On Solution of Boundary Value Problems via Weak Contractions

The aim is to present a new relational variant of fixed point result that generalizes various fixed point results of the existing theme for contractive type mappings. As an application, we solve a periodic boundary value problem and validate all assertions with the help of nontrivial examples. We also highlight the close connections of the fixed point results equipped with a binary relation to that of graph related metrical fixed point results. Radically, these investigations unify the theory of metrical fixed points for contractive type mappings.

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Nonmonotone impulse effects in second-order periodic boundary value problems

Abstract and Applied Analysis ◽

10.1155/s1085337504306299 ◽

2004 ◽

Vol 2004 (7) ◽

pp. 577-590 ◽

Cited By ~ 3

Author(s):

Irena Rachůnková ◽

Milan Tvrdý

Keyword(s):

Boundary Value Problem ◽

Boundary Value Problems ◽

Periodic Boundary ◽

Boundary Value ◽

Existence Results ◽

Second Order ◽

Periodic Boundary Value Problem ◽

Lower Upper ◽

Periodic Boundary Value Problems

We deal with the nonlinear impulsive periodic boundary value problemu″=f(t,u,u′),u(ti+)=Ji(u(ti)),u′(ti+)=Mi(u′(ti)),i=1,2,…,m,u(0)=u(T),u′(0)=u′(T). We establish the existence results which rely on the presence of a well-ordered pair(σ1,σ2)of lower/upper functions(σ1≤σ2 on [0,T])associated with the problem. In contrast to previous papers investigating such problems, the monotonicity of the impulse functionsJi,Miis not required here.

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On the solvability of a semi-periodic boundary value problem for the nonlinear Goursat equation

BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS ◽

10.31489/2021m4/110-117 ◽

2021 ◽

Vol 104 (4) ◽

pp. 110-117

Author(s):

N.T. Orumbayeva ◽

◽

T.D. Tokmagambetova ◽

Zh.N. Nurgalieva ◽

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...

Keyword(s):

Differential Equations ◽

Boundary Value Problem ◽

Ordinary Differential Equations ◽

Boundary Value Problems ◽

Periodic Boundary ◽

Hyperbolic Equations ◽

Boundary Value ◽

Periodic Problem ◽

Periodic Boundary Value Problem ◽

Functional Relations

In this paper, by means of a change of variables, a nonlinear semi-periodic boundary value problem for the Goursat equation is reduced to a linear gravity problem for hyperbolic equations. Reintroducing a new function, the obtained problem is reduced to a family of boundary value problems for ordinary differential equations and functional relations. When solving a family of boundary value problems for ordinary differential equations, the parameterization method is used. The application of this approach made it possible to establish the coefficients of the unique solvability of the semi-periodic problem for the Goursat equation and to propose constructive algorithms for finding an approximate solution.

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Existence of Multiple Positive Solutions for Even Order Multi-Point Boundary Value Problems

Georgian Mathematical Journal ◽

10.1515/gmj.2007.775 ◽

2007 ◽

Vol 14 (4) ◽

pp. 775-792

Author(s):

Youyu Wang ◽

Weigao Ge

Keyword(s):

Fixed Point ◽

Fixed Point Theorem ◽

Boundary Value Problem ◽

Boundary Value Problems ◽

Positive Solutions ◽

Boundary Value ◽

Sufficient Conditions ◽

Multiple Positive Solutions ◽

Point Boundary ◽

Even Order

Abstract In this paper, we consider the existence of multiple positive solutions for the 2𝑛th order 𝑚-point boundary value problem: where (0,1), 0 < ξ 1 < ξ 2 < ⋯ < ξ 𝑚–2 < 1. Using the Leggett–Williams fixed point theorem, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem. The associated Green's function for the above problem is also given.

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Fixed point theorems via Generalized WF-Contractions with applications

SeMA Journal ◽

10.1007/s40324-021-00261-2 ◽

2021 ◽

Author(s):

Rosana Rodríguez-López ◽

Rakesh Tiwari

Keyword(s):

Fixed Point ◽

Differential Equations ◽

Boundary Value Problem ◽

Fixed Point Theorems ◽

Boundary Value ◽

Second Order ◽

Point Boundary ◽

New Class ◽

Second Order Differential Equations ◽

Fixed Point Result

AbstractThe aim of this paper is to introduce a new class of mixed contractions which allow to revise and generalize some results obtained in [6] by R. Gubran, W. M. Alfaqih and M. Imdad. We also provide an example corresponding to this class of mappings and show how the new fixed point result relates to the above-mentioned result in [6]. Further, we present an application to the solvability of a two-point boundary value problem for second order differential equations.

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