Search Results (24)

This paper proposes and studies a new class of nonlinear nonlocal problem driven by a tempered Caputo-type fractional derivative with respect to an arbitrary smooth kernel. The novelty lies in treating a single nonlocal integro-delay setting that simultaneously couples an arbitrary kernel, exponential tempering, a delayed state, a lower-order distributed fractional memory term, and multipoint nonlocal initial data, rather than introducing a new fractional operator. The resulting problem can be viewed as a rigorous well-posedness prototype for hereditary systems with delayed feedback, tempered memory, and nonlocal initialization. First, an equivalent Volterra integral equation is derived. Then, the existence and uniqueness of solutions are obtained by the Banach contraction principle in a suitable Banach space of continuous functions. Next, a Picard successive approximation procedure is introduced and shown to converge uniformly to the unique solution, together with an explicit a priori error estimate. Moreover, a continuous dependence result is proved with respect to perturbations in the initial constants, the multipoint coefficients, and the nonlinear term. Finally, the main results are illustrated with two examples enhanced by graphs of explicit Picard approximations and convergence tables. Full article

We provide sufficient conditions for the existence of $(\omega ,c)$-periodic solutions of a general class of nonlinear functional integral equations. This study extends and generalizes previous contributions in the literature. As an application of the developed theory, we establish the existence of $(\omega ,c)$-periodic solutions for recurrent neural networks with time-varying coefficients and mixed delays, as well as for a class of nonlinear Volterra–Stieltjes integral equations with infinite delay. Full article

We introduce and study a new class of nonlinear operators on metric spaces, called rational $(a,p)-$quasicontractions. Within this framework, we establish Greguš-type fixed-point theorems for closed, convex subsets of Banach spaces. The results establish the existence and uniqueness of fixed points, as well as the convergence of the Picard iteration for every initial guess. We show that rational $(a,p)-$quasicontractions strictly extend several classical contractive classes, including Hardy-Rogers, Kannan, Chatterjea, and rational contractions, and we provide explicit examples exhibiting the properness of these inclusions. As an application, we consider a nonlocal boundary value problem for a Caputo fractional differential equation of order $\alpha \in (1,2)$ with distributed delay and mixed nonlocal boundary conditions. By rewriting the problem as a Hammerstein-Volterra integral equation on a cone, and imposing natural growth and rational Lipschitz conditions on the delayed nonlinearity, we show that the associated Hammerstein operator is a rational $(a,p)-$quasicontraction. This yields the existence, uniqueness, and global attractivity of a positive solution. Two model fractional nonlinearities with delayed feedback are discussed in detail, along with a numerical scheme that illustrates the predicted geometric convergence of the discrete Picard iteration in the Caputo fractional setting. Full article

This paper investigates a class of implicit neutral fractional integro-differential equations of Volterra–Fredholm type. The equations incorporate a tempered fractional derivative in the Caputo sense, along with both retarded (delay) and advanced arguments. The problem is formulated on a time domain segmented into past, present, and future intervals and includes nonlinear mixed integral operators. Using Banach’s contraction mapping principle and Schauder’s fixed point theorem, we establish sufficient conditions for the existence and uniqueness of solutions within the space of continuous functions. The study is then extended to general Banach spaces by employing Darbo’s fixed point theorem combined with the Kuratowski measure of noncompactness. Ulam–Hyers–Rassias stability is also analyzed under appropriate conditions. To demonstrate the practical applicability of the theoretical framework, explicit examples with specific nonlinear functions and integral kernels are provided. Furthermore, detailed numerical simulations are conducted using MATLAB-based specialized algorithms, illustrating solution convergence and behavior in both finite-dimensional and Banach space contexts. Full article

The purpose of this article is to introduce the concept of elliptic-valued suprametric spaces and to establish some common fixed point theorems within this newly proposed framework. The development of elliptic-valued suprametric spaces, along with the main results presented here, is illustrated through non-trivial examples. As applications, we employ our fixed point results to study the nonlinear Volterra integral equation of the second kind, demonstrating the existence and uniqueness of solutions under suitable conditions. In particular, we highlight the role of such equations in age-structured population models, where they serve as powerful tools for describing hereditary effects, density-dependent interactions, and delayed responses in population dynamics. This connection bridges the abstract theory with applied contexts in mathematical biology, ecology, and epidemiology, emphasizing the relevance of elliptic-valued suprametric structures in both theoretical analysis and real-world applications. Furthermore, we derive corresponding fixed point results in elliptic-valued suprametric spaces, complex-valued suprametric spaces, elliptic-valued metric spaces, and complex-valued metric spaces as corollaries of our main findings. Full article

In this article, we investigate a nonlinear $\psi$-Hilfer fractional order Volterra integro-delay differential equation ($\psi$-Hilfer FRVIDDE) and a nonlinear $\psi$-Hilfer fractional Volterra delay integral equation ($\psi$-Hilfer FRVDIE), both of which incorporate multiple variable time delays. We establish sufficient conditions for the existence of a unique solution and the Ulam–Hyers stability (U-H stability) of both the $\psi$-Hilfer FRVIDDE and $\psi$-the Hilfer FRVDIE through two new main results. The proof technique relies on the Banach contraction mapping principle, properties of the Hilfer operator, and some additional analytical tools. The considered $\psi$-Hilfer FRVIDDE and $\psi$-Hilfer FRVDIE are new fractional mathematical models in the relevant literature. They extend and improve some available related fractional mathematical models from cases without delay to models incorporating multiple variable time delays, and they also provide new contributions to the qualitative theory of fractional delay differential and fractional delay integral equations. We also give two new examples to verify the applicability of main results of the article. Finally, the article presents substantial and novel results with new examples, contributing to the relevant literature. Full article

This study utilizes approximation techniques to address a boundary value problem involving a differential equation with a delayed argument. The problem is approached through analytical techniques by transforming it firstly into an equivalent integral equation. Specifically, we derive a Fredholm–Volterra integral equation that encapsulates the delayed behavior inherent in the original differential equation. The Fredholm operator in this equation features a degenerate kernel, which enables simplification and facilitates the construction of successive approximations. To solve this integral equation, we employ the two-sided convergence method, a powerful iterative technique that generates two sequences of approximate solutions—lower and upper bounds—that converge monotonically toward the exact solution. This method is particularly well-suited for problems with delayed arguments, as it ensures both stability and convergence under appropriate conditions on the kernel functions. The main objective of the study is to demonstrate the applicability and accuracy of the Simple Two-Sided Convergence Method for this class of boundary value problems. A numerical example is presented to illustrate the theoretical results, and the obtained approximations are compared with the exact analytical solution. All computations were carried out using Maple, ensuring precise numerical evaluation. Full article

This work analyses the existence, uniqueness, and Ulam-type stability of neutral fractional functional differential equations with recursively defined state-dependent delays. Employing the Caputo fractional derivative of order $\alpha \in (0,1)$ with respect to a strictly increasing function $\phi$, the analysis extends classical results to nonuniform memory. The neutral term and delay chain are defined recursively by the solution, with arbitrary continuous initial data. Existence and uniqueness of solutions are established using Krasnosel’skii’s fixed point theorem. Sufficient conditions for Ulam–Hyers stability are obtained via the Volterra-type integral form and a $\phi$-fractional Grönwall inequality. Examples illustrate both standard and nonlinear time scales, including a Hopfield neural network with iterated delays, which has not been previously studied even for integer-order equations. Fractional neural networks with iterated state-dependent delays provide a new and effective model for the description of AI processes—particularly machine learning and pattern recognition—as well as for modelling the functioning of the human brain. Full article

This study investigates extremal solutions for fractional-order delayed difference equations, utilizing the Caputo nabla operator to establish mild lower and upper approximations via discrete fractional calculus. A new approach is employed to demonstrate the uniform convergence of the sequences of lower and upper approximations within the monotone iterative scheme using the summation representation of the solutions, which serves as a discrete analogue to Volterra integral equations. This research highlights practical applications through numerical simulations in discrete bidirectional associative memory neural networks. Full article

In recent decades, many researchers have pointed out that derivatives and integrals of the non-integer order are well suited for describing various real-world materials, for example, polymers. It has also been shown that fractional-order mathematical models are more effective than integer-order mathematical models. Thereby, given these considerations, the investigation of qualitative properties, in particular, Ulam-type stabilities of fractional differential equations, fractional integral equations, etc., has now become a highly attractive subject for mathematicians, as this represents an important field of study due to their extensive applications in various branches of aerodynamics, biology, chemistry, the electrodynamics of complex media, polymer science, physics, rheology, and so on. Meanwhile, the qualitative concepts called Ulam–Hyers–Mittag-Leffler (U-H-M-L) stability and Ulam–Hyers–Mittag-Leffler–Rassias (U-H-M-L-R) stability are well-suited for describing the characteristics of fractional Ulam-type stabilities. The Banach contraction principle is a fundamental tool in nonlinear analysis, with numerous applications in operational equations, fractal theory, optimization theory, and various other fields. In this study, we consider a nonlinear fractional Volterra integral equation (FrVIE). The nonlinear terms in the FrVIE contain multiple variable delays. We prove the U-H-M-L stability and U-H-M-L-R stability of the FrVIE on a finite interval. Throughout this article, new sufficient conditions are obtained via six new results with regard to the U-H-M-L stability or the U-H-M-L-R stability of the FrVIE. The proofs depend on Banach’s fixed-point theorem, as well as the Chebyshev and Bielecki norms. In the particular case of the FrVIE, an example is delivered to illustrate U-H-M-L stability. Full article

This paper examines fractional multi-time scale stochastic functional differential equations that, in addition, are driven by fractional noises. Based on a specially crafted fixed-point principle for the so-called “local operators”, we prove a Peano-type theorem on the existence of weak solutions, that is, those defined on an extended stochastic basis. To encompass all commonly used particular classes of fractional multi-time scale stochastic models, including those with random delays and impulses at random times, we consider equations with nonlinear random Volterra operators rather than functions. Some crucial properties of the associated integral operators, needed for the proofs of the main results, are studied as well. To illustrate major findings, several existence theorems, generalizing those known in the literature, are offered, with the emphasis put on the most popular examples such as ordinary stochastic differential equations driven by fractional noises, fractional stochastic differential equations with variable delays and fractional stochastic neutral differential equations. Full article

The Chebyshev cardinal functions based on the Lobatto grid are introduced and used for the first time to solve the fractional delay differential equations. The presented algorithm is based on the collocation method, which is applied to solve the corresponding Volterra integral equation of the given equation. In the employed method, the derivative and fractional integral operators are expressed in the Chebyshev cardinal functions, which reduce the computational load. The method is characterized by its simplicity, adherence to boundary conditions, and high accuracy. An exact analysis has been provided to demonstrate the convergence of the scheme, and illustrative examples validate our investigation. Full article

We study certain Volterra integral equations that extend and recover first order ordinary differential equations (ODEs). We formulate the former equations from the latter by replacing classical derivatives with nonlocal integral operators with anti-symmetric kernels. Replacements of spatial derivatives have seen success in fracture mechanics, diffusion, and image processing. In this paper, we consider nonlocal replacements of time derivatives which contain future data. To account for the nonlocal nature of the operators, we formulate initial “volume” problems (IVPs) for these integral equations; the initial data is prescribed on a time interval rather than at a single point. As a nonlocality parameter vanishes, we show that the solutions to these equations recover those of classical ODEs. We demonstrate this convergence with exact solutions of some simple IVPs. However, we find that the solutions of these nonlocal models exhibit several properties distinct from their classical counterparts. For example, the solutions exhibit discontinuities at periodic intervals. In addition, for some IVPs, a continuous initial profile develops a measure-valued singularity in finite time. At subsequent periodic intervals, these solutions develop increasingly higher order distributional singularities. Full article

In this manuscript, we investigate some convergence and stability results for reckoning fixed points using a faster iterative scheme in a Banach space. Also, weak and strong convergence are discussed for close contraction mappings in a Banach space and for Suzuki generalized nonexpansive mapping in a uniformly convex Banach space. Our method opens the door to many expansions in the problems of monotone variational inequalities, image restoration, convex optimization, and split convex feasibility. Moreover, some experimental examples were conducted to gauge the usefulness and efficiency of the technique compared with the iterative methods in the literature. Finally, the proposed approach is applied to solve the nonlinear Volterra integral equation with a delay. Full article

In this study, we developed a new faster iterative scheme for approximate fixed points. This technique was applied to discuss some convergence and stability results for almost contraction mapping in a Banach space and for Suzuki generalized nonexpansive mapping in a uniformly convex Banach space. Moreover, some numerical experiments were investigated to illustrate the behavior and efficacy of our iterative scheme. The proposed method converges faster than symmetrical iterations of the S algorithm, Thakur algorithm and $K^{*}$ algorithm. Eventually, as an application, the nonlinear Volterra integral equation with delay was solved using the suggested method. Full article