Boundary value problems for Hilfer type sequential fractional differential equations and inclusions involving Riemann\u2013Stieltjes integral multi-strip boundary conditions

In this article, we investigate the existence of solutions for boundary value problems of fractional differential equations and inclusions with semiperiodic and three-point boundary conditions. The existence results for equations are obtained by applying Banach’s contraction mapping principle, Schaefer-type fixed point theorem, Leray-Schauder degree theory, Krasnoselskii’s fixed point theorem, and Leray-Schauder nonlinear alternative, whereas the existence of solutions for convex and nonconvex set-valued maps (inclusion case) is shown via nonlinear alternative of Leray-Schauder type for multivalued maps and Wegrzyk’s fixed point theorem for generalized contractions, respectively. We emphasize that a variety of fixed point theorems are used to obtain different existence criteria for the problems at hand. Several examples are discussed for illustration of the obtained results. Moreover, an interesting observation related to symmetric second-order three-point boundary value problems is presented.

The following Reprint contains a total of 19 articles accepted and published in the Special Issue "Fractional Differential Equations, Inclusions and Inequalities with Applications II" of the MDPI journal Mathematics and covers various aspects of recent developments in the theory and applications of fractional differential equations, inclusions, inequalities, and systems of fractional differential equations and inclusions. Topics include, but are not limited to, the following themes: Oscillation for fractional-order differential equations, (p,q)-difference equations, q-difference inclusions, Integral inequalities, Fractional integro-differential equations, Sequential fractional differential equations, Sequential fractional differential inclusions, Numerical fractional differential equations, -fractional differential equations, Positive solutions for fractional boundary value problems, Stochastic fractional differential equations, Fractional-order systems, Caputo fractional differential equations, Delay difference equations, and BlackScholes fractional equations. In total, 80 manuscripts were submitted, and 19 papers by 68 authors from 21 countries were successfully published. It is hoped that the Reprint together with the ideas and publications therein will be of interest to readers and will inspire new studies on fractional differential equations, inclusions and inequalities.

This paper is a survey of the recent results of the author for various classes of boundary value problems for Hilfer fractional differential equations and inclusions of fractional order in (1,2] supplemented with different kinds of nonlocal boundary conditions.

We study a boundary value problem of sequential fractional differential equations equipped with nonlocal integral boundary conditions (strip conditions of finite arbitrary size) involving the first-order derivative of the unknown function. As a variant problem, we discuss a case when nonlocal integral boundary conditions are governed by the unknown function. The existence of solutions for the given problems is investigated by means of some standard tools of fixed point theory. We emphasize that existence results obtained in this paper are new and enrich the existence theory for sequential fractional differential equations. The obtained results are well illustrated with the aid of examples.

In this paper, we study the existence of solutions for a new class of boundary value problems for nonlinear fractional differential inclusions with mixed type integral boundary conditions. The cases when the multifunction has convex as well as non-convex values are considered. Our results rely on the standard tools of fixed point theory and are well illustrated with the aid of an example.

This study discussed the existence of solutions for a nonlinear fractional integro-differential equation of order with four-point nonlocal integral boundary conditions. The given problem is transformed to an equivalent fixed point problem in terms of an operator equation. Then, by means of Banach contraction principle and a fixed point theorem due to Krasnoselskii, the existence results are obtained. The last existence result is based on nonlinear alternative of Leray-Schauder type. An illustrative example is also presented. Key words: Integro-differential equations, fractional order, four-point integral boundary conditions, fixed point theorems, Leray-Schauder degree.

Some boundary value problems of fractional differential equations and inclusions

In this paper, we establish sufficient conditions for the existence and uniqueness of solutions for a boundary value problem of fractional differential equations with nonlocal and average type integral boundary conditions. The Leray–Schauder nonlinear alternative, Krasnoselskii’s fixed point theorem and Banach’s fixed point theorem together with Holder inequality are applied to construct proofs for the main results. Examples illustrating the obtained results are also presented.

In this paper, we establish sufficient conditions for the existence and uniqueness of solutions for boundary value problems of Hadamard-type fractional functional differential equations and inclusions involving both retarded and advanced arguments. We make use of the standard tools of fixed point theory to obtain the main results.

This paper investigates the existence and uniqueness of solutions to a class of sequential fractional differential equations and inclusions involving the (k,ψ)-Hilfer and (k,ψ)-Caputo derivatives under non-separated boundary conditions. By reformulating the problems into equivalent fixed-point systems, several classical fixed-point theorems, including those of Banach, Krasnosel’skii˘’s, Schaefer, and the Leray–Schauder alternative, are employed to derive rigorous results. The study is further extended to the multi-valued setting, where existence results are established for both convex- and nonconvex-valued multi-functions using appropriate fixed-point techniques. Numerical examples are provided to illustrate the applicability and effectiveness of the theoretical findings.

This paper is concerned with Hadamard fractional Langevin differential equation subject to fractional integral and derivative boundary conditions and which involves three different fractional orders. By using Schaefer’s fixed point theorem and Banach contraction principle, existence and uniqueness results of solutions for the proposed equation are obtained. An example demonstrating the consistency to the theoretical findings is also presented.

In this paper, we study the existence and uniqueness of solutions for Riemann–Stieltjes integral boundary value problems of nonlinear implicit Hadamard fractional differential equations. The investigation of the main results depends on Schauder’s fixed point theorem and Banach’s contraction principle. An illustrative example is given to show the applicability of theoretical results.

<abstract><p>In this work, we are concerned with a boundary value problem of fractional orders differential inclusion with nonlocal, integral and infinite points boundary conditions. We prove some existence results for that nonlocal boundary value problem. Next, the existence of maximal and minimal solutions is proved. Finally, the sufficient condition for the uniqueness and continuous dependence of solution are studied.</p></abstract>

In this paper, we study a new class of one-dimensional semi-linear problems of fractional differential equations supplemented with nonlocal flux type integral boundary conditions. New existence and uniqueness results are obtained for the given problem by using some standard fixed point theorems. The obtained results are illustrated with the aid of examples.

The Green’s function is widely used in solving boundary value problems for differential equations, to which many mathematical and physical problems are reduced. In particular, solutions of partial differential equations by the Fourier method are reduced to boundary value problems for ordinary differential equations. Using the Green's function for a homogeneous problem, one can calculate the solution of an inhomogeneous differential equation. Knowing the Green's function makes it possible to solve a whole class of problems of finding eigenvalues in quantum field theory. The developed method for constructing the Green’s function of boundary value problems for ordinary linear differential equations is described. An algorithm and program for calculating the Green's function of boundary value problems for differential equations of the second and third orders in an explicit analytical form are presented. Examples of computing the Green's function for specific boundary value problems are given. The fundamental system of solutions of ordinary differential equations with singular points needed to construct the Green's function is calculated in the form of generalized power series with the help of the developed programs in the Maple environment. An algorithm is developed for constructing the Green's function in the form of power series for second-order and third-order differential equations with given boundary conditions. Compiled work programs in the Maple environment for calculating the Green functions of arbitrary boundary value problems for differential equations of the second and third orders. Calculations of the Green's function for specific third-order boundary value problems using the developed program are presented. The obtained approximate Green’s function is compared with the known expressions of the exact Green’s function and very good agreement is found