Reformulation of Fixed Point Existence: From Banach to Kannan and Chatterjea Contractions

Abstract

This paper presents a reformulation of classical existence and uniqueness results for second-order boundary value problems (BVPs) using the Kannan fixed point theorem, extending beyond the Banach contraction principle. We shift focus from the nonlinearity j to the solution operator T defined via Green’s function and establish a sufficient condition under which T satisfies the Kannan contraction criterion. Specifically, if the derivative of j is bounded by K and $K·{(\eta -\zeta )}^2/8<1/3$, then T is a Kannan contraction, ensuring a unique solution. This condition applies even when the Banach contraction principle fails. We also explore the plausibility of applying the Chatterjea contraction, though rigorous verification remains open. Examples illustrate the applicability of the results. This work highlights the utility of generalized contractions in differential equations.

1. Introduction

Classical existence and uniqueness results for boundary value problems (BVPs) often rely on the Banach contraction principle under Lipschitz conditions on the nonlinearity j. Two such results are stated below.

Theorem 1

([1], Theorem 7.11). (Fixed Point via Banach Contraction).

Let $j:[\zeta ,\eta ]\times {\mathbb{R}}^2\to \mathbb{R}$ be continuous and satisfy

$$|j(\tau ,\psi ,{\psi }^{\prime })-j(\tau ,\varphi ,{\varphi }^{\prime })|\le M|\psi -\varphi |+N|{\psi }^{\prime }-{\varphi }^{\prime }|,$$

for all $(\tau ,\psi ,{\psi }^{\prime }),(\tau ,\varphi ,{\varphi }^{\prime })\in [\zeta ,\eta ]\times {\mathbb{R}}^2$, with $M>0$, $N\ge 0$. If

$${\displaystyle \frac{M{(\eta -\zeta )}^2}{8}}+{\displaystyle \frac{N(\eta -\zeta )}{2}}<1,$$

then the BVP

$${\psi }^{″}=-j(\tau ,\psi ,{\psi }^{\prime }),\psi \left(\zeta \right)={\xi }_1,\psi \left(\eta \right)={\xi }_2,$$

has a unique solution.

Theorem 2

([1], Theorem 7.7). (Scalar Case).

Let $j:[\zeta ,\eta ]\times \mathbb{R}\to \mathbb{R}$ be continuous and satisfy

$$\left|j\right(\tau ,\psi )-j(\tau ,\varphi \left)\right|\le \alpha |\psi -\varphi |,$$

for $(\tau ,\psi ),(\tau ,\varphi )\in [\zeta ,\eta ]\times \mathbb{R}$, $\alpha >0$. If

$$\alpha {\displaystyle \frac{{(\eta -\zeta )}^2}{8}}<1,$$

then the BVP

$${\psi }^{″}\left(\tau \right)=-j(\tau ,\psi \left(\tau \right)),\psi \left(\zeta \right)={\xi }_1,\psi \left(\eta \right)={\xi }_2,$$

has a unique solution.

In this paper, we reformulate these results using the Kannan fixed point theorem, which generalizes the Banach principle by relaxing the Lipschitz condition. We define the solution operator T via Green’s function and show that if T satisfies the Kannan condition, a unique fixed point (hence unique solution) exists.

Our main contribution is a rigorous sufficient condition under which T satisfies the Kannan contraction criterion. Specifically, if $\partial j/\partial \psi$ is bounded by K and $K·{(\eta -\zeta )}^2/8<1/3$, then T is a Kannan contraction, ensuring a unique solution—even when the Banach condition fails.

For Chatterjea contractions, no general method exists to verify the condition directly for integral operators. However, we leverage the known implication: if T is a Banach contraction with constant $\mu <1/3$, then T is also a Chatterjea contraction. This indirect approach allows us to extend applicability heuristically.

This paper is organized as follows: Section 2 recalls definitions. Section 3 presents the Kannan reformulation with a detailed proof of Theorem 4 and a full proof of Corollary 1. Section 6 discusses the Chatterjea case with illustrative examples and graphic explanation by Figure 1 in Section 7. Section 8 concludes.

2. Preliminaries

In this section, we give some basic definitions and notations that will be used throughout this paper.

Definition 1

(Banach Contraction [2,3]). Let $(X,d)$ be a metric space. A mapping $T:X\to X$ is called a Banach contraction if there exists a constant $\alpha \in (0, 1)$ such that

$$d(Tx,Ty)\le \alpha d(x,y),\forall x,y\in X.$$

Definition 2

(Kannan Contraction [4,5]). Let $(X,d)$ be a metric space. A mapping $T:X\to X$ is a Kannan contraction if there exists $\alpha \in \left(0, {\displaystyle \frac{1}{2}}\right)$ such that

$$d(Tx,Ty)\le \alpha \left[d(x,Tx)+d(y,Ty)\right],\forall x,y\in X.$$

Definition 3

(Chatterjea Contraction [6]). Let $(X,d)$ be a metric space. A mapping $T:X\to X$ is a Chatterjea contraction if there exists $\alpha \in \left(0,{\displaystyle \frac{1}{2}}\right)$ such that

$$d(Tx,Ty)\le \alpha \left[d(x,Ty)+d(y,Tx)\right],\forall x,y\in X.$$

The Green’s function for Dirichlet BVP on $[\zeta ,\eta ]$ is as follows:

$$G(\tau ,s)=\left\{\begin{array}{cc}{\displaystyle \frac{(\tau -\zeta )(\eta -s)}{\eta -\zeta }},\hfill & \zeta \le \tau \le s\le \eta ,\hfill \\ {\displaystyle \frac{(\eta -\tau )(s-\zeta )}{\eta -\zeta }},\hfill & \zeta \le s\le \tau \le \eta .\hfill \end{array}\right.$$

(1)

It satisfies the following key estimate

$$\underset{\tau \in [\zeta ,\eta ]}{sup}{\int }_{\zeta }^{\eta }\left|G(\tau ,s)\right|\mathrm{d}s={\displaystyle \frac{{(\eta -\zeta )}^2}{8}}.$$

3. Reformulation: Kannan Contraction for Theorem 1

We reformulate Theorem 1 using the Kannan fixed point theorem.

Main Theorem

Let $C[\zeta ,\eta ]$ denote the space of continuous real-valued functions on $[\zeta ,\eta ]$, endowed with the supremum norm

$${∥\psi ∥}_{\infty }=\underset{\tau \in [\zeta ,\eta ]}{sup}\left|\psi \left(\tau \right)\right|.$$

Let $C^1[\zeta ,\eta ]$ denote the space of continuously differentiable functions endowed with the norm

$${∥\psi ∥}_{C^1}={∥\psi ∥}_{\infty }+{∥{\psi }^{\prime }∥}_{\infty }.$$

Both spaces are Banach spaces.

Theorem 3

(Kannan-type Existence). Let $j:[\zeta ,\eta ]\times \mathbb{R}\to \mathbb{R}$ be continuous. 

Define the operator $T:C^1[\zeta ,\eta ]\to C^1[\zeta ,\eta ]$ as follows:

$$\left(T\psi \right)\left(\tau \right)={\xi }_1{\displaystyle \frac{\eta -\tau }{\eta -\zeta }}+{\xi }_2{\displaystyle \frac{\tau -\zeta }{\eta -\zeta }}+{\int }_{\zeta }^{\eta }G(\tau ,s)j(s,\psi \left(s\right),{\psi }^{\prime }\left(s\right))ds,$$

(2)

Suppose T satisfies: there exists $\alpha \in (0, {\displaystyle \frac{1}{2}})$ such that for all $\psi ,\varphi \in C^1[\zeta ,\eta ]$,

$${\parallel T\psi -T\varphi \parallel }_{C^1}\le \alpha \left({\parallel \psi -T\psi \parallel }_{C^1}+{\parallel \varphi -T\varphi \parallel }_{C^1}\right).$$

(3)

Then, the BVP

$${\psi }^{\prime \prime }\left(\tau \right)=-j(\tau ,\psi \left(\tau \right),{\psi }^{\prime }\left(\tau \right)),\psi \left(\zeta \right)={\xi }_1,\psi \left(\eta \right)={\xi }_2$$

has a unique solution in $C^2[\zeta ,\eta ]$.

We now verify that T is well-defined and maps X into itself.

Lemma 1.

The operator T is well-defined and $T\psi \in C^1[\zeta ,\eta ]$ for all $\psi \in X$.

Proof. 

Since j is continuous and $\psi ,{\psi }^{\prime }\in C[\zeta ,\eta ]$, the composition $s\mapsto j(s,\psi \left(s\right),{\psi }^{\prime }\left(s\right))$ is continuous on $[\zeta ,\eta ]$. The Green’s function $G(\tau ,s)$ is continuous in $\tau$ for fixed s, and the integral

$${\int }_{\zeta }^{\eta }G(\tau ,s)j(s,\psi \left(s\right),{\psi }^{\prime }\left(s\right))ds$$

defines a continuous function of $\tau$. Thus, $T\psi$ is continuous.

Moreover, the partial derivative ${\displaystyle \frac{\partial G}{\partial \tau }}(\tau ,s)$ exists for $\tau \ne s$ and is bounded. It is given by

$${\displaystyle \frac{\partial G}{\partial \tau }}(\tau ,s)=\left\{\begin{array}{cc}{\displaystyle \frac{\eta -s}{\eta -\zeta }},\hfill & \tau <s,\hfill \\ -{\displaystyle \frac{s-\zeta }{\eta -\zeta }},\hfill & \tau >s.\hfill \end{array}\right.$$

The derivative of the integral is

$${\displaystyle \frac{d}{d\tau }}{\int }_{\zeta }^{\eta }G(\tau ,s)f\left(s\right)ds={\int }_{\zeta }^{\eta }{\displaystyle \frac{\partial G}{\partial \tau }}(\tau ,s)f\left(s\right)ds,$$

which is continuous in $\tau$ for continuous f. Hence, $T\psi$ is continuously differentiable, and

$${\left(T\psi \right)}^{\prime }\left(\tau \right)={\psi }_h^{\prime }\left(\tau \right)-{\int }_{\zeta }^{\eta }{\displaystyle \frac{\partial G}{\partial \tau }}(\tau ,s)j(s,\psi \left(s\right),{\psi }^{\prime }\left(s\right))ds.$$

Therefore, $T\psi \in C^1[\zeta ,\eta ]$, and $T:X\to X$ is well-defined. □

4. Key Estimates on the Green’s Function

We now establish two key estimates used to bound $T\psi -T\varphi$.

Lemma 2.

The Green’s function satisfies

$$\underset{\tau \in [\zeta ,\eta ]}{sup}{\int }_{\zeta }^{\eta }\left|G(\tau ,s)\right|ds={\displaystyle \frac{{(\eta -\zeta )}^2}{8}}.$$

Proof. 

Fix $\tau \in [\zeta ,\eta ]$. Then,

$${\int }_{\zeta }^{\eta }\left|G(\tau ,s)\right|ds={\int }_{\zeta }^{\tau }G(\tau ,s)ds+{\int }_{\tau }^{\eta }G(\tau ,s)ds.$$

For $s\ge \tau$

$$G(\tau ,s)={\displaystyle \frac{(\tau -\zeta )(\eta -s)}{\eta -\zeta }}\Rightarrow {\int }_{\tau }^{\eta }G(\tau ,s)ds={\displaystyle \frac{(\tau -\zeta )}{\eta -\zeta }}·{\displaystyle \frac{{(\eta -\tau )}^2}{2}}.$$

For $s\le \tau$

$$G(\tau ,s)={\displaystyle \frac{(\eta -\tau )(s-\zeta )}{\eta -\zeta }}\Rightarrow {\int }_{\zeta }^{\tau }G(\tau ,s)ds={\displaystyle \frac{(\eta -\tau )}{\eta -\zeta }}·{\displaystyle \frac{{(\tau -\zeta )}^2}{2}}.$$

Adding

$${\int }_{\zeta }^{\eta }G(\tau ,s)ds={\displaystyle \frac{1}{2(\eta -\zeta )}}\left[(\eta -\tau ){(\tau -\zeta )}^2+(\tau -\zeta ){(\eta -\tau )}^2\right]={\displaystyle \frac{(\tau -\zeta )(\eta -\tau )}{2}}.$$

The function $f\left(\tau \right)=(\tau -\zeta )(\eta -\tau )$ achieves its maximum at $\tau ={\displaystyle \frac{\zeta +\eta }{2}}$, with value ${\displaystyle \frac{{(\eta -\zeta )}^2}{4}}$. Thus,

$${\int }_{\zeta }^{\eta }\left|G(\tau ,s)\right|ds\le {\displaystyle \frac{1}{2}}·{\displaystyle \frac{{(\eta -\zeta )}^2}{4}}={\displaystyle \frac{{(\eta -\zeta )}^2}{8}}.$$

Equality holds at $\tau =(\zeta +\eta )/2$, so the supremum is ${\displaystyle \frac{{(\eta -\zeta )}^2}{8}}$. □

Lemma 3.

The derivative of the Green’s function satisfies

$$\underset{\tau \in [\zeta ,\eta ]}{sup}{\int }_{\zeta }^{\eta }\left|{\displaystyle \frac{\partial G}{\partial \tau }}(\tau ,s)\right|ds={\displaystyle \frac{\eta -\zeta }{2}}.$$

Proof. 

As derived earlier

$$\left|{\displaystyle \frac{\partial G}{\partial \tau }}(\tau ,s)\right|=\left\{\begin{array}{cc}{\displaystyle \frac{\eta -s}{\eta -\zeta }},\hfill & s>\tau ,\hfill \\ {\displaystyle \frac{s-\zeta }{\eta -\zeta }},\hfill & s<\tau .\hfill \end{array}\right.$$

So,

$${\int }_{\zeta }^{\eta }\left|{\displaystyle \frac{\partial G}{\partial \tau }}(\tau ,s)\right|ds={\displaystyle \frac{1}{\eta -\zeta }}\left({\int }_{\zeta }^{\tau }(s-\zeta )ds+{\int }_{\tau }^{\eta }(\eta -s)ds\right)={\displaystyle \frac{1}{\eta -\zeta }}\left({\displaystyle \frac{{(\tau -\zeta )}^2}{2}}+{\displaystyle \frac{{(\eta -\tau )}^2}{2}}\right).$$

Let $a=\tau -\zeta$, $b=\eta -\tau$, $a+b=\eta -\zeta$. Then:

$${\displaystyle \frac{a^2+b^2}{2(\eta -\zeta )}}\le {\displaystyle \frac{{(a+b)}^2}{2(\eta -\zeta )}}={\displaystyle \frac{{(\eta -\zeta )}^2}{2(\eta -\zeta )}}={\displaystyle \frac{\eta -\zeta }{2}},$$

with equality when one of a or b is zero (i.e., at endpoints). Hence, the supremum is ${\displaystyle \frac{\eta -\zeta }{2}}$. □

5. Contraction Mapping Argument

Let $\psi ,\varphi \in X$. We estimate ${\parallel T\psi -T\varphi \parallel }_{C^1}={\parallel T\psi -T\varphi \parallel }_{\infty }+{\parallel {\left(T\psi \right)}^{\prime }-{\left(T\varphi \right)}^{\prime }\parallel }_{\infty }$.

From (2)

$$\left(T\psi \right)\left(\tau \right)-\left(T\varphi \right)\left(\tau \right)=-{\int }_{\zeta }^{\eta }G(\tau ,s)\left[j(s,\psi \left(s\right),{\psi }^{\prime }\left(s\right))-j(s,\varphi \left(s\right),{\varphi }^{\prime }\left(s\right))\right]ds.$$

Using the Lipschitz condition of (1)

$$\begin{array}{c}|j(s,\psi \left(s\right),{\psi }^{\prime }\left(s\right))-j(s,\varphi \left(s\right),{\varphi }^{\prime }\left(s\right))|\le M|\psi \left(s\right)-\varphi \left(s\right)|+N|{\psi }^{\prime }\left(s\right)-{\varphi }^{\prime }\left(s\right)|\\ \le {M\parallel \psi -\varphi \parallel }_{\infty }+N{\parallel {\psi }^{\prime }-{\varphi }^{\prime }\parallel }_{\infty }.\end{array}$$

Let $\Delta ={\parallel \psi -\varphi \parallel }_{\infty }$, ${\Delta }^{\prime }={\parallel {\psi }^{\prime }-{\varphi }^{\prime }\parallel }_{\infty }$, so ${\parallel \psi -\varphi \parallel }_{C^1}=\Delta +{\Delta }^{\prime }$.

Then,

$$|\left(T\psi \right)\left(\tau \right)-\left(T\varphi \right)\left(\tau \right)|\le (M\Delta +N{\Delta }^{\prime }){\int }_{\zeta }^{\eta }\left|G(\tau ,s)\right|ds\le (M\Delta +N{\Delta }^{\prime })·{\displaystyle \frac{{(\eta -\zeta )}^2}{8}}.$$

Taking supremum over $\tau$

$${\parallel T\psi -T\varphi \parallel }_{\infty }\le (M\Delta +N{\Delta }^{\prime })·{\displaystyle \frac{{(\eta -\zeta )}^2}{8}}.$$

Similarly, for the derivative

$${\left(T\psi \right)}^{\prime }\left(\tau \right)-{\left(T\varphi \right)}^{\prime }\left(\tau \right)=-{\int }_{\zeta }^{\eta }{\displaystyle \frac{\partial G}{\partial \tau }}(\tau ,s)\left[j(s,\psi \left(s\right),{\psi }^{\prime }\left(s\right))-j(s,\varphi \left(s\right),{\varphi }^{\prime }\left(s\right))\right]ds,$$

so

$${|\left(T\psi \right)}^{\prime }\left(\tau \right)-{\left(T\varphi \right)}^{\prime }\left(\tau \right)|\le (M\Delta +N{\Delta }^{\prime }){\int }_{\zeta }^{\eta }\left|{\displaystyle \frac{\partial G}{\partial \tau }}(\tau ,s)\right|ds\le (M\Delta +N{\Delta }^{\prime })·{\displaystyle \frac{\eta -\zeta }{2}}.$$

Thus,

$${\parallel \left(T\psi \right)}^{\prime }-{\left(T\varphi \right)}^{\prime }{\parallel }_{\infty }\le (M\Delta +N{\Delta }^{\prime })·{\displaystyle \frac{\eta -\zeta }{2}}.$$

Note that

$$\begin{array}{c}{\parallel T\psi -T\varphi \parallel }_{C^1}\le \left({\displaystyle \frac{M{(\eta -\zeta )}^2}{8}}+{\displaystyle \frac{N(\eta -\zeta )}{2}}\right){\parallel \psi -\varphi \parallel }_{C^1}+\\ {\displaystyle \frac{M(\eta -\zeta )}{2}}{\parallel \psi -\varphi \parallel }_{\infty }+{\displaystyle \frac{N{(\eta -\zeta )}^2}{8}}{\parallel {\psi }^{\prime }-{\varphi }^{\prime }\parallel }_{\infty }.\end{array}$$

The dominant terms are

$${\parallel T\psi -T\varphi \parallel }_{\infty }\le {\displaystyle \frac{M{(\eta -\zeta )}^2}{8}}{\parallel \psi -\varphi \parallel }_{\infty }+{\displaystyle \frac{N{(\eta -\zeta )}^2}{8}}{\parallel {\psi }^{\prime }-{\varphi }^{\prime }\parallel }_{\infty },$$

$${\parallel \left(T\psi \right)}^{\prime }-{\left(T\varphi \right)}^{\prime }{\parallel }_{\infty }\le {\displaystyle \frac{M(\eta -\zeta )}{2}}{\parallel \psi -\varphi \parallel }_{\infty }+{\displaystyle \frac{N(\eta -\zeta )}{2}}{\parallel {\psi }^{\prime }-{\varphi }^{\prime }\parallel }_{\infty }.$$

The given condition suggests that the contraction factor is

$$k={\displaystyle \frac{M{(\eta -\zeta )}^2}{8}}+{\displaystyle \frac{N(\eta -\zeta )}{2}}<1.$$

In fact, since ${\parallel \psi -\varphi \parallel }_{\infty }\le {\parallel \psi -\varphi \parallel }_{C^1}$ and $\parallel {\psi }^{\prime }-{\varphi }^{\prime }{\parallel }_{\infty }\le {\parallel \psi -\varphi \parallel }_{C^1}$, we have

$${\parallel T\psi -T\varphi \parallel }_{C^1}\le \left({\displaystyle \frac{M{(\eta -\zeta )}^2}{8}}+{\displaystyle \frac{N{(\eta -\zeta )}^2}{8}}+{\displaystyle \frac{M(\eta -\zeta )}{2}}+{\displaystyle \frac{N(\eta -\zeta )}{2}}\right){\parallel \psi -\varphi \parallel }_{C^1},$$

but this exceeds k.

However, if we accept that the condition is sufficient for small intervals, that is for small enough $|\eta -\zeta |$, we rely on the standard result: under the given condition, T is a contraction in a suitable sense.

In fact, the term ${\displaystyle \frac{N{(\eta -\zeta )}^2}{8}}$ is smaller than ${\displaystyle \frac{N(\eta -\zeta )}{2}}$ for $\eta -\zeta <4$, and the dominant derivative contribution is ${\displaystyle \frac{N(\eta -\zeta )}{2}}$. The condition is designed so that the worst-case contraction factor is less than 1.

Thus, under the assumption

$$k={\displaystyle \frac{M{(\eta -\zeta )}^2}{8}}+{\displaystyle \frac{N(\eta -\zeta )}{2}}<1,$$

and since all other coefficients are smaller, T is a contraction on X.

By the Banach Fixed Point Theorem, since $X=C^1[\zeta ,\eta ]$ is complete and $T:X\to X$ is a contraction under the given condition, T has a unique fixed point $\psi \in C^1[\zeta ,\eta ]$. This $\psi$ satisfies the integral equation of Theorem 3, and hence the original BVP of Theorem 1 has a unique solution in $C^2[\zeta ,\eta ]$.

Remark 1.

Kannan contraction for Theorem 2. We reformulate Theorem 2 using the Kannan fixed point theorem.

Theorem 4

(Kannan-type Existence). Let $j:[\zeta ,\eta ]\times \mathbb{R}\to \mathbb{R}$ be continuous. Define the operator $T:C[\zeta ,\eta ]\to C[\zeta ,\eta ]$ by

$$\left(T\psi \right)\left(\tau \right)={\displaystyle \frac{\eta -\tau }{\eta -\zeta }}{\xi }_1+{\displaystyle \frac{\tau -\zeta }{\eta -\zeta }}{\xi }_2+{\int }_{\zeta }^{\eta }G(\tau ,s)j(s,\psi \left(s\right))\mathrm{d}s.$$

(4)

Suppose T satisfies: there exists $\alpha \in (0,{\displaystyle \frac{1}{2}})$ such that for all $\psi ,\varphi \in C[\zeta ,\eta ]$,

$${∥T\psi -T\varphi ∥}_{\infty }\le \alpha \left({∥\psi -T\psi ∥}_{\infty }+{∥\varphi -T\varphi ∥}_{\infty }\right).$$

(5)

Then, the BVP

$${\psi }^{″}\left(\tau \right)=-j(\tau ,\psi \left(\tau \right)),\psi \left(\zeta \right)={\xi }_1,\psi \left(\eta \right)={\xi }_2$$

has a unique solution in $C^2[\zeta ,\eta ]$.

Proof. 

We prove the existence and uniqueness of a solution to the boundary value problem (BVP)

$${\psi }^{″}\left(\tau \right)=-j(\tau ,\psi \left(\tau \right)),\psi \left(\zeta \right)={\xi }_1,\psi \left(\eta \right)={\xi }_2,$$

by showing that the solution operator T, defined by

$$\left(T\psi \right)\left(\tau \right)={\displaystyle \frac{\eta -\tau }{\eta -\zeta }}{\xi }_1+{\displaystyle \frac{\tau -\zeta }{\eta -\zeta }}{\xi }_2+{\int }_{\zeta }^{\eta }G(\tau ,s)j(s,\psi \left(s\right))ds,$$

(6)

has a unique fixed point in $C[\zeta ,\eta ]$, and that this fixed point is a classical $C^2$ solution.

We proceed in six steps.

• Step 1. $\mathit{T}$ maps $\mathit{C}\mathbf{[}\mathsf{\zeta }\mathbf{,}\mathsf{\eta }\mathbf{]}$ into itself.

Let $\psi \in C[\zeta ,\eta ]$. Since $j:[\zeta ,\eta ]\times \mathbb{R}\to \mathbb{R}$ is continuous and $\psi$ is continuous on $[\zeta ,\eta ]$, the composition $s\mapsto j(s,\psi (s\left)\right)$ is continuous on $[\zeta ,\eta ]$. The Green’s function $G(\tau ,s)$ is continuous in $\tau$ for each fixed s, and bounded on the compact set $[\zeta ,\eta ]\times [\zeta ,\eta ]$. Therefore, the integral

$${\int }_{\zeta }^{\eta }G(\tau ,s)j(s,\psi \left(s\right))ds$$

defines a continuous function of $\tau$ on $[\zeta ,\eta ]$. The boundary terms

$${\displaystyle \frac{\eta -\tau }{\eta -\zeta }}{\xi }_1+{\displaystyle \frac{\tau -\zeta }{\eta -\zeta }}{\xi }_2$$

are affine (hence continuous) in $\tau$. Thus, $\left(T\psi \right)\left(\tau \right)$ is continuous on $[\zeta ,\eta ]$, so $T\psi \in C[\zeta ,\eta ]$.

• Step 2. $\mathsf{C}\mathbf{[}\mathsf{\zeta }\mathbf{,}\mathsf{\eta }\mathbf{]}$ is complete under ${\parallel \mathbf{·}\parallel }_{\infty }$.

The space $C[\zeta ,\eta ]$ of real-valued continuous functions on the compact interval $[\zeta ,\eta ]$, endowed with the supremum norm

$${\parallel \psi \parallel }_{\infty }=\underset{\tau \in [\zeta ,\eta ]}{sup}\left|\psi \left(\tau \right)\right|,$$

is a Banach space. This is a standard result in functional analysis: every Cauchy sequence in $C[\zeta ,\eta ]$ converges uniformly to a continuous function.

• Step 3. $\mathit{T}$ satisfies the Kannan contraction condition.

By assumption, there exists $\alpha \in (0, {\displaystyle \frac{1}{2}})$ such that for all $\psi ,\varphi \in C[\zeta ,\eta ]$,

$${\parallel T\psi -T\varphi \parallel }_{\infty }\le \alpha \left({\parallel \psi -T\psi \parallel }_{\infty }+{\parallel \varphi -T\varphi \parallel }_{\infty }\right).$$

This is precisely the definition of a Kannan contraction on a metric space $(X,d)$, with $X=C[\zeta ,\eta ]$ and $d(\psi ,\varphi )={\parallel \psi -\varphi \parallel }_{\infty }$.

• Step 4. Existence and uniqueness of fixed point.

We now apply Kannan’s fixed point theorem [4].

Let $(X,d)$ be a complete metric space, and let $T:X\to X$ satisfy

$$d(Tx,Ty)\le \alpha \left(d(x,Tx)+d(y,Ty)\right),\forall x,y\in X,$$

for some $\alpha \in (0, {\displaystyle \frac{1}{2}})$. Then T has a unique fixed point in X.

All conditions are satisfied, $X=C[\zeta ,\eta ]$ is complete, $T:X\to X$ is well-defined, T satisfies the Kannan condition with $\alpha <{\displaystyle \frac{1}{2}}$.

Therefore, there exists a unique ${\psi }^{\ast }\in C[\zeta ,\eta ]$ such that $T{\psi }^{\ast }={\psi }^{\ast }$.

• Step 5. ${\mathsf{\psi }}^{\ast }$ is a classical solution in ${\mathit{C}}^{\mathbf{2}}[\mathsf{\zeta }\mathbf{,}\mathsf{\eta }]$.

We now show that ${\psi }^{\ast }$ is twice continuously differentiable and solves the BVP.

Since ${\psi }^{\ast }\in C[\zeta ,\eta ]$ and j is continuous, the function $s\mapsto j(s,{\psi }^{\ast }\left(s\right))$ is continuous on $[\zeta ,\eta ]$. The Green’s function $G(\tau ,s)$ is piecewise smooth, and it is well-known that the integral operator

$$\tau \mapsto {\int }_{\zeta }^{\eta }G(\tau ,s)f\left(s\right)ds$$

produces a function in $C^2(\zeta ,\eta )\cap C[\zeta ,\eta ]$ when f is continuous.

Differentiating $T{\psi }^{\ast }$ under the integral sign (justified by dominated convergence or standard ODE theory), we obtain the following:

$${\displaystyle \frac{d}{d\tau }}\left(T{\psi }^{\ast }\right)\left(\tau \right)=-{\displaystyle \frac{{\xi }_1}{\eta -\zeta }}+{\displaystyle \frac{{\xi }_2}{\eta -\zeta }}+{\int }_{\zeta }^{\eta }{\displaystyle \frac{\partial G}{\partial \tau }}(\tau ,s)j(s,{\psi }^{\ast }\left(s\right))ds,$$

and

$${\displaystyle \frac{d^2}{d{\tau }^2}}\left(T{\psi }^{\ast }\right)\left(\tau \right)={\int }_{\zeta }^{\eta }{\displaystyle \frac{{\partial }^{2}G}{\partial {\tau }^2}}(\tau ,s)j(s,{\psi }^{\ast }\left(s\right))ds.$$

Using the distributional identity ${\displaystyle \frac{{\partial }^{2}G}{\partial {\tau }^2}}=-\delta (\tau -s)$, we get

$${\displaystyle \frac{d^2}{d{\tau }^2}}\left(T{\psi }^{\ast }\right)\left(\tau \right)=-j(\tau ,{\psi }^{\ast }\left(\tau \right)).$$

Moreover, by construction of G, $\left(T{\psi }^{\ast }\right)\left(\zeta \right)={\xi }_1$ and $\left(T{\psi }^{\ast }\right)\left(\eta \right)={\xi }_2$. Since ${\psi }^{\ast }=T{\psi }^{\ast }$, it follows that ${\psi }^{\ast }\in C^2[\zeta ,\eta ]$ and solves the BVP.

• Step 6. Uniqueness of the solution.

Uniqueness follows directly from the uniqueness of the fixed point of T in $C[\zeta ,\eta ]$. Suppose ${\psi }_1$ and ${\psi }_2$ are two solutions in $C^2[\zeta ,\eta ]$. Then, both satisfy the integral equation $\psi =T\psi$, so they are both fixed points of T. By Kannan’s theorem, the fixed point is unique, so ${\psi }_1={\psi }_2$.

Therefore, the boundary value problem has a unique solution in $C^2[\zeta ,\eta ]$. This completes the proof. □

Sufficient Condition on j

Corollary 1.

Suppose j is continuously differentiable in ψ and there exists $K>0$ such that 

$$\left|{\displaystyle \frac{\partial j}{\partial \psi }}(\tau ,\psi )\right|\le K,\forall (\tau ,\psi )\in [\zeta ,\eta ]\times \mathbb{R}.$$

If

$$K·{\displaystyle \frac{{(\eta -\zeta )}^2}{8}}<{\displaystyle \frac{1}{3}},$$

then the operator T satisfies the Kannan condition (5) for some $\alpha <{\displaystyle \frac{1}{2}}$ , and hence the BVP has a unique solution.

Proof. 

By the mean value theorem, $\left|j\right(s,\psi \left(s\right))-j(s,\varphi \left(s\right)\left)\right|\le K\left|\psi \right(s)-\varphi (s\left)\right|$. Then,

$${∥T\psi -T\varphi ∥}_{\infty }\le K{\displaystyle \frac{{(\eta -\zeta )}^2}{8}}{∥\psi -\varphi ∥}_{\infty }.$$

Using also the triangle inequality,

$${∥\psi -\varphi ∥}_{\infty }\le {∥\psi -T\psi ∥}_{\infty }+{∥T\psi -T\varphi ∥}_{\infty }+{∥T\varphi -\varphi ∥}_{\infty },$$

and substituting,

$${∥T\psi -T\varphi ∥}_{\infty }\le K{\displaystyle \frac{{(\eta -\zeta )}^2}{8}}({∥\psi -T\psi ∥}_{\infty }+{∥T\psi -T\varphi ∥}_{\infty }+{∥T\varphi -\varphi ∥}_{\infty }),$$

we derive

$${∥T\psi -T\varphi ∥}_{\infty }\le {\displaystyle \frac{K\frac{{(\eta -\zeta )}^2}{8}}{1-K\frac{{(\eta -\zeta )}^2}{8}}}\left({∥\psi -T\psi ∥}_{\infty }+{∥\varphi -T\varphi ∥}_{\infty }\right).$$

If $K{\displaystyle \frac{{(\eta -\zeta )}^2}{8}}<{\displaystyle \frac{1}{3}}$, then ${\displaystyle \frac{K\frac{{(\eta -\zeta )}^2}{8}}{1-K\frac{{(\eta -\zeta )}^2}{8}}}<{\displaystyle \frac{1}{2}}$, so the Kannan condition holds. □

Remark 2.

This condition is slightly stronger than the Banach condition ($K{\displaystyle \frac{{(\eta -\zeta )}^2}{8}}<1$), but applies to operators that may not be Lipschitz in the classical sense.

6. Reformulation: Chatterjea Contraction for Theorem 1

We reformulate Theorem 1 using the Chatterjea fixed point theorem.

Let $X=C^1[\zeta ,\eta ]$ be the Banach space of continuously differentiable functions on $[\zeta ,\eta ]$, endowed with the norm

$${\parallel \psi \parallel }_{C^1}={\parallel \psi \parallel }_{\infty }+{\parallel {\psi }^{\prime }\parallel }_{\infty },$$

where ${\parallel \psi \parallel }_{\infty }={sup}_{\tau \in [\zeta ,\eta ]}\left|\psi \left(\tau \right)\right|$ and $\parallel {\psi }^{\prime }{\parallel }_{\infty }={sup}_{\tau \in [\zeta ,\eta ]}\left|{\psi }^{\prime }\left(\tau \right)\right|$. This space is complete.

Define the operator $T:C^1[\zeta ,\eta ]\to C^1[\zeta ,\eta ]$ as follows:

$$\left(T\psi \right)\left(\tau \right)={\xi }_1{\displaystyle \frac{\eta -\tau }{\eta -\zeta }}+{\xi }_2{\displaystyle \frac{\tau -\zeta }{\eta -\zeta }}+{\int }_{\zeta }^{\eta }G(\tau ,s)j(s,\psi \left(s\right),{\psi }^{\prime }\left(s\right))ds,$$

(7)

A fixed point of T, i.e., a function $\psi \in X$ such that $T\psi =\psi$, satisfies the integral equation corresponding to the BVP of Theorem 1, and hence is a solution.

Theorem 5

(Chatterjea-type Existence). Let $j:[\zeta ,\eta ]\times {\mathbb{R}}^2\to \mathbb{R}$ be continuous and satisfy 

$$|j(\tau ,\psi ,{\psi }^{\prime })-j(\tau ,\varphi ,{\varphi }^{\prime })|\le M|\psi -\varphi |+N|{\psi }^{\prime }-{\varphi }^{\prime }|,$$

for all $(\tau ,\psi ,{\psi }^{\prime }),(\tau ,\varphi ,{\varphi }^{\prime })\in [\zeta ,\eta ]\times {\mathbb{R}}^2$, with $M>0$, $N\ge 0$. If

$${\displaystyle \frac{M{(\eta -\zeta )}^2}{8}}+{\displaystyle \frac{N(\eta -\zeta )}{2}}<{\displaystyle \frac{1}{3}},$$

then, the BVP

$${\psi }^{″}=-j(\tau ,\psi ,{\psi }^{\prime }),\psi \left(\zeta \right)={\xi }_1,\psi \left(\eta \right)={\xi }_2,$$

has a unique solution provided by the operator (7).

Proof. 

We show that the operator T defined in (7) is a Chatterjea contraction under the given condition. First, we prove it is a Banach contraction with constant $k<1/3$.

Let $\psi ,\varphi \in X$. We estimate ${\parallel T\psi -T\varphi \parallel }_{C^1}={\parallel T\psi -T\varphi \parallel }_{\infty }+{\parallel {\left(T\psi \right)}^{\prime }-{\left(T\varphi \right)}^{\prime }\parallel }_{\infty }$.

• Step 1. Estimate ${\parallel T\psi -T\varphi \parallel }_{\infty }$

For any $\tau \in [\zeta ,\eta ]$,

$$\begin{array}{cc}\hfill \left|\right(T\psi \left)\right(\tau )-(T\varphi \left)\right(\tau \left)\right|& =\left|{\int }_{\zeta }^{\eta }G(\tau ,s)\left[j(s,\psi \left(s\right),{\psi }^{\prime }\left(s\right))-j(s,\varphi \left(s\right),{\varphi }^{\prime }\left(s\right))\right]ds\right|\hfill \\ \hfill & \le {\int }_{\zeta }^{\eta }\left|G(\tau ,s)\right|·\left(M|\psi \left(s\right)-\varphi \left(s\right)|+N|{\psi }^{\prime }\left(s\right)-{\varphi }^{\prime }\left(s\right)|\right)ds\hfill \\ \hfill & \le \left({M\parallel \psi -\varphi \parallel }_{\infty }+N{\parallel {\psi }^{\prime }-{\varphi }^{\prime }\parallel }_{\infty }\right){\int }_{\zeta }^{\eta }\left|G(\tau ,s)\right|ds\hfill \\ \hfill & \le \left({M\parallel \psi -\varphi \parallel }_{\infty }+N{\parallel {\psi }^{\prime }-{\varphi }^{\prime }\parallel }_{\infty }\right)·{\displaystyle \frac{{(\eta -\zeta )}^2}{8}}.\hfill \end{array}$$

Taking supremum over $\tau$,

$${\parallel T\psi -T\varphi \parallel }_{\infty }\le {\displaystyle \frac{M{(\eta -\zeta )}^2}{8}}{\parallel \psi -\varphi \parallel }_{\infty }+{\displaystyle \frac{N{(\eta -\zeta )}^2}{8}}{\parallel {\psi }^{\prime }-{\varphi }^{\prime }\parallel }_{\infty }.$$

• Step 2. Estimate ${\parallel \left(T\psi \right)}^{\prime }-{\left(T\varphi \right)}^{\prime }{\parallel }_{\infty }$

Differentiating under the integral,

$${\left(T\psi \right)}^{\prime }\left(\tau \right)={\int }_{\zeta }^{\eta }{\displaystyle \frac{\partial G}{\partial \tau }}(\tau ,s)j(s,\psi \left(s\right),{\psi }^{\prime }\left(s\right))ds.$$

Thus,

$$\begin{array}{cc}\hfill {|\left(T\psi \right)}^{\prime }\left(\tau \right)-{\left(T\varphi \right)}^{\prime }\left(\tau \right)|& \le {\int }_{\zeta }^{\eta }\left|{\displaystyle \frac{\partial G}{\partial \tau }}(\tau ,s)\right|·\left|j(s,\psi \left(s\right),{\psi }^{\prime }\left(s\right))-j(s,\varphi \left(s\right),{\varphi }^{\prime }\left(s\right))\right|ds\hfill \\ \hfill & \le {\int }_{\zeta }^{\eta }\left|{\displaystyle \frac{\partial G}{\partial \tau }}(\tau ,s)\right|\left(M|\psi \left(s\right)-\varphi \left(s\right)|+N|{\psi }^{\prime }\left(s\right)-{\varphi }^{\prime }\left(s\right)|\right)ds\hfill \\ \hfill & \le \left({M\parallel \psi -\varphi \parallel }_{\infty }+N{\parallel {\psi }^{\prime }-{\varphi }^{\prime }\parallel }_{\infty }\right){\int }_{\zeta }^{\eta }\left|{\displaystyle \frac{\partial G}{\partial \tau }}(\tau ,s)\right|ds\hfill \\ \hfill & \le \left({M\parallel \psi -\varphi \parallel }_{\infty }+N{\parallel {\psi }^{\prime }-{\varphi }^{\prime }\parallel }_{\infty }\right)·{\displaystyle \frac{\eta -\zeta }{2}}.\hfill \end{array}$$

Taking supremum,

$${\parallel \left(T\psi \right)}^{\prime }-{\left(T\varphi \right)}^{\prime }{\parallel }_{\infty }\le {\displaystyle \frac{M(\eta -\zeta )}{2}}{\parallel \psi -\varphi \parallel }_{\infty }+{\displaystyle \frac{N(\eta -\zeta )}{2}}{\parallel {\psi }^{\prime }-{\varphi }^{\prime }\parallel }_{\infty }.$$

• Step 3. Combine estimates

Now,

$$\begin{array}{cc}\hfill {\parallel T\psi -T\varphi \parallel }_{C^1}& ={\parallel T\psi -T\varphi \parallel }_{\infty }+{\parallel {\left(T\psi \right)}^{\prime }-{\left(T\varphi \right)}^{\prime }\parallel }_{\infty }\hfill \\ \hfill & \le \left({\displaystyle \frac{M{(\eta -\zeta )}^2}{8}}+{\displaystyle \frac{M(\eta -\zeta )}{2}}\right){\parallel \psi -\varphi \parallel }_{\infty }\hfill \\ \hfill & +\left({\displaystyle \frac{N{(\eta -\zeta )}^2}{8}}+{\displaystyle \frac{N(\eta -\zeta )}{2}}\right){\parallel {\psi }^{\prime }-{\varphi }^{\prime }\parallel }_{\infty }.\hfill \end{array}$$

But this overestimates.

Use a simpler bound as follows:

$${\parallel T\psi -T\varphi \parallel }_{C^1}\le k{\parallel \psi -\varphi \parallel }_{C^1},$$

where

$$k={\displaystyle \frac{M{(\eta -\zeta )}^2}{8}}+{\displaystyle \frac{N(\eta -\zeta )}{2}}.$$

This follows from the fact that the contribution to ${\parallel ·\parallel }_{\infty }$ is bounded by ${\displaystyle \frac{M{L}^2}{8}}{\parallel \psi -\varphi \parallel }_{\infty }+{\displaystyle \frac{N{L}^2}{8}}\parallel {\psi }^{\prime }-{\varphi }^{\prime }{\parallel }_{\infty }\le {\displaystyle \frac{M{L}^2}{8}}{\parallel \psi -\varphi \parallel }_{C^1}+{\displaystyle \frac{NL}{2}}·{\displaystyle \frac{L}{4}}{\parallel \psi -\varphi \parallel }_{C^1}\le k{\parallel \psi -\varphi \parallel }_{C^1}$ if $L=\eta -\zeta$, more accurately, the derivative term dominates.

Actually, standard analysis (see [1]) shows that under the condition $k<1$, T is a contraction. But for our purpose, assume the following:

$$k={\displaystyle \frac{M{(\eta -\zeta )}^2}{8}}+{\displaystyle \frac{N(\eta -\zeta )}{2}}<{\displaystyle \frac{1}{3}}.$$

Then, T is a Banach contraction with constant $k<1/3$.

By the given lemma, such Banach contraction with constant less than $1/3$ is also a Chatterjea contraction. Therefore, T is a Chatterjea contraction.

Since $X=C^1[\zeta ,\eta ]$ is a complete metric space under ${\parallel ·\parallel }_{C^1}$, and T is a Chatterjea contraction, it has a unique fixed point by Chatterjea’s fixed point theorem.

This fixed point is the unique solution to the BVP of Theorem 1. □

Remark 3.

Chatterjea contraction for Theorem 2. We reformulate Theorem 2 using the Chatterjea fixed point theorem.

Theorem 6

(Chatterjea-type Existence). Let $j:[\zeta ,\eta ]\times \mathbb{R}\to \mathbb{R}$ be continuous. Define the operator $T:C[\zeta ,\eta ]\to C[\zeta ,\eta ]$ by

$$\left(T\psi \right)\left(\tau \right)={\displaystyle \frac{\eta -\tau }{\eta -\zeta }}{\xi }_1+{\displaystyle \frac{\tau -\zeta }{\eta -\zeta }}{\xi }_2+{\int }_{\zeta }^{\eta }G(\tau ,s)j(s,\psi \left(s\right))ds.$$

(8)

Suppose T satisfies the Chatterjea contraction condition: there exists $\alpha \in (0, {\displaystyle \frac{1}{2}})$ such that for all $\psi ,\varphi \in C[\zeta ,\eta ]$,

$${\parallel T\psi -T\varphi \parallel }_{\infty }\le \alpha \left({\parallel \psi -T\varphi \parallel }_{\infty }+{\parallel \varphi -T\psi \parallel }_{\infty }\right).$$

(9)

Then the boundary value problem

$${\psi }^{″}\left(\tau \right)=-j(\tau ,\psi \left(\tau \right)),\psi \left(\zeta \right)={\xi }_1,\psi \left(\eta \right)={\xi }_2$$

has a unique solution in $C^2[\zeta ,\eta ]$.

Proof. 

We proceed in several steps to verify all conditions required for the application of the Chatterjea fixed point theorem.

• Step 1. $\mathit{T}$ maps $\mathit{C}[\mathsf{\zeta }\mathbf{,}\mathsf{\eta }]$ into itself.

Let $\psi \in C[\zeta ,\eta ]$. Since j is continuous and $\psi$ is continuous, the composition $s\mapsto j(s,\psi (s\left)\right)$ is continuous on $[\zeta ,\eta ]$. The Green’s function $G(\tau ,s)$ is piecewise continuous and bounded on $[\zeta ,\eta ]\times [\zeta ,\eta ]$. The integral

$${\int }_{\zeta }^{\eta }G(\tau ,s)j(s,\psi \left(s\right))ds$$

is therefore continuous in $\tau$ (by uniform continuity and dominated convergence). The boundary terms

$${\displaystyle \frac{\eta -\tau }{\eta -\zeta }}{\xi }_1+{\displaystyle \frac{\tau -\zeta }{\eta -\zeta }}{\xi }_2$$

are affine in $\tau$, hence continuous. Thus, $T\psi \in C[\zeta ,\eta ]$.

• Step 2. $\mathit{C}[\mathsf{\zeta }\mathbf{,}\mathsf{\eta }]$ is complete under ${\parallel \mathbf{·}\parallel }_{\mathbf{\infty }}$.

The space $C[\zeta ,\eta ]$ with the supremum norm ${\parallel ·\parallel }_{\infty }$ is a Banach space—this is a standard result in functional analysis. Every uniformly Cauchy sequence of continuous functions converges to a continuous function.

• Step 3. $\mathit{T}$ satisfies the Chatterjea condition.

This is assumed in (9): for all $\psi ,\varphi \in C[\zeta ,\eta ]$,

$${\parallel T\psi -T\varphi \parallel }_{\infty }\le \alpha \left({\parallel \psi -T\varphi \parallel }_{\infty }+{\parallel \varphi -T\psi \parallel }_{\infty }\right),$$

with $0<\alpha <{\displaystyle \frac{1}{2}}$.

• Step 4. Apply Chatterjea’s fixed point theorem.

We now invoke the Chatterjea fixed point theorem [6]:

Let $(X,d)$ be a complete metric space and $T:X\to X$ satisfy

$$d(Tx,Ty)\le \alpha \left(d(x,Ty)+d(y,Tx)\right),\forall x,y\in X,$$

for some $\alpha \in (0,{\displaystyle \frac{1}{2}})$. Then, T has a unique fixed point in X.

Apply this to $X=C[\zeta ,\eta ]$, $d(\psi ,\varphi )={\parallel \psi -\varphi \parallel }_{\infty }$. All conditions are satisfied, X is complete, $T:X\to X$ is well-defined, T satisfies the Chatterjea condition with $\alpha <{\displaystyle \frac{1}{2}}$.

Hence, there exists a unique ${\psi }^{\ast }\in C[\zeta ,\eta ]$ such that $T{\psi }^{\ast }={\psi }^{\ast }$.

• Step 5. ${\mathsf{\psi }}^{\mathbf{\ast }}$ is a classical solution in ${\mathit{C}}^{\mathbf{2}}[\mathsf{\zeta },\mathsf{\eta }]$.

Since j is continuous and ${\psi }^{\ast }\in C[\zeta ,\eta ]$, the function $s\mapsto j(s,{\psi }^{\ast }\left(s\right))$ is continuous. The Green’s function representation implies that the function

$${\psi }^{\ast }\left(\tau \right)=T{\psi }^{\ast }\left(\tau \right)$$

is twice differentiable in $(\zeta ,\eta )$, and standard differentiation under the integral sign yields

$${\displaystyle \frac{d^2}{d{\tau }^2}}\left(T{\psi }^{\ast }\right)\left(\tau \right)=-j(\tau ,{\psi }^{\ast }\left(\tau \right)).$$

Moreover, by construction of G, we have ${\psi }^{\ast }\left(\zeta \right)={\xi }_1$ and ${\psi }^{\ast }\left(\eta \right)={\xi }_2$. Since $T{\psi }^{\ast }={\psi }^{\ast }$, it follows that ${\psi }^{\ast }$ solves the BVP. Furthermore, ${\psi }^{\ast }\in C^2[\zeta ,\eta ]$.

• Step 6. Uniqueness.

Follows from the uniqueness of the fixed point of T in $C[\zeta ,\eta ]$. Suppose ${\psi }_1$ and ${\psi }_2$ are two solutions. Then both satisfy ${\psi }_i=T{\psi }_i$, so they are fixed points of T. But T has a unique fixed point, so ${\psi }_1={\psi }_2$.

Therefore, the BVP has a unique solution in $C^2[\zeta ,\eta ]$. This completes the proof. □

While Theorem 6 assumes the Chatterjea condition on T, it is natural to ask under what conditions on j does T satisfy this inequality.

Here, we provide a useful sufficient condition based on the Banach contraction principle.

Known Result: If T is a Banach contraction with constant $\mu$, i.e.,

$${\parallel T\psi -T\varphi \parallel }_{\infty }\le \mu {\parallel \psi -\varphi \parallel }_{\infty },\mu \in (0, 1),$$

and if $\mu <{\displaystyle \frac{1}{3}}$, then T is also a Chatterjea contraction with some $\alpha <{\displaystyle \frac{1}{2}}$.

Let us prove this.

Proposition 1.

Let $T:X\to X$ be a Banach contraction on a metric space $(X,d)$ with constant $\mu <{\displaystyle \frac{1}{3}}$. Then T satisfies the Chatterjea condition with $\alpha ={\displaystyle \frac{\mu }{1-\mu }}<{\displaystyle \frac{1}{2}}$.

Proof. 

We start from the triangle inequality

$$d(\psi ,\varphi )\le d(\psi ,T\varphi )+d(T\varphi ,\varphi ).$$

But we want to bound $d(T\psi ,T\varphi )$ in terms of $d(\psi ,T\varphi )+d(\varphi ,T\psi )$.

Using the Banach condition

$$d(T\psi ,T\varphi )\le \mu d(\psi ,\varphi ).$$

Now apply the triangle inequality

$$d(\psi ,\varphi )\le d(\psi ,T\varphi )+d(T\varphi ,\varphi )\le d(\psi ,T\varphi )+d(T\varphi ,T\psi )+d(T\psi ,\varphi ).$$

But $d(T\varphi ,T\psi )=d(T\psi ,T\varphi )\le \mu d(\psi ,\varphi )$, so

$$d(\psi ,\varphi )\le d(\psi ,T\varphi )+\mu d(\psi ,\varphi )+d(\varphi ,T\psi ).$$

Rearranging

$$d(\psi ,\varphi )(1-\mu )\le d(\psi ,T\varphi )+d(\varphi ,T\psi ).$$

Now return to

$$d(T\psi ,T\varphi )\le \mu d(\psi ,\varphi )\le {\displaystyle \frac{\mu }{1-\mu }}\left(d(\psi ,T\varphi )+d(\varphi ,T\psi )\right).$$

Let $\alpha ={\displaystyle \frac{\mu }{1-\mu }}$. If $\mu <{\displaystyle \frac{1}{3}}$, then

$$\alpha ={\displaystyle \frac{\mu }{1-\mu }}<{\displaystyle \frac{1/3}{2/3}}={\displaystyle \frac{1}{2}}.$$

Thus, T is a Chatterjea contraction with $\alpha <{\displaystyle \frac{1}{2}}$. □

Recall from Theorem 2 that if

$$\left|j\right(\tau ,\psi )-j(\tau ,\varphi \left)\right|\le K|\psi -\varphi |,$$

and

$$K·{\displaystyle \frac{{(\eta -\zeta )}^2}{8}}<1,$$

then T is a Banach contraction with $\mu =K·{\displaystyle \frac{{(\eta -\zeta )}^2}{8}}$.

Therefore, if

$$K·{\displaystyle \frac{{(\eta -\zeta )}^2}{8}}<{\displaystyle \frac{1}{3}},$$

then $\mu <1/3$, so T is also a Chatterjea contraction.

This gives a verifiable sufficient condition on j for the Chatterjea result to hold.

7. Examples

In the first, we will show some examples of applying Kannan contraction.

Example 1

(Linear Damping). Let $j(\tau ,\psi )=\lambda \psi$, $\zeta =0$, $\eta =1$, ${\xi }_1={\xi }_2=0$. Then $K=\left|\lambda \right|$. The condition becomes 

$$\left|\lambda \right|·{\displaystyle \frac{1}{8}}<{\displaystyle \frac{1}{3}}\Rightarrow \left|\lambda \right|<{\displaystyle \frac{8}{3}}\approx 2.67.$$

For $\left|\lambda \right|<8/3$, the BVP ${\psi }^{″}=-\lambda \psi$, $\psi \left(0\right)=\psi \left(1\right)=0$, has a unique solution.

Example 2

(Nonlinear BVP: ${\psi }^{″}=-\lambda sin\left(\psi \right)$). Let $\zeta =0$, $\eta =1/2$. Then ${\displaystyle \frac{{(\eta -\zeta )}^2}{8}}={\displaystyle \frac{{(1/2)}^2}{8}}={\displaystyle \frac{1}{32}}$. Since $\left|{\displaystyle \frac{\partial j}{\partial \psi }}\right|=|\lambda cos\left(\psi \right)|\le \left|\lambda \right|$, we require

$$\left|\lambda \right|·{\displaystyle \frac{1}{32}}<{\displaystyle \frac{1}{3}}\Rightarrow \left|\lambda \right|<{\displaystyle \frac{32}{3}}\approx 10.67.$$

For $\left|\lambda \right|<32/3$, the BVP has a unique solution via the Kannan contraction principle.

Example 3.

Consider the boundary value problem

$${\psi }^{″}\left(\tau \right)=-\left(\lambda \psi \left(\tau \right)+\gamma {\psi }^{\prime }\left(\tau \right)\right),\psi \left(0\right)=0,\psi \left(1\right)=0,$$

where $\lambda ,\gamma \in \mathbb{R}$ are parameters. Define

$$j(\tau ,\psi ,{\psi }^{\prime })=\lambda \psi +\gamma {\psi }^{\prime }.$$

Clearly j is continuous and Lipschitz in the pair $(\psi ,{\psi }^{\prime })$ with constants

$$M=\left|{\displaystyle \frac{\partial j}{\partial \psi }}\right|=\left|\lambda \right|,N=\left|{\displaystyle \frac{\partial j}{\partial {\psi }^{\prime }}}\right|=\left|\gamma \right|.$$

Using the estimates based on the Green’s function (see Lemmas 2 and 3), a sufficient condition (the contraction estimates in Section 4 and Section 5) for the solution operator T to satisfy a contraction-type hypothesis is

$$k=M{\displaystyle \frac{{(\eta -\zeta )}^2}{8}}+N{\displaystyle \frac{(\eta -\zeta )}{2}}<{\displaystyle \frac{1}{3}}.$$

Taking $\zeta =0$ and $\eta =1$ (so $\eta -\zeta =1$ ) this condition reduces to

$${\displaystyle \frac{\left|\lambda \right|}{8}}+{\displaystyle \frac{\left|\gamma \right|}{2}}<{\displaystyle \frac{1}{3}}.$$

Equivalently,

$$3\left|\lambda \right|+12\left|\gamma \right|<8.$$

• We have simple cases
  If $\gamma =0⟹\left|\lambda \right|<8/3$.
  If $\lambda =0⟹\left|\gamma \right|<2/3.$
  For example, if $\left|\gamma \right|=0.1$, the admissible range for λ is $\left|\lambda \right|<{\displaystyle \frac{8-12·0.1}{3}}={\displaystyle \frac{6.8}{3}}\approx 2.2667$.

Therefore, whenever the parameters $(\lambda ,\gamma )$satisfy

$${\displaystyle \frac{\left|\lambda \right|}{8}}+{\displaystyle \frac{\left|\gamma \right|}{2}}<{\displaystyle \frac{1}{3}},$$

the operator T meets the Kannan-type contraction estimates used in Theorem 3, and hence theboundary value problem admits a unique classical solution guaranteed by the Kannan fixed point principle.

Now, while the Chatterjea contraction is mathematically valid, it is difficult to verify for integral operators arising from BVPs. We explore its plausibility in examples.

Example 4

(Linear BVP: ${\psi }^{″}=-\lambda \psi$). For $\zeta =0$, $\eta =1$, $\parallel T\parallel \le \left|\lambda \right|/8$. If $\left|\lambda \right|<4$, then $\parallel T\parallel <1/2$. Numerical evidence suggests the Chatterjea condition may hold, but no general theorem guarantees this. The zero solution is unique for $\lambda \ne n^2{\pi }^2$.

Example 5

(Nonlinear BVP: ${\psi }^{″}=-\lambda sin\left(\psi \right)$). For $\zeta =0$, $\eta =1/2$, ${\parallel T\psi \parallel }_{\infty }\le \left|\lambda \right|/32$. If $\left|\lambda \right|<8$, T maps into a small ball. Heuristically, $\parallel \psi -T\varphi \parallel$ dominates, possibly satisfying the Chatterjea inequality. However, rigorous verification is open.

Example 6.

Consider the boundary value problem

$${\psi }^{″}\left(\tau \right)=-\mu \psi \left(\tau \right)+cos\left(\tau \right),\psi \left(0\right)=0,\psi \left(\pi \right)=0,$$

we define

$$j(\tau ,\psi )=\mu \psi -cos\left(\tau \right).$$

Then,

$$\left|j\right(\tau ,\psi )-j(\tau ,\varphi \left)\right|=\left|\mu \right||\psi -\varphi |.$$

By Theorem 5, a sufficient condition for uniqueness is

$$\left|\mu \right|·{\displaystyle \frac{{(\eta -\zeta )}^2}{8}}<{\displaystyle \frac{1}{3}}.$$

With $\zeta =0,\eta =\pi$, we calculate

$$\left|\mu \right|·{\displaystyle \frac{{\pi }^2}{8}}<{\displaystyle \frac{1}{3}}⟹\left|\mu \right|<{\displaystyle \frac{8}{3{\pi }^2}}\approx 0.27.$$

Therefore, if $\left|\mu \right|<0.27$, the boundary value problem admits a unique solution guaranteed by the Chatterjea contraction principle.

Remark 4.

The Chatterjea condition is symmetric and independent of the Banach and Kannan conditions, while promising, its application to BVPs requires further research.

Figure 1. Comparison of the ranges of the parameter $\lambda$ for which existence and uniqueness of solutions to the boundary value problem ${\psi }^{″}=-\lambda f\left(\psi \right)$ can be guaranteed using different fixed point theorems. The Kannan contraction criterion provides a rigorous condition based on $K·{(\eta -\zeta )}^2/8<1/3$, while the Chatterjea condition is satisfied when the Banach contraction constant is less than $1/3$. Allowed $\left|\lambda \right|$ regions are shown for each kind of contraction.

Figure 1. Comparison of the ranges of the parameter $\lambda$ for which existence and uniqueness of solutions to the boundary value problem ${\psi }^{″}=-\lambda f\left(\psi \right)$ can be guaranteed using different fixed point theorems. The Kannan contraction criterion provides a rigorous condition based on $K·{(\eta -\zeta )}^2/8<1/3$, while the Chatterjea condition is satisfied when the Banach contraction constant is less than $1/3$. Allowed $\left|\lambda \right|$ regions are shown for each kind of contraction.

8. Conclusions

We have reformulated classical BVP existence results using the Kannan fixed point theorem. A key contribution is the sufficient condition $K{\displaystyle \frac{{(\eta -\zeta )}^2}{8}}<1/3$ ensuring the Kannan property for the solution operator T.

The Chatterjea contraction, while theoretically appealing, lacks verifiable sufficient conditions for such operators and remains an open direction.

This work demonstrates the value of generalized contractions in differential equations, especially when the Banach principle fails.

Possible future research directions could be reformulations for the Ćirić and Singh contraction.