A Fixed-Point Chatterjea–Singh Mapping Approach: Existence and Uniqueness of Solutions to Nonlinear BVPs

Abstract

This paper introduces an application of the Chatterjea–Singh fixed-point theorem to nonlinear boundary value problems (BVPs). We define a Chatterjea–Singh mapping as one whose iterate $T^p$ satisfies a Chatterjea-type contractive condition. Under this weaker assumption than classical Banach or Chatterjea contractions, we prove the existence and uniqueness of solutions to second-order BVPs. Our method applies even when T itself does not satisfy a contraction property. Examples illustrate how iteration can recover convergence where standard conditions fail. This work extends generalized fixed-point theory in differential equations and highlights the flexibility of delayed contraction criteria.

1. Introduction

The study of the existence and uniqueness of solutions to boundary value problems (BVPs) often relies on fixed-point theorems applied to integral operators defined via Green’s functions. Classical approaches employ the Banach contraction principle under global Lipschitz conditions [1,2]. However, when such strong conditions fail, generalized contractions—such as Kannan [3,4], Chatterjea [5], and their extensions—are valuable alternatives.

In their classical work, Singh [6,7] extended Kannan’s idea by requiring only that some iterate $T^p$ ($p\ge 1$) satisfies the contraction condition. This allows mappings that are not themselves contractive to still admit unique fixed points (see also [8]).

Building upon this, the present paper introduces and applies a class, Chatterjea–Singh mappings, where $T^p$ satisfies the Chatterjea-type inequality

$$d(T^{p}x,T^{p}y)\le \alpha \left[d(x,T^{p}y)+d(y,T^{p}x)\right],\alpha \in [0,\frac{1}{2}),$$

(1)

even if T does not. We apply this framework to reformulate classical existence results for second-order BVPs, providing greater flexibility than direct Chatterjea or Singh–Kannan conditions.

Specifically, we consider the Dirichlet BVP

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

(2)

and define the associated solution operator T using Green’s function. We show that if $T^p$ satisfies the Chatterjea–Singh condition, then the BVP has a unique solution in $C^2[\zeta ,\eta ]$, regardless of whether T is contractive.

This approach is particularly useful when the nonlinearity j grows moderately fast, making T non-contractive but allowing higher iterates $T^p$ to exhibit sufficient smoothing for convergence.

It is worth mentioning that the concept we refer to as the Chatterjea–Singh contraction is entirely new. The idea of considering iterates of a contraction to generate new classes of mappings appeared in the literature several decades ago.

In particular, Rhoades [8] further examined this relationship in the context of comparing various definitions of contractive mappings, introducing notation where corresponds to the Chatterjea contraction and corresponds to what can be interpreted as the Chatterjea–Singh contraction.

Subsequently, Fisher [9] proved that if T is a Banach contraction, then $T^p$ satisfies the Chatterjea-type contractive condition. This underlying mechanism connects Banach and Chatterjea contractions through iteration.

More recently, Cvetković [10] revisited the iterates of Chatterjea mappings, providing modern analytical perspectives and applications.

Therefore, in this paper, our goal is to extend the Chatterjea–Singh contraction and analyze its behavior in the framework of nonlinear boundary value problems using a fixed-point approach.

Section 2 recalls key definitions and establishes notations. Section 3 presents the main fixed-point theorem in the context of BVPs, with complete proof. Section 4 provides illustrative examples showing the advantage of the proposed method over classical methods. Concluding remarks are given in Section 5.

2. Preliminaries

Let $(\zeta ,\eta )$ be a finite interval. Denote as $C[\zeta ,\eta ]$ the space of continuous real-valued functions on $[\zeta ,\eta ]$, equipped with the supremum norm:

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

This is a Banach space. Similarly, let $C^1[\zeta ,\eta ]$ denote the space of continuously differentiable functions on $[\zeta ,\eta ]$, with norm

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

Green’s function for the Dirichlet problem on $[\zeta ,\eta ]$ is given by

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

(3)

It satisfies the following key estimates:

Lemma 1

([1]).

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

(4)

Lemma 2

([1]).

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

(5)

We now introduce the central definitions used throughout this paper.

Definition 1

(Banach Contraction [2]). 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 for all $x,y\in X$,

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

(6)

Definition 2

(Kannan Contraction [3,4]). 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 for all $x,y\in X$,

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

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Definition 3

(Chatterjea Contraction [5]). 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 for all $x,y\in X$,

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

(8)

Definition 4

(Singh Contraction [6,7]). T is a Singh contraction if there exists a positive integer p and a constant $a\in \left(0,{\displaystyle \frac{1}{2}}\right)$ such that for all $x,y\in X$

$$,d(T^{p}x,T^{p}y)\le a\left[d(x,T^{p}x)+d(y,T^{p}y)\right].$$

(9)

Definition 5

(Chatterjea–Singh Mapping [9,10]). Let $(X,d)$ be a metric space and $T:X\to X$. We say T is a Chatterjea–Singh mapping if there exists an integer $p\ge 1$ and a constant $\alpha \in [0,\frac{1}{2})$ such that for all $x,y\in X$,

$$d(T^{p}x,T^{p}y)\le \alpha \left[d(x,T^{p}y)+d(y,T^{p}x)\right].$$

(10)

When $p=1$, this reduces to the classical Chatterjea contraction [5]. This lies in allowing $p>1$, which significantly broadens the proposed method’s applicability.

Our goal is to apply this idea to the solution operator of a BVP, proving existence and uniqueness without assuming that T itself is contractive.

3. Main Result: Chatterjea–Singh Framework for BVPs

We begin with the vector case involving first derivatives.

3.1. Vector Case: First-Order Dependence

Theorem 1

(Proving Existence via Chatterjea–Singh Condition). Let $j:[\zeta ,\eta ]\times {\mathbb{R}}^2\to \mathbb{R}$ be continuous. Define the operator $T:C^1[\zeta ,\eta ]\to C^1[\zeta ,\eta ]$ as

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

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Suppose there exist $p\in \mathbb{N}$ and $\alpha \in [0,\frac{1}{2})$ such that for all $\psi ,\varphi \in C^1[\zeta ,\eta ]$,

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

(12)

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 $C^2[\zeta ,\eta ]$.

Proof. 

Let $S=T^p$. Then $S:C^1[\zeta ,\eta ]\to C^1[\zeta ,\eta ]$ satisfies

$${\parallel S\psi -S\varphi \parallel }_{C^1}\le \alpha \left({\parallel \psi -S\varphi \parallel }_{C^1}+{\parallel \varphi -S\psi \parallel }_{C^1}\right),\forall \psi ,\varphi \in C^1[\zeta ,\eta ].$$

We proceed with several steps.

Step 1. 

T maps $C^1[\zeta ,\eta ]$ into itself.

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. Green’s function and its partial derivative $\partial G/\partial \tau$ are bounded piecewise smooth functions. Hence, $T\psi$ and ${\left(T\psi \right)}^{\prime }$ are continuous, so $T\psi \in C^1[\zeta ,\eta ]$. Through induction, $S\psi =T^p\psi \in C^1[\zeta ,\eta ]$.

Step 2. 

$C^1[\zeta ,\eta ]$ is complete under ${\parallel ·\parallel }_{C^1}$.

Standard result: $C^1[\zeta ,\eta ]$ with norm ${\parallel \psi \parallel }_{C^1}={\parallel \psi \parallel }_{\infty }+{\parallel {\psi }^{\prime }\parallel }_{\infty }$ is a Banach space.

Step 3. 

Construct a Cauchy sequence.

Fix arbitrary ${\psi }_0\in C^1[\zeta ,\eta ]$. Define ${\psi }_{n+1}=S{\psi }_n=T^p{\psi }_n$.

We estimate that $d_n={\parallel {\psi }_n-{\psi }_{n-1}\parallel }_{C^1}$ for $n\ge 1$.

Apply the Chatterjea–Singh condition with $\psi ={\psi }_n$, $\varphi ={\psi }_{n-1}$

$${\parallel {\psi }_{n+1}-{\psi }_n\parallel }_{C^1}={\parallel S{\psi }_n-S{\psi }_{n-1}\parallel }_{C^1}\le \alpha \left(\parallel {\psi }_n-S{\psi }_{n-1}{\parallel }_{C^1}+{\parallel {\psi }_{n-1}-S{\psi }_n\parallel }_{C^1}\right).$$

But $S{\psi }_{n-1}={\psi }_n$, $S{\psi }_n={\psi }_{n+1}$, so

$${\parallel {\psi }_{n+1}-{\psi }_n\parallel }_{C^1}\le \alpha \left(\parallel {\psi }_n-{\psi }_n\parallel +\parallel {\psi }_{n-1}-{\psi }_{n+1}\parallel \right)=\alpha ·{\parallel {\psi }_{n-1}-{\psi }_{n+1}\parallel }_{C^1}.$$

Now,

$${\parallel {\psi }_{n-1}-{\psi }_{n+1}\parallel }_{C^1}\le {\parallel {\psi }_{n-1}-{\psi }_n\parallel }_{C^1}+{\parallel {\psi }_n-{\psi }_{n+1}\parallel }_{C^1}=d_n+d_{n+1}.$$

Thus,

$$d_{n+1}\le \alpha (d_n+d_{n+1})\Rightarrow d_{n+1}(1-\alpha )\le \alpha d_n\Rightarrow d_{n+1}\le {\displaystyle \frac{\alpha }{1-\alpha }}{d}_n.$$

Let $\beta ={\displaystyle \frac{\alpha }{1-\alpha }}$. Since $\alpha <\frac{1}{2}$, $\beta <1$. So

$$d_{n+1}\le {\beta }^n{d}_1.$$

For $m>n$,

$${\parallel {\psi }_m-{\psi }_n\parallel }_{C^1} \le \sum _{k=n}^{m-1}{\parallel {\psi }_{k+1}-{\psi }_k\parallel }_{C^1}=\sum _{k=n}^{m-1}{d}_{k+1}\le d_1\sum _{k=n}^{\infty }{\beta }^k=d_1{\displaystyle \frac{{\beta }^n}{1-\beta }}\to 0\mathrm{as}n\to \infty .$$

Hence, $\left\{{\psi }_n\right\}$ is Cauchy.

Step 4. 

Existence of fixed point for S.

Through completeness, ${\psi }_n\to {\psi }^{\ast }\in C^1[\zeta ,\eta ]$. This shows that $S{\psi }^{\ast }={\psi }^{\ast }$.

Apply (12) with $\psi ={\psi }_n$, $\varphi ={\psi }^{\ast }$

$${\parallel S{\psi }_n-S{\psi }^{\ast }\parallel }_{C^1}\le \alpha \left({\parallel {\psi }_n-S{\psi }^{\ast }\parallel }_{C^1}+{\parallel {\psi }^{\ast }-S{\psi }_n\parallel }_{C^1}\right).$$

As $n\to \infty$, $S{\psi }_n={\psi }_{n+1}\to {\psi }^{\ast }$, ${\psi }_n\to {\psi }^{\ast }$, so the right-hand side tends to

$$\alpha \left(\parallel {\psi }^{\ast }-S{\psi }^{\ast }\parallel +\parallel {\psi }^{\ast }-{\psi }^{\ast }\parallel \right)=\alpha {\parallel {\psi }^{\ast }-S{\psi }^{\ast }\parallel }_{C^1}.$$

The left-hand side is ${\to \parallel {\psi }^{\ast }-S{\psi }^{\ast }\parallel }_{C^1}$. Thus,

$${\parallel {\psi }^{\ast }-S{\psi }^{\ast }\parallel }_{C^1}{\le \alpha \parallel {\psi }^{\ast }-S{\psi }^{\ast }\parallel }_{C^1}\Rightarrow (1-\alpha ){\parallel {\psi }^{\ast }-S{\psi }^{\ast }\parallel }_{C^1}\le 0\Rightarrow {\psi }^{\ast }=S{\psi }^{\ast }.$$

Step 5. 

${\psi }^{\ast }$ is a fixed point of  T .

Since $T^p{\psi }^{\ast }={\psi }^{\ast }$, apply T: $T^{p+1}{\psi }^{\ast }=T{\psi }^{\ast }$. But also, $T^p\left(T{\psi }^{\ast }\right)=T\left(T^p{\psi }^{\ast }\right)=T{\psi }^{\ast }$, so $T{\psi }^{\ast }$ is a fixed point of $S=T^p$.

Now apply (12) with $\psi ={\psi }^{\ast }$, $\varphi =T{\psi }^{\ast }$

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

But $S{\psi }^{\ast }={\psi }^{\ast }$, $S\left(T{\psi }^{\ast }\right)=T{\psi }^{\ast }$, so

$${\parallel {\psi }^{\ast }-T{\psi }^{\ast }\parallel }_{C^1} \le \alpha \left({\parallel {\psi }^{\ast }-T{\psi }^{\ast }\parallel }_{C^1}+{\parallel T{\psi }^{\ast }-{\psi }^{\ast }\parallel }_{C^1}\right)=2\alpha {\parallel {\psi }^{\ast }-T{\psi }^{\ast }\parallel }_{C^1}.$$

Thus,

$$(1-2\alpha ){\parallel {\psi }^{\ast }-T{\psi }^{\ast }\parallel }_{C^1}\le 0.$$

Since $\alpha <\frac{1}{2}$, $1-2\alpha >0$, so ${\parallel {\psi }^{\ast }-T{\psi }^{\ast }\parallel }_{C^1}=0\Rightarrow T{\psi }^{\ast }={\psi }^{\ast }$.

Step 6. 

${\psi }^{\ast }\in C^2[\zeta ,\eta ]$ solves the BVP.

Since ${\psi }^{\ast }=T{\psi }^{\ast }$ and $j(·,{\psi }^{\ast }(·),{\left({\psi }^{\ast }\right)}^{\prime }(·))$ is continuous, standard regularity theory implies that ${\psi }^{\ast }\in C^2(\zeta ,\eta )\cap C^1[\zeta ,\eta ]$, and satisfies the differential equation and boundary conditions.

Step 7. 

Uniqueness.

Suppose ${\varphi }^{\ast }$ is another solution. Then $T{\varphi }^{\ast }={\varphi }^{\ast }\Rightarrow T^p{\varphi }^{\ast }={\varphi }^{\ast }$. Apply (12)

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

$$=\alpha \left(\parallel {\psi }^{\ast }-{\varphi }^{\ast }\parallel +\parallel {\varphi }^{\ast }-{\psi }^{\ast }\parallel \right)=2\alpha {\parallel {\psi }^{\ast }-{\varphi }^{\ast }\parallel }_{C^1}.$$

But $S{\psi }^{\ast }={\psi }^{\ast }$, $S{\varphi }^{\ast }={\varphi }^{\ast }$, so

$${\parallel {\psi }^{\ast }-{\varphi }^{\ast }\parallel }_{C^1}\le 2\alpha {\parallel {\psi }^{\ast }-{\varphi }^{\ast }\parallel }_{C^1}\Rightarrow (1-2\alpha ){\parallel {\psi }^{\ast }-{\varphi }^{\ast }\parallel }_{C^1}\le 0\Rightarrow {\psi }^{\ast }={\varphi }^{\ast }.$$

This completes the proof. □

3.2. Scalar Case

An analogous result holds in $C[\zeta ,\eta ]$

Theorem 2

(Scalar Chatterjea–Singh BVP). Let $j:[\zeta ,\eta ]\times \mathbb{R}\to \mathbb{R}$ be continuous. Define $T:C[\zeta ,\eta ]\to C[\zeta ,\eta ]$ as

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

(13)

If there exists $p\in \mathbb{N}$, $\alpha \in [0,\frac{1}{2})$ such that

$${∥T^p\psi -T^p\varphi ∥}_{\infty }\le \alpha \left({∥\psi -T^p\varphi ∥}_{\infty }+{∥\varphi -T^p\psi ∥}_{\infty }\right),\forall \psi ,\varphi \in C[\zeta ,\eta ],$$

(14)

then the BVP

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

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

Proof. 

Identical to Theorem 1, replace ${\parallel ·\parallel }_{C^1}$ with ${\parallel ·\parallel }_{\infty }$, and omit derivative terms. All steps are performed verbatim due to the structural similarity of the Chatterjea–Singh condition. □

Corollary 1.

Suppose j is continuously differentiable in ψ, and $|j_{\psi }(\tau ,\psi )|\le K$. Let $\mu =K·{\displaystyle \frac{{(\eta -\zeta )}^2}{8}}$. If ${\mu }^p<{\displaystyle \frac{1}{3}}$, then $T^p$ satisfies the Chatterjea–Singh condition for some $\alpha <\frac{1}{2}$; hence, the BVP has a unique solution.

Proof. 

Since T is a Banach contraction with constant $\mu$, $S=T^p$ is a Banach contraction with constant ${\mu }^p<{\displaystyle \frac{1}{3}}$. Then

$$\parallel S\psi -S\varphi \parallel \le {\mu }^p\parallel \psi -\varphi \parallel \le {\mu }^p\left(\parallel \psi -S\varphi \parallel +\parallel S\varphi -S\psi \parallel +\parallel S\psi -\varphi \parallel \right).$$

Using triangle inequalities and rearranging as before yields

$$\parallel S\psi -S\varphi \parallel \le {\displaystyle \frac{{\mu }^p}{1-{\mu }^p}}\left(\parallel \psi -S\varphi \parallel +\parallel \varphi -S\psi \parallel \right).$$

Let $\alpha ={\displaystyle \frac{{\mu }^p}{1-{\mu }^p}}$. Since ${\mu }^p<{\displaystyle \frac{1}{3}}$, $\alpha <{\displaystyle \frac{1/3}{2/3}}={\displaystyle \frac{1}{2}}$. So the Chatterjea–Singh condition holds. □

4. Examples and Applications

Example 1

(Nonlinear Oscillator). Consider ${\psi }^{\prime \prime }=-\lambda sin\left(\psi \right)$, $\psi \left(0\right)=0$, $\psi (\pi /2)=1$. Then $j\left(\psi \right)=\lambda sin\left(\psi \right)$, so $|j_{\psi }|\le |\lambda |$. With $\zeta =0$, $\eta =\pi /2$, ${\displaystyle \frac{{(\eta -\zeta )}^2}{8}}={\pi }^2/32\approx 0.308$. For $p=1$, we need $\left|\lambda \right|·0.308<{\displaystyle \frac{1}{3}}\Rightarrow \left|\lambda \right|<1.08$. For $p=2$, ${\left(0.308\right|\lambda \left|\right)}^2<{\displaystyle \frac{1}{3}}\Rightarrow \left|\lambda \right|<\sqrt{1/(3·0.{308}^2)}\approx 1.87$. For $p=3$, up to $\left|\lambda \right|\approx 2.38$. Thus, larger nonlinearities are admissible via iteration.

Example 2

(Polynomial Nonlinearity). Let ${\psi }^{\prime \prime }=-\lambda {\psi }^3$, $\psi \left(0\right)=\psi \left(1\right)=0$. Then $j\left(\psi \right)=\lambda {\psi }^3$, $|j_{\psi }|=3|\lambda |{\psi }^2$. On a ball $\left|\psi \right|\le R$, $K=3\left|\lambda \right|R^2$. ${\displaystyle \frac{{(\eta -\zeta )}^2}{8}}=1/8$. We need $\left(3\right|\lambda |R^2{/8)}^p<1/3$. Even if $3\left|\lambda \right|R^2/8\ge 1$, choosing large p makes the power small, recovering existence within invariant sets.

Example 3

(Vector Nonlinearity with Derivative: Damped Nonlinear Oscillator). Consider the boundary value problem

$${\psi }^{\prime \prime }\left(\tau \right)=-\lambda \psi {\left(\tau \right)}^3-\mu {\left({\psi }^{\prime }\left(\tau \right)\right)}^2,\psi \left(0\right)=0,\psi \left(1\right)=1,$$

on the interval $[0,1]$. Here, the nonlinearity depends on both ψ and ${\psi }^{\prime }$

$$j(\tau ,\psi ,{\psi }^{\prime })=\lambda {\psi }^3+\mu {\left({\psi }^{\prime }\right)}^2.$$

We assume that $\lambda ,\mu >0$. This models a nonlinear oscillator with state-dependent stiffness (${\psi }^3$) and quadratic damping (${\left({\psi }^{\prime }\right)}^2$).

Define the solution operator $T:C^1[0,1]\to C^1[0,1]$ as

$$\left(T\psi \right)\left(\tau \right)=(1-\tau )·0+\tau ·1+{\int }_0^{1}G(\tau ,s)\left[\lambda \psi {\left(s\right)}^3+\mu {\left({\psi }^{\prime }\left(s\right)\right)}^2\right]ds,$$

where Green’s function for $[0,1]$ is

$$G(\tau ,s)=\left\{\begin{array}{cc}\tau (1-s),\hfill & 0\le \tau \le s\le 1,\hfill \\ s(1-\tau ),\hfill & 0\le s\le \tau \le 1.\hfill \end{array}\right.$$

Note that j is continuous but not globally Lipschitz in ψ or ${\psi }^{\prime }$ due to the cubic and quadratic terms. Hence, the Banach contraction principle may fail on the whole space. Moreover, direct Chatterjea or Kannan conditions on T are difficult to verify because of the asymmetry introduced by the derivative term.

However, suppose we restrict T to a bounded closed subset $\mathcal{B}\subset C^1[0,1]$, say,

$$\mathcal{B}=\left\{\psi \in C^1[0,1]:{\parallel \psi \parallel }_{\infty }\le R,{\parallel {\psi }^{\prime }\parallel }_{\infty }\le L\right\},$$

for some $R,L>0$. On this set, we can estimate

$$\left|j(s,\psi \left(s\right),{\psi }^{\prime }\left(s\right))-j(s,\varphi \left(s\right),{\varphi }^{\prime }\left(s\right))\right|\le \lambda |{\psi }^3-{\varphi }^3|+\mu |{\left({\psi }^{\prime }\right)}^2-{\left({\varphi }^{\prime }\right)}^2|.$$

Using mean value estimates $|{\psi }^3-{\varphi }^3| \le 3R^2|\psi -\varphi |$, $|{\left({\psi }^{\prime }\right)}^2-{\left({\varphi }^{\prime }\right)}^2|=|{\psi }^{\prime }-{\varphi }^{\prime }|·|{\psi }^{\prime }+{\varphi }^{\prime }| \le 2L|{\psi }^{\prime }-{\varphi }^{\prime }|$,

so

$$|j(s,\psi ,{\psi }^{\prime })-j(s,\varphi ,{\varphi }^{\prime })| \le 3\lambda R^2|\psi \left(s\right)-\varphi \left(s\right)| + 2\mu L|{\psi }^{\prime }\left(s\right)-{\varphi }^{\prime }\left(s\right)|.$$

Let $M=3\lambda R^2$, $N=2\mu L$. Then, from standard estimates, ${sup}_{\tau }{\int }_0^1\left|G(\tau ,s)\right|ds={\displaystyle \frac{1}{8}}$, ${sup}_{\tau }{\int }_0^1\left|{\displaystyle \frac{\partial G}{\partial \tau }}(\tau ,s)\right|ds={\displaystyle \frac{1}{2}}$, we obtain

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

Thus,

$${\parallel T\psi -T\varphi \parallel }_{C^1}\le \underset{={\displaystyle \frac{5(M+N)}{8}}}{\underbrace{\left({\displaystyle \frac{M+N}{8}}+{\displaystyle \frac{M+N}{2}}\right)}}{\parallel \psi -\varphi \parallel }_{C^1}.$$

Let ${\mu }_T={\displaystyle \frac{5}{8}}(3\lambda R^2+2\mu L)$. If ${\mu }_T<1$, then T is a Banach contraction. But even if ${\mu }_T\ge 1$, if T maps $\mathcal{B}$ into itself, then $T^p$ may still become contractive for large p.

Now suppose $T\left(\mathcal{B}\right)\subseteq \mathcal{B}$ (which can be ensured by choosing a large enough $R,L$ depending on $\lambda ,\mu$). Then $S=T^p$ is a Banach contraction with constant ${\mu }_T^p$. As shown in Corollary 1, if ${\mu }_T^p<{\displaystyle \frac{1}{3}}$, then S satisfies the Chatterjea–Singh condition

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

for some $\alpha ={\displaystyle \frac{{\mu }_T^p}{1-{\mu }_T^p}}<{\displaystyle \frac{1}{2}}$.

Therefore, according to Theorem 1, the BVP has a unique solution in $C^2[0,1]$, even if T itself is not contractive—provided the p value chosen is sufficiently large that ${\mu }_T^p<1/3$.

This illustrates the power of the Chatterjea–Singh approach: it allows us to handle coupled nonlinearities involving both $\psi$ and ${\psi }^{\prime }$, where classical fixed-point methods may fail due to lack of global Lipschitz continuity or immediate contraction.

Remark 1.

Unlike Banach or Chatterjea conditions, the Chatterjea–Singh framework does not require global contractivity of T. It suffices that $T^p$ contracts in a symmetric way. This makes it ideal for numerical schemes where few iterations may suffice to stabilize.

5. Conclusions

We have successfully adapted the Chatterjea–Singh fixed-point framework to boundary value problems. By requiring only that an iterate $T^p$ satisfies a Chatterjea-type condition, we extend the domain of applicability beyond classical contraction principles. The resulting theorems guarantee the existence and uniqueness of solutions even when T is not contractive.

This approach combines the symmetry advantages of Chatterjea contractions with the flexibility of Singh’s iterate-based generalization. It is especially effective for mildly super-linear nonlinearities, where high iterates naturally damp oscillations.

Future research directions include applying this method to systems of BVPs, fractional differential equations, and numerical analyses of iterative solvers.