Home Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis
Article Open Access

Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis

  • Saleh S. Almuthaybiri and Christopher C. Tisdell EMAIL logo
Published/Copyright: September 28, 2020
Download Article (PDF)
Open Mathematics
From the journal Open Mathematics Volume 18 Issue 1

Abstract

We examine the existence and uniqueness of solutions to two-point boundary value problems involving fourth-order, ordinary differential equations. Such problems have interesting applications to modelling the deflections of beams. We sharpen traditional results by showing that a larger class of problems admit a unique solution. We achieve this by drawing on fixed-point theory in an interesting and alternative way via an application of Rus’s contraction mapping theorem. The idea is to utilize two metrics on a metric space, where one pair is complete. Our theoretical results are applied to the area of elastic beam deflections when the beam is subjected to a loading force and the ends of the beam are either both clamped or one end is clamped and the other end is free. The existence and uniqueness of solutions to the models are guaranteed for certain classes of linear and nonlinear loading forces.

Keywords: fourth-order boundary value problem; existence and uniqueness of solutions; beam analysis; Rus’s contraction mapping theorem; two metrics
MSC 2010: 34B15

1 Introduction

This paper studies the existence and uniqueness of solutions to the following nonlinear, fourth-order differential equation:

(1.1) y ( i v ) = f ( x , y , y ′ , y ″ , y ‴ ) , x ∈ [ 0 , 1 ] ,

where f : Ω ⊆ [ 0 , 1 ] × ℝ 4 → ℝ is assumed to be continuous, and (1.1) is subject to either of the two-point boundary conditions:

(1.2) y ( 0 ) = a , y ′ ( 0 ) = c , y ( 1 ) = b , y ′ ( 1 ) = d ;

(1.3) y ( 0 ) = a , y ′ ( 0 ) = c , y ″ ( 1 ) = e , y ‴ ( 1 ) = h ;

and a , b , c , d , e , h are given constants in ℝ .

A natural motivation for the investigation of fourth-order boundary value problems (BVPs) arises in the analysis of elastic beam deflections. Consider a beam occupying the interval [0, 1] with x denoting the position along the beam. The beam is subjected to certain forces, and if y = y ( x ) represents the resultant deflection of the beam at position x, then the equation of motion leads to differential Eq. (1.1). In simplified situations, the problem is subjected to either of the boundary conditions:

(1.4) y ( 0 ) = 0 , y ′ ( 0 ) = 0 ; y ( 1 ) = 0 , y ′ ( 1 ) = 0 ;

(1.5) y ( 0 ) = 0 , y ′ ( 0 ) = 0 ; y ″ ( 1 ) = 0 , y ‴ ( 1 ) = 0 .

The boundary condition (1.4) may be interpreted in a physical sense as the beam having clamped ends at x = 0 and at x = 1 , while (1.5) may be interpreted in a material sense as the beam having a clamped end at x = 0 and a free end at x = 1 .

Fourth-order BVPs and their application to elastic beam deflections have been studied by many researchers. Indeed, entire monographs have appeared regarding the field such as [1,2]. While an exhaustive literature survey is beyond the scope of the present paper due to space constraints, let us situate our current work within the field of research by discussing its differences and connections with recent and noteworthy publications in the area. We analyse the following four dimensions: the type of problem under consideration, the assumptions imposed, the methods employed, and the nature of the results obtained.

Observe that the differential equation under consideration (1.1) in our present paper features a scalar-valued, fully nonlinear right-hand side that depends on each of the lower-order derivatives. This is in contrast to such works as [3,4,5,6], where either f does not depend in a nonlinear way on each of the derivatives y, y ′ , y ″ and y ‴ or a system of equations were considered. The two sets of boundary conditions (1.2) and (1.3) that we consider herein differ in form from those considered by [4,5,6,7,8,9,10]. Thus, the problem under consideration in our present paper is distinct from the aforementioned works.

On the other hand, the fully nonlinear problem (1.1), (1.5) was analysed in [11,12,13]. The assumptions on f in [11,12] are of a local nature, that is, the domain of f is restricted to closed and bounded sets. While these types of assumptions are quite wide-ranging, the very nature of localized assumptions means that only limited, localized information about solutions can be necessarily obtained. For instance, in the context of localized assumptions, nothing can be concluded about the existence and uniqueness of solutions that may lie outside of the closed and bounded set that is under consideration. This is especially important as multiple solutions to fourth-order BVPs have been shown to exist [14].

Herein, we provide an analysis in both global and local settings. In doing so, we obtain more complete and balanced knowledge in our conclusions regarding the existence and uniqueness of solutions.

In [13], the approach involved obtaining the existence of solutions via the application of fixed point index theory in cones. In contrast, our methods herein involve a suitable application of the Rus fixed point theorem [15] via two metrics. Thus, our assumptions and methods herein are different from the aforementioned works.

Indeed, a range of authors have pursued a spectrum of approaches to the existence and/or uniqueness of solution to fourth-order BVPs. This includes methods such as Schauder fixed-point theorem [4,9] and topological degree [16], monotone iteration [6], reduction to second order systems [11], lower and upper solutions [5], fixed point theorems in cones [3,6,7,13,17], fixed points of general α-concave operators, and fixed point theorems in partially ordered metric spaces [18]. Thus, we can see that our approach of applying the Rus fixed point theorem appears to occupy a unique position within the literature as a strategy to ensure the existence and uniqueness of solutions to fourth-order BVPs.

The results herein form an advancement over traditional approaches such as applications of Banach’s fixed point theorem. This is achieved through the use of two metrics and the Rus fixed point theorem. As we will discover, this enables a greater class of problems to be better understood regarding the existence and uniqueness of solutions. This includes sharpening the Lipschitz constants involved within a global (unbounded) context and within closed and bounded domains.

The reader is referred to [14,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50] for some additional developments in the field of fourth-order BVPs and applications to beam analysis. In some of these works, the fourth-order BVP under consideration is reducible to a larger system of second-order BVPs and thus the underlying analysis for the fourth-order problem is closely linked with that of second-order problems. Herein, we make no such reduction, preferring to work directly on the original form of the problems.

As flagged in [29, p. 108], no reduction of order (to become a second-order problem) is available for BVP (1.1), (1.4) due to the nature of the boundary conditions. This realization may partially explain Yao’s position [48, p. 237] regarding reasons for the slow progress of research into (1.1), (1.2) and (1.1), (1.4). Thus, new methods and perspectives are needed [51,52] to advance the associated existence and uniqueness theory and its application to beam deflection analysis.

Sufficiently motivated by the above discussion and drawing on the position that “knowing an equation has a unique solution is important from both a modelling and theoretical point of view” [53, p. 794], the purpose of the present work then is to address the aforementioned challenges by examining the existence and uniqueness of solutions to BVPs (1.1), (1.2) and (1.1), (1.3). We sharpen traditional results by showing that a larger class of problems admit a unique solution and illustrate the nature of the advancement through exemplification. We achieve this by drawing on the fixed-point theory in an interesting and alternative way via an application of Rus’s contraction mapping theorem [15] which utilizes two metrics on a metric space. Our existence and uniqueness results are then applied to the area of elastic beam deflections when the beam is subjected to a loading force and the ends of the beam are either both clamped or one end is clamped and the other end is free. The existence and uniqueness of solutions to the models are guaranteed under linear and nonlinear loading forces – that is, the models are identified to be well-posed.

By a solution to (1.1), (1.2) we mean a function y : [ 0 , 1 ] → ℝ such that y is four times differentiable, with a continuous fourth-order derivative on [ 0 , 1 ] , which we denote by y ∈ C 4 ( [ 0 , 1 ] ) , and our y satisfies both (1.1) and (1.2). A solution to (1.1), (1.3) is defined similarly.

An important result that we shall draw on is the following contraction mapping theorem of Rus.

Theorem 1.1

[15] Let Y be a nonempty set and let ρ and δ be two metrics on Y such that ( Y , ρ ) forms a complete metric space. If the mapping T : Y → Y is continuous with respect to ρ on Y and

ρ ( T y , T z ) ≤ κ δ ( y , z ) , f o r s o m e κ > 0 a n d a l l y , z ∈ Y ;

δ ( T y , T z ) ≤ α δ ( y , z ) , f o r s o m e 0 < α < 1 a n d a l l y , z ∈ Y ;

then there exists a unique y ∈ Y such that T y = y .

Theorem 1.1 relies on the existence of two metrics (which are not necessarily equivalent). The space Y is assumed to be complete with respect to ρ , but Y is not necessarily complete under δ . The operator T is assumed to be continuous with respect to the metric ρ and contractive with respect to δ , but not necessarily contractive with respect to ρ . As we shall see, these kinds of assumptions can be applied to operators associated with BVPs, sharpening traditional results by showing that a larger class of problems admit a unique solution. Recently, Theorem 1.1 and its variants have been successfully applied to the study of third-order BVPs [55] and nonlinear initial value problems [56]. For recent extensions of Theorem 1.1 see [57].

Our paper is organized as follows.

In Section 2, we present our main existence and uniqueness results for solutions to our BVPs via applications of Rus’s contraction mapping theorem through two metrics.

In Section 3, the new results of Section 2 are applied to the area of elastic beam deflections with either the ends of the beam both being clamped or when one end is clamped and the other end is free. Through discussions of simple applications, we illustrate the nature of the advancements over traditional results and approaches such as Banach’s fixed point theorem in bounded and unbounded settings.

Finally, we include some discussion in Section 4 and identify some possible areas for future research.

2 Existence and uniqueness

In this section, we formulate our main results regarding the existence and uniqueness of solutions via fixed point methods under Rus’s contraction mapping theorem.

Let us construct a range of metrics in an appropriate metric space. Consider the set of real-valued functions that are defined on [ 0 , 1 ] and are thrice continuously differentiable therein. Denote this space by C 3 ( [ 0 , 1 ] ) . For functions y , z ∈ C 3 ( [ 0 , 1 ] ) and appropriate non-negative constants L i and W i to be determined a little later, consider the following metrics on C 3 ( [ 0 , 1 ] ) :

(2.1) d ∞ ( y , z ) ≔ ∑ i = 0 3 max x ∈ [ 0 , 1 ] | y ( i ) ( x ) − z ( i ) ( x ) | ; δ p ( y , z ) ≔ ∑ i = 0 3 L i ∫ 0 1 | y ( i ) ( x ) − z ( i ) ( x ) | p d x 1 / p , p > 1 ; d ( y , z ) ≔ max i ∈ { 0 , 1 , 2 , 3 } W i max x ∈ [ 0 , 1 ] | y ( i ) ( x ) − z ( i ) ( x ) | .

It is well known that each of the pairs ( C 3 ( [ 0 , 1 ] ) , d ∞ ) and ( C 3 ( [ 0 , 1 ] ) , d ) form a complete metric space. The pair ( C 3 ( [ 0 , 1 ] ) , δ p ) also forms a metric space; however, it is not complete. A helpful relationship between our metrics on C 3 ( [ 0 , 1 ] ) that we will draw on involves the inequalities

(2.2) δ p ( y , z ) ≤ max i ∈ { 0 , 1 , 2 , 3 } { L i } d ∞ ( y , z ) ;

(2.3) δ p ( y , z ) ≤ ∑ i = 0 3 L i W i d ( y , z ) .

2.1 On (1.1) and (1.2)

In order to construct an appropriate operator T and the corresponding fixed-point problem for (1.1), (1.2) we note that BVP (1.1), (1.2) is equivalent to the integral equation [48, p. 238], [40]

(2.4) y ( x ) = ∫ 0 1 G ( x , s ) f ( s , y ( s ) , y ′ ( s ) , y ″ ( s ) , y ‴ ( s ) ) d s + ϕ ( x ) , x ∈ [ 0 , 1 ] ,

where G ( x , s ) is the Green function for the following BVP:

y ( i v ) = 0 , x ∈ [ 0 , 1 ] ,

y ( 0 ) = 0 , y ′ ( 0 ) = 0 , y ( 1 ) = 0 , y ′ ( 1 ) = 0 ,

and is given explicitly by

(2.5) 0 ≤ G ( x , s ) = 1 6 s 2 ( 1 − x ) 2 [ ( x − s ) + 2 ( 1 − s ) x ] , for 0 ≤ s ≤ x ≤ 1 , x 2 ( 1 − s ) 2 [ ( s − x ) + 2 ( 1 − x ) s ] , for 0 ≤ x ≤ s ≤ 1 ;

and ϕ is the unique solution to the BVP

y ( i v ) = 0 , x ∈ [ 0 , 1 ] ,

y ( 0 ) = a , y ′ ( 0 ) = c , y ( 1 ) = b , y ′ ( 1 ) = d ,

which is given explicitly by

(2.6) ϕ ( x ) = a ( 2 x 3 − 3 x 2 + 1 ) + b ( − 2 x 3 + 3 x 2 ) + c ( x 3 − 2 x 2 + x ) + d ( x 3 − x 2 ) .

To avoid the repeated use of complicated expressions, we define the following constants to simplify certain notation.

For i = 0 , 1 , 2 , 3 , define positive constants β i via

(2.7) β i ≥ max x ∈ [ 0 , 1 ] ∫ 0 1 ∂ i ∂ x i G ( x , s ) d s .

Such choices are always possible due to the smoothness of G.

Let p > 1 and q > 1 be constants such that 1 / p + 1 / q = 1 . For i = 0 , 1 , 2 , 3 , define:

(2.8) c i ≔ max x ∈ [ 0 , 1 ] ∫ 0 1 ∂ i ∂ x i G ( x , s ) q d s 1 / q ;

and

(2.9) γ i ≔ ∫ 0 1 ∫ 0 1 ∂ i ∂ x i G ( x , s ) q d s p / q d x 1 / p .

We are now in a position to state and prove our first result for the existence and uniqueness of solutions to (1.1), (1.2).

Theorem 2.1

Let f : [ 0 , 1 ] × ℝ 4 → ℝ be continuous and let L i be non-negative constants for i = 0 , 1 , 2 , 3 (not all zero) such that

(2.10) f ( x , u 0 , u 1 , u 2 , u 3 ) − f ( x , v 0 , v 1 , v 2 , v 3 ) ≤ ∑ i = 0 3 L i | u i − v i | , f o r a l l ( x , u 0 , u 1 , u 2 , u 3 ) , ( x , v 0 , v 1 , v 2 , v 3 ) ∈ [ 0 , 1 ] × ℝ 4 .

If there are constants p > 1 and q > 1 such that 1 / p + 1 / q = 1 with γ i defined in (2.9) such that

(2.11) ∑ i = 0 3 L i γ i < 1 ,

then BVP (1.1), (1.2) has a unique solution in C 4 ( [ 0 , 1 ] ) .

Proof

Based on form (2.4), we define the operator T : C 3 ( [ 0 , 1 ] ) → C 3 ( [ 0 , 1 ] ) by

( T y ) ( x ) ≔ ∫ 0 1 G ( x , s ) f ( s , y ( s ) , y ′ ( s ) , y ″ ( s ) , y ‴ ( s ) ) d s + ϕ ( x ) , x ∈ [ 0 , 1 ] .

We wish to show that there exists a unique y ∈ C 3 ( [ 0 , 1 ] ) such that

T y = y ,

which is equivalent to proving that BVP (1.1), (1.2) has a unique solution. (Any solutions lying in C 3 ( [ 0 , 1 ] ) will also lie in C 4 ( [ 0 , 1 ] ) as repeatedly differentiating (2.4) will show.)

To prove that our T has a unique fixed point, we show that the assumptions of Theorem 2.1 ensure that the conditions of Theorem 1.1 hold.

Consider the pair ( Y , ρ ) ≔ ( C 3 ( [ 0 , 1 ] ) , d ∞ ) to form a complete metric space and consider the metric δ = δ p on Y, where p > 1 .

For y , z ∈ C 3 ( [ 0 , 1 ] ) and x ∈ [ 0 , 1 ] , consider

(2.12) ( T y ) ( x ) − ( T z ) ( x ) ≤ ∫ 0 1 G ( x , s ) f ( s , y ( s ) , y ′ ( s ) , y ″ ( s ) , y ‴ ( s ) ) − f ( s , z ( s ) , z ′ ( s ) , z ″ ( s ) , z ‴ ( s ) ) d s ≤ ∫ 0 1 | G ( x , s ) | ∑ i = 0 3 ( L i | y ( i ) ( s ) − z ( i ) ( s ) | ) d s ≤ ∫ 0 1 | G ( x , s ) | q d s 1 / q ∑ i = 0 3 L i ∫ 0 1 | y ( i ) ( x ) − z ( i ) ( x ) | p d x 1 / p ≤ c 0 δ p ( y , z ) ,

where we have invoked our assumption (2.10) and Hőlder’s inequality for p > 1 and q > 1 such that 1 / p + 1 / q = 1 and c 0 is defined in (2.8).

By repeating the aforementioned approach on derivatives we can obtain

max x ∈ [ 0 , 1 ] ( T y ) ( i ) ( x ) − ( T z ) ( i ) ( x ) ≤ c i δ p ( y , z )

for i = 0 , 1 , 2 , 3 . Thus, defining

κ ≔ ∑ i = 0 3 c i

we see that

d ∞ ( T y , T z ) ≤ κ δ p ( y , z ) , for some κ > 0 and all y , z ∈ C 3 ( [ 0 , 1 ] ) .

Now, from (2.2), for all y , z ∈ C 3 ( [ 0 , 1 ] ) , we have

d ∞ ( T y , T z ) ≤ κ δ p ( y , z ) ≤ κ max i ∈ { 0 , 1 , 2 , 3 } { L i } d ∞ ( y , z ) .

Thus, given any ε > 0 we can choose Δ = ε / κ max i ∈ { 0 , 1 , 2 , 3 } { L i } so that d ∞ ( T y , T z ) < ε whenever d ∞ ( y , z ) < Δ . Hence, T is continuous on C 3 ( [ 0 , 1 ] ) with respect to the d ∞ metric.

Finally, we show that T is contractive on C 3 ( [ 0 , 1 ] ) with respect to the δ p metric. From (2.12), for each y , z ∈ C 3 ( [ 0 , 1 ] ) and i = 0 , 1 , 2 , 3 , we have

L i ∫ 0 1 | ( T y ) ( i ) ( x ) − ( T z ) ( i ) ( x ) | p d x 1 / p ≤ L i γ i δ p ( y , z ) ,

where γ i are defined in (2.9). Summing both sides of the previous inequality over i we obtain

δ p ( T y , T z ) ≤ ∑ i = 0 3 L i γ i δ p ( y , z )

for all y , z ∈ C 3 ( [ 0 , 1 ] ) . From our assumption (2.11), we have ensured

δ p ( T y , T z ) ≤ α δ p ( y , z )

for some α < 1 and all y , z ∈ C 3 ( [ 0 , 1 ] ) .

Thus, Theorem 1.1 is applicable and yields the existence of a unique fixed point to T that lies in C 3 ( [ 0 , 1 ] ) . This solution is also in C 4 ( [ 0 , 1 ] ) as can be verified by differentiating the integral Eq. (2.4). Thus, we have equivalently shown that BVP (1.1), (1.2) has a unique solution.□

Remark 2.1

Lipschitz condition (2.10) will be satisfied if, for example, our

f ( x , u 0 , u 1 , u 2 , u 3 )

has partial derivatives ∂ f / ∂ u i that are uniformly bounded and continuous on [ 0 , 1 ] × ℝ 4 for each corresponding i = 0 , 1 , 2 , 3 . In this case, each bound M i can form L i for i = 0 , 1 , 2 , 3 .

For example, for an f such as

f ( x , u 0 , u 1 , u 2 , u 3 ) = [ sin ( u 0 ) + cos ( u 1 ) + sin ( u 2 ) + cos ( u 3 ) ] / 10

we could choose M i = L i = 1 / 10 .

Although Lipschitz condition (2.10) is more difficult to be satisfied on the unbounded domain [ 0 , 1 ] × ℝ 4 when compared with a closed and bounded subset of this region, the conclusion of Theorem 2.1 provides robust, global information about the solutions to the problem under consideration.

In comparison, the very nature of localized assumptions means that only limited, localized information about solutions can be necessarily obtained. For instance, in the context of localized assumptions, nothing can be concluded about the existence and uniqueness of solutions that may lie outside of the closed and bounded set that is under consideration. This is especially important as multiple solutions to fourth-order BVPs have been shown to exist [14].

Motivated by the above discussion let us now provide some balance by examining questions of existence and uniqueness of solutions to (1.1), (1.2) within subsets of [ 0 , 1 ] × ℝ 4 . Essentially, we impose less restrictive conditions on f in exchange for obtaining less information regarding the solutions.

Theorem 2.2

Let f : B → ℝ be continuous and uniformly bounded by M > 0 on the set

B ≔ ( x , u , v , w , z ) ∈ ℝ 5 : x ∈ [ 0 , 1 ] , | u − ϕ ( x ) | ≤ R , | v − ϕ ′ ( x ) | ≤ β 1 β 0 R , | w − ϕ ′ ′ ( x ) | ≤ β 2 β 0 R , | z − ϕ ′ ′ ′ ( x ) | ≤ β 3 β 0 R ,

where R > 0 , ϕ is defined in (2.6) and β i are defined in (2.7). Assume M β 0 ≤ R . For i = 0 , 1 , 2 , 3 , let L i be non-negative constants (not all zero) such that

(2.13) f ( x , u 0 , u 1 , u 2 , u 3 ) − f ( x , v 0 , v 1 , v 2 , u 3 ) ≤ ∑ i = 0 3 L i | u i − v i | , f o r a l l ( x , u 0 , u 1 , u 2 , u 3 ) , ( x , v 0 , v 1 , v 2 , v 3 ) ∈ B .

If there are constants p > 1 and q > 1 such that 1 / p + 1 / q = 1 with γ i defined in (2.9) and

(2.14) ∑ i = 0 3 L i γ i < 1 ,

then BVP (1.1), (1.2) has a unique solution in C 4 ( [ 0 , 1 ] ) such that

( x , y ( x ) , y ′ ( x ) , y ″ ( x ) , y ‴ ( x ) ) ∈ B , for all x ∈ [ 0 , 1 ] .

Proof

Consider the pair ( C 3 ( [ a , b ] ) , d ) where the constants W i in our d in (2.1) are chosen such that W 0 = 1 , W 1 = β 0 / β 1 , W 2 = β 0 / β 2 and W 3 = β 0 / β 3 . Our pair forms a complete metric space. Now, for the constant R > 0 and function ϕ in the definition of B consider the following set ℬ R ⊂ C 3 ( [ 0 , 1 ] ) defined via

ℬ R ≔ { y ∈ C 3 ( [ 0 , 1 ] ) : d ( y , ϕ ) ≤ R } .

Since ℬ R is a closed subspace of C 3 ( [ 0 , 1 ] ) , the pair ( ℬ R , d ) forms a complete metric space.

Consider the operator T : ℬ R → C 3 ( [ 0 , 1 ] ) defined as in the proof of Theorem 2.1.

To establish the existence and uniqueness to T x = x , we show that the conditions of Theorem 1.1 hold with Y = ℬ R .

Let us show T : ℬ R → ℬ R . For y ∈ ℬ R and x ∈ [ 0 , 1 ] , consider

( T y ) ( x ) − ϕ ( x ) ≤ ∫ 0 1 | G ( x , s ) | | f ( s , y ( s ) , y ′ ( s ) , y ″ ( s ) , y ‴ ( s ) ) | d s ≤ M ∫ 0 1 | G ( x , s ) | d s ≤ M β 0 .

Similarly,

( T y ) ′ ( x ) − ϕ ′ ( x ) ≤ ∫ 0 1 ∂ ∂ x G ( x , s ) f ( s , y ( s ) , y ′ ( s ) , y ″ ( s ) , y ‴ ( s ) ) d s ≤ M ∫ 0 1 ∂ ∂ x G ( x , s ) d s ≤ M β 1 .

Thus, β 0 | ( T y ) ′ ( x ) − ϕ ′ ( x ) | / β 1 ≤ M β 0 .

In addition, via similar arguments, we obtain

| ( T y ) ″ ( x ) − ϕ ″ ( x ) | ≤ M β 2 , | ( T y ) ‴ ( x ) − ϕ ‴ ( x ) | ≤ M β 3 ,

so that β 0 | ( T y ) ″ ( x ) − ϕ ″ ( x ) | / β 2 ≤ M β 0 and β 0 | ( T y ) ‴ ( x ) − ϕ ‴ ( x ) | / β 3 ≤ M β 0 .

Thus, for all y ∈ ℬ R we have

d ( T y , ϕ ) ≤ max { M β 0 , M β 0 , M β 0 , M β 0 } = M β 0 ≤ R ,

where the final inequality holds by assumption. Thus, for all y ∈ ℬ R we have T y ∈ ℬ R so that T : ℬ R → ℬ R .

Similar to the proof of Theorem 2.1 we have, for i = 0 , 1 , 2 , 3 , and all x ∈ [ 0 , 1 ] , that

( T y ) ( i ) ( x ) − ( T z ) ( i ) ( x ) ≤ c i d p ( y , z )

via applications of Hőlder’s inequality.

Combining the aforementioned inequalities we obtain

(2.15) d ( T y , T z ) ≤ c δ p ( y , z ) , for some c > 0 and all y , z ∈ ℬ R ,

where

c ≔ max c 0 , β 0 β 1 c 1 , β 0 β 2 c 2 , β 0 β 3 c 3 .

Thus, the first inequality of Theorem 1.1 holds.

Furthermore, T is continuous on ℬ R with respect to the d metric as can be shown from the following arguments. For all y , z ∈ ℬ R , we may apply (2.3) to (2.15) to obtain

d ( T y , T z ) ≤ c δ p ( y , z ) ≤ c L 0 + L 1 β 1 β 0 + L 2 β 2 β 0 + L 3 β 3 β 0 d ( x , y ) .

Thus, given any ε > 0 we can choose

Δ = ε L 0 + L 1 β 1 β 0 + L 2 β 2 β 0 + L 3 β 3 β 0

so that d ( T x , T y ) < ε whenever d ( x , y ) < Δ . Hence, T is continuous on ℬ R with respect to the d metric.

Finally, the contraction condition of T on ℬ R follows from essentially the same arguments as those used in the proof of Theorem 2.1 and so we do not repeat it here.

Thus, we conclude that all the conditions of Theorem 1.1 hold for T on ℬ R and so the unique fixed point of T in ℬ R is guaranteed to exist.□

The following three new results are variations on the theme of Theorem 2.2.

Theorem 2.3

Let f : C → ℝ be continuous and uniformly bounded by M > 0 on the set

C ≔ ( x , u , v , w , z ) ∈ ℝ 5 : x ∈ [ 0 , 1 ] , | u − ϕ ( x ) | ≤ β 0 β 1 R , | v − ϕ ′ ( x ) | ≤ R , | w − ϕ ″ ( x ) | ≤ β 2 β 1 R , | z − ϕ ‴ ( x ) | ≤ β 3 β 1 R ,

where R > 0 , ϕ is defined in (2.6) and β i are defined in (2.7). Assume M β 1 ≤ R . For i = 0 , 1 , 2 , 3 , let L i be non-negative constants (not all zero) such that

(2.16) f ( x , u 0 , u 1 , u 2 , u 3 ) − f ( x , v 0 , v 1 , v 2 , u 3 ) ≤ ∑ i = 0 3 L i | u i − v i | , f o r a l l ( x , u 0 , u 1 , u 2 , u 3 ) , ( x , v 0 , v 1 , v 2 , v 3 ) ∈ C .

If there are constants p > 1 and q > 1 such that 1 / p + 1 / q = 1 with γ i defined in (2.9) such that

(2.17) ∑ i = 0 3 L i γ i < 1 ,

then BVP (1.1), (1.2) has a unique solution in C 4 ( [ 0 , 1 ] ) such that

( x , y ( x ) , y ′ ( x ) , y ′ ′ ( x ) , y ′ ′ ′ ( x ) ) ∈ C , for all x ∈ [ 0 , 1 ] .

Proof

The proof follows similar lines of argument to that of Theorem 2.2 and so we just sketch the basic idea.

Consider the pair ( C 3 ( [ a , b ] ) , d ) where the constants W i in our d in (2.1) are chosen such that W 0 = β 1 / β 0 , W 1 = 1 , W 2 = β 1 / β 2 and W 3 = β 1 / β 3 . Our pair forms a complete metric space. Now, for the constant R > 0 and function ϕ in the definition of C, consider the following set L 0 λ 0 = 8 ( 11 / 1,680 ) 1 / 2 < 1 defined via

C R ≔ { y ∈ C 3 ( [ 0 , 1 ] ) : d ( y , ϕ ) ≤ R } .

Since C R is a closed subspace of C 3 ( [ 0 , 1 ] ) , the pair ( C R , d ) forms a complete metric space.

Following similar steps to that of Theorem 2.2, the condition M β 1 ≤ R ensures T : C R → C R . Furthermore, (2.16) and (2.17) guarantee that T is contractive on C R with respect to δ p .□

Theorem 2.4

Let f : D → ℝ be continuous and uniformly bounded by M > 0 on the set

D ≔ ( x , u , v , w , z ) ∈ ℝ 5 : x ∈ [ 0 , 1 ] , | u − ϕ ( x ) | ≤ β 0 β 2 R , | v − ϕ ′ ( x ) | ≤ β 1 β 2 R , | w − ϕ ′ ′ ( x ) | ≤ R , | z − ϕ ′ ′ ′ ( x ) | ≤ β 3 β 2 R ,

where R > 0 , ϕ is defined in (2.6) and β i are defined in (2.7). Assume M β 2 ≤ R . For i = 0 , 1 , 2 , 3 , let L i be non-negative constants (not all zero) such that

(2.18) f ( x , u 0 , u 1 , u 2 , u 3 ) − f ( x , v 0 , v 1 , v 2 , u 3 ) ≤ ∑ i = 0 3 L i | u i − v i | , f o r a l l ( x , u 0 , u 1 , u 2 , u 3 ) , ( x , v 0 , v 1 , v 2 , v 3 ) ∈ D .

If there are constants p > 1 and q > 1 such that 1 / p + 1 / q = 1 with γ i defined in (2.9) such that

(2.19) ∑ i = 0 3 L i γ i < 1 ,

then BVP (1.1), (1.2) has a unique solution in C 4 ( [ 0 , 1 ] ) such that

( x , y ( x ) , y ′ ( x ) , y ′ ′ ( x ) , y ′ ′ ′ ( x ) ) ∈ D , f o r a l l x ∈ [ 0 , 1 ] .

Proof

The proof follows similar lines of argument to that of Theorem 2.2 and so we just sketch the basic idea.

Consider the pair ( C 3 ( [ a , b ] ) , d ) where the constants W i in our d in (2.1) are chosen such that W 0 = β 2 / β 0 , W 1 = β 2 / β 1 , W 2 = 1 and W 3 = β 2 / β 3 . Our pair forms a complete metric space. Now, for the constant R > 0 and function ϕ in the definition of D, consider the following set D R ⊂ C 3 ( [ 0 , 1 ] ) defined via

D R ≔ { y ∈ C 3 ( [ 0 , 1 ] ) : d ( y , ϕ ) ≤ R } .

Since D R is a closed subspace of C 3 ( [ 0 , 1 ] ) , the pair ( D R , d ) forms a complete metric space.

Following similar steps to that of Theorem 2.2, the condition M β 2 ≤ R ensures T : D R → D R . Furthermore, (2.18) and (2.19) guarantee that T is contractive on D R with respect to δ p .□

Theorem 2.5

Let f : E → ℝ be continuous and uniformly bounded by M > 0 on the set

E ≔ ( x , u , v , w , z ) ∈ ℝ 5 : x ∈ [ 0 , 1 ] , | u − ϕ ( x ) | ≤ β 0 β 3 R , | v − ϕ ′ ( x ) | ≤ β 1 β 3 R , | w − ϕ ″ ( x ) | ≤ β 2 β 3 R , | z − ϕ ‴ ( x ) | ≤ R ,

where R > 0 , ϕ is defined in (2.6) and β i are defined in (2.7). Assume M β 3 ≤ R . For i = 0 , 1 , 2 , 3 , let L i be non-negative constants (not all zero) such that

(2.20) f ( x , u 0 , u 1 , u 2 , u 3 ) − f ( x , v 0 , v 1 , v 2 , u 3 ) ≤ ∑ i = 0 3 L i | u i − v i | , f o r a l l ( x , u 0 , u 1 , u 2 , u 3 ) , ( x , v 0 , v 1 , v 2 , v 3 ) ∈ E .

If there are constants p > 1 and q > 1 such that 1 / p + 1 / q = 1 with γ i defined in (2.9) such that

(2.21) ∑ i = 0 3 L i γ i < 1 ,

then BVP (1.1), (1.2) has a unique solution in C 4 ( [ 0 , 1 ] ) such that

( x , y ( x ) , y ′ ( x ) , y ″ ( x ) , y ‴ ( x ) ) ∈ E , f o r a l l x ∈ [ 0 , 1 ] .

Proof

Omitted due to brevity.□

2.2 On (1.1), (1.3)

BVP (1.1), (1.3) is equivalent to the integral equation [48, p. 238], [40]

(2.22) y ( x ) = ∫ 0 1 G ( x , s ) f ( s , y ( s ) , y ′ ( s ) , y ″ ( s ) , y ‴ ( s ) ) d s + ψ ( x ) , x ∈ [ 0 , 1 ] ,

where G ( x , s ) is the Green function [46, p. 2] for the following BVP:

y ( i v ) = 0 , x ∈ [ 0 , 1 ] ,

y ( 0 ) = 0 , y ′ ( 0 ) = 0 , y ″ ( 1 ) = 0 , y ‴ ( 1 ) = 0 ,

and is given explicitly by

(2.23) 0 ≤ G ( x , s ) ≔ 1 6 s 2 ( 3 x − s ) , for 0 ≤ s ≤ x ≤ 1 ; x 2 ( 3 s − x ) , for 0 ≤ x ≤ s ≤ 1 ,

and ψ is the unique solution to the BVP

y ( i v ) = 0 , x ∈ [ 0 , 1 ] ,

y ( 0 ) = a , y ′ ( 0 ) = c , y ″ ( 1 ) = e , y ‴ ( 1 ) = h ,

which is given explicitly by

(2.24) ψ ( x ) = h x 3 / 6 + ( e − h ) x 2 / 2 + c x + a .

Once again, to avoid the repeated use of complicated expressions, we define the following constants to simplify certain notation.

For i = 0 , 1 , 2 , 3 , define positive constants θ i via

(2.25) θ i ≥ max x ∈ [ 0 , 1 ] ∫ 0 1 ∂ i ∂ x i G ( x , s ) d s .

Such choices are always possible due to the smoothness of G .

Let p > 1 and q > 1 be constants such that 1 / p + 1 / q = 1 . For i = 0 , 1 , 2 , 3 , define:

(2.26) η i ≔ max x ∈ [ 0 , 1 ] ∫ 0 1 ∂ i ∂ x i G ( x , s ) q d s 1 / q ;

and

(2.27) λ i ≔ ∫ 0 1 ∫ 0 1 ∂ i ∂ x i G ( x , s ) q d s p / q d x 1 / p .

Let us now state and prove a new result for the existence and uniqueness of solutions to (1.1), (1.3) on the set [ 0 , 1 ] × ℝ 4 .

Theorem 2.6

Let f : [ 0 , 1 ] × ℝ 4 → ℝ be continuous and let L i be non-negative constants for i = 0 , 1 , 2 , 3 (not all zero) such that

(2.28) | f ( x , u 0 , u 1 , u 2 , u 3 ) − f ( x , v 0 , v 1 , v 2 , v 3 ) | ≤ ∑ i = 0 3 L i | u i − v i | , for all ( x , u 0 , u 1 , u 2 , u 3 ) , ( x , v 0 , v 1 , v 2 , v 3 ) ∈ [ 0 , 1 ] × ℝ 4 .

If there are constants p > 1 and f ( x , y ) such that 1 / p + 1 / q = 1 with λ i defined in (2.27) such that

(2.29) ∑ i = 0 3 L i λ i < 1 ,

then BVP (1.1), (1.3) has a unique solution in C 4 ( [ 0 , 1 ] ) .

Proof

Consider the operator T : C 3 ( [ 0 , 1 ] ) → C 3 ( [ 0 , 1 ] ) constructed from the form (2.22), namely

( T y ) ( x ) ≔ ∫ 0 1 G ( x , s ) f ( s , y ( s ) , y ′ ( s ) , y ″ ( s ) , y ‴ ( s ) ) d s + ψ ( x ) , x ∈ [ 0 , 1 ] .

We want to show that there exists a unique y ∈ C 3 ( [ 0 , 1 ] ) such that

T y = y .

To prove this, we shall show that the conditions of Theorem 1.1 hold. Consider the pair ( Y , ρ ) = ( C 3 ( [ 0 , 1 ] ) , d ∞ ) which forms a complete metric space. In addition, consider the metric δ = δ p on Y, where p > 1 .

For y , z ∈ C 3 ( [ 0 , 1 ] ) and x ∈ [ 0 , 1 ] , consider

( T y ) ( x ) − ( T z ) ( x ) ≤ ∫ 0 1 | G ( x , s ) | | f ( s , y ( s ) , y ′ ( s ) , y ″ ( s ) , y ‴ ( s ) ) − f ( s , z ( s ) , z ′ ( s ) , z ″ ( s ) , z ‴ ( s ) ) | d s ≤ ∫ 0 1 | G ( x , s ) | ∑ i = 0 3 ( L i | y ( i ) ( s ) − z ( i ) ( s ) | ) d s ≤ ∫ 0 1 | G ( x , s ) | q d s 1 / q ∑ i = 0 3 L i ∫ 0 1 | y ( i ) ( s ) − z ( i ) ( s ) | p d x 1 / p ≤ max x ∈ [ 0 , 1 ] ∫ 0 1 | G ( x , s ) | q d s 1 / q δ p ( y , z ) = η 0 δ p ( y , z ) ,

where η 0 is defined in (2.26). Repeating the aforementioned argument to derivatives of the operator T yields

( T y ) ( i ) ( x ) − ( T z ) ( i ) ( x ) ≤ η i δ p ( y , z ) ,

for i = 0 , 1 , 2 , 3 . Thus, defining

κ 1 ≔ ∑ i = 0 3 η i

we see that

d ∞ ( T y , T z ) ≤ κ 1 δ p ( y , z ) , for some κ 1 > 0 and all y , z ∈ Y .

Now, for all y , z ∈ C 3 ( [ 0 , 1 ] ) consider

d ∞ ( T y , T z ) ≤ κ 1 δ p ( y , z ) ≤ κ 1 max i ∈ { 0 , 1 , 2 , 3 } { L i } d ∞ ( y , z ) .

Thus, given any ε > 0 we can choose Δ = ε / κ 1 max i ∈ { 0 , 1 , 2 , 3 } { L i } so that d ∞ ( T y , T z ) < ε whenever d ∞ ( y , z ) < Δ . Hence, T is continuous on C 3 ( [ 0 , 1 ] ) with respect to the d ∞ metric.

Finally, we show that T is contractive on C 3 ( [ 0 , 1 ] ) with respect to the δ p metric. For each y , z ∈ C 3 ( [ 0 , 1 ] ) and i = 0 , 1 , 2 , 3 , we have

L i ∫ 0 1 | ( T y ) ( i ) ( x ) − ( T z ) ( i ) ( x ) | p d x 1 / p ≤ L i ∫ 0 1 ∫ 0 1 ∂ i ∂ x i G ( x , s ) q d s p / q d x 1 / p δ p ( y , z )

and so summing the previous inequality over i we obtain

δ p ( T y , T z ) ≤ ∑ i = 0 3 L i λ i δ p ( y , z ) .

Thus, by (2.29), we have

δ p ( T y , T z ) ≤ α 1 δ p ( y , z )

for some α 1 < 1 and all y , z ∈ C 3 ( [ 0 , 1 ] ) .

Thus, Theorem 1.1 is applicable and the operator T has a unique fixed point in C 3 ( [ 0 , 1 ] ) . This solution is also in C 4 ( [ 0 , 1 ] ) as differentiating the integral Eq. (2.22) shows. Our conclusion is equivalent to showing that BVP (1.1), (1.3) has a unique solution.□

Now, let us now explore the existence and uniqueness of solutions to (1.1), (1.3) on subsets of [ 0 , 1 ] × ℝ 4 .

Theorem 2.7

Let f : F → ℝ be continuous and uniformly bounded by M > 0 on the set

F ≔ ( x , u , v , w , z ) ∈ ℝ 5 : x ∈ [ 0 , 1 ] , | u − ψ ( x ) | ≤ R , | v − ψ ′ ( x ) | ≤ θ 1 θ 0 R , | w − ψ ′ ′ ( x ) | ≤ θ 2 θ 0 R , | z − ψ ′ ′ ′ ( x ) | ≤ θ 3 θ 0 R ,

where R > 0 , ψ is defined in (2.24) and θ i are defined in (2.25). Assume M θ 0 ≤ R . For i = 0 , 1 , 2 , 3 , let L i be non-negative constants (not all zero) such that

(2.30) | f ( x , u 0 , u 1 , u 2 , u 3 ) − f ( x , v 0 , v 1 , v 2 , u 3 ) | ≤ ∑ i = 0 3 L i | u i − v i | , f o r a l l ( x , u 0 , u 1 , u 2 , u 3 ) , ( x , v 0 , v 1 , v 2 , v 3 ) ∈ F .

If there are constants p > 1 and q > 1 such that 1 / p + 1 / q = 1 with λ i defined in (2.27) such that

(2.31) ∑ i = 0 3 L i λ i < 1 ,

then BVP (1.1), (1.3) has a unique (nontrivial) solution in C 4 ( [ 0 , 1 ] ) such that

( x , y ( x ) , y ′ ( x ) , y ″ ( x ) , y ‴ ( x ) ) ∈ F , for all x ∈ [ 0 , 1 ] .

Proof

The proof is very similar to that of the proof of Theorem 2.2 by making the appropriate modifications (e.g., G instead of G , etc.). Thus, we omit the proof for brevity.□

Similarly, we have the following three results which we state without proof due to brevity and concerns of repetition.

Theorem 2.8

Let f : H → ℝ be continuous and uniformly bounded by M > 0 on the set

H ≔ ( x , u , v , w , z ) ∈ ℝ 5 : x ∈ [ 0 , 1 ] , | u − ψ ( x ) | ≤ θ 0 θ 1 R , | v − ψ ′ ( x ) | ≤ R , | w − ψ ″ ( x ) | ≤ θ 2 θ 1 R , | z − ψ ‴ ( x ) | ≤ θ 3 θ 1 R ,

where R > 0 , ψ is defined in (2.24) and θ i are defined in (2.25). Assume M θ 1 ≤ R . For i = 0 , 1 , 2 , 3 , let L i be non-negative constants (not all zero) such that

(2.32) f ( x , u 0 , u 1 , u 2 , u 3 ) − f ( x , v 0 , v 1 , v 2 , u 3 ) ≤ ∑ i = 0 3 L i | u i − v i | , f o r a l l ( x , u 0 , u 1 , u 2 , u 3 ) , ( x , v 0 , v 1 , v 2 , v 3 ) ∈ H .

If there are constants p > 1 and q > 1 such that 1 / p + 1 / q = 1 with γ i defined in (2.9) such that

(2.33) ∑ i = 0 3 L i λ i < 1 ,

then BVP (1.1), (1.3) has a unique (nontrivial) solution in C 4 ( [ 0 , 1 ] ) such that

( x , y ( x ) , y ′ ( x ) , y ″ ( x ) , y ‴ ( x ) ) ∈ H , f o r a l l x ∈ [ 0 , 1 ] .

Theorem 2.9

Let f : J → ℝ be continuous and uniformly bounded by M > 0 on the set

J ≔ ( x , u , v , w , z ) ∈ ℝ 5 : x ∈ [ 0 , 1 ] , | u − ψ ( x ) | ≤ θ 0 θ 2 R , | v − ψ ′ ( x ) | ≤ θ 1 θ 2 R , | w − ψ ″ ( x ) | ≤ R , | z − ψ ‴ ( x ) | ≤ θ 3 θ 2 R ,

where R > 0 , ψ is defined in (2.24) and θ i are defined in (2.25). Assume M θ 2 ≤ R . For i = 0 , 1 , 2 , 3 , let L i be non-negative constants (not all zero) such that

(2.34) | f ( x , u 0 , u 1 , u 2 , u 3 ) − f ( x , v 0 , v 1 , v 2 , u 3 ) | ≤ ∑ i = 0 3 L i | u i − v i | , f o r a l l ( x , u 0 , u 1 , u 2 , u 3 ) , ( x , v 0 , v 1 , v 2 , v 3 ) ∈ J .

If there are constants p > 1 and q > 1 such that 1 / p + 1 / q = 1 with γ i defined in (2.9) such that

(2.35) ∑ i = 0 3 L i λ i < 1 ,

then BVP (1.1), (1.3) has a unique (nontrivial) solution in C 4 ( [ 0 , 1 ] ) such that

( x , y ( x ) , y ′ ( x ) , y ″ ( x ) , y ‴ ( x ) ) ∈ J , f o r a l l x ∈ [ 0 , 1 ] .

Theorem 2.10

Let f : K → ℝ be continuous and uniformly bounded by M > 0 on the set

K ≔ ( x , u , v , w , z ) ∈ ℝ 5 : x ∈ [ 0 , 1 ] , | u − ψ ( x ) | ≤ θ 0 θ 3 R , | v − ψ ′ ( x ) | ≤ θ 1 θ 3 R , | w − ψ ″ ( x ) | ≤ θ 2 θ 3 R , | z − ψ ‴ ( x ) | ≤ R ,

where R > 0 , ψ is defined in (2.24) and θ i are defined in (2.25). Assume M θ 3 ≤ R . For i = 0 , 1 , 2 , 3 , let L i be non-negative constants (not all zero) such that

(2.36) | f ( x , u 0 , u 1 , u 2 , u 3 ) − f ( x , v 0 , v 1 , v 2 , u 3 ) | ≤ ∑ i = 0 3 L i | u i − v i | , f o r a l l ( x , u 0 , u 1 , u 2 , u 3 ) , ( x , v 0 , v 1 , v 2 , v 3 ) ∈ K .

If there are constants p > 1 and q > 1 such that 1 / p + 1 / q = 1 with λ i defined in (2.27) such that

(2.37) ∑ i = 0 3 L i λ i < 1 ,

then BVP (1.1), (1.3) has a unique (nontrivial) solution in C 4 ( [ 0 , 1 ] ) such that

( x , y ( x ) , y ′ ( x ) , y ″ ( x ) , y ‴ ( x ) ) ∈ K , f o r a l l x ∈ [ 0 , 1 ] .

3 Application to beam deflections

In this section, we analyse fourth-order simplified BVPs arising from the deflection of elastic beams subject to linear and nonlinear loading forces. Through this discussion, we illustrate the nature of the advancement of this paper when compared with traditional approaches.

3.1 Nonlinear loading force

If we consider a loading force on the beam given by f ( x , y ) , which may be nonlinear, then we obtain the following fourth-order differential equation:

(3.1) y ( i v ) = f ( x , y ) , x ∈ [ 0 , 1 ] ,

where y = y ( x ) represents the resultant deflection of the beam at position x. This form is similar to that of [14].

In this case, the Lipschitz condition of our theorems reduce to

(3.2) | f ( x , u ) − f ( x , v ) ≤ L 0 | u − v |

on [ 0 , 1 ] × ℝ or on suitable subsets.

A standard approach to problem (3.1) with either fully clamped ends (1.4) or clamped/free ends (1.5) could employ Banach’s contraction mapping theorem within the space of continuous functions C ( [ 0 , 1 ] ) coupled with the maximum metric

ρ ( y , z ) ≔ max x ∈ [ 0 , 1 ] | y ( x ) − z ( x ) | , for all y , z ∈ C ( [ 0 , 1 ] ) .

To obtain a contraction for the operator T with respect to our d as per Banach’s contraction mapping theorem, the standard condition for clamped ends takes the form:

(3.3) L 0 max x ∈ [ 0 , 1 ] ∫ 0 1 | G ( x , s ) | d s = L 0 / 384 < 1

see [18, Theorem 3.1 Condition c] or [12].

For clamped/free ends, to obtain a contraction for the operator T with respect to our ρ as per Banach’s contraction mapping theorem, the condition

(3.4) L 0 max x ∈ [ 0 , 1 ] ∫ 0 1 | G ( x , s ) | d s = L 0 / 8 < 1

is involved, see, for example, [11, p. 58] or [58, Remark 1].

In comparison, the simplified version of Theorem 2.1 (or related theorems) with p = q = 2 involves (2.11) taking the form:

(3.5) L 0 ∫ 0 1 ∫ 0 1 | G ( x , s ) | 2 d s d x 1 / 2 = L 0 71 / 17,463,600 1 / 2 < 1 .

The left-hand side of (3.5) can be verified by direct integration or through use of a suitable computing package, via

∫ 0 1 | G ( x , s ) | 2 d s = ∫ 0 x s 2 ( 1 − x ) 2 [ ( x − s ) + 2 ( 1 − s ) x ] 6 2 d s + ∫ x 1 x 2 ( 1 − s ) 2 [ ( s − x ) + 2 ( 1 − x ) s ] 6 2 d s = [ ( x − 1 ) 4 x 7 ( 20 x 2 − 50 x + 33 ) / 1 , 260 ] − x 4 ( 20 x 2 + 10 x + 3 ) ( x − 1 ) 7 / 1 , 260

and another integration yields

∫ 0 1 ∫ 0 1 | G ( x , s ) | 2 d s d x = ∫ 0 1 ( x − 1 ) 4 x 7 ( 20 x 2 − 50 x + 33 ) / 1 , 260 − x 4 ( 20 x 2 + 10 x + 3 ) ( x − 1 ) 7 / 1 , 260 d x = 71 / 17 , 463 , 600 .

One can observe that our condition (3.5) is sharper than condition (3.3) and this illustrates one aspect of how the new results of this paper represent an advancement over traditional approaches, and how they are applicable to a wider class of problems.

The simplified version of Theorem 2.6 (or related theorems) with p = q = 2 involves (2.29) taking the form:

(3.6) L ∫ 0 1 ∫ 0 1 | G ( x , s ) | 2 d s d x 1 / 2 = L ( 11 / 1 , 680 ) 1 / 2 < 1 .

The left-hand side of (3.6) can be verified via the following steps.

∫ 0 1 | G ( x , s ) | 2 d s = ∫ 0 x s 2 ( 3 x − s ) 6 2 d s + ∫ x 1 x 2 ( 3 s − x ) 6 2 d s = [ 11 x 7 / 420 ] + x 4 / 12 − x 5 / 12 + x 6 / 36 − x 7 / 36

and another integration yields

∫ 0 1 ∫ 0 1 | G ( x , s ) | 2 d s d x = ∫ 0 1 11 x 7 / 420 + x 4 / 12 − x 5 / 12 + x 6 / 36 − x 7 / 36 d x = 11 / 1,680 .

Again, observe that our condition (3.6) is sharper than condition (3.4) and this illustrates another element of our advancement.

Let us further demonstrate the more aspects of our sharpened assumptions above through the discussion of some concrete cases.

Remark 3.1

If we consider the special case of f in (3.1) in the form

(3.7) f ( y ) ≔ 10 sin y + 1 ,

then we see that the smallest Lipschitz constant that we can calculate so that (3.2) holds on [ 0 , 1 ] × ℝ is L 0 = 10 , which is formulated from the bound on | f ′ ( y ) | . Such an L 0 satisfies (3.6) but not (3.4). Thus, our f satisfies the conditions of Theorem 2.6 (with appropriate boundary conditions) and we can obtain information in unbounded sets regarding the existence, uniqueness and approximation of solutions.

On the other hand, if we consider f on a closed and bounded domain [ − R , R ] , for any constant R > 0 , then we still cannot get (3.4) to hold for the above choice of L 0 = 10 . Hence, the results of [11] do not apply to the f in this example.

Furthermore, we can rerun the same argument as above for the example

f ( y ) = 385 sin y + 1

to show that (3.5) holds but (3.3) does not. Hence, the results of [12] do not apply to this f with L 0 = 385 .

Remark 3.2

Furthermore, we consider the special case

(3.8) f ( x , y ) ≔ x + 1 + 8 y 2 ,

subject to clamped/free ends (1.5).

Choose R = 1 / 2 to form the set F in the statement of Theorem 2.7. Here, choose θ 0 = 1 / 8 (with the remaining θ i not coming in to play) and a bound M on f over F can be chosen to be M = 4 .

In addition, we see ∂ f / ∂ y is continuous and bounded on F by 8. Thus, we can choose our Lipschitz constant to be L 0 = 8 . (In the notation of Theorem 2.7, we have L 0 = 8 and the remaining L i are zero.)

Choosing p = q = 2 , our λ 0 is contained in (3.6) (with the other λ i not coming in to play). Thus, we have

L 0 λ 0 = 8 ( 11 / 1 , 680 ) 1 / 2 < 1 .

Hence, all of the conditions of Theorem 2.7 are satisfied and its conclusion may be applied to our problem.

On the other hand, if we try to verify or apply the conditions in [11] to our problem, then we run into an impossibility. For all x ∈ [ 0 , 1 ] and | y | ≤ M / 8 , where M > 0 , the assumption in [11] becomes

| f ( x , y ) | ≤ 2 + 8 ( M / 8 ) 2 ≤ M ,

which has only the solution M = 4 . However, (3.4) takes the form

L 0 / 8 = 8 / 8 < 1 ,

which is clearly impossible. Thus, the results in [11] do not apply to this example.

The aforementioned examples and discussion illustrate how the new results of this paper represent an advancement over traditional approaches, and how they are applicable to a wider class of problems.

3.2 Linear loading force

If the loading force on the beam is linear and given by f ( x , y ) = h ( x ) y + j ( x ) , then the following fourth-order ordinary differential equation is obtained by:

(3.9) y ( i v ) = h ( x ) y + j ( x ) , x ∈ [ 0 , 1 ]

and we can form the following corollaries.

Corollary 3.1

Let h and j be continuous. If | h ( x ) | < ( 1 , 74 , 63 , 600 / 71 ) 1 / 2 for all x ∈ [ 0 , 1 ] , then the elastic beam deflection BVP (3.9), (1.4) with linear loading force has a unique solution in C 4 ( [ 0 , 1 ] ) .

Proof

This is a special case of Theorem 2.1 with p = q = 2 and f ( x , y ) = h ( x ) y + j ( x ) .□

Corollary 3.2

Let h and j be continuous. If | h ( x ) | < ( 1 , 680 / 11 ) 1 / 2 for all x ∈ [ 0 , 1 ] , then the elastic beam deflection BVP (3.9), (1.5) with linear loading force has a unique solution in C 4 ( [ 0 , 1 ] ) .

Proof

This is a special case of Theorem 2.6 with p = q = 2 .□

4 Discussion and conclusion

Let us briefly discuss our results and also identify some potential open problems for further research.

This paper generated new advancements into the existence and uniqueness of solutions to BVPs involving nonlinear, fourth-order differential equations. We sharpened traditional results by showing that a larger class of problems admit a unique solution and achieved this by drawing on the fixed-point theory in an interesting and alternative way via an application of Rus’s contraction mapping theorem. Our results add to the recent literature on BVPs [54,59,60,61] and collectively move towards a more complete understanding of their underlying theory and application.

There are many potential ways forward regarding the ideas of the present paper in both pure and applied forms. The application of Theorem 1.1 to all kinds of problems still appears to be underutilized and so there are opportunities to improve our understanding of solutions to a range of differential, integral and difference equations via Theorem 1.1. For example, important work into the area of nanoscaled beams has been carried out in [62,63], where nonlocal and higher order theories have been used to capture effect sizes with nonlocal forms of boundary conditions. It is unknown if Theorem 1.1 can be used in these kinds of problems and this remains an open question.

Acknowledgments

This research was supported by the Australian Research Council’s Discovery Project DP0450752.

References

[1] Ravi P. Agarwal, Donal O’Regan, and Patricia J. Y. Wong, Positive solutions of differential, difference and integral equations, Kluwer Academic Publishers, Dordrecht, 1999, 10.1007/978-94-015-9171-3.Search in Google Scholar

[2] Da Jun Guo and V. Lakshmikantham, Nonlinear problems in abstract cones, Notes and Reports in Mathematics in Science and Engineering, 5, Academic Press, Inc., Boston, MA, 1988.Search in Google Scholar

[3] Alberto Cabada and Lorena Saavedra, Existence of solutions for nth-order nonlinear differential boundary value problems by means of fixed point theorems, Nonlinear Anal. Real World Appl. 42 (2018), 180–206, 10.1016/j.nonrwa.2017.12.008.Search in Google Scholar

[4] Yongxiang Li and Yabing Gao, Existence and uniqueness results for the bending elastic beam equations, Appl. Math. Lett. 95 (2019), 72–77, 10.1016/j.aml.2019.03.025.Search in Google Scholar

[5] Yongfang Wei, Qilin Song, and Zhanbing Bai, Existence and iterative method for some fourth order nonlinear boundary value problems, Appl. Math. Lett. 87 (2019), 101–107, 10.1016/j.aml.2018.07.032.Search in Google Scholar

[6] Xiaoping Zhang, Existence and iteration of monotone positive solutions for an elastic beam equation with a corner, Nonlinear Anal. Real World Appl. 10 (2009), no. 4, 2097–2103, 10.1016/j.nonrwa.2008.03.017.Search in Google Scholar

[7] Daqing Jiang, Huizhao Liu, and Lili Zhang, Optimal existence theory for single and multiple positive solutions to fourth-order periodic boundary value problems, Nonlinear Anal. Real World Appl. 7 (2006), no. 4, 841–852, 10.1016/j.nonrwa.2005.05.003.Search in Google Scholar

[8] Eric R. Kaufmann and Nickolai Kosmatov, Elastic beam problem with higher order derivatives, Nonlinear Anal. Real World Appl. 8 (2007), no. 3, 811–821, 10.1016/j.nonrwa.2006.03.006.Search in Google Scholar

[9] Feliz Minhós and Infeliz Coxe, Systems of coupled clamped beams equations with full nonlinear terms: existence and location results, Nonlinear Anal. Real World Appl. 35 (2017), 45–60, 10.1016/j.nonrwa.2016.10.006.Search in Google Scholar

[10] Man Xu and Ruyun Ma, Nonlinear elastic beam problems with the parameter near resonance, Open Math. 16 (2018), no. 1, 1176–1186, 310.1515/math-2018-0097.Search in Google Scholar

[11] Quang A. Dang and Thi Kim Quy Ngo, Existence results and iterative method for solving the cantilever beam equation with fully nonlinear term, Nonlinear Anal. Real World Appl. 36 (2017), 56–68, 10.1016/j.nonrwa.2017.01.001.Search in Google Scholar

[12] Quang A. Dang and Thanh Huong Nguyen, Solving the Dirichlet problem for fully fourth order nonlinear differential equation, Afr. Mat. 30 (2019), no. 3–4, 623–641, 10.1007/s13370-019-00671-6.Search in Google Scholar

[13] Yongxiang Li, Existence of positive solutions for the cantilever beam equations with fully nonlinear terms, Nonlinear Anal. Real World Appl. 27 (2016), 221–237, 10.1016/j.nonrwa.2015.07.016.Search in Google Scholar

[14] Zhanbing Bai and Haiyan Wang, On positive solutions of some nonlinear fourth-order beam equations, J. Math. Anal. Appl. 270 (2002), no. 2, 357–368, 10.1016/S0022-247X(02)00071-9.Search in Google Scholar

[15] Ioan A. Rus, On a fixed point theorem of Maia, Studia Univ. Babe-Bolyai Math. 22 (1977), 40–42.Search in Google Scholar

[16] Haixia Lu, Li Sun, and Jingxian Sun, Existence of positive solutions to a non-positive elastic beam equation with both ends fixed, Bound. Value Probl. 2012 (2012), 56, 10.1186/1687-2770-2012-56.Search in Google Scholar

[17] Chengbo Zhai, Ruipeng Song, and Qianqian Han, The existence and the uniqueness of symmetric positive solutions for a fourth-order boundary value problem, Comput. Math. Appl. 62 (2011), no. 6, 2639–2647, 10.1016/j.camwa.2011.08.003.Search in Google Scholar

[18] J. Caballero, J. Harjani, and K. Sadarangani, Uniqueness of positive solutions for a class of fourth-order boundary value problems, Abstr. Appl. Anal. 2011 (2011), 543035, 10.1155/2011/543035.Search in Google Scholar

[19] Chengbo Zhai and Douglas R. Anderson, A sum operator equation and applications to nonlinear elastic beam equations and Lane-Emden-Fowler equations, J. Math. Anal. Appl. 375 (2011), no. 2, 388–400, 10.1016/j.jmaa.2010.09.017.Search in Google Scholar

[20] Wenxia Wang, Yanping Zheng, Hui Yang, and Junxia Wang, Positive solutions for elastic beam equations with nonlinear boundary conditions and a parameter, Bound. Value Probl. 2014 (2014), 80, 10.1186/1687-2770-2014-80.Search in Google Scholar

[21] Yongxiang Li and Yabing Gao, The method of lower and upper solutions for the cantilever beam equations with fully nonlinear terms, J. Inequal. Appl. 2019 (2019), 136, 10.1186/s13660-019-2088-5.Search in Google Scholar

[22] Yongxiang Li and Xuechun Chen, Solvability for fully cantilever beam equations with superlinear nonlinearities, Bound. Value Probl. 2019 (2019), 83, 10.1186/s13661-019-1200-6.Search in Google Scholar

[23] Qun Gao, The solvability and numerical simulation for the elastic beam problems with nonlinear boundary conditions, Abstr. Appl. Anal. 2014 (2014), 454658, 10.1155/2014/454658.Search in Google Scholar

[24] Shunyong Li and Xiaoqin Zhang, Existence and uniqueness of monotone positive solutions for an elastic beam equation with nonlinear boundary conditions, Comput. Math. Appl. 63 (2012), no. 9, 1355–1360, 10.1016/j.camwa.2011.12.065.Search in Google Scholar

[25] Babak Azarnavid, Kourosh Parand, and Saeid Abbasbandy, An iterative kernel based method for fourth order nonlinear equation with nonlinear boundary condition, Commun. Nonlinear Sci. Numer. Simul. 59 (2018), 544–552, 10.1016/j.cnsns.2017.12.002.Search in Google Scholar

[26] G. Engel, K. Garikipati, T. J. R. Hughes, M. G. Larson, L. Mazzei, and R. L. Taylor, Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity, Comput. Methods Appl. Mech. Engrg. 191 (2002), no. 34, 3669–3750, 10.1016/S0045-7825(02)00286-4.Search in Google Scholar

[27] Vincenzo Ferone and Bernd Kawohl, Rearrangements and fourth-order equations, Quart. Appl. Math. 61 (2003), no. 2, 337–343, 10.1090/qam/1976374.Search in Google Scholar

[28] Daniel Franco, Donal O’Regan, and Juan Perán, Fourth-order problems with nonlinear boundary conditions, J. Comput. Appl. Math. 174 (2005), no. 2, 315–327, 10.1016/j.cam.2004.04.013.Search in Google Scholar

[29] Andrzej Granas, Ronald Guenther, and John Lee, Nonlinear boundary value problems for ordinary differential equations, Dissertationes Math. (Rozprawy Mat.) 244 (1985).Search in Google Scholar

[30] Maria do Rosário Grossinho, Luis Sanchez, and Stepan A. Tersian, On the solvability of a boundary value problem for a fourth-order ordinary differential equation, Appl. Math. Lett. 18 (2005), no. 4, 439–444, 10.1016/j.aml.2004.03.011.Search in Google Scholar

[31] Zhaocai Hao and Zhong-Rui Wang, Positive solutions for fourth-order singular boundary value problems, Nonlinear Funct. Anal. Appl. 10 (2005), no. 1, 41–53.Search in Google Scholar

[32] Daqing Jiang, Huizhao Liu, and Xiaojie Xu, Nonresonant singular fourth-order boundary value problems, Appl. Math. Lett. 18 (2005), no. 1, 69–75, 10.1016/j.aml.2003.05.016.Search in Google Scholar

[33] Xiu-Fen Jiang and Qing-Liu Yao, An existence theorem of positive solution for fourth-order superlinear semipositone eigenvalue problems, Chinese Quart. J. Math. 16 (2001), no. 2, 64–68.Search in Google Scholar

[34] Basant K. Karna, Eric R. Kaufmann, and Jason Nobles, Comparison of eigenvalues for a fourth-order four-point boundary value problem, Electron. J. Qual. Theory Differ. Equ. 2005 (2005), no. 15, 1–9, 10.14232/ejqtde.2005.1.15.Search in Google Scholar

[35] A. Khan, M. A. Noor, and T. Aziz, Parametric quintic-spline approach to the solution of a system of fourth-order boundary-value problems, J. Optim. Theory Appl. 122 (2004), no. 2, 309–322, 10.1023/B:JOTA.0000042523.83186.4c.Search in Google Scholar

[36] Nickolai Kosmatov, Countably many solutions of a fourth-order boundary value problem, Electron. J. Qual. Theory Differ. Equ. 2004 (2004), no. 12, 1–15, 10.14232/ejqtde.2004.1.12.Search in Google Scholar

[37] Yongxiang Li, Two-parameter nonresonance condition for the existence of fourth-order boundary value problems, J. Math. Anal. Appl. 308 (2005), no. 1, 121–128, 10.1016/j.jmaa.2004.11.021.Search in Google Scholar

[38] Xi-Lan Liu and Wan-Tong Li, Positive solutions of the nonlinear fourth-order beam equation with three parameters, J. Math. Anal. Appl. 303 (2005), no. 1, 150–163, 10.1016/j.jmaa.2004.08.026.Search in Google Scholar

[39] Ruyun Ma, Multiple positive solutions for a semipositone fourth-order boundary value problem, Hiroshima Math. J. 33 (2003), no. 2, 217–227, 10.32917/hmj/1150997947.Search in Google Scholar

[40] Ruyun Ma and C. C. Tisdell, Positive solutions of singular sublinear fourth-order boundary value problems, Appl. Anal. 84 (2005), 1199–1220, 10.1080/00036810500047634.Search in Google Scholar

[41] Ru Yun Ma and Hong Ping Wu, Positive solutions of a fourth-order two-point boundary value problem, (Chinese) Acta Math. Sci. Ser. A Chin. Ed. 22 (2002), no. 2, 244–249.Search in Google Scholar

[42] Kamil Orucoglu, A new Green function concept for fourth-order differential equations, Electron. J. Differ. Equ. 2005 (2005), no. 28, 1–12.Search in Google Scholar

[43] Jolanta Przybycin, Existence and uniqueness theorems for fourth-order boundary value problems, Ann. Polon. Math. 67 (1997), no. 1, 59–64, 10.4064/ap-67-1-59-64.Search in Google Scholar

[44] Hong Ping Wu and Ru Yun Ma, Positive solutions of fourth-order two-point boundary value problems, (Chinese) Acta Anal. Funct. Appl. 2 (2000), no. 4, 342–348.Search in Google Scholar

[45] Xiaojie Xu, Daqing Jiang, Donal O’Regan, and R. P. Agarwal, Multiple positive solutions of fourth-order boundary value problems, Math. Inequal. Appl. 8 (2005), no. 1, 79–88, 10.7153/mia-08-08.Search in Google Scholar

[46] Bo Yang, Positive solutions for a fourth-order boundary value problem, Electron. J. Qual. Theory Differ. Equ. 2005 (2005), no. 3, 1–17, 10.14232/ejqtde.2005.1.3.Search in Google Scholar

[47] Yisong Yang, Fourth-order two-point boundary value problems, Proc. Amer. Math. Soc. 104 (1988), no. 1, 175–180, 10.2307/2047481.Search in Google Scholar

[48] Qingliu Yao, Positive solutions for eigenvalue problems of fourth-order elastic beam equations, Appl. Math. Lett. 17 (2004), no. 2, 237–243, 10.1016/S0893-9659(04)90037-7.Search in Google Scholar

[49] Qing-Liu Yao, Solvability of an elastic beam equation with Caratheodory function, Math. Appl. (Wuhan) 17 (2004), no. 3, 389–392.Search in Google Scholar

[50] Qing Liu Yao and Li Shun Ren, Solutions and positive solutions to a class of nonlinear fourth-order boundary value problems, (Chinese) Xiamen Daxue Xuebao Ziran Kexue Ban 43 (2004), no. 6, 765–768.Search in Google Scholar

[51] Christopher C. Tisdell, Critical perspectives of pedagogical approaches to reversing the order of integration in double integrals, Internat. J. Math. Ed. Sci. Tech. 48 (2017), no. 8, 1285–1292, 10.1080/0020739X.2017.1329559.Search in Google Scholar

[52] Christopher C. Tisdell, On Picard’s iteration method to solve differential equations and a pedagogical space for otherness, Internat. J. Math. Ed. Sci. Tech. 50 (2019), no. 5, 788–799, 10.1080/0020739X.2018.1507051.Search in Google Scholar

[53] Christopher C. Tisdell, Rethinking pedagogy for second-order differential equations: a simplified approach to understanding well-posed problems, Internat. J. Math. Ed. Sci. Tech. 48 (2017), no. 5, 794–801, 10.1080/0020739X.2017.1285062.Search in Google Scholar

[54] Saleh S. Almuthaybiri and Christopher C. Tisdell, Establishing existence and uniqueness of solutions to the boundary value problem involving a generalized Emden equation, embracing Thomas – text Fermi-like theories, J. Eng. Math., (2020) 10.1007/s10665-020-10055-6.Search in Google Scholar

[55] Saleh S. Almuthaybiri and Christopher C. Tisdell, Sharper existence and uniqueness results for solutions to third-order boundary value problems, Math. Modell. Anal. 25 (2020), no. 3, 409–420, 10.3846/mma.2020.11043.Search in Google Scholar

[56] Saleh S. Almuthaybiri and Christopher C. Tisdell, Global existence theory for fractional differential equations: new advances via continuation methods for contractive maps, Analysis 39 (2019), no. 4, 117–128, 10.1515/anly-2019-0027.Search in Google Scholar

[57] Charles P. Stinson, Saleh S. Almuthaybiri, and Christopher C. Tisdell, A note regarding extensions of fixed point theorems involving two metrics via an analysis of iterated functions, ANZIAM J. (EMAC 2019) 61 (2019), C15–C30, 10.21914/anziamj.v61i0.15048.Search in Google Scholar

[58] Edson Alves, To Fu Ma, and Maurício Luciano Pelicer, Monotone positive solutions for a fourth order equation with nonlinear boundary conditions, Nonlinear Anal. 71 (2009), no. 9, 3834–3841, 10.1016/j.na.2009.02.051.Search in Google Scholar

[59] Liping Wei and Ruyun Ma, Global structure of sign-changing solutionsr for discrete Dirichlet problems, Open Math. 18 (2020), no. 1, 572–581, 10.1515/math-2020-0180.Search in Google Scholar

[60] Jinxia Cen, Chao Min, Van Thien Nguyen, and Guo-ji Tang, On the well-posedness of differential quasi-variational-hemivariational inequalities, Open Math. 18 (2020), no. 1, 540–551, 10.1515/math-2020-0028.Search in Google Scholar

[61] Muhammad Nazam, Anam Arif, Hasan Mahmood, and Choonkil Park, Some results in cone metric spaces with applications in homotopy theory, Open Math. 18 (2020), no. 1, 295–306, 10.1515/math-2020-0025.Search in Google Scholar

[62] Őmer Civalek and Cigdem Demir, Buckling and bending analyses of cantilever carbon nanotubes using the Euler-Bernoulli beam theory based on non-local continuum model, Asian J. Civil Eng. 12 (2001), 651–661.Search in Google Scholar

[63] Cigdem Demir and Őmer Civalek, A new nonlocal FEM via Hermitian cubic shape functions for thermal vibration of nano beams surrounded by an elastic matrix, Compos. Struct. 168 (2017), 33–50, 10.1016/j.compstruct.2017.02.091.Search in Google Scholar
Received: 2019-11-06
Revised: 2020-03-11
Accepted: 2020-06-30
Published Online: 2020-09-28

© 2020 Saleh S. Almuthaybiri and Christopher C. Tisdell, published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

  1. Regular Articles
  2. Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians
  3. Strong and weak convergence of Ishikawa iterations for best proximity pairs
  4. Curve and surface construction based on the generalized toric-Bernstein basis functions
  5. The non-negative spectrum of a digraph
  6. Bounds on F-index of tricyclic graphs with fixed pendant vertices
  7. Crank-Nicolson orthogonal spline collocation method combined with WSGI difference scheme for the two-dimensional time-fractional diffusion-wave equation
  8. Hardy’s inequalities and integral operators on Herz-Morrey spaces
  9. The 2-pebbling property of squares of paths and Graham’s conjecture
  10. Existence conditions for periodic solutions of second-order neutral delay differential equations with piecewise constant arguments
  11. Orthogonal polynomials for exponential weights x2α(1 – x2)2ρe–2Q(x) on [0, 1)
  12. Rough sets based on fuzzy ideals in distributive lattices
  13. On more general forms of proportional fractional operators
  14. The hyperbolic polygons of type (ϵ, n) and Möbius transformations
  15. Tripled best proximity point in complete metric spaces
  16. Metric completions, the Heine-Borel property, and approachability
  17. Functional identities on upper triangular matrix rings
  18. Uniqueness on entire functions and their nth order exact differences with two shared values
  19. The adaptive finite element method for the Steklov eigenvalue problem in inverse scattering
  20. Existence of a common solution to systems of integral equations via fixed point results
  21. Fixed point results for multivalued mappings of Ćirić type via F-contractions on quasi metric spaces
  22. Some inequalities on the spectral radius of nonnegative tensors
  23. Some results in cone metric spaces with applications in homotopy theory
  24. On the Malcev products of some classes of epigroups, I
  25. Self-injectivity of semigroup algebras
  26. Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales
  27. On the symmetrized s-divergence
  28. On multivalued Suzuki-type θ-contractions and related applications
  29. Approximation operators based on preconcepts
  30. Two types of hypergeometric degenerate Cauchy numbers
  31. The molecular characterization of anisotropic Herz-type Hardy spaces with two variable exponents
  32. Discussions on the almost 𝒵-contraction
  33. On a predator-prey system interaction under fluctuating water level with nonselective harvesting
  34. On split involutive regular BiHom-Lie superalgebras
  35. Weighted CBMO estimates for commutators of matrix Hausdorff operator on the Heisenberg group
  36. Inverse Sturm-Liouville problem with analytical functions in the boundary condition
  37. The L-ordered L-semihypergroups
  38. Global structure of sign-changing solutions for discrete Dirichlet problems
  39. Analysis of F-contractions in function weighted metric spaces with an application
  40. On finite dual Cayley graphs
  41. Left and right inverse eigenpairs problem with a submatrix constraint for the generalized centrosymmetric matrix
  42. Controllability of fractional stochastic evolution equations with nonlocal conditions and noncompact semigroups
  43. Levinson-type inequalities via new Green functions and Montgomery identity
  44. The core inverse and constrained matrix approximation problem
  45. A pair of equations in unlike powers of primes and powers of 2
  46. Miscellaneous equalities for idempotent matrices with applications
  47. B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces
  48. Rate of convergence of uniform transport processes to a Brownian sheet
  49. Curves in the Lorentz-Minkowski plane with curvature depending on their position
  50. Sequential change-point detection in a multinomial logistic regression model
  51. Tiny zero-sum sequences over some special groups
  52. A boundedness result for Marcinkiewicz integral operator
  53. On a functional equation that has the quadratic-multiplicative property
  54. The spectrum generated by s-numbers and pre-quasi normed Orlicz-Cesáro mean sequence spaces
  55. Positive coincidence points for a class of nonlinear operators and their applications to matrix equations
  56. Asymptotic relations for the products of elements of some positive sequences
  57. Jordan {g,h}-derivations on triangular algebras
  58. A systolic inequality with remainder in the real projective plane
  59. A new characterization of L2(p2)
  60. Nonlinear boundary value problems for mixed-type fractional equations and Ulam-Hyers stability
  61. Asymptotic normality and mean consistency of LS estimators in the errors-in-variables model with dependent errors
  62. Some non-commuting solutions of the Yang-Baxter-like matrix equation
  63. General (p,q)-mixed projection bodies
  64. An extension of the method of brackets. Part 2
  65. A new approach in the context of ordered incomplete partial b-metric spaces
  66. Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis
  67. Remark on subgroup intersection graph of finite abelian groups
  68. Detectable sensation of a stochastic smoking model
  69. Almost Kenmotsu 3-h-manifolds with transversely Killing-type Ricci operators
  70. Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms
  71. Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion
  72. On surrounding quasi-contractions on non-triangular metric spaces
  73. SEMT valuation and strength of subdivided star of K 1,4
  74. Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems
  75. Gradient estimates for a weighted nonlinear parabolic equation and applications
  76. On the equivalence of three-dimensional differential systems
  77. Free nonunitary Rota-Baxter family algebras and typed leaf-spaced decorated planar rooted forests
  78. The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices
  79. Explicit determinantal formula for a class of banded matrices
  80. Dynamics of a diffusive delayed competition and cooperation system
  81. Error term of the mean value theorem for binary Egyptian fractions
  82. The integral part of a nonlinear form with a square, a cube and a biquadrate
  83. Meromorphic solutions of certain nonlinear difference equations
  84. Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces
  85. Some integral curves with a new frame
  86. Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation
  87. Towards a homological generalization of the direct summand theorem
  88. A standard form in (some) free fields: How to construct minimal linear representations
  89. On the determination of the number of positive and negative polynomial zeros and their isolation
  90. Perturbation of the one-dimensional time-independent Schrödinger equation with a rectangular potential barrier
  91. Simply connected topological spaces of weighted composition operators
  92. Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions
  93. A study of uniformities on the space of uniformly continuous mappings
  94. The strong nil-cleanness of semigroup rings
  95. On an equivalence between regular ordered Γ-semigroups and regular ordered semigroups
  96. Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow
  97. Noetherian properties in composite generalized power series rings
  98. Inequalities for the generalized trigonometric and hyperbolic functions
  99. Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions
  100. A new characterization of a proper type B semigroup
  101. Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields
  102. Estimates of entropy numbers in probabilistic setting
  103. Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory
  104. S-shaped connected component of positive solutions for second-order discrete Neumann boundary value problems
  105. The logarithmic mean of two convex functionals
  106. A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation
  107. Approximation properties of tensor norms and operator ideals for Banach spaces
  108. A multi-power and multi-splitting inner-outer iteration for PageRank computation
  109. The edge-regular complete maps
  110. Ramanujan’s function k(τ)=r(τ)r2(2τ) and its modularity
  111. Finite groups with some weakly pronormal subgroups
  112. A new refinement of Jensen’s inequality with applications in information theory
  113. Skew-symmetric and essentially unitary operators via Berezin symbols
  114. The limit Riemann solutions to nonisentropic Chaplygin Euler equations
  115. On singularities of real algebraic sets and applications to kinematics
  116. Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type
  117. New (p, q)-estimates for different types of integral inequalities via (α, m)-convex mappings
  118. Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions
  119. Boundary layer analysis for a 2-D Keller-Segel model
  120. On some extensions of Gauss’ work and applications
  121. A study on strongly convex hyper S-subposets in hyper S-posets
  122. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis
  123. Special Issue on Graph Theory (GWGT 2019), Part II
  124. On applications of bipartite graph associated with algebraic structures
  125. Further new results on strong resolving partitions for graphs
  126. The second out-neighborhood for local tournaments
  127. On the N-spectrum of oriented graphs
  128. The H-force sets of the graphs satisfying the condition of Ore’s theorem
  129. Bipartite graphs with close domination and k-domination numbers
  130. On the sandpile model of modified wheels II
  131. Connected even factors in k-tree
  132. On triangular matroids induced by n3-configurations
  133. The domination number of round digraphs
  134. Special Issue on Variational/Hemivariational Inequalities
  135. A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion
  136. On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions
  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
  138. An efficient approach for the numerical solution of fifth-order KdV equations
  139. Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function
  140. Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function
  141. An equivalent quasinorm for the Lipschitz space of noncommutative martingales
  142. Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation
  143. Special Issue on Problems, Methods and Applications of Nonlinear analysis
  144. Generalized Picone inequalities and their applications to (p,q)-Laplace equations
  145. Positive solutions for parametric (p(z),q(z))-equations
  146. Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation
  147. (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group
  148. Quasilinear Dirichlet problems with competing operators and convection
  149. Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations
  150. Special Issue on Evolution Equations, Theory and Applications
  151. Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation
  152. Three classes of decomposable distributions
https://doi.org/10.1515/math-2020-0056
Keywords for this article
fourth-order boundary value problem; existence and uniqueness of solutions; beam analysis; Rus’s contraction mapping theorem; two metrics
Creative Commons
BY 4.0

Articles in the same Issue

  1. Regular Articles
  2. Non-occurrence of the Lavrentiev phenomenon for a class of convex nonautonomous Lagrangians
  3. Strong and weak convergence of Ishikawa iterations for best proximity pairs
  4. Curve and surface construction based on the generalized toric-Bernstein basis functions
  5. The non-negative spectrum of a digraph
  6. Bounds on F-index of tricyclic graphs with fixed pendant vertices
  7. Crank-Nicolson orthogonal spline collocation method combined with WSGI difference scheme for the two-dimensional time-fractional diffusion-wave equation
  8. Hardy’s inequalities and integral operators on Herz-Morrey spaces
  9. The 2-pebbling property of squares of paths and Graham’s conjecture
  10. Existence conditions for periodic solutions of second-order neutral delay differential equations with piecewise constant arguments
  11. Orthogonal polynomials for exponential weights x2α(1 – x2)2ρe–2Q(x) on [0, 1)
  12. Rough sets based on fuzzy ideals in distributive lattices
  13. On more general forms of proportional fractional operators
  14. The hyperbolic polygons of type (ϵ, n) and Möbius transformations
  15. Tripled best proximity point in complete metric spaces
  16. Metric completions, the Heine-Borel property, and approachability
  17. Functional identities on upper triangular matrix rings
  18. Uniqueness on entire functions and their nth order exact differences with two shared values
  19. The adaptive finite element method for the Steklov eigenvalue problem in inverse scattering
  20. Existence of a common solution to systems of integral equations via fixed point results
  21. Fixed point results for multivalued mappings of Ćirić type via F-contractions on quasi metric spaces
  22. Some inequalities on the spectral radius of nonnegative tensors
  23. Some results in cone metric spaces with applications in homotopy theory
  24. On the Malcev products of some classes of epigroups, I
  25. Self-injectivity of semigroup algebras
  26. Cauchy matrix and Liouville formula of quaternion impulsive dynamic equations on time scales
  27. On the symmetrized s-divergence
  28. On multivalued Suzuki-type θ-contractions and related applications
  29. Approximation operators based on preconcepts
  30. Two types of hypergeometric degenerate Cauchy numbers
  31. The molecular characterization of anisotropic Herz-type Hardy spaces with two variable exponents
  32. Discussions on the almost 𝒵-contraction
  33. On a predator-prey system interaction under fluctuating water level with nonselective harvesting
  34. On split involutive regular BiHom-Lie superalgebras
  35. Weighted CBMO estimates for commutators of matrix Hausdorff operator on the Heisenberg group
  36. Inverse Sturm-Liouville problem with analytical functions in the boundary condition
  37. The L-ordered L-semihypergroups
  38. Global structure of sign-changing solutions for discrete Dirichlet problems
  39. Analysis of F-contractions in function weighted metric spaces with an application
  40. On finite dual Cayley graphs
  41. Left and right inverse eigenpairs problem with a submatrix constraint for the generalized centrosymmetric matrix
  42. Controllability of fractional stochastic evolution equations with nonlocal conditions and noncompact semigroups
  43. Levinson-type inequalities via new Green functions and Montgomery identity
  44. The core inverse and constrained matrix approximation problem
  45. A pair of equations in unlike powers of primes and powers of 2
  46. Miscellaneous equalities for idempotent matrices with applications
  47. B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces
  48. Rate of convergence of uniform transport processes to a Brownian sheet
  49. Curves in the Lorentz-Minkowski plane with curvature depending on their position
  50. Sequential change-point detection in a multinomial logistic regression model
  51. Tiny zero-sum sequences over some special groups
  52. A boundedness result for Marcinkiewicz integral operator
  53. On a functional equation that has the quadratic-multiplicative property
  54. The spectrum generated by s-numbers and pre-quasi normed Orlicz-Cesáro mean sequence spaces
  55. Positive coincidence points for a class of nonlinear operators and their applications to matrix equations
  56. Asymptotic relations for the products of elements of some positive sequences
  57. Jordan {g,h}-derivations on triangular algebras
  58. A systolic inequality with remainder in the real projective plane
  59. A new characterization of L2(p2)
  60. Nonlinear boundary value problems for mixed-type fractional equations and Ulam-Hyers stability
  61. Asymptotic normality and mean consistency of LS estimators in the errors-in-variables model with dependent errors
  62. Some non-commuting solutions of the Yang-Baxter-like matrix equation
  63. General (p,q)-mixed projection bodies
  64. An extension of the method of brackets. Part 2
  65. A new approach in the context of ordered incomplete partial b-metric spaces
  66. Sharper existence and uniqueness results for solutions to fourth-order boundary value problems and elastic beam analysis
  67. Remark on subgroup intersection graph of finite abelian groups
  68. Detectable sensation of a stochastic smoking model
  69. Almost Kenmotsu 3-h-manifolds with transversely Killing-type Ricci operators
  70. Some inequalities for star duality of the radial Blaschke-Minkowski homomorphisms
  71. Results on nonlocal stochastic integro-differential equations driven by a fractional Brownian motion
  72. On surrounding quasi-contractions on non-triangular metric spaces
  73. SEMT valuation and strength of subdivided star of K 1,4
  74. Weak solutions and optimal controls of stochastic fractional reaction-diffusion systems
  75. Gradient estimates for a weighted nonlinear parabolic equation and applications
  76. On the equivalence of three-dimensional differential systems
  77. Free nonunitary Rota-Baxter family algebras and typed leaf-spaced decorated planar rooted forests
  78. The prime and maximal spectra and the reticulation of residuated lattices with applications to De Morgan residuated lattices
  79. Explicit determinantal formula for a class of banded matrices
  80. Dynamics of a diffusive delayed competition and cooperation system
  81. Error term of the mean value theorem for binary Egyptian fractions
  82. The integral part of a nonlinear form with a square, a cube and a biquadrate
  83. Meromorphic solutions of certain nonlinear difference equations
  84. Characterizations for the potential operators on Carleson curves in local generalized Morrey spaces
  85. Some integral curves with a new frame
  86. Meromorphic exact solutions of the (2 + 1)-dimensional generalized Calogero-Bogoyavlenskii-Schiff equation
  87. Towards a homological generalization of the direct summand theorem
  88. A standard form in (some) free fields: How to construct minimal linear representations
  89. On the determination of the number of positive and negative polynomial zeros and their isolation
  90. Perturbation of the one-dimensional time-independent Schrödinger equation with a rectangular potential barrier
  91. Simply connected topological spaces of weighted composition operators
  92. Generalized derivatives and optimization problems for n-dimensional fuzzy-number-valued functions
  93. A study of uniformities on the space of uniformly continuous mappings
  94. The strong nil-cleanness of semigroup rings
  95. On an equivalence between regular ordered Γ-semigroups and regular ordered semigroups
  96. Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow
  97. Noetherian properties in composite generalized power series rings
  98. Inequalities for the generalized trigonometric and hyperbolic functions
  99. Blow-up analyses in nonlocal reaction diffusion equations with time-dependent coefficients under Neumann boundary conditions
  100. A new characterization of a proper type B semigroup
  101. Constructions of pseudorandom binary lattices using cyclotomic classes in finite fields
  102. Estimates of entropy numbers in probabilistic setting
  103. Ramsey numbers of partial order graphs (comparability graphs) and implications in ring theory
  104. S-shaped connected component of positive solutions for second-order discrete Neumann boundary value problems
  105. The logarithmic mean of two convex functionals
  106. A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation
  107. Approximation properties of tensor norms and operator ideals for Banach spaces
  108. A multi-power and multi-splitting inner-outer iteration for PageRank computation
  109. The edge-regular complete maps
  110. Ramanujan’s function k(τ)=r(τ)r2(2τ) and its modularity
  111. Finite groups with some weakly pronormal subgroups
  112. A new refinement of Jensen’s inequality with applications in information theory
  113. Skew-symmetric and essentially unitary operators via Berezin symbols
  114. The limit Riemann solutions to nonisentropic Chaplygin Euler equations
  115. On singularities of real algebraic sets and applications to kinematics
  116. Results on analytic functions defined by Laplace-Stieltjes transforms with perfect ϕ-type
  117. New (p, q)-estimates for different types of integral inequalities via (α, m)-convex mappings
  118. Boundary value problems of Hilfer-type fractional integro-differential equations and inclusions with nonlocal integro-multipoint boundary conditions
  119. Boundary layer analysis for a 2-D Keller-Segel model
  120. On some extensions of Gauss’ work and applications
  121. A study on strongly convex hyper S-subposets in hyper S-posets
  122. On the Gevrey ultradifferentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis
  123. Special Issue on Graph Theory (GWGT 2019), Part II
  124. On applications of bipartite graph associated with algebraic structures
  125. Further new results on strong resolving partitions for graphs
  126. The second out-neighborhood for local tournaments
  127. On the N-spectrum of oriented graphs
  128. The H-force sets of the graphs satisfying the condition of Ore’s theorem
  129. Bipartite graphs with close domination and k-domination numbers
  130. On the sandpile model of modified wheels II
  131. Connected even factors in k-tree
  132. On triangular matroids induced by n3-configurations
  133. The domination number of round digraphs
  134. Special Issue on Variational/Hemivariational Inequalities
  135. A new blow-up criterion for the N – abc family of Camassa-Holm type equation with both dissipation and dispersion
  136. On the finite approximate controllability for Hilfer fractional evolution systems with nonlocal conditions
  137. On the well-posedness of differential quasi-variational-hemivariational inequalities
  138. An efficient approach for the numerical solution of fifth-order KdV equations
  139. Generalized fractional integral inequalities of Hermite-Hadamard-type for a convex function
  140. Karush-Kuhn-Tucker optimality conditions for a class of robust optimization problems with an interval-valued objective function
  141. An equivalent quasinorm for the Lipschitz space of noncommutative martingales
  142. Optimal control of a viscous generalized θ-type dispersive equation with weak dissipation
  143. Special Issue on Problems, Methods and Applications of Nonlinear analysis
  144. Generalized Picone inequalities and their applications to (p,q)-Laplace equations
  145. Positive solutions for parametric (p(z),q(z))-equations
  146. Revisiting the sub- and super-solution method for the classical radial solutions of the mean curvature equation
  147. (p,Q) systems with critical singular exponential nonlinearities in the Heisenberg group
  148. Quasilinear Dirichlet problems with competing operators and convection
  149. Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations
  150. Special Issue on Evolution Equations, Theory and Applications
  151. Instantaneous blow-up of solutions to the Cauchy problem for the fractional Khokhlov-Zabolotskaya equation
  152. Three classes of decomposable distributions
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 30.10.2025 from https://www.degruyterbrill.com/document/doi/10.1515/math-2020-0056/html
Scroll to top button