Hilfer-Hadamard Nonlocal Integro-Multipoint Fractional Boundary Value Problems

This paper is concerned with the existence and uniqueness of solutions for a new class of boundary value problems, consisting by Hilfer-Hadamard fractional differential equations, supplemented with nonlocal integro-multipoint boundary conditions. The existence of a unique solution is obtained via Banach contraction mapping principle, while the existence results are established by applying Schaefer and Krasnoselskii fixed point theorems as well as Leray-Schauder nonlinear alternative. Examples illustrating the main results are also constructed.

Uniqueness and existence of solutions for nonlinear fractional differential equations with two fractional orders

We study the existence and uniqueness of solutions for integro-differential equations involving two fractional orders. By using the Banach’s fixed point theorem, Leray-Schauder’s nonlinear alternative and Leray-Schauder’s degree theory, the existence and uniqueness of solutions are obtained. Some illustrative examples are also presented.

An inverse problem approach to determine possible memory length of fractional differential equations

It is a fundamental problem to determine a starting point in fractional differential equations which reveals the memory length in real life modeling. This paper describes it by an inverse problem. Fixed point theorems such as Krasnoselskii’s and Schauder type’s and nonlinear alternative for single–valued mappings are presented. Through existence analysis of the inverse problem, the range of the initial value points and the memory length of fractional differential equations are obtained. Finally, three examples are demonstrated to support the theoretical results and numerical solutions are provided.

EXISTENCE RESULTS FOR A COUPLED SYSTEM OF NONLINEAR FRACTIONAL q-INTEGRO-DIFFERENCE EQUATIONS WITH q-INTEGRAL-COUPLED BOUNDARY CONDITIONS

In this paper, we introduce and investigate a new class of coupled fractional [Formula: see text]-integro-difference equations involving Riemann–Liouville fractional [Formula: see text]-derivatives and [Formula: see text]-integrals of different orders, equipped with [Formula: see text]-integral-coupled boundary conditions. The given problem is converted into an equivalent fixed-point problem by introducing an operator whose fixed-points coincide with solutions of the problem at hand. The existence and uniqueness results for the given problem are, respectively, derived by applying Leray–Schauder nonlinear alternative and Banach contraction mapping principle. Illustrative examples for the obtained results are constructed. This paper concludes with some interesting observations and special cases dealing with uncoupled boundary conditions, and non-integral and integral types nonlinearities.

On the existence and uniqueness of a nonlinear q-difference boundary value problem of fractional order

In this research collection, we estimate the existence of the unique solution for the boundary value problem of nonlinear fractional [Formula: see text]-difference equation having the given form [Formula: see text] [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] represents the Caputo-type nonclassical [Formula: see text]-derivative of order [Formula: see text]. We use well-known principal of Banach contraction, and Leray–Schauder nonlinear alternative to vindicate the unique solution existence of the given problem. Regarding the applications, some examples are solved to justify our outcomes.

194Nonlinear isocaloric substitution analysis in nutritional epidemiology: An example using the UK Biobank data

Background Intake of macronutrients and its components are associated with mortality and morbidity. Isocaloric substitution analysis is a tool to examine how changing the source of energy intake (e.g. from saturated to monounsaturated fat) is associated with health. However, conventional methods assume linearity, which may be untrue in many cases. This paper presents a nonlinear alternative, using UK Biobank data as an example data set. Methods Nonlinear isocaloric substitution analysis was conducted using penalised cubic splines in Cox proportional hazard models. In the UK Biobank, 195,658 participants completed at least one dietary questionnaire and were included in the analyses. Diet was assessed using the Oxford WebQ, a web-based 24-hour recall questionnaire. Prospective all-cause mortality was derived from on linked death records. Results More than half of the associations between macronutrient intake and all-cause mortality were nonlinear. Nonlinear isocaloric substitution analysis provides effect sizes conditional on the current intake, while conventional analysis provides an average over the whole range of intake. For example, conventional isocaloric substitution estimated no effect of replacing sugar with starch. However, the nonlinear method revealed that replacing sugar with starch was associated with a higher risk when the current starch intake was greater than 30% of total energy, and with a lower risk when current intake was less than 25%. Conclusions Nonlinear isocaloric substitution provides an alternative to conventional isocaloric substitution analysis when the underlying association is nonlinear. Key messages Nonlinear associations are common in nutritional epidemiology. In such cases, nonlinear isocaloric substitution analyses should be used.

Multiple Positive Solutions of Second-Order Nonlinear Difference Systems with Repulsive Singularities

We study the existence of positive solutions for second-order nonlinear repulsive singular difference systems with periodic boundary conditions. Our nonlinearity may be singular in its dependent variable. The proof of the main result relies on a fixed point theorem in cones and a nonlinear alternative principle of Leray-Schauder; the result is applicable to the case of a weak singularity as well as the case of a strong singularity. An example is given; some recent results in the literature are improved and generalized.

Non-Instantaneous Impulsive Boundary Value Problems Containing Caputo Fractional Derivative of a Function with Respect to Another Function and Riemann–Stieltjes Fractional Integral Boundary Conditions

In the present article we study existence and uniqueness results for a new class of boundary value problems consisting by non-instantaneous impulses and Caputo fractional derivative of a function with respect to another function, supplemented with Riemann–Stieltjes fractional integral boundary conditions. The existence of a unique solution is obtained via Banach’s contraction mapping principle, while an existence result is established by using Leray–Schauder nonlinear alternative. Examples illustrating the main results are also constructed.

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

In this paper, we study boundary value problems for sequential fractional differential equations and inclusions involving Hilfer fractional derivatives, supplemented with Riemann–Stieltjes integral multi-strip boundary conditions. Existence and uniqueness results are obtained in the single-valued case by using the classical Banach and Krasnosel’skiĭ fixed point theorems and the Leray–Schauder nonlinear alternative. In the multi-valued case an existence result is proved by using nonlinear alternative for contractive maps. Examples illustrating our results are also presented.

The existence of nonnegative solutions for a nonlinear fractional q-differential problem via a different numerical approach

This paper deals with the existence of nonnegative solutions for a class of boundary value problems of fractional q-differential equation ${}^{c}\mathcal{D}_{q}^{\sigma }[k](t) = w (t, k(t), {}^{c} \mathcal{D}_{q}^{\zeta }[k](t) )$ D q σ c [ k ] ( t ) = w ( t , k ( t ) , c D q ζ [ k ] ( t ) ) with three-point conditions for $t \in (0,1)$ t ∈ ( 0 , 1 ) on a time scale $\mathbb{T}_{t_{0}}= \{ t : t =t_{0}q^{n}\}\cup \{0\}$ T t 0 = { t : t = t 0 q n } ∪ { 0 } , where $n\in \mathbb{N}$ n ∈ N , $t_{0} \in \mathbb{R}$ t 0 ∈ R , and $0< q<1$ 0 < q < 1 , based on the Leray–Schauder nonlinear alternative and Guo–Krasnoselskii theorem. Moreover, we discuss the existence of nonnegative solutions. Examples involving algorithms and illustrated graphs are presented to demonstrate the validity of our theoretical findings.