This paper investigates the approximate controllability of a system of fractional control differential equations. The system involves Caputo fractional derivatives, Volterra–Fredholm integral equations, and impulsive effects. The analysis is carried out in the framework of Banach spaces under nonlocal conditions of order r ∈ (1, 2). The main objective is to establish sufficient conditions for the approximate controllability of the proposed control problem. Assuming that the associated linear system is approximately controllable, the analysis relies on tools from fractional calculus, Krasnoselskii’s fixed point theorem, and the theory of resolvent operators. The obtained results extend and improve several existing results in the related literature. Finally, two illustrative examples are presented to demonstrate the applicability of the theoretical findings.

In this paper, we present the concept of rational theta-Reich and theta-Sehgal contraction mappings in usual metric spaces by integrating the classical Reich and Sehgal contractions with Singh's iterative framework. We establish unified fixed-point theorems that guarantee existence and uniqueness under both constant and functional parameters. Our proposed rational theta-Reich and theta-Sehgal formulations provide a comprehensive synthesis of foundational results, including those of Banach, Kannan, and Chatterjea. Furthermore, we provide a rigorous analysis of the Picard iteration's convergence, offering new insights into the stability and asymptotic behavior of nonlinear operators. The versatility of the proposed framework is demonstrated through its application across diverse domains, including fractal geometry, fractional calculus, and the modeling of financial markets and economic growth trajectories.

This paper investigates the existence of mild solutions and optimal controls for a class of second-order delayed neutral nonautonomous evolution systems involving hemivariational inequalities, history-dependent operators, and Clarke's generalized subdifferential in Hilbert spaces. The novelty of the proposed framework lies in the simultaneous treatment of second-order dynamics, delay effects, memory-dependent operators, and nonsmooth multivalued nonlinearities within a unified mathematical setting, a combination that has received little attention in the existing literature. By integrating the theory of strongly continuous cosine families with fixed-point techniques for condensing multivalued mappings, new sufficient conditions are derived for the existence of mild solutions and optimal control pairs. In addition, a representative second-order partial differential model is presented to illustrate the applicability of the theoretical results. Consequently, the obtained results substantially generalize and extend several previously reported solvability and optimal control results for delayed nonsmooth evolution systems.

This paper investigates a class of nonlinear implicit neutral integrodifferential equations involving Hilfer–Katugampola fractional operators, which provide an effective framework for describing dynamical systems with memory and hereditary properties. By converting the considered problem into an equivalent fractional integral equation, we show that the associated operator satisfies an appropriate contraction condition in a suitable complete metric space. Consequently, the Banach fixed-point theorem is employed to establish sufficient conditions ensuring the existence and uniqueness of solutions. Furthermore, the qualitative behavior of solutions is analyzed through several notions of Ulam-type stability, including Ulam–Hyers and Ulam–Hyers–Rassias stability, by means of a generalized fractional Grönwall's inequality. In addition, a numerical procedure based on the Volterra integral representation and the Picard iterative technique is proposed to approximate the solutions. Numerical simulations demonstrate the convergence, stability, and effectiveness of the developed method, confirming the applicability of the obtained theoretical results to nonlinear fractional integrodifferential models.

This paper is devoted to the analysis of controllability for a class of backward fractional integro-differential equations involving history-dependent operators, which arise naturally in systems with memory effects. The study begins with the formulation of an appropriate functional framework, within which the concept of approximate controllability is established by means of Schauders fixed-point theorem, without imposing restrictive compactness or Lipschitz-type conditions. Building on this foundation, the investigation is extended to exact controllability through a more refined analytical approach, where M & ouml;nch ' s fixed-point theorem is employed in conjunction with suitable operator-theoretic techniques to derive verifiable sufficient conditions. The results obtained significantly generalize existing contributions in the literature by accommodating the combined effects of fractional dynamics and history dependence. Finally, a representative example is presented to illustrate the applicability, validity, and effectiveness of the proposed theoretical framework in practical settings.

This research investigates the crucial concept of approximate controllability for a class of complex dynamical systems: second-order neutral non-autonomous evolution systems in Hilbert spaces. These systems, characterized by inherent memory effects (due to the neutral term) and non-smooth behavior (modeled by hemivariational inequalities), pose significant analytical challenges. We begin by rigorously establishing the existence of mild solutions for this intricate system. This crucial step relies on a powerful combination of mathematical tools, including cosine functions, a robust fixed-point technique, and the generalized Clarke's subdifferential, which effectively handles the non-smoothness arising from the hemivariational inequalities. Building upon this foundation, we delve into the core objective: approximate controllability. This fundamental property explores the system's ability to be arbitrarily close to any desired target state through judicious selection of control inputs. We derive sufficient conditions for approximate controllability, providing valuable insights into the system's controllability characteristics. Finally, to underscore the practical significance of our theoretical findings, we present a concrete application demonstrating the developed theory's efficacy in addressing real-world problems.

The existence of solutions for non-autonomous integrodifferential evolution equations with nonlocal conditions is investigated in this article. Initially, existence results for mild solutions of the proposed equation are established through the leveraging of the theory of resolvent operators, fixed point theorems, and an estimation technique grounded in the measure of noncompactness. Finally, the applicability of the findings is illustrated by means of an example concerning a class of non-autonomous nonlocal partial integrodifferential equations.

In this manuscript, a fixed-point method is developed to address challenging classes of equations. The main contribution lies in the construction of a generalized contraction operator, which serves as the foundation for extending Darbo’s fixed-point theorem. This advancement significantly broadens the applicability of the theorem, enabling it to handle both hybrid differential equations and fractional hybrid differential equations formulated in Banach spaces. By establishing a rigorous and flexible analytical framework, the study provides an effective tool for examining these complex mathematical models and facilitates further investigation into their properties and solutions.

In this article, we investigate damped elastic inclusion systems characterized by nonconvex-valued perturbations, the summation of dual multivalued mappings, and history-dependent operators. By extending standard convex frameworks, these perturbations facilitate the modeling of discontinuous and complex dynamics often found in real-world systems. Our primary contribution is the derivation of a novel existence theorem for mild solutions under suitable conditions. Lastly, the proposed theoretical results are validated through a detailed examination of elastic feedback control systems as an application.

In this article, a detailed study is conducted on an infinite system of two-variable functional integral equations driven by a Fredholm-type Riemann-Liouville operator. The main objective is to address the solvability of this system within the framework of the Banach sequence space ℓ ^p (p>1) , which presents significant analytical challenges due to its infinite-dimensional structure. The existence of solutions is established by employing the Meir–Keeler condensing operator in conjunction with the theory of measures of noncompactness, providing a robust and generalized approach. In addition to the theoretical results, a new iterative scheme is developed to obtain approximate solutions, combining the modified homotopy perturbation method with Adomian’s decomposition method to enhance both accuracy and computational efficiency. A rigorous convergence analysis is then carried out, ensuring convergence in the ℓ ^p -norm, along with a stability analysis demonstrating the continuous dependence of solutions on initial data. To support the theoretical findings, numerical experiments are performed on the closed-form sequence {φ _m(κ ,τ )} , which reveal key properties such as strict positivity, monotone decay with respect to m, and progressively decreasing sup-norm bounds. These results remain consistent under grid refinement, thereby confirming the reliability, stability, and effectiveness of the proposed analytical and numerical framework.

This work is devoted to the study of a nonlinear tripled system of Langevin‐type fractional differential equations involving generalized ψ ‐Caputo derivatives in the setting of Banach spaces. The proposed framework extends several existing results from finite‐dimensional and specialized functional spaces to more general infinite‐dimensional environments. Sufficient conditions for the existence of solutions are established by applying Mönch′s fixed point theorem together with the technique of measure of noncompactness. In addition, uniqueness results are obtained through the Banach contraction principle under suitable Lipschitz‐type assumptions. Furthermore, illustrative examples, including systems defined in the space of null sequences, are presented to verify the applicability and effectiveness of the obtained results. The presented analysis enriches the theory of fractional differential systems and broadens the scope of solvability results for tripled Langevin‐type models.

In this study, optimal and approximate control strategies are explored for a class of neutral integrodifferential systems. These systems are characterized by resolvent and history-dependent operators, operate within Hilbert spaces, and incorporate nonlocal conditions and hemivariational inequalities. To achieve the desired results, a combination of techniques was employed, including fixed-point theory, resolvent operator methods, semigroup theory, and generalized Clarke’s subdifferential. Initially, the existence of mild solutions for the system is established through the application of the Bohnenblust–Karlin fixed-point theorem. Subsequently, sufficient conditions for the approximate controllability of the system are derived. The existence of optimal control solutions for the system, along with the associated cost functional, is then established using Balder’s theorem. Finally, to illustrate the theoretical findings, the existence of a solution to a partial control system is presented as an application.

This manuscript investigates the exact controllability of a class of stochastic impulsive fractional integro-differential evolution equations with nonlocal conditions and history-dependent operators in a Hilbert space framework, without imposing the compactness of the semigroup generated by the linear part, thereby significantly enlarging the class of admissible infinite-dimensional systems. The model incorporates fractional dynamics, stochastic perturbations, impulsive effects, and memory terms under suitable mild assumptions, providing a realistic and comprehensive description of complex evolution processes. The main results are derived by combining M & ouml;nch's fixed-point theorem with the measure of noncompactness and stochastic analysis techniques, which effectively overcome difficulties caused by noncompactness and hereditary effects. Moreover, suitable control functions are constructed to ensure exact controllability over a finite time interval. The main advantage of this work lies in relaxing restrictive compactness assumptions while simultaneously handling stochasticity, impulses, and memory effects within a unified framework. Finally, illustrative examples are provided to demonstrate the applicability and effectiveness of the obtained results and to highlight their improvement over existing literature.

This paper delves into the theoretical investigation of extremal solutions for a coupled sequential Caputo fractional differential system. We employ functional analysis to rigorously prove the existence of these solutions, combining the monotone iterative technique with the method of upper and lower solutions to establish sufficient conditions for finding the minimal and maximal solutions. To validate our theoretical results, we provide two illustrative examples.

This paper addresses a class of Hilfer fractional stochastic nonlinear integro-differential equations incorporating impulsive effects and optimal control in Hilbert spaces. We first establish the existence of mild solutions, ensuring the solvability of the system through the application of fractional calculus, stochastic analysis, and fixed-point techniques. The analytical framework effectively manages the combined difficulties arising from nonlocal operators, stochastic perturbations, and impulsive dynamics. Subsequently, we formulate the associated optimal control problem and derive the necessary conditions for optimality. An illustrative example is provided to demonstrate the practicality and robustness of the theoretical results.

In this paper, the existence of mild solutions for a class of noninstantaneous impulsive Caputo fractional integro-differential equations subject to nonlocal conditions in Banach spaces is investigated. The problem is formulated in a general infinite-dimensional setting so that systems involving memory effects and impulses acting over finite intervals can be described. The analysis is carried out through the combined application of Mönch's fixed-point theorem, Grimmer's resolvent operator framework, and measures of noncompactness, by which the lack of compactness is effectively handled and suitable existence conditions are established. The obtained results are regarded as an extension of several existing theories since more general assumptions and dynamical behaviors are accommodated. Moreover, an explicit example is provided to illustrate the applicability of the theoretical results and to demonstrate how the proposed assumptions can be verified in concrete cases.

This article investigates the solvability and stability of a nonlinear tripled system of fractional integrodifferential equations via integral boundary conditions. We prove the fundamental existence and uniqueness results for the solution using powerful methods from nonlinear functional analysis, including the Banach fixed-point theorem. Crucially, we gauge the system's resistance to small perturbations using Ulam-Hyers stability analysis. Our results confirm that approximate solutions are close to the true solution, providing an essential resilience measure for complex models in which memory and hereditary traits are captured by fractional derivatives. Finally, an illustrative example has been presented to validate the stated theoretical results.

This study investigates the demanding problem of proving the existence of solutions for a tripled system that couples quantum integral equations with quadratic integral equations. To tackle the intrinsic nonlinear and noncompact features of the model, we employ Petryshyn’s fixed-point theorem, a significant generalization of Darbo’s theorem formulated within the framework of measures of noncompactness. Based on this approach, we derive rigorous and verifiable existence conditions applicable to a broad class of quantum integral systems. The theoretical findings are supported by a comprehensive illustrative example that confirms the validity of the proposed criteria. In addition, we develop a constructive collocation method founded on barycentric interpolation and Jackson quadrature for q-integrals, and we verify the required assumptions within the same example. Numerical experiments are finally presented to confirm the practical applicability of the existence results and to demonstrate the accuracy, stability, and robustness of the proposed discretization scheme.

This paper presents and analyzes a class of self-mappings in metric spaces which are endowed of mixed characteristics between that of contractive cyclic self-mappings on union of subsets and that of contractive mappings within the individual subsets. Stability analysis of dynamic systems can be focused on by formalizing the relationship between cyclic contractive mappings over non-necessarily intersecting subsets and the stability criteria for switched systems with cyclic dynamics. The case of disjoint set structures necessitates the application of best proximity theory. Therefore, this manuscript introduces the mentioned class of hybrid cyclic/non-cyclic bivalued contractive self-mappings which can operate iteration-by-iteration in either a cyclic operation mode (the selected image lies in the next adjacent subset) or in an intra mode operation (the selected image lies in the current subset). Unlike cyclic operators, these proposed self-mappings allow dynamic switches to either the current subset or the next adjacent one in a cyclic disposal, governed by an iteration-dependent image selection sequence. This “modus-operandi” is possible since one of the images of each point at each iteration is activated for the cyclic operation mode while the other one is activated for an intra mode mode within some subset. Under the key assumptions that the subsets are closed, at least one best proximity set is a singleton, and the subset itself is boundedly compact, new boundedness and convergence theorems are established. It is emphasized how cyclic contractive properties are useful in the context of asymptotic stability of hybrid systems, validating the theoretical framework through illustrative examples.