Fixed Point Methods for Rational-Type Contractions: Applications to Nonlinear Integral and Boundary Value Problems

This paper is concerned with the nonlinear boundary value problem (1) $\beta u''-u'+f(u)=0$, (2) $u'(0)-au(0)=0,u'(1)=0$, where $f(u)=b(c-u)\exp(-k/(1+u))$ and $\beta,a,b,c,k$ are constants. First a formal singular perturbation procedure is applied to reveal the possibility of multiple solutions of (1) and (2). Then an iteration procedure is introduced which yields sequences converging to the maximal solution from above and the minimal solution from below. A criterion for a unique solution of (1), (2) is given. It is mentioned that for certain values of the parameters multiple solutions have been found numerically. Finally, the stability of solutions of (1), (2) is discussed for certain values of the parameters. A solution $u(x)$ of (1), (2) is said to be stable if the first eigenvalue $\sigma$ of the variational equations $(1)' \beta v''-v'+[\sigma\beta+f'(u)]v=0$ and $(2)' v'(0)-av(0)=0, v'(1)=0$, is positive.

Previous article Next article On the Transformation of a Class of Boundary Value Problems into Initial Value Problems for Ordinary Differential EquationsMurray S. KlamkinMurray S. Klamkinhttps://doi.org/10.1137/1004006PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] S. Goldstein, Modern Developments in Fluid Dynamics, Vol. I, Oxford, London, 1957, 135–136 Google Scholar[2] H. P. Greenspan and , G. F. Carrier, The magnetohydrodynamic flow past a flat plate, J. Fluid Mech., 6 (1959), 77–96 MR0108191 0088.19203 CrossrefISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails A note on the transformation of boundary value problems to initial value problems: The iterative transformation methodApplied Mathematics and Computation, Vol. 415 | 1 Feb 2022 Cross Ref Sawtooth patterns in flexural force curves of structural biological materials are not signatures of toughness enhancement: Part IIJournal of the Mechanical Behavior of Biomedical Materials, Vol. 124 | 1 Dec 2021 Cross Ref Group invariant solution for a pre-existing fracture driven by a power-law fluid in permeable rockInternational Journal of Modern Physics B, Vol. 30, No. 28n29 | 20 Nov 2016 Cross Ref Free Boundary Formulation for BVPs on a Semi-infinite Interval and Non-iterative Transformation MethodsActa Applicandae Mathematicae, Vol. 140, No. 1 | 23 October 2014 Cross Ref Propagation of a pre-existing turbulent fluid fractureInternational Journal of Non-Linear Mechanics, Vol. 54 | 1 Sep 2013 Cross Ref An investigation of an Emden-Fowler equation from thin film flowActa Mechanica Sinica, Vol. 28, No. 2 | 28 February 2012 Cross Ref A group invariant solution for a pre-existing fluid-driven fracture in permeable rockNonlinear Analysis: Real World Applications, Vol. 12, No. 1 | 1 Feb 2011 Cross Ref On the equivalence of non-iterative transformation methods based on scaling and spiral groupsMathematical Methods in the Applied Sciences, Vol. 33, No. 5 | 19 June 2009 Cross Ref Analysis of the constant B-number assumption while modeling flame spreadCombustion and Flame, Vol. 152, No. 3 | 1 Feb 2008 Cross Ref Upward flame spread on a vertically oriented fuel surface: The effect of finite widthProceedings of the Combustion Institute, Vol. 31, No. 2 | 1 Jan 2007 Cross Ref Автомодельные решения и степенная геометрияУспехи математических наук, Vol. 55, No. 1 | 1 Jan 2000 Cross Ref BibliographyPower Geometry in Algebraic and Differential Equations | 1 Jan 2000 Cross Ref Self-similar solutionsPower Geometry in Algebraic and Differential Equations | 1 Jan 2000 Cross Ref A Similarity Approach to the Numerical Solution of Free Boundary ProblemsSIAM Review, Vol. 40, No. 3 | 2 August 2006AbstractPDF (387 KB)A Novel Approach to the Numerical Solution of Boundary Value Problems on Infinite IntervalsSIAM Journal on Numerical Analysis, Vol. 33, No. 4 | 12 July 2006AbstractPDF (1468 KB)Nonlinear boundary value problems on semi-infinite intervalsComputers & Mathematics with Applications, Vol. 28, No. 10-12 | 1 Nov 1994 Cross Ref Numerical transformation methods: a constructive approachJournal of Computational and Applied Mathematics, Vol. 50, No. 1-3 | 1 May 1994 Cross Ref The falkneer-skan equation: Numerical solutions within group invariance theoryCalcolo, Vol. 31, No. 1-2 | 1 Mar 1994 Cross Ref Non-iterative Transformation Methods EquivalenceModern Group Analysis: Advanced Analytical and Computational Methods in Mathematical Physics | 1 Jan 1993 Cross Ref Optimal numerical algorithmsApplied Numerical Mathematics, Vol. 10, No. 3-4 | 1 Sep 1992 Cross Ref Special TopicsNumerical Methods for Partial Differential Equations | 1 Jan 1992 Cross Ref A noniterative transformation method applied to two-point boundary-value problemsApplied Mathematics and Computation, Vol. 39, No. 1 | 1 Sep 1990 Cross Ref A non‐iterative solution of a system of ordinary differential equations arising from boundary layer theoryInternational Journal of Mathematical Education in Science and Technology, Vol. 21, No. 2 | 1 Mar 1990 Cross Ref ReferencesNonlinear Boundary Value Problems in Science and Engineering | 1 Jan 1989 Cross Ref A SOLUTION METHOD FOR A CLASS OF NONLINEAR BOUNDARY VALUE PROBLEMSChemical Engineering Communications, Vol. 43, No. 1-3 | 27 April 2007 Cross Ref Transformation of a Boundary Value Problem to an Initial Value ProblemGroup Invariance in Engineering Boundary Value Problems | 1 Jan 1985 Cross Ref Multiplicity and Stability in Distributed-Parameter Systems (DPS)Computational Methods in Bifurcation Theory and Dissipative Structures | 1 Jan 1983 Cross Ref Supplementary ReferencesGroup Analysis of Differential Equations | 1 Jan 1982 Cross Ref Optimization of nonlinear kinetic equation computationNumerical Integration of Differential Equations and Large Linear Systems | 25 August 2006 Cross Ref Gas discharges in planar low‐pressure thermionic diodes. 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Positive L^TP and Continuous Solutions for Fredholm Integral Inclusions. A Note on the Structure of the Solution Set for the Cauchy Differential Inclusion in Banach Spaces. Fixed Point Theory for Acylic Maps between Topological Vector Spaces having Sufficiently many Linear Functionals, and Generalized Contractive Maps with Closed Values between Complete Metric Spaces. Using the Integral Manifolds to Solvability of Boundary Value Problems. On the Semicontinuity of Nonlinear Spectra. Existence Results for Two-Point Boundary Value Problems. Generalized Strongly Nonlinear Implicit Quasi-Variational Inequalities for Fuzzy Mappings. Vector Variational Inequalities, Multi-Objective Optimizations, Pareto Optimality and Applications. Variational Principle and Fixed Points. On the Baire Category Method in Existence Problems for Ordinary and Partial Differential Inclusions. Maximal Element Principles on Generalized Convex Spaces and their Applications. Fixed Point Results for Multi-valued Contractions on Gauge Spaces. The Study of Variational Inequalities and Applications to Generalized Complementarity Problems, Fixed Point Theorems of Set-Valued Mappings and Minimization Problems. Remarks on the Existence of Maximal Elements with Respect to a Binary Relation in Non-Compact Topological Spaces. Periodic Solutions of a Singularly Perturbed System of Differential Inclusions in Banach Space. Constrained Differential Inclusions. Nonlinear Boundary Value Problems with Multi-valued Terms. Optimal Control of a Class of Nonlinear Parabolic Problems with Multi-valued Terms. Continuation Theory for A-Proper Mappings and their Uniform Limits and Nonlinear Perturbations of Fredholm Mappings. Existence Theorems for Strongly Accretive Operators in Banch Spaces. A Kneser Type Property for the Solution Set of a Semilinear Differential Inclusion with Lower Semicontinuous Nonlinearity. A Nonlinear Multi-valued Problem with Nonlinear Boundary Conditions. Extensions of Monotone Sets. Convergence of Iterates of Nonexpansive Set-Valued Mappings. A Remark on the Intersection of a Lower Semicontinuous Multi-function and Fixed Set. Random Approximations and Random Fixed Point Theorems for Set-Valued Random Maps.Existence Theorems for Two-Variable Functions and Fixed Point Theorems for Set-Valued Mappings. An Extension Theorem and Duals of Gale-Mas-Colell's and Shafer-Sonnenschein's Theorems. Iterative Algorithms for Nonlinear Variational Inequalities Involving Set-Valued H-Cocoercive Mappings.

Some general foundational issues of quantum mechanics are considered and are related to aspects of quantum computation. The importance of quantum entanglement and quantum information is discussed a...

This paper presents the application of a smeared fixed crack approach for nonlinear finite element analysis of shear-critical reinforced concrete beams. The experimental data was adopted from tests undertaken on twelve reinforced concrete beams by Bresler and Scordelis in 1963, and from duplicate tests undertaken by Vecchio and Shim in 2004. To this end, all beams were modeled in 3D using the software package ATENA-GiD. In the modeling, the nonlinear behaviors of the concrete were represented by fracture-plastic constitutive models, which were formulated within the smeared crack and crack/crush band approaches. The applicability of nonlinear analysis was demonstrated through accurate simulations of the full load-deflection responses, underlying mechanisms, crack patterns, and failure modes of all 24 beams. Detailed documentation of the results is presented to demonstrate the potential and practical value of nonlinear finite element analysis in providing an informed assessment of the safety and performance of reinforced concrete structures.

In order to accurately perceive user intention and improve the reliability of user intention information mining, the existence of solution of integral boundary value problem of fuzzy partial fractional differential equation is used to mine the information of user intention. Firstly, the periodic boundary value problem of fractional order fuzzy linear differential equation, periodic boundary value problem of fractional order fuzzy nonlinear differential equation and periodic boundary value problem of fractional order fuzzy coupled differential system are described in this research. The existence of the solution of the integral boundary value problem of the fuzzy partial fractional order differential equation is proved. Then a user-oriented information mining framework and an evaluation model based on the existence of solutions for the integral boundary value problem of fuzzy partial fractional differential equations are constructed. Finally, this research makes a case study and a comprehensive evaluation of user-oriented information mining based on the existence of solutions of integral boundary value problems for fuzzy partial fractional differential equations. The results show that the method based on the existence of the solution of the integral boundary value problem of fuzzy partial fractional order differential equation is feasible and scientific for the information mining of user intention. It is also concluded that this method is suitable for the modelling and solving of intelligence mining with complicated and unclear user intention. The research provides a good guidance for information mining oriented to user intention.

On antiperiodic boundary value problem for a semilinear differential inclusion of a fractional order q. The investigation of control systems with nonlinear units forms a complicated and very important part of contemporary mathematical control theory and harmonic analysis, which has numerous applications and attracts the attention of a number of researchers around the world. In turn, the development of the theory of differential inclusions is associated with the fact that they provide a convenient and natural tool for describing control systems of various classes, systems with discontinuous characteristics, which are studied in various branches of the optimal control theory, mathematical physics, radio physics, acoustics etc. One of the best approaches to the study of this kind of problems is provided by the methods of multivalued and nonlinear analysis, which are distinguished as very powerful, effective and useful. However, the solving of these problems within the frameworks of existing theories is often a very difficult problem, since many of them find sufficiently adequate description in terms of differential equations and inclusions with fractional derivatives. The theory of differential equations of fractional order originates from the ideas of Leibniz and Euler, but only by the end of the XX century, interest in this topic increased significantly. In the 70s - 80s, this direction was greatly developed by the works of A.A. Kilbas, S.G. Samko, O.I. Marichev, I. Podlubny, K.S. Miller, B. Ross, R. Hilfer, F. Mainardi, H. M. Srivastava. Notice that the research in this direction will open up prospects and new opportunities for the studying of non-standard systems that specialists encounter while describing the development of physical and chemical processes in porous, rarefied and fractal media. It is known that a periodic boundary value problem is one of the classical boundary value problems of differential equations and inclusions. At the same time, in recent years, along with periodic boundary value problems, antiperiodic boundary value problems are of great interest due to their applications in physics and interpolation problems.
 In this paper, we study an antiperiodic boundary value problem for semilinear differential inclusions with Caputo fractional derivative of order q in Banach spaces. We assume that the nonlinear part is a multivalued map obeying the conditions of the Caratheodory type, boundedness on bounded sets, and the regularity condition expressed in terms of measures of noncompactness. In the first section, we present a necessary information from fractional analysis, Mittag -- Leffler function, theory of measures of noncompactness, and multivalued condensing maps. In the second section, we construct the Green's function for the given problem, then, we introduce into consideration a resolving multivalued integral operator in the space of continuous functions. The solutions to the boundary value problem are fixed points of the resolving multioperator. Therefore, we use a generalization of Sadovskii type theorem to prove their existence. Then, we first prove that the resolving multioperator is upper semicontinuous and condensing with respect to the two-component measure of noncompactness in the space of continuous functions. In a proof of a main theorem of the paper, we show that a resolving multioperator transforms a closed ball into itself. Thus, we obtain that the resolving multioperator obeys all the conditions of the fixed point theorem and we prove the existence of solutions to the antiperiodic boundary value problem.

In the last two decades, the theory of variational analysis including variational inequalities (VI) emerged as a rapidly growing area of research because of its applications in nonlinear analysis, optimization, economics, game theory, and so forth; see, for example, [1] and the references therein. In the recent past, many authors devoted their attention to study the VI defined on the set of fixed points of a mapping, called hierarchical variational inequalities. Very recently, several iterative methods have been investigated to solve VI, hierarchical variational inequalities, and triple hierarchical variational inequalities. Since the origin of the VI, it has been used as a tool to study optimization problems. Hierarchical variational inequalities are used to study the bilevel mathematical programming problems. A triple level mathematical programming problem can be studied by using triple hierarchical variational inequalities. The Ekeland’s variational principle provides the existence of an approximate minimizer of a bounded below and lower semicontinuous function. It is one of the most important results from nonlinear analysis and it has applications in different areas of mathematics and mathematical sciences, namely, fixed point theory, optimization, optimal control theory, game theory, nonlinear equations, dynamical systems, and so forth, for example, [2–9] and the references therein. During the last decade, it has been used to study the existence of solutions of equilibrium problems in the setting of metric spaces, for example, [2, 3] and the references therein. Banach’s contraction principle is remarkable in its simplicity, yet it is perhaps the most widely applied fixed point theory in all of analysis. This is because the contractive condition on the mapping is simple and easy to verify, and because it requires only completeness of the metric space. Although, the basic idea was known to others earlier, the principle first appeared in explicit form inBanach’s 1922 thesis where it was used to establish the existence of a solution to an integral equation. Caristi’s fixed point theorem [10, 11] has found many applications in nonlinear analysis. It is shown, for example, that this theorem yields essentially all the known inwardness results of geometric fixed point theory in Banach spaces. Recall that inwardness conditions are the ones which assert that, in some sense, points from the domain are mapped toward the domain. This theorem is amazing equivalent to Ekeland’s variational principle.

A shooting reproducing kernel Hilbert space method for multiple solutions of nonlinear boundary value problems

and Applied Analysis 3 1-homogeneous operator in a Banach space and then demonstrate its application in establishing the existence of positive solutions for p-Laplacian boundary value problems under certain conditions. (xi) In the paper titled “Existence of solutions for nonhomogeneous A-harmonic equations with variable growth,” the authors establish a theorem for the existence of weak solutions for nonhomogeneous A-harmonic equations in subspace and then give three examples to demonstrate its application. (xii) In the paper titled “Multiple solutions for degenerate elliptic systems near resonance at higher eigenvalues,” the authors study the degenerate semilinear elliptic system in an open bounded domain with smooth boundary, and some multiplicity results of solutions are obtained for the system near resonance at certain eigenvalues by the classical saddle point theorem and a local saddle point theorem in critical point theory. (xiii) In the paper titled “A regularity criterion for the Navier-Stokes equations in the multiplier spaces,” the authors establish a regularity criterion in terms of the pressure gradient for weak solutions to the NavierStokes equations in a special class. The third set of papers, including four papers, deal with several boundary value problems for highly nonlinear ordinary differential equations. (i) In the paper titled “Positive solutions for second-order singular semipositone differential equations involving Stieltjes integral conditions,” the authors investigate the existence of positive solutions for second-order singular differential equations with a negatively perturbed term, by means of the fixed-point theory in cones. (ii) In the paper titled “Positive solutions for Sturm-Liouville boundary value problems in a Banach Space,” the sufficient conditions for the existence of single and multiple positive solutions for a second-order SturmLiouville boundary value problem are established in a Banach space, by using the fixed-point theorem of strict set contraction operators in the frame of the ODE technique. (iii) In the paper titled “Positive solutions of a nonlinear fourth-order dynamic eigenvalue problem on time scales,” the authors study a nonlinear fourth-order dynamic eigenvalue problem on time scales and obtain the existence and nonexistence of positive solutions when 0 λ, respectively, for some λ, by using the Schauder fixed-point theorem and the upper and lower solution method. (iv) In the paper titled “Bifurcation analysis for a predatorprey model with time delay and delay-dependent parameters,” a class of stage-structured predator-prey model with time delay and delay-dependent parameters is considered. By using the normal form theory and center manifold theory, some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifur-cating from Hopf bifurcations are obtained. The fourth set of papers focus on finding the approximate and numerical solutions of various complex nonlinear boundary value problems. (i) In the paper titled “On spectral homotopy analysis method for solving linear Volterra and Fredholm integrodifferential equations,” a spectral homotopy analysis method (SHAM) is proposed to solve linear Volterra integrodifferential equations, and some examples are given to test the efficiency and the accuracy of the proposed method. (ii) In the paper titled “The solution of a class of singularly perturbed two-point boundary value problems by the iterative reproducing kernel method,” the authors establish an iterative reproducing kernel method (IRKM) for solving singular perturbation problems with boundary layers and give two numerical examples to demonstrate the effectiveness of the method. (iii) In the paper titled “A Galerkin solution for Burgers’ equation using cubic B-spline finite elements,” a Galerkin method using cubic B-splines is set up to find the numerical solutions of Burgers’ equation, and the method is shown to be capable of solving Burgers’ equation accurately for values of viscosity ranging from very small to very large. (iv) In the paper titled “Forward-backward splitting methods for accretive operators in Banach spaces,” the authors introduce two iterative forward-backward splitting methods with relaxations to find zeros of the sum of two accretive operators in Banach spaces and prove the weak and strong convergence of these methods under mild conditions, and also discuss applications of these methods to variational inequalities, the split feasibility problem, and a constrained convex minimization problem. Yong Hong Wu Lishan Liu Benchawan Wiwatanapataphee Shaoyong Lai Submit your manuscripts at http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematics Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Problems in Engineering Hindawi Publishing Corporation http://www.hindawi.com Differential Equations International Journal of Volume 2014 Applied Mathematics Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Probability and Statistics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Mathematical Physics Advances in Complex Analysis Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Optimization Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Operations Research Advances in

This paper investigates a mixed-type partial differential equation involving the Caputo fractional derivative of order ρ ∈ (0,1) for t > 0, and a classical parabolic equation for t < 0. The problem is studied in an arbitrary N-dimensional domain Ω with smooth boundary, subject to Dezin-type non-local boundary and gluing conditions. For the forward problem, existence and uniqueness of the classical solution are established under suitable assumptions on the data, employing the Fourier method. The influence of the parameter λ in the non-local boundary condition on solvability is analyzed. Additionally, an inverse problem is considered, where the source term is separable as F(x,t) = f(x)g(t), with known g(t) and unknown spatial function f(x). Under certain conditions on g(t), the uniqueness and existence of the solution are proven. This work extends previous results on mixed-type equations, highlighting the role of fractional derivatives and nonlocal conditions in both forward and inverse settings. The findings contribute to the theory of mixed-type and fractional differential equations, with potential applications in subdiffusion and related processes.

In the article we obtained sufficient conditions of the existence of the nonlinear Noetherian boundary value problem solution for the system of differential-algebraic equations which are widely used in mechanics, economics, electrical engineering, and control theory. We studied the case of the nondegenerate system of differential algebraic equations, namely: the differential algebraic system that is solvable relatively to the derivative. In this case, the nonlinear system of differential algebraic equations is reduced to the system of ordinary differential equations with an arbitrary continuous function. The studied nonlinear differential-algebraic boundary-value problem in the article generalizes the numerous statements of the non-linear non-Gath boundary value problems considered in the monographs of А.М. Samoilenko, E.A. Grebenikov, Yu.A. Ryabov, A.A. Boichuk and S.M. Chuiko, and the obtained results can be carried over matrix boundary value problems for differential-algebraic systems. The obtained results in the article of the study of differential-algebraic boundary value problems, in contrast to the works of S. Kempbell, V.F. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and A.A. Boychuk, do not involve the use of the central canonical form, as well as perfect pairs and triples of matrices. To construct solutions of the considered boundary value problem, we proposed the iterative scheme using the method of simple iterations. The proposed solvability conditions and the scheme for finding solutions of the nonlinear Noetherian differential-algebraic boundary value problem, were illustrated with an example. To assess the accuracy of the found approximations to the solution of the nonlinear differential-algebraic boundary value problem, we found the residuals of the obtained approximations in the original equation. We also note that obtained approximations to the solution of the nonlinear differential-algebraic boundary value problem exactly satisfy the boundary condition.

We prove the local existence of strong solutions of the periodic Hunter--Saxton equation, and we show that all strong solutions except space-independent solutions blow up in finite time.

Based on the theory of lower and upper solutions, we study the monotone iterative method for the nonlinear integral boundary value problems of fractionalp-Laplacian equations with delay, which involves both Riemann–Liouville derivative and Caputo derivative. Some new results on the existence of positive solutions are established and the iterative methods for finding approximate solutions of the boundary value problem are obtained. Finally, two examples are given out to illustrate the numerical solution and the related graphic simulations are also provided.

One parameter operator imbedding to modify Newton method for solution of nonlinear equations