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In this paper we deal with the initial boundary value problem of the following non-linear wave equation of Kirchhoff type ... where M is a continuous function on [0,+∞) such thatM(s) ≥ m0 > 0 for all s ≥ 0. By assuming ρ > 0 is such that H1 0 (∩),→ Lρ+2 and g > 0 is exponentially decreasing, we discuss the global existence and asymptotic behavior of solutions. MSC 2020. 35J05. [ABSTRACT FROM AUTHOR]

The aim of this study is to prove the existence of solutions for Caputo boundary value problems of nonlinear fractional integro-differential equations with integral boundary conditions, by using the measure of non compactness combined with M¨onch's fixed point theorem. Two examples are offered to demonstrate our outcomes. [ABSTRACT FROM AUTHOR]

By using the fixed point theorems, we give suficient conditions for the existence and uniqueness of solutions for the nonlocal fractional boundary value problem of nonlinear Riesz-Caputo differential equation. The boundedness assumption on the nonlinear term is replaced by growth conditions or by a continuous function. Finally, some examples are presented to illustrate the applications of the obtained results. [ABSTRACT FROM AUTHOR]

This document is a preface to a special issue of the journal "Mathematics in Engineering, Science & Aerospace" focused on mathematical problems in science, engineering, and aerospace. The preface explains that mathematical analysis encompasses various disciplines and is applicable to fields such as engineering, economics, and biology. The special issue includes research articles on mathematical analysis with applications in engineering and applied sciences. The preface also acknowledges the contributors and reviewers who helped develop the issue. [Extracted from the article]

The aim of this paper is to establish the asymptotic analysis of nonlinear boundary value problems. The non-stationary motion is given by the elastic constructive law. The contact is described with a version of Tresca's law of friction. A variational formulation of the model, in the form of a coupled system for the displacements and the nonlinear source terms, is derived. The existence of a unique weak solution of the model is established. We also give the problem in transpose form, and we demonstrate different estimates of the displacement and of the source term independently of the small parameter. The main corresponding convergence results are stated in the different theorems of the last section. [ABSTRACT FROM AUTHOR]

This paper discusses the problem of the existence and uniqueness of solutions to the boundary value problem for the nonlinear fractional-order pantograph equation, using the fractional derivative of variable order of Hadamard type. The main results are proved through the application of fractional calculus and Krasnoselskii's fixed-point theorem. Moreover, the Ulam–Hyers–Rassias stability of the nonlinear fractional pantograph equation is analyzed. To conclude this paper, we provide an example illustrating our findings and approach. [ABSTRACT FROM AUTHOR]

We consider autonomous differential inclusions with nonlinear boundary conditions. Sufficient conditions for the existence of solutions in the class of absolutely continuous functions are obtained for these inclusions. It is shown that the corresponding existence theorem applies to the Cauchy problem and the antiperiodic boundary value problem. The result is used to derive a new mean value inequality for continuously differentiable functions. [ABSTRACT FROM AUTHOR]

This paper considers a periodic boundary value problem for a nonlinear partial differential equation with a deviating spatial variable. It is called the nonlocal erosion equation and was proposed as a model for the formation of dynamic patterns on the semiconductor surface. As is demonstrated below, the formation of a spatially inhomogeneous relief is a self-organization process. An inhomogeneous relief appears due to local bifurcations in the neighborhood of homogeneous equilibria when they change their stability. The analysis of this problem is based on modern methods of the theory of infinite-dimensional dynamic systems, including such branches as the theory of invariant manifolds, the apparatus of normal forms, and asymptotic methods for studying dynamic systems. [ABSTRACT FROM AUTHOR]

We discuss the existence of solutions for a boundary value problem of nonlinear coupled Riemann–Liouville fractional integro-differential equations equipped with coupled nonlocal fractional integro-multipoint boundary conditions. The standard tools of the modern functional analysis are employed to derive the desired results for the problem at hand. The case of nonlinearities depending on the Riemann–Liouville fractional integrals is also discussed. Examples illustrating the obtained results are presented. [ABSTRACT FROM AUTHOR]

In the paper, we develop two novel iterative methods to determine the solution of a second-order nonlinear boundary value problem (BVP), which precisely satisfies the specified non-separable boundary conditions by taking advantage of the property of the corresponding boundary shape function (BSF). The first method based on the BSF can exactly transform the BVP to an initial value problem for the new variable with two given initial values, while two unknown terminal values are determined iteratively. By using the BSF in the second method, we derive the fractional powers exponential functions as the bases, which automatically satisfy the boundary conditions. A new splitting and linearizing technique is used to transform the nonlinear BVP into linear equations at each iteration step, which are solved to determine the expansion coefficients and then the solution is available. Upon adopting those two novel methods very accurate solution for the nonlinear BVP with non-separable boundary conditions can be found quickly. Several numerical examples are solved to assess the efficiency and accuracy of the proposed iterative algorithms, which are compared to the shooting method. [ABSTRACT FROM AUTHOR]

Purpose. To develop linear and nonlinear mathematical models of heat conduction for isotropic heterogeneous media with internal heating. This will allow for an increased accuracy in determining temperature fields, which will subsequently impact the effectiveness of designing mechanisms, devices, and individual components of structures that have a layered structure and are subjected to heat stress. Methodology. For the development of linear and nonlinear mathematical models of the temperature field and the analysis of temperature regimes in layered media with internal thermal heating, the coefficient of thermal conductivity is described as a whole using asymmetric unit functions. This makes it possible to solve a differential equation with singular coefficients in both linear and nonlinear boundary value problems of heat conduction with appropriate boundary conditions. Findings. Quadratic equations are obtained to determine the analytical solutions of linear and nonlinear boundary problems of heat conduction for a layered plate with internal heat load. Originality. The scientific novelty lies in the given method of linearization of the nonlinear mathematical model of heat conduction and obtaining analytical solutions, in a closed form, of the corresponding linear and nonlinear boundary value problems for isotropic layered media subjected to internal heating. Practical value. The developed linear and nonlinear mathematical models for determining the temperature distribution in layered structures with internal heating make it possible to analyze heat exchange processes and ensure the thermal stability of such structures. This also makes it possible to increase the heat resistance of structures and protect them from overheating, which can lead to damage to individual components and elements of mechanisms, as well as to the entire structure as a whole. The resulting analytical solutions can be used to predict temperature fields in mine shafts, underground environments and mechanisms of mining equipment, in particular, in drilling and underground compressor stations, ventilation systems and other equipment, which improves work efficiency and reduces useful energy consumption. [ABSTRACT FROM AUTHOR]

We consider the two-point boundary value problems for a nonlinear one-dimensional second-order differential equation with involution in the second derivative and in lower terms. The questions of existence and uniqueness of the classical solution of two-point boundary value problems are studied. The definition of the Green's function is generalized for the case of boundary value problems for the second-order linear differential equation with involution, indicating the points of discontinuities and the magnitude of discontinuities of the first derivative. Uniform estimates for the Green's function of the linear part of boundary value problems are established. Using the contraction mapping principle and the Schauder fixed point theorem, theorems on the existence and uniqueness of solutions to the boundary value problems are proved. The results obtained in this paper cover the boundary value problems for one-dimensional differential equations with and without involution in the lower terms. [ABSTRACT FROM AUTHOR]

Based on the reproducing kernel theory, we solve the nonlinear fourth order boundary value problem in the reproducing kernel space W5 2 [0,1]. Its approximate solution is obtained by truncating the n-term of the exact solution and using the "-best approximate method. Meanwhile, the approximate solution u(i) n (x) converges uniformly to the exact solution u(i)(x); (i,0,1,2,3,4). The validity and accuracy of this method are verified by some examples. [ABSTRACT FROM AUTHOR]

In this article, we present a novel enhancement to the α -parameterized differential transform method (PDTM) for solving nonlinear boundary value problems. The proposed method is applied to solve the generalized Gardner equation by utilizing genetic algorithms to obtain optimal parameter values. Our proposed approach extends the general differential transformation method, allowing for the use of various values for the coefficient α. Our solution procedure offers a distinct advantage by allowing the original differential transformation method to be divided into multiple steps, thereby illustrating specific solution properties for nonlinear boundary value problems. Additionally, possible alternative solutions based on varying parameter values are also explored and discussed. The results with those obtained through the DTM method and exact solutions are compared to confirm the accuracy of our method and its efficiency in reaching the exact solution quickly. [ABSTRACT FROM AUTHOR]

Introducing a new generalized multivariate Mittag-Leffler function which is a generalization of the multivariate Mittag-Leffler function, we derive a sufficient condition for the uniqueness of solutions to a brand new boundary value problem of the fractional nonlinear partial integro-differential equation using Banach's fixed point theorem and Babenko's technique. This has many potential applications since uniqueness is an important topic in many scientific areas, and the method used clearly opens directions for studying other types of equations and corresponding initial or boundary value problems. In addition, we use Python which is a high-level programming language efficiently dealing with the summation of multi-indices to compute approximate values of the generalized Mittag-Leffler function (it seems impossible to do so by any existing integral representation of the Mittag-Leffler function), and provide an example showing applications of key results derived. [ABSTRACT FROM AUTHOR]

In this paper, we study the cyclic anti-periodic boundary value problem of a nonlinear tripled system of fractional Langevin equations. By means of Krasnoselskii fixed point theorem and Banach contraction mapping theorem, we obtain sufficient conditions for the existence and uniqueness of results. Moreover, we investigate the Ulam-Hyers and Ulam-Hyers-Rassias stabilities of the proposed problem. Finally, two illustrative examples are presented. [ABSTRACT FROM AUTHOR]

Using variable-order fractional derivatives in differential equations is essential. It enables more precise modeling of complex phenomena with varying memory and long-range dependencies, improving our ability to describe real-world processes reliably. This study investigates the properties of solutions for a two-point boundary value problem associated with φ -Caputo fractional derivatives of variable order. The primary objectives are to establish the existence and uniqueness of solutions, as well as explore their stability through the Ulam-Hyers concept. To achieve these goals, Banach's and Krasnoselskii's fixed point theorems are employed as powerful mathematical tools. Additionally, we provide numerical examples to illustrate results and enhance comprehension of theoretical findings. This comprehensive analysis significantly advances our understanding of variable-order fractional differential equations, providing a strong foundation for future research. Future directions include exploring more complex boundary value problems, studying the effects of varying fractional differentiation orders, extending the analysis to systems of equations, and applying these findings to real-world scenarios, all of which promise to deepen our understanding of Caputo fractional differential equations with variable order, driving progress in both theoretical and applied mathematics. [ABSTRACT FROM AUTHOR]

This paper deals with the design and analysis of a robust numerical scheme based on an improvised quartic B-spline collocation (IQBSC) method for a class of nonlinear derivative dependent singular boundary value problems (DDSBVP). The convergence analysis of the method is studied by means of Green's function approach. It should be pointed out that the numerical order of convergence of standard quartic B-spline collocation (SQBC) scheme for second-order boundary value problems (BVPs) is four, however, our proposed IQBSC method is shown to be sixth order convergence. To illustrate the applicability and accuracy of the method, we consider eight test problems. The obtained results are compared to those from some existing numerical schemes in order to show the advantage of present method. It is shown that the rate of convergence of present numerical scheme is higher than that of some of existing numerical methods. The CPU time of the present numerical method is provided. [ABSTRACT FROM AUTHOR]

The local dynamics of systems of two equations with delay is considered. The main assumption is that the delay parameter is large enough. Critical cases in the problem of the stability of the equilibrium state are identified, and it is shown that they are of infinite dimension. Methods of infinite-dimensional normalization are used and further developed. The main result is the construction of special nonlinear boundary value problems that play the role of normal forms. Their nonlocal dynamics determine the behavior of all solutions of the original system in a neighborhood of the equilibrium state. [ABSTRACT FROM AUTHOR]

We propose a new class of SBVPs which deals with exothermic reactions. We also propose four computationally stable methods to solve singular nonlinear BVPs by using Hermite wavelet collocation which are coupled with Newton's quasilinearization and Newton-Raphson method. We compare the results which are obtained by using Hermite wavelets with the results obtained by using Haar wavelets. The efficiency of these methods are verified by applying these four methods on Lane-Emden equations. Convergence analysis is also presented. [ABSTRACT FROM AUTHOR]

The global solvability of the boundary value problem for the nonlinear mass transfer equations is proved under inhomogeneous Dirichlet boundary conditions for the velocity given on the entire boundary, and for the substance's concentration, given on the part of the boundary. It is assumed that the reaction coefficient in one of the equations of the model depends nonlinearly on the substance's concentration, and also depends on spatial variables. The local uniqueness of a weak solution of the boundary value problem is proved, the principle of maximum and minimum for the substance's concentration is established. Several types of conditions for the reaction coefficient are considered, each of them has its own mathematical apparatus. [ABSTRACT FROM AUTHOR]

In this paper, we construct a new generalized result to study the existence of solutions of nonlinear fractional boundary value problems (FBVPs). The proposed results unify the existence criteria of certain FBVPs including periodic and antiperiodic as special cases that have been previously studied separately in the literature. The method we employ is topological in its nature and manifests themselves in the forms of differential inequalities (lower and upper solutions, and coupled lower and upper solutions (CLUSs)). Two examples are given to demonstrate the applicability of the developed theoretical results. [ABSTRACT FROM AUTHOR]

Chains of Van der Pol equations with a large delay in coupling are considered. It is assumed that the number of chain elements is also sufficiently large. In a natural manner, a chain is replaced by a Van der Pol equation with an integral term in the space variable and with periodic boundary conditions. Primary attention is given to the local dynamics of chains with unidirectional and bidirectional coupling. For sufficiently large values of the delay parameter, parameters are explicitly determined for which critical cases occur in the stability problem for the zero equilibrium state. It is shown that the problems under consideration have an infinite-dimensional critical case. The well-known methods of invariant integral manifolds and the methods of normal forms are inapplicable in these problems. Proposed by this paper's author, the method of infinite normalization—the method of quasi-normal forms—is used to show that the leading terms of the asymptotics of the original system are determined by solutions of (nonlocal) quasi-normal forms, i.e., special nonlinear boundary value problems of the parabolic type. As the main results, corresponding quasi-normal forms are constructed for the considered chains. [ABSTRACT FROM AUTHOR]

The Hermite wavelet method (HWM) is introduced in this study to solve a nonlinear differential equation determining the human corneal morphology. The changes in curvature of the human cornea in hypotony, normal intraocular pressure, glaucoma, and other conditions are discussed. The Hermite wavelet operational matrices of derivatives are used to generate wavelet solutions based on this technique. The solutions of the nonlinear differential equation are determined for various values of constant parameters that can appear in the diverse physical situations. The proposed wavelet solutions are more accurate than the other approximate analytical solutions listed in the literature. The HWM solutions are compared to homotopy perturbation method, Taylor series, perturbation technique and artificial neural network solutions. There is broad consensus. This illustrates that HWM is a useful and appropriate strategy for handling difficulties with nonlinear boundary value problems that emerge in corneal geometry. [ABSTRACT FROM AUTHOR]

A boundary value problem for a non-linear difference equation of order three is considered. We show that this equation can be interpreted as the equation satisfied by the value function in a stochastic optimal control problem. We thus obtain an expression for the solution of the non-linear difference equation that can be used to find an explicit solution to this equation. An example is presented. [ABSTRACT FROM AUTHOR]

In this paper, we present a reliable numerical algorithm to determine approximate solutions of the two-point boundary value problems having Robin boundary conditions that naturally occur in the investigation of distinct tumor growth issues, the dispersal of heat sources in the person head and steady state oxygen diffusion in spherical cell possessing Michaelis–Menten uptake kinetics. This approach is based on a modified concept of Adomian polynomials (AP), and the two-step Adomian decomposition method (TSADM) merged with Padé approximants. Furthermore a Maple package ADMP is applied to solve various problems, which is very easy to use and efficient and needed to input the system of equations with initial or boundary conditions and diverse essential parameters to deliver the analytic approximate solutions within a few seconds. The suggested scheme does not require linearization, perturbations, guessing the initial terms, a set of basis function or other limiting presumptions, which yields the solutions in closed form. Many examples are examined to make clear the scope and validity of the package ADMP. [ABSTRACT FROM AUTHOR]

In this paper we consider a fourth order nonlinear differential equation with two integral boundary conditions. We establish the existence, uniqueness and positivity of solutions and propose iterative methods both on continuous and on discrete levels for finding the solution. The numerical solution is proved to be of second order accuracy. Many examples demonstrate the validity of the obtained theoretical results, the efficiency of the numerical method and the applicability of it to a wide class of problems. [ABSTRACT FROM AUTHOR]

Sufficient conditions of blow up in finite time of solutions to initial boundary value problems for nonlinear systems of equations of thermoelasticity type are obtained. It is shown that solutions even with large enough initial energies of the considered problems may blow up in a finite time. [ABSTRACT FROM AUTHOR]

A huge number of mathematical models are called Boussinesq-type equations; therefore, a wide range of sixth-order Boussinesq-type equations attracts a lot of attention from outside researchers around the world. The paper studies the classical solution of one nonlinear inverse boundary value problem for linearized Boussinesq equation of the sixth order with an additional integral condition. One method is based on the application of the Fourier method. The other method is the application of the method of contracted mappings. The essence of the problem is that it is required, together with the solution, to determine the unknown kernel. The problem is considered in a rectangular area. When solving the original inverse boundary value problem, a transition from the original inverse problem to some auxiliary inverse problem is carried out. The existence and uniqueness of a solution to an auxiliary problem are proved with the help of contracted mappings. Then a transition to the original inverse problem is made again; as a result, a conclusion is made about the solvability of the original inverse problem. The proposed methods for finding solutions to the inverse problem can be used in the study of solvability for various problems of mathematical physics. [ABSTRACT FROM AUTHOR]

The purpose of this study is to investigate an impulsive dynamic singular nonlinear Sturm-Liouville problem on infinite intervals. The existence and uniqueness of the solutions of such problem will be investigated by considering Weyl's limit-circle case. [ABSTRACT FROM AUTHOR]

We consider the first mixed boundary value problem for a nonlinear differential-difference parabolic equation. We give some sufficient conditions for the nonlinear differential-difference operator to be radially continuous and coercive as well as has the property of (V,W)-semibounded variation (in this case we provide the algebraic condition of strong ellipticity for an essentially nonlinear differential-difference operator). We also justify the existence theorems for a generalized solution. [ABSTRACT FROM AUTHOR]

A fourth-order nonlinear differential equation on a network that is a model of a system of Euler–Bernoulli rods is considered. Based on the monotone iteration method, the existence of a solution of a boundary value problem on a graph for this equation is established using the positiveness of the Green's function and the maximum principle for the corresponding linear differential equation. An example is given to illustrate the results. [ABSTRACT FROM AUTHOR]

The paper concerns a novel concept based on the iterative approach applied for solving the nonlinear satellite-fixed geodetic boundary value problem (NSFGBVP) using the finite element method (FEM). We formulate the NSFGBVP that consists of the Laplace equation defined in the 3D bounded domain outside the Earth, the nonlinear boundary condition (BC) prescribed on the disretized Earth's surface, and the Dirichlet BC given on a spherical boundary placed approximately at the altitude of GOCE satellite mission and additional four side boundaries. The iterative approach is based on determining the direction of actual gravity vector together with the value of the disturbing potential. Such a concept leads to the first iteration where the oblique derivative boundary value problem is solved, and the last iteration represents the approximation of the actual disturbing potential and the direction of gravity vector. As a numerical method for our approach, we implement the FEM with triangular prisms. High-resolution numerical experiments deal with the local gravity field modelling in the Andean, Himalayan and Alpine mountain range, where we focus on the contribution to the disturbing potential solution by solving the nonlinear geodetic boundary value problem in comparison with the solution to the oblique derivative geodetic boundary value problem. The obtained results showed the maximal contribution of the presented approach 0.0571 [ m 2 s - 2 ] (Andes), 0.0702 [ m 2 s - 2 ] (Himalayas), 0.0066 [ m 2 s - 2 ] (Alps) in the disturbing potential, that is located in the areas of the highest values of the deflection of vertical. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some existence results for the integral boundary value problems of nonlinear fractional q-difference equations. The differential operator is taken in the Riemann-Liouville sense. [ABSTRACT FROM AUTHOR]

This article reports a theoretical examination of two-dimensional laminar and incompressible fluid flow due to the linear stretching of a sheet. The viscous fluid flow is analyzed under the impact of the transverse uniform magnetic field. The flow is generated due to the stretching surface and its momentum and temperature distributions have been studied. The features of the flow are modeled mathematically and the nature of framed equations are fully coupled nonlinear partial differential equations. The system of governing equations is operated using an iterative finite difference scheme. This nonlinear boundary value problem is reduced to linear form using the quasilinearization technique and it is further solved using the Thomas algorithm. The numerical results are extracted and the physical behaviors of the flow are found to overlap with the outcomes. Additionally, the findings are extended to the computations of friction factor and heat transfer rate on the boundary for various pertinent parameters. The streamlines of the fluid flow are also depicted for flow controlling parameters. From the contours, the parabolic structure of the streamlines is approaching linear nature near the core of the channel with the slight increment of the physical parameters (Reynolds number, magnetic parameter, and Grashof number). [ABSTRACT FROM AUTHOR]

The fractional order p -Laplacian differential equation model is a powerful tool for describing turbulent problems in porous viscoelastic media. The study of such models helps to reveal the dynamic behavior of turbulence. Therefore, this article is mainly concerned with the periodic boundary value problem (BVP) for a class of nonlinear Hadamard fractional differential equation with p -Laplacian operator. By virtue of an important fixed point theorem on a complete metric space with two distances, we study the solvability and approximation of this BVP. Based on nonlinear analysis methods, we further discuss the generalized Ulam-Hyers (GUH) stability of this problem. Eventually, we supply two example and simulations to verify the correctness and availability of our main results. Compared to many previous studies, our approach enables the solution of the system to exist in metric space rather than normed space. In summary, we obtain some sufficient conditions for the existence, uniqueness, and stability of solutions in the metric space. [ABSTRACT FROM AUTHOR]

In this paper, linear and nonlinear fourth-order Fractional Integro Differential Equations (FIDEs) with boundary value problems are solved by Laplace Transform with Modified Variational IterationMethod (LT-MVIM). A new technique based on the VIM is introduced to remove the random choice of initial guess by setting a specific rule depends on unknown parameters. These parameters contributed to the increase in the number of terms of the polynomial approximation and its degree, which, in turn, accelerates the convergence and increases the accuracy from one iteration compared to the standard method, where the initial approximation is still randomly chosen. Moreover, the standard method requires an infinite number of iterations, which need massive calculations in each iteration. Some examples are given in order to show the accuracy of the solutions obtained by the proposed method. Furthermore, comparisons are made between the solutions obtained by the proposed method and Laplace Transform Variational Iteration Method (LT-VIM) based on the exact solutions, revealing that the LT-MVIM contributes to accelerating the convergence of approximate solution to the exact solution by reducing the computational work to obtain the approximate solution using one iteration. Whereas, LT-VIM needs more iterations to obtain a suitable approximate solution, which results in an increase in the computational workload. [ABSTRACT FROM AUTHOR]

In this paper, we study the time-asymptotic behavior of solutions to an outflow problem for the one-dimensional bipolar Navier–Stokes–Poisson system in the half space. First, we make some suitable assumptions on the boundary data and space-asymptotic states such that the time-asymptotic state of the solution is a nonlinear wave, which is the superposition of the transonic stationary solution and the 2-rarefaction wave. Next, we show the stability of this nonlinear wave under a class of large initial perturbation, provided that the strength of the transonic stationary solution is small enough, while the amplitude of the 2-rarefaction wave can be arbitrarily large. The proof is completed by a delicate energy method and a continuation argument. The key point is to derive the positive upper and lower bounds of the particle densities. [ABSTRACT FROM AUTHOR]

This study introduces a novel and very accurate method for implementing Robin boundary conditions while solving boundary value problems with compact finite difference schemes. Most studies numerically approximate Neumann and Robin boundary conditions with a first-order approximation, however, this reduces the accuracy in general. Our implementation produces highly accurate results in solving nonlinear singular boundary value problems. Other reported results achieved utilizing various techniques are used for comparison. Tables and graphs showing the results are given. [ABSTRACT FROM AUTHOR]

In this paper we study a solution existence problem for the singular nonlinear Burgers equation where and , with small parameter ε under the boundary conditions and the oscillating initial condition The first derivative of the operator F at the solution point is assumed to be degenerate, i.e., is not surjective. The existence of a continuous solution to this nonlinear problem is proved by applying p-regularity theory and the Michael selection theorem. [ABSTRACT FROM AUTHOR]

We prove the unique solvability of the nonlinear boundary value problem for an odd order nonlinear equation with multiple characteristics in a curvilinear domain. The uniqueness of a solution is established by the method of energy integrals by using some elementary inequalities and Friedrichs type inequalities. To prove the existence of a solution to this problem, an auxiliary problem is considered whose Green function is constructed. With the help of this auxiliary problem, the original problem is reduced to a system of Hammerstein integral equations. The solvability of the nonlinear system is proved by the contraction mapping principle. [ABSTRACT FROM AUTHOR]

In this article, we prove some results on existence and uniqueness of fixed points for an almost -contraction mapping defined on a metric space endowed with an amorphous relation. Our results generalize and improve several well known fixed point theorems of the existing literature. To substantiate the credibility of our results, we construct some examples. We also apply our results to determine a unique solution of a boundary value problem associated with nonlinear elastic beam equations. [ABSTRACT FROM AUTHOR]

In a paper from 1960, Felix Browder established a theorem concerning the continuation of the fixed points of a family of continuous functions ft:X→X$f_t:X\rightarrow X$ depending continuously on a parameter t∈[0,1]$t\in [0,1]$, where X$X$ is a convex and compact subset of Rn$\mathbb {R}^n$. Here, the result is presented for a compact mapping f:A×X→X$f:A\times X\rightarrow X$ where X$X$ is a convex, closed, and bounded subset of an arbitrary normed space and A$A$ is an arcwise connected topological space. Applications to nonlinear boundary value problems are given; specifically, we shall present new viewpoints of known results, introduce some novel results, and exhibit some open problems. [ABSTRACT FROM AUTHOR]