On two pursuit differential game problems with state and geometric constraints in a Hilbert space

This paper concerns with the study of two pursuit differential game problems of many pursuers and many evaders on a nonempty closed convex subset of R^n. Throughout the period of the games, players must stay within the given closed convex set. Players’ laws of motion are defined by certain first order differential equations. Control functions of the pursuers and evaders are subject to geometric constraints. Pursuit is said to be completed if the geometric position of each of the evader coincides with that of a pursuer. We proved two theorems each of which is solution to a problem. Sufficient conditions for the completion of pursuit are provided in each of the theorems. Moreover, we constructed strategies of the pursuers that ensure completion of pursuit.

Halpern–Ishikawa type iterative schemes for approximating fixed points of multi-valued non-self mappings

Let C be a nonempty closed convex subset of a real Hilbert space H and $$T: C\rightarrow CB(H)$$ T : C → C B ( H ) be a multi-valued Lipschitz pseudocontractive nonself mapping. A Halpern–Ishikawa type iterative scheme is constructed and a strong convergence result of this scheme to a fixed point of T is proved under appropriate conditions. Moreover, an iterative method for approximating a fixed point of a k-strictly pseudocontractive mapping $$T: C\rightarrow Prox(H)$$ T : C → P r o x ( H ) is constructed and a strong convergence of the method is obtained without end point condition. The results obtained in this paper improve and extend known results in the literature.

A Purusit Differential Game Problem on a Closed Convex Subset of a Hilbert Space

We study a pursuit differential game problem with finite number of pursuers and one evader on a nonempty closed convex subset of the Hilbert space l2. Players move according to certain first order ordinary differential equations and control functions of the pursuers and evader are subject to integral constraints. Pursuers win the game if the geometric positions of a pursuer and the evader coincide. We formulated and prove theorems that are concern with conditions that ensure win for the pursuers. Consequently, wining strategies of the pursuers are constructed. Furthermore, illustrative example is given to demonstrate the result.

Stability And Data Dependence Results For The Mann Iteration Schemes on n-Banach Space

Let be an n-Banach space, M be a nonempty closed convex subset of , and S:M→M be a mapping that belongs to the class mapping. The purpose of this paper is to study the stability and data dependence results of a Mann iteration scheme on n-Banach space

Mann Iteration Processes on Uniform Convex n-Banach Space

Let Y be a"uniformly convex n-Banach space, M be a nonempty closed convex subset of Y, and S:M→M be adnonexpansive mapping. The purpose of this paper is to study some properties of uniform convex set that help us to develop iteration techniques for1approximationjof"fixed point of nonlinear mapping by using the Mann iteration processes in n-Banachlspace.

On nonconvex retracts in normed linear spaces

Let E be a real normed linear space. A subset X ⊂ E is called a retract of E if there exists a continuous mapping r : E → X, a retraction, satisfying r(x) = x, x ∈ X. It is well known that every nonempty closed convex subset of E is a retract of E. Nonconvex retracts are studied in this paper.

Strong convergence results for nonlinear mappings in real Banach spaces

Let X be a real Banach space, K be a nonempty closed convex subset of X, T : K → K be a nearly uniformly L-Lipschitzian mapping with sequence {an}. Let kn ⊂ [1, ∞) and En be sequences with limn→∞ kn = 1, limn→∞ En = 0 and F(T) = {ρ ∈ K : T ρ = ρ} 6= ∅. Let {αn}n≥0 be real sequence in [0, 1] satisfying the following conditions: (i)P n≥0 αn = ∞ (ii) limn→∞ αn = 0. For arbitrary x0 ∈ K, let {xn}n≥0 be iteratively defined by xn+1 = (1 − αn)xn + αnT nxn, n ≥ 0. If there exists a strictly increasing function Φ : [0, ∞) → [0, ∞) with Φ(0) = 0 such that < T nx − T nρ, j(x − ρ) >≤ knkx − ρk 2 − Φ(kx − ρk) + En for all x ∈ K, then, {xn}n≥0 converges strongly to ρ ∈ F(T). It is also proved that the sequence of iteration {xn} defined by xn+1 = (1 − bn − dn)xn + bnT nxn + dnwn, n ≥ 0, where {wn}n≥0 is a bounded sequence in K and {bn}n≥0, {dn}n≥0 are sequences in [0,1] satisfying appropriate conditions, converges strongly to a fixed point of T.

A convergence theorem of multi-step iterative scheme for nonlinear maps

Let K be a nonempty closed convex subset of a real Banach space X,T:K ? K a nearly uniformly L-Lipschitzian (with sequence {rn}) asymptotically generalized ?-hemicontractive mapping (with sequence kn ? [1,?), lim n?? kn = 1) such that F(T) = {p?K:Tp=p}. Let {?n}n?0, {?kn}n?0 be real sequences in [0,1] satisfying the conditions: (i) ?n?0 ?n = 1 (ii) limn?? ?n, ?kn = 0, k = 1, 2,..., p?1. For arbitrary x0 ? K, let {xn}n?0 be a multi-step sequence iteratively defined by xn+1=(1??n)xn + ?nTny1n, n?0, ykn = (1 ? ?kn )xn + ?kn Tnyk+1n, k = 1,2,..., p?2 (0.1), yp?1n=(1? ?p?1n)xn + ?p?1n Tnxn, n ? 0, p ? 2. Then, {xn}n?0 converges strongly to p ? F(T). The result proved in this note significantly improve the results of Kim et al. [2].

Convergence Theorems for Fixed Points of Multivalued Mappings in Hilbert Spaces

Let H be a real Hilbert space and K a nonempty closed convex subset of H. Suppose T:K→CB(K) is a multivalued Lipschitz pseudocontractive mapping such that F(T)≠∅. An Ishikawa-type iterative algorithm is constructed and it is shown that, for the corresponding sequence {xn}, under appropriate conditions on the iteration parameters, lim infn→∞⁡ d (xn,Txn)=0 holds. Finally, convergence theorems are proved under approximate additional conditions. Our theorems are significant improvement on important recent results of Panyanak (2007) and Sastry and Babu (2005).

Invariant Means and Reversible Semigroup of Relatively Nonexpansive Mappings in Banach Spaces

The purpose of this paper is to study modified Halpern type and Ishikawa type iteration for a semigroup of relatively nonexpansive mappingsI={T(s):s∈S}on a nonempty closed convex subsetCof a Banach space with respect to a sequence of asymptotically left invariant means{μn}defined on an appropriate invariant subspace ofl∞(S), whereSis a semigroup. We prove that, given some mild conditions, we can generate iterative sequences which converge strongly to a common element of the set of fixed pointsF(I), whereF(I)=⋂{F(T(s)):s∈S}.