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In [11], the author discussed a new class of nearly weak uniformly L-Lipschitzian mappings and prove some strong convergence results of the modified Ishikawa iteration with errors in real Banach spaces. And the author has given the open problem as follows: Are there any difference on convergence between the Mann iteration and Ishikawa iteration? Can we prove the equivalence on convergence between these two iterations? In this paper, we given an affirmative answer to the open problem.
This paper is devoted to study the existence of solutions for a class of initial value problems for non-instantaneous impulsive fractional differential equations involving the Caputo fractional derivative in a Banach space. The arguments are based upon Monch's fixed point theorem and the technique of measures of noncompactness.
Content available remote Approximate property of a functional equation with a general involution
In this paper, we prove the Hyers-Ulam stability of the functional equation f(x + y, z + w) + f(x + σ (y), z + τ(w)) = 2f(x, z) + 2f(y, w), where σ, τ are involutions.
Content available remote The Knaster-Tarski theorem versus monotonne nonexpansive mappings
Let X be a partially ordered set with the property that each family of order intervals of the form [a, b], [a,→) with the finite intersection property has a nonempty intersection. We show that every directed subset of X has a supremum. Then we apply the above result to prove that if X is a topological space with a partial order ⪯ for which the order intervals are compact, F is a nonempty commutative family of monotone maps from X into X and there exists c ∈ X such that c ⪯ Tc for every T ∈ F, then the set of common fixed points of F is nonempty and has a maximal element. The result, specialized to the case of Banach spaces, gives a general fixed point theorem for monotone mappings that drops many assumptions from several recent results in this area. An application to the theory of integral equations of Urysohn’s type is also given.
Content available remote An abstract nonlocal functional-differential second order evolution problem
The aim of the paper is to prove two theorems on the existence and uniqueness of mild and classical solutions of a semilinear functional-differential evolution second order equation together with nonlocal conditions. The theory of strongly continuous cosine families of linear operators in a Banach space is applied. The paper is based on publications [1–9] and is a generalization of paper [6].
W artykule udowodniono dwa twierdzenia o istnieniu i jednoznaczności całkowych i klasycznych rozwiązań semiliniowego funkcjonalno-różniczkowego zagadnienia ewolucyjnego rzędu drugiego z warunkami nielokalnymi. W tym celu zastosowano teorię rodziny cosinus liniowych operatorów w przestrzeni Banacha. Artykuł bazuje na publikacjach [1–9] i jest pewnym uogólnieniem publikacji [6].
The fractional derivative of the Riemann-Liouville and Caputo types played an important role in the development of the theory of fractional derivatives, integrals and for its applications in pure mathematics ([18], [21]). In this paper, we study the existence of weak solutions for fractional differential equations of Riemann-Liouville and Caputo types. We depend on converting of the mentioned equations to the form of functional integral equations of Volterra-Stieltjes type in reflexive Banach spaces.
Content available On regulated functions
The main purpose of this review article is to present the concept of a regulated function and to indicate the connection of the class of regulated functions with other significant classes of functions. In particular, we give a characterization of regulated functions in terms of step functions and we show that the linear space of regulated functions forms a Banach space under the classical supremum norm.
In the present paper we provide some existence results and Ulam’s type stability concepts for the Darboux problem of partial fractional random differential equations in Banach spaces, by applying the measure of noncompactness and a random fixed point theorem with stochastic domain.
The main objective of this article is to present the state of the art concerning approximate controllability of dynamic systems in infinite-dimensional spaces. The presented investigation focuses on obtaining sufficient conditions for approximate controllability of various types of dynamic systems using Schauder’s fixed-point theorem. We describe the results of approximate controllability for nonlinear impulsive neutral fuzzy stochastic differential equations with nonlocal conditions, impulsive neutral functional evolution integro-differential systems, stochastic impulsive systems with control-dependent coefficients, nonlinear impulsive differential systems, and evolution systems with nonlocal conditions and semilinear evolution equation.
The main aim of this article is to review the existing state of art concerning the complete controllability of semilinear dynamical systems. The study focus on obtaining the sufficient conditions for the complete controllability for various systems using the Banach fixedpoint theorem. We describe the results for stochastic semilinear functional integro-differential system, stochastic partial differential equations with finite delays, semilinear functional equations, a stochastic semilinear system, a impulsive stochastic integro-differential system, semilinear stochastic impulsive systems, an impulsive neutral functional evolution integro-differential system and a nonlinear stochastic neutral impulsive system. Finally, two examples are presented.
We propose a method of solving the problem with non-homogeneous integral condition for homogeneous evolution equation with abstract operator in a linear space H. For right-hand side of the integral condition which belongs to the special subspace H ⊆ L, in which the vectors are represented using Stieltjes integrals over a certain measure, the solution of the problem is represented in the form of Stieltjes integral over the same measure.
Content available About differentiability and V BG * class
Let X be a finite dimensional real Banach space. We show that if the contingent of the curve Γ : (a, b) → X fulfils some conditions then each parametrization of that curve is V BG * . Stanisław Saks proved that each V BG * function is differentiable at a set of full Lebesgue measure. The result of this paper is a partial converse of that theorem.
Content available remote On derivations of operator algebras with involution
The purpose of this paper is to prove the following result. Let X be a complex Hilbert space, let L(X) be an algebra of all bounded linear operators on X and let A(X) (…) L(X) be a standard operator algebra, which is closed under the adjoint operation. Suppose there exists a linear mapping D : A(X) → L(X) satisfying the relation 2D(AA*A) = D(AA*)A + AA*D(A) + D(A)A*A + AD(A*A) for all A (…) A(X). In this case, D is of the form D(A) = [A,B] for all A (…) A(X) and some fixed B (…) L(X), which means that D is a derivation.
We say that a function f from [0,1] to a Banach space X is increasing with respect to E⊂X* if x∗∘f is increasing for every x*∈E. A function f:[0,1]m→X is separately increasing if it is increasing in each variable separately. We show that if X is a Banach space that does not contain any isomorphic copy of c0 or such that X* is separable, then for every separately increasing function f:[0,1]m→X with respect to any norming subset there exists a separately increasing function g:[0,1]m→R such that the sets of points of discontinuity of f and g coincide.
Content available remote Solvability of certain sequence spaces inclusion equations with operators
In this paper, we deal with sequence spaces inclusion equations (SSIE), which are determined by an inclusion where each term is a sum or a sum of products of sets of the form a(T) and f(x)(T) where f maps U+ to itself, and (...), the sequence x is the unknown and T is a given triangle. Here, we determine the set of all sequences x with positive entries such that (…) and (…) where (...). We are led to study, among other things, the inclusion equations (…) and (…) where (…) is the operator of first differences defined by (…) for (…) with (…). The first (SSIE) leads to determine the set of all sequences x such that (…) and (…) implies (…). These results generalize some of the results given in [1].
Content available remote Fixed point iterations of three asymptotically pseudocontractive mappings
In this paper, we establish the strong convergence for a modified three-step iterative scheme with errors associated with three mappings in real Banach spaces. Moreover, our technique of proofs is of independent interest. Remark at the end simplifies many known results.
Content available An abstract nonlocal second order evolution problem
The aim of the paper is to prove theorems on the existence and uniqueness of mild and classical solutions of a semilinear evolution second order equation together with nonlocal conditions. The theory of strongly continuous cosine families of linear operators in a Banach space is applied.
Content available remote Existence of solutions of the dynamic Cauchy problem in Banach spaces
In this paper we obtain the existence of solutions and Carathéodory type solutions of the dynamic Cauchy problem in Banach spaces for functions defined on time scales (…), where f is continuous or f satisfies Carathéodory conditions and some conditions expressed in terms of measures of noncompactness. The Mönch fixed point theorem is used to prove the main result, which extends these obtained for real valued functions.
Let f be a martingale taking values in a Banach space B and let S(f) be its square function. We show that if B is a Hilbert space, then P(S(f) ≥1)≤√e∥f∥1and the constant √e is the best possible. This extends the result of Cox, who established this bound in the real case. Next, we show that this inequality characterizes Hilbert spaces in the following sense: if B is not a Hilbert space, then there is a martingale f for which the above weak-type estimate does not hold.
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