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In this paper, we discuss some extensions of the Maligranda–Orlicz lemma. It deals with the problem of constructing a norm in a subspace of the space of bounded functions, for which it becomes a normed algebra so that the norm introduced is equivalent to the initial norm of the subspace. This is done by satisfying some inequality between these norms. We show in this paper how this inequality is relevant to the study of operator equations in Banach algebras. In fact, we study how to equip a subspace of the space of bounded functions with a norm equivalent to a given one so that it is a normed algebra. We give a general condition for the construction of such norms, which allows us to easily check whether a space with a given norm is an algebra with a pointwise product and the consequences of such a choice for measures of noncompactness in such spaces. We also study quasi-normed spaces. We introduce a general property of measures of noncompactness that allows the study of quadratic operator equations, prove a fixed-point theorem suitable for such problems, and complete the whole with examples and applications. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a comprehensive and expanded framework for generalized calculus and generalized polynomials in discrete calculus. Our focus is on (q ; h) -time scales. Our proposed approach encompasses both difference and quantum problems, making it highly adoptable. Our framework employs forward and backward jump operators to create a unique approach. We use a weighted jump operator α that combines both jump operators in a convex manner. This allows us to generate a time scale α , which provides a new approach to discrete calculus. This beneficial approach enables us to define a general symmetric derivative on time scale α , which produces various types of discrete derivatives and forms a basis for new discrete calculus. Moreover, we create some polynomials on α -time scales using the α -operator. These polynomials have similar properties to regular polynomials and expand upon the existing research on discrete polynomials. Additionally, we establish the α -version of the Taylor formula. Finally, we discuss related binomial coefficients and their properties in discrete cases. We demonstrate how the symmetrical nature of the derivative definition allows for the incorporation of various concepts and the introduction of fresh ideas to discrete calculus. [ABSTRACT FROM AUTHOR]

We propose here a general framework covering a wide range of fractional operators for vector-valued functions. We indicate to what extent the case in which assumptions are expressed in terms of weak topology is symmetric to the case of norm topology. However, taking advantage of the differences between these cases, we emphasize the possibly less-restrictive growth conditions. In fact, we present a definition and a serious study of generalized Hilfer fractional derivatives. We propose a new version of calculus for generalized Hilfer fractional derivatives for vector-valued functions, which generalizes previously studied cases, including those for real functions. Note that generalized Hilfer fractional differential operators in terms of weak topology are studied here for the first time, so our results are new. Finally, as an application example, we study some n-point boundary value problems with just-introduced general fractional derivatives and with boundary integral conditions expressed in terms of fractional integrals of the same kind, extending all known cases of studies in weak topology. [ABSTRACT FROM AUTHOR]

In this paper, we present the definitions of fractional integrals and fractional derivatives of a Pettis integrable function with respect to another function. This concept follows the idea of Stieltjes-type operators and should allow us to study fractional integrals using methods known from measure differential equations in abstract spaces. We will show that some of the well-known properties of fractional calculus for the space of Lebesgue integrable functions also hold true in abstract function spaces. In particular, we prove a general Goebel–Rzymowski lemma for the De Blasi measure of weak noncompactness and our fractional integrals. We suggest a new definition of the Caputo fractional derivative with respect to another function, which allows us to investigate the existence of solutions to some Caputo-type fractional boundary value problems. As we deal with some Pettis integrable functions, the main tool utilized in our considerations is based on the technique of measures of weak noncompactness and Mönch's fixed-point theorem. Finally, to encompass the full scope of this research, some examples illustrating our main results are given. [ABSTRACT FROM AUTHOR]

Here, we propose a general framework covering a wide variety of fractional operators. We consider integral and differential operators and their role in tempered fractional calculus and study their analytic properties. We investigate tempered fractional integral operators acting on subspaces of L 1 [ a , b ] , such as Orlicz or Hölder spaces. We prove that in this case, they map Orlicz spaces into (generalized) Hölder spaces. In particular, they map Hölder spaces into the same class of spaces. The obtained results are a generalization of classical results for the Riemann–Liouville fractional operator and constitute the basis for the use of generalized operators in the study of differential and integral equations. However, we will show the non-equivalence differential and integral problems in the spaces under consideration. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the existence of integrable solutions of Gripenberg-type equations with m-product of fractional operators on a half-line R + = [ 0 , ∞) . We prove the existence of solutions in some weighted spaces of integrable functions, i.e., the so-called L 1 N -solutions. Because such a space is not a Banach algebra with respect to the pointwise product, we cannot follow the idea of the proof for continuous solutions, and we prefer a fixed point approach concerning the measure of noncompactness to obtain our results. Appropriate measures for this space and some of its subspaces are introduced. We also study the problem of uniqueness of solutions. To achieve our goal, we utilize a generalized Hölder inequality on the noted spaces. Finally, to validate our results, we study the solvability problem for some particularly interesting cases and initial value problems. [ABSTRACT FROM AUTHOR]

The application of fractional calculus-based mathematical models in physics is a well-established practice. However, a challenge arises due to the variety of fractional operators used in these models and the search for solutions in different function spaces. This paper proposes a unified approach to this issue by applying general fractional operators that can capture and extend previous attempts, with appropriate parameters that align with the features of the phenomena being modeled. The main goal is to establish invariant spaces for these operators. This will simplify the application of various proof techniques. • Fractional problems with a parametrized definition of fractional integral operators. • An appropriate invariant domain for this class of operators is constructed. • A general class of fractional-order operators used to model nonlinear phenomena. • The invariance of the operator on the proposed space is demonstrated. • The method is applicable to differential problems with data of non-polynomial growth. [ABSTRACT FROM AUTHOR]

We provide existence results for semilinear differential inclusions involving measures: [ABSTRACT FROM AUTHOR]

We study the quadratic integral equation in the space of Orlicz space Eφ in the most important case when φ satisfies the D2-condition. Considered operators are not compact and then we use the technique of measure of noncompactness associated with the Darbo fixed point theorem to prove the existence of a monotonic, but discontinuous solution. Our present work allows to generalize both previously proved results for quadratic integral equations, as well as, that for classical equations. Due to different continuity properties of considered operators in Orlicz spaces, we distinguish different cases and we study the problem in the most important case - in such a way to cover all Lebesgue spaces Lp (p ≥ 1). [ABSTRACT FROM AUTHOR]

The main goal of the paper is to present a new approach to the problem of continuous dependence of solutions of differential or dynamic problems on their domains. This is of particular interests when we use dynamic (difference, in particular) equations as discretization of a given one. We cover a standard construction based of difference approximations for the continuous one, but we are not restricted only to this case. For a given differential equation we take a sequence of time scales and we study the convergence of time scales to the domain of the considered problem. We choose a kind of convergence of such approximated solutions to the exact solution. This is a step for creating numerical analysis on time scales and we propose to replace in such a situation the difference equations by dynamic ones. In the proposed approach we are not restricted to the case of classical numerical algorithms. Moreover, this allows us to find an exact solution for considered problems as a limit of a sequence of solutions for appropriate time scales instead of solving it analytically or calculating approximated solutions for the original problems. [ABSTRACT FROM AUTHOR]

The paper is devoted to the study of the measure-driven differential inclusions dx(t) ∈ G(t, x(t)) dμ(t), x(0) = x0 for arbitrary finite Borel measure μ. This type of results allows one to treat in a similar manner differential and difference inclusions, as well as impulsive problems and therefore to study the evolution of hybrid systems with very complex (including Zeno) behavior. Our method is based on viewing the Borel measures as Lebesgue-Stieltjes measures. We thus obtain, under very general assumptions, the existence of regulated or bounded variation solutions of the considered problem and we indicate some advantages of our approach. [ABSTRACT FROM AUTHOR]

In this paper we consider the nonlocal (nonstandard) Cauchy problem for differential inclusions in Banach spaces x'(t) ∊ F(t, x(t)); x(0) = g(x), t ∊ [0, T] = I. Investigation over some multivalued integrals allow us to prove the existence of solutions for considered problem. We concentrate on the problems for which the assumptions are expressed in terms of the weak topology in a Banach space. We recall and improve earlier papers of this type. The paper is complemented by a short survey about multivalued integration including Pettis and Henstock-Kurzweil-Pettis multivalued integrals. [ABSTRACT FROM AUTHOR]

The object of this paper is to investigate the existence of a class of solutions for some boundary value problems of fractional order with integral boundary conditions.The considered problems are very interesting and important from an application point of view. They include two, three, multipoint, and nonlocal boundary value problems as special cases. We stress on single and multivalued problems for which the nonlinear term is assumed only to be Pettis integrable and depends on the fractional derivative of an unknown function. Some investigations on fractional Pettis integrability for functions and multifunctions are also presented. An example illustrating the main result is given. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study the quadratic integral equation of the form Several existence theorems for a.e. monotonic solutions in Orlicz spaces are proved for strongly nonlinear functions f. The presented method of the proof can be easily extended to different classes of solutions. [Copyright &y& Elsevier]

Abstract: We propose a non-standard approach to impulsive differential equations in Banach spaces by embedding this type of problems into differential (dynamic) problems on time scales. We give an existence result for dynamic equations and, as a consequence, we obtain an existence result for impulsive differential equations. [Copyright &y& Elsevier]

The main goal of the paper is to define new type of integrals for vector-valued functions on time scales. This allows to make possible the advantages of dynamic equations also for vector-valued functions i.e. for dynamic modeling in Banach spaces. To do it we define some appropriate integrals for vector-valued functions on time scales and we prove their properties. We emphasize on the particular ones, which are useful for solving dynamic equations in Banach spaces. [ABSTRACT FROM AUTHOR]

Using the Darbo fixed point theorem associated with the measure of noncompactness, we establish the existence of monotonic integrable solution on a half-line R+ for a nonlinear quadratic functional integral equation. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we investigate the dynamic Cauchy problem in Banach spaces. We check how dense a time scale must be in such a way that Peano’s Theorem holds and we present a counterexample to Peano’s Theorem on a time scale with only one right dense point. [Copyright &y& Elsevier]

Abstract: In this paper, we investigate the global solvability in of a set-valued system of nonlinear fractional differential equations with hysteresis. Some existence theorems for both single and multivalued systems are proved. [Copyright &y& Elsevier]

Abstract: This paper is devoted to unify and extend the results of the existence of the weak solutions of continuous and discrete Cauchy problem in Banach spaces. We offer the existence of the weak solution of dynamic Cauchy problem on an infinite time scale. The measure of weak noncompactness and the fixed point theorem of Kubiaczyk are used to prove the main result. [Copyright &y& Elsevier]

Abstract: We prove some existence theorems for the Cauchy problemusing different types of integrals and its properties. The requirements on the function f are depended on types of solutions, and as possible as, close to necessary conditions. We present a brief survey of classes of solutions and we give some comparison results for them. [Copyright &y& Elsevier]

In this paper we investigate the unified theory for solutions of differential equations without impulses and with impulses, even at variable times, allowing the presence of beating phenomena, in the space of regulated functions. One of the aims of the paper is to give sufficient conditions to ensure that a regulated solution of an impulsive problem is globally defined. [ABSTRACT FROM AUTHOR]