Search Results (123)

In this paper, we investigate a $(k,{\rm Y})$ fractional quadratic integral equation in the Banach space of real-valued continuous functions on $[0,1]$. By using a measure of noncompactness associated with monotonicity and Darbo’s fixed point theorem, we provide sufficient conditions for the existence of at least one monotonic solution and analyze its stability. Finally, an illustrative example is presented to demonstrate the theoretical results, including several particular cases. Full article

This paper investigates a class of coupled $\psi$-Hilfer fractional pantograph–Langevin equations with nonlocal integral boundary conditions. By reformulating the problem as an equivalent fixed point equation and employing Mönch’s fixed point theorem together with the Kuratowski measure of noncompactness, we establish sufficient conditions for the existence of at least one solution. Under additional Lipschitz-type assumptions, we prove Ulam–Hyers stability on a suitable closed ball and derive explicit, computable stability constants. A concrete numerical example is presented in which all hypotheses are verified and the stability constants are explicitly computed (e.g., $K_1\approx 3.811$, $K_2\approx 2.761$), illustrating the applicability of the theoretical results. The study contributes additional qualitative results to the analysis of fractional pantograph–Langevin systems within the unified framework of $\psi$-Hilfer fractional derivatives. 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

This paper establishes new fixed-point theorems in the framework of complete p-normed spaces, where $p\in (0,1]$. By extending the classical Banach, Schauder, and Krasnosel’skii fixed-point theorems, we derive several results for the sum of contraction and compact operators acting on s-convex subsets. The analysis is further generalized to multivalued upper semi-continuous operators by employing Kuratowski and Hausdorff measures of noncompactness. These results lead to new Darbo–Sadovskii-type fixed-point theorems and global versions of Krasnosel’skii’s theorem for multifunctions in p-normed spaces. The theoretical findings are then applied to demonstrate the existence of solutions for nonlinear integral equations formulated in p-normed settings. A section on numerical applications is also provided to illustrate the effectiveness and applicability of the proposed results. Full article

This paper establishes a rigorous analytical framework for a nonlinear multi-order fractional differential system governed by the generalized $\sigma$-Hilfer operator in weighted Banach spaces. In contrast to existing studies that often treat specific kernels or fixed fractional orders in isolation, our approach provides a unified treatment that simultaneously handles multiple fractional orders, a tunable kernel $\sigma \left(\varsigma \right)$, weighted integral conditions, and a nonlinearity depending on a fractional integral of the solution. By converting the hierarchical differential structure into an equivalent Volterra integral equation, we derive sufficient conditions for the existence and uniqueness of solutions using the Banach contraction principle and Mönch’s fixed-point theorem with measures of non-compactness. The analysis is extended to Ulam–Hyers stability, ensuring robustness under modeling perturbations. A principal contribution is the systematic classification of the system’s symmetric reductions—specifically the Riemann–Liouville, Caputo, Hadamard, and Katugampola forms—all governed by a single spectral condition dependent on $\sigma \left(\varsigma \right)$. The theoretical results are illustrated by numerical examples that highlight the sensitivity of solutions to the memory kernel and the fractional orders. This work provides a cohesive analytical tool for a broad class of fractional systems with memory, thereby unifying previously disparate fractional calculi under a single, consistent framework. Full article

This paper is devoted to the study of existence results for a nonlinear Langevin-type fractional $(p,q)$-difference equation in Banach space. The considered model extends the fractional q-difference Langevin equation by introducing two parameters p and q, which provide additional flexibility in describing discrete fractional processes. By using the Kuratowski measure of noncompactness together with Mönch’s fixed-point theorem, we derive sufficient conditions that guarantee the existence of at least one solution. The main idea consists in converting the boundary value problem into an equivalent fractional $(p,q)$-integral equation and verifying that the corresponding operator is continuous, bounded, and condensing. An illustrative example is presented to demonstrate the applicability of the obtained results. Full article

This paper is devoted to the investigation of existence and approximate controllability results for a class of fractional integro-differential equations formulated in Fréchet spaces. The analysis is carried out using a generalized version of Darbo’s fixed point theorem adapted to Fréchet spaces, combined with the concept of the measure of noncompactness. To demonstrate the validity and applicability of the theoretical findings, an illustrative example is presented to demonstrate the applicability and validity of the theoretical findings. Full article

In this paper, we study majorization-type inequalities for n-convex (specifically 3-convex) functions. Numerous papers deal with such integral inequalities, in which n-convex functions are defined on compact intervals and nonnegative measures are used in the integrals. The main goal of this paper is to formulate similar results for noncompact intervals and signed measures. We follow a well-known method often used for compact intervals: approximation of n-convex functions with simple n-convex functions. After some preliminary results, we present new approximation theorems, some of which extend classical results, while others are completely unique approximations. Then we obtain some novel majorization-type inequalities, which can be applied under more general conditions than those currently known. Finally, we illustrate the applicability of our results by answering problems from different areas: discrete majorization-type inequalities, specifically one-dimensional inequality of Sherman for n-convex functions; characterization of Steffensen–Popoviciu measures for nonnegative, continuous, and increasing 3-convex functions; Hermite–Hadamard-type inequalities for 3-convex functions. Full article

Introduction: Changes in left ventricular (LV) wall thickness serve as important diagnostic and prognostic indicators in various cardiovascular diseases. To date, no automated software exists for the measurement of myocardial segmental wall thickness in cardiac MRI (CMR), which leads to reliance on manual caliper measurements that carry risks of inaccuracy. Aims: This paper aims to present a new automated segmental wall thickness measurement software, OptiLayer, developed to address this issue and to compare it with the conventional manual measurement method. Methods: In our pilot study, the algorithm of the OptiLayer software was tested on 50 HEALTHY individuals, and 50 excessively trabeculated noncompaction (LVET) subjects with preserved LV function, whose morphology makes it more challenging to measure left ventricular wall thickness, although often occurring with myocardial thinning. Measurements were performed by two independent investigators who assessed LV wall thicknesses in 16 segments, both manually using the Medis Suite QMass program and automatically with the new OptiLayer method, which enables high-density sampling across the distance between the epicardial and endocardial contours. Results: The results showed that the segmental wall thickness measurement values of the OptiLayer algorithm were significantly higher than those of the manual caliper. In comparisons of the HEALTHY and LVET subgroups, OptiLayer measurements demonstrated differences at several points than manual measurements. Between the investigators, manual measurements showed low intraclass correlations (ICC below 0.6 on average), while measurements with OptiLayer gave excellent agreement (ICC above 0.9 in 75% of segments). Conclusions: Our study suggests that OptiLayer, a new automated wall thickness measurement software based on high-precision anatomical segmentation, offers a faster, more accurate, and more reproducible alternative to manual measurements. Full article

In this paper, we study the problem of uniqueness of fixed points for operators acting from a Banach space X into a subspace Y with a stronger norm. Our main objective is to preserve the expected regularity of fixed points, as determined by the norm of Y, while analyzing their uniqueness without imposing the classical or generalized contraction condition on Y. The results presented here provide generalized uniqueness theorems that extend existing fixed-point theorems to a broader class of operators and function spaces. The results are used to study fractional initial value problems in generalized Hölder spaces. Full article

Let $\mathcal{E}$ and $\mathcal{F}$ be nonempty disjoint subsets of a metric space $(M,d)$. For a non-self-mapping $\phi :\mathcal{E}\to \mathcal{F}$, which is fixed-point free, a point $\varkappa \in \mathcal{E}$ is said to be a best proximity point for the mapping $\phi$ whenever the distance of the point $\varkappa$ to its image under $\phi$ is equal to the distance between the sets, $\mathcal{E}$ and $\mathcal{F}$. In this article, we establish new best proximity point theorems and obtain real extensions of Edelstein’s fixed point theorem in metric spaces, Krasnoselskii’s fixed point theorem in strictly convex Banach spaces, Dhage’s fixed point theorem in strictly convex Banach algebras, and Sadovskii’s fixed point problem in strictly convex Banach spaces. We then present applications of these best proximity point results to complex function theory, as well as the existence of a solution of a nonlinear functional integral equation and the existence of a mutually nearest solution for a system of integral equations. Full article

This paper investigates the structural properties of solutions of vector equilibrium systems and mixed variational inequalities in topological vector spaces. Based on Himmelberg-type fixed point theorem, combined with the analysis of set-valued mapping and quasi-monotone conditions, the existence criteria of solutions for two classes of generalized equilibrium problems with weak compactness constraints are constructed. This work introduces an innovative application of the measurable selection theorem of semi-continuous function space to eliminate the traditional compactness constraints, and provides a more universal theoretical framework for game theory and the economic equilibrium model. In the analysis of mixed variational problems, the topological stability of the solution set under the action of generalized monotone mappings is revealed by constructing a new KKM class of mappings and introducing the theory of pseudomonotone operators. In particular, by replacing the classical compactness assumption with pseudo-compactness, this study successfully extends the research boundary of scholars on variational inequalities, and its innovations are mainly reflected in the following aspects: (1) constructing a weak convergence analysis framework applicable to locally convex topological vector spaces, (2) optimizing the monotonicity constraint of mappings by introducing a semi-continuous asymmetric condition, and (3) in the proof of the nonemptiness of the solution set, the approximation technique based on the family of relatively nearest neighbor fields is developed. The results not only improve the theoretical system of variational analysis, but also provide a new mathematical tool for the non-compact parameter space analysis of economic equilibrium models and engineering optimization problems. This work presents a novel combination of measurable selection theory and pseudomonotone operator theory to handle non-compact constraints, advancing the theoretical framework for economic equilibrium analysis. Full article

Abnormal aortic elasticity serves as a marker for cardiovascular mortality and has a negative impact on the left ventricular (LV) afterload. Noncompaction cardiomyopathy (NCCM) is characterized by hypertrabeculation of the LV endomyocardial wall, with an underdeveloped endocardial helix. This may result in absence of LV twist, disturbed aortic elasticity, LV dysfunction, and ultimately premature heart failure (HF). This study compared the aortic stiffness and clinical outcome in patients with NCCM to that of a control group with dilated cardiomyopathy (DCM). Sixty NCCM patients, matched by age and sex, were compared with 60 DCM controls. Transthoracic echocardiography was performed to measure the systolic (SD) and diastolic diameters (DD) of the ascending aorta. These measurements, along with systolic (SBP) and diastolic blood pressure (DBP), were utilized to calculate the aortic stiffness index defined as ln(SBP/DBP)/[(SD-DD)/DD]. This index was then compared to clinical features and outcome. The mean age was 49 ± 16 years (55% males) in the NCCM group and 49 ± 16 years (55% male) in the DCM group. Aortic stiffness index (ASI) was significantly higher in the NCCM group than in the DCM group (7.0 [5.8–10.2] vs. 6.2 [4.8–7.7], p = 0.011). This difference remained statistically significant after adjustment for established risk factors associated with aortic stiffness (β = 1.771; 95% CI [0.253–3.289], p = 0.023). Patients with NCCM demonstrated increased aortic stiffness when compared to those with DCM, which may reflect the underlying pathophysiological processes. Additional research is necessary to evaluate the impact of aortic stiffness on the advancement of LV dysfunction, the onset of heart failure, and long-term outcomes. Full article

This article is a survey, and its aim is to provide a concise introduction to the topic of measures of noncompactness and measures of weak noncompactness in Banach spaces. These measures constitute a useful tool in nonlinear analysis, for example, in studies on the existence of solutions to nonlinear differential and integral equations. Recently, they have also been applied to the analysis of infinite systems of such equations. Throughout the paper, particular attention is given to highlighting the symmetry that exists between these concepts. Some open problems are also included at the end of the paper. Full article