞ ; (iii) lim n →∞ (( θ n − /θ n ) − /λ n θ n = 0 , ∞ X n =1 λ n < ∞ .Suppose that u + KF u = 0 has a solution in E . There exists a real constant γ > with ψ ( λ n M ) ≤ γ , n ∈ N for some constant M > . Then, the sequence { u n } converges strongly to the solution of u + KF u .Proof.
Let W := E × E ∗ with norm k x k pW := k u k pE + k v k pE ∗ ∀ w = ( u, v ) ∈ W and define ∧ p : W × W → R by ∧ p ( w , w ) = φ p ( u , u ) + φ p ( v , v ) , where respectively w = ( u , v ) and w = ( u , v ). Let u ∗ ∈ E be a solution of u + KF u = 0. Observethat setting v ∗ := F u ∗ and w ∗ := ( u ∗ , v ∗ ), we have that u ∗ = − Kv ∗ .We divide the proof into two parts. Part 1:
We prove that { w n } is bounded, where w n := ( u n , v n ) . Let r > r ≥ max (cid:26) ∧ p ( w ∗ , w ) , δ p + pq k x ∗ k q (cid:27) . The proof is by induction. By construction, ∧ p ( w ∗ , w ) ≤ r . Suppose that ∧ p ( w ∗ , w n ) ≤ r for some n ∈ N . We show that ∧ p ( w ∗ , w n +1 ) ≤ r. Suppose this is not the case, then ∧ p ( w ∗ , w n +1 ) > r. From inequality (2.1), we have k w n k ≤ r p + k w ∗ k . Let B := { w ∈ E : ∧ p ( w ∗ , w ) ≤ r } and notice thatby Lemma (2.9 and 2.11), J q and J p are uniformly continuous on bounded subsets. Consequently, since F and K are bounded, we define(3.8) M := sup {k F u + θ n ( J p u − J p u ) k : θ n ∈ (0 , , u ∈ B } + 1 , N GENERALIZED Φ-STRONGLY MONOTONE MAPPINGS AND ALGORITHMS 9 (3.9) M := sup {k Kv + θ n ( J p v − J p v ) k : θ n ∈ (0 , , v ∈ B } + 1 . Let ψ : [0 , ∞ ) → [0 , ∞ ) be the modulus of continuity of J q and ψ : [0 , ∞ ) → [0 , ∞ ) be the modulusof continuity of J p . Recall that by the uniform continuity of J q and J p on bounded subsets of E ∗ and E respectively. Then we have k J q ( J p u n ) − J q ( J p u n − λ n ( F u n + θ n ( J p u n − J p u ))) k ≤ ψ ( λ n M ) , (3.10) k J p ( J q v n ) − J p ( J q v n − λ n ( Kv n + θ n ( J q v n − J q v ))) k ≤ ψ ( λ n M ) . (3.11)Let M := M + M , Φ := min { Φ , Φ } and define γ := min ( , Φ( δ )2 M ) where ψ ( λ n M ) ≤ γ with ψ ( λ n M ) ≥ δ , and ψ := ψ + ψ . Applying Lemma 2.3 with y ∗ := λ n ( F u n + θ n ( J p u n − J p u )) and by using thedefinition of u n +1 , we compute as follows, φ p ( u ∗ , u n +1 ) = φ p ( u ∗ , J q ( J p u n − λ n ( F u n + θ n ( J p u n − J p u ))))= V p ( u ∗ , J p u n − λ n ( F u n + θ n ( J p u n − J p u ))) ≤ V p ( u ∗ , J p u n ) − pλ n h J q ( J p u n − λ n ( F u n + θ n ( J p u n − J p u ))) − u ∗ , F u n + θ n ( J p u n − J p u ) i = φ p ( u ∗ , u n ) − pλ n h u n − u ∗ , F u n + θ n ( J p u n − J p u ) i− pλ n h J q ( J p u n − λ n ( F u n + θ n ( J p u n − J p u ))) − u n , F u n + θ n ( J p u n − J p u ) i . By Schwartz inequality and uniform continuity property of J q on bounded sets of E ∗ (Lemma 2.9), weobtain φ p ( u ∗ , u n +1 ) ≤ φ p ( u ∗ , u n ) − pλ n h u n − u ∗ , F u n + θ n ( J p u n − J p u ) i + pλ n ψ ( λ n M ) M (By applying inequality (3.10)) ≤ φ p ( u ∗ , u n ) − pλ n h u n − u ∗ , F x n − F u ∗ i since u ∗ ∈ N ( F )) − pλ n θ n h u n − u ∗ , J p u n − J p u i + pλ n ψ ( λ n M ) M . By Lemma 2.5, p h u n − u ∗ , J p u − J p u n i ≤ φ p ( u ∗ , u ) − φ p ( u ∗ , u n ) . Also, since F is generalized Φ-strongly monotone, we have, φ p ( u ∗ , u n +1 ) ≤ φ p ( u ∗ , u n ) − pλ n Φ ( k u n − u ∗ k ) + pλ n θ n h u n − u ∗ , J p u − J p u n i + pλ n ψ ( λ n M ) M ≤ φ p ( u ∗ , u n ) − pλ n Φ ( k u n − u ∗ k ) + pλ n θ n ( φ p ( u ∗ , u ) − φ p ( u ∗ , u n )) + pλ n ψ ( λ n M ) M . (3.12)By the uniform continuity property of J q on bounded sets of E ∗ , we have k u n +1 − u n k = k J q ( J p u n +1 ) − J q ( J p u n ) k ≤ ψ ( λ n M ) , such that k u n +1 − u ∗ k − k u n − u ∗ k ≤ ψ ( λ n M ) , which gives k u n − u ∗ k ≥ k u n +1 − u ∗ k − ψ ( λ n M ) . (3.13) AND O.T. MEWOMO From Lemma 2.4, k u n +1 − u ∗ k p ≥ φ p ( u ∗ , u n +1 ) − pq k u ∗ k≥ r − pq k u ∗ k≥ (cid:18) δ p + pq k u ∗ k (cid:19) − pq k u ∗ k≥ δ p . So, k u n +1 − u ∗ k ≥ δ. Therefore, the inequality (3.13) becomes, k u n − u ∗ k ≥ δ − ψ ( λ n M ) ≥ δ . Thus, Φ ( k u n − u ∗ k ) ≥ Φ ( δ . (3.14)Substituting (3.14) into (3.12) gives φ p ( u ∗ , u n +1 ) ≤ φ p ( u ∗ , u n ) − pλ n Φ ( δ pλ n θ n ( φ p ( u ∗ , u ) − φ p ( u ∗ , u n ))+ pλ n ψ ( λ n M ) M . (3.15)Similarly, φ q ( v ∗ , v n +1 ) = φ p ( v ∗ , J q ( J p v n − λ n ( Kv n + θ n ( J q v n − J q v ))))= V p ( v ∗ , J q v n − λ n ( Kv n + θ n ( J q v n − J q v ))) ≤ V p ( v ∗ , J q v n ) − pλ n h J p ( J q v n − λ n ( Kv n + θ n ( J q v n − J q v ))) − v ∗ , Kv n + θ n ( J q v n − J q v ) i = φ p ( v ∗ , v n ) − pλ n h v n − v ∗ , Kv n + θ n ( J q v n − J q v ) i− pλ n h J p ( J q v n − λ n ( Kv n + θ n ( J q v n − J q v ))) − v n , Kv n + θ n ( J q v n − J q v ) i . By Schwartz inequality and uniform continuity property of J on bounded subsets of E (Lemma 2.11),we obtain φ p ( v ∗ , v n +1 ) ≤ φ p ( v ∗ , v n ) − pλ n h v n − v ∗ , F x n + θ n ( J p u n − J p u ) i + pλ n ψ ( λ n M ) M (By applying inequality (3.11)) ≤ φ p ( u ∗ , u n ) − pλ n h u n − u ∗ , Kv n − Kv ∗ i since v ∗ ∈ N ( K )) − pλ n θ n h v n − v ∗ , J q v n − J q v i + pλ n ψ ( λ n M ) M . By Lemma 2.5, p h v n − v ∗ , J q v − J q v n i ≤ φ p ( v ∗ , v ) − φ p ( v ∗ , v n ) . Also, since K is generalized Φ-stronglymonotone, we have, φ p ( v ∗ , v n +1 ) ≤ φ p ( v ∗ , v n ) − pλ n Φ ( k v n − v ∗ k ) + pλ n θ n h v n − v ∗ , J q v − J q v n i + pλ n ψ ( λ n M ) M ≤ φ p ( v ∗ , v n ) − pλ n Φ ( k v n − v ∗ k ) + pλ n θ n ( φ p ( v ∗ , v ) − φ p ( v ∗ , v n )) + pλ n ψ ( λ n M ) M . (3.16)By the uniform continuity property of J p on bounded sets of E ∗ , we have k v n +1 − v n k = k J p ( J q v n +1 ) − J p ( J q v n ) k ≤ ψ ( λ n M ) , such that k v n +1 − v ∗ k − k v n − v ∗ k ≤ ψ ( λ n M ) , N GENERALIZED Φ-STRONGLY MONOTONE MAPPINGS AND ALGORITHMS 11 which gives k v n − v ∗ k ≥ k v n +1 − v ∗ k − ψ ( λ n M ) . (3.17)From Lemma 2.4, k v n +1 − v ∗ k p ≥ φ p ( v ∗ , v n +1 ) − pq k v ∗ k≥ r − pq k u ∗ k≥ (cid:18) δ p + pq k v ∗ k (cid:19) − pq k v ∗ k≥ δ p . So, k v n +1 − v ∗ k ≥ δ. Therefore, the inequality (3.17) becomes, k v n − v ∗ k ≥ δ − ψ ( λ n M ) ≥ δ . Thus, Φ ( k v n − v ∗ k ) ≥ Φ ( δ . (3.18)Substituting (3.18) into (3.16) gives φ p ( v ∗ , v n +1 ) ≤ φ p ( v ∗ , v n ) − pλ n Φ ( δ pλ n θ n ( φ p ( v ∗ , v ) − φ p ( v ∗ , v n ))+ pλ n ψ ( λ n M ) M . (3.19)Add (3.15) and (3.19) gives r < ∧ p ( w ∗ , w n +1 ) ≤ ∧ p ( w ∗ , w n ) − pλ n Φ( δ pλ n θ n ( ∧ p ( w ∗ , w ) − ∧ p ( w ∗ , w n )) + pλ n ψ ( λ n M ) M ≤ ∧ p ( w ∗ , w n ) − pλ n Φ( δ pλ n θ n ( ∧ p ( w ∗ , w ) − ∧ p ( w ∗ , w n )) + pλ n γ M ≤ ∧ p ( w ∗ , w n ) − pλ n δ pλ n θ n ( ∧ p ( w ∗ , w ) − ∧ p ( w ∗ , w n )) ≤ r − pλ n δ < r, a contradiction. Hence, ∧ p ( w ∗ , w n +1 ) ≤ r. By induction, ∧ p ( w ∗ , w n ) ≤ r ∀ n ∈ N . Thus, from inequality(2.1), { w n } is bounded. Part 2:
Define A : W → W ∗ by Aw = ( F u − v, Kv + u ) , ∀ w = ( u, v ) ∈ W. We show that { w n } strongly converges to a solution of Aw = 0 . Since A satisfies the range condition (Lemma 3.3) and bythe strict convexity of X (Lemma 2.7), we obtain for every t >
0, and w ∈ W , there exists a unique w t ∈ D ( A ), where D ( A ) is the domain of A such that J Wp w ∈ J Wp w t + tAw t . Taking J t w = w t , then we define a single-valued mapping J t : E → D ( A ) by J t = ( J Wp + tA ) − J Wp .Such a J t is called the resolvent of A . Therefore, by Theorem 2.13, for each n ∈ N , there exists aunique x n ∈ D ( A ) such that, x n = ( J Wp + 1 θ n A ) − J Wp w . AND O.T. MEWOMO Then, setting x n := ( y n , z n ) ∈ E × E ∗ and w := ( u , v ) ∈ E × E ∗ , we have( y n , z n ) = ( J Wp + 1 θ n A ) − J Wp ( u , v ) , which is equivalent to ( J Wp + 1 θ n A )( y n , z n ) = J Wp ( u , v ) . Since A ( y n , z n ) = ( F y n − z n , Kz n + y n ), then, J p y n + 1 θ n ( F y n − z n ) = J p u ,J q z n + 1 θ n ( Kz n + y n ) = J q v , and these lead to(3.20) θ n ( J p y n − J p u ) + F y n − z n = 0 , (3.21) θ n ( J q z n − J q v ) + Kz n + y n = 0 . Notice that the sequences { y n } and { z n } are bounded because they are convergent sequences byTheorem 2.13. Moreover, by Theorem 2.13, lim x n ∈ A −
0. Let y n → u ∗ and z n → v ∗ , then u ∗ in E solves the equation u + KF u = 0 if and only if x ∗ = ( u ∗ , v ∗ ) is a solution of Ax = 0 in W for v ∗ = F u ∗ ∈ E ∗ . The implication is that
F u ∗ − v ∗ = 0 ,Kv ∗ + u ∗ = 0 . Following the same arguments as in part 1, we get,(3.22) φ p ( y n , u n +1 ) ≤ φ p ( y n , u n ) − pλ n h u n − y n , F u n − v n + θ n ( J u n − J u ) i + pλ n ψ ( λ n M ) M and(3.23) φ p ( z n , v n +1 ) ≤ φ p ( z n , v n ) − pλ n h v n − z n , Kv n + u n + θ n ( J q v n − J q v ) i + pλ n ψ ( λ n M ) M . By Theorem 2.10 and Eq. (3.20), we obtain by the generalized Φ-strongly monotonicity of F, and h u n − y n , F u n − v n + θ n ( J p u n − J p u ) i = h x n − y n , F u n − v n + θ n ( J p u n − J p y n + J p y n − J p u ) i = θ n h u n − y n , J p u n − J p y n i + h u n − y n , F u n − v n + θ n ( J p y n − J p u ) i = θ n h u n − y n , J p u n − J p y n i + h u n − y n , F u n − v n − ( F y n − z n ) i≥ θ n g ( k u n − y n k ) + Φ( k u n − y n k ) + h u n − y n , z n − v n i≥ p θ n φ p ( y n , u n ) + h u n − y n , z n − v n i (by Lemma 2.4 for some real constants p > . This makes the inequality (3.22) to become(3.24) φ p ( y n , u n +1 ) ≤ (1 − λ n θ n ) φ p ( y n , u n ) − pλ n h u n − y n , z n − v n i + pλ n ψ ( λ n M ) M . From Lemma 2.5, we obtain that φ p ( y n , u n ) ≤ φ p ( y n − , u n ) − p h y n − u n , J p y n − − J p y n i = φ p ( y n − , u n ) + p h u n − y n , J p y n − − J p y n i≤ φ p ( y n − , u n ) + k J p y n − − J p y n kk u n − y n k . (3.25)Let R > k x k ≤ R, k y n k ≤ R for all n ∈ N . Then the estimates below follows from (3.20), J p y n − − J p y n + 1 θ n ( F y n − − z n − − ( F y n − z n ) = θ n − − θ n θ n ( J p u − J p y n − ) . N GENERALIZED Φ-STRONGLY MONOTONE MAPPINGS AND ALGORITHMS 13
Taking the duality pairing of each side of this equation with respect to y n − − y n and using thegeneralized Φ-strongly monotonicity property of F , then h J p y n − − J p y n , y n − − y n i ≤ θ n − − θ n θ n k J p u − J p y n − kk y n − − y n k , which gives,(3.26) k J p y n − − J p y n k ≤ (cid:18) θ n − θ n − (cid:19) k J p y n − − J p u k . Using (3.25) and (3.26), the inequality (3.22) becomes(3.27) φ p ( y n , u n +1 ) ≤ (1 − λ n θ n ) φ p ( y n − , u n ) + C (cid:18) θ n − θ n − (cid:19) − pλ n h u n − y n , z n − v n i + pλ n ψ ( λ n M ) M , for some constant C > . Similar analysis gives that(3.28) φ p ( z n , v n +1 ) ≤ (1 − λ n θ n ) φ p ( z n − , v n ) + C (cid:18) θ n − θ n − (cid:19) − pλ n h v n − z n , u n − y n i + pλ n ψ ( λ n M ) M , for some constant C >
0. Since ψ := ψ + ψ , M := M + M and ψ ( λ n M ) ≤ γ , adding (3.26) and(3.28) generates ∧ ( x n , w n +1 ) ≤ (1 − λ n θ n ) ∧ ( x n − , w n ) + C (cid:18) θ n − θ n − (cid:19) + pλ n γ M , where C := C + C > . By Lemma 2.6, φ ( x n − , w n ) → n → ∞ and using Lemma 2.12, we havethat w n − x n − → n → ∞ . Since by Theorem 2.13, x n → w ∗ ∈ N ( A ), we obtain that w n → w ∗ as n → ∞ . But w n = ( u n , v n ) and w ∗ = ( u ∗ , v ∗ ), this implies that u n → u ∗ with u ∗ the solution of theHammerstein equation. (cid:3) Corollary 3.6.
Let E be a uniformly smooth and uniformly convex real Banach space with the dualspace E ∗ . Suppose F : E → E ∗ and K : E ∗ → E are bounded and strongly monotone mappings. Define { u n } and { v n } iteratively for arbitrary u ∈ E and v ∈ E ∗ by (3.29) u n +1 = J q ( J p u n − λ n ( F u n − v n + θ n ( J p u n − J p u ))) , n ∈ N , (3.30) v n +1 = J p (cid:0) J ∗ q v n − λ n (cid:0) Kv n + u n + θ n ( J ∗ q v n − J ∗ q v ) (cid:1)(cid:1) , n ∈ N , where J p : E → E ∗ is the generalized duality mapping with the inverse, J q : E ∗ → E and the realsequences { λ n } and { θ n } in (0 , are such that, (i) lim θ n = 0 and { θ n } is decreasing; (ii) ∞ X n =1 λ n θ n = ∞ ; (iii) lim n →∞ (( θ n − /θ n ) − /λ n θ n = 0 , ∞ X n =1 λ n < ∞ .Suppose that u + KF u = 0 has a solution in E . There exists a real constant γ > with ψ ( λ n M ) ≤ γ , n ∈ N for some constant M > . Then, the sequence { u n } converges strongly to the solution of u + KF u .Proof.
Define Φ ( k u − u k ) := k k u − u k and Φ ( k v − v k ) := k k v − v k for some constants k , k ∈ (0 ,
1) and let W := E × E ∗ with norm k w k W := k u k E + k v k E ∗ ∀ w = ( u, v ) ∈ W. The resultfollows from Theorem 3.5. (cid:3) AND O.T. MEWOMO Corollary 3.7.
Chidume and Idu [10] . Let E be a uniformly convex and uniformly smooth realBanach space and F : E → E ∗ , K : E ∗ → E be maximal monotone and bounded maps, respectively.For ( x , y ) , ( u , v ) ∈ E × E ∗ , define the sequences { u n } and { v n } in E and E ∗ respectively, by (3.31) u n +1 = J − ( J u n − λ n ( F u n − v n ) − λ n θ n ( J u n − J x )) , n ∈ N , (3.32) v n +1 = J (cid:0) J − v n − λ n ( Kv n + u n ) − λ n θ n ( J − v n − J − y ) (cid:1) , n ∈ N , where { λ n } and { θ n } are real sequences in (0 , satisfying the following conditions: (i) ∞ X n =1 λ n θ n = ∞ , (ii) λ n M ∗ ≤ γ θ n ; δ − E ( λ n M ∗ ) ≤ γ θ n , (iii) δ − E (cid:16) θn − − θnθn K (cid:17) λ n θ n → ; δ − E ∗ (cid:16) θn − − θnθn K (cid:17) λ n θ n → as n → ∞ , (iv) θ n − − θ n θ n K ∈ (0 , ,for some constants M ∗ > and γ > , where δ E : (0 , ∞ ) → (0 , ∞ ) is the modulus of convexity of E and K := 4 RL sup {k J x − J y k : k x k ≤ R, k y k ≤ R } + 1 , x, y ∈ E, R > . Assume that the equation u + KF u = 0 has a solution. Then the sequences { u n } ∞ n =1 and { v n } ∞ n =1 converge strongly to u ∗ and v ∗ , respectively, where u ∗ is the solution of u + KF u = 0 with v ∗ = F u ∗ .Proof. From Lemma 2.8, we see that T : E × E ∗ → E ∗ × E defined by T ( u, v ) = ( J u − F u + v, J − v − Kv − u ) for all ( u, v ) ∈ E × E ∗ is J -pseudocontractive and A := ( J − T ) is maximal monotone.Therefore, the iterative sequences (3.31) and (3.32) are respectively equivalent to(3.33) u n +1 = J − ( J u n − λ n ( F u n + θ n ( J u n − J x ))) , n ∈ N and(3.34) v n +1 = J (cid:0) J − v n − λ n (cid:0) Kv n + θ n ( J − v n − J − y ) (cid:1)(cid:1) , n ∈ N , where J : E → E ∗ is the normalized duality mapping with the inverse, J − : E ∗ → E. Hence, theresult follows from Theorem 3.5. (cid:3)
Remark 3.8.
Prototype for our iteration parameters in Theorem 3.5 are, λ n = n +1) a and θ n = n +1) b ,where 0 < b < a and a < Conclusion 3.9.
We have considered the class of generalized Φ-strongly monotone mappings in Banachspaces. This is the class of monotone-type mappings such that if a solution of the equation 0 ∈ Ax exists, it is necessarily unique. Our results generalize and improve the recent and important results ofChidume and Idu [10]. Also, our results show extention and application of the main results of Aibinuand Mewomo [1, 2]. Acknowledgment:
The first author acknowledges with thanks the bursary and financial support from Department ofScience and Technology and National Research Foundation, Republic of South Africa Center of Ex-cellence in Mathematical and Statistical Sciences (DST-NRF CoE-MaSS) Doctoral Bursary. Opinionsexpressed and conclusions arrived at are those of the authors and are not necessarily to be attributedto the CoE-MaSS.
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