we first obtain some properties of a fundamentally nonexpansive self-mapping on a nonempty subset of a banach space and next show that if the banach space is having the opial condition, then the fixed points set of such a mapping with the convex range is nonempty. in particular, we establish that if the banach space is uniformly convex, and the range of such a mapping is bounded, closed and con...

<p style='text-indent:20px;'>This paper presents a survey for some recent research on the controllability of nonlinear fractional evolution systems (FESs) in Banach spaces. The prime focus is exact and approximate several types FESs, which include basic with classical initial nonlocal conditions, FESs time delay or impulsive effect. In addition, results via resolvent operator are reviewed...

Let and A be the generator of an -times resolvent family 1 α < < 2 α ( ) { } 0 t S t α ≥ on a Banach space X. It is shown that the fractional Cauchy problem , ( ) ( ) ( ) t u t Au t f t α = + D ( ] 0, t r ∈ ; has maximal regularity on ( ) ( 0 , u u′ ) ( ) 0 D A ∈ [ ] ( 0, ; C r ) X if and only if is of bounded semivariation on ( ) ⋅ Sα [ ] 0, r .

In this paper, we use the theory of resolvent operators, the fractional powers of operators, fixed point technique and the Gelfand–Shilov principle to establish the existence and uniqueness of local mild and then local classical solutions of a class of nonlinear fractional evolution integro-differential systems with nonlocal conditions in Banach space. As an application that illustrates the abs...

In this paper, we consider a system of generalized resolvent equations involving generalized pseudocontractive mapping with corresponding system of variational inclusions in real Banach spaces. We establish an equivalence between the system of generalized resolvent equations and the system of variational inclusions using the concept of H(.,.)-co-accretive mapping. Furthermore, we prove the exis...

A new generalized Yosida inclusion problem, involving A-relaxed co-accretive mapping, is introduced. The resolvent and associated approximation operator construed a few of its characteristics are discussed. existence result quantified in q-uniformly smooth Banach spaces. four-step iterative scheme proposed convergence analysis Our theoretical assertions illustrated by numerical example. In addi...

We consider a second order regular differential operator whose coefficients are nonselfadjoint bounded operators acting in a Hilbert space. An estimate for the resolvent and a bound for the spectrum are established. An operator is said to be stable if its spectrum lies in the right half-plane. By the obtained bounds, stability and instability conditions are established.

We consider real spaces only. Definition. An operator T : X → Y between Banach spaces X and Y is called a Hahn-Banach operator if for every isometric embedding of the space X into a Banach space Z there exists a norm-preserving extension T̃ of T to Z. A geometric property of Hahn-Banach operators of finite rank acting between finite-dimensional normed spaces is found. This property is used to ch...

We show that each power bounded operator with spectral radius equal to one on a reflexive Banach space has a nonzero vector which is not supercyclic. Equivalently, the operator has a nontrivial closed invariant homogeneous subset. Moreover, the operator has a nontrivial closed invariant cone if 1 belongs to its spectrum. This generalizes the corresponding results for Hilbert space operators. Fo...