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Generating numerical solutions for boundary value problems (BVPs) using numerical methods can be a difficult task since the solutions may involve complex mathematical formulations. Hence, various types of B-splines collocation methods were developed to produce better numerical approximations with much simpler approaches. The present study introduced two new types of extended Cubic Hybrid B-spline Collocation Method (CHBSM) namely the Extended Cubic Hybrid B-spline Collocation Method (ECHBSM) and Extended Trigonometric Cubic Hybrid B-spline Collocation Method (ETCHBSM) for solving second order linear two-point BVPs. These methods were tested on three examples of linear two-point BVPs of order two. For comparison purposes, three established collocation methods which are the Cubic B-spline Collocation Method (CBSM), Cubic Trigonometric B-spline Collocation Method (CTBSM) and Cubic Hybrid B-spline Collocation Method (CHBSM) were also applied on these examples. The numerical results were tabulated and analysed to make comparisons with the analytical solutions and the conventional methods from past literatures. For CHBSM, ECHBSM and ETCHBSM, optimization was applied to the free parameter, γ using a simple proposed approach. The result demonstrated that CHBSM yields the best approximation with the analytical solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study a two-point boundary value problem involving p-Laplacian and its relationship with the associated discrete ones. Specifically, by using discrete variational methods, we first obtain a sequence of positive solutions for the associated discrete two-point boundary value problems corresponding to different step lengths. Then, utilizing the sequence of positive solutions, we construct a sequence of continuous functions which can be shown to be precompact. Finally, we prove that the limit function of the convergent subsequence is the desired positive solution. In particular, our result covers the cases when the nonlinear function in the equation is (p - 1) -superlinear and asymptotically (p - 1) -linear. [ABSTRACT FROM AUTHOR]

This article presents a comprehensive literature survey on fractional-order optimal control problems. Fractional-order differential equation is extensively used nowadays to model real-world systems accurately, which exhibit fractal dimensions, memory effects, as well as chaotic behaviour. These versatile features attract engineers to concentrate more on this, and it is widely used in the broad domain of science and technology. The mentioned numerical tools take the necessary optimal conditions into account, which makes it a two-point boundary value problem of non-integer order. In this review article, some numerical approaches for the approximation have been stated for obtaining the solution to fractional optimal control problems (FOCPs). Here, few numerical approaches including Grunwald-Letnikov approximation, Adams type predictor-corrector method, generalized Euler’s method, Caputo-Fabrizio method Bernoulli and Legendre polynomials method, Legendre operational method, and Ritz’s and Jacobi’s method are treated as an advanced method to obtain the solution of FOCP. Fractional delayed optimal control is selected for our investigation. It refers to a type of control problem where the control action is delayed by a fractional amount of time. In other words, the control input at a given time depends not only on the current state of the system but also on its past state at fractional times. The fractional delayed optimal control problem is formulated as an optimization problem that seeks to minimize a cost function subject to a set of constraints that represent the dynamics of the system and the fractional delay in the control input. The solution to this problem typically involves the use of fractional polynomials types, i.e. Chebyshev and Bassel polynomials.

This paper presents an enhanced version of the standard shooting method that enables problems with two unknown parameters to be solved. A novel approach is applied to the analysis of the natural vibrations of Euler-Bernoulli beams. The proposed algorithm, named as two-parameter multiple shooting method, is a new powerful numerical tool for calculating the natural frequencies and modes of multi-segment prismatic and non-prismatic beams with different boundary conditions. The impact of the axial force and additional point masses is also taken into account. Due to the fact that the method is based directly on the fourth-order ordinary differential equation, the structures do not have to be divided into many small elements to obtain an accurate enough solution, even though the geometry is very complex. To verify the proposed method, three different examples are considered, i.e., a three-segment non-prismatic beam, a prismatic column subject to non-uniformly distributed compressive loads, and a two-segment beam with an additional point mass. Numerical analyses are carried out with the software MATHEMATICA. The results are compared with the solutions computed by the commercial finite element program SOFiSTiK. Good agreement is achieved, which confirms the correctness and high effectiveness of the formulated algorithm. [ABSTRACT FROM AUTHOR]

The uniform trigonometrization method (UTM) faces two difficulties in solving the Mars entry problem with maximum terminal altitude. One is that two possible solutions need to be evaluated by Pontryagin's minimum principle (PMP). The other one is that using numerical continuation logic to solve the Hamiltonian two-point boundary value problem (TPBVP) involves a significant amount of computation. To overcome these two challenges, indirect sequential convex programming (ISCP), which combines the UTM and sequential convex programming (SCP), is developed in this article. The proposed method first modifies the two control options into a single solution, which eliminates the requirement for PMP evaluation. The nonlinear TPBVP resulted from the improved UTM is then relaxed into a convex programming problem that is established by introducing the virtual control and buffer. Subsequently, the SCP process generates a numerical solution to the Mars entry problem within a short time. Compared to the general pseudospectral method and UTM, simulations demonstrate the effectiveness and efficiency of the ISCP method for longitudinal Mars entry missions with or without path constraints. • The two solutions of the uniform trigonometrization method are improved to one. • The two-point boundary value problem is solved by sequential convex programming. • The bang-bang solution is obtained by indirect sequential convex programming. [ABSTRACT FROM AUTHOR]

In this article, we look at a class of two-point nonlinear boundary value problems and transform this into derivative-dependent Fredholm–Hammerstein integral equations, i.e., the integral equation, where the kernel is of Green's type, and the nonlinear function inside the integral is dependent on the derivative. We obtain the error analysis by replacing all the integrals in the Galerkin method with numerical quadrature. We propose the discrete Galerkin and iterated discrete Galerkin methods by piecewise polynomials to obtain the convergence analysis of these derivative-dependent Fredholm–Hammerstein integral equations. By choosing the numerical quadrature rule appropriately, the convergence rates in Galerkin and iterated Galerkin methods are preserved. We show that the iterated discrete Galerkin method improves over the discrete Galerkin method in terms of the order of convergence. To demonstrate the theoretical results, several numerical examples are provided. [ABSTRACT FROM AUTHOR]

A quasilinear Fredholm integro-differential equation is considered on a finite interval. By using Dzhumabaev parametrization method the integro-differential equation is reduced to a special Cauchy problem. The sufficient conditions of the existence of a unique solution to the special Cauchy problem are established, and a new general solution to the integro-differential equation is constructed using this solution. The new general solution is used to solve a boundary value problem, the conditions for the existence of a unique solution to the boundary value problem are obtained. [ABSTRACT FROM AUTHOR]

In this paper, a new computational approach is presented to solve a boundary-value problem for a differential equation with piecewise constant argument of generalized type (DEPCAG). The presented technique is based on the Dzhumabaev parametrization method. A useful numerical algorithm is developed to obtain the numerical values from the problem. Numerical experiments are conducted to demonstrate the accuracy and efficiency.

Autonomous berthing is a crucial technology for autonomous ships, requiring optimal trajectory planning to prevent collisions and minimize time and control efforts. This paper presents a two-phase, two-point boundary value problem (TPBVP) strategy for creating an optimal berthing trajectory for a twin-propeller, twin-rudder ship with autonomous berthing capabilities. The process is divided into two phases: the approach and the terminal. Tunnel thruster use is limited during the approach but fully employed during the terminal phase. This strategy permits concurrent optimization of the total trajectory duration, individual phase trajectories, and phase transition time. The efficacy of the proposed method is validated through two simulations. The first explores a scenario with phase transition, and the second generates a trajectory relying solely on the approach phase. The results affirm our algorithm's effectiveness in deciding transition necessity, identifying optimal transition timing, and optimizing the trajectory accordingly. The proposed two-phase TPBVP approach holds significant implications for advancements in autonomous ship navigation, enhancing safety and efficiency in berthing operations.

Abstract The novelty of this paper is that, based on Mawhin’s continuation theorem, we present some sufficient conditions that ensure that there is at least one solution to a particular kind of a boundary value problem with the p-Laplacian and generalized fractional Caputo derivative.

In this article, we use an effective computational method for solving the two-point boundary value problems (BVPs). In order to use the weak-form integral equation method, we firstly introduce the suitable adjoint test functions. Then, we derive weak form of the given BVP, by utilizing suitable trial functions. Finally, we impose a combination of exponential functions as the basis of solution space. Different types of boundary conditions (of Dirichlet and robin types) are investigated and stability of the method are numerically considered.

The encouragement of presented writing is to acquire various theorems for α-(Ƒ, ℏ)-Geraghty contraction type mappings having a mixed monotone property in partial order on metric spaces. An instance is preferred to display the straightness of the obtained findings. Further, we propel the applicability of our conclusion in finding the solution for a two-point boundary value problem. Our results not only extend and generalize several results in the fixed point theory literature but also unify most of them. [ABSTRACT FROM AUTHOR]

• Optimal Strategies for the evader are designed under various conditions. • The effects of orbit energy and orbital radius on escape strategy are illustrated. • Pursuer's optimal delta-V's asymmetrical distribution law is unveiled. • A correction method is designed to reduce pursuer's delta-V. • Effects of incomplete information on strategies are discussed. This research investigates the orbital pursuit-evasion games with impulsive maneuvers in long-distance interception scenarios. Specifically, this paper examines the influences of escape impulses, with different directions, applied by the evader on the intercept-correction delta-v (Δ V Corr ) of the pursuer that is already on an interception orbit. After obtaining the evader's strategies based on maximizing the Δ V Corr under different circumstances, it is found that, with the knowledge of the evader's best strategy, the pursuer could make Δ V Corr smaller by adjusting its initial interception orbit. Then, a method to modify the initial orbit with energy constraints is designed so that the pursuer can match the evader's escape maneuver with less delta-v to prepare. Finally, this paper summarizes the best strategies of both sides after considering the initial trajectory adjustment and discusses the impact of incomplete information on tactics in orbital pursuit-evasion games. [ABSTRACT FROM AUTHOR]

In the present study, the necessary and sufficient conditions of equilibrium control for general optimal control problems with time inconsistency are established in sense of open-loop. As an application, the linear quadratic optimal control problems with time inconsistency were also explored and an explicit equilibrium control is constructed.

The novelty of this paper is that, based on Mawhin's continuation theorem, we present some sufficient conditions that ensure that there is at least one solution to a particular kind of a boundary value problem with the p-Laplacian and generalized fractional Caputo derivative. [ABSTRACT FROM AUTHOR]

The problem of processing the results of radar measurements of the parameters of motion of a target under conditions of perturbations due to aircraft maneuvers is examined. In order to adapt to the perturbations of measurement procedures, the method of invariant immersion has been modified. The use of the modified method in the case of a linear observation equation results in the widespread form of the discrete measurement procedure, derived in this article as a special case of a modified equation of invariant immersion. Based on statistical modeling of the digital processing of the results of radar measurements of the movement parameters of a maneuvering target, performance indicators of radar stations were calculated — the averaged absolute and relative measurement errors of the motion parameters, the wiring ratio, and the duration of path breaks. The analysis of specific indicators confirmed the enhanced performance of using the measurement procedures of radar stations in informational-measurement procedures, synthesized using acceleration models in the form of white noise and the Singer acceleration model. Enhancement of the performance was due to the inclusion of the coefficient of adaptation in the structure of the modified equations of invariant immersion. The proposed measurement procedures based on the modified method of invariant immersion were compared with the Kalman measurement procedures, synthesized using specified acceleration models. Calculations of the performance indicators confirm the reliability of the results obtained, since the well-known fact of an increase in the error of measurement of the height of an aircraft by monopulse radar stations near the surface is demonstrated. The results of the study will be useful in the development of adaptive measurement procedures for information-measurement systems operating under conditions of perturbations of measurement processes, such as for survey locators. [ABSTRACT FROM AUTHOR]

In this paper, modification of Dzhumabaev parameterization method is developed to a boundary value problem for systems of loaded differential equations with piecewise constant argument of generalized type (EPCAG). The method is based on reducing the considering problem to an equivalent multi-point boundary value problem for ordinary differential equations with parameters. An equivalent boundary value problem with parameters consists of the Cauchy problem for a system of ordinary differential equations with parameters, a two-point condition, a continuity condition, and additional conditions for a piecewise constant argument. The solution of the Cauchy problem for a system of ordinary differential equations with parameters is constructed using the fundamental matrix of the differential equation. A system of linear algebraic equations for the parameters is compiled using the values of the solution at the corresponding points and substituting them into the two-point condition, the continuity condition, and the conditions for the piecewise constant argument. A modification of Dzhumabaev parameterization method for solving the considering boundary value problem is proposed which is based on solving the constructed system and the 4th order Runge-Kutta method for solving the Cauchy problem on subintervals. The obtained results are verified by a numerical example. Numerical analysis showed high efficiency of the constructed modification of Dzhumabaev parameterization method.

Abstract This paper provides conditions for determining the sign of all the partial derivatives of the Green functions of n-th order boundary value problems subject to a wide set of homogeneous two-point boundary conditions, removing restrictions of previous results about the distance between the two extremes that define the problem. To do so, it analyzes the sign of the derivatives of the solutions of related two-point n-th order boundary value problems subject to n − 1 $n-1$ boundary conditions by introducing a new property denoted by ‘hyperdisfocality’.

A two-point boundary value problem for Volterra–Fredholm integro-differential equations is considered. To solve this problem, Dzhumabaev parametrization method is used. Conditions for the existence and uniqueness of two-point boundary value problem for Volterra–Fredholm integro-differential equations are established. An algorithm for finding solution of this problem is proposed. Then, numerical approach for analysis of two-point boundary value problem for Volterra–Fredholm integro-differential equations is offered. Results are illustrated by numerical example. [ABSTRACT FROM AUTHOR]

In the present study, the necessary and sufficient conditions of equilibrium control for general optimal control problems with time inconsistency are established in sense of open-loop. As an application, the linear quadratic optimal control problems with time inconsistency were also explored and an explicit equilibrium control is constructed. [ABSTRACT FROM AUTHOR]

In this article, a two-point boundary-value problem for a differential equation with piecewise constant argument of generalized type is solved. We develop a numerical algorithm that is based on the parametrization method proposed by Dzhumabaev. The proposed algorithm is expanded to find a numerical solution to the boundary-value problem for differential equations with piecewise constant argument of generalized type. The results are illustrated by numerical example. [ABSTRACT FROM AUTHOR]

The paper studies a two-point boundary value problem with unbounded boundary conditions for a linear singularly perturbed differential equation. Asymptotic estimates are given for a linearly independent system of solutions of a homogeneous perturbed equation. Auxiliary, so-called boundary functions, the Cauchy function are defined. For sufficiently small values of the parameter, estimates for the Cauchy function and boundary functions are found. An algorithm for constructing the desired solution of the boundary value problem has been developed. A theorem on the solvability of a solution to a boundary value problem is proved. For sufficiently small values of the parameter, an asymptotic estimate for the solution of the inhomogeneous boundary value problem is established. The initial conditions for the degenerate equation are determined. The formula is determined; the phenomena of the initial jump are studied.

The linear convex log-homotopy has been used in the derivation of particle flow filters. One natural question is whether it is beneficial to consider other forms of homotopy. We revisit this question by considering a general linear form of log-homotopy for which we derive particle flow filters, validate the distribution of flows, and obtain conditions for the stability of particle flows. We then formulate the problem of stiffness mitigation as an optimal control problem by minimizing the condition number of the Hessian matrix of the posterior density function. The optimal homotopy can be efficiently obtained by solving a 1-D second-order two-point boundary value problem. Compared with traditional matrix analysis based approaches to improving condition numbers, this approach explicitly exploits the special structure of the stochastic differential equations in particle flow filters. The effectiveness of the proposed approach is demonstrated by numerical examples. [ABSTRACT FROM AUTHOR]

This paper provides conditions for determining the sign of all the partial derivatives of the Green functions of n-th order boundary value problems subject to a wide set of homogeneous two-point boundary conditions, removing restrictions of previous results about the distance between the two extremes that define the problem. To do so, it analyzes the sign of the derivatives of the solutions of related two-point n-th order boundary value problems subject to n − 1 boundary conditions by introducing a new property denoted by 'hyperdisfocality'. [ABSTRACT FROM AUTHOR]

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In this paper we develop an iterative numerical method based on Bernstein splines for solving two-point boundary problems associated to differential equations of fractional order α ∈ (0; 1). The convergence of the method is proved by providing the error estimate and it is tested on a numerical example. [ABSTRACT FROM AUTHOR]

Two-point boundary value problem for Fredholm integro-differential equation with singular kernel is considered. We propose a method for the investigation and solving of problem for the Fredholm integro-differential equation with weakly singular kernel based on the partition of the interval and introduction of additional parameters. Every partition of the interval is associated with a homogeneous Fredholm integral equation with weakly singular kernel of the second kind. The definition of regular partitions is presented. It is shown that the set of regular partitions is nonempty. A criterion for the solvability of the investigated problem is established and an algorithm for finding its solutions is constructed. [ABSTRACT FROM AUTHOR]

Optimal in a certain sense conditions guaranteeing the unique solvability of the boundary value problem... [ABSTRACT FROM AUTHOR]

This study is aimed to relate nonlinear infinite-horizon optimal control problems (NLIHOCPs) with open-loop information. The difficulties of solving the two-point boundary value problems (TPBVPs) arising from NLIHOCPs can be assigned to the nonlinearity of differential equations, the combination of split boundary conditions, and how the transversality conditions in infinite-horizon are treated. In this paper, we propose a combined quasilinearization method (QLM) with the exponential Bernstein functions (EBFs) for solving nonlinear TPBVPs on the semi-infinite domain. First, the QLM is used to reduce the nonlinear TPBVP to a sequence of linear differential equations. Then, a collocation method based on the EBFs is utilized to find the approximate solution of the resulting linear differential equations. By applying the EBFs, the transversality conditions for TPBVP on the semi-infinite domain are satisfied. The convergence of the QLM + EBFs is proved. Some numerical experiments are performed to confirm the validity of the proposed computational scheme. [ABSTRACT FROM AUTHOR]

The nonlinear two-point boundary value problem (TPBVP) is converted to a new form of the problem including integral terms. Then the combination of weak-form integral equation method (WFIEM) and special functions create a robust and applicable numerical scheme to solve the problem. To display the accuracy of our method, some examples are investigated. Also, the fractional Boussinesq-like equation involving the β-derivative has been considered that describes the propagation of small amplitude long capillary-gravity waves on the surface of shallow water.

The solvability of second order differential systems with the classical separated or periodic boundary conditions is considered. The proofs use special classes of curvature bound sets or bound sets together with the simplest version of the Leray-Schauder continuation theorem. The special cases where the bound set is a ball, a parallelotope or a bounded convex set are considered.

This paper's focal point is on the nonstandard Optimal Control (OC) problem. In this matter, the value of the final state variable, y(T) is said to be unknown. Moreover, the Lagrangian integrand in the function is in the form of a piecewise constant integrand function of the unknown state value y(T). In addition, the Lagrangian integrand depends on the y(T) value. Thus, this case is considered as the nonstandard OC problem where the problem cannot be resolved by using Pontryagin’s Minimum Principle along with the normal boundary conditions at the final time in the classical setting. Furthermore, the free final state value, y(T) in the nonstandard OC problem yields a necessary boundary condition of final costate value, p(T) which is not equal to zero. Therefore, the new necessary condition of final state value, y(T) should be equal to a certain continuous integral function of y(T)=z since the integrand is a component of y(T). In this study, the 3-stage piecewise constant integrand system will be approximated by utilizing the continuous approximation of the hyperbolic tangent (tanh) procedure. This paper presents the solution by using the computer software of C++ programming and AMPL program language. The Two-Point Boundary Value Problem will be solved by applying the indirect method which will involve the shooting method where it is a combination of the Newton and the minimization algorithm (Golden Section Search and Brent methods). Finally, the results will be compared with the direct methods (Euler, Runge-Kutta, Trapezoidal and Hermite-Simpson approximations) as a validation process.

In this study, a class of direct numerical integrators for solving special second-order ordinary differential equations (ODEs) is proposed and studied. The method is multistage and multistep in nature. This class of integrators is called "two-step Runge-Kutta-Nyström", denoted by TSRKN. The direct approach to higher-order ODEs is desirable to avoid tedious computational work caused by converting the higherorder ODEs into the system of first-order equations. The order conditions for the TSRKN are derived using Taylors series expansion and according to the order conditions, a three-stage TSRKN method which is convergent of order four is constructed. The convergence analysis of the method is discussed and the performance of the newly derived method is compared with existing methods. The numerical results show the superiority of the TSRKN method in terms of number of function evaluations and demonstrate that the TSRKN can also be used to solve linear second-order boundary value problems (BVPs) since Runge-Kutta-Nyström (RKN) approach is practically used to only solve higher-order initial value problems (IVPs) directly. [ABSTRACT FROM AUTHOR]

An analysis carried out to study unsteady flow through porous media. The non-linear equation governing the flow problem is transformed into linear differential equation using quasi linearization technique and solved by applying cubic spline collocation method. The obtained numerical results are compared with finite difference method to show the effectiveness and accuracy of the method. [ABSTRACT FROM AUTHOR]

In this paper, we study a two-point boundary value problem consisting of the heat equation on the open interval (0,1) with boundary conditions which relate first and second spatial derivatives at the boundary points. Moreover, the unique solution to this problem can be represented probabilistically in terms of a sticky Brownian motion. This probabilistic representation is attained from the stochastic differential equation for a sticky Brownian motion on the bounded interval [0,1]. [ABSTRACT FROM AUTHOR]

This paper deals with the existence of positive solutions for a system of nonlinear singular fourth-order differential equations with a parameter $\lambda$ subject two-point boundary conditions. Our analysis relies on the Krasnoselskii fixed point theorem and under suitable conditions, we derive explicit eigenvalue intervals of $\lambda$ for the existence of at least one positive solution for the system.

Chebyshev spectral methods for two-point boundary value problems describing the processes of counter interaction of optical waves in media with cubic nonlinearity and linear media with periodic modulation of the refractive index are considered. On the example of a linear problem, it is shown that the spectral method for achieving a given accuracy requires of two-three orders less time in comparison with the spline collocation method of the 5th accuracy order. Moreover, Chebyshev mesh has natural adaptive properties for the considered problems of the nonlinear interaction of optical waves. A conservative iterative algorithm for implementation of the nonlinear spectral model is proposed. The proposed method has a lower sensitivity to the choice of an appropriate initial guess and provides a higher rate of convergence in comparison with Newton’s method under conditions of strong coupling of interacting waves.

The asymptotic output tracking problem is studied for a class of non-minimum phase nonlinear systems without requiring the a priori knowledge, or even the existence, of a finite-dimensional exosystem that generates the prescribed periodic reference signal. The design strategy is illustrated in two steps. First, it is shown that the knowledge of a solution to a certain two-point boundary value problem involving the underlying zero-dynamics of the plant is instrumental and sufficient to construct a state feedback regulator that achieves boundedness of the trajectories and (exact) asymptotic tracking, hence completely circumventing the need for solving partial differential equations. Then, since the computation of the latter solution may be affected by numerical errors that are particularly detrimental in the presence of unstable zero-dynamics, the above architecture is robustified by means of an additional hybrid feedback loop whose trajectories converge to a solution of the two-point boundary value problem. Once the latter scheme has been established and discussed, the extension to the case of output feedback is presented, firstly in the specially structured case in which the input vector field depends only on the measured output and then extended to the generic case. The theory is then corroborated by means of a physically motivated numerical example involving an inverted pendulum on a cart. Interestingly, it is also shown that the solution provided by the hybrid scheme above coincides with the limiting solution of a suitably defined cheap optimal control problem. [ABSTRACT FROM AUTHOR]

Two-point boundary value problem for fourth order partial integro-differential equation is considered. By new unknown function the problem is reduced to an equivalent family of two-point boundary value problems for the Volterra integro-differential equations of second order. By the method of introduction functional parameters are constructed the algorithms for finding of solution to family of two-point boundary value problems for the Volterra integro-differential equations of second order. Conditions of existence unique solution to family of two-point boundary value problems for Volterra integro-differential equations of second order are obtained in the terms of initial data. Criteria of unique solvability to the two-point boundary value problem for fourth order partial integro-differential equation is established. [ABSTRACT FROM AUTHOR]

In this study, the two-point boundary value problem is considered for the variable fractional order differential equation with causal operator. Under the definition of the Caputo-type variable fractional order operators, the necessary inequality and the existence results of the solution are obtained for the variable order fractional linear differential equations according to Arzela–Ascoli theorem. Then, based on the proposed existence results and the monotone iterative technique, the existence of the extremal solution is studied, and the relative results are obtained based on the lower and upper solution. Finally, an example is provided to illustrate the validity of the theoretical results. [ABSTRACT FROM AUTHOR]

This paper focuses on two-point boundary value problem for first order causal difference equations. We will start with two new comparison theorems. Then, by utilizing these theorems and fixed point theorems, we obtain the existence of solutions for the corresponding linear problem. By applying monotone iterative technique, sufficient conditions for the existence of extremal solutions are also established. The results of this paper extend some existing results in the literature. Finally, two examples to show the usefulness of our results are exhibited. [ABSTRACT FROM AUTHOR]

A general theorem concerning the three critical points for some classes of coercive functionals depending on a real parameter is established, which may derive existence's results of three solutions with various sufficient conditions for some classes of two-point semi-linear boundary value problems. Moreover, by applying known three existence theorems, we derive multiple existence results for a class of quasi-linear differential equation. [ABSTRACT FROM AUTHOR]

Abstract This paper is concerned with the two-point boundary value problems of a nonlinear fractional q-difference equation with dependence on the first order q-derivative. We discuss some new properties of the Green function by using q-difference calculus. Furthermore, by means of Schauder’s fixed point theorem and an extension of Krasnoselskii’s fixed point theorem in a cone, the existence of one positive solution and of at least one positive solution for the boundary value problem is established.

This paper provides a method to bound and calculate any eigenvalues and eigenfunctions of n-th order boundary value problems with sign-regular kernels subject to two-point boundary conditions. The method is based on the selection of a particular type of cone for each eigenpair to be determined, the recursive application of the operator associated to the equivalent integral problem to functions belonging to such a cone, and the calculation of the Collatz–Wielandt numbers of the resulting functions.