Iterative algorithms for infinite accretive mappings and applications to p-Laplacian-like differential systems

Fixed Point Theory and Applications, Jan 2016

Some new iterative algorithms with errors for approximating common zero point of an infinite family of m-accretive mappings in a real Banach space are presented. A path convergence theorem and some new weak and strong convergence theorems are proved by means of some new techniques, which extend the corresponding works by some authors. As applications, an infinite p-Laplacian-like differential system is investigated, from which we construct an infinite family of m-accretive mappings and discuss the connections between the equilibrium solution of the differential systems and the zero point of the m-accretive mappings.

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Iterative algorithms for infinite accretive mappings and applications to p-Laplacian-like differential systems

Wei and Agarwal Fixed Point Theory and Applications Iterative algorithms for infinite accretive mappings and applications to p-Laplacian-like differential systems Li Wei 2 Ravi P Agarwal 0 1 0 Department of Mathematics, Faculty of Science, King Abdulaziz University , Jeddah, 21589 , Saudi Arabia 1 Department of Mathematics, Texas A&M University-Kingsville , Kingsville, TX 78363 , USA 2 School of Mathematics and Statistics, Hebei University of Economics and Business , Shijiazhuang, 050061 , China Some new iterative algorithms with errors for approximating common zero point of an infinite family of m-accretive mappings in a real Banach space are presented. A path convergence theorem and some new weak and strong convergence theorems are proved by means of some new techniques, which extend the corresponding works by some authors. As applications, an infinite p-Laplacian-like differential system is investigated, from which we construct an infinite family of m-accretive mappings and discuss the connections between the equilibrium solution of the differential systems and the zero point of the m-accretive mappings. accretive mapping; gauge function; contraction; common zero point; retraction; p-Laplacian-like differential systems 1 Introduction Let E be a real Banach space with norm · and let E∗ denote the dual space of E. We use ‘→’ and ‘ ’ (or ‘w-lim’) to denote strong and weak convergence, respectively. We denote the value of f ∈ E∗ at x ∈ E by x, f . Define a function ρE : [, +∞) → [, +∞) called the modulus of smoothness of E as follows: ρE(t) = sup x + y + x – y  –  : x, y ∈ E, x = , y ≤ t . A Banach space E is said to be uniformly smooth if ρEt(t) → , as t → . A Banach space E is said to be strictly convex if and only if x = y = ( – λ)x + λy for x, y ∈ E and  < λ <  implies that x = y. A Banach space E is said to be uniformly convex if for any ε ∈ (, ] there exists δ >  such that x = y = , x – y ≥ ε ⇒ x + y  ≤  – δ. It is well known that a uniformly convex Banach space is reflexive and strictly convex. An operator B : E → E∗ is said to be monotone if u – v, Bu – Bv ≥ , for all u, v ∈ D(B). The monotone operator B is said to be maximal monotone if the graph of B, G(B), is not contained properly in any other monotone subset of E × E∗. A single-valued mapping F : D(F) = E → E∗ is said to be hemi-continuous [] if w-limt→ F(x + ty) = Fx, for any x, y ∈ E. A single-valued mapping F : D(F) = E → E∗ is said to be demi-continuous [] if w-limn→∞ Fxn = Fx, for any sequence {xn} strongly convergent to x in E. Following from [] or [], the function h is said to be a proper convex function on E if h is defined from E onto (–∞, +∞], h is not identically +∞ such that h(( – λ)x + λy) ≤ ( – λ)h(x) + λh(y), whenever x, y ∈ E and  ≤ λ ≤ . h is said to be strictly convex if h(( – λ)x + λy) < ( – λ)h(x) + λh(y), for all  < λ <  and x, y ∈ E with x = y, h(x) < +∞ and h(y) < +∞. The function h : E → (–∞, +∞] is said to be lower-semi-continuous on E if lim infy→x h(y) ≥ h(x), for any x ∈ E. A continuous strictly increasing function ϕ : [, +∞) → [, +∞) is called a gauge function [] if ϕ() =  and ϕ(t) → ∞, as t → ∞. The duality mapping Jϕ : E → E∗ associated with the gauge function ϕ is defined by [] Jϕ(x) = f ∈ E∗ : x, f = x ϕ x , f = ϕ x , x ∈ E. It can be seen from [] that the duality mapping Jϕ has the following properties: (i) Jϕ (–x) = –Jϕ(x) and Jϕ (kx) = ϕϕ(( kxx )) Jϕ (x), for ∀x ∈ E and k > ; (ii) if E∗ is uniformly convex, then Jϕ is uniformly continuous on each bounded subset in E; (iii) the reflexivity of E and strict convexity of E∗ imply that Jϕ is single-valued, monotone and demi-continuous. In the case ϕ(t) ≡ t, we call Jϕ the normalized duality mapping, which is usually denoted by J . For the gauge function ϕ, the function : [, +∞) → [, +∞) defined by is a continuous convex strictly increasing function on [, +∞). Following the result in [], a Banach space E is said to have a weakly continuous duality mapping if there is a gauge ϕ for which the duality mapping Jϕ(x) is single-valued and weak-to-weak∗ sequentially continuous (i.e., if {xn} is a sequence in E weakly convergent to a point x, then the sequence Jϕ(xn) converges weakly∗ to Jϕ (x)). It is well known that lp has a weakly continuous duality mapping with a gauge function ϕ(t) = tp– for all  < p < +∞. Let C be a nonempty, closed and convex subset of E and Q be a mapping of E onto C. Then Q is said to be sunny [] if Q(Q(x) + t(x – Q(x))) = Q(x), for all x ∈ E and t ≥ . A mapping Q of E into E is said to be a retraction [] if Q = Q. If a mapping Q is a retraction, then Q(z) = z for every z ∈ R(Q), where R(Q) is the range of Q. A mapping f : C → C is called a contraction with contractive constant k ∈ (, ) if f (x) – f (y) ≤ k x – y , for ∀x, y ∈ C. A mapping T : C → C is said to be nonexpansive if Tx – Ty ≤ x – y , for ∀x, y ∈ C. We use Fix(T ) to denote the fixed point set of T . That is, Fix (...truncated)


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Li Wei, Ravi P Agarwal. Iterative algorithms for infinite accretive mappings and applications to p-Laplacian-like differential systems, Fixed Point Theory and Applications, 2016, pp. 5, Volume 2016, Issue 1, DOI: 10.1186/s13663-015-0492-1