Study of Non-negative Solution for a Tenth-order Boundary Value Problems via Fixed Point Theory

Abstract

In this paper, we explore the existence of non-negative solutions
for a boundary value problem associated with a tenth-order
differential equation
$$-\aleph^{(10)}(\zeta) &= \digamma\left(\zeta, \aleph(\zeta), \aleph'(\zeta), \aleph''(\zeta), \aleph'''(\zeta), \aleph^{(4)}(\zeta), \aleph^{(5)}(\zeta), \aleph^{(6)}(\zeta), \aleph^{(7)}(\zeta), \aleph^{(8)}(\zeta), \aleph^{(9)}(\zeta)\right), $$
$$\aleph(0) &= \aleph'(0) = \aleph''(0) = \aleph'''(0) = \aleph^{(4)}(0) = \aleph^{(5)}(1)=$$
$$\aleph^{(6)}(1) = \aleph^{(7)}(1) = \aleph^{(8)}(1) = \aleph^{(9)}(1) = 0.$$
Where $\zeta\in[0,1]$ and $\digamma\in C([0,1]\times[0,\infty)\times[0,\infty)\times[0,\infty)\times[0,\infty)\times[0,\infty)\times[0,\infty)\times(-\infty,0]
\times[0,\infty)\times(-\infty,0]\times[0,\infty)\rightarrow[0,\infty))$
is non-negative continuous function. 

We employ the nonlinear Leray-Schauder alternative and the Leray-Schauder fixed point theorem to prove the existence of at least one non-negative solution. The complete continuity of the integral operator is established rigorously via the Arzel`a–Ascoli theorem, with explicit verification of uniform boundedness and equicontinuity. The non-decreasingness and convexity of the solution are derived from the sign
structure of the Green function derivatives. Sharp L∞–L1 kernel estimates are obtained and their optimality is discussed. The theoretical results are validated through an explicit example with full numerical verification of all hypotheses.

 In our analysis, we use the nonlinear Leray-Schauder alternative and the fixed point theorem of Leray-Schauder to prove the existence of at least one non-negative solution. As a numerical application, we provide an example to verify the accuracy of the obtained results.