Banach Spaces Of Type Research Articles

Nonlocal Conditions for Semilinear Fractional Differential Equations With Hilfer Derivative

This paper studies the existence of solutions and some topological proprieties of solution sets for nonlocal semi fractional differential equations of Hilfer type in Banach space by using noncompact measure method in the weighted space of continuous functions. The main result is illustrated with the aid of an example

Topological Structure of Solutions Sets for Semilinear Evolution Inclusions

This paper deals with a semilinear evolution inclusion involving a nondensely defined closed linear operator satisfying the Hille–Yosida condition and source term of multivalued type in Banach spaces. The topological structure of the set of solutions is investigated in the case that semigroup is noncompact. It is shown that the solution set is nonempty, compact and an R_\delta -set. It is proved on compact intervals and then, using the inverse limit method, obtained on non-compact intervals. As a sample of application, we consider a parabolic partial differential inclusion at end of the paper.

A characterization of alternatively convex or smooth Banach spaces

In this paper, we give a characterization of alternatively convex or smooth Banach spaces. In fact we prove that every normaloid numerical radius attaining operator on a Banach space $X$ is radialoid if and only if $X$ is alternatively convex or smooth. In addition, we show that every compact normaloid operator on $X$ is radialoid if and only if every rank one normaloid operator on X is radialoid. Finally we present some types of Banach spaces on which the compact normaloid operators are radialoid.

On Nonhomogeneous Second Order Difference Inclusions of Accretive Type in Banach Spaces

ABSTRACTIn this paper, we investigate the existence, uniqueness and asymptotic behavior of solutions to a nonhomogeneous second order difference inclusion of accretive type in Banach spaces. We develop new methods and significantly extend and improve some known results in the literature. Our conclusions are new even for Hilbert spaces.

On multi-singular integral equations involving n weakly singular kernels

We deal with some sources of Banach spaces which are closely related to an important issue in applied mathematics i.e. the problem of existence and uniqueness of the solution for the very applicable weakly singular integral equations. In the classical mode, the uniform space (C[a,b], ||.||?) is usually applied to the related discussion. Here, we apply some new types of Banach spaces, in order to extend the area of problems we could discuss. We consider a very general type of singular integral equations involving n weakly singular kernels, for an arbitrary natural number n, without any restrictive assumption of differentiability or even continuity on engaged functions. We show that in appropriate conditions the following multi-singular integral equation of weakly singular type has got exactly a solution in a defined Banach space x(t) = ?p,i=1 ?i/?(^?i) ?t,0 fi(s,x(s)) (tn-tn-1)1-?i,n...(t1-s)1-?i,1 dt + ?(t). In particular we consider the famous fractional Langevin equation and by the method we could extend the region of variations of parameter ?+ ? from interval [0,1) in the earlier works to interval [0,2).

Globally attracting solutions to impulsive fractional differential inclusions of Sobolev type

AbstractWe study a generalized Cauchy problem associated with a class of impulsive fractional differential inclusions of Sobolev type in Banach spaces. Our aim is to prove the existence of a compact set of globally attracting solutions to the problem in question. An application to fractional partial differential equations subject to impulsive effects is given to illustrate our results.

Approximate Controllability for Fractional Differential Equations of Sobolev Type Via Properties on Resolvent Operators

Abstract This paper treats the approximate controllability of fractional differential systems of Sobolev type in Banach spaces. We first characterize the properties on the norm continuity and compactness of some resolvent operators (also called solution operators). And then via the obtained properties on resolvent operators and fixed point technique, we give some approximate controllability results for Sobolev type fractional differential systems in the Caputo and Riemann-Liouville fractional derivatives with order 1 < α < 2, respectively. Particularly, the existence or compactness of an operator E −1 is not necessarily needed in our results.

When the Karush–Kuhn–Tucker Theorem Fails: Constraint Qualifications and Higher-Order Optimality Conditions for Degenerate Optimization Problems

In this paper, we present higher-order analysis of necessary and sufficient optimality conditions for problems with inequality constraints. The paper addresses the case when the constraints are not assumed to be regular at a solution of the optimization problems. In the first two theorems derived in the paper, we show how Karush–Kuhn–Tucker necessary conditions reduce to a specific form containing the objective function only. Then we present optimality conditions of the Karush–Kuhn–Tucker type in Banach spaces under new regularity assumptions. After that, we analyze problems for which the Karush–Kuhn–Tucker form of optimality conditions does not hold and propose necessary and sufficient conditions for those problems. To formulate the optimality conditions, we introduce constraint qualifications for new classes of nonregular nonlinear optimization. The approach of p-regularity used in the paper can be applied to various degenerate nonlinear optimization problems due to its flexibility and generality.

Topological properties of solution sets for partial functional evolution inclusions

This paper deals with functional evolution inclusions of neutral type in Banach space when the semigroup is compact as well as noncompact. The topological properties of the solution set is investigated. It is shown that the solution set is nonempty, compact and an Rδ-set which means that the solution set may not be a singleton but, from the point of view of algebraic topology, it is equivalent to a point, in the sense that it has the same homology group as one-point space. As a sample of application, we consider a partial differential inclusion at end of the paper.

Analysis of Fractional Functional Differential Equations of Neutral Type with Nonlocal Conditions

This work deals with the existence of solutions for a class of nonlinear nonlocal fractional functional differential equations of neutral type in Banach spaces. In particular, we prove the existence of solutions with the assumptions that the nonlinear parts satisfy locally Lipschitz like conditions and closed linear operator \(-A(t)\) generates analytic semigroup for each \(t \ge 0\). We also investigate global existence of solution and study the continuous dependence of solution on initial data. We conclude the article with an application to the developed results.

Existence of Coupled Quasi-solutions of Nonlinear Integro-Differential Equations of Volterra Type in Banach Spaces

We study an initial value problem for a class of integro-differential equations of Volterra type in a real Banach space. Using method of upper and lower solutions and M¨ onch and Von Harten theorem, we obtain an existence theorem of coupled quasi-solutions, which is an extension of those established by Y. Chen and W. Zhuang in [1].

First-Order Regular and Degenerate Identification Differential Problems

We are concerned with both regular and degenerate first-order identification problems related to systems of differential equations of weakly parabolic type in Banach spaces. Several applications to partial differential equations and systems will be given in a subsequent paper to show the fullness of our abstract results.

Function Spaces and Applications

Function spaces and its various applications are subjects of investigation in many research centers and we are glad that this special issue has received the remarkable attention by researchers in these areas. We have received papers from wide range of topics of function spaces such as interpolation methods for stochastic processes spaces, vector optimization problems in Banach spaces, Marcinkiewicz integral operator in Morrey spaces, Hardy’s operator in the variable exponent Lebesgue spaces, and differential equations of weakly parabolic type in Banach spaces. A number of papers are devoted to investigating different kinds of problems in Hardy spaces or Hardy subspaces, for example, Calderon-Zygumnd singular integral operators in Hardy spaces, weighted Hardy spaces related to the Schrodinger operator, and composition operator in hyperbolic Bloch-type spaces. Geometric problems such as modulus of convexity in some Banach sequence spaces, p-uniform convexity, and quniform smoothness in C spaces are also covered by this issue.

On the Convergence of Successive Approximations for a Fractional Differential Equation in Banach Spaces

In this paper we present a convergence result for the successive approximations for a nonlinear fractional differential equation of Caputo type in Banach spaces. Also an example is given to illustrate our result.

Sobolev-Type Volterra-Fredholm Functional Integrodifferential Equations in Banach Spaces

This paper deals with the problems such as the existence, uniqueness, continuous dependence and boundedness of mild solution of Volterra-Fredholm func- tional integrodifferential equations of sobolev type in Banach spaces. Our analysis is based on Banach fixed point principle, the integral inequality established by B. G. Pachpatte, Gronwall-Bellman inequality and the semigroup theory.

Fixed Point Theorems for Contractions of Rational Type with PPF Dependence in Banach Spaces

We prove the existence of the PPF dependent fixed point in the Razumikhin class for contractions of rational type in Banach spaces, by using a general class of pairs of functions. Our result has as particular cases a great number of interesting consequences which extend and generalize some results appearing in the literature.

On the Complex Inversion Formula and Admissibility for a Class of Volterra Systems

This paper studies Volterra integral evolution equations of convolution type from the point of view of complex inversion formula and the admissibility in the Salamon-Weiss sens. We first present results on the validity of the inverse formula of the Laplace transform for the resolvent families associated with scalar Volterra integral equations of convolution type in Banach spaces, which extends and improves the results in Hille and Philllips (1957) and Cioranescu and Lizama (2003, Lemma 5), respectively, including the stronger version for a class of scalar Volterra integrodifferential equations of convolution type on unconditional martingale differences UMD spaces, provided that the leading operator generates aC0-semigroup. Next, a necessary and sufficient condition forLp-admissibilityp∈1,∞of the system's control operator is given in terms of the UMD-property of its underlying control space for a wider class of Volterra integrodifferential equations when the leading operator is not necessarily a generator, which provides a generalization of a result known to hold for the standard Cauchy problem (Bounit et al., 2010, Proposition 3.2).

Nonlocal Cauchy Problem for Integrodifferential Equations of Sobolev Type in Banach Space

In this paper, we prove the existence and uniqueness of mild and strong solutions of nonlinear integrodifferential equations of Sobolev type in Banach space. The results are obtained by using compact semigroups and the Schauder fixed point theorem. Keywords: Mild and Strong solutions; Nonlocal condition; Integrodifferential equations; Compact semigroup; Schauder fixed point theorem. 2010 Mathematics Subject Classification: 45N05; 47H10; 47B38; 34G20

Krasnoselskii‐type fixed point theorems with applications to Hammerstein integral equations in spaces

In this paper, we study fixed point theorems and new variants of some nonlinear altenatives of Krasnoselskii type in Banach spaces by using measures of weak noncompactness. Then we give an application to solve a nonlinear Hammerstein integral equation in L1 spaces.

Existence of Solutions for Integro-differential Equations of Mixed Type in Banach Spaces

The initial value problem of differential equation can be reduced to the equivalent integral equation.Recently,this method is used successfully to discuss the existence of the solution to initial value problem of nonlinear differential equation.Using the fixed point theorem of convexpower condensing operator,we study the existence of solutions to initial value problems for nonlinear second order integro-differential equations of mixed type in Banach spaces.