We investigate the controllability and Ulam–Hyers stability of a class of conformable fractional differential systems with time delays and impulsive effects. Specifically, we analyze the movement rules before and after the impulse, utilizing both the Banach and Schauder fixed point theorem to derive controllability. Furthermore, we employ nonlinear functional analysis methods to study Ulam–Hyers stability. To demonstrate the applicability and feasibility of our main conclusions, an illustrative example is provided.

We investigate the controllability and Ulam–Hyers stability of a class of conformable fractional differential systems with time delays and impulsive effects. Specifically, we analyze the movement rules before and after the impulse, utilizing both the Banach and Schauder fixed point theorem to derive controllability. Furthermore, we employ nonlinear functional analysis methods to study Ulam–Hyers stability. To demonstrate the applicability and feasibility of our main conclusions, an illustrative example is provided.

This paper deals with the controllability and Hyers–Ulam(HU) stability results of an implicit coupled fractional q-difference system with order α ∈ ( 1 , 2 ) . By employing tools from fixed point theory, nonlinear functional analysis and the theory of q-calculus, we establish sufficient conditions for the existence and uniqueness of solutions, as well as controllability and stability criteria for the considered system. In particular, the Banach fixed point theorem is used to derive controllability results, while the Schauder fixed point theorem is applied to study the existence of at least one solution and HU stability. In the end, some illustrative examples are presented to demonstrate the applicability of the obtained theoretical results.

The overarching goal of this paper is to introduce and study a class of random generalized inverse quasi-variational inequalities (RGIQVIs), which captures the desired properties of both random inverse quasi-variational inequalities (RIQVIs) and random generalized quasi-variational inequalities (RGQVIs) within the same framework. By using the randomization principle for coincidence points, the Banach contraction principle for set-valued mapping, and the Schauder fixed point theorem, we obtain some new existence theorems of random solutions for RGIQVIs under mild conditions. Moreover, we give some applications concerning the random generalized Nash game and random road pricing problem to illustrate and to support our main results.

In this manuscript, we examine impulsive evolution systems in Hilbert spaces. Using a resolvent-like operator, we first establish the finite-approximate controllability for linear systems. Subsequently, by applying the Schauder fixed-point theorem, we prove the existence of a solution and demonstrate the finite-approximate controllability of semilinear impulsive systems in Hilbert spaces. Finally, we extend these results to a broader application, specifically to the heat equation.

This paper investigates a class of initial value problems arising in the modeling of systems with memory and delayed feedback, described by nonlinear fractional differential equations with finite delay, governed by the generalized Caputo-Katugampola fractional derivative. The presence of the parameter [Formula: see text] in this operator allows interpolation between different fractional behaviors and provides additional flexibility in modeling the intensity and scaling of memory effects in delay systems. By employing ρ-Laplace transform techniques, we first derive an equivalent integral formulation of the considered problem. We then establish the existence and uniqueness of solutions to the proposed Cauchy problem by employing the Banach contraction principle and Schauder's fixed point theorem. The use of both fixed-point approaches enables us to address existence and uniqueness under complementary sets of assumptions, thereby enlarging the class of admissible nonlinearities. Moreover, we examine the Ulam-Hyers stability of the solutions under suitable conditions, demonstrating that small variations in the initial data result in proportionally small deviations in the solution. This stability property reflects the robustness of the model with respect to perturbations and is closely related to the contraction condition imposed on the associated operator. To illustrate the theoretical results and confirm the applicability of the method, numerical examples are provided and discussed. The numerical simulations are carried out using the L1 scheme, which is known for its stability and effectiveness in approximating Caputo-type fractional derivatives.

This paper investigates the existence of nonnegative weak solutions to a class of degenerate elliptic equations with singular nonlinearities. The problem under consideration is of the form [Formula: see text] with homogeneous Dirichlet boundary conditions, where [Formula: see text] is a bounded domain, [Formula: see text], [Formula: see text], [Formula: see text] is a nonnegative element of the dual Sobolev space [Formula: see text], and [Formula: see text] is a continuous function that may blow up at zero but remains bounded at infinity. The degeneracy of the principal part, controlled by the parameter [Formula: see text], adds significant difficulty to the analysis. Using a double approximation scheme (regularizing both the degeneracy and the singularity), truncation arguments, monotonicity methods, and the Schauder fixed point theorem, we establish the existence of a solution [Formula: see text] under appropriate conditions on the data. Our main contribution lies in the simultaneous treatment of degeneracy and singularity, extending classical results to a broader class of non-uniformly elliptic operators. The proofs rely on uniform a priori estimates, compactness arguments, and a careful passage to the limit in the approximate problems.

ABSTRACT This paper is devoted to the study of a coupled system of fractional Langevin equations involving the ‐Hilfer fractional derivative and nonlocal Riemann–Stieltjes integral boundary conditions. By transforming the problem into an equivalent system of integral equations, sufficient conditions for the existence of solutions are established via Schauder's fixed point theorem and the Leray–Schauder nonlinear alternative. In addition, uniqueness results are obtained by applying the Banach contraction principle. Several illustrative examples are provided to demonstrate the applicability of the theoretical results and to validate the proposed approach.

In this work, we investigate the global asymptotic behavior of an age-structured epidemic model with diffusion, under Dirichlet boundary conditions and a protection phase of limited duration. We consider a bounded n-dimensional spatial domain with homogeneous Dirichlet boundary conditions. The system is directly transformed to a coupled system of reaction-diffusion equations and a continuous difference equation with a time-delay and a non-local spatial term due to mobility during the protection phase. The Basic Reproduction Number (BRN) is determined for this system and some properties are obtained according to this parameter. Using Schauder's fixed point theorem, it is shown that disease-free equilibrium always exists and is globally attractive when BRN is less than one. Otherwise, we use the method of monotonic iterations for a particular elliptic problem to show that the endemic steady-state exists when BRN is greater than one. This condition on BRN follows naturally from the construction of an upper- and a lower-solution. Moreover, we prove that the system is uniformly strongly persistent. Finally, we perform numerical simulations that confirm and complete our theoretical results.

This work deals with two mathematical aspects of subsurface flow problems within aquifer systems, namely Mathematical Modeling and Theoretical Analysis. Concerning the Mathematical Modeling, the classical challenges for this class of problems are a rigorous description of diverse interactions that may take place between different involved aquifers. Recall that the main challenge at this stage is a realistic description of the flows from one aquifer to another passing necessarily through an aquitard which is a porous layer with small permeability coefficient and small thickness (compared with the mean thickness of involved aquifers. In the same way as most of flow phenomena, the governing equations of subsurface flows are based upon conservation laws.) To address the mathematical modeling of water exchange between different aquifers separated by aquitards we expose a mathematical approach based upon the Taylor expansion. Introducing the concept of observers located inside the aquitard and the neighboring aquifers, the mass conservation law has been applied and has led to one mass balance equation for each aquifer. Thanks to this original approach we have recovered the well-known mass balance equations exposed in the literature for flow problems in aquifer systems. Due to the assumptions of small thickness and homogeneity of the absolute permeability of aquitards for our framework the water flow is supposed vertical in aquitards and so we deal with one-dimensional flows there. This is the reason why the Taylor expansion deployed there concerns only the vertical space variable. The flux continuity has been applied to get the coupling of flow equations in the two aquifers. Since the flows in aquifers are supposed horizontal it is clear that the interface aquitard/aquifer flux acts as an additional source-term for each aquifer (and not a boundary term). Concerning the Theoretical Analysis of the global system of elliptic equations (as the flow is supposed to be submitted to a steady state) the Schauder Fixed Point Theorem has been applied for facing the nonlinearity of the right-hand sides of the system. This is the way we have got the existence of a solution to the system, but not the uniqueness. Thanks to a monotonicy assumption on the right-hand side vector-function we get the uniqueness of the solution. Finally the stability of that solution has been established under appropriate conditions.

In this paper, we investigate the existence of forced waves for an asymptotic KPP equation with anisotropic nonlocal dispersal in a shifting habitat. Without requiring the sign of intrinsic growth rate at negative infinity, we prove the existence of zero-valued forced waves at + ∞ or − ∞ for s in different ranges. The main method is based on the construction of suitable upper and lower solutions combined with the monotone iteration principle and Schauder's fixed point theorem. Furthermore, we investigate the asymptotic behavior of forced waves via contracting rectangles. Finally, to illustrate our results, we consider forced waves in three classical models.

In this article, we investigate some properties such as the existence, uniqueness, and Ulam–Hyers–Rassias stability for the fractional Volterra–Fredholm integrodifferential equations of Ψ ‐Hilfer type with boundary conditions. We prove the desired results by using the Banach fixed point theorem and the Schauder fixed point theorem. Certain inequalities are also used to obtain estimates for solutions to the given problem, which is also discussed. We provide an example to illustrate our abstract results. MSC2020 Classification: 26A33, 26D10, 47H10

The goal of the paper is to study the existence of mild solutions and the exact controllability of an abstract Caputo-fractional system with semilinear boundary control in Banach spaces. Sufficient criteria for the existence of mild solutions and controllability for the desired issue are provided, based on Schauder's fixed point theorem. Moreover, for fractional linear boundary control, we present a characterization of controllability using a standard fractional linear system with internal control. We provide an example to demonstrate the viability of our theoretical findings.

This study primarily focuses on establishing the sufficient conditions for the existence and uniqueness of the mild solution along with the approximate and trajectory controllability results for a new class of the nonlinear $ \Psi $-Caputo fractional neutral-type integro-differential system with finite delay and nonlocal conditions in a Hilbert space. A key advantage of the $ \Psi $-Caputo fractional derivative is that it allows to choose a suitable kernel function $ \Psi $. First, we derive the existence of the mild solution for the proposed control system by using a fixed point approach. For this purpose, the proposed control system is transformed into an equivalent fixed point problem using the $ \Psi $-Riemann-Liouville fractional integral operator. Then, the existence of the mild solution is established by Schauder's fixed point theorem. Then, the uniqueness of the mild solution is studied with the help of the Banach contraction principle. Moreover, the approximate controllability result of the proposed control system is established under the consideration that the corresponding linear system is approximate controllable. Further, the trajectory controllability result is studied by using the Grönwall's inequality. The set of sufficient conditions is derived by using the concepts of fractional calculus, Laplace transform, fixed point techniques, and semigroup theory of bounded linear operators. Finally, an illustrative example is presented to validate and demonstrate the applicability of the theoretical results.

This study explores the existence of traveling wave solutions connecting the disease-free equilibrium and the endemic equilibrium in the Kermack-McKendrick epidemic model. By constructing a pair of upper and lower solutions and applying Schauder's fixed point theorem, we establish the existence of non-negative traveling wave solutions. It is demonstrated that such traveling wave solutions persist even when considering nonlocal spatio-temporal delays and nonlinear saturated incidence. The approach primarily utilizes dynamical systems theory, including geometric singular perturbation theory, the center manifold theorem, and Fredholm theory.

We investigate traveling wave solutions in the two-species reaction-diffusion Lotka–Volterra competition system under weak competition. For the strict weak competition regime $ (b<a<1/c, \, d>0) $, we construct refined upper and lower solutions combined with the Schauder fixed point theorem to establish the existence of traveling waves for all wave speeds $ s\geq s^*: = \max\{2, 2\sqrt{ad}\} $, and provide verifiable sufficient conditions for the emergence of non-monotone waves. Such conditions for non-monotone waves have not been explicitly addressed in previous studies. It is interesting to point out that our result for non-monotone waves also holds for the critical speed case $ s = s^* $. In addition, in the critical strong-weak competition case $ (b<a = 1/c, \, d>0) $, we rigorously prove, for the first time, the existence of front-pulse traveling waves.

This paper is concerned with the spreading speed and time-periodic travelling wave solutions of a time-periodic reaction-diffusion two-group SI epidemic model with demographic structure (uses the logistic growth rate), which is non-monotonic. Based on comparisons with relevant periodic equations of KPP-type, we study the spreading speed of this model in terms of the basic reproduction number of the corresponding periodic kinetic system and three critical wave speeds, exhibiting a more complex and new propagating pattern. On this basis, the existence and non-existence of a time-periodic traveling wave solution of this model can be established by using the Schauder fixed point theorem associated with the limit argument. Finally, through numerical simulation, the transmission mode of infectious diseases and the profile of the periodic traveling wave solutions of the system are presented.

In this paper, we study on a half-line and demonstrate the existence of unbounded or bounded solutions of the following three-point fourth-order boundary value problem: For all $\xi\in(0,+\infty)$, ${\Phi}''''(\xi)+p(\xi) g(\xi, {\Phi}(\xi), {\Phi}'(\xi), {\Phi}''(\xi),{\Phi}'''(\xi))=0$ with ${\Phi}''(0)= \Lambda$, ${\Phi}(\rho)=B_1$, ${\Phi}'(0)=B_2$, and ${\Phi}'''(+\infty)=\Omega$, where $\rho$ is fixed and $\rho\in(0,+\infty)$, and $g:[0,+\infty)\times \mathbb{R}^4\rightarrow\mathbb{R}$ provides the condition of Nagumo. In order to address this objective, we employ various mathematical techniques, including the upper and lower solution method, Schauder's fixed point theorem, and topological degree theory. By utilizing these methods, we establish sufficient conditions that guarantee the existence of at least one solution, as well as at least three solutions, for the aforesaid problem. To illustrate the significance of the obtained findings, we provide an example demonstrating the practical implications of the results herein.

We will study the well-posedness of a nonlinear p-Laplacian Hadamard-type fractional order differential equation in this article. The solution of the boundary value problem is obtained in an integral form, while the Schauder fixed point theorem coupled with some properties of the Green functions is used to obtain conditions for the existence of solutions. The uniqueness of the solution is also considered for boundary value problem while the stability of the solutions is also considered using the generalized Hyers-Ulam and Hyers-Ulam stability analysis. Examples are used to demonstrate the applicability of the results that were obtained.

This paper investigates the controllability of a class of fractional-order composite systems subject to a constant control delay. To address such systems, we apply a generalized integral transform belonging to the Laplace family, which unifies several classical transforms as special cases. By employing this transform, we derive explicit solution representations and establish necessary and sufficient conditions for the controllability of the associated linear delayed system via the Gramian matrix approach. The analysis is then extended to the semilinear case by applying Schauder's fixed point theorem. As a concrete application, we formulate a fractional-order epidemiological model with vaccination delay, where the Caputo derivative captures long-memory effects and the control delay represents the time required for immunity to take effect. Numerical simulations confirm that the proposed method effectively steers the system to prescribed terminal states, demonstrating both the theoretical validity and practical relevance of the results.